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Home , Abductive reasoning

Seeking

Ab duction in

Philosophy of

and Artical Intelligence

Ato cha AlisedaLLera

Seeking Explanations

Ab duction in Logic

Philosophy of Science

and Artical Intelligence

ILLC Dissertation Series

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SEEKING EXPLANATIONS

ABDUCTION IN LOGIC

AND

a dissertation

submitted to the department of philosophy

interdepartmental program in philosophy and symbolic systems

and the committee on graduate studies

of stanford university

in partial fulfillment of the requirements

for the degree of

doctor of philosophy

By

Ato cha AlisedaLLera

August

Promotor Profdr J van Benthem

Faculteit Wiskunde en Informatica

Universiteit van Amsterdam

Plantage Muidergracht

TV Amsterdam

The investigations were supp orted by the Universidad Nacional Autonoma de Mexico

Instituto de Investigaciones Filosocas

c

Copyright by Ato cha AlisedaLLera

ISBN

Contents

Acknowledgements ix

What is Ab duction

Overview and Prop osal for Investigation

What is Ab duction

The Founding Father CS Peirce

Philosophy of Science

Articial Intelligence

Further Fields of Application

A Taxonomy for Ab duction

Thesis Aim and Overview

Ab duction as Logical

Intro duction

Directions in Reasoning

Forward and Backward

Formats of Inference

Premises and Background Theory

Inferential Strength A Parameter

Requirements for Ab ductive Inference

Styles of Inference and Structural Rules

Structural Rules For Ab duction

Further Logical Issues vii

Discussion and Conclusions

Further Questions

Related Work

Ab duction as Computation

Intro duction

Pro cedural Ab duction

Intro duction to Semantic Tableaux

Ab duction with Tableaux

Generating Ab ductions in Tableaux

Tableaux Extensions and Closures

Computing Plain Ab ductions

Consistent Ab ductive Explanations

Explanatory Ab duction

Quality of Ab ductions

Further Logical Issues

Discussion and Conclusions

Further Questions

Related Work

Scientic and Epistemic Change

Intro duction

Scientic Explanation as Ab duction

Ab duction as Epistemic Change

Explanation and Revision

AGM Postulates for Contraction

A Algorithms for Chapter

Abstract

Bibliography viii

Acknowledgements

It is a privilege to have ve professors on my reading committee representing the

dierent areas of my PhD program in Philosophy and Symb olic Systems Computer

Science Linguistics Logic Philosophy and

Tom Wasow was the rst p erson to point me in the direction of ab duction He

gave me useful comments on earlier versions of this dissertation always insisting that

it b e readable to non exp erts It was also a pleasure to work with him this last year

co ordinating the undergraduate Symb olic Systems program

Dagnn Fllesdal encouraged me to continue exploring connections between ab

y of science YoavShohamandPat Supp es gavemevery go o d duction and philosoph

advice ab out future expansions of this work Jim Greeno chaired my defense and also

gave me helpful suggestions For their help with my dissertation and for the classes

in which they taught me I am very grateful

Tomy advisor Johan van Benthem I oer my deep est gratitude He is a wonderful

teacher and mentor Among many things Johan taught me that basic notions in logic

may give rise to the most interesting questions Working with him meant amazement

and challenge and his passion for logic is an endless inspiration to my work

I received help from numerous other p eople and institutions For inspiring con

versations invaluable bibliography and constructive criticism in early stages of this

pro ject I wish to thank Carmina Curco Pablo Gervas Dinda Gorlee Jerry Hobbs

Marianne Kalsb eek GeertJan Kruij Ralf Muller David Pearce and Vctor Sanchez

Valencia I would also liketo thank the organizers and participants of the ECAI

Workshop on Ab ductive and to whom I had the opp ortunity

to present a small part of this dissertation Moreover I thank the students in the ix

senior seminar on Explanatory Reasoning which I led last winter for challenging

many ideas of my favorite authors

I gladly acknowledge the nancial help I received from the Universidad Nacional

Autonoma de Mexico UNAM which sp onsored my education The former director

of the Institute of Philosophical Research I IF Leon Olive supp orted and approved

my pro ject and the current director Olb eth Hansb erg help ed me ab ove and b eyond

the initial commitment

I hadawonderful oce at the Center for the Study of Language and Information

CSLI I thank the director John Perry for that privilege and esp ecially for allowing

me to use the sp eech recognition system of his Archimedes pro ject when I suered

buildings with unusual an injury caused by extensive typing CSLI is housed in two

Spanish I am happy to say that after all these years of working in Cordura

I had my defense in Ventura fortune

On a more personal note my situation called for extensive traveling to London

where my husband was doing his PhD at the London Scho ol of Economics This

involved many people who kept their home op en for us to stay and their basements

empty for our multiple b oxes They provided us with their love supp ort and many

pleasant sobremesas after dinner conversations Thanks a million to our parents

and siblings to Aida and Martn in Mexico to Vivian and Eduardo in Oakland to

Marty in Palo Alto to Julienne in Amsterdam and sp eciallyto Rosie in London I

would also like to mention my friends you know who you are who kept me sane and

happy while at planet Stanford and abroad

Finally I would like to mention the key people who motivated me to pursue

a graduate degree in the United States I thank Adolfo Garca de la Sienra for

encouraging me to come to Stanford His intellectual generosity sent me to my rst

international congress to present our work I also thank Carlos Torres Alcaraz who

intro duced me to the b eautiful framework of semantic tableaux and directed my

undergraduate thesis and Elisa Viso Gurovich my rst teacher

who has given me her unconditional supp ort all along Iwould also like to express my

gratitude to my late father who inculcated in me the will to learn and the satisfaction

of doing what you like He would b e very happy to know that I nally nished scho ol x

Ab ove all I thank Ro dolfo who shares with me his passion for music and his

adventurous spirit He kept all to o well his promise to take care of me through

graduate scho ol even though he had to write his own thesis and live in another

country I test many of my ideas with him and I hop e we will some day write that

enty years book we have talked ab out I am delighted that we complement our tw

of friendship with nine of marriage This dissertation is of course dedicated to him

with all my love xi

Chapter

What is Ab duction

Overview and Prop osal for

Investigation

What is Ab duction

A central theme in the study of human reasoning is the construction of explanations

that give us an understanding of the world we live in Broadly sp eaking abduction

is a reasoning pro cess invoked to explain a puzzling observation A typical example

is a practical comp etence like medical diagnosis When a do ctor observes a symptom

in a patient she hyp othesizes ab out its p ossible causes based on her knowledge of

the causal relations between diseases and symptoms This is a practical setting

Ab duction also o ccurs in more theoretical scientic contexts For instance it has

been claimed HanCP that when Kepler discovered that Mars had an

elliptical orbit his reasoning was ab ductive But ab duction may also be found in

our daytoday common sense reasoning If wewake up and the lawn is wet we might

explain this observation by assuming that it must have rained or by assuming that the

sprinklers have b een on Ab duction is thinking from to explanation a typ e

of reasoning characteristic of many dierent situations with incomplete information

The history of this typ e of reasoning go es backtoAntiquity It has b een compared

Chapter What is Ab duction

with Aristotles apagoge CP which intended to capture a nonstrictly

deductivetyp e of reasoning whose conclusions are not necessary but merely p ossible

not to be confused with epagoge the Aristotelian term for induction Later on

ab duction as reasoning from eects to causes is extensively discussed in Laplaces

famous memoirs Lap Sup as an imp ortant metho dology in the In

the mo dern age this reasoning was put on the intellectual agenda under the

ab duction by CS Peirce CP

Tostudyatyp e of reasoning that o ccurs in contexts as varied as scientic discov

ust b e provided ery medical diagnosis and common sense suitably broad features m

that cover a lot of cases and yet leave some signicant substance to the notion of

ab duction The purp ose of this preliminary chapter is to intro duce these which will

lead to the more sp ecic questions treated in subsequent chapters But b efore we

start with a more general analysis let us expand our sto ck of examples

Examples

The term ab duction is used in the literature for a variety of explanatory pro cesses

We list a few partly to show what we must cover and partly to show what we will

ve aside lea

Common Sense Explaining observations with simple

All you know is that the lawn gets wet either when it rains or when the sprin

klers are on You wake up in the morning and notice that the lawn is wet

Therefore you hyp othesize that it rained during the night or that the sprinklers

had been on

Common Sense Laying causal connections between facts

You observe that a certain typ e of clouds nimb ostratus usually precede rain

fall You see those clouds from your window at night Next morning you see

that the lawn is wet Therefore you infer a causal connection between the

nimb ostratus at night and the lawn b eing wet

What is Ab duction

Common Sense Facing a

You know that rain causes the lawn to get wet and that it is indeed raining

However you observe that the lawn is not wet How could you explain this

anomaly

Statistical Reasoning Medical Diagnosis

Jane Jones recovered rapidly from a strepto co ccus infection after she was given a

dose of p enicillin Almost all strepto co ccus infections clear up quickly up on ad

ministration of p enicillin unless they are p enicillinresistant in which case the

probabilityofquick recovery is rather small The do ctor knew that Janes infec

tion is of the p enicillinresistanttyp e and is completely puzzled by her recovery

a Jane Jones then confesses that her grandmother had given her Belladonna

homeopatic which stimulates the immune system by strengthening the

physiological resources of the patient to ght infectious diseases

The examples so far are fairly typical of what our later analysis can deal with

But actual explanatory reasoning can be more complicated than this For instance

even in common sense settings there may be various options which are considered

in some sequence dep ending on your memory and computation strategy

Common Sense When something do es not work

You come into your house late at night and notice that the lightinyour ro om

which is always left on is o It has b eing raining very heavily and so you think

some power line went do wn but the lights in the rest of the house work ne

Then you wonder if you left b oth heaters on something which usually causes

the breakers to cut o so you check them but they are OK Finally a simpler

explanation crosses your mind Mayb e the light bulb of your lamp which you

last saw working well is worn out and needs replacing

This is an adaptation of Hemp els famous illustration of his InductiveStatistical mo del of ex

planation as presented in Sal The part ab out homeopathyisentirely mine however

Chapter What is Ab duction

So ab duction involves computation over various candidates dep ending on your

background knowledge In a scientic setting this means that ab ductions will dep end

on the relevant background theory as well as ones metho dological working habits

We mention one oftencited example even though we should state clearly at this

pointthatit go es far b eyond what we shall eventually deal with in our analysis



Scientic Reasoning Keplers discovery

One of Johannes Keplers great discoveries was that the orbit of the planets is

elliptical rather than circular What initially led to this discovery was his obser

vation that the longitudes of Mars did not t circular orbits However b efore

even dreaming that the best explanation involved ellipses instead of circles he

tried several other forms Moreover Kepler had to make several other assump

tions ab out the planetary system without which his discovery do es not work

His helio centric view allowed him to think that the sun so near to the center

of the planetary system and so large must somehow cause the planets to move

as they do In addition to this strong conjecture he also had to generalize his

y assuming that the same physical conditions ndings for Mars to all planets b

obtained throughout the solar system This whole pro cess of explanation to ok

manyyears

It will be clear that the Kepler example has a lo ose end so to sp eak How we

construct the explanation dep ends on what we take to be his scientic background

theory This is a general feature of ab ductions explanation is always explanation

wrt some b o dy of b eliefs But even this is not the only parameter that plays a role

One could multiply the ab ove examples and nd still further complicating factors

Sometimes no single obvious explanation is available but rather several comp eting

ones and wehave to select Sometimes the explanation involves not just advancing

facts or rules in our current conceptual frame but rather the creation of new concepts

that allow for new of the relevant phenomena Evidently we must draw

a line somewhere in our present study



This example is a simplication of one in Han

What is Ab duction

All our examples were instances of reasoning in which an explanation is needed to

account for a certain phenomenon Is there more unity than this At rst glance the

only clear common feature is that these are not cases of ordinary

and this for a number of In particular the explanations pro duced mightbe

defeated Mayb e the lawn is wet b ecause children have b een playing with water Co

o ccurrence of clouds and the lawn b eing wet do es not necessarily link them in a causal

way Janes recovery might after all be due to a normal pro cess of the body What

we learn subsequently can invalidate an earlier ab ductive conclusion Moreover the

reasoning involved in these examples seems to go in reverse to ordinary deduction as

yp othesis and not from data to conclusion as all these cases run from evidence to h

it is usual in deductive patterns Finally describing the wayinwhich an explanation

is found do es not seem to follow sp ecic rules Indeed the precise nature of Keplers



discovery remains under intensive debate

What we would like to do is the following Standard deductive logic cannot

account for the ab ove typ es of reasoning In this century under the inuence of

foundational research in mathematics there has been a contraction of concerns in

logic to this deductive core The result was a loss in breadth but also a clear gain

in depth By now an impressive body of results has been obtained ab out deduction

and we would like to study the wider eld of ab duction while hanging on to these

standards of rigor and clarity Of course we cannot do everything at once and achieve

the whole agenda of logic in its traditional op enended form We want to nd some

features of ab duction that allow for concrete logical analysis thereby extending the

scop e of standard metho ds In the next section we discuss three main features that

o ccur across all of the ab ove examples prop erly viewed which will b e imp ortantin

our investigation



For Peirce Keplers reasoning was a prime piece of ab duction CP whereas for Mill

it was merely a description of the facts Mill BkIIIch I I CP Even nowadays one

nds very dierent reconstructions While Hanson presents Keplers helio centric view as an essential

assumption Han Thagard thinks he could make the discovery assuming instead that the earth

was stationary and the sun moves around it Tha Still a dierent accountofhow this discovery

can b e made is given in SLB LSB

Chapter What is Ab duction

Three Faces of Ab duction

We shall nowintro duce three broad opp ositions that help in clarifying what ab duction

is ab out At the end we indicate how these will b e dealt with in this thesis

Ab duction Pro duct or Pro cess

The logical key words of judgment and are nouns which denote either an activity

indicated by their corresp onding verb or the result of that activity In just the same

way the common word explanation which we treat as largely synonymous with

ab duction may be used both to refer to a nished pro duct the explanation of a

phenomenon or to an activity the pro cess that led to that explanation These two

uses are closely related The pro cess of explanation pro duces explanations as its

pro duct but the two are not the same

One can relate this distinction to more traditional ones An example is Reichen

bachs Rei wellknown opp osition of context of discovery versus context of jus

tication Keplers explanationpro duct the orbit of the planets is elliptical which

justies the observed facts do es not include the explanationpro cess of how he came

to make this discovery The context of discovery has often been taken to be purely

psychological but this do es not preclude its exact study Cognitive psychologists

study mental patterns of discovery theorists in AI study formal hyp othesis

formation and one can even work with concrete computational algorithms that pro

duce explanations To be sure it is a matter of debate whether Keplers reasoning

may b e mo deled by a computer For a computer program that claims to mo del this

particular discovery cf SLB However this may b e one can certainly write sim

ple programs that pro duce common sense explanations of why the lawn is wet as we

will show later on On the other hand once pro duced explanations are public ob jects

of justication that can be checked and tested by indep endent logical criteria

The pro ductpro cess distinction has b een recognized by logicians Bet vBe

by of science Rub in the context of deductive reasoning as well as

Sal in the context of scientic explanation Both lead to interesting questions by

themselves and so do es their interplay Likewise these two faces of ab duction are

What is Ab duction

both relevant for our study On the pro duct side our fo cus will be on conditions

that give a piece of information explanatory force and on the pro cess side we will

be concerned with the design of algorithms that pro duce explanations

Ab duction Construction or Selection

Given a to be explained there are often several p ossible explanations but only

one or a few that counts as the best one Pending subsequent testing in our com

mon sense example of light failure several explanations account for the unexp ected

the ro om p ower line down breakers cut o bulb worn out But only darkness of

one may be considered as b est explaining the event namely the one that really

happ ened But other preference criteria may b e appropriate to o esp ecially when we

have no direct test available

Thus ab duction is connected to b oth hyp othesis construction and hyp othesis

selection Some authors consider these pro cesses as two separate steps construction

dealing with what counts as a p ossible explanation and selection with applying some

preference criterion over p ossible explanations to select the b est one Other authors

regard ab duction as a single pro cess by which a single b est explanation is constructed

Our p osition is an intermediate one We will split ab duction into a rst phase of

hyp othesis construction but also acknowledge a next phase of hyp othesis selection

We shall mainly fo cus on a characterization of p ossible explanations We will argue

that the notion of a b est explanation necessarily involves contextual asp ects varying

from application to application There is at least a new parameter of preference

ranking here Although there exist b oth a philosophical tradition on the logic of

preference Wri and logical systems in AI for handling preferences that may be

ould take used to single out b est explanationsSho DPb the resulting study w

us to o far aeld

Ab duction vs Induction

Once b eyond deductive logic diverse terminologies are b eing used Perhaps the most

widely used term is inductive reasoning Mill Sal HHN Tha Fla Mic

Chapter What is Ab duction

Ab duction is another fo cus and it is imp ortant at least to clarify its relationship to

induction For CS Peirce as we shall see deduction induction and ab duction

formed a natural triangle but the literature in general shows many overlaps and

even confusions

Since the time of John Stuart Mill the technical name given to all

kinds of nondeductive reasoning has b een induction though several methods for

discovery and demonstration of causal relationships Mill were recognized These

included generalizing from a sample to a general prop erty and reasoning from data to

a causal hyp othesis the latter further divided into metho ds of agreement dierence

residues and concomitantvariation A more rened and mo dern terminology is enu

e induction and explanatory induction of which inductive generalization merativ

inductive pro jection statistical concept formation are some instances

Such a broad connotation of the term induction continues to the present day For

instance in the computational philosophy of science induction is understo o d in the

broad sense of any kind of inference that expands know ledge in the faceofuncertainty

Tha

Another heavy term for nondeductive reasoning is statistical reasoning intro

ducing a probabilistic avour like our example of medical diagnosis in which p os

sible explanations are not certain but only probable Statistical reasoning exhibits

the same diversity as ab duction First of all just as the latter is strongly identied

with backwards deduction as we shall see later on in this chapter the former nds

its reverse notion in The problem in probabilityis given an sto chastic

mo del what can wesay ab out the outcomes The problem is statistics is the reverse

given a set of outcomes what can we say ab out the mo del Both ab duction and

statistical reasoning are closely linked with notions like conrmation the testing of

hyp othesis and likeliho o d a measure for alternativehyp otheses

On the other hand some authors take induction as an instance of ab duction

Ab duction as inference to the best explanation is considered by Harman Har as

the basic form of nondeductive inference which includes enumerative induction as

a sp ecial case

This confusion returns in articial intelligence Induction is used for the pro cess

What is Ab duction

of learning from examples but also for creating a theory to explain the observed

facts Sha thus making ab duction an instance of induction Ab duction is usually

restricted to pro ducing explanations in the form of facts When the explanations are

rules it is regarded as part of induction The relationship between ab duction and

induction prop erly dened has b een the topic for recentworkshops in AI conferences

ECAI

To clear up all these conicts one mightwant to coin new terminology altogether

Many authors write as if there were preordained reasonably clear notions of ab duc

tion and its rivals which we only have to analyze to get a clear picture But these

technical terms may be irretrievably confused in their full generality burdened with

Ihave argued for a new term the debris of defunct philosophical theories Therefore

of explanatory reasoning in Alia trying to describ e its fundamental asp ects

without having to decide if they are instances of either ab duction or induction In

this broader p ersp ective we can also capture explanation for more than one instance

or for generalizations which we have not mentioned at all and intro duce further

nestructure For example given two observed events in order to nd an explana

tion that accounts for them it must b e decided whether they are causally connected

eg entering the copier ro om and the lights going on correlated with a common

cause eg observing b oth the barometric pressure and the temp erature dropping at

the same time or just coincidental without any link you reading this paragraph in

place A while I revise it somewhere in place B But in this dissertation we shall

concentrate on explanatory reasoning from simple facts giving us enough variety for

now Hop efully this case study of ab duction will lead to broader clarity of denition

as well

More precisely we shall understand ab duction as reasoning from a single obser

vation to its explanations and induction as enumerative induction from samples to

general statements While induction explains a set of observations ab duction ex

plains a single one Induction makes a for further observations ab duction

for later observations While induction needs no back do es not directly account

ground theory per se ab duction relies on abackground theory to construct and test

its explanations Note that this ab ductive formulation do es not commit ourselves to

Chapter What is Ab duction

any sp ecic logical inference kind of observation or form of explanation

As for their similarities induction and ab duction are b oth nonmonotonic typ es

of inference and both run in opp osite direction to standard deduction In non

monotonic inference new may invalidate a previous valid In

the terminology of philosophers of science nonmonotonic are not erosion

proof Sal Qua direction induction and ab duction b oth run from evidence to

explanation In logical terms this maybeviewed as going from a conclusion to part

of its premises in reverse of ordinary deduction We will return to these issues in

more detail in our logical chapter much

The Founding Father CS Peirce

The literature on ab duction is so vast that we cannot undertake a complete survey

here What we shall do is survey some highlights starting with the historical sources

of the mo dern use of the term In this eld all th century roads lead back to the

work of CS Peirce For a much more elab orate historical analysis cfAli Together

with the other sources to be discussed the coming sections will lead up to further

parameters for the general taxonomy of ab duction that we prop ose toward the end

of this chapter

Understanding Peirces Ab duction

Charles Sanders Peirce the founder of American was the rst

to give to ab duction a and hence his to our study

However his notion of ab duction is a dicult one to unravel On the one hand it is

entangled with many other asp ects of his philosophy and on the other hand several

dierent conceptions of ab duction evolved in his thought We will p oint out a few

general asp ects of his theory of and later concentrate on some of its more

logical asp ects

The notions of logical inference and of validitythatPeirce puts forward go b eyond

our present understanding of what logic is ab out They are linked to his epistemology

The Founding Father CS Peirce

a dynamic view of thought as logical inquiry and corresp ond to a deep philosophical

concern that of studying the nature of synthetic reasoning

In his early theory Peirce prop osed three mo des of reasoning deduction induc

tion and ab duction each of which corresp onds to a syllogistic form illustrated by

the following often quoted example CP

DEDUCTION

Rule All the b eans from this bag are white

Case These b eans are from this bag

Result These beans are white

INDUCTION

Case These b eans are from this bag

Result These beans are white

Rule All the b eans from this bag are white

ABDUCTION

Rule All the b eans from this bag are white

Result These beans are white

Case These b eans are from this bag

Of these deduction is the only reasoning which is completely certain infering its

a necessary conclusion Induction pro duces a Rule validated only in the Result as

long run CP and ab duction merely suggests that something may be the

Case CP

Later on Peirce prop osed these typ es of reasoning as the stages comp osing a

metho d for logical inquiryofwhich ab duction is the b eginning

From its abductive suggestion deduction can draw a prediction which

can be tested by induction CP

Ab duction plays a role in direct p erceptual judgments in which

The abductive suggestion comes to us as a ash CP

Chapter What is Ab duction

As well as in the general pro cess of invention

It abduction is the only logical operation which introduces any new

ideas CP

In all this ab duction is both an act of insight and an inference as has b een

suggested in And These explications do not x one unique notion Peirce rened

his views on ab duction throughout his work He rst identied ab duction with the

syllogistic form ab ove to later enrich this idea by the more general conception of

the process of forming an explanatory hypothesisCP

also refering to it as And

The process of choosing a hypothesis CP

Something which suggests that he did not always distinguish clearly b etween the

construction and the selection of a hyp othesis as was pointed out in Fan The

evolution of his theory is also reected in the varied terminology he used to refer to

ab duction b eginning with presumption and hypothesis CP then using

abduction and retroduction interchangeably CP

A nice concise account of the development of ab duction in Peirce which clearly

distinguishes three stages in the evolution of his thought is given in Fan Another

key on Peirces ab duction in its relation to creativity in art and science is

found in And

The Key Features of Peircean Ab duction

For Peirce three asp ects determine whether a hyp othesis is promising it must be

explanatory testable and economic A hyp othesis is an explanation if it accounts

for the facts Its status is that of a suggestion until it is veried which explains the

motivation for the economic criterion need for the testability criterion Finally the

is twofold a resp onse to the practical problem of having innumerable explanatory

The Founding Father CS Peirce

hyp otheses to test as well as the need for a criterion to select the best explanation

amongst the testable ones

For the explanatory asp ect Pierce gave the following oftenquoted logical formu

lation CP

The surprising fact C is observed

ButifAwere true C would be a matter of course

Hence there is to susp ect that A is true

This formulation has played a key role in Peirce scholarship and it has been the

point of departure of recent studies on ab ductive reasoning in articial intelligence

However no one seems to agree on its interpretation We KKT HSAM PG

will also give our own for details cf Ali

Interpreting Peirces Ab duction

The notion of ab duction in Peirce has puzzled scholars all along Some have concluded

that Peirce held no coherent view on ab duction at all Fra others have tried to give

a joint account with induction Rei and still others claim it is a form of inverted

mo dus p onens And A more mo dern view is found in Kap who interprets

Peirces ab duction as a form of An account that tries to make sense of

the two extremes of ab duction b oth as a instinct and as a rational activity

is found in Ayi I have argued in Ali that this diversity suggests that Peirce

recognized not only dierenttyp es of reasoning but also several degrees within each

one and even merges b etween the typ es In the context of p erception he writes

The p erceptual judgements are to be regarded as extreme cases of ab

ductive inferences CP

Abductory induction on the other hand is suggested when some kind of guess

work is involved in the reasoning CP Anderson And also recognizes

several degrees in Peirces notion of creativity

This multiplicity returns in articial intelligence Flab suggests that some

confusions in mo dern accounts of ab duction in AI can b e traced backtoPeirces two

Chapter What is Ab duction

theories of ab duction the earlier syllogistic one and the later inferential one As to

more general semiotic asp ects of Peirces philosophy another prop osal for character

izing ab duction in AI is found in Kru

Our own understanding of ab ductive reasoning reects this Peircean diversity in

part taking ab duction as a style of that o ccurs at dierent levels

and in several degrees These will be reected in our prop osal for a taxonomy with

several parameters for ab ductive reasoning This scheme will be richer than the

ulation often encountered in the literature logical form for Peirces ab ductive form

C

A C

A

where the status of A is that of a tentative explanation Though simple and

intuitive this formulation captures neither the fact that C is surprising nor the addi

tional asp ects of testability and economythatPeirce prop osed For instance existing

computational accounts of ab duction do not capture that C is a surprising fact In

our logical treatment of ab duction in subsequent chapters we bring out at least two

asp ects of Peirces formulation that go beyond the preceding schema namely C

is a surprising fact A is an explanation For further asp ects we refer to subse

quent sections The additional criteria of testability and economy are not part of our

framew ork Testability as understo o d byPeirce is an extralogical empirical criterion

while economy concerns the selection of explanations which we already put aside as

a further stage of ab duction requiring a separate study

Philosophy of Science

Peirces work stands at the crossroads of many traditions including logic epistemol

ogy and philosophy of science Esp eciallythe latter eld has continued many of his

central concerns Ab duction is clearly akin to core notions of mo dern metho dology

such as explanation induction discovery and heuristics We have already discussed

Philosophy of Science

a connection b etween pro cesspro duct asp ects of ab duction and the wellknown divi

sion b etween contexts of discovery and justication Rei We shall discuss several

further p oints of contact in chapter below But for the moment we start with a

rst foray

The Received View explanation as a pro duct

The dominant trend in philosophy has fo cused on ab duction as pro duct rather than

a pro cess just as it has done for other epistemic notions Aristotle Mill and in

this century the inuential philosopher of science Carl Hemp el all based their ac

counts of explanation on prop osing criteria to characterize its pro ducts These ac

counts generally classify into argumentative and nonargumentative typ es of expla

nation Rub Sal Nag Of particular imp ortance to us is the argumentative

Hemp elian tradition Its followers aim to mo del empirical whyquestions whose an

swers are scientic explanations in the form of In these arguments the

explanandum the fact to b e explained is derived deductively or inductively from

the explananda that which do es the explaining supplemented with relevant laws

general or statistical and initial conditions F or instance the fact that an explo

sion o ccurred may b e explained by mylighting the match given the laws of physics

and initial conditions to the eect that oxygen was present the match was not wet

et cetera

In its deductiveversion the Hemp elian account found in many standard texts on

the philosophy of science Nag Sal is called deductivenomological for obvious

reasons But its engine is not just standard deduction Additional restrictions must

b e imp osed on the relation b etween explananda and explanantia as neither deduction

nor induction is a sucient condition for genuine explanation To mention a simple

ula is derivable from itself but it seems counterintuitive example every form

or at least very uninformative to explain anything by itself

Other nondeductive approaches to explanation exist in the literature For in

stance Rub p oints at these two

Chapter What is Ab duction

Sal p takes them to b e an assemblage of factors that are statis

tical ly relevant

while vFr p makes them simply an answer

For Salmon the question is not how probable the explanans renders the explanan

dum but rather whether the facts adduced make a dierence to the probability of

the explanandum Moreover this relationship need not be in the form of an argu

ment For van Fraassen a representative of pragmatic approaches to explanation

the explanandum is a contrastive whyquestion Thus rather than asking why

one asks why rather than The pragmatic view seems closer to ab duction as a

indeed the fo cus on questions intro duces some dynamics of explaining pro cess and

Still it do es not tell us how to pro duce explanations

There are also alternative deductive approaches An example is the work of

Rescher Res which intro duces a direction of thought Interestingly this estab

lishes a temp oral distinction b etween prediction and retro duction Reschers term

for ab duction by marking the precedence of the explanandum over the hyp othesis

in the latter case Another and quite famous deductivist tradition is Popp ers logic

of scientic discovery Pop Its metho d of conjectures and refutations prop oses the

testing of hyp otheses b y attempting to refute them

The actual procedure of science is to operate with conjectures to jump

to conclusions often after one single observation Pop p

Thus science starts from problems and not from observations though

observations may give rise to a problem special ly if they are unexpected

that is to say if they clash with our expectations or theories Pop

p

rather than explanation Popp ers deductive fo cus is on refutation of falseho o ds

of One might sp eculate ab out a similar negative counterpart to ab duction

Although Popp ers metho d claims to be a logic of scientic discovery he views the

actual construction of explanations as an exclusive matter for psychology and hence

his trial and error philosophy oers no further clues for our next topic

Philosophy of Science

What is common to all these approaches in the philosophy of science is the imp or

tance of a hidden parameter in ab duction Whether with Hemp el Salmon or Popp er

scientic explanation never takes place in isolation but always in the context of some

background theory This additional parameter will b ecome part of our general scheme

to b e prop osed below

The Neglected View explanation as a pro cess

Much more marginal in the philosophy of science are accounts of explanation that

fo cus on explanatory pro cesses as such One early author emphasizing explanation

as a pro cess of discovery is Hanson Han who gave an account of patterns of

retro duction discovery recognizing a central role for ab duction which he calls

Also relevant here is the work by Lakatos Lak a critical resp onse to Popp ers

logic of scientic discovery For Lakatos there is only a fallibilistic logic of discovery

which is neither psychology nor logic but an indep endent discipline the logic of

He pays particular attention to pro cesses that created new concepts in

mathematics often referring to Polya Pol as the founding father of heuristics



in mathematical discovery We will come back to this issue later in the c hapter

when presenting further elds of application

What these examples reveal is that in science explanation involves the invention of

new concepts just as much as the p ositing of new statements in some xed conceptual

framework So far this has not led to extensive formal studies of concept formation

similar to what is known ab out deductive logic Exceptions that prove the rule

are o ccasional uses of Beths Denability Theorem in the philosophical literature A

similar lacuna visavis concept revision exists in the current theory of b elief revision

in AI

Thus philosophy of science oers little for our interests in ab ductive pro cesses

We are certainly in sympathy with the demand for conceptual change in explanation

but this topic will remain b eyond the technical scop e of this thesis



In fact Polyacontrasts twotyp es of arguments A demonstrative syllogism in which from A B

and B false A is concluded and a heuristic syl logism in whichfromA B andB true it follows

that A is more credible The latter of course recalls Peirces ab ductiveformulation

Chapter What is Ab duction

Articial Intelligence

Our next area of comparison is articial intelligence The transition with the previous

section is less abrupt than it may be seem It has often b een noted by lo oking at

the resp ective research agendas that articial intelligence is philosophy of science

pursued by other means cf Tan Research on ab ductive reasoning in AI dates

back to Pop but it is only fairly recently that it has attracted great interest

in areas like KKT knowledge assimilation KM and diag

nosis PG to name just a few Ab duction is also coming up in the context of data

bases and knowledge bases that is in mainstream computer science

In this setting the pro ductpro cess distinction has a natural counterpart namely

ab duction While the former in logicbased vs computationalbased approaches to

fo cuses on giving a to the logic of ab duction usually dened as back

wards deduction plus additional conditions the latter is concerned with providing

algorithms to pro duce ab ductions

It is imp ossible to giveanoverview here of this explo ding eld Therefore welimit

ourselves to a brief description of ab duction as logical inference a presentation

of ab duction in logic programming and a sketch of the relevance of ab ductive

thinking in knowledge There is much more in this eld of p otential

philosophical interest however For ab duction in bayesian networks connectionism

and many other angles the reader is advised to consult Jos PR Kon Pau

AAA

Ab duction as Logical Inference

The general trend in logic based approaches to ab duction in AI interprets ab duc

tion as backwards deduction plus additional conditions This brings it very close to

deductivenomological explanation in the Hemp el style witness the following format

What follows is the standard version of ab duction as deduction via some consistent

additional assumption satisfying certain extra conditions It combines some com

mon requirements from the literature cf Kon KKT MP and chapter for

further motivation

Articial Intelligence

Given a theory a set of formulae and a formula an atomic formula

is an explanation if

j

is consistent with

is minimal there are several ways to characterize minimalityto

be discussed in chapter

has some restricted syntactical form usually an atomic formula or

a conjunction of them

An additional condition not always made explicit is that j This says that

the fact to b e explained should not already follow from the background theory alone

Sometimes the latter condition gures as a precondition for an abductive problem

What can one say in general ab out the prop erties of such an enriched notion

of consequence As we have mentioned b efore a new logical trend in AI studies

variations of classical consequence via their structural rules which govern the com

bination of basic inferences without referring to any sp ecial logical connectives Cf

the analysis of nonmonotonic consequence relations in AI of Gaba KLM and

the analysis of dynamic styles of inference in linguistics and cognition in vBe

Perhaps the rst example of this approach in ab duction is the work in Fla and

indeed our analysis in chapter will follow this same pattern

Ab duction in Logic Programming

Logic Programming LLoKow was intro duced by Kowalski and Colmerauer in

and is implemented as amongst others the programming language Prolog

It is inspired by rstorder logic and it consists of logic programs queries and a



underlying inferential mechanism known as resolution



Roughly sp eaking a Prolog program P is an ordered set of rules and facts Rules are restricted

to hornclause form A L L in which each L is either an atom A or its negation A

n i i i

A query q theorem is p osed to program P to be solved proved If the query follows from the

program a p ositive answer is pro duced and so the query is said to be succesful Otherwise a

Chapter What is Ab duction

Ab duction emerges naturally in logic programming as a repair mechanism com

pleting a program with the facts needed for a query to succeed This may b e illustrated

by our rain example from the intro duction in Prolog

Program P

lawnwet rain

lawnwet sprinklerson

Query q lawnwet

Given program P query q do es not succeed b ecause it is not derivable from

the program For q to succeed either one or all of the facts rain sprinklers

on lawnwet would have to be added to the program Ab duction is the pro cess

via an extension of the by which these additional facts are pro duced This is done

resolution mechanism that comes into play when the backtracking mechanism fails

In our example ab ove instead of declaring failure when either of the ab ove facts is

not found in the program they are marked as hyp othesis and prop osed as those

formulas which if added to the program would make the query succeed

In actual Prolog ab duction for these facts to b e counted as ab ductions they have

to b elong to a predened set of ab ducibles and to be veried by additional condi

tions socalled integrity constraints in order to prevent a combinatorial explosion

of p ossible explanations

In logic programming the pro cedure for constructing explanations is left entirely

to the resolution mechanism which aects not only the order in which the p ossible

explanations are pro duced but also restricts the form of explanations Notice that

rules cannot o ccur as ab ducibles since explanations are pro duced out of subgoal

literals that fail during the backtracking mechanism Therefore our common sense

negativeanswer is pro duced indicating that the query has failed However the interpretation of

negation is by failure That is no means it is not derivable from the available information in P

without implying that the negation of the query q is derivable instead Resolution is an inferential

mechanism based on refutation working backwards from the negation of the query to the data in

the program In the course of this pro cess valuable bypro ducts app ear the socalled computed

answer substitutions which give more detailed information on the ob jects satisfying given queries

Articial Intelligence

example in which a causal connection is ab duced to explain why the lawn is wet



cannot b e implemented in logic programming The additional restrictions select the

b est hyp othesis Thus pro cesses of both construction and selection of explanations

are clearly marked in logic programming Another relevant connection here is to

recent research in inductive logic programming Mic whichintegrates ab duction

and induction

Logic programming do es not use blind deduction Dierent control mechanisms

for pro of search determine how queries are pro cessed This additional degree of

freedom is crucial to the eciency of the enterprise Hence dierent control p olicies

will vary in the ab ductions pro duced their form and the order in which they app ear

us this variety suggests that the pro cedural notion of ab duction is intensional To

and must be identied with dierent practices rather than with one deterministic

xed pro cedure

Ab duction and Theories of Epistemic Change

Most of the logicbased and computationbased approaches to ab duction reviewed

in the preceding sections assume that neither the explanandum nor its negation is

derivable from the background theory j j This leaves no ro om

to represent problems like our common sense light example in which the theory

exp ects the contrary of our observation Namely that the light in my ro om is on

These are cases where the theory needs to be revised in order to account for the

observation Such cases arise in practical settings like diagnostic reasoning PR

b elief revision in databases AD and theory renement in machine learning SL

Gin

When imp orting revision into ab ductive reasoning an obvious related territory is

theories of b elief change in AI Mostly inspired by the work of Gardenfors Gar

a work whose ro ots lie in the philosophy of science these theories describ e how to

incorp orate a new piece of information into a database a scientic theory or a set



At least this is how the implementation of ab duction in logic programming stands as of now

It is of course p ossible to write extended programs that pro duce these typ e of explanations

Chapter What is Ab duction

of common sense b eliefs The three main typ es of belief change are op erations of

expansion contraction and revision A theory may be expanded by adding new

formulas contracted by deleting existing formulas or revised by rst b eing contracted

and then expanded These op erations are dened in suchaway as to ensure that the

theory or b elief system remains consistent and suitably closed when incorp orating

new information

Our earlier cases of ab duction may be describ ed now as expansions where the

background theory gets extended to account for a new fact What is added are cases

where the surprising observation in Peirces sense calls for revision Either way

ts the essential role of the background theory in explanation this p ersp ective highligh

Belief revision theories provide an explicit calculus of mo dication for b oth cases It

must be claried however that changes o ccur only in the theory as the situation

or world to be mo delled is supp osed to be static only new information is coming

in Another imp ortant typ e of epistemic change studied in AI is that of update the

pro cess of keeping b eliefs uptodate as the world changes We will not analyze this

second pro cess here even though we are condent that it can be done in the same

here Evidently all this ties in very neatly with our earlier ndings style prop osed

For instance the theories involved in ab ductive b elief revision might be structured

like those provided by our discussion or by cues from the philosophy of science We

will explore this connection in more detail in chapter

Further Fields of Application

The ab ovesurvey is by no means exhaustive Ab duction o ccurs in many other research

areas of whichwe will mention three linguistics cognitive science and mathematics

the former was indeed an early motivation of this dissertation Although we will not

pursue these lines elsewhere in this dissertation they do provide a b etter p ersp ective

on the broader interest of our topic For instance ab duction in cognitive science is

an interdisciplinary theme relating ab out all areas relevant to a PhD in Symbolic Systems

Further Fields of Application

Ab duction in Linguistics

In linguistics ab duction has been prop osed as a pro cess for natural language inter

pretation HSAM where our observations are the utterances that we hear or

read More precisely interpreting a sentence in discourse is viewed as providing a

b est explanation of why the sentence would b e true For instance a listener or reader

ab duces assumptions in order to resolve for denite the cat

is on the mat invites you to assume that there is a cat and a mat and dynamically

accommo dates them as presupp osed information for the sentence b eing heard or read

Ab duction also nds a place in theories of language acquisition Most prominently

Chomsky prop osed that learning a language is a pro cess of theory construction A

by her innate knowledge of language child ab duces the rules of grammar guided

universals Indeed in Cho he refers to Peirces justication for the logic of ab duc

tion based on the human capacity for guesing the right hyp otheses to reinforce

his claim that language acquisition from highly restricted data is p ossible

Ab duction has also b een used in the semantics of questions Questions are then

the input to ab ductive pro cedures generating answers to them Some work has been

done in this direction in natural language as a mechanism for dealing with indirect

replies to yesno questions GJK Of course the most obvious case where ab duction

is explicitly called for are Why questions inviting the other p erson to provide a

reason or cause

Ab duction also connects up with linguistic presupp ositionswhich are progressively

accommo dated during a discourse Ali treats accommo dation as a nonmonotonic

pro cess in which presupp ositions are not direct up dates for explicit utterances but

rather ab ductions that can b e refuted b ecause of later information Accommo dation

can b e describ ed as a repair strategy in which the presupp ositions to b e accommo dated

are not part of the background In fact the linguistic literature has ner views

of typ es of accommo dation cf the lo calglobal distinction in Hei which

might corresp ond to the two ab ductive triggers prop osed in the next section A

broader study on presupp ositions which considers ab ductive mechanisms and uses

the framework of semantic tableaux to represent the context of discourse is found in

Ger

Chapter What is Ab duction

More generally we feel that the taxonomy prop osed later in this chapter might

correlate with the linguistic diversity of presupp osition triggered by denite descrip

tions verbs adverbs et cetera but we must leave this as an op en question

Ab duction in Cognitive Science

In cognitive science ab duction is a crucial ingredient in pro cesses like inference learn

ing and discovery p erformed by people to build theories of the world surrounding

them There is a growing literature on computer programs mo deling these pro cesses

cf HHN Tha SL Gie

An imp ortant pilot reference is Simon SLB whose authors claim that scientic

Although there is no precise discovery can be viewed as a problemsolving activity

metho d by which scientic discovery is achieved as a form of problem solving it

can be pursued via several metho dologies The authors distinguish between weak

and strong metho ds of discovery The former is the typ e of problem solving used in

novel domains It is characterized by its generality since it do es not require indepth

knowledge of its particular domain In contrast strong metho ds are used for cases

in which our domain knowledge is rich and are sp ecially designed for one sp ecic

structure

Weak metho ds include generation testing heuristic metho ds and meansends

analysis to build explanations and solutions for given problems These metho ds have

proved useful in AI and cognitive simulation and are used by several programs An

example is the BACON system which mo dels explanations and descriptive scientic

laws such as Keplers law Ohms law etcetera It is a matter of debate if BACON

really makes discoveries since it pro duces theories new to the program but not to the

world and its discoveries seem sp o onfed rather than created Be that as it may our

analysis in this dissertation do es not claim to mo del these higher typ es of explanations

and discoveries

orthy reference is found in the new eld of computational philoso Another notew

phy of science Tha and in broader computational cognitive studies of inductive

Further Fields of Application

reasoning HHN These studies distinguish several relevant pro cesses simple ab

duction existential ab duction ab duction to rules and analogical ab duction Wewill

prop ose a similar multiplicityinwhat follows

Ab duction in Mathematics

Ab duction in mathematics is usually identied with notions like discovery and heuris

tics A key reference in this area is the work by the earlier mentioned G Polya

Pol Pol Pol In the context of number theory for example a general prop

erty may b e guessed by observing some relation as in

Notice that the numbers are all o dd primes and that the sum of any of

two of them is an even number An initial observation of this kind eventually led

Goldbach with the help of Euler to formulate his famous conjecture Every even

number greater than two is the sum of two odd primes

Another example is found in a conguration of numb ers suchasinthewellknown

Pascals triangle Pol

There are several hidden prop erties in this triangle which the reader may or

may not discover dep ending on her previous training and mathematical knowledge

A simple one is that any number dierent from is the sum of two other numb ers

in the array namely of its northwest and northeast neighb ors eg A

more complex relationship is this the numb ers in anyrow sum to a p ower of More

precisely

n n

n

   

n

See Pol for more details on ab ducing laws ab out the binomial co ecients in the

Triangle Pascal

Chapter What is Ab duction

The next step is to ask why these prop erties hold and then pro ceed to prove

them Goldbachs conjecture remains unsolved it has not been p ossible to prove

it or to refute it it has only been veried for a large number of cases the latest

news is that it is true for all integers less than cf Rib The results

regarding Pascals triangle on the other hand have many dierent pro ofs dep ending

ones particular taste and knoweldge of geometrical recursive and computational

metho ds Cf Pol for a detailed discussion of alternative pro ofs

According to Polya a mathematician discovers just as a naturalist do es by ob

serving the collection of his sp ecimens numb ers or birds and then guessing their

connection and relationship Polp However the two dier in that verication

by observation for the naturalist is enough whereas the mathematician requires a

pro of to accept her ndings This points to a unique feature of mathematics once a

theorem nds a pro of it cannot b e defeated Thus mathematical truths are eternal

with p ossibly many ways of being explained On the other hand some ndings may

er Ab duction in mathematics shows very well that observ remain unexplained forev

ing go es beyond visual p erception as familiarity with the eld is required to nd

surprising facts Moreover the relationship b etween observation and pro of need not

be causal it is just pure mathematical structure that links them together

Much more complex cases of mathematical discovery can be studied in which

concept formation is involved A recent and interesting approach along these lines is

found in Vis which prop oses a catalogue of pro cedures for creating concepts when

solving problems These include redescription and transp osition

which are explicitly related to Peirces treatment of ab duction

A Taxonomy for Ab duction

What we have seen so far may be summarized as follows Ab duction is a general

pro cess of explanation whose pro ducts are sp ecic explanations with a certain in

ferential structure We consider these two asp ects of equal imp ortance Moreover on

the pro cess side we distinguished b etween constructing p ossible explanations and se

lecting the b est one amongst these This thesis is mainly concerned with the structure

A Taxonomy for Ab duction

of explanations as pro ducts and with the pro cess of constructing these

As for the logical form of ab duction we have found that it may be viewed as a

threefold relation

between an observation an ab duced item andabackground theory Other

parameters are p ossible here such as a preference ranking but these would rather

concern the further selection pro cess Against this background we prop ose three

main parameters that determine typ es of ab duction i An inferential parameter

explananda background theory sets some suitable logical relationship among

and explanandum ii Next triggers determine what kind of ab duction is to be

p erformed may be a novel phenomenon or it may be in conict with the theory

iii Finally outcomes are the various pro ducts of an ab ductive pro cess

facts rules or even new theories

Ab ductive Parameters

Varieties of Inference

In the ab ove schema the notion of explanatory inference is not xed It can be

Instead classical derivability or semantic entailment j but it do es not havetobe

we regard it as a parameter which can be set indep endently It ranges over such

diverse values as probable inference in which the explanandum

pr obabl e

renders the explanandum only highly probable or as the inferential mechanism of logic

programming Further interpretations include dynamic inference

pr ol og

cf vBea replacing by information change p otential

dy namic

along the lines of belief up date or revision Our point here is that ab duction is not

one sp ecic nonstandard logical inference mechanism but rather a way of using any

one of these

Chapter What is Ab duction

Dierent Triggers

According to Peirce as we saw ab ductive reasoning is triggered by a surprising

phenomenon The notion of surprise however is a relative one for a fact is

surprising only with resp ect to some backgound theory providing exp ectations

What is surprising to me eg that the lights go on as I enter the copier ro om

might not b e surprising to you Weinterpret a surprising fact as one which needs an

is not already explanation From a logical p oint of view this assumes that the fact

explained by the background theory

Moreover our claim is that one also needs to consider the status of the negation

of Do es the theory explain the negation of observation instead Thus

we identify at least two triggers for ab duction novelty and anomaly

Ab ductive Novelty

is novel It cannot b e explained but it is consistent with the theory

Ab ductive Anomaly

is anomalous The theory explains rather its negation

In the computational literature on ab duction novelty is the condition for an ab

ductive problem KKT My suggestion is to incorp orate anomaly as a second basic

typ e

Of course nonsurprising facts where should not be candidates for

explanation Even so one might sp eculate if facts which are merely probable on the

basis of might still need explanation of some sort to further cement their status

Dierent Outcomes

Ab ducibles themselves come in various forms facts rules or even theories Some

times one simple fact suces to explain a surprising phenomenon such as rain ex

plaining why the lawn is wet In other cases a rule establishing a causal connection

might serve as an explanation as in our case connecting cloud typ es with rainfall And

A Taxonomy for Ab duction

many cases of ab duction in science provide new theories to explain surprising facts

These dierent options may sometimes exist for the same observation dep ending on

how seriously wewant to take it In this thesis we shall mainly consider explanations

in the forms of atomic facts conjunctions of them and simple conditionals but wedo

make o ccasional excursions to more complex kinds of statements

Moreover we are aware of the fact that genuine explanations sometimes intro duce

new concepts over and ab ove the given vo cabulary For instance th eventual expla

nation of planetary motion was not Keplers but Newtons who intro duced a new

Except notion of force and then derived elliptic motion via the Law of Gravity

for passing references in subsequent chapters ab duction via new concepts will be

outside the scop e of our analysis

Ab ductive Pro cesses

Once the ab ove parameters get set several kinds of ab ductive pro cesses arise For

example ab duction triggered bynovelty with an underlying deductive inference calls

for a pro cess by which the theory is expanded with an explanation The fact to be

explained is consistent with the theory so an explanation added to the theory accounts

deductively for the fact However when the underlying inference is statistical in a

case of novelty theory expansion might not be enough The added might

lead to a marginally consistent theory with low probability which would not yield

a strong explanation for the observed fact In such a case theory revision is needed

ie removing some data from the theory to account for the observed fact with high

probability For a sp ecic example of this latter case cf chapter

Our aim is not to classify ab ductive pro cesses but rather to p oint out that several

kinds of these are used for dierent combinations of the ab ove parameters In the

coming chapters we explore in detail some pro cedures for computing dierent typ es

of outcomes in a deductiv e format those triggered by novelty chapter and those

triggered by anomaly chapter

Chapter What is Ab duction

Examples Revisited

Given our taxonomy for ab ductive reasoning we can now see more patterns across

our earlier list of examples Varying the inferential parameter we cover not only

cases of deduction but also statistical inferences Thus Hemp els statistical mo del of

explanation Hem also b ecomes a case of ab duction Our example of medical

diagnosis Janes quickrecovery was an instance Logic programming inference seems

more appropriate to an example like whose overall structure is the similar to

As for triggers novelty drives b oth rain examples and as well as the medical

diagnosis one A trigger by anomaly o ccurs in example where the theory

contrary of our observation the lightso example and the Kepler predicts the

example since his initial observation of the longitudes of Mars contradicted the

previous rule of circular orbits of the planets As for dierent outcomes examples

and ab duce facts examples and pro duce rules as their forms

of explanantia Dierent forms of outcomes will play a role in dierent typ es of

pro cedures for pro ducing explanations In computer science jargon triggers and

outcomes are resp ectively preconditions and outputs of ab ductive devices whether

be computational pro cedures or inferential ones these

This taxonomy gives us the big picture of ab ductive reasoning In the remainder

of this thesis we are going to investigate various of its asp ects which give rise to

more sp ecic logical and computational questions Indeed more than we have b een

able to answer Before embarking up on this course however we need to discuss

one more general strategic issue which explains the separation of concerns between

chapters and that are to follow

Ab ductive Logic Inference Search Strategy

Classical logical systems ha ve two comp onents a semantics and a pro of theory The

former aims at characterizing what it is for a formula to be true in a mo del and it

is based on the notions of truth and interpretation The latter characterizes what

counts as a valid pro of for a formula by providing the inference rules of the system

having for its main notions pro of and derivability These two formats can b e studied

A Taxonomy for Ab duction

indep endently but they are closely connected At least in classical rstorder logic

the completeness theorem tells us that all true formulas have a pro of and vice versa

Many logical systems have b een prop osed that follow this pattern prop ositional logic

predicate logic mo dal logic and various typ ed

From a mo dern p ersp ective however there is much more to reasoning than this

Computer science has p osed a new challenge to logic that of providing automatic

pro cedures to op erate logical systems This requires a further nestructure of rea

soning In fact recent studies in AI give precedence to a control strategy for a logic

theory In particular the heart of logic programming lies in over its complete pro of

its control strategies which lead to much greater sensitivityas to the order in which

premises are given the avoidance of search lo ops or the p ossibility to cut pro of

searches using the extralogical op erator when a solution has b een found These

features are extralogical from a classical p ersp ective but they do have a clear formal

structure which can b e brought out and has indep endentinterest as a formal mo del

for broader patterns of argumentation cf vBe Kal Kow

Several contemp orary authors stress the imp ortance of control strategies and a

more nelystructured algorithmic description of logics This concern is found b oth in

the logical tradition Gabb vBe and in the philosophical tradition Gil

the latter arguing for a conception of logic as inferencecontrol Note the shift here

away from Kowalskis famous dictum Algorithm Logic Control In line with

this philosophywe wish to approach ab duction with two things in mind First there

is the inference parameter already discussed whichmayhave several interpretations

But given any sp ecic choice there is still a signicant issue of a suitable search

strategy over this inference which mo dels some particular ab ductive practice The

former parameter may be dened in semantic or pro oftheoretic terms The search

pro cedure on the other hand deals with concrete mechanisms for pro ducing valid

inferences It is then p ossible to control which kinds of outcome are pro duced with a

certain eciency In particular in ab duction we may want to pro duce only useful

or interesting formulas preferably even just some minimal set of these

In this light the aim of an ab ductive search pro cedure is not necessarily com

pleteness with resp ect to some semantics A pro cedure that generates all p ossible

Chapter What is Ab duction

explanations mightbeof no practical use and might also miss imp ortant features of

human ab ductive activity In chapter we are going to exp eriment with semantic

tableaux as a vehicle for attractive ab ductive strategies that can be controlled in

various ways

Thesis Aim and Overview

The main aim in this dissertation is to study the notion of ab duction that is rea

soning from an observation to its p ossible explanations from a logical p oint of view

This approach to ab ductive reasoning naturally leads to connections with theories of

explanation in the philosophy of science and to computationally oriented theories of

b elief change in Articial Intelligence

Many dierent approaches to ab duction can b e found in the literature as well as

a b ewildering variety of instances of explanatory reasoning To delineate our sub ject

more precisely and create some order a general taxonomy for ab ductive reasoning

was prop osed in this chapter Several forms of ab duction are obtained by instanti

ating three parameters the kind of reasoning involved eg deductive statistical

the kind of observation triggering the ab duction novelty or anomaly wrt some

background theory and the kind of explanations pro duced facts rules or theories

In chapter I cho ose a number of ma jor variants of ab duction thus conceived and

investigate their logical prop erties A convenient measure for this purp ose are so

called structural rules of inference Ab duction deviates from classical consequence

in this resp ect much like many current nonmonotonic consequence relations and dy

namic styles of inference As a result we can classify forms of ab duction by dierent

structural rules A more computational analysis of pro cesses pro ducing ab ductive

inferences is then presented in chapter using the framework of semantic tableaux

I show how to implement various search strategies to generate various forms of ab

ab ductive pro cesses should be ductive explanations Our eventual conclusion is that

our primary concern with ab ductive inferences their secondary pro ducts Finally

chapter is a confrontation of the previous analysis with existing themes in the phi

losophy of science and articial intelligence In particular I analyse two wellknown

Thesis Aim and Overview

mo dels for scientic explanation the deductivenomological one and the inductive

statistical one as forms of ab duction This then provides them with a structural

logical analysis in the style of chapter Moreover I argue that ab duction can mo del

dynamics of b elief revision in articial intelligence For this purp ose an extended

version of the semantic tableaux of chapter provides a new representation of the

op erations of expansion and contraction

Chapter

Ab duction as Logical Inference

Intro duction

In the preceding overview chapter we have seen how the notion of ab duction arose

in the last century out of philosophical reection on the nature of human reasoning

as it interacts with patterns of explanation and discovery Our analysis brought out

a number of salient asp ects to the ab ductive pro cess which we shall elab orate in a

ed as a kind of number of successive chapters For a start ab duction may be view

logical inference and that is how we will approach it in the analysis to follow here

Evidently though as wehave already p ointed out it is not standard logical inference

and that for a numb er of reasons Intuitively ab duction runs in a backward direction

rather than the forward one of standard inference and moreover being sub ject to

revision it exhibits nonstandard nonmonotonic features ab ductive conclusions may

have to b e retracted in the light of further evidence that are more familiar from the

literature on nonstandard forms of reasoning in articial intelligence Therefore

we will discuss ab duction as a broader notion of consequence in the latter sense

develop ed already for nonmonotonic using some general metho ds that have been

and dynamic logics such as systematic classication in terms of structural rules

This is not a mere technical convenience Describing ab duction in an abstract general

waymakes it comparable to b etterstudied styles of inference thereby increasing our

understanding of its contribution to the wider area of what may be called natural

Chapter Ab duction as Logical Inference

reasoning

The outcomes that we obtain in this rst systematic chapter naturally divided

in three parts are as follows In the rst part sections we prop ose a general

logical format for ab duction involving more parameters than in standard inference

allowing for genuinely dierent roles of premises Wendanumb er of natural styles

of ab duction rather than one single candidate These ab ductive versions are classi

ed by dierent structural rules of inference and this issue o ccupies the second part

sections As a small contribution to the logical literature in the eld wegivea

y also b e viewed complete characterization of one simple style of ab duction whichma

as the rst structural characterization of a natural style of explanation in the philos

ophy of science In the third part of this chapter sections we turn to discuss

further logical issues such as how those representations are related to more familiar

completeness theorems and nally we show how ab duction tends to involve higher

complexity than we stand to gain more explanatory p ower than what

is provided by standard inference but this surplus comes at a price Despite these

useful insights pure logical analysis do es not exhaust all there is to ab duction In

particular its more dynamic pro cess asp ects and its interaction with broader concep

tual change must be left for subsequent chapters that will deal with computational

asp ects as well as further connections with the philosophy of science and articial

intelligence What we do claim however is that our logical analysis provides a sys

tematic framework for studying the intuitive notion of ab duction which gives us a

view of its variety and complexity and whichallows us to raise some interesting new

questions

Here are some preliminary remarks ab out the logical nature of ab ductive inference

which set the scene for our subsequent discussion The standard textb o ok pattern of

logical inference is this Conclusion C fol lows from a set of premises P

Moreover there are at least two ways of thinking ab out in this setting

one semantic based on the notions of mo del and interpretation every mo del in which

P is true makes C true the other syntactic based on a pro oftheoretic derivation of

C from P Both explications suggest forward chaining from premises to conclusions

P C and the conclusions generated are undefeasible We briey recall some features

Directions in Reasoning Forward and Backward

that make ab duction a form of inference which do es not t easily into this format

All of them emerged in the course of our preceding chapter Most prominently in

ab duction the conclusion is the given and the premises or part of them are the

output of the inferential pro cess P C Moreover the abduced has to be

consistent with the background theory of the inference as it has to be explanatory

h explanations may undergo change as we mo dify our background theory And suc

Finally when dierent sets of premises can be ab duced as explanations we need

a notion of preference between them allowing us to cho ose a best or minimal one

These various features though nonstandard when compared with classical logic

are familiar from neighb ouring areas For instance there are links with classical

accounts of explanation in the philosophy of science Car Hem as well as recent

research in artical intelligence on various notions of common sense reasoning McC

for Sho Gaba It has been claimed that this is an appropriate broader setting

general logic as well vBe gaping back to the original program by Bernard Bolzano

in his Wissenschaftslehre Bol Indeed our discussion of ab duction

in Peirce in the preceding chapter reected a typical feature of preFregean logic

b oundaries between logic and general metho dology were still rather uid In our

view current p ostFregean logical research is slowly moving back towards this same

broader agenda More concretelywe shall review the mentioned features of ab duction

in some further detail now making a few strategic references to this broader literature

Directions in Reasoning

Forward and Backward

Intuitively a valid inference from say premises P P to a conclusion C allows for



various directions of thought In a forward direction given the premises we may

want to draw some strongest or rather some most appropriate conclusion Notice

incidentally that the latter notion already intro duces a certain dep endence on context

and good sense the strongest conclusion is simply P P but this will often be



unsuited Classical logic also has a backward direction of thought when engaged

Chapter Ab duction as Logical Inference

in refutation If we know that C is false then at least one of the premises must be

false And if we know more say the truth of P and the falsityofthe conclusion we

may even refute the sp ecic premise P Thus in classical logic there is a duality



between forward pro of and backward refutation This duality has been noted by

many authors It has even b een exploited systematically by Beth when developing

refutation metho d of semantic tableaux Bet Read in one direction closed his

tableaux are eventualy failed analyses of p ossible counterexamples to an inference

read in another they are Gentzenstyle sequent derivations of the inference We shall

be using tableaux in our next chapter on computing ab duction Beth himself to ok

this as a formal mo del of the historical opp osition b etween metho ds of analysis and

synthesis in the developmentofscientic argument Metho dologically the directions

are dierent sides of the same coin namely some appropriate notion of inference

Likewise in ab duction we see an interplay of dierent directions This time

though the backward direction is not meant to provide refutations but rather con

e are lo oking for suitable premises that would supp ort the conclusion rmations W

Our view of the matter is the following In the nal analysis the distinction be

tween directions is a relative one What matters is not the direction of ab duction

but rather an interplay of two things As we have argued in chapter one should

distinguish b etween the choice of an underlying notion of inference and the inde

p endent issue as to the search strategy that we use over this Forward reasoning is a

b ottom up use of while backward reasoning is a topdo wn use of In line with

this in this chapter we shall concentrate on notions of inference leaving further

search pro cedures to the next more computational chapter In this chapter the

intuitively backward direction of ab duction is not crucial to us except as a pleasant

manner of sp eaking Instead we concentrate on appropriate underlying notions of

consequence for ab duction

In this case a corresp onding refutation would rather b e a forward pro cess if the ab duced premise

turns out false it is discarded and an alternativehyp othesis must b e prop osed Interestingly Tij

a recent practical account of ab duction in diagnostic reasoning mixes b oth p ositive conrmation

of the observation to b e explained with refutation of alternatives

Formats of Inference Premises and Background Theory

Formats of Inference

Premises and Background Theory

The standard format of logical inference is essentially binary giving a transition from

premises to a conclusion

P P

n

C

These are lo cal steps which take place in the context of some implicit or ex

plicit background theory as we have seen in chapter In this standard format

the background theory is either omitted or lump ed together with the other premises

Often this is quite appropriate esp ecially when the background theory is under

sto o d But sometimes wedowant to distinguish b etween dierent roles for dierent

typ es of premise and then a richer format b ecomes appropriate The latter have

been prop osed not so much in classical logic but in the philosophy of science arti

cial in telligence and informal studies on These often make

a distinction between explicit premises and implicit background assumptions More

drastically premise sets and even background theories themselves often have a hi

erarchical structure which results in dierent access for prop ositions in inference

This is a realistic picture witness the work of cognitive psychologists like Joh

In Hemp els account of scientic explanation premises play the role of either sci

the entic laws or of initial conditions or of sp ecic explanatory items suggesting

following format

Scientic laws initial conditions explanatory facts

Observation

Further examples are found on the b orderline of the philosophy of science and

in the study of conditionals The famous Ramsey Test presup

p oses revision of explicit b eliefs in the background assumptions Sos vBe which

Chapter Ab duction as Logical Inference

again have to be suitably structured More elab orate hierarchical views of theories

have been prop osed in articial intelligence and computer science Rya denes

ordered theory presentations which can b e arbitrary rankings of principles involved

in some reasoning practice Other implementations of similar ideas use lab els for for

mulas as in the lab elled deductive systems of Gaba While in Hemp els account

structuring the premises makes sure that scientic explanation involves an interplay

of laws and facts Ryans motivation is resolution of conicts between premises in

reasoning where some sentences are more resistant than others to revision This

ation is close to that of the Gardenfors theory to be discussed in chapter A motiv

working version of these ideas is a recent study of ab duction in diagnosis Tij

which can b e viewed as a version of our later account in this chapter with some pref

erence structure added More structured views of premises and theories can also

be found in situation semantics with its dierent typ es of constraints that govern

inference cf PB

In all these prop osals the theory over which inference takes place is not just a

bag into which formulas are thrown indiscriminately but an organized structure in

which premises have a place in a hierarchy and play sp ecic dierent roles These

additional features need to b e captured in richer inferential formats for more compli

cated reasoning tasks Intuitivevalidity may b e partly based on the typ e and status

of the premises that o ccur We cite one more example to elab orate what we have in

mind

In argumentation theoryaninteresting prop osal was made in Tou Toulmins

general notion of consequence was inspired on the kind of reasoning done bylawyers

whose claims need to b e defended according to juridicial pro cedures which are richer

than pure mathematical pro of Toulmins format of reasoning contains the necessary

tags for these pro cedures

Formats of Inference Premises and Background Theory

Qualier

Data Claim

Rebuttal

Warrant

j

Backing

Every claim is defended from certain relevant data by citing if pressed the

background assumptions ones warrant that supp ort this transition There is a

dynamic pro cess here If the warrant itself is questioned then one has to pro duce

ones further backing Moreover indicating the purp orted strength of the inference

is part of making any claim whence the qualier with a rebuttal listing some

main typ es of p ossible exception rebuttal whichwould invalidate the claim vBe

relates this format to issues in articial intelligence as it seems to describ e common

sense reasoning rather well Toulmins mo del has also b een prop osed as a mechanism

for intelligent systems p erforming explanation Ant

Thus once again to mo del reasoning outside of mathematics a richer format is

needed Notice that the ab ove prop osals are syntactic It may be much harder to

nd purely semantic correlates to some of the ab ove distinctions as they seem to

involve reasoning pro cedure rather than prop ositional content For instance even

the distinction between individual facts and universal laws is not as straightforward

as it might seem Various asp ects of the Toulmin schema will return in what follows

Toulmin inferential strength is a parameter to be set in accordance with the For

sub ject matter under discussion Interestingly contentdep endence of reasoning is

also a recurrent nding of cognitive psychologists cf the earliermentioned Joh

In chapter we have already defended exactly the same strategy for ab duction

Moreover the pro cedural avor of the Toulmin schema ts well with our pro duct

pro cess distinction

As for the basic building blo cks of ab ductive inference in the remainder of this

thesis we will conne ourselves to a ternary format

j

Chapter Ab duction as Logical Inference

This mo dest step already enables us to demonstrate a number of interesting depar

tures from standard logical systems Let us recall some considerations from chapter

motivating this move The theory needs to be explicit for a number of reasons

Validityof an ab ductive inference is closely related to the background theory as the

presence of some other explanation in may actually disqualify as an explana

tion Moreover what we called triggers of explanation are sp ecic conditions on a

with resp ect to one theory and an observation A fact may need explanation

theory but not with resp ect to another Making a distinction b etween and allows

us to highlight the sp ecic explanation which we did not have b efore and control

dierent forms of explanation facts rules or even new theories But certainly our

accounts would b ecome yet more sensitiveifweworked with some of the ab overicher

formats

Inferential Strength A Parameter

At rst glance once we have Tarskis notion of truth seems an

obvious dened notion A conclusion follows if it is true in all mo dels where the

premises are true But the contemp orary philosophical and computational traditions

haveshown that natural notions of inference may need more than truth in the ab ove

sense or may even hinge on dierent prop erties altogether For example among the

candidates which revolve around truth statistical inference requires not total inclu

sion of premise mo dels in conclusion mo dels but only a signicantoverlap resulting

in a high degree of certainty Other approaches intro duce new semantic primitives

Notably Shohams notion of causal and default reasoning Sho intro duces a pref

erence order on mo dels requiring only that the most preferredmodels of b e included

in the mo dels of

More radically dynamic semantics replaces the notion of truth by that of in

move leads to a formation change aiming to mo del the ow of information This

redesign for Tarski semantics with eg quantiers b ecoming actions on assignments

vBC This logical paradigm has ro om for many dierent inferential notions

Gro vBea An example is up datetotestconsequence

Inferential Strength A Parameter

process the succesive premisses in thereby absorbing their informational con

tent into the initial information state At the end check if the resulting state is rich

enough to satisfy the conclusion

Informational content rather than truth is also the key semantic prop erty in situ

ation theory PB In addition to truthbased and informationbased approaches

there are of course also various pro oftheoretic variations on standard consequence

Examples are deafult reasoning is provable unless and until is disproved

of scientic inference Rei and indeed Hemp els hyp otheticodeductive mo del

itself

All these alternatives agree with our analysis of ab duction On our view ab duction

is not a new notion of inference It is rather a topicdep endent practice of explanatory

reasoning which can b e supp orted byvarious memb ers of the ab ovefamily In fact it

is app ealing to think of ab ductive inference in several resp ects as inference involving

preservation of b oth truth and In fact appropriately dened

both might turn out equivalent It has also b een argued that since ab duction is a

form of reversed deduction just as deduction is truthpreserving ab duction must b e

falsitypreserving Mic However Flagives convincing arguments against this

particular move Moreover as wehave already discussed intuitively ab duction is not

just deduction in reverse

Our choice here is to study ab ductive inference in more depth as a strengthened

form of classical inference This is relevant it oers nice connections with articial in

telligence and the philosophy of science and it gives a useful simple start for a broader

systematic study of ab ductive inference One can place this choice in a historical con

text namely the work of Bernard Bolzano a nineteenth century philosopher and

mathematician and theologian engaged in the study of dierent varieties of infer

ence Weprovide a brief excursion providing some p ersp ective for our later technical

considerations

Bolzanoss Program

Bolzanos notion of deducibility Ableitbarkeit has long b een recognized as a prede

cessor of Tarskis notion of logical consequence Cor However the two dier in

Chapter Ab duction as Logical Inference

several resp ects and in our broader view of logic they even app ear radically dierent

These dierences have been studied b oth from a philosophical Tho and from a

logical p ointofview vBea

One of Bolzanos goals in his theory of science Bol was to show why the

claims of science form a theory as opp osed to an arbitrary set of prop ositions For

this purp ose he denes his notion of deducibility as a logical relationship extract

ing conclusions from premises forming compatible propositions those for which some

In set of ideas make all prop ositions true when uniformly substitued throughout

addition compatible prop ositions must share common ideas Bolzanos use of sub

stitutions is of interest by itself but for our purp oses here we will identify these

somewhat roughly with the standard use of mo dels Thompson attributes the dif

ference b etween Bolzanos consequence and Tarskis to the fact that the former notion

is epistemic while the latter is ontological These dierences have strong technical

eects With Bolzano the premises must b e consistent sharing at least one mo del

with Tarski they need not Therefore from a contradiction everything follows for

Tarski and nothing for Bolzano

Restated for our ternary format then Bolzanos notion of deducibility reads as

follows cf vBea

j if

The conjunction of and is consistent

Every mo del for plus veries

Therefore Bolzanos notion may be seen anachronistically as Tarskis conse

quence plus the additional condition of consistency Bolzano do es not stop here A

ner grain to deducibility o ccurs in his notion of exact deducibility which imp oses

greater requirements of relevance A mo dern version involving inclusionminimality

for sets of ab ducibles may b e transcrib ed again with some historical injustice as



j if

j

such that j There is no prop er subset of

Requirements for Ab ductive Inference

That is in addition to consistency with the background theory the premise set

must be fully explanatory in that no subpart of it would do the derivation Notice

that this leads to nonmonotonicity Here is an example



j a b a b



j a b a b c b

Bolzanos agenda for logic is relevant to our study of ab ductive reasoning and

the study of general nonmonotonic consequence relations for several reasons It

suggests the metho dological point that what we need is not so much proliferation

dierent logics as a better grasp of dierent styles of consequence Moreover of

his work reinforces an earlier claim that truth is not all there is to understanding

explanatory reasoning More sp ecically his notions still haveinterest For example

exact deducibility has striking similarities to explanation in philosophy of science cf

chapter

Requirements for Ab ductive Inference

In this section we dene ab duction as a strengthened form of classical inference Our

prop osal will be in line with ab duction in articial intelligence as well as with the

Hemp elian account of explanation We will motivate our requirements with our very

simple rain example presented here in classical prop ositional logic

r w s w

w

The rst condition for a formula to count as an explanation for with resp ect

to is the inference requirement Many formulas would satisfy this condition In

addition to earliermentioned obvious explanations r rain s sprinklerson one

mighttake their conjunction with any other formula even if the latter is inconsistent

with eg r w One can take the fact itself w or one can intro duce entirely

new facts and rules say there are children playing with water and this causes the

lawn to get wet

Chapter Ab duction as Logical Inference

Inference j

s r s r s r z r w s w w c c w w

Some of these explanations must be ruled out from the start We therefore

imp ose a consistency requirementon the left hand side leaving only the following as

p ossible explanations

Consistency is consistent

s r s r s r z w c c w w

An explanation is only necessaryif is not already entailed by Otherwise

any consistent formula will count as an explanation Thus we rep eat an earlier trigger

for ab duction j By itself this do es not rule out any p otential ab ducibles on

the ab ove list as it do es not involve the argument But also in order to avoid

what wemaycallexternal explanations those that do not use the background theory

at all like the explanation involving children in our example it must be required

that be insucient for explaining by itself j In particular this condition

avoids the trivial reexive explanation Then only the following explanations

are left in our list of examples

Explanation j j

s r s r s r z w

Now b oth and contribute to explaining However we are still left with some

formulas which do not seem to be genuine explanations r z w Therefore

we explore a more sensitive criterion admitting only the best explanation

Selecting the Best Explanation

Intuitively a reasonable ground for cho osing a statement as the b est explanation is

its It should be minimal ie as weak as p ossible in p erforming its job

This would lead us to prefer r over r z in the preceding example As Peirce puts it

wewant the explanation that adds least to what has been observed cf CP

Requirements for Ab ductive Inference

The criterion of simplicityhas b een extensively considered b oth in the philosophyof

science and in articial intelligence But its precise formulation remains controversial

as measuring simplicity can b e a tricky matter One attempt to capture simplictyin

a logical way is as follows

Weakest Explanation

is the weakest explanation for with resp ect to i

i j

ii For all other formulas such that j j

This denition makes the explanations r and s almost the weakest in the ab ove

example just as wewant Almost but not quite For the explanation w a triv

ial solution turns outtobe the minimal one The following is a folklore observation

to this eect

Fact Given any theory and observation to be explained from it

is the weakest explanation

Proof Obviously we have i j Moreover let be any

other explanation This says that j But then we also have by

conditionalizing that j and hence j a

That is a solution that will always count as an explanation in a deductive

format was noticed by several philosophers of science Car It has b een used as

an argumenttoshowhow the issue would imp ose restrictions on the syntactic form of

ab ducibles Surely in this case the explanation seems to o complex to count Wewill

therefore reject this prop osal noting also that it fails to recognize let alone compare

intuitively minimal explanations like r and s in our running example

Other criteria of minimality exist in the literature One of them is based on pref

erence orderings The b est explanation is the most preferred one given an explicit

ordering of available assertions In our example we could dene an order in which

inconsistent explanations are the least preferred and the simplest the most These

Chapter Ab duction as Logical Inference

preference approaches are quite exible and can accommo date various working in

tuitions However they may still dep end on many factors including the background

theory This seems to fall outside a logical framework referring rather to further eco

nomic decision criteria like utilities AcaseinpointisPeirces economy of research

in selecting a most promising hyp othesis What makes a hyp othesis go o d or b est has

no easy answer One may app eal to criteria of simplicity likeliho o d or predictive

power To complicate matters even further we often do not compare lo cally quality

of explanations given a xed theory but rather globally whole packages of theory

h greater space of options As we have explanation This p ersp ective gives a muc

not b een able to shed a new light from logic up on these matters we will ignore these

dimensions here

Further study would require more rened views of theory structure and reasoning



practice in line with some of the earlier references oreven more ambitiously follow

ing current approaches to verisimilitude in the philosophy of science cf Kui

We conclude with one nal observation p erhaps one reason why the notion of

minimality has proved so elusive is again our earlier pro ductpro cess distinction

Philosophers have tried to dene minimality in terms of intrinsic prop erties of state

ments and inferences as pro ducts But it may rather b e a pro cessfeature having to do

with computational eort in some particular pro cedure p erforming ab duction Thus

one and the same statementmight b e minimal in one ab duction and nonminimal in

another

Ab ductive Styles

Following our presentation of various requirements for ab ductive reasoning wemake

things more concrete for further reference We consider ve versions of ab duction

plain consistent explanatory minimal and preferential dened as follows

Given a set of formulae and a sentence is an ab ductive expla

nation if



Preferences over mo dels though not over statements will b e mentioned briey as providing one

p ossible inference mechanism for ab duction

Requirements for Ab ductive Inference

Plain

i j

Consistent

i j

ii consistent

Explanatory

i j

ii j

iii j

Minimal

i j

ii is the weakest such explanation

Preferential

i j

ii is the b est explanation according to some given preferential

ordering

We can form other combinations of course but these will already exhibit many

characteristic phenomena Note that these requirements do not dep end on classical

consequence For instance in Chapter the consistency and the explanatory re

quirements work just as well for statistical inference The former then also concerns

the explanandum For in probabilistic reasoning it is p ossible to infer twocontra

dictory conclusions even when the premises are consistent The latter helps capture

when an explanation helps raise the probabilityofthe explanandum

A full version of ab duction would make the formula to be ab duced part of the

derivation consistent explanatory and the best p ossible one However instead of

incorp orating all these conditions at once we shall consider them one by one Do

ing so claries the kind of restriction each requirement adds to the notion of plain

ab duction Our standard versions will base these requirements on classical conse

quence underneath But we also lo ok briey toward the end at versions involving

Chapter Ab duction as Logical Inference

other notions of consequence We will nd that our various notions of ab duction have

advantages but also drawbacks such as an increase of complexity for explanatory

reasoning as compared with classical inference

Our more systematic analysis of dierent ab ductive styles uses a logical metho d

ology that has recently b ecome p opular across a range of nonstandard logics

Styles of Inference and Structural Rules

The basic idea of logical structural analysis is the following

A notion of logical inference can be completely characterized by its basic

combinatorial prop erties expressed by structural rules

Structural rules are instructions which tell us eg that a valid inference remains

valid when we insert additional premises monotonicity or that wemay safely chain

valid inferences transitivity or cut This typ e of analysis started in Sco de

scrib es a style of inference at a very abstract structural level giving its pure combi

natorics It has proved very successful in artical intelligence for studying dierent

typ es of plausible reasoning KLM and indeed as a general framework for non

monotonic consequence relations Gab A new area where it has proved itself is

dynamic semantics where not one but many new notions of dynamic consequences

are to b e analyzed vBea

To understand this p ersp ective in more detail one must understand how it char

follows we use logical sequents with a nite acterizes classical inference In what

sequence of premises to the left and one conclusion to the right of the sequentarrow

Classical Inference

The structural rules for classical inference are the following

Reexivity C C

Styles of Inference and Structural Rules

Contraction

X A Y A Z C

X A Y Z C

X A Y A Z C

X Y A Z C

Permutation

X A B Y C

X B A Y C

Monotonicity

X Y C

X A Y C

Cut Rule

X A Y C Z A

X Z Y C

These rules state the following prop erties of classical consequence Any premise

implies itself no trouble is caused by deleting rep eated premises premises may be

p ermuted without altering validity adding new information do es not invalidate pre

vious conclusions and premises may be replaced by sequences of premises implying

them In all these rules allow us to treat the premises as a mere set of data without

further relevant structure This plays an imp ortant role in classical logic witness

what intro ductory textb o oks have to say ab out simple prop erties of the notion of

Chapter Ab duction as Logical Inference

 

consequence Structural rules are also used extensively in completeness pro ofs

These rules are structural in that they mention no sp ecic symb ols of the log

ical language In particular no connectives or quantiers are involved Thus one

rule may t many logics prop ositional rstorder mo dal typ etheoretic etc This

makes them dierent from inference rules like say Conjunction of Consequents or

Disjunction of Antecedents which also x the meaning of conjnction and disjunction

Each rule in the ab ove list reects a prop erty of the settheoretic denition of clas

sical consequence Gro which with some abuse of notation calls for inclusion

of the intersection of the mo dels for the premises in the mo dels for the conclusion

C i P P C P P

n n

Now in order to prove that a set of structural rules completely characterizes a

style of reasoning representation theorems exist For classical logic one version was

proved by van Benthem in vBe

Prop osition Monotonicity Contraction Reexivity and Cut completely

determine the structural properties of classical consequence

Proof Let Rbe any abstract relation b etween nite sequences of ob jects

and single ob jects satisfying the classical structural rules Now dene

a fA j A is a nite sequence of ob jects such that ARag



In MenPage the following simple prop erties of classical logic are intro duced

If and then

i there is a nite subset of such that

If x for all i and x x then

i n

Notice that the rst is a form of Monotonicity and the third one of Cut



As noted in Gro page In the Henkin construction for rstorder logic or prop ositional

mo dal logic the notion of maximal consistent set plays a ma jor part but it needs the classical

structural rules For example Permutation Contraction and Expansion enable you to think of the

premises of an argumentasa set Reexivityis needed to show that for maximal consistent sets

memb ership and derivability coincide

Styles of Inference and Structural Rules

Then it is easy to show the following two assertions

If a a Rb then a a b

k

k

using Cut and Contraction

If a a b then a a Rb

k

k

using Reexivity and Monotonicity a

Permutation is omitted in this theorem And indeed it turns out to be derivable

from Monotonicityand Contraction

NonClassical Inference

For nonclassical consequences classical structural rules may fail A wellknown ex

ample are the ubiquitous nonmonotonic logics However this is not to say that

no structural rules hold for them The point is rather to nd appropriate reformu

lations of classical principles or even entirely new structural rules that t other

styles of consequence For example many nonmonotonic typ es of inference satisfy a

weaker form of monotonicity Additions to the premises are allowed only when these

premisses imply them

Cautious Monotonicity

X A X C

X A C

Dynamic inference is nonmonotonic inserting arbitrary new pro cesses into a

premise sequence can disrupt earlier eects But it also quarrels with other classical

structural rules such as Cut But again representation theorems exist Thus the

earlier dynamic style known of up datetotest is characterized by the following re

stricted forms of monotonicity and cut in which additions and omissions are licensed

only to the left side

Left Monotonicity

X C

A X C

Chapter Ab duction as Logical Inference

Left Cut

X C X C Y D

X Y D

For a broader survey and analysis of dynamic styles see Gro vBea For

sophisticated representation theorems in the broader eld of nonclassical inference

in articial intelligence see Mak KLM Yet other uses of nonclassical structural

rules o ccur in relevant logic linear logic and categorial logics cf DH vBe

Characterizing a notion of inference in this way determines its basic rep ertoire for

handling arguments Although this do es not provide a more ambitious semantics or

even a full pro of theory it can at least provide valuable hints The suggestiveGentzen

style format of the structural rules turns into a if appropriately

extended with intro duction rules for connectives However it is not always clear how

to do so in a natural manner as we will discuss later on in connection with ab duction

We will lo ok at these matters for ab duction in a moment But since this p ersp ec

tive may still b e unfamiliar to many readers we provide a small excursion

Are nonclassical inferences really logical

Structural analysis of consequence relations go es back to Bolzanos program of chart

ing dierent styles of inference It has even been prop osed as a distinguished enter

prise of Descriptive Logic in Fla However many logicians remain doubtful and

withhold the status of b ona de logical inference to the pro ducts of nonstandard

styles

This situation is somewhat reminiscent of the emergence of noneuclidean geome

tries in the nineteenth century Euclidean geometry was thought of as the one and

only geometry until the fth p ostulate the parallel axiom w as rejected giving rise

to new geometries Most prominentlytheoneby Lobachevsky which admits of more

than one parallel and the one by Riemann admitting none The legitimacy of these



geometries was initially doubted but their impact gradually emerged In our con

text it is not geometry but styles of reasoning that o ccupy the space and there is



The with logic can be carried even further as these new geometries were sometimes lab eled metageometries

Structural Rules For Ab duction

not one p ostulate under critical scrutiny but several Rejecting monotonicity gives

rise to the family of nonmonotonic logics and rejecting p ermutation leads to styles

of dynamic inference Linear logics on the other hand are created by rejecting con

traction All these alternative logics might get their empirical vindication to o as

reecting dierent modes of human reasoning

Whether nonclassical mo des of reasoning are really logical is like asking if non

euclidean geometries are really geometries The issue is largely terminological and

decide as Quine did on another o ccasion cfQui to just give con we might

servatives the word logic for the more narrowly describ ed variety using the word

reasoning or some other suitable substitute for the wider brands In any case an

analysis in terms of structural rules do es help us to bring to lightinteresting features

of ab duction logical or not

Structural Rules For Ab duction

In this section we provide structural rules for dierent versions of ab duction with

classical consequence underneath Plain ab duction is characterized by classical in

ference A complete characterization for consistent ab duction is provided For the

explanatory and preferential versions we just give some structural rules and sp eculate

ab out their complete characterization

Consistent Ab duction

We recall the denition

j i

i j

ii are consistent

The rst thing to notice is that the two items to the left b ehave symmetrical ly

j i j

Chapter Ab duction as Logical Inference

Indeed in this case wemay technically simplify matters to a binary format after

all X C in which X stands for the conjunction of and and C for To bring

these in line with the earliermentioned structural analysis of nonclassical logics we

view X as a nite sequence X X of formulas and C as a single conclusion

k

Classical Structural Rules

Of the structural rules for classical consequence contraction and permutation hold

for consistent ab duction But reexivity monotonicity and cut fail witness by the

following counterexamples

Reexivity p p p p

Monotonicity p p but p p p

Cut p q pand p q q but p q q q

New Structural Rules

Here are some restricted versions of the ab ove failed rules and some others which are

valid for consistent ab duction

Conditional Reexivity CR

X B

i k

X X

i

Simultaneous Cut SC

U A U A A A B

k k

U B

Conclusion Consistency CC

U A U A

k

i k

A A A

k i

Structural Rules For Ab duction

These rules state the following Conditional Reexivity requires that the sequence

X derive something else X B as this ensures consistency Simultaneous Cut is a

combination of Cut and Contraction in which the sequent A A may b e omitted

k

in the conclusion when each of its elements A is consistently derived by U and this

i

one in its turn consistently derives B Conclusion Consistency says that a sequent

A A implies its elements if each of these are implied consistently by something

k

U arbitrary which is another form of reexivity

Prop osition These rules are sound for consistent abduction

of In each of these three cases it is easy to check by simple set Pro

theoretic reasoning that the corresp onding classical consequence holds

Therefore the only thing to be checked is that the premises mentioned

in the conclusions of these rules must be consistent For Conditional Re

exivity this is because X already consistently implied something For

Simultaneous Cut this is b ecause U already consistently implied some

thing Finally for Conclusion Consistency the reason is that U must be

consistent and it is contained in the intersection of all the A which is

i

therefore consistent to o a

A Representation Theorem

The given structural rules in fact characterize consistent ab duction

Prop osition Aconsequencerelation satises the structural rules CR

SC CC i it is representable in the form of consistent abduction

Proof Soundness of the rules was proved ab ove Now consider the com

pleteness direction Let be any abstract relation satisfying

Dene for any prop osition A

A fX j X Ag

Wenowshow the following statement of adequacy for this representation

Chapter Ab duction as Logical Inference

Claim A A B i A A B

k

Proof Only if Since A A B by Rule CR we have

k

T

A A A i k Therefore A A A for each

k i k

i

i with i k which gives the prop er inclusion Next let U be any

T

A i k That is U A U A By Rule sequence in

k

i

SC U B ie U B and we have shown the second inclusion

T

If Using the assumption of nonemptiness let say U A

i

i k ie U A U A By Rule CC A A

k k

B By A i k By the second inclusion then A A

k i

the denition of the function this means that A A B a

k

More Familiar Structural Rules

The ab ove principles characterize consistent ab duction Even so there are more famil

iar structural rules which are valid as well including mo died forms of Monotonicity

and Cut For instance it is easy to see that satises a form of mo died mono

tonicity B may b e added as a premise if this addition do es not endanger consistency

And the latter may be shown by their implying any conclusion

Mo died Monotonicity

X A X B C

X B A

As this was not part of the ab ove list we exp ect some derivation from the ab ove

principles And indeed there exists one

Mo died Monotonicity Derivation

X B C

X B X s X A

i

X B A

Structural Rules For Ab duction

These derivations also help in seeing how one can reason p erfectly well with non

classical structural rules Another example is the following valid form of Mo died

Cut

Mo died Cut

X A U A V B U X V C

U X V B

This maybe derived as follows

Mo died Cut Derivation

U X V C

U X V X s X A

i

U X V C

U X V A U A V B

U X V U s V s

U X V B

Finally we check some classically structural rules that do remain valid as they

stand showing the power of Rule

Permutation

X A B Y C

X A B Y X A B Y separately

X B A Y X B A Y separately

X A B Y C

X B A Y C

Contraction one sample case

X A A Y B

X A A Y X s A Y s

i i

X A Y X s A Y s

X A A Y B

i i

X A Y B

Thus consistent ab duction dened as classical consequence plus the consistency

requirement has appropriate forms of reexivity monotonicity and cut for which it

Chapter Ab duction as Logical Inference

is assured that the premises remain consistent Permutation and contraction are not

aected by the consistency requirement therefore the classical forms remain valid

More generally the preceding examples show simple ways of mo difying all classical

structural principles by putting in one extra premise ensuring consistency

Simple as it is our characterization of this notion of inference do es provide a

complete structural description of Bolzanos notion of deducibilityintro duced earlier

in this chapter section

Explanatory Ab duction

Explanatory ab duction was dened as plain ab duction plustwo conditions

However we will consider a weaker of necessity and insuciency

version which only considers the former condition and analyze its structural rules

This is actually somewhat easier from a technical viewp oint The full version remains

of general interest though as it describ es the necessary collab oration of two premises

set to achieve a conclusion It will b e be analyzed further in chapter in connection

with philosophical mo dels of scientic explanation We rephrase our notion as

Weak Explanatory Ab duction

j i

i j

ii

The rst thing to notice is that we must leave the binary format of premises and

conclusion This notion is nonsymmetric as and have dierent roles Given

such a ternary format we need a more nely grained view of structural rules For

instance there are now two kinds of monotonicity one when a formula is added to

the explanations and the other one when it is added to the theory

MonotonicityforExplanations

j

j A

Structural Rules For Ab duction

Monotonicityfor Theories

j

A j

The former is valid but the latter is not A counterexample is p j q r q but

p q j r q Monotonicity for explanations states that an explanation for a fact do es

not get invalidated when we strengthen it as long as the theory is not mo died

Here are some valid principles for weak explanatory ab duction

Weak Explanatory Reexivity

j

j

Weak Explanatory Cut

j j

j

In addition the classical forms of contraction and permutation are valid on each

side of the bar Of course one should not p ermute elements of the theory with

those in the explanation slot or vice versa We conjecture that the given principles

completely characterize the weak explanatory ab duction notion when used together

with the ab ove valid form of monotonicity

Structural Rules with Connectives

Pure structural rules involve no logical connectives Nevertheless there are natural

be used in the setting of ab ductive consequence For instance connectives that may

all Bo olean op erations can be used in their standard meaning These to o will give

rise to valid principles of inference In particular the following wellknown classical

laws hold for all notions of ab ductive inference studied so far

Chapter Ab duction as Logical Inference

Disjunction of antecedents

j A j A



j A



Conjunction of Consequents

j A j A



j A



These rules will play a role in our prop osed calculus for ab duction as we will show

later on

We conclude a few brief p oints on the other versions of ab duction on our list We

have not undertaken to characterize these in any technical sense

Minimal and Preferential Ab duction

Consider our versions of minimal ab duction One said that j and is the

weakest such explanation By contrast preferential ab duction said that j

and is the best explanation according to some given preferential ordering For the

former with the exception of the ab ove disjunction rule for antecedents no other rule

that we have seen is valid But it do es satisfy the following form of transivity

TransitivityforMinimal Ab duction

j j

j

For preferential ab duction on the other hand no structural rule formulated so

far is valid The reason is that the relevant preference order amongst formulas itself

needs to be captured in the formulation of our inference rules A valid formulation

of monotonicitywould then b e something along the following lines

Structural Rules For Ab duction

Monotonicityfor Preferential Ab duction

j

j

In our this is no longer a structural rule since it adds a mathematical

relation that cannot in general b e expressed in terms of the consequence itself This

is a p ointof debate however and its solution dep ends on what each logic artesan is

willing to represent in a logic In any case this format is b eyond what we will study

in this thesis It would be a good source though for the heterogeneous inference

that is coming to the fore these days BR

Structural Rules for Nonstandard Inference

underneath In this section All ab ductive versions so far had classical consequence

we briey explore structural behaviour when the underlying notion of inference is

non standard as in preferential entailment Moreover we throwinsomewords ab out

structural rules for ab duction in logic programming and for induction

Preferential Reasoning

Interpreting the inferential parameter as preferential entailment means that

if only the most preferred mo dels of are included in the mo dels of This

leads to a completely dierent set of structural rules Here are some valid examples

transcrib ed into our ternary format from KLM

Reexivity

Cautious Monotonicity

j j

j

Cut

j

j

Chapter Ab duction as Logical Inference

Disjunction

j j

j

We have also investigated in greater detail what happ ens to these rules when we

add our further conditions of consistency and explanation cf Ali for a reasoned

table of outcomes In all what happ ens is merely that we get structural mo dications

similar to those found earlier on for classical consequence Thus a choice for a

preferential pro of engine rather than classical consequence seems orthogonal to the

behavior of ab duction

Structural rules for Prolog Computation

An analysis via structural rules may b e also p erformed for notions of with a more

pro cedural avor In particular the earliermentioned case of Prolog computation

ob eys clear structural rules cf vBe Kal Min Their format is somewhat

as one needs to represent more of the Prolog program dierent from classical ones

structure for premises including information on rule heads Also Kalsb eek Kal

gives a complete calculus of structural rules for logic programming including such

control devices as the cut op erator The characteristic expressions of a Gentzen

style sequent calculus for these systems in the reference ab ove are sequents of the

form P where P is a prop ositional Horn clause program and is an atom A

failure of a goal is expressed as P meaning that nitely fails In this case

valid monotonicity rules must take account of the place in which premises are added

as Prolog is sensitive to the order of its program clauses Thus of the follo wing rules

the rst one is valid but the second one is not

Right Monotonicity

P

P

Left Monotonicity

P

P

Counterexample

Structural Rules For Ab duction

The question of complete structural calculi for ab ductive logic programming will

not be addressed in this thesis we will just mention that a natural rule for an ab

ductive up date is as follows

Atomic Ab ductive Up date

P

P

We will briey return to structural rules for ab duction as a pro cess in chapter

Structural Rules For Induction

Unlike ab duction enumerative induction is a typ e of inference that explains a set of

observations and makes a prediction for further ones cf our discussion in chapter

Our previous rule for conjunction of consequents already suggests how to give

an account for further observations provided that weinterpret the commas b elowas

conjunction amongst formulae in the usual Gentzen calculus commas to the right

are interpreted rather as disjunctions





That is an inductive explanation for remains an explanation when a formula

is added provided that also accounts for it separately Note that this rule is a



kind of monotonicity but this time the increase is on the conclusion set rather than

on the premise set More generally an inductive explanation for a set of formulae

remains valid for more input data when it explains it

Inductive Monotonicityon Observations

j j

n

j

n

In order to put forward a set of rules characterizing inductive explanation a

further analysis of its prop erties should be made and this falls beyond the scop e of

Chapter Ab duction as Logical Inference

this thesis What we anticipate however is that a study of enumerative induction

from a structural p oint of view will bring yet another twist to the standard structural

analysis that of giving an account of changes in conclusions

Further Logical Issues

Our analysis so far has only scratched the surface of a broader eld In this section

we discuss a number of more technical logical asp ects of ab ductivestyles of inference

This identies further issues that seem relevant to understanding the logical prop erties

of ab duction

Completeness

The usual completeness theorems have the following form

j i

a p os With our ternary format we would exp ect some similar equivalence with

sibly dierent treatment of premises on dierent sides of the comma

j i

Can we get such completeness results for any of the ab ductive versions we have

describ ed so far Here are two extremes

The representation arguments for the ab ove characterizations of ab duction may

be reworked into completeness theorems of a very simple kind This works just as

in vBea chapter In particular for consistent ab duction our earlier argument

essentially shows that follows from a set of ternary sequents iitcan be

derived from using only the derivation rules CR SC CC ab ove

These representation arguments may be viewed as p o or mans completeness

pro ofs for a language without logical op erators Richer languages arise by adding

op erators and completeness arguments need corresp onding upgrading of the repre

sentations used Cf Kur for an elab orate analysis of this upward route for the

Further Logical Issues

case of categorial and relevant logics Gro considers the same issue in detail for

dynamic styles of inference At some level no more completeness theorems are to

b e exp ected The complexity of the desired pro of theoretical notion will usually b e



recursively enumerable But our later analysis will show that with a predicate

logical language the complexity of semantic ab duction j will b ecome higher than

that The reason is that it mixes derivability with nonderivability b ecause of the

consistency condition

So our b est chance for achieving signicant completeness is with an intermediate

is still decidable language like that of prop ositional logic In that case ab duction

and we may hop e to nd simple pro of rules for it as well Cf Tam for the

technically similar enterprise of completely axiomatizing simultaneous pro ofs and

in prop ositional logic Can we convert our representation arguments into

fulledged completeness pro ofs when we add prop ositional op erators We

have already seen that we do get natural valid principles like disjunction of antecedents

and conjunction of consequents However there is no general metho d that connects

a representational result into more familiar prop ositional completeness arguments A

case of succesful though nontrivial transfer is in Kan but essential diculties

are identied in Gro

Instead of solving the issue of completeness here we merely prop ose the following

axioms and rules for a sequent calculus for consistent ab duction

Axiom p j p

Rules for Conjunction

j j



j



The following are valid provided that are formulas with only p ositive prop o

sitional letters

j j



j

Chapter Ab duction as Logical Inference

j



j

Rules For Disjunction

j j



j



j



j

j



j

Rules for Negation

A j

j A

j A A j



A j

It is easy to see that these rules are sound on the interpretation of j as consistent

ab duction This calculus is already unlike most usual logical systems though First of

all there is no substitution rule as p j p is an axiom whereas in general j unless

has only p ositive prop ositional letters in which case it is proved to be consistent

By itself this is not dramatic for instance several mo dal logics exist without a valid

substitution rule but it is certainly uncommon Moreover note that the rules which

move things to the left are dierent from their classical counterparts and



others are familiar but here a condition to ensure consistency is added Even so



one can certainly do practical work with a calculus like this

Further Logical Issues

For instance all valid principles of classical prop ositional logic that do not involve

negations are derivable here Semantically this makes sense as p ositive formulas are

always consistent without sp ecial precautions On the other hand it is easy to check

that the calculus provides no pro of for a typically invalid sequentlike p p j p p

Digression A general semantic view of ab ductive consequence

Sp eaking generally we can view a ternary inference relation j as a ternary

relation C T A F between sets of mo dels for resp ectively and What

structural rules do is constrain these relations to just a sub class of all p ossibilities

This typ e of analysis has with the theory of generalized quantiers in

natural language semantics It may be found in vBea on the mo del theory of

verisimilitude or in vBeb on general consequence relations in the philosophy of

science When enough rules are imp osed wemay represent a consequence relation by



means of simpler notions involving only part of the a priori relevant regions

of mo dels induced by our three argument sets

In this light the earlier representation arguments might even be enhanced by

including logical op erators We merely provide an indication It can be seen easily

that in the presence of disjunction our explanatory ab duction satises full Bo olean

Distributivity for its ab ducible argument

i

W

j i for some i j

i i

i

Principles like this can b e used to reduce the complexity of a consequence relation

For instance the predicate argument A may now be reduced to a pointwise one as

any set A is the union of all singletons fag with a A

Complexity

Our next question addresses the complexity of dierent versions of ab duction Non

monotonic logics may be b etter than classical ones for mo delling common sense rea

soning and scientic inquiry But their gain in expressive power usually comes at

the price of higher complexity and ab duction is no exception Our interest is then

Chapter Ab duction as Logical Inference

to briey compare the complexity of ab duction to that of classical logic We have

no denite results here but we do have some conjectures In particular we lo ok at

consistent ab duction b eginning with predicate logic

Predicatelogical validity is undecidable byChurchs Theorem Its exact complex



ityis the validities are recursively enumerable but not recursive To understand

this outcome think of the equivalent assertion of derivability there exists aPPis

a pro of for More generally or notation refers to the usual prenex forms

Complexity is measured for denability of notions in the Arithmetical Hierarchy

here by lo oking at the quantier prenex followed by a decidable matrix predicate A

subscript n indicates n quantier changes in the prenex If a notion is b oth and

n

it is called The complementary notion of satisability is also undecidable

n n



b eing denable in the form Now ab ductive consequence moves further up in this

hierarchy



Prop osition Consistent Abduction is complete





Proof The statementthat is consistent is while the statement



that j is cf the ab ove observ ations Therefore their

conjunction may b e written using wellknown prenex op erations in either

of the following forms

DE C or DE C



Hence consistent ab duction is in This analysis gives an upp er bound



only But we cannot do b etter than this So it is also a lower b ound For



the sake of reductio supp ose that consistent ab duction were Then we

could reduce satisability of any formula B eectively to the ab ductive

consequence B B B and hence we would have that satisability is

 

also But then Posts Theorem says that a notion which is b oth



and must b e decidable This is a contradiction and hence is

 

not Likewise consistent ab duction cannot be because of another

reduction this time from the validity of any formula B to True True

B a

Further Logical Issues

By similar arguments we can show that the earlier weak explanatory ab duction is



also and the same holds for other variants that we considered Therefore our



strategy in this chapter of adding amendments to classical consequence is costly as

it increases its complexity On the other hand we seem to pay the price just once

It makes no dierence with resp ect to complexity whether we add one or all of the

ab ductive requirements at once Wedonothave similar results ab out the cases with

complexity of our minimalityand preference as their complexity will dep end on the

unsp ecied preference order

Complexitymay be lower in a number of practically imp ortant cases First con

sider p o orer languages In particular for propositional logic all our notions of ab duc

tion remain obviously decidable Nevertheless their nestructure will be dierent

Prop ositional satisability is NPcomplete while validity is CoNPcomplete We con

jecture that consistent ab duction will be complete this time in the Polynomial



Hierarchy

Another direction would restrict attention to useful fragments of predicate logic

For example universal clauses without function symbols have a decidable consequence

problem Therefore we have the following

notions of abduction are decidable over universal Prop osition All our

clauses

Finally complexity as measured in the ab ove sense may miss out on some good

features of ab ductive reasoning such as p ossible natural bounds on search space for

ab ducibles A very detailed study on the complexity of logicbased ab duction which

takes into account dierent kinds of theories prop ositional clausal Horn as well as

several minimality measures is found in EG

The Role of Language

Our notions of ab duction all work for arbitrary formulas and hence they have no

toward any sp ecial formal language But in practice we can often do with simpler

forms Eg observations will often b e atoms and the same holds for explanations

Here are a few observations showing what may happ en

Chapter Ab duction as Logical Inference

Syntactic restrictions may make for sp ecial eects For instance our discussion

of minimal ab duction contained Carnaps trick which shows that the choice of

will always do for a minimal solution But notice that this trivialization

no longer works when only atomic explanations are allowed

Here is another example Let consist of prop ositional Horn clauses only In

that case we can determine the minimal ab duction for an atomic conclusion directly

A simple example will demonstrate the general metho d

Let fq r s p s q p t q g and fq g

q r s p s q p t q q

i jp s q p t q q

ii jp s p t q

That is rst make the conjunction of all formulas in having q for head and

construct the implication to q i obtaining a formula whichisalready an ab ductive

solution a slightly simpler form than Then construct an equivalent simpler

formula ii of which each disjunct is also an ab ductive solution Note that one of

them is the trivial one Thus it is relatively easier to p erform this pro cess over a

simple theory rather than having to engage in a complicated reasoning pro cess to

pro duce ab ductive explanations

Finallywe mention another partly linguistic partly ontological issue that comes

up naturally in ab duction As philosophers of science have observed there seems to

be a natural distinction b etween individual facts and general laws in explanation

Roughly sp eaking the latter b elong to the theory while the former o ccur as ex

plananda and explanantia But intuitively the logical basis for this distinction do es

not seem to lie in but rather in the nature of things How could we make

such a distinction Fla mentions this issue as one of the ma jor op en questions in

understanding ab duction and even its implementations Here is what we think has

to be the way to go Explanations are sought in some sp ecic situation where we

can check sp ecic facts Moreover we adduce general laws not tied to this situation

Discussion and Conclusions

whichinvolve general reasoning ab out the kind of situation that we are in The latter

picture is not what is given to us by classical logic Wewould rather have to think of a

mixed situation as in say the computer program Tarskis World cf BE where

we have two sources of information One is direct querying of the current situation

the other general deduction provided that it is sound with resp ect to this situation

The prop er format for ab duction then b ecomes a mixture of theorem proving and

mo del checking cf SUM Unfortunatelythiswould go far b eyond the b ounds

of this dissertation

Discussion and Conclusions

Studying ab duction as a kind of logical inference has provided much more detail to

the broad schema in the previous chapter Dierent conditions for a formula to count

a genuine explanation gave rise to dierent ab ductive styles of inference More as

over the latter can b e used over dierent underlying notions of consequence classical

preferential statistical The resulting ab ductive logics have links with existing pro

posals in the philosophy of science and even further back in time with Bolzanos

notion of deducibility They tend to be nonmonotonic in nature Further logical

analysis of some key examples revealed many further structural rules In particular

consistent ab duction was completely characterized Finally we have discussed pos

sible complete systems for sp ecial kinds of ab duction as well as the complexity of

ab duction in general

Here is what we consider the main outcomes of our analysis We can see ab duc

tive inference as a more structured form of consequence whose behavior is dierent

from classical logic but which still has clear inferential structure The mo dications

of classical structural rules which arise in this pro cess may even be of interest by

themselves and we see this whole area as a new challenge to logicians Note that

y sp ecic set of mo died we did not lo cate the logical character of ab duction in an

structural rules If pressed we would say that some mo died versions of Reexivity

Monotonicity and Cut seem essential but we have not been able to nd a single

formulation that would stand once and for all Cf Gabb for a fuller discussion

Chapter Ab duction as Logical Inference

of the latter p oint Another noteworthy point was our ternary format of inference

whichgives dierent roles to the theory and explanation on the one hand and to the

conclusion on the other This leads to nergrained views of inference rules whose

interest has b een demonstrated

Summarizing we have shown that ab duction can be studied with prot as a

purely logical notion of inference Of course we have not exhausted this viewp oint

here but we must leave its full exploration to real logicians Also we do not claim

hapter that this analysis exhausts all essential features of ab duction as discussed in c

To the contrary there are clear limitations to what our present p ersp ective can

achieve While wewere successful in characterizing what an explanation is and even

showhow it should b ehave inferentially under addition or deletion of information the

generation of ab ductions was not discussed at all The latter pro cedural enterprise is

the topic of our next chapter Another clear limitation is our restriction to the case

of novelty where there is no conict between the theory and the observation For

the case of anomaly we need to go into theory revision as will happ en in chapter

That chapter will also resume some threads from the present one including a full

version of ab duction in which all our cumulative conditions are incorp orated The

latter will be needed for our discussion of Hemp els deductivenomological mo del of

explanation

Further Questions

We nally indicate a few further issues that wehave considered in our work but that

did not make it into our main exp osition These take the form of op en questions or

merely promising directions

To provide complete structural characterizations of all ab ductive styles put

forward in this chapter In particular to characterize full explanatory ab duction

which accumulates all our constraints

To relate our deviant structural rules to sp ecic search strategies for ab duction

Toprovide complete calculi for our styles of ab duction with additional logical

connectives

Related Work

To provide another analysis of ab duction not via the amendment strategy

of this chapter but via some new semantic primitivesasisdoneinintuitionistic or

relevant logic This might also decrease complexity

To explore purely pro oftheoretic approaches where ab duction serves to ll

gaps in given arguments The relevant parameters will then also include some argu

ment and not just sets of assertions

In line with the previous suggestion to give a full exploration of ab duction in

the setting of Toulmins argumentation theory

To analyze ab ductions where the explanation involves changing vo cabulary

interp o Even Bolzano already considered such inferences which may be related to

lation theorems in standard logic More ambitiously the anomaly version of this

would lead to logical theories of concept revision

Related Work

Ab duction has b een recognized as a nonmonotonic logic but with few exceptions no

study has been made to characterize it as a logical inference In Kon a general

theory of ab duction is dened as classical inference with the additional conditions of

consistency and minimality and it is proved to b e implied by Reiters causal theories

Rei in which a diagnosis is a minimal set of abnormalities that is consistent with

the observed behaviour of a system Another approach closer to our own though

develop ed indep endentlyisfoundinPeter Flachs PhD dissertation Conjectures an

inquiry concerning the logic of induction Fla whichwewillnow briey describ e

and compare to our work some of what follows is based on the more recent version

of his prop osal Flaa

Flachs logic of induction

Flachs thesis is concerned with a logical study of conjectural reasoning comple

mented with an application to relational databases An inductive consequence rela

tion LxL L a prop ositional language is a set of formulae interpreted

Chapter Ab duction as Logical Inference

as is a p ossible inductivehyp othesis that explains or as is a p ossible induc

tive hyp othesis conrmed by The main reason for this distinction is to dissolve

the paradoxical situation p osed by Hemp els adequacy conditions for conrmatory

reasoning Hem Hem namely that in which a piece of evidence E could con



rm any hyp othesis whatso ever Therefore two systems are prop osed one for the

logic of conrmation and the other for the logic of explanation each one provided

representation theorem for its characterization These two sys with an appropriate

tems share a set of inductive principles and dier mainly in that explanations may

be strengthened without ceasing to b e explanations H and conrmed hyp otheses

may be weakened without b eing disconrmed H To give an idea of the kind of

principles these systems share we show two of them the wellknown principles of

verication and falsication in Philosophy of Science

I If and j then

I If and j then

They state that when a hyp othesis is tentatively concluded on the basis of

evidence and a prediction drawn from and is observed then counts as a

hyp othesis for both and I and not for and I a consequence of the

latter is that reexivity is only valid for consistent formulae

Comparison to our work

Despite dierences in notation and terminology Flachs approach is connected to

ours in several ways Its philosophical motivation is based on Peirce and Hemp el

its metho dology is also based on structural rules and we agree that the relationship

between explananda and explanandum is a logical parameter rather than xed to



This situation arises from accepting reexivity H any observation rep ort is conrmed by

itself and stating on the one hand that if an observation rep ort conrms a hyp othesis then it also

conrms every consequence of it H and on the other that if an observation rep ort conrms a

hyp othesis then it also conrms every formula logically entailing it H

Related Work

deduction and on the need for complementing the logical approach with a computa

tional p ersp ective Once wegetinto the details however our prop osals present some

fundamental dierences from a philosophical as well as a logical p ointofview

Flach departs from Hemp els work on conrmation Hem Hem while ours

is based on later prop osals on explanation HO Hem This leads to a dis

crepancy in our basic principles One example is consistent reexivity a general

inductive principle for Flach but rejected by us for explanatory ab duction since one

of Hemp els explanatory adequacy conditions implies that it is invalid cf chapter

Note that this prop erty reects a more fundamental dierence between conrmation

not explain and explanation than H and H evidence conrms itself but it do es



itself There are also dierences in the technical setup of our systems Although

Flachs notion of inductive reasoning may be viewed as a strengthened form of logi

cal entailment the representation of the additional conditions is explicit in the rules

rather than within the consequence relation For example in his setting consistency

is enforced by adding the condition of reexivity to the rules which require it

recall that reexivity is only allowed for consistent formulae a style reecting the

metho dology of KLM

Nevertheless there are interesting analogies between the t wo approaches which

we must leave to future work We conclude with a general remark A salient p oint

in b oth our approaches is the imp ortance of consistency also crucial in Hemp els

adequacy conditions both for conrmation and explanation and in AI approaches

to ab duction Thus Bolzanos notion of deducibility comes back as capturing an

intrinsic prop erty of conjectural reasoning in general



Flach correctly p oints out that Hemp els own solution to the was to drop condition

H from his logic of conrmation Our observation is that the fact that Hemp el later develop ed

an indep endent account for the logic of explanation HO Hem suggests he clearly separated

conrmation from explanation In fact his logic for the latter diers in more principles than the

ones mentioned ab ove

Chapter

Ab duction as Computation

Intro duction

Our logical analysis of ab duction in the previous chapter is in a sense purely struc

tural It was p ossible to state how ab ductive logic b ehaves but not how ab ductions

are generated In this chapter we turn to the question of ab duction as a computa

tional pro cess There are several frameworks for computing ab ductions two of which

are logic programming and semantic tableaux The former is a p opular one and it

has op ened a whole eld of ab ductive logic programming KKT The latter has

also been prop osed for handling ab duction MP and it is our preference here

Semantic tableaux are a wellmotivated standard logical framework But over these

structures dierent search strategies can compute several versions of ab duction with

the nonstandard b ehaviour that we observed in the preceding chapter Moreover

we can naturally compute various kinds of ab ducibles atoms conjunctions or even

conditionals This go es beyond the framework of ab ductive logic programming in

which ab ducibles are atoms from a sp ecial set of ab ducibles

This chapter is naturally divided into three parts We rst prop ose ab duction as

h ab ductive version corresp onding to some a pro cess of tableau expansion with eac

appropriate tableau extension for the background theory In the second part we

put forward an algorithm to compute these dierent ab ductive versions In particu

lar explanations with complex forms are constructed from simpler ones This allows

Chapter Ab duction as Computation

us to identify cases without consistent atomic explanations whatso ever It also sug

gests that in practical implementations of ab duction one can implement our views on

dierent ab ductive outcomes in chapter The third part discusses various logical as

p ects of tableau ab duction including further semantic analysis validity of structural

rules as studied in chapter plus soundness and completeness of our algorithms

Generally sp eaking this chapter shows how to implement ab duction how to pro

vide pro cedural counterparts to the ab ductiveversions describ ed in chapter There

are still further uses though which go beyond our analysis so far Ab duction as

revision can also be implemented in semantic tableaux Chapter will demonstrate

AI A detailed this when elab orating a connection with theories of belief change in

description of our algorithms as well as an implementation in Prolog co de followin

App endix A

Pro cedural Ab duction

Computational Persp ectives

There are several options for treating ab duction from a pro cedural p ersp ective One

is standard pro of analysis as in logical pro of theory cf Tro or in sp ecial logi

cal systems that dep end very much on pro oftheoretic motivations such as relevant

logic or linear logic Pro of search via the available rules would then be the driving

force for nding ab ducibles Another approachwould program purely computational

algorithms to pro duce the various typ es of ab duction that wewant An intermediate

h combines pro of theory with an algorithmic p ossibility is logic programming whic

avor The latter is more in line with our view of ab ductive logic as inference plus

acontrol strategy cf chapter Although we will eventually cho ose yet a dierent

route toward the latter end wedosketch a few features of this practically imp ortant

approach

Pro cedural Ab duction

Ab ducing in Logic Programming

Computation of ab ductions in logic programming can b e formulated as the following

pro cess We wish to pro duce literals which when added to the current

n

program P as new facts make an earlier failed goal the surprising fact succeed

after all via Prolog computation

p

is an ab ductive explanation for query given program P

if P while P

p p

Notice that we insert the ab ducibles as facts into the program here as an aid

h mechanism however that other p ositions It is a feature of the Prolog pro of searc

might give dierent derivational eects In this chapter we merely state these and

other features of resolutionstyle theorem proving without further explanation as our

main concerns lie elsewhere

Two Ab ductive Computations

Ab ductions are pro duced via the PROLOG resolution mechanism and then checked

against a set of p otential ab ducibles But as we just noted there are several ways

to characterize an ab ductive computation and several ways to add a fact to a Prolog

program An approach which distinguishes between two basic ab ductive pro cedures

is found in Sti who denes most specic abduction MSA and least specic ab

duction LSA These dier as follows Via MSA only pure literals are pro duced as

ab ductions and via LSA those that are not The following example illustrates these

two pro cedures it is a combination of our earlier common sense rain examples

Program P r c w r w s

Query q w

MSA c s

LSA r

A literal is a pure literal for program P if it cannot b e resolved via any clause in the program

A nonpure literal on the other hand is one which only o ccurs in the b o dy of a clause but never as ahead

Chapter Ab duction as Computation

This distinction is useful when we want to identify those ab ductions which are

nal causes MSA from indirect causes whichmay b e explained by something else

LSA

Structural Rules

This typ e of framework also lends itself to a study of logical structural rules like in

chapter This time nonstandard eects may reect computational p eculiarities

of our pro of search pro cedure Cf Min Kal vBe for more on this general

phenomenon As for Monotonicitywehave already shown cf chapter that right

for Prolog computation but not leftinsertion of new clauses in a program is valid

Thus adding an arbitrary formula at the b eginning of a program may invalidate

earlier programs With inserting atoms we can be more lib eral but cf Kal for

pitfalls even there For another imp ortant structural rule consider Reexivity It is

valid in the following form

Reexivity P

p

but invalid in the form

Reexivity P

p

Moreover these outcomes reect the downward computation rule of Prolog Other



algorithms can have dierent structural rules

Intro duction to Semantic Tableaux

Construction Tableau

The logical framework of semantic tableaux is a refutation metho d intro duced in the

s indep endently by Beth Bet and Hintikka Hin A more mo dern version



In particular success or failure of Reexivitymay dep end on whether a lo op clause is

present in the program Also Reexivityisvalid under LSA but not via MSA computation since

aformula implied by itself is not a pure literal

Intro duction to Semantic Tableaux

is found in Smu and it is the one presented here The general idea of semantic

tableaux is as follows

To test if a formula follows from a set of premises a tableau tree for

the sentences in fg is constructed The tableau itself is a binary tree

built from its initial set of sentences by using rules for each of the logical

connectives that sp ecify how the tree branches If the tableau closes the

initial set is unsatisable and the entailment j holds Otherwise

tableau has op en branches the formula is not a valid if the resulting

consequence of A tableau closes if every branch contains an atomic

formula and its negation

The rules for constructing the tableau tree are the following Double negations

are suppressed True conjunctions add b oth conjuncts negated conjunctions branch

into two negated conjuncts True disjunctions branch into two true disjuncts while

negated disjunctions add b oth negated disjuncts Implications a b are treated as

disjunctions a b

Negation

X X

Conjunction

X

X Y

Y

X Y X jY

Disjunction

X Y X j Y

X

X Y

Y

Chapter Ab duction as Computation

Implication

X Y X j Y

X

X Y

Y

These seven rules for tableaux construction reduce to two general typ es one con

junctive typ e and one disjunctive typ e The former for a true conjunction

and the latter for a true disjunction suce if every formula to b e incorp orated into the

tableau is transformed rst into a prop ositional conjunctive or a disjunctive normal

form

Rule A



Rule B

j



A Simple Example

To show how this works we give an extremely simple example More elab orate

tableaux will b e found in the course of this chapter

Let fr w g Set w We ask whether j The tableau is as follows

N

Here an empty circle indicates that a branch is op en and a crossed circle that

the branch is closed

Intro duction to Semantic Tableaux

T fg

r w

r w

w w

N

The resulting tableau is op en showing that j The op en branch indicates

a counterexample that is a case in which is true while is false rw false

More generally the construction principle is this A tableau has to be expanded by

applying the construction rules until formulas in the no des have no connectives and

have b ecome literals atoms or their negations Moreover this construction pro cess

ensures that each no de of the tableau can only carry a subformula of or

Logical Prop erties

The tableau metho d as sketched so far has the following general prop erties These can

be established by simple analysis of the rules and their motivation as providing an

exhaustive search for a counterexample In what follows we concentrate on verifying

the top formulas disregarding the initial motivation of nding counterexamples to

consequence problems This presentation incurs no loss of generality Given a tableau

for a theory T

If T has op en branches is consistent Each op en branch corresp onds to

averifying mo del

If T has all branches closed is inconsistent

Another more computational feature is that given some initial verication prob

lem the order of rule application in a tableau tree do es not aect the result The

Chapter Ab duction as Computation

structure of the tree may b e dierent but the outcome as to consistency is the same

Moreover returning to the formulation with logical consequence problems we have

that semantic tableaux are a sound and complete system

j i there is a closed tableau for fg

Given the workings of the ab ove rules which decrease complexity tableaux are

a decision metho d for prop ositional logic This is dierent with predicate logic not

treated here where quantier rules may lead to unb ounded rep etitions In the latter

case the tableau metho d is only semidecidable If the initial set of formulas is

unsatisable the tableau will close in nitely many steps But if it is satisable the

tableau may b ecome innite without terminating recording an innite mo del In

this chapter we shall only consider the prop ositional case

A more combinatorial observation is that there are two faces of tableaus When

an entailment do es not hold read upside down op en branches are records of counter

examples When the entailment do es hold read b ottom up a closed tableau is easily

reconstructed as a Gentzen sequent calculus pro of This is no accident of the metho d

In fact Beths motivation for inventing tableaux was his desire to nd a combination

of pro of analysis and pro of synthesis as we already observed in chapter

Tableaux are widely used in logic and they havemany further interesting prop er

ties For a more detailed presentation the reader is invited to consult Smu Fit

Ab duction with Tableaux

For convenience in what follows we give a quick reference list of some ma jor

notions concerning tableaus

Closed Branch Abranch of a tableau is closed if it contains some formula and its

negation

Atomically Closed Branch A branch is atomically closed if it is closed by an

atomic formula or a negation thereof

Op en branch A branch of a tableau is op en if it is not closed

Complete branch A branch B of a tableau is complete if referring to the earlier

mentioned two main formula typ es for every which o ccurs in B both and

for every which o ccurs in B at least one of o ccurs o ccur in B and

 

in B

Completed Tableau A tableau T is completed if every branch of T is either

closed or complete

Pro of of X A pro of of a formula X is a closed tableau for X

Pro of of j A pro of of j is a closed tableau for fg

Ab duction with Tableaux

In this section we will show the main idea for p erforming ab duction as a kind of

tableau extension First of all a tableau itself can represen t nite theories Weshow

this by a somewhat more elab orate example To simplify the notation from now on

we write for fg that is we omit the brackets

Example

Let fb c rr w s w g and let fw g

A tableauforisas follows

Chapter Ab duction as Computation

T

b

c r

r w

s w

c r

r w r w

N

s w s w s w

The result is an op en tableau Therefore the theory is consistent and each op en

branch corresp onds to a verifying mo del For example the second branch from left

to right indicates that a mo del for is given by making c r false and b w true so

we get two p ossible mo dels out of this branch one in which s is true the other in

which it is false Generally sp eaking when constructing the tableau the p ossible

branches either c or r makes the valuations for the formulas are depicted by the

rst split then for each of these either r or w and so on

When formulas are added thereby extending the tableau some of these p ossible

mo dels may disapp ear as branches start closing For instance when is added

ie w the result is the following

Ab duction with Tableaux

T w

b

c r

r w

s w

c r

r w r w

N

s w s w s w

w w w w w w

N N N N N

Notice that although the resulting theory remains consistent all but one branch

has closed In particular most mo dels we had b efore are no longer valid as w is

no longer true There is still an op en branch indicating there is a mo del satisfying

w c rsw false b true which indicates that j w

The Main Ideas

An attractive feature of the tableau metho d is that when isnotavalid consequence

of we get all cases in which the consequence fails graphically represented by the

ed as descriptions of mo dels op en branches as shown ab ove the latter may be view

for

This fact suggests that if these counterexamples were corrected by amending the

theory through adding more premises we could p erhaps make avalid consequence

Chapter Ab duction as Computation

of some minimally extended theory This is indeed the whole issue of ab duc

tion Accordingly ab duction may be formulated in this framework as a pro cess of

expansion extending a tableau with suitable formulas that close the op en branches

In our example ab ove the remaining op en branch had the following relevantlit

eral part

b

c

r

s

w

The following are some formulas whose addition to the tableau would close this

branch and hence the whole tableau

fb c rswc rr w s w s w c w g

Note that several forms of statementmay count here as ab ductions In particular

those in disjunctive form eg c w create two branches which then both close

We will consider these various cases in detail later on

Generating Ab ductions in Tableaux

In principle we can compute ab ductions for all our earlier ab ductive versions cf

chapter A direct way of doing so is as follows

Generating Ab ductions in Tableaux

First compute ab ductions according to the plain version and then elimi

nate all those which do not comply with the various additional require

ments

This strategy rst translates our ab ductive formulations to the setting of semantic

tableaux as follows

Given a set of formulae and a sentence is an ab ductive expla

nation if

Plain

T is closed j

Consistent Plain Ab duction

T is op en j

Explanatory Plain Ab duction

i T is op en j

ii T is op en j

In addition to the ab ductive conditions wemust state constraints over our search

space for ab ducibles as the set of formulas fullling any of the ab ove conditions is in

principle innite Therefore we imp ose restrictions on the vo cabulary as well as on

the form of the ab duced formulas

Restriction on Vo cabulary

is in the vo cabulary of the theory and the observation

Vo c fg

Restriction on Form

The syntactic form of is either a literal a conjunction of literals without

rep eated conjuncts or a disjunction of literals without rep eated disjuncts

Chapter Ab duction as Computation

Once it is clear what our search space for ab ducibles is we continue with our

discussion Note that while computation of plain ab ductions involves only closed

branches the other versions use insp ection of b oth closed and op en branches For ex

ample an algorithm computing consistent ab ductions would pro ceed in the following

two steps

Generating Consistent Abductions First Version

Generate all plain ab ductions b eing those formulas such that

T is closed

Take out all those for which T is closed

In particular an algorithm pro ducing consistent ab ductions along these lines must

pro duce all explanations that are inconsistent with This means many ways of

closing T which will then have to b e removed in Step This is of course wasteful

Even worse when there are no consistent explanations b esides the trivial one so

that we would want to give up our pro cedure still pro duces the inconsistent ones

The same p oint holds for our other versions of ab duction

Of course there is a preference for pro cedures that generate ab ductions in a

reasonably ecient way We will show how to devise these making use of the repre

sentation structure of tableaux in a waywhichavoids the pro duction of inconsistent

formulae Here is our idea

Generating Consistent Abductions Second Version

Generate all formulas which close some but not all op en branches of

T

Check which of the formulas pro duced are such that

T is closed

That is rst pro duce formulas which extend the tableau for the background theory

in a consistentway and then checkwhich of these are ab ductive explanations In our

Tableaux Extensions and Closures

example ab ove the dierence b etween the two algorithmic versions is as follows taking

into account only the atomic formulas pro duced Version pro duces a formula b

which is removed for b eing inconsistent and version pro duces a consistent formula

w which is removed for not b eing explanatory As wewillshow later the consistent

formulae pro duced by the second pro cedure are not necessarily wasteful They might

b e partial explanations an ingredient for explanations in conjunctive form or part

of explanations in disjunctive form

In other words consistent ab ductions are those formulas which if they had been

in the theory before they would have closed those branches which remain open after

is incorporated into the tableau

In order to implement our second algorithm we need to intro duce some further

distinctions into the tableau framework More precisely we need dierent ways in

which a tableau may be extended by a formula In the next section we dene such

extensions

Tableaux Extensions and Closures

Tableaux extensions informal explanation

A tableau is extended with a formula via the usual expansion rules explained in

section An extension may mo dify a tableau in several ways These dep end b oth

on the form of the formula to be added and on the other formulas in the theory

represented in the original tableau If an atomic formula is added the extended

tableau is just like the original with this formula app ended at the b ottom of its op en

branches If the formula has a more complex form the extended tableau may lo ok

quite dierent eg disjunctions cause every op en branch to split into two In total

however when expanding a tableau with a formula the eect on the op en branches

typ es Either i the added formula closes no op en branch or can only be of three

ii it closes all op en branches or iii it may close some op en branches while leaving

others op en In order to compute consistent and explanatory ab ductions we need to

clearly distinguish these three ways of extending a tableau We lab el them as op en

Chapter Ab duction as Computation

closed and semiclosed extensions resp ectively In what follows we dene these

notions more precisely

Formal Denitions

A prop ositional language is assumed with the usual connectives whose formulas are of

three typ es literals atoms or their negations typ e conjunctive form or typ e

disjunctive form cf section

Completed Tableaux

A completed tableau for a theory T is represented as the union of its branches

Each set of branches is the set of formulas which lab el that branch

T where each maybe op en or closed

k i

Our treatment of tableaux will b e always on completed tableau cf section so

we just refer to them as tableaux from now on

The Extension Op eration

Given T the addition of a formula to each of its branches is dened by the

following op eration

closed

isa completed op en branch

Case is a literal

f g

Case is an typ e



f g



Case is a typ e



f f g f g g



Tableaux Extensions and Closures

That is the addition of a formula to a branch is either itself when it is closed

or it is the union of its resulting branches The op eration is dened over branches

but it easily generalizes to tableaux as follows

Tableau Extension

S

f j T g T f g

def

Our notation allows also for emb eddings Note that op eration

is just another way of expressing the usual tableau expansion rules cf section

Therefore each tableau maybeviewed as the result of a suitable series of extension

steps starting from the empty tableau

Branch Extension Typ es

Given an op en branch and a formula wehave the follo wing p ossibilities to extend

it

Op en Extension

is op en if each is op en

n i

Closed Extension

is closed i each is closed

n i

SemiClosed Extension

is semiclosed i at least one is op en and at least one

n i j

is closed

Extensions can also be dened over whole tableaux by generalizing the ab ove

denitions Afew examples will illustrate the dierent situations that may o ccur

Examples

Let fa b cg

Chapter Ab duction as Computation

Op en Extension d d closes no branch

d

a b

c c

d d

SemiClosed Extension a a closes only one branch

a

a b

c c

a a

N

Closed Extension c c closes all branches

c

a b

c c

c c

N N

Tableaux Extensions and Closures

Finally to recapitulate an earlier point these typ es of extension are related to

consistency in the following way

Consistent Extension

If is op en or semiclosed then is a consistent extension

Inconsistent Extension

If is closed then is an inconsistent extension

Branch and Tableau Closures

As we have stated in section given a theory and a formula plain ab ductive

T Fur explanations are those formulas which close the op en branches of

thermore we suggested that consistent ab ductive explanations are amongst those

formulas which close some but not all op en branches of T

In order to compute b oth kinds we need to dene total and partial closures of a

tableau The rst is the set of all literals which close every op en branch of the tableau

the second of those literals which close some but not all op en branches For technical

convenience we dene total and partial closures for b oth branches and tableaux We

also need an auxiliary notion The negation of a literal is either its ordinary negation

if the literal is an atom or else the underlying atom if the literal is negative

Given T f g here the are just the op en branches of T

n i

Branch Total Closure BTC

The set of literals which close an op en branch

i

BTC fx jx g where x ranges over literals

i i

Chapter Ab duction as Computation

Tableau Total Closure TTC

The set of those literals which close all branches at onceie the intersection of

the BTCs

i n

TTC BT C

i

i

Branch Partial Closure BPC

The set of those literals which close the branch but do not close all the other

op en branches

BP C BT C TTC

i i

Tableau Partial Closure TPC

The set formed by the union of BPC ie all those literals which partially close

the tableau

i n

TPC BP C

i

i

In particular the denition of BPC may lo ok awkward as it denes partial clo

sure in terms of branch and tableau total closures Its motivation lies in a way to

compute what we will later call partial explanations b eing formulas which do close

some branches so they do explain without closing all so they are partial We

will use the latter to construct explanations in conjunctive form

Having dened all we need to exploit the framework of semantic tableau for our

purp oses we pro ceed to the construction of ab ductive explanations

Computing Plain Ab ductions

Computing Plain Ab ductions

Our strategy for computing plain ab duction in semantic tableaux will b e as follows

We will b e using tableaux as an ordinary consequence test while b eing careful ab out

the search space for p otential ab ducibles The computation is divided into dierent

forms of explanations Atomic explanations come rst followed by conjunctions of

literals to end with those in disjunctive form Here wesketch the main ideas for their

construction and give an example for each kind The detailed algorithms for each

case are describ ed in App endix A to the thesis

Varieties of Ab duction

Atomic Plain Ab duction

The idea b ehind the construction of atomic explanations is very simple One just

computes those atomic formulas which close every op en branchofT corre

sp onding precisely to its Total Tableaux Closure TTC Here is an example

Let fa bg b

a b

a b

b b

N

The two p ossible atomic plain ab ductions are fa bg

Conjunctive Plain Ab duction

Single atomic explanations may not always exist or they may not b e the only ones of

interest The case of explanations in conjunctive form is similar

n

to the construction of atomic explanations We lo ok for literals that close branches

Chapter Ab duction as Computation

but in this case we want to get the literals that close some but not all of the op en

branches These are the conjuncts of a conjunctive explanation and they b elong

to the tableau partial closure of ie to TPC Each of these partial

explanations make the fact less surprising by closing some of the op en branches

Together they constitute an ab ductive explanation

As a consequence of this characterization no partial explanation is an atomic

conjunctive explanation must be a conjunction of partial explanation That is a

explanations The motivation is this We want to construct explanations which are

nonredundant in whichevery literal do es some explaining Moreover this condition

allows us to b ound the pro duction of explanations in our algorithm We do not want

to create what are intuitively redundant combinations For example if p and q are

ab ductive explanations then p q should not be pro duced as explanation Thus we

imp ose the following condition

NonRedundancy

is Given an ab ductive explanation for a theory and a formula

nonredundant if it is either atomic or no subformula of dierent from

is an ab ductive explanation

The following example gives an ab ductive explanation in conjunctive form which

is nonredundant

Let fa c bg and b The corresp onding tableau is as follows

fa c bg

a c b

a c b

N

b b

Computing Plain Ab ductions

The only atomic explanation is the trivial one fbg The conjunctive explanation

is fa cg of which neither one is an atomic explanation

Disjunctive Plain Ab ductions

To stay in line with computational practice we shall sometimes regard ab ductive

explanations in disjunctive form as implications This is justied by the prop ositional

equivalence b etween and These sp ecial explanations close a branch

i j i j

by splitting it rst into two Disjunctive explanations are constructed from atomic

vide an and partial explanations We will not analyze this case in full detail but pro

example of what happ ens

Let fag b

The tableau structure for T b is as follows

b

a

b

Notice rst that the p ossible atomic explanations are fa bg of which the rst

is inconsistent and the second is the trivial solution Moreover there are no partial

explanations as there is only one op en branch An explanation in disjunctive form is

constructed bycombining the atomic explanations fa bg The eect of adding it

to the tableau is as follows

Chapter Ab duction as Computation

fbg fa bg

a

b

a b

N N

This examples serves as a representation of our example in chapter in which a

causal connection is found between certain typ e of clouds a and rain b namely

that a causes b a b

Algorithm for Computing Plain Ab ductions

The general p oints of our algorithm for computing plain ab ductions is displayed here

Cf App endix A for the more detailed description

Input

A set of prop ositional formulas separated by commas representing the theory

A literal formula representing the fact to b e explained

Preconditions are such that j j

Output

Pro duces the set of ab ductive explanations suchthat

n

i T is closed

i

ii complies with the vo cabulary and form restrictions cf section

i

Pro cedure

Calculate f g

k

Take those which are op en branches

i n

Atomic Plain Explanations

Consistent Ab ductive Explanations

Compute TTC f g

n m

f g is the set of atomic plain ab ductions

m

Conjunctive Plain Explanations

For each op en branch construct its partial closure BPC

i i

Check if all branches have a partial closure for otherwise there cannot be a

i

conjunctive solution in which case goto END

Each BPC contains those literals which partially close the tableau Con

i

junctive explanations are constructed by taking one literal of each BPC and

i

making their conjunction Atypical solution is a formula as follows

a b z a BP C b BP C z BP C

 n

Each conjunctive solution is reduced there may be rep eated literals The

set of solutions in conjunctive form is

l

END

Disjunctive Plain Explanations

Construct disjunctive explanations bycombining atomic explanations amongst

themselves conjunctive explanations amongst themselves conjunctive with atomic

and each of atomic and conjunctive with We just showtwo of these construc

tions

Generate pairs from set of atomic explanations and construct their disjunctions

i j

For each atomic explanation construct the disjunction with as follows

i

binations ab ove is the set of explanations in disjunctive The result of all com

form

END

Consistent Ab ductive Explanations

The issue now is to compute ab ductive explanations with the additional requirements

of b eing consistent For this purp ose we will follow the same presentation as for plain

Chapter Ab duction as Computation

ab ductions atomic conjunctive and disjunctive and will give the key points for

their construction Our algorithm follows version of the strategies sketched earlier

cf section That is it rst constructs those consistent extensions on the original

tableau for which do some closing and then checks which of these is in fact an

explanation ie closes the tableau for This wayweavoid the pro duction of

any inconsistency whatso ever It turns out that in the atomic and conjunctive cases

explanations are sometimes necessarily inconsistent therefore we identify these cases

prevent our algorithm from doing anything at all so that we do not pro duce and

formulae which are discarded afterwards

Atomic Consistent Ab ductions

When computing plain atomic explanations we now want to avoid any computation

when there are only inconsistent atomic explanations b esides the trivial one Here

is an observation which helps us get one ma jor problem out of the way Atomic

explanations are necessarily inconsistent when is an op en extension So we

can prevent our algorithm from pro ducing anything at all in this case

Fact Whenever is an open extension and a nontrivial atomic

abductive explanation dierent from it fol lows that is inconsis

tent

Proof Let b e an op en extension and an atomic explanation

The latter implies that is a closed extension Therefore

must be a closed extension to o since closes no branches But

then is an inconsistent extension Ie is inconsistent a

This result cannot be generalized to more complex forms of ab ducibles We will

lead to see later that for explanations in disjunctive form op en extensions need not

inconsistency In case is a semiclosed extension we have to do real work

however and follow the strategy sketched ab ove The key p oint in the algorithm is

this Instead of building the tableau for directly and working with its op en

branches we must start with the op en branches of

Consistent Ab ductive Explanations

Atomic Consistent Explanations

Given construct T and compute its op en branches

k

Compute If it is an op en extension then there are no atomic consistent

explanations by fact GOTO END

Else compute TPC f g which gives those literals which par

n

k

tially close T

Next check whichof close the tableau for T

i

ConsistentAtomic Explanations f jT is closed g

i i

END

Conjunctive Consistent Explanations

For conjunctive explanations we can also avoid any computation when there are only

blatant inconsistencies essentially by the same observation as b efore

Fact Whenever is an open extension and isaconjunctive

n

abductive explanation it holds that is inconsistent

The pro of is analogous to that for the atomic case

The mo dication for the algorithm in case is semiclosed works as follows

For each op en branch of T construct its partial closure BPC

i i

Construct conjunctions of the ab ove as in the plain case without taking into account

those literals closing the branches where app ears to ensure all of them are consistent

extensions Lab el these conjunctive extensions as resp ectiv ely

l

Check whichofthe ab ove close the tableau for T

i

Consistent Conjunctive Explanations f jT is closedg

i i

END

Chapter Ab duction as Computation

Disjunctive Consistent Explanations

As for disjunctive explanations unfortunately we no longer have the clear cut dis

tinction between op en and semiclosed extensions to know when there are only in

consistent explanations The reason is that for explanations in disjunctive form

openand closed do es not imply that is closed b ecause

generates two branches

We will not write here the algorithm to compute disjunctive consistent explana

tions found in app endix A but instead just presentthekey issue in its construction

y combining the atomic and conjunctive consistent Construct disjunctiveformulas b

ones ab ove These are all consistent

Check which of the ab oveformulas close the tableau for

What this construction suggests is that there are always consistent explanations

in disjunctive form provided that the theory is consistent

Fact Given that is consistent there exists an abductive consistent expla

nation in disjunctive form

The key point to prove this fact is that an explanation may be constructed as

X for any X

Explanatory Ab duction

As for explanatory ab ductions recall these are those formulas which are only con

structed when the theory do es not explain the observation already j and that

cannot do the explaining by themselves j but do so in combination with the

theory j

Given our previous algorithmic constructions it turns out that the rst condi

tion is already builtin since all our pro cedures start with the assumption that the

Qualityof Ab ductions



tableau for is op en As for the second condition its implementation actually

amounts to preventing the construction of the trivial solution Except for

this solution our algorithms never pro duce an such that j as proved below

Fact Given any and our algorithm never produces abductive ex

planations with j except for

Proof Recall our situation j j while we have made

sure and are consistent Now rst supp ose that is a literal If

j then which is the trivial solution Next supp ose that

is a conjunction pro duced by our algorithm If j then one of the

conjuncts must be itself The only other p ossibility is that is an

inconsistent conjunction but this is ruled out by our consistency test

e pro duced the relevant But then our nonredundancy lter would hav

conjunct by itself and then rejected it for triviality Finally supp ose that

is a disjunctive explanation Given the ab ove conditions tested in our

algorithm we know that is consistent with at least one disjunct But

i

also this disjunct by itself will suce for deriving in the presence of

and it will imply by itself Therefore by our redundancy test we

would have pro duced this disjunct by itself rather than the more complex

explanation and we are in one of the previous cases a

Therefore it is easy to mo dify anyofthe ab ove algorithms to handle the compu

tation of explanatory ab ductions We just need to avoid the trivial solution when

and this can be done in the mo dule for atomic explanations

Quality of Ab ductions

A question to ask at this p oint is whether our algorithms pro duce intuitively good

explanations for observed phenomena One of our examples cf disjunctive plain



It would have b een p ossible to take out this condition for the earlier versions However note

that in the case that is closed the computation of ab ductions is trivialized since as the

tableau is already closed any formula counts as a plain explanation and any consistent formula as

a consistent ab duction

Chapter Ab duction as Computation

ab ductions section suggested that ab ductive explanations for a fact and a

theory with no causal knowledge with no formulas in conditional form must be in

disjunctive form if such a fact is to b e explained in a consistentway Moreover Fact

stated that consistent explanations in disjunctive form are always available provided

that the original theory is consistent However pro ducing consistent explanations

do es not guarantee that these are go o d or relevant These further prop erties may

it will dep end up on the nature of the theory itself If the theory is a bad theory

pro duce bad or weird explanations The following example illustrates this point

A Bad Theory

Let fr w r g w r w

The tableau structure for is depicted as follows

r w

r

r w

w w

N

r w

N N

Interpreted in connection with our rain example our algorithm will pro duce the

following consistent explanation of why the lawn is wet w given that rain causes

the lawn to get wet r w and that it is not raining r One explanation is that

the absence of rain causes the lawn to get wet r w But this explanation

seems to trivialize the fact of the lawn being wet as it seems to be so regardless of rain

Qualityof Ab ductions

A b etter though more complex way of explaining this fact would b e to conclude

that the theory is not rich enough to explain why the lawn is wet and then lo ok for

some external facts to the theory eg sprinklers are on and they make the lawn

wet But this would amount to dropping the vo cabulary assumption

Therefore pro ducing go o d or bad explanations is not just a business of prop erly

dening the underlying notion of consequence or of giving an adequate pro cedure

An inadequate theory like the one ab ove can be the real cause of bad explanations

makes a go o d explanation is not the ab ducible itself but the In other words what

interplay of the ab ducible with the background theory Bad theories pro duce bad

explanations Our algorithm cannot remedy this only record it

Discussion

Having a mo dule in each ab ductiveversion that rst computes only atomic explana

tions already gives us some account of minimal explanations see chapter when

minimality is regarded as simplicity As for conjunctive explanations as we have

noted b efore their construction is one way of computing nontrivial partial expla

nations which make a fact less surprising b y closing some though not all op en

branches for its refutation One might tie up this approach with a more general issue

namelyweaker notions of approximative logical consequence Finally explanations

in disjunctive form can b e constructed in various ways Eg one can combine atomic

explanations or form the conjunction of all partial explanations and then construct

a conditional with This reects our view that ab ductive explanations are built in

a comp ositional fashion complex solutions are constructed from simpler ones

Notice moreover that we are not constructing all p ossible formulas which close

the op en branches as we have b een taking care not to pro duce what we have called

we have noted bad redundant explanations Finally despite these precautions as

explanations may slip through when the background theory is inappropriate

Chapter Ab duction as Computation

Further Logical Issues

Our algorithmic tableau analysis suggests a number of further logical issues which

we briey discuss here

Rules Soundness and Completeness

The ab ductive consequences pro duced by our tableaux can be viewed as a ternary

notion of inference Its structural prop erties can be studied in the same way as we

did for the more abstract notions of chapter But the earlier structural rules lose

some of their p oint in this algorithmic setting For instance it follows from our

tableau algorithm that consistent ab duction do es not allow monotonicity in either its

or its argument One substitute which we had in chapter was as follows If

and where is any conclusion at all then In

our algorithm we have to make a distinction here We pro duce ab ducibles and if

we already found solving then the algorithm may not pro duce stronger

ab ducibles than that It might happ en due to the closure patterns of branches in the

initial tableau that we pro duce one solution implying another but this do es not have

to b e As for strengthening the theory this might result in an initial tableau with

p ossibly few er op en branches over which our pro cedure may then pro duce weaker

ab ducibles invalidating the original choice of co op erating with to derive

More relevant therefore is the traditional question whether our algorithms are

sound and complete Again wehavetomake sure what these prop erties mean in this

setting First Soundness should mean that anycombination which gets out

of the algorithm do es indeed presentavalid case of ab duction as dened in chapter

For plain ab duction it is easy to see that we have soundness as the tableau closure

condition guarantees classical consequence which is all we need Next consider

of now is also that all ab ducibles consistent ab duction What we need to makesure

are consistent with the background theory But this is what happ ened by our

use of partial branch closures These are sure by denition to leave at least one

branch for op en and hence they are consistent with it Finally the conditions of

the explanatory algorithm ensure likewise that the theory do es not explain the fact

Further Logical Issues

already and that could not do the job on its own

Next we consider completeness Here we merely make some relevant observa

tions demonstrating the issues for our motives cf the end of this paragraph

Completeness should mean that any valid ab ductive consequence should actually be

pro duced by it This is trickier Obviously we can only exp ect completeness within

the restricted language employed by our algorithm Moreover the algorithm weeds

t conjuncts et cetera whichcutsdown outcomes even more As a more out irrelevan

signicant source of incompleteness however we can lo ok at the case of disjunctive

explanations The implications pro duced always involve one literal as a consequent

This is not enough for a general ab ductive conclusion which might involve more

What we can say for instance is this By simple insp ection of the algorithm one can

see that every consistent atomic explanation that exists for an ab ductive problem

will be pro duced by the algorithm In any case we feel that completeness is less

computational approaches to ab duction What comes rst is whether of an issue in

a given ab ductive pro cedure is natural and simple Whether its yield meets some

preassigned goal is only a secondary concern in this setting

We can also take another lo ok at issues of soundness and completeness relatively

indep endently from our axiom The following analysis of closure on tableaux is

inspired by our algorithm but it pro duces a less pro cedural logical view of what is

going on

An Alternative Semantic Analysis

Our strategy for pro ducing ab ductions in tableaux worked as follows One starts with

a tableau for the background theory i then adds the negation of the new ob

servation to its op en branches ii and one also closes the remaining op en branches

iii sub ject to certain constraints In particular for the explanatory version one

do es not allow itself as a closure atom as it is regarded as a trivial solution This

op eration maybe expressed via a kind of closure op eration

CLOSE

Chapter Ab duction as Computation

Wenowwanttotake an indep endent in a sense tableaufree lo ok at the situa

tion in the most general case allowing disjunctive explanations If is particularly

simple we can make do as shown b efore with atoms or their conjunctions First

we recall that tableaus may b e represented as sets of op en branches Wemay assume

that all branches are completed and hence all relevant information resides in their

literals This leads to the following observation

Fact Let be a set of sentences and let T be a complete tableau for with

the conjunction of al l sentences in is running over its open branches Then

equivalent to

W V

l

open in T

l a literal

This fact is easy to show The conjunctions are the total descriptions of each

op en branch and the disjunction says that any mo del for must cho ose one of them

This amounts to the usual Distributive Normal Form theorem for prop ositional logic

Now we can give a description of our CLOSE op eration in similar terms In its most

general form our way of closing an op en tableau is really dened by putting

W V

l S l

op en in T

S a set of literals

The inner part of this says that the set of relevant literals S closes every branch

The disjunction states the weakest combination that will still close the tableau Now

we ha ve a surprisingly simple connection

Fact CLOSE is equivalent to

Proof By Fact plus the prop ositional De Morgan laws is equiv

W V

alent to l But then a simple argument

op en in T

l a literal

involving choices for each op en branch shows that the latter assertion is

equivalent to CLOSE a

In the case of ab duction we pro ceeded as follows There is a theory and a

not follow from it The latter shows b ecause we surprising fact q say which do es

Further Logical Issues

have an op en tableau for followed by q We close up its op en branches without

using the trivial explanation q What this involves as said ab ove is a mo died

op eration that we can write as

CLOSE q

Example

Let fp q r q g

T q q

p q

r q r q

q q q q

N N N

The ab ductions pro duced are p or r

Again we can analyze what the new op eration do es in more direct terms

Fact CLOSE q is equivalent to falseq

Proof From its denition it is easy to see that CLOSE q is equivalent

with falseq CLOSE But then we have that

CLOSE q q i by Fact

falseq q i by prop ositional logic

falseq a

This rule maybechecked in the preceding example Indeed wehave that falseq

r whose negation is equivalent to our earlier outcome p r is equivalenttop

Chapter Ab duction as Computation

This analysis suggests ab ductive variations that we did not consider b efore For

instance we need not forbid all closures involving q but only those which involve

q in nal p osition ie negated forms of the surprising fact to be explained

There might be other harmless o ccurrences of q on a branch emanating from the

background theory itself

Example

Let fq p p q g

T

q p

p q p q

N N

q q

In this case our earlier strategy would merely pro duce an outcome p as can be

checked by our falsecomputation rule The new strategy however would compute

an ab duction p or q which maybe just as reasonable

This analysis do es not extend to complex conclusions We leave its p ossible ex

tensions op en here

Tableaux and Resolution

In spite of the logical equivalence between the metho ds of tableau and resolution

Fit Gal in actual implementations the generation of ab ductions turns out to

be very dierent The metho d of resolution used in logic programming do es not handle

negation explicitly in the language and this fact restricts the kind of ab ductions to

be pro duced In addition in logic programming only atomic formulas are pro duced

Discussion and Conclusions

as ab ductions since they are identied as those literals which make the computation

fail In semantic tableaux on the other hand it is quite natural to generate ab ductive

formulas in conjunctive or disjunctive form as we have shown

As for similarities b etween these two metho ds as applied to ab duction b oth frame

works have one of the explanatory conditions already built in In logic

programming the ab ductive mechanism is put to work when a query fails and in

semantic tableaux ab duction is triggered when the tableau for is op en

Discussion and Conclusions

Exploring ab duction as a form of computation gave us further insight into this phe

nomenon Our concrete algorithms implement some earlier points from chapter

which did not quite t the abstract structural framework Ab ductions come in dier

ent degrees atomic conjunctive disjunctiveconditional and each ab ductive condi

tion corresp onds to new pro cedural complexity In practice though it turned out

easy to mo dify the algorithms accordingly Indeed these ndings reect an intuitive

feature of explanation While it is sometimes dicult to describ e what an explana

tion is in general it may be easier to construct a set of explanations for a particular

problem

As for the computational framework semantic tableaux are a natural vehicle for

implementing ab duction They allow for a clear formulation of what counts as an

ab ductive explanation while being exible and suggestive as to p ossible mo dica

tions and extensions Derivabilityand consistency the ingredients of consistent and

explanatory ab duction are indeed a natural blend in tableaux b ecause we can ma

nipulate op en and closed branches with equal ease Hence it is very easy to check if

the consistency of a theory is preserved when adding a formula By the same token

this typ e of conditions on ab duction app ears rather natural in this light

Even so our actual algorithms were more than a straight transcription of the log

hapter which might be very inecient Our computational ical formulations in c

strategy provided an algorithm which pro duces consistent formulas selecting those

which count as explanations and this pro cedure turns out to be more ecient than

Chapter Ab duction as Computation

the other way around Nevertheless ab duction in tableaux has no unique form as

we showed by some alternatives A nal insight emerging from a pro cedural view

of ab duction is the imp ortance of the background theory when computing explana

tions Bad theories will pro duce bad explanations Sophisticated computation cannot

improve that

Our tableau approach also has clear limitations It is hard to treat notions of

ab duction in which is some nonstandard consequence In particular with an

underlying statistical inference it is unclear how probabilistic entailments should

represented Our computation of ab ductions relies on tableaux being op en or be

closed which represent only the two extremes of probable inference Wedohaveone

sp eculation though The computation of what we called partial explanations which

close some but not all op en branches might provide a notion of partial entailment

in which explanations only make a fact less surprising without explaining it in full

Cf Tij for other approaches to approximative deduction in ab ductive diagnostic

settings As for other p ossible uses of the tableau framework the case of ab duction

as revision was not addressed here In the following chapter we shall see that we do

get further mileage in that direction to o

F urther Questions

While working on the issues discussed in this chapter the following questions emerged

which we list here for further research

Finestructure of tableaux Our algorithms raise various side issues For in

stance consider tableau expansion When do es a new incoming statement A

close no branches of T at all Such syntactic observations may have inter

esting semantic counterparts The preceding is equivalent to T implies the

existential closure of A wrt those prop osition letters that do not occur in

a general theory of tableaux as a concrete T More generally what would be

system of constructive up date logic in the sense of vBea

Related Work

Pro of theory revisited Develop a complete pro of theory of ab duction which

stands to our use of semantic tableaux as Gentzen sequent calculus stands to

ordinary tableaus The dual sequent presentation of MP may b e suggestive

here but our algorithms are dierent

From prop ositional to predicate logic Extend our algorithm for tableau ab duc

tion to rstorder predicate logic with quantiers

Structural rules One would like to have a characterization of the valid struc

tural rules for our ab ductive algorithms Do these have additional features

not encountered in chapter At a tangent cf Kal for a similar theme

in Prolog computation could one correlate deviant structural rules with new

pro cedures for cut elimination

Related Work

The framework of semantic tableaux has recently been used beyond its traditional

logical purp oses esp ecially in computationally oriented approaches One example

is found in Nie implementing circumscription In connection with ab duction

semantic tableaux are used in Ger to mo del natural language presupp ositions cf

chapter ab duction in linguistics A b etterknown approach is found in MP a

source of inspiration for our work whichwe pro ceed to briey describ e and compare

The following is a mere sketch which cannot do full justice to all asp ects of their

prop osal

Ma yer and Pirris Tableau Ab duction

Mayer and Pirris article presents a mo del for computing minimal ab duction in

prop ositional and rstorder logic For the prop ositional case they prop ose twochar

acterizations The rst corresp onds to the generation of all consistent and minimal

explanations where minimality means logically weakest cf chapter The second

generates a single minimal and consistent explanation by a nondeterministic algo

rithm The rstorder case constructs ab ductions by reversed skolemization making

Chapter Ab duction as Computation

use of unication and what they call dynamic herbrandization of formulae To give

an idea of their pro cedures for generating explanations we merely note their main

steps construction of minimal closing sets construction of ab ductive solu

tions as literals which close all branches of those sets elimination of inconsistent

solutions The resulting systems are presented in two fashions once as semantic

tableaux and once as sequent calculi for ab ductive reasoning There is an elegant

treatment of the parallelism b etween these two Moreover a predicatelogical exten

sion is given which is probably the rst signicant treatment of rstorder ab duction

h go es beyond the usual resolution framework In subsequent work the au whic

thors have b een able to extend this approach to mo dal logic and default reasoning

A nal interesting p oint of the Mayer and Pirri presentation is that it shows very

well the contrast between computing prop ositional and rstorder ab duction While

the former is easy to compute even considering minimality the latter is inherently

undecidable

Comparison to our work

Our work has been inspired by MP But it has gone in dierent directions b oth

concerning strategy and output i Mayer and Pirri compute explanations in line with

version of the general strategies that we sketched earlier That is they calculate

all closures of the relevant tableau to later eliminate the inconsistent cases We do

the opp osite following version That is we rst compute consistentformulas which

close at least one branch of the original tableau and then check which of these are

explanations Our reasons for this had to do with greater computational eciency ii

As for the typ e of explanations pro duced Mayer and Pirris prop ositional algorithms

basically pro duce minimal atomic explanations or nothing at all while our approach

provides explanations in conjunctive and disjunctive form as well iii Mayers and

Pirris approach stays closer to the classical tableau framework while ours gives it

several new twists We prop ose several new distinctions and extensions eg for

the purp ose of identifying when there are no consistent explanations at all iv

Eventually we go even further cf chapter and prop ose semantic tableaux as a

vehicle for revision which requires a new contraction algorithm

Chapter

Scientic Explanation and

Epistemic Change

Intro duction

In the preceding chapters notions from the philosophy of science and articial intelli

gence have b een a source of inspiration In this chapter we aim at explicit comparison

with these traditions

In the philosophy of science we confront our logical account of chapter with

the notion of scientic explanation as prop osed by Hemp el in two of his mo dels

of scientic inference deductivenomological and inductivestatistical Hem We

show that both can be viewed as forms of ab duction the former with deductive

underlying inference the latter with statistical inference As for articial intelligence

we confront our computational account of chapter with the notion of epistemic

change as mo deled by Gardenfors Gar

In this confrontation we hop e to learn something ab out the scop e and limitations

of our analysis so far We nd anumb er of analogies as wellasnew challenges The

notion of statistical inference gives us an opp ortunity to expand the logical analysis at

a p ointleftopen in chapter and the dynamics of b elief change allows us to extend

our treatment of tableaux to the case of revision We will also encounter further

natural desiderata however which our current analysis cannot handle

Chapter Scientic Explanation and Epistemic Change

Our selection of topics in these two elds is by no means complete Many other

connections exist relating our prop osal to philosophy of science and knowledge rep

resentation in AI Some of these concerns such as cognitive pro cessing of natural

language involve ab ductive traditions beyond the scop e of our analysis We have

already identied some of these in chapter Nevertheless the connections that

tention We view all three elds as naturally related we do develop have a clear in

comp onents in cognitive science and we hop e to show that ab duction is one common

theme making for cohesion

Scientic Explanation as Ab duction

At a general level our discussion in chapter already showed that scientic reasoning

can be analyzed as an ab ductive pro cess This reasoning comes in dierent kinds

reecting amongst others various patterns of discovery with dierent triggers

A discovery may be made to explain a novel phenomenon which is consistent with

our current theory but it may also p ointatan anomaly b etween the theory and the

er the results of scientic reasoning vary in their degree phenomenon observed Moreov

of novelty and complexity Some discoveries are simple empirical generalizations from

observed phenomena others are complex scientic theories intro ducing sweeping new

notions We shall concentrate on rather lo cal scientic explanations which can be

taken to b e some sort of logical arguments ab ductive ones in our view Weareaware

of the existence of divergent on this cf chapter That scientic inference

can be viewed as ab ductive should not be surprising Both Peirce and Bolzano were

inspired in their logical systems by the way reasoning is done in science Indeed

Peirces ab ductiveformulations may b e regarded as precursors of Hemp els notion of

explanation as will b ecome clear shortly

center of our discussion on the logic of explanation lies the prop osal by At the

Hemp el and Opp enheimer HO Hem Their aim was to mo del explanations of

empirical whyquestions For this purp ose they distinguished several kinds of ex

planation based on the logical relationship b etween the explanans and explanandum

deductive or inductive as well as on the form of the explanandum singular events

Scientic Explanation as Ab duction

or general regularities These two distinctions generate four mo dels altogether two

deductivenomological ones DN and two statistical ones InductiveStatistical I

S and DeductiveStatistical DS

We analyze the two mo dels for singular events and present them as forms of

ab duction ob eying certain structural rules

The DeductiveNomological Mo del

The general schema of the DN mo del is the following

L L

m

C C

n

E

L L are general laws which constitute a scientic theory T and together

m

with suitable antecedent conditions C C constitute a potential explanation

n

hT Ci for some observed phenomenon E The relationship between explanandum

and explananda is deductive signaled by the horizontal line in the schema Addi

tional conditions are then imp osed on the explananda

hT Ci is a potential explanation of E i

T is essentially general and C is singular

E is derivable from T and C jointlybut not from C alone

The rst condition requires T to b e a general theory having at least one univer

sally quantied formula A singular sentence C has no quantiers or variables but

just closed atoms and Bo olean connectives The second condition further constrains

the derivability relation Both T and C are required for the derivation of E

Finallythe following requirement is imp osed

Chapter Scientic Explanation and Epistemic Change

hT Ci is an explanation of E i

hT Ci is apotential explanation of E

C is true

The sentences constituting the explananda must be true This is an empirical

condition on the status of the explananda hT Ci remains a potential explanation for

E until C is veried

From our logical p ersp ective of chapter the ab ove conditions dene a form

of ab duction In p otential explanation we encounter the derivability requirement

for the plain version T C E plus one of the conditions for our explanatory

ab ductive style C E The other condition that we had T E is not explicitly

required ab ove It is implicit however since a signicant singular sentence cannot

be derived solely from quantied laws which are usually taken to be conditional

An earlier ma jor ab ductive requirement that seems absent is consistency T C

Our reading of Hemp el is that this condition is certainly presupp osed Inconsistencies

certainly never count as scientic explanations Finally the DN account do es not

require minimality for explanations it relo cates suchissuestochoices b etween b etter

or worse explanations which fall outside the scop e of the mo del Wehaveadvo cated

the same p olicy for ab duction in general leaving minimal selection to our algorithms

of chapter

There are also dierences in the opp osite direction Unlike our ab ductive notions

of chapter the DN account crucially involves restrictions on the form of the ex

planantia Also the truth requirement is a ma jor dierence Nevertheless it ts well

with our discussion of Peirce in chapter an ab ducible has the status of a suggestion

until it is veried

It seems clear that the Hemp elian deductive mo del of explanation is ab ductive

in that it complies with most of the logical conditions discussed in chapter If we

t DN explanation into a deductive format the rst thing to notice is that laws and

initial conditions play dierent roles in explanation Therefore we need a ternary

format of notation T j C E A consequence of this fact is that this inference is HD

Scientic Explanation as Ab duction

nonsymmetric T j C E C j T E Thus we keep T and C separate

HD HD

in our further discussion Here is the resulting notion once more

Hemp elian DeductiveNomological Inference

HD

T j C E i

HD

i T C E

ii T C is consistent

iii T E C E

iv T consists of universally quantied sentences

C has no quantiers or variables

Structural Analysis

We now analyze this notion once more in terms of structural rules F or a general

motivation of this metho d see chapter We merely lo ok at a number of crucial

rules discussed earlier which tell us what kind of explanation patterns are available

and more imp ortantlyhow dierent explanations may be combined

Reexivity

Reexivity is one form of the classical Law of Identity every statement implies itself

This might assume two forms in our ternary format

E j C E T j E E

HD HD

However Hemp elian inference rejects b oth as they would amount to irrelevant

ve neither the phenomenon E should count explanations Given condition iii ab o

as an explanation for itself nor should the theory contain the observed phenomenon

b ecause no explanation would then be needed in the rst place In addition left

reexivity violates condition iv since E is not a universal but an atomic formula

Thus Reexivity has no place in a structural account of explanation

Chapter Scientic Explanation and Epistemic Change

Monotonicity

Monotonic rules in scientic inference provide means for making additions to the

theory or the initial conditions while keeping scientic arguments valid Although

deductive inference by itself is monotonic the additional conditions on invali

HD

date classical forms of monotonicityas we have shown in detail in chapter So we

have to be more careful when preserving explanations adding a further condition

Unfortunately some of the tricks from chapter do not work in this case b ecause of

Consider our additional language requirements Moreover outcomes can be tricky

the following monotonicity rule which lo oks harmless

HD Monotonicity on the Theory

T jA E T BjD E

HD HD

T BjA E

HD

This says that if we have an explanation A for E from a theory as well as

another explanation for the same fact E from a strengthened theory then the original

explanation will still work in the strengthened theory This sounds convincing but it

will fail if the strengthened theory is inconsistent with A Indeed we have not b een

able to nd any convincing monotonicity rules at all

What we learn here is a genuine limitation of the approach in chapter With

notions of DN complexity pure structural analysis may not be the best way to

We might also just bring in the additional conditions explicitly rather than go

enco ding them in ab ductive sequent form Thus a theory may be strengthened in

an explanation provided that this happ ens consistently and without implying the

observed phenomenon without further conditions It is imp ortant to observe that

the complexities of our analysis are no pathologies but rather reect the true state

of aairs They show that there is much more to the logic of Hemp elian explanation

than might be thought and that this can be brought out in a precise manner For

instance the failure of monotonicity rules means that one has to be very careful as

a matter of logic in lifting scientic explanations to broader settings

Scientic Explanation as Ab duction

Cut

The classical Cut rule allows us to chain derivations and replace temp orary assump

tions by further premises implying them Thus it is essential to standard reasoning

Can explanations be chained Again there are many p ossible cut rules some of

which aect the theory and some the conditions We consider one natural version

HD Cut

T jA B T jB E

HD HD

T jA E

HD

Our rain example of chapter gives an illustration of this rule Nimb ostratus

clouds A explain rain B and rain explains wetness E therefore nimb ostratus

clouds A explain wetness E But is this principle generally valid It almost lo oks

that way but again there is a catch In case A implies E by itself the prop osed

conclusion do es not follow Again we would have to add this constraint separately

Logical Rules

A notion of inference with as many side conditions as has considerably re

HD

stricts the forms of valid structural rules one can get Indeed this observation shows

that there are clear limits to the utilityof the purely structural rule analysis which

has b ecome so p opular in a broad eld of contemp orary logic To get at least some

interesting logical principles which govern explanation we must bring in logical con

nectives Here are some valid rules

Disjunction of Theories

T j C E T j C E



T T j C E



Chapter Scientic Explanation and Epistemic Change

Conjunction of two Explananda

T j C E T j C E



T j C E E



Conjunction of Explanandum and Theory

T j C E

T j C T E

Disjunction of Explanans and Explanandum

T j C E

T j C E E

Weakening Explanans by Theory

T F j A E

T F j F A E

The rst rules shows that wellknown classical inferences for disjunction and con

junction carry over to explanation The third says that explananda can b e strength

ened with any consequence of the background theory The fourth shows how ex

plananda can be weakened up to the explanandum The fth rule states that ex

plananda may be weakened provided that the background theory can comp ensate

for this The last rule actually highlights a aw in Hemp els mo dels which he him

self recognized It allows for a certain trivialization of minimal explanations which

tactic restrictions see Hem Sal might b e blo cked again by imp osing further syn

p

More generally it may be said that the DN mo del has been under continued

criticism through the decades after its emergence No generally accepted formal

mo del of deductive explanation exists But at least we hop e that our style of analysis

has something fresh to oer in this ongoing debate if only to bring simple but

essential formal problems into clearer fo cus

Scientic Explanation as Ab duction

The InductiveStatistical Mo del

Hemp els IS mo del for explaining particular events E has essentially the same form

as the DN mo del The fundamental dierence is the status of the laws While in the

DN mo del laws are universal generalizations in the IS mo del they are statistical

regularities This dierence is reected in the outcomes In the IS mo del the

phenomenon E is only derived with high probability r relative to the explanatory

facts

L L

m

C C

n

r

E

In this schema the double line expresses that the inference is statistical rather than

deductive This mo del retains all adequacy conditions of the DN mo del But it adds a

further requirement on the statistical laws known as maximal specicity RMS This

requirement resp onds to a problem which Hemp el recognized as the of IS

explanation As opp osed to classical deduction in statistical inference it is p ossible

to infer contradictory conclusions from consistent premises One of our examples from

chapter demonstrates this

The Ambiguity of IS Explanation

Supp ose that theory T makes the following statements Almost all cases of strepto

co ccus infection clear up quickly after the administration of p enicillin L Almost

no cases of p enicillinresistant strepto co ccus infection clear up quickly after the ad

ministration of p enicillin L Jane Jones had strepto co ccus infection C Jane

Jones received treatmen t with p enicillin C Jane Jones had a p enicillinresistant

strepto co ccus infection C From this theory it is p ossible to construct two con

tradictory arguments one explaining why Jane Jones recovered quickly E and the

other one explaining its negation why Jane Jones did not recover quickly E

Chapter Scientic Explanation and Epistemic Change

Argument Argument

L L



C C C C

  

r r

E E

The premises of b oth arguments are consistent with each other they could all b e

true However their conclusions contradict each other making these arguments rival

ones Hemp el hop ed to solve this problem by forcing all statistical laws in an argument

to be maximal ly specic That is they should contain all relevant information with

resp ect to the domain in question In our example then premise C of the second

argument invalidates the rst argument since the law L is not maximally sp ecic

with resp ect to all information ab out Jane in T So theory T can only explain E

but not E

notion of IS explanation relative to a knowledge situation The RMS makes the

something describ ed by Hemp el as epistemic relativity This requirement helps

but it is neither a denite nor a complete solution Therefore it has remained

controversial

These problems may b e understo o d in logical terms Conjunction of Consequents

was a valid principle for DN explanation It also seems a reasonable principle for

explanation generally But its implementation for IS explanation turns out to be

highly nontrivial The RMS may be formulated semiformally as follows

Requirement of Maximal Sp ecicity

Auniversal statistical law A B is maximally sp ecic i for all A such

that A A A B

We should note however that while there is consensus of what this requirement

means on an intuitive level there is no agreement as to its precise formalization

One of its problems is that it is not always p ossible to identify a maximal sp ecic lawgiven two

rival arguments Examples are cases where the twolaws in conict have no relation whatso ever as

in the following example due to Stegmuller Ste Most philosophers are not millionaires Most

mine owners are millionaires John is b oth a philosopher and a mine owner Is he a millionaire

Scientic Explanation as Ab duction

cf Sal for a brief discussion on this With this caveat we give a version of

IS explanation in our earlier format presupp osing some underlying notion of

i

inductive inference

Hemp elian Inductive Statistical Inference

HI

T C E i

HI

i T C E

i

ii T C is consistent

iii T E C E

i i

iv T is comp osed of statistical quantied formulas which may include

forms like Most A are B C has no quantiers

v RMS All la ws in T are maximally sp ecic with resp ect to TC

The ab ove formulation complies with our earlier ab ductive requirements but the

RMS further complicates matters Moreover there is another source of

Hemp els DN mo del xes predicate logic as its language for making form distinctions

and classical consequence as its underlying engine But in the IS mo del the precise

logical nature of these ingredients is left unsp ecied

Some Interpretations of

i

Statistical inference may b e understo o d in a qualitative or a quantitative fashion

i

The former might read as is inferredfrom with high probability while

i

the latter might read it as most of the models are models These are dieren t

ways of setting up a calculus of inductive reasoning

In addition wehave similar options as to the language of our background theories

The general statistical statements A B that may o ccur in the background theory

maybeinterpreted as either The probability of B conditioned on A is close to or

as statements of the form most of the Aobjects are Bobjects We will not pursue

these options here except for noting that the last interpretation would allow us to

Chapter Scientic Explanation and Epistemic Change

use the theory of generalized quantiers Many structural prop erties have already

been investigated for probabilistic generalized quantiers cf vLa vBeb for a

number of p ossible approaches

Finally the statistical approach to inference mightalso simplify some features of

the DN mo del based on ordinary deduction In statistical inference the DN notion

of nonderivabilty need not be a negative statement but rather a positive one of

inference with admittedly low probabilty This interpretation has some interesting

consequences that will b e discussed at the end of this chapter It might also decrease

complexity since the notion of explanation b ecomes more uniformly derivational in

its formulation

Structural Analysis Revisited

Again we briey consider some structural prop erties of IS explanation Our discus

sion will be informal since we do not x any formal explanation of the key notions

discussed b efore

As for Reexivity of this principle fails for much the same reasons as for

i

DN explanation Next consider Monotonicity This time new problems arise for

strengthening theories in explanations due to the RMS A law L might b e maximally

sp ecic with resp ect to T jA but not necessarily so with resp ect to T BjA or to

erse previous T jA B Worse still adding premises to a statistical theory may rev

conclusions In the ab ove p enicillin example the theory without C explains p erfectly

why Jane Jones recovered quickly But adding C reverses the inference explaining

instead why she did not recover quickly If we then add that she actually to ok some

homeopathic medicine with high chances of recovery cf chapter the inference

reverses again and we will b e able to explain once more why she recovered quickly

These examples show once again that inductive statistical explanations are epis

changing temically relative There is no guarantee of preserving consistency when

premises therefore making monotonicity rules hop eless And stating that the addi

tions must be maximally sp ecic seems to beg the question Nevertheless we can

salvage some monotonicity provided that we are willing to combine deductive and

Scientic Explanation as Ab duction

inductive explanations There is no logical reason for sticking to pure formulations

Here is a principle that we nd plausible mo dulo further sp ecication of the precise

inductive inference used recall that stands for deductivenomological inference

HD

and for inductivestatistical

HI

Monotonicityon the theory

T jA C T BjD C

HI HD

T BjA C

HI

This rule states that statistical arguments are monotonic in their background

theory at least when what is added explains the relevant conclusion deductively with

some other initial condition This would indeed b e a valid formulation provided that

we can take care of consistency for the enlarged theory T B j A In particular all

maximal sp ecic laws for T j A remain sp ecic for T B j A For by inferring C in a

deductive and consistent way there is no place to add something that would reverse

the inference or alter the maximal sp ecicity of the rules

Here is a simple illustration of this rule If on the one hand Jane has high

chances of recovering quickly from her infection by taking p enicillin and on the other

she would recover by taking some medicine providing a sure cure B she still has

high chances of recovering quickly when the assertion ab out this cure is added to the

theory

Not surprisingly there is no obvious Cut rule in this setting either Here it is

not the RMS causing the problem as the theory do es not change but rather the

wellknown fact that statistical implications are not transitive Again we have a

rule combining statistical and deductive explanation which might prop osal for a

form a substitute The following is our prop osed formulation

Deductive cut on the explanans

T jA B T jB C

HI HD

T jA C HI

Chapter Scientic Explanation and Epistemic Change

Again here is a simple illustration of this rule If the administration of p enicillin

do es explain the recovery of Jane with high probability and this in turn explains

deductively her good mood p enicillin explains with high probability Janes go o d

mo o d This reects the wellknown observation that most A are B all B are C

implies most A are C Note that the converse chaining is invalid

Patrick Supp es has asked whether one can formulate a more general representation

theorem of the kind we gave for consistent ab duction cf chapter which would

ve ro om for statistical interpretations This might account for the uid b oundary lea

between deduction and induction in common sense reasoning Exploring this question

however has no unique and easy route As wehave seen it is very hard to formulate a

sound monotonic rule for IS explanation But this failure is partly caused by the fact

that this typ e of inference requires to o many conditions the same problem arose in

the HD mo del So we could explore a calculus for a simpler notion of probabilistic

consequence or rather work with a combination of deductive and statistical inferences

Still we would have to formulate precisely the notion of probabilistic consequence

and we have suggested there are several qualitative and quantitative options for

doing it Thus characterizing an ab ductive logical system that could accommo date

statistical reasoning is a question that deserves careful analysis beyond the connes

of this thesis

Further Relevant Connections

In this section we briey present Salmons notion of statistical relevance as a sample

of more sophisticated p ostHemp elian views of explanation Moreover we briey

discuss a p ossible computational implementation of Hemp els mo dels more along the

lines of our chapter and nally we present Thagards notion of explanation within

his account of computational philosophy of science

Salmons Statistical Relevance

Despite the many conditions imp osed the DN mo del still allows explanations ir

relevant to the explanandum The problem of characterizing when an explanans is

Scientic Explanation as Ab duction

relevant to an explanandum is a deep one beyond formal logic It is a key issue in

the general philosophy of science

One noteworthy approach regards relevance as causality W Salmon rst analyzed

explanatory relevance in terms of statistical relevance Sal For him the issue in

inductive statistical explanation is not how probable the explanans T jA renders the

explanandum C but rather whether the facts cited in the explanans make a dierence

to the probability of the explanandum Thus it is not high probability but statistical

relevance that makes a set of premises statistically entail a conclusion

Now recall that we said that statistical nonderivability might b e rened

i

to state that follows with low probability from With this reinterpretation one

the probability of a conclusion and can indeed measure if an added premise changes

thereby count as arelevant explanation This is not all there is to causality Salmon

himself found problems with his initial prop osal and later develop ed a causal theory of

explanation Sal whichwas rened in Sal Here explanandum and explananda

are related through a causal nexus of causal pro cesses and causal interactions Even

this last version is still controversial cf Hit

A Computational Account of Hemp els mo dels

Hemp el treats scientic explanation as a given pro duct without dealing with the

pro cesses that pro duce such explanations But just as we did in chapter it seems

natural to supplement this inferential analysis with a computational search pro cedure

for pro ducing scientic explanations Can we mo dify our earlier algorithms to do such

ajob For the DN mo del indeed easy mo dications to our algorithm would suce

But for the full treatment of universal laws we would need a predicatelogical ver

sion of the tableau algorithm For the inductive statistical case however semantic

tableaux seem inappropriate We would need to nd a way of representing statis

tical information in tableaux and then characterize inductive inference inside this

framework Some more promising formalisms for computing inductive explanations

are lab eled deductive systems with weights by Dov Gabbay Gaba and recent

systems of dep endencedriven qualitative probabilistic reasoning by W Meyer Viol

cf Mey and van Lambalgen Alechina AL

Chapter Scientic Explanation and Epistemic Change

Thagards Computational Explanation

An alternative route toward a pro cedural account of scientic explanation is taken by

Thagard in Tha Explanation cannot b e captured by a Hemp elian syntactic struc

ture Instead it is analyzed as a processofproviding understandingachieved through

a mechanism of lo cating and matching The mo del of explanation is the program

PI pro cesses of induction which computes explanations for given phenomena by

pro cedures such as ab duction analogy concept formation and generalization and

then accounts for a b est explanation comparing those which the program is able to

construct

This approach concerns ab duction in cognitive science cf chapter in which

y computer programs explanation is regarded as a problemsolving activity mo deled b

The style of programming here is quite dierent from ours in chapter involving

complex mo dules and record structures Thisviewgives yet another p ointofcontact

between contemp orary philosophyof science and articial intelligence

Discussion

Our analysis has tested the logical to ols develop ed in chapter on Hemp els mo dels of

scientic explanation The deductivenomological mo del is indeed a form of ab ductive

inference Nevertheless structural rules provide limited information esp ecially once

we move to inductivestatistical explanation In discussing this situation we found

that the pro of theory of combinations of DN and IS explanation may actually be

b etterb ehaved than either by itself Even in the absence of sp ectacular p ositive

insights we can still observe that the ab ductive inferential view of explanation do es

bring it in line with mo dern nonmonotonic logics

Our negative structural ndings also raise interesting issues by themselves From

a logical p oint of view having an inferential notion like without reexivityand

HD

even mo died monotonicity challenges a claim made in Gab Gabbay argues

that reexivity restricted monononicity and cut are the three minimal conditions

whichany consequence relation should satisfy to b e a bona de nonmonotonic logic

Later in Gabb however Gabbay admits this is not such a strong approach after

Scientic Explanation as Ab duction

all as Other systems such as relevance logic do not satisfy even reexivity We

do not consider this failure an argument against the inferential view of explanation

As wehave suggested in chapter there are no universal logical structural rules that

t every consequence relation What counts rather is that such relations t a logical

format

Nonmonotonic logics have been mainly develop ed in AI As a further point of

interest we mention that the ab ove ambiguity of statistical explanation also o ccurs

acists is just another in default reasoning The famous example of Quakers and P

version of the one by Stegmuller cited earlier in this chapter More sp ecically one of



its prop osed solutions by Reiter Reiisinfactavariation of the RMS These cases

were central in Stegmuller criticisms of the p ositivist view of scientic explanation

He concluded that there is no satisfactory analysis using logic But these very same

examples are a source of inspiration to AI researchers developing nonstandard logics

There is a lot more to these connections than what we can cover in this disser

tation With a grain of salt contemp orary logicbased AI research may be viewed

orks b etter as logical p ositivism pursued by other means One reason why this w

nowadays than in the past is that Hemp el and his contemp oraries thought that clas

sical logic was the logic to mo del and solve their problems By present lights it may

have b een their logical apparatus more than their research program which hamp ered

them Even so they also grappled with signicant problems that are as daunting to

mo dern logical systems as to classical ones including a variety of pragmatic factors

In all there seem to b e ample reasons for philosophers of science AI researchers and

logicians for making common cause

Analyzing scientic reasoning in this broader p ersp ective also sheds light on the

limitations of this dissertation We have not really accounted for the distinction

between laws and individual facts we have no go o d account of relevance and we

have not really fathomed the depths of probability Moreover much of scientic

explanation involves conceptual change where the theory is mo died with new notions

in the pro cess of accounting for new phenomena So far neither our logical framework



In Tan the author shows this fact in detail relating current default theories to Hemp els IS mo del

Chapter Scientic Explanation and Epistemic Change

of chapter nor our algorithmic one of chapter has anything to oer in this more

ambitious realm

Ab duction as Epistemic Change

Notions related to explanation have also emerged in theories of b elief change in AI

One do es not just want to incorp orate new b eliefs but often also to justify them

Indeed the work of Peter Gardenfors a key gure in this tradition cf Gar

contains many explicit sources in the earlier philosophy of science Our discussion of

epistemic change will b e in the same spirit taking a numb er of cues from his analysis

will concentrate on where changes o ccur only in the theory The We

situation or world to be mo deled is supp osed to be static only new information is

coming in The typ e of epistemic change which accounts for a changing world is

called update We leave its connection to ab duction for future research and only

briey discuss it at the end of this chapter

Theories of Belief Revision in AI

In this section we shall expand on the brief intro duction given in chapter high

lighting asp ects that distinguish dierent theories of b elief revision This sets the



scene for approaching ab duction as a similar enterprise

Given a consistent theory called the b elief state and a sentence the incoming

b elief there are three epistemic attitudes for with resp ect to either is accepted

is rejected or is undetermined Given

these attitudes three main op erations may incorp orate into thereby eecting

an epistemic change in our currently held b eliefs

Expansion

sentence is added to An accepted or undetermined



The material of this section is mainly based on Gar with some mo dications taken from

other approaches In particular in our discussion b elief revision op erations are not required to handle incoming b eliefs together with all their logical consequences

Ab duction as Epistemic Change

Revision

In order to incorp orate a rejected into and maintain consistency in the

resulting b elief system enough sentences in conict with are deleted from

in some suitable manner and only then is added

Contraction Some sentence is retracted from together with enough

sentences implying it

Of these op erations revision is the most complex one It may indeed be dened

as a comp osition of the other two First contract those b eliefs of that are in conict

with and then expand the mo died theory with sentence While expansion can

as several formulas be uniquely dened this is not so with contraction or revision

can be retracted to achieve the desired eect These op erations are intuitively non

deterministic Let me illustrate this point with example from chapter where we

observed that the lawn is not wet which was in conict with our theory

Example

r w r

w

In order to incorp orate w into and maintain consistency the theory must be

l or r revised But there are two p ossibilities for doing this deleting either of r

allows us to then expand the contracted theory with l consistently The contraction

op eration p er se cannot state in purely logical or settheoretical terms which of these

two options should b e chosen Therefore an additional criterion must b e incorp orated

in order to x which formula to retract

Here the general intuition is that changes on the theory should b e kept minimal

in some sense of informational economy Various ways of dealing with the latter issue

o ccur in the literature We mention only that in Gar It is based on the notion

of entrenchment a preferential ordering which lines up the formulas in a b elief state

according to their imp ortance Thus we can retract those formulas which are least

entrenched rst In practice however fulledged AI systems of b elief revision can b e

Chapter Scientic Explanation and Epistemic Change

quite diverse Here are some asp ects that help to classify them In our terminology

these are

Representation of Belief States

Op erations for Belief Revision

Epistemological Stance

Regarding the rst we nd there are essentially three ways in which the back

ground knowledge is represented i b elief sets ii belief bases or iii p ossible

world mo dels A b elief set is a set of sentences from a logical language L closed under

logical consequence In this classical approach expanding or contracting a sentence

in a theory is not just a matter of addition and deletion as the logical consequences

tence in question should also be taken into account The second approach of the sen

emerged in reaction to the rst It represents the theory as a base for a belief set

B where B is a nite subset of satisfying ConsB That is the set of

logical consequences of B is the classical belief state The intuition b ehind this is

that some of the agents b eliefs have no indep endent status but arise only as infer

ences from more basic beliefs Finally the more semantic approach iii moves away

from syntactic structure and represents theories as sets W of p ossible worlds ie

their mo dels Various equivalences between these approaches have been established

in the literature cf GR

As for the second asp ect op erations of b elief revision can be given either con

structively or merely via p ostulates The former approach is more appropriate for

algorithmic mo dels of b elief revision the latter serves as a logical description of the

prop erties that anysuch op erations should satisfy The two can also b e combined An

algorithmic contraction pro cedure maybechecked for correctness according to given

p ostulates Say one which states that the result of contracting with should be

included in the original state

Finally each approach takes an epistemological stance with resp ect to justi

cation of the incoming b eliefs Here are two ma jor paradigms A foundationalist

approach argues one should keep track of the justication for ones b eliefs whereas

Ab duction as Epistemic Change

a coherentist p ersp ective sees no need for this as long as the changing theory stays

consistent and keeps its overall coherence

Therefore each theory of epistemic change may be characterized by its repre

sentation of b elief states its description of b elief revision op erations and its stand

on the main prop erties of b elief one should be lo oking for These choices may be

interdep endent Say a constructive approach might favor a representation by b elief

base rather than bases and hence dene b elief revision op erations on some nite

the whole background theory Moreover the epistemological stance determines what

constitutes rational epistemc change The foundationalist accepts only those b eliefs

which are justied thus having an additional challenge of computing the reasons for

an incoming b elief On the other hand the coherentist must maintain coherence and

hence make only those minimal changes which do not endanger at least consistency

In particular the theory prop osed in Garknown as the AGM paradigm after

its original authors Alchourron Gardenfors and Makinson represents b elief states

as theories closed under logical consequence while providing rationality p ostulates

to characterize the belief revision op erations and nally it advocates a coherentist

view The latter is based on the empirical claim that p eople do not keep track of

justications for their b eliefs as some psychological exp eriments seem to indicate

Har

Ab duction as Belief Revision

Ab ductive reasoning may be seen as an epistemic pro cess for b elief revision In this

context an incoming sentence is not necessarily an observation but rather a b elief

for which an explanation is sought Existing approaches to ab duction usually do not

deal with the issue of incorp orating into the set of b eliefs Their concern is just how

to give an accountfor If the underlying theory is closed under logical consequence

however then should be automatically added once we have added its explanation

which a foundationalist would then keep tagged as such

to theories of belief revision have often been Practical connections of ab duction

noted Of many references in the literature we mention ADwhich uses ab ductive

Chapter Scientic Explanation and Epistemic Change

pro cedures to realize contractions over theories with immutability conditions and

Wil which studies the relationship b etween explanations based on ab duction and

Sp ohnian reasons

Our claim will b e stronger Ab duction can function in a mo del of theory revision

as a means of pro ducing explanations for incoming b eliefs But also more gener

ally ab ductive reasoning as dened in this dissertation itself provides a mo del for

epistemic change Let us discuss some reasons for this recalling our architecture of

chapter

First what were called the two triggers for ab ductive reasoning corresp ond to

the two epistemic attitudes of a formula b eing undetermined or rejected We did not

these do not call for explanation consider accepted b eliefs since

is a novelty j j is undetermined

is an anomaly j j is rejected

is an accepted b elief j

The epistemic attitudes are presented in Gar in terms of memb ership eg a

formula is accepted if We dened them in terms of entailment since our

theories are not closed under logical consequence In our account of ab duction b oth

anovel phenomenon and an anomalous one involved achange in the original theory

The latter calls for a revision and the former for either expansion or revision So

the basic op erations for ab duction are expansion and contraction Therefore b oth

epistemic attitudes and changes in them are reected in an ab ductive mo del

However our main concern is not the incoming belief itself We rather want

to compute and add its explanation But since is a logical consequence of the

revised theory it could easily b e added Here then are our ab ductive op erations for

epistemic change

Ab ductive Expansion

Given a no vel formula for a consistent explanation for is computed

and then added to

Ab duction as Epistemic Change

Ab ductive Revision

Given a novel or an anomalous formula for a consistent explanation for

is computed which will involve mo dication of the background theory into

some suitably new Again intuitively this involves b oth contraction and

expansion

In its emphasis on explanations our ab ductive mo del for b elief revision is richer

than many theories of b elief revision Admittedly though not all cases of b elief

revision involve explanation so our greater richness also reects our restriction to a

sp ecial setting

our ab ductive mo del as a theory of epistemic change the next Once we interpret

question is what kind of theory Our main motivation is to nd an explanation

for an incoming b elief This fact places ab duction in the ab ove foundationalist line

which requires that b eliefs have a justication Often ab ductive beliefs are used

by a scientic community where the earlier claim that individuals do not keep

track of the justications of their b eliefs do es not apply On the other hand an

imp ortant feature of ab ductive reasoning is maintaining consistency of the theory

Otherwise explanations would be meaningless Therefore ab duction is committed

to the coherentist approach as well This is not a case of opp ortunism Ab duction

rather demonstrates that the earlier philosophical stances are not incompatible In

deed Haa argues for an intermediate stance of foundherentism Combinations

of foundationalist and coherentist approaches are also found in the AI literature cf

Gal Moreover taking into account the computation of explanations of incoming

b eliefs makes an espistemic mo del closer to theories of theory renement in machine

learning SL Gin

Semantic Tableaux Revisited

Toward An Ab ductive Mo del for Belief Revision

The combination of stances that we just describ ed naturally calls for a pro cedural

approach to ab duction as an activity But then the same motivations that wegavein

chapter apply Semantic tableaux provided an attractive constructive representation

Chapter Scientic Explanation and Epistemic Change

of theories and ab ductive expansion op erations that work over them So here is a

further challenge for this framework Can we extend our ab ductive tableau pro cedures

to also deal with revision

What we need for this purp ose is an accountofcontraction on tableaux Revision

will then b e forthcoming through combination with expansion as has b een mentioned

b efore

Revision in Tableaux

Our main idea is extremely straightforward In semantic tableaux contraction of a

theory so as to give up some earlier consequences translates into the opening of

a closed branch of T Let us explain this in more detail for the case of revision

T isclosed In order to revise The latter pro cess starts with for which

the rst goal is to stop from b eing a consequence of This is done by op ening a

closed branchofT not closed by thus transforming it into T This rst step

solves the problem of retracting inconsistencies The next step is muchasinchapter

to nd an explanatory formula for by extending the mo died as to make

it entail Therefore revising a theory in the tableau format can b e formulated as a

combination of two equally natural moves namely opening and closing of branches

Given for which T is closed is an ab ductive explanation if

There isaset of formulas suc h that

l i

T is op en

l

Moreover let We also require that

l

T isclosed

How to implement this technically To op en a tableau it may be necessary



to retract several formulas and not just one The second item in this

l

formulation is precisely the earlier pro cess of ab ductive extension which has b een



The reason is that sets of formulas whichentail should be removed Eg given f

g and in order to make consistent one needs to remove either f g or

f g

Ab duction as Epistemic Change

develop ed in chapter Therefore from now on we concentrate on the rst p oint of

the ab ove pro cess namely how to contract a theory in order to restore consistency

Our discussion will b e informal Through a series of examples we discover several

key issues of implementing contraction in tableaux We explore some complications

of the framework itself as well as several strategies for restoring consistency and the

eects of these in the pro duction of explanations for anomalous observations

Contraction in Tableaux

The general case of contraction that we shall need is this We have a currently

inconsistent theoryofwhichwewant to retain some prop ositions and from whichwe

want to reject some others In the ab ove case the observed anomalous phenomenon

was to be retained while the throwaways were not given in advance and must be

computed by some algorithm We start by discussing the less constrained case of any

inconsistent theory seeing howitmay b e made consistent through contraction using

its semantic tableau as a guide

As is wellknown a contraction op eration is not uniquely dened as there may

b e several options for removing formulas from a theory so as to restore consistency

Supp ose fp q p g We can remove either p q or p the choice of which

dep ends as wehave noticed on preferential criteria aiming at p erforming a minimal

change over

We start by noting that op ening a branchmay not suce for restoring consistency

Consider the following example

Example

Let fp q p q g

Chapter Scientic Explanation and Epistemic Change

T

p q

p

q

p

N

By removing p the closed branch is op ened However note that this is not

sucient to restore consistency in b ecause q was never incorp orated to the tableau

Thus even up on removal of p from we have to recompute the tableau and we

will nd another closure this time b ecause of q This phenomenon reects a certain

design decision for tableaux which seemed harmless as long as we are merely testing

for standard logical consequence When constructing a tableau as so on as a literal

l may close a branch ie l app ears somewhere higher up or vice versa it do es

op ening a branch we are so and no formula is added thereafter Therefore when

not sure that all formulas of the theory are represented on it Thus considerable

reconguration or even total reconstruction may be needed b efore we can decide

that a tableau has really b een op ened Of course for our purp oses wemightchange

the format of tableaux and compute closed branches b eyond inconsistency so as to

make all sources of closure explicit

Recomputation is a complication arising from the sp ecic tableau framework

that we use suggesting that we need to do more work in this setting than in other

approaches to ab duction Moreover it also illustrates that hidden conventions con

cerning tableau construction may ha ve unexp ected eects once we use tableaux for

new purp oses beyond their original motivation Granting all this we feel that such

phenomena are of some indep endentinterest and wecontinue with further examples

Ab duction as Epistemic Change

demonstrating what tableaux have to oer for the study of contraction and restoring

consistency

Global and Lo cal Strategies for Contraction

Consider the following variantofour preceding example

Example

Let fp q pg

T

p q

p

q

p

N

In this tableau we can op en the closed branch by removing either p or p How

ever while p is indeed a formula of p is not Here if we follow standard accounts

of contraction in theories of belief revision we should trace back the source of

this subformula p q in this case and remove it But tableaus oer another route

Alternativelywe could explore removing subformulas from a theory by merely mo d

ifying their source formulas as a more delicate kind of minimal change These two

and alternatives suggest two strategies for contracting theories whichwe lab el global

local contraction resp ectively Notice in this connection that each o ccurrence of a

formula on a branchhasaunique history leading up to one sp ecic corresp onding sub

formula o ccurrence in some formula from the theory b eing analyzed by the tableau

Chapter Scientic Explanation and Epistemic Change

The following illustrates what each strategy outputs as the contracted theory in our

example

Global Strategy

BranchOp ening fp pg

i Contract with p

p corresp onds to p in

fpg fp q g

ii Contract with p

p corresp onds to p q in

fp q g fpg

Lo cal Strategy

BranchOp ening fp pg

i Contract with p

Replace in the branch all connected o ccurrences of p following its upward

history by the atom true T

T

p q

p

q

T

Ab duction as Epistemic Change

fpg fp q T g

ii Contract with p

Replace in the branch all connected o ccurrences of p following its history by

the atom true T

T

T q

T

q

p

fpg fT q pg

Here we have a case in which the two strategies dier When contracting with

p the lo cal strategy gives a revised theory which is equivalent to fq pg with less

change than the global one Indeed if p is the source of inconsistencywhyremove the

whole formula p q when we could mo dify it by T q This simple example shows that

removing subformulas from branches and mo difying their source formulas gives a

more minimal change than removing the latter

However the choice is often less clearcut Sometimes the lo cal strategy pro duces

contracted theories which are logically equivalent to their globally contracted coun

terparts Consider the following illustration a variation on example from chapter

Example

fr l r l g

Chapter Scientic Explanation and Epistemic Change

T

r l

r

r l

N

l

N

Again we briey note the obvious outcomes of b oth lo cal and global contraction

strategies

Global Strategy

Left BranchOp ening fr r g

i Contract with r

fr g fr l l g

ii Contract with r

fr l g fr l g

Right BranchOp ening fl lg

i Contract with l

fr l g fr l g

ii Contract with l

fl g frr l g

Lo cal Strategy

Left BranchOp ening fr r g

i Contract with r

Ab duction as Epistemic Change

fr g fr l l g

ii Contract with r

fT l r l g

RightBranchOp ening fr r g

i Contract with l

fr T r l g

ii Contract with l

fl g frr l g

Now the only deviant case in the lo cal strategy is this Lo cally contracting with

r makes the new theory fT l r l g Given that the rst formula is a the

output is logically equivalent to its global counterpart fr l g Therefore mo difying

versus deleting conicting formulae makes no dierence in this whole example

A similar indierence shows up in computations with simple disjunctions although

more complex theories with disjunctions of conjunctions may again show dierences

between the two strategies We refrain from sp elling out these examples here which

can easily b e supplied by the reader Also we leave the exact domain of equivalence

of the two strategies as an op en question Instead we survey a useful practical case

again in the form of an example

Computing Explanations

explanation for an anomaly We want to keep the Let us now return to ab ductive

latter xed in what follows it is precisely what needs to b e accommo dated mo difying

merely the background theory As it happ ens this constraint involves just an easy

mo dication of our contraction pro cedure so far

Example

fp q rp q g r

Chapter Scientic Explanation and Epistemic Change

There are ve p ossibilities on what to retract r do es not count since it is the

anomalous observation In the following descriptions of lo cal output note that a

removed literal l will lead to a substitution of F the falsum for its source l in a

formula of the input theory

Contracting with p

Global Strategy fp q r g

Lo cal Strategy fp q rT q g

Contracting with p

Global Strategy fp q g

Lo cal Strategy fF q rp q g

Contracting with q

Global Strategy fp q r g

Lo cal Strategy fp q rp T g

Contracting with q

Global Strategy fp q g

Lo cal Strategy fp F rp q g

Contracting with r

Global Strategy fp q g

Lo cal Strategy fp q T p q g

The only case in which the revised theories are equiv alent is when contracting

with r so we have several cases in which we can compare the dierent explanations

pro duced by our two strategies To obtain the latter we need to p erform p ositive

standard ab duction over the contracted theory Let us lo ok rst at the case when the

theory was contracted with p Following the global strategy the only explanation for

r with resp ect to the revised theory fp q r g is the trivial solution

Ab duction as Epistemic Change

r itself On the other hand following the lo cal strategy there is another p ossible

explanation for r with resp ect to its revised theory fp q rT q g namely

q r Moreover if wecontract with pwe get the same set of p ossible explanations

in b oth strategies Thus again the lo cal strategy seems to allow for more p ointed

explanations of anomalous observations

We do not claim that either of our strategies is denitely b etter than the other one

Wewould rather p oint at the fact that tableaux admit of many plausible contraction

op erations which we take to be a vindication of our framework Indeed tableaux

bitious approach We outline yet another strategy also suggest a slightly more am

to restore consistency It addresses a point mentioned in earlier chapters viz that

explanation often involves changing ones conceptual framework

Contraction by Revising the Language

Supp ose we have fp q g and we observe or learn that p Following our

global contraction strategy would leave the theory empty while following the lo cal

one would yield a revised theory with T q as its single formula But there is another

option equally easy to implement in our tableau algorithms After all in practice we

often resolve by making a distinction Mark the prop osition inside

the anomalous formula p by some new proposition letter say p and replace

its o ccurrences if any in the theory by the latter In this case we obtain a new

consistent theory consisting of p q p And other choice points in the ab ove

examples could b e marked by suitable new prop osition letters as well

ariants of the same prop osition where some Wemay think of the pair p p as twov

distinction has been made Here is a simple illustration of this formal manipulation

p q Rich and Famous

p Materially p o or

p Po or in spirit

In a dialogue the anomalous statement mightthen b e defused as follows

Chapter Scientic Explanation and Epistemic Change

A X is a richand famous p erson but X is p o or

B Why is X p o or

Possible answers

A Because Xis poor in spirit

A Because b eing rich makes X poor in spirit

A Because b eing famous makes X poor in spirit

Over the new contracted and reformulated theories our ab ductive algorithms of

chapter can easily pro duce these three consistent explanations as p p p q

p The idea of reinterpreting the language to resolve inconsistencies suggests that

there is more to belief revision and contraction than removing or mo difying given

be a source of the anomaly and hence it needs formulas The language itself may

revision to o For a more delicate study of linguistic change in resolving

and anomalies cf Wei For related work in argumentation theory cf vBe

This mightbe considered as a simple case of conceptual change

What we have shown is that at least some language changes are easily incor

p orated into tableau algorithms and are even suggested by them Intuitively any

inconsistent theory can b e made consistentbyintro ducing enough distinctions into its

vo cabularyandtaking apart relev ant assertions We must leavethe precise extent

and algorithmic content of this folklore fact to further research

Another app ealing consequence of to accommo dating inconsistencies via language

change concerns structural rules of chapter We will only mention that structural

rules would acquire yet another parameter in their notation namely the vo cabulary

over which the formulas are to b e interpreted eg pjV qjV p q jV V Interest

 

inglythis format was also used in Bolzano Bol An immediate side eect of this

move are rened notions of consistency in the spirit of those prop osed by Hofstadter

in which consistency is relativetoan interpretation in Hof

Ab duction as Epistemic Change

Outline of Contraction Algorithms

Global Strategy

Input for which T is closed

Output contracted for which T isopen

Pro cedure CONTRACT

Construct T and lab el its closed branches

n

IF

Cho ose a closed branch not closed by if there are none then cho ose

i

any op en branch

Calculate the literals which op en it BranchOp ening f g Cho ose

i 

one of them say

Find a corresp onding formula for in higher up in the branch is either

itself or a formula in conjunctive or disjunctiveforminwhich o ccurs

Assign

ELSE

Assign

IF T is op en AND all formulas from are represented in the op en

branch then go to END

ELSE

IF T is OPEN

Add remaining formulas to the op en branches until there are no more formulas

to add or until the tableau closes If the resulting tableau is op en reassign

and goto END else CONTRACT

ELSE CONTRACT

This is the earlierdiscussed iteration clause for tableau recomputation

END

is the contracted theory with resp ect to such that T is op en

Chapter Scientic Explanation and Epistemic Change

Lo cal Strategy

Input for which T is closed

Output contracted for which T isopen

Pro cedure CONTRACT

Construct T and lab el its closed branches

n

Cho ose a closed branch not closed by if there are none then cho ose

i

anyopenbranch

Calculate the literals which op en it BranchOp ening f g Cho ose

i 

one of them say

Replace by T together with all its o ccurrences up in the branch is either

itself or a formula in conjunctive or disjunctive form in which o ccurs

Assign T

IF T is op en AND all formulas from are represented in the op en

branch then go to END

ELSE

IF T isOPEN

Add remaining formulas to the op en branches until there are no more formulas

to add or until the tableau closes If the resulting tableau is op en reassign

and goto END else CONTRACT

ELSE CONTRACT

END

is the contracted by T substitution theory with resp ect to such that

T is op en

Rationality Postulates

To conclude our informal discussion of contraction in tableaux we briey discuss the

AGM rationality p ostulates We list these p ostulates at the end of this chapter

These are often taken to be the hallmark of any reasonable op eration of contraction

Ab duction as Epistemic Change

and revision and many pap ers show lab orious verications to this eect What do

these p ostulates state in our case and do they make sense

To b egin with we recall that theories of epistemic change diered amongst other

things in the way their op erations were dened These can b e given either construc

tively as we have done or via p ostulates The former pro cedures might then be

checked for correctness according to the latter However in our case this is not as

straightforward as it may seem The AGM p ostulates take b elief states to b e theories

closed under logical consequence But our tableaux analyze nondeductively closed

changes in nite sets of formulas corresp onding with b elief bases This will lead to

the p ostulates themselves

Here is an example Postulate for contraction says that If the formula to be

retracted do es not o ccur in the b elief set K nothing is to b e retracted

K If K then K K

In our framework we cannot just replace belief states by b elief bases here Of

course the intuition b ehind the p ostulate is still correct If is not a consequence of

that we encounter in the tableau then it will never b e used for contraction by our

algorithms Another point of divergence is that our algorithms do not put the same

emphasis on contracting one sp ecic item from the bac kground theory as the AGM

p ostulates This will vitiate further discussion of even more complex p ostulates such

as those breaking down contractions for complex formulas into successive cases

One more general reason for this mismatch is the following Despite their op er

ational terminology and ideology the AGM p ostulates describ e expansions con

tractions and revisions as in the terminology of chapter epistemic products rather

than processes in their own right This gives them a static avor which may not

always be appropriate

Therefore we conclude that the AGM p ostulates as they stand do not seem to

apply to contraction and revision pro cedures like ours Evidently this raises the

issue of which general features are present in our algorithmic approach justifying it

as a legitimate notion of contraction There is still a chance that a relatively slight

revision of the revision p ostulates will do the job An alternative more pro cedural

Chapter Scientic Explanation and Epistemic Change

approach might b e to view these issues rather in the dynamic logic setting of dRi

vBea We must leave this issue to further investigation

Discussion and Questions

The preceding was our brief sketch of a p ossible use of semantic tableaux for p erform

ing all op erations in an ab ductive theory of b elief revision Even in this rudimentary

state it presents some interesting features For a start expansion and contraction

are not reverses of each other The latter is essentially more complex Expanding a

tableau for with formula merely hangs the latter to the op en branches of T

But retracting from often requires complete reconguration of the initial tableau

and the contraction pro cedure needs to iterate as a cascade of formulas mayhave to

ed The advantage of contraction over expansion however is that we need be remov

not run any separate consistency checks as we are merely weakening a theory

We have not come down in favor of any of the strategies presented The lo cal

strategy tends to retain more of the original theory thus suggesting a more minimal

change than the global one Moreover its substitution approach is nicely in line with

our alternative analysis of tableau ab duction in section and it may lend itself

to similar results But in many practical cases the more standard global ab duction

works just as well We must leave a precise comparison to future research

Regarding dierent choices of formulas to be contracted our algorithms are bla

tantly nondeterministic If we want to be more fo cused we would have to exploit

the tableau structure itself to represent say some entrenchment order The more

fundamental formulas would lie closer to the ro ot of the tree In this way instead of

constructing all op enings for each branch we could construct only those closest to

the leaves of the tree These corresp ond to the less imp ortant formulas leaving the

inner core of the theory intact

It would b e of interest to have a pro oftheoretic analysis of our contraction pro ce

dure In a sense the AGM rationality p ostulates maybe compared with the struc

tural rules of chapter And the more general question we have come up against is

this what are the appropriate logical constraints on a pro cess view of contraction

Ab duction as Epistemic Change

and revision by ab duction

Finally regarding other epistemic op erations in AI and their connection to ab

duction we have briey mentioned update the pro cess of keeping b eliefs uptodate

as the world changes Its connection to ab duction p oints to an interesting area of

research the changing world mightbe full of new surprises or existing b eliefs might

have lost their explanations Thus appropriate op erations would have to be dened

to keep the theory up dated with resp ect to these changes

Conclusions

t of ab duction as a theory of b elief In this second part of the chapter wegave an accoun

revision We prop osed semantic tableaux as a logical representation of belief bases

over which the ma jor op erations of epistemic change can b e p erformed The resulting

theory combines the foundationalist with the coherentist stand in b elief revision

We claimed that b eliefs need justication and used our ab ductive machinery to

construct explanations of incoming b eliefs when needed The result is richer than

standard theories in AI but it comes at the price of increased complexity In fact it

has b een claimed in Doy that a realistic workable system for b elief revision must

not only trade deductive closed theories for b elief bases but also drop the consis

tency requirement As w e saw in chapter the latter is undecidable for suciently

expressive predicatelogical languages And it may still requires exp onential time for

sentences in prop ositional logic Given our analysis in chapter we claim that any

system which aims at pro ducing genuine explanations for incoming b eliefs must main

tain consistency What this means for workable systems of belief revision remains a

mo ot p oint

The tableau analysis conrms the intuition that revision is more complex than

expansion and that it admits of more variation Several choices are involved for

which there seem to be various natural options even in this constrained logical set

ting What wehave not explored in full is the way in which tableaux might generate

entrenchment orders that we can prot from computationally As things stand dier

ent tableau pro cedures for revision may output very dierent explanations ab ductive

Chapter Scientic Explanation and Epistemic Change

revision is not one unique pro cedure but a family

Even so the preceding analysis mayhave shown that the standard logical to ol of

semantic tableaux has more uses than those for which they were originally designed

Explanation and Belief Revision

We conclude with a short discussion and an example which relates both themes of

this chapter ab duction as scientic explanation and ab duction as a mo del for epis

temic change Theories of belief revision are mainly concerned with common sense

reasoning while theories of explanation in the philosophy of science mainly concern

scientic inquiry Nevertheless some ideas by Gardenfors on explanation chapter

of his book Gar turn out illuminating in creating a connection Moreover

how explanations are computed for incoming b eliefs makes a dierence in the typ e

of op eration required to incorp orate the b elief We give an example relating to our

medical diagnosis case in the rst part of this chapter to illustrate this p oint

Explanation in Belief Revision

Gardenfors basic idea is that an explanation is that which makes the explanandum

less surprising by raising its probability The relationship bet ween explananda and

explanandum is relative to an epistemic state based on a probabilistic mo del in

volving a set of p ossible worlds a set of probability measures and a b elief function

Explanations are prop ositions which eect a sp ecial epistemic change increasing the

b elief value of the explanandum Explananda must also convey information whichis

relevant to the b eliefs in the initial state This prop osal is very similar to Salmons

views on statistical relevance which we discussed in connection with our statistical

reinterpretation of nonderivability as derivability with low probability The main

wo is this While Gardenfors requires that the change is in dierence between the t

raising the probability Salmon admits just any change in probability Gardenfors

notion of explanation is closer to our partial explanations of chapter whichclosed

some but not all of the available op en tableau branches A natural research topic

Explanation and Belief Revision

will be to see if we can implement the idea of raising probability in a qualitative

manner in tableaux Gardenfors prop osal involves degrees of explanation suggesting

a measure for explanatory power with resp ect to an explanandum

Combining explanation and b elief revision into one logical endeavor also has some

broader attractions This coexistence and p ossible interaction seems to reect

actual practice b etter than Hemp els mo dels which presupp ose a view of scientic

progress as mere accumulation This again links up with philosophical traditions that

have traditionally been viewed as a nonlogical or even antilogical These include

historic and pragmatic accounts KuhvFr that fo cus on analyzing explanations

evolutionary scientic change Scientic rev of anomalous instances as cases of r

olutions are taken to be those noncumulative developmental episo des in which an

older paradigm is replaced in whole or in part by an incompatible new one Kuh

p In our view the Hemp elian and Kuhnian scho ols of thought far from b eing

enemies emphasize rather two sides of the same coin

Finally our analysis of explanation as compassing b oth scientic inference and

general epistemic change shows that the philosophy of science and articial intelli

gence share central aims and goals Moreover these can be pursued with to ols from

logic and computer science which help to clarify the phenomena and show their

complexities

Belief Revision in Explanation

As we have shown computing explanations for incoming b eliefs gives a richer mo del

than many theories of belief revision but this mo del is necessarily more complex

After all it gives a much broader p ersp ective on the pro cesses of expansion and

revision We illustrate this p ointby an example whichgoesbeyond our conservative

algorithms of chapter

In standard b elief revision in AI given an undetermined b elief our case of nov

elty the natural op eration for mo difying a theory is expansion The reason is that the

incoming b elief is consistent with the theory so the minimal change criterion dictates

Once ab duction is considered however the that it is enough to add it to the theory

Chapter Scientic Explanation and Epistemic Change

explanation for the fact has to b e incorp orated as well and simple theory expansion

might not be always appropriate Consider our previous example of statistical rea

soning in medical diagnosis cf chapter and of this chapter concerning the

quick recovery of Jane Jones which we briey repro duce as follows



L L L C

 

E

Given theory wewant to explain why Jane Jones recovered quickly Clearly

the theory neither claims with high probability that she recovered quickly

Wehave a case of novelty the observed fact is consis nor that she did not

tent with the theory Now supp ose a do ctor comes with the following explanation for

her quick recovery After careful examination I have come to the conclusion that

Jane Jones recovered quickly b ecause although she received treatment with p enicillin

and was resistant her grandmother had given her Belladonna This is a p erfectly

sound and consistent explanation However note that having the fact that Jane

Jones was resistant to p enicillin as part of the explanation do es lower the probability

of explaining her quick recovery to the p oint of statistically implying the contrary

Therefore in order to make sense of the do ctors explanation the theory needs to

be revised as well deleting the statistical rule L and replacing it with something



along the following lines Almost no cases of p enicillinresistant strepto co ccus infec

tion clear up quickly after the administration of p enicillin unless they are cured by

something else L



Thus we have shown with this example that for the case of novelty in statistical

explanation theory expansion maynot be the appropriate op eration to p erform let

alone the minimal one and theory revision might be the only way to salv age b oth

consistency and high probability



Almost all cases of strepto co ccus infection clear up quickly after the administration of p enicillin

L Almost no cases of p enicillinresistant strepto co ccus infection clear up quickly after the ad

ministration of p enicillin L Almost all cases of strepto co ccus infection clear up quickly after the

administration of Belladonna a homeopathic medicine L Jane Jones had strepto co ccus infection

C Jane Jones recovered quickly E

AGM Postulates for Contraction

AGM Postulates for Contraction

K For any sentence and any b elief set K K is a b elief set

K No new b eliefs o ccur in K K K

K If the formula to be retracted do es not occur in the b elief set nothing is to be

retracted

If K then K K

K The formula to b e retracted is not a logical consequence of the b eliefs retained

unless it is a tautology

If not then K

K It is p ossible to undo contractions Recovery Postulate

If K then K K

K The result of contracting logically equivalent formulas must b e the same

If then K K

K Two separate contractions may be p erformed by contracting the relevant con

junctive formula

K K K

K If K then K K

App endix A

Algorithms for Chapter

In this app endix we give a description of our tableaux algorithms for computing

plain and consistent ab ductions We concentrate on the overall structure and main

pro cedure calls Moreover an actual prolog co de written in Arity Prolog for some

of these pro cedures maybe found at the end of this app endix

We begin by recalling the four basic op erations for constructing ab ductive expla

nations in tableau cf chapter for their formal denitions

BTC The set of literals closing an op en branch

TTC The set of literals closing all branches at once

BPC The set of literals which close a branch but not all the rest

TPC The set of literals partially closing the tableau

Plain Ab ductive Explanations

INPUT

The theory is given as a set of prop ositional formulas separated by commas

The fact to b e explained isgiven as a literal formula

Preconditions are suchthatj j

App endix A Algorithms for Chapter

OUTPUT

Pro duces the set of ab ductive explanations such that

n

i T is closed

i

ii complies with the vo cabulary and form restrictions cf chapter section

i

MAIN PROCEDURE

BEGIN

constructtableau

k

getop enbranches

n

k

constructatomicexps Atomic exps

n

i emptysol f al se

Check if all branches have a nonempty BPC and calculate them

RE P E AT

ysol tr ue IF BP C THEN empt

i

ELSE i i

UNT IL emptysol OR i n

IF notemptysol THEN

constructconjunctiveexps BP C BPC C onj exps

n

ELSE C onj exps

constructdisjunctiveexpsAtomic exps C onj exps D isj exps

writeThe following are the output ab ductive explanations

writeAtomic Explanations Atomic exps

writeConjunctive Explanations C onj exps

writeDisjunctive Explanations D isj exps EN D

App endix A Algorithms for Chapter

SUBPROCEDURES

Atomic Explanations

constructatomicexps Atomic exps

n

BEGIN

TTC f g

n m

Atomicexps f g

m

EN D

Conjunctive Explanations

constructconjunctiveexpsBP C BPC Conjexps

n

EachBPC contains those literals which partially close the tableau Conjunctive

i

explanations are constructed by taking one literal of each of these and making their

conjunction This pro cess can b e easily illustrated by the following tree structure

a a

j

b b b b

k k

z z z z z z z z

z z z z

Each tree level represents the BPC for a branch eg BPC fa a g

i j

Each tree branch is a conjunctive ab ductive solution

a b z

a b z

 

a b z

 

a b z

n z

k

a b z

p j z k

App endix A Algorithms for Chapter

Some of these solutions are rep eated so further reduction is needed to get the set

of ab ductive conjunctive explanations

Conjexpsf g

l

Disjunctive Explanations

constructdisjunctiveexpsAtomicexpsConjexpsDisjexps

Disjunctive explanations are constructed bycombining atomic explanations among

themselves conjunctive explanations among themselves atomic with conjunctive and

each of atomic and conjunctivewith

i Atomic Atomic

 m    m m m m

ii Conjunctive Conjunctive



l l l l

iii Atomic Conjunctive

m m

l l

iv Atomic

m

v Conjunctive

l

The union of all these make up the set of ab ductive disjunctive explanations

Disjexpsf g

r

Consistent Ab ductive Explanations

The input is the same as for plain ab ductive explanations The output requires the

additional condition of consistency

OUTPUT

Pro duces the set of ab ductive explanations such that i and ii from ab ove

n

plus

iii T isopen i

App endix A Algorithms for Chapter

MAIN PROCEDURE

BEGIN

constructtableau

k

getop enbranches

n

k

numb erelements N

n

addformulatableau

n

m

numb erelements N

m

extensiontyp eN N Type

THEN IF extensiontyp eN N semi cl osed

constructatomicexps Atomic exps

n

constructconjunctiveexps BP C BPC C onj exps

n

ELSE

C onj exps

Disj exps

writeThere are neither atomic nor conjunctive explanations

EN DIF

constructdisjunctiveexpsAtomic exps C onj exps T emp D isj exps

eliminateinconsistentdisjunctiveexpsTemp Disj exps D isj exps

Explanations Atomic exps C onj exps Disj exps

writeThe following are the output ab ductive explanations

writeExplanations

EN D

Prolog Co de

What follows is a Prolog implementation for some of the pro cedures ab ove written

for Arity Prolog

This implmentation shows that ab duction is not particularly hard to use in prac

tice which may explain its app eal to programmers It may be a complex notion in

App endix A Algorithms for Chapter

general but when welldelimited it p oses a feasible programming task

BEGIN PROLOG CODE

CLAUSES FOR THE CONSTRUCTION OF ABDUCTIVE EXPLANATIONS

Atomic Explanations

Consists of intersecting the tableau partial

closure with the tableau totalclosure

atomicexplanationsTableau At omic set

constructatomicexpsTab leau At omic exp

tableautotclosureTablea uB ranc hes clos ure TTC

tableauparclosureTablea u TP C

intersectionTTCTPCAto mic exp

Conjuntive Explanations

This predicate first constructs the partial closures

of the open branches and then checks that indeed every

branch has a set of partial closures for otherwise

there are no conjunctive explanations Then a tree like

structure is formed for these partial explanations in

which each branch is a solution Repeated literals are

removed from every solution and what is left is the set

of conjunctive explanations

conjunctiveexplanationsBr anc hes clos ure Ato mic exp Conjunctiveexp

App endix A Algorithms for Chapter

constructconjunctiveexps Ta blea uCo nje xp

openbranchesTableauOp en bran ches

closedbranchesTableau Clo sed bran ches

tableauparclosureOpen br anch es TP C

numberelementsOpenbra nch esN

numberelementsOpenbra nch esN

NN

constructtableauTPC So luti onP aths

reducesolutionsSolutio nP aths Con jex p

In case some branch partial closure is empty

the result of conjunctive explanations returns

false meaning there are no explanations in

conjunctive form

constructconjunctiveexps Ta blea u

Disjunctive Explanations

Here we show the construction of disjunctive

explanations between the atomic explanations

and the observation and between conjunctive

explanations and observation

conscondexpXFnot X v F not F v not X

conssetcondexpXR F SolRest

conscondexpXFSol

conssetcondexpR F Rest

App endix A Algorithms for Chapter

conssetcondexp F Result

disjexplanationsAtomic Partial Obs Part Part

conssetcondexpAtomic F Part

conssetcondexpXR F Part

CLAUSES FOR THE CONSTRUCTION

OF BRANCH AND TABLEAU CLOSURES

Branch Total Closure BTC For a given

Branch it computes those literals which

close it

branchtotclosureBranchL ist BTC

branchtotclosureXR List BT C

literalX

branchtotclosureRList not XBTC

branchtotclosureXR List BT C

branchtotclosureRListB TC

branchtotclosureBTC BTC

Generates the sets of branch closures

for a given tableau

tableauclosureTableauLis tR esul t

tableauclosureXRLis tso fa rC los Res branchtotclosureXClo s

App endix A Algorithms for Chapter

tableauclosureRClos Res t

tableauclosureTableau cl osur eTa blea uc losu re

Tableau Total Closure TTC

Intersection of all branch total closures

tableautotclosureTablea uB ranc hes clos ure Tcl osur e

tableauclosureTableau Br anch esc los ure

intersectionBranchesclo sure Tcl osur e

Branch Partial Closure BPC

For a given branch constructs those literals

which close it but do not close the whole tableau

branchparclosureTableau Br anch Lis tBP clo sure

branchtotclosureBranch BTC

tableautotclosureTable au TT C

BPclosure BTC TTC

Tableau Partial Closure TPC

Constructs th union of all branch partial closures

tableauparclosureTabl eau Lis tso far TP clos ure

tableauparclosureXR Lis tso far BP CX Res branchparclosureXR X B PCX

App endix A Algorithms for Chapter

tableauparclosureRBPCX Re st

tableauparclosureTP clos ure TPc losu re

CLAUSES FOR THE CONSTRUCTION OF

TABLEAU AND ADDITION OF FORMULAS

Tableau Construction

constructtableauTheoryTr ee sof arT able au Resu lt

constructtableauXRT ree so far Tabl eau Res ult

addformulatableauTreeso fa rX Tree Res ult

constructtableauRTreeRe sul tTa blea uRe sul t

constructtableauTabl eau Res ult Tabl eau Res ult

Adding a Formula to a Tableau

addformulatableauTableau Fo rmul aNe wTa ble au

addformulatableauBR For mul aN ewB ranc hR est

addformulabranchBForm ula New Bra nch

addformulatableauRFor mul aRe st

addformulatableauFo rmul aR esul t

Add formula to a Branch

addformulabranchBranchF orm ula Resu lt

Case branch is closed nothing is to be added

App endix A Algorithms for Chapter

addformulabranchBranch For mula Bra nch

closedBranchtrue

Case A is a literal just appended at the end

of the branch path

addformulabranchBranch A Bran chA literalA

Case AB is a conjunction first clause is the

condition when both are literals the second when

they are not

addformulabranchBranch AB Br anch AB

literalAliteralB

addformulabranchBranch AB Res ult

addformulabranchBranc hA BA Res

addformulabranchResB Resu lt

Case AvB is a disjunction first clause is the

condition when both are literals the second when

they are not

addformulabranchBranch AvB B ranc hA B ranc hB

literalAliteralB

addformulabranchBranch AvB Br anch Br anc h

addformulabranchBranc hAv BA Bra nch

addformulabranchBranc hAv BB Bra nch

END OF PROLOG CODE

Abstract

In this dissertation I study ab duction that is reasoning from an observation to its

p ossible explanations from a logical p oint of view This approach naturally leads to

connections with theories of explanation in the philosophyof science and to compu

tationally oriented theories of b elief change in Articial Intelligence

Many dierent approaches to ab duction can b e found in the literature as well as

a b ewildering variety of instances of explanatory reasoning To delineate our sub ject

more precisely and create some order a general taxonomy for ab ductive reasoning is

prop osed in chapter Several forms of ab duction are obtained by instantiating three

parameters the kind of reasoning involved eg deductive statistical the kind of

observation triggering the ab duction novelty or anomaly wrt some background

theory and the kind of explanations pro duced facts rules or theories In chapter

I cho ose anumber of ma jor variants of ab duction thus conceived and investigate

their logical prop erties A convenient measure for this purp ose are socalled structural

rules of inference Ab duction deviates from classical consequence in this resp ect

much like many current nonmonotonic consequence relations and dynamic styles

of inference As a result we can classify forms of ab duction by dierent structural

rules A more computational analysis of pro cesses pro ducing ab ductive inferences

is then presented in chapter using the framework of semantic tableaux I show

how to implement various searc h strategies to generate various forms of ab ductive

explanations

Our eventual conclusion is that ab ductive pro cesses should be our primary con

cern with ab ductive inferences their secondary pro ducts Finally chapter is a

Abstract

confrontation of the previous analysis with existing themes in the philosophy of sci

ence and articial intelligence In particular I analyse two wellknown mo dels for

scientic explanation the deductivenomological one and the inductivestatistical

one as forms of ab duction This then provides them with a structural logical anal

ysis in the style of chapter Moreover I argue that ab duction can mo del dynamics

of b elief revision in articial intelligence For this purp ose an extended version of

tation of the op erations the semantic tableaux of chapter provides a new represen

of expansion and contraction

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Titles in the ILLC Dissertation Series

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Ecient

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