Tenacious Tortoises: A Formalism for Argument over Rules of Inference
Peter McBurney and Simon Parsons Department of Computer Science University of Liverpool
Liverpool L69 7ZF U.K. g fP.J.McBurney,S.D.Parsons @csc.liv.ac.uk
May 31, 2000
Abstract Eighty years later, philosopher Susan Haack [9] took up the question of how one justifies the use of MP as a As multi-agent systems proliferate and employ different deductive rule of inference. If one does so by means of and more sophisticated formal logics, it is increasingly examples of its valid application, then this is in essence likely that agents will be reasoning with different rules a form of induction, which (as she remarks) seems too of inference. Hence, an agent seeking to convince an- weak a means of justification for a rule of deduction. If, other of some proposition may first have to convincethe on the other hand, one uses a deductive means of justi- latter to use a rule of inference which it has not thus far fication, such as demonstrating the preservation of truth adopted. We define a formalism to represent degrees across the inference step in a truth-table, one risks us- of acceptability or validity of rules of inference, to en- ing the very rule being justified. So how can one person able autonomous agents to undertake dialogue concern- convince another of the validity of a rule of deductive ing inference rules. Even when they disagree over the inference? acceptability of a rule, two agents may still use the pro- That rules of inference may be the subject of fierce posed formalism to reason collaboratively. argument is shown by the debate over Constructivism in pure mathematics in the twentieth century [21]: here the rule of inference being contested was double negation
1 Introduction elimination in a Reductio Ad Absurdum (RAA) proof:
:È ! ɵ ´:È ! :ɵ
FROM ´ and È
In 1895, the logician Charles Dodgson (aka Lewis Car- INFER :: È roll) famously imagined a dialogue between Achilles FROM ::
and a tortoise, in which the application of Modus Po- INFER È nens (MP) was contested as a valid rule of inference Although the choice of inference rules in purely for-
mal mathematics may be arbitrary,1 the question of ac- É [4]. Given arbitrary propositions È and , and the two
ceptability of rules of inference is important for Arti-
´È ! ɵ É premises È and , one can only conclude from these premises if one accepts that Modus Ponens ficial Intelligence for a number of reasons. Firstly, it is a valid rule of inference. This the tortoise refuses is relevant to modeling scientific reasoning. Construc- to do, much to the exasperation of Achilles. Instead, tivism, for example, has been proposed as a formalism the tortoise insists that a new premise be added to the for modern physics [3], as have other, non-standard log-
ics. In the propositional calculus proposed for quantum
È ^ ´È ! ɵµ ! É argument, namely: ´ . When Achilles does this, the tortoise still refuses to accept mechanics by Birkhoff and von Neumann [2], for ex- ample, the distributive laws did not hold:
É as the conclusion, insisting on yet another premise:
È ^ ´È ! ɵ ^ ´´È ^ ´È ! ɵµ ! ɵµ ! É ´ . The 1Goguen [8], for example, argues that standards of mathematical tortoise continues in this vein, ad infinitum. proof are socially constructed.
1
A _ B µ ^ ´A _ C µ ` A _ ´B ^ C µ Ê
´ does not include. For example, may be the use of the
A _ B µ ^ C ` ´A ^ C µ _ ´B ^ C µ ´ contrapositive or RAA. There are three ways in which Indeed, it is possible to view scientific debates over al- the dialogue between A and B could then proceed.
ternative causal theories as concerned with the validity First, A could attempt to demonstrate that Ê can be of particularmodes of inference, as we have shown with derived from the rules of inference which are contained
regard to claims of carcinogenicity of chemicals based in B’s logic. Similarly, A could attempt to demonstrate Ê on animal evidence[13]. Intelligent systems which seek that Ê is admissible in B’s logic [20], i.e. that is to formally model such domains will need to represent an element of that set of inference rules under which these arguments [14]. the theorems of B’s logic remain unchanged.2 In ei- Secondly, it is not obvious that one logical formal- ther of these two cases, it would then be rational for ism is appropriate for all human reasoning, a subject B to accept , being a proposition whose proof com- of much past debate in philosophy (e.g. see [10]). A mences from agreed assumptions and which uses infer- many-valued logic proposed for quantum physics, for ence rules equivalent (in the sense of derivability or ad- instance, has also been suggested to describe religious missibility) to those B has adopted. In such a case, the reasoning in Azande and Nuer societies, reasoning difference of opinion is resolved, to the satisfaction of which appeared to contravene Modus Ponens [5]. In- both agents.
deed, some anthropologists have argued that formal hu- Suppose then that A is unable to prove that Ê is man reasoning processes are culturally-dependent and derivable from or admissible in B’s logic. The second hence different across cultures [18]. To the extent that approach which A may pursue is to attempt to give non-
this is the case, systems of autonomous software agents deductive reasons for B to adopt Ê. Examples of such acting on behalf of humans will need to reflect the di- reasons could include: scientific evidence for the causal
versity of formal processes. In such circumstances, it mechanism possibly represented by Ê, where the rea- is possible that interacting agents may be using logics soning is in a scientific domain; instances of its valid ap- with different rules of inference, as is possible in the plication (e.g. the use of precedents in legal arguments); agent negotiation system of [15]. If one agent seeks the (possibly non-deductive) positive consequences for
to convince another of a particular proposition then that B of adopting Ê (e.g. that doing so will improve the first agent may have to demonstratethe validity of a rule welfare of B, of A and/or of third parties); the (possi- of inference used to prove the proposition. Our objec- bly non-deductive) negative consequences for B of not
tive in this work is to develop a formalism in which such adopting Ê (e.g. that not doing so will be to the detri- a debate between agents could be conducted. ment of B, of A and/or of third parties); or empirical evidence which would impact the choice of a particular logic.3 The precise nature of such arguments will de- 2 Arguments over rules of infer- pend upon the domain represented by the multi-agent
system, and the nature of the proposition . Moreover, ence for A to successfully convince B using such arguments, B would require some formal means of assessing them, We begin by noting that a dialogue between two agents perhaps using a logic of values as outlined in [7]. Al- in which one only asserts, and the other only denies, a though currently being explored, these ideas are not rule of inferencewill not likely lead very far. A dialogue pursued further here. between agents concerning a rule of inference will need Suppose, however, that A exhausts all such argu- to express more than simply their respective positions
ments, and still fails to convince B to adopt either Ê if either agent is to be persuaded to change its position.
or . Then, a third approach which A could pursue is What more may be expressed?
to represent B’s misgivings over the use of Ê in an ap- Suppose we have two agents, denoted A and B, and propriate formalism and use this to seek compromise
that A seeks to convince B of a proposition . For ex- 2 ample, this may be a joint intention which A desires Note that Ê could be admissible in B’s logic yet not derivable from the axioms and inference rules of that logic. All derivable rules both agents to adopt. B asks for a proof of . Suppose that A provides a proof which commences from axioms are admissible, however [20]. 3Theory change in logic on the basis of empirical evidence has which are all accepted by B. Assume, however, that this been much discussed in philosophy, typically in a context of holist
proof uses a rule of inference Ê which B says its logic epistemology [17].
2 between the two of them. We term such a formalism for the probability that the application of the inference an Acceptability Formalism (AF) and see it as akin to rule is invalid. Thus, we cannot say that the application formalisms for representing uncertainty regarding the of the inference rule is valid in any one case, but we truth of propositions. Note that while B’s misgivings can say that, if applied to repeated samples drawn from
concerning rule Ê may arise from uncertainty as to its the same population, it will be invalidly applied (say) at
validity, they need not: B may be quite certain in reject- most 5% of the time. Thus, the calculation of Ô-values ing the rule. for statistical hypothesis tests, which is common prac- What would be an appropriate formalism for repre- tice in the biological and medical sciences [19], effec-
senting degrees of acceptability of a rule of inference? tively associates each inference with a value from the
fÔ : Ô ¾ ´¼; ½µg ½¼¼´½ Ôµ± At this point, A has adopted Ê and B has not, so that, in set . The label “ ” is thus a effect, A (or, strictly, A’s designer) has decided that the measure of confidence in the validity of application of rule is an acceptable rule and B has not so decided. In the inference rule.4
other words, A has assigned Ê the label Acceptable to Once the two agents have agreed to adopt such a dic-
Ê, and B has not assigned this label. Thus, a very sim- tionary, the labels could then be applied to multiple con-
ple representation of their views of Ê would be assign- tested rules of inference, and used in successive proofs.
ing labels from the qualitative dictionary: fAcceptable, To do this will require a calculus for combining labels f Unacceptableg or from the dictionary Acceptable, No for different rules, and for propagating labels through
opinion, Unacceptableg. Such simple dictionaries leave chains of reasoning, which is the subject of the next little room for compromise; so it behooves A to request Section. B to assign a label froma moregranulardictionary,such as the five-element set:
fAlways acceptable, Mostly acceptable but some- 3 Terrapin Logic TL times unacceptable, Acceptable and unacceptable to the same extent, Sometimes acceptable but mostly un- 3.1 Formalization
acceptable, Always unacceptableg. We now present a formal description of the logic, which Were B to assign any but the final label, Always unac- we call TL (for “Terrapin Logic”, from the Algonquian ceptable, then A has the opportunity to demonstrate to
for tortoise), to enable reasoning about acceptability la- B that the current use of Ê in the proof of is an ac- bels for rules of inference. Our formalization is similar ceptable application of the rule, and thus achieve some to that for the Logic of Argumentation LA presented in form of compromise between the two. [7], itself influenced by labelled deductive systems and To formalize this third approach we therefore assume earlier formalizations of argumentation. that A and B agree a dictionary D of labels to be as-
signed to rules of inference. The elements of such an We start with a set of atomic propositions including ? AF dictionary could be linguistic qualifiers, as in the > and , the ever true and ever false propositions. We assume this set of well-formed formulae (wff s), labeled
examples above, but they need not be. For example, D
f:; !; ^; _g Ä may be the set of integers between 1 and 100 (inclu- Ä, is closed under the connectives . may
sive), where larger numbers represent greater relative then be used to create a database ¡ whose elements are
~
: G : Ê : d µ G =
acceptability of the rule. It is possible to view stan- 4-tuples, ´ , in which is a wff,
´ ; ; : : : ; µ
½ Ò ½
dard statistical hypothesis-testing procedures, Neyman- ¼ is an ordered sequence of wff s, with
Ò ½ Ê = ´` ; ` ;:::; ` µ
¾ Ò Pearson theory [6], in this way. Here, for a proposition , and where ½ is an ordered sequence of inference rules, such that:
concerning unknown parameters, the inference rule
` ` ::: `
½ ½ ¾ ¾ Ò ½ Ò
is: ¼ .
¾ G
In other words, each element k is derived from the
FROM is true of a sample
½ preceding element k as a result of the application of
INFER is true of the population from which that
` ; ´k = ½;:::;Ò ½µ sample arises. the k-th rule of inference, k . The
Under assumptions regarding the manner in which the rules of inference in any such sequence may be non-
~
d = ´d ; d ; : : : ; d µ
¾ Ò sample was obtained from the population (e.g. that it distinct. The element ½ is an or- was randomly selected) and assumptions regarding the dered sequence of elements from a Dictionary D , being distribution of the parameters of interest in the popula- an assignment of AF labels to the sequence of inference tion, Neyman-Pearson theory estimates an upper bound 4This interpretation is akin to Pollock’s statistical syllogism [16].
3
Ð ¾ Ä = :
rules Ê . We also permit wff s to be elements of tion , these rules permit the (possibly con-
´Ð : ; : ; : ;µ ¡, by including tuples of the form , where tested) assertion of a Law of the Excluded Middle
each ; indicates a null term. Note that the assignment (LEM). d
of AF labels may be context-dependent, i.e. the i as- _
¯ The rule -E allows the elimination of a disjunc-
`
i ½ signed to i may also depend on . This is the case tion and its replacement by tuple when that tuple
for statistical inference, where the Ô-value depends on is a TL-consequence of each disjunct.
characteristics of the sample from which the inference :
is made, such as its size. ¯ The rule -I allows the introduction of negation. ?
With this formal system, we can take a database and The rule :-E allows the derivation of , the ever- ` use the consequence relation ÌCÊ defined in Figure 1 false proposition, from a contradiction. The rule
to build arguments for propositions of interest. This ::-E allows the elimination of a double negation.
consequence relation is defined in terms of rules for
! ´ : ; :
building new arguments from old. The rules are written ¯ The rule -I says that if on adding a tuple
¾ Ä : ;µ
in a style similar to standard Gentzen proof rules, with ; to a database, where , it is possible
!
the antecedents of the rule above the horizontal line and to conclude , then there is an argument for ! the consequent below. In Figure 1, we use the notation . The rule thus allows the introduction of into
arguments.
ª À G to refer to that ordered sequence created from ap-
pending the elements of sequence À after the elements
! ¯ The rule -E says that from an argumentfor and
of sequence G, each in their respective order. The rules
! an argument for it is possible to build an ar- are: gument for . The rule thus allows the elimination
of ! from arguments and is analogous to MP in
´ : G : Ê : ¯ The rule Ax says that if the tuple
standard propositional logic.
~ µ
d is in the database, then it is possible to build
~
: G : Ê : dµ the argument ´ from the database. Our purpose in this paper is to propose a formal syntax The rule thus allows the construction of arguments and proof rules for argument over rules of inference, from database items. and so we do not consider semantic issues. Interpreta-
tions of TL would be defined with respect to a specified
^ ´ : G : ¯ The rule -I says that if the arguments
AF dictionary or dictionary-class, and may assign !
~
: dµ ´ : À : Ë : e~µ
Ê and may be built from to represent a relationship between propositions other
^ the database, then an argument for may also than material implication. A virtue of our initial focus be built. The rule thus shows how to introduce ar- on syntactical elements is that, once defined, the proof
guments about conjunctions; using it requires an rules may be applied in different semantic contexts. We
` ´ ^ µ
inference of the form: ; , which we are currently exploring alternative semantic interpreta- `
denote ^-I in Figure 1. This inference is then as- tions for TL, along with the issue of its consistency and d
signed an AF dictionary value of ^-I . completeness relative to these. ^
¯ The rule -E1 says that if it is possible to build an 3.2 Negotiation within TL
^ argument for from the database, then it is
also possible to build an argument for . Thus the Given the formalism TL just defined, how may this be rule allows the elimination of one conjunct from used by two agents, A and B, in dialogue over a con-
an argument, and its use requires an inference of tested rule of inference? We assume the agents have
^ `
the form: . This inference is denoted agreed a common set of assumptions to which they both
D d
` adhere, and have agreed a common AF dictionary of ^ by ^-E1 , and is assigned an AF value of -E1 .
labels to assign to inference rules. We assume the el- ^ The rule ^-E2 is analogous to -E1 but allows the
ements of D are partially ordered under a relation de-
elimination of the other conjunct. D
noted <. We further assume that contains an element
d d ¾ D d < d
½ ½ _
¯ The rule -I1 allows the introduction of a disjunc- such that for all other , we have ,
d ½ tion from the left disjunct. The rule _-I2 allows and that the assignment of to a rule of inference the introduction of a disjunction from the right dis- by an agent marks it as always and completely unac-
junct. If instantiated with a wff and its nega- ceptable.
4
~
´ : G : Ê : dµ ¾ ¡
Ax
~
¡ ` ´ : G : Ê : d µ
ÌCÊ
~
¡ ` ´ : G : Ê : dµ ¡ ` ´ : À : Ë : e~µ ÌCÊ ÌCÊ and
^-I
~
¡ ` ´ ^ : G ª À ª ´ ^ µ : Ê ª Ë ª ´` µ : d ª e~ ª ´d µµ
ÌCÊ ^
^-I -I
~
¡ ` ´ ^ : G : Ê : dµ ÌCÊ
^-E1
~
¡ ` ´ : G ª ´ µ : Ê ª ´` µ : d ª ´d µµ
ÌCÊ ^
^-E1 -E1
~
¡ ` ´ ^ : G : Ê : d µ ÌCÊ
^-E2
~
¡ ` ´ : G ª ´µ : Ê ª ´` µ : d ª ´d µµ
ÌCÊ ^
^-E2 -E2
~
dµ ¡ ` ´ : G : Ê : ÌCÊ
_-I1
~
¡ ` ´ _ : G ª ´ _ µ : Ê ª ´` µ : d ª ´d µµ
ÌCÊ _
_-I1 -I1
¡ ` ´ : À : Ë : e~µ ÌCÊ
_-I2
¡ ` ´ _ : À ª ´ _ µ : Ë ª ´` µ : e~ ª ´e µµ
ÌCÊ _
_-I2 -I2
~
¡ ` ´ _ : G : Ê : d µ
ÌCÊ and
~
¡; ´ : ; : ; : ;µ ` ´ : À : Ë : e~µ ¡; ´ : ; : ; : ;µ ` ´ : Â : Ì : f µ: ÌCÊ ÌCÊ and
_-E
~ ~
¡ ` ´ : G ª À ª Â ª ´ µ : Ê ª Ë ª Ì ª ´` µ : d ª e~ ª f ª ´d µµ
ÌCÊ _
_-E -E
~
¡; ´ : ; : ; : ;µ ` ´? : G : Ê : d µ ÌCÊ
:-I
~
¡ ` ´: : G ª ´: µ : Ê ª ´` µ : d ª ´d µµ
ÌCÊ :
:-I -I
~
¡ ` ´ : G : Ê : dµ ¡ ` ´: : À : Ë : e~µ ÌCÊ ÌCÊ and
:-E
~
¡ ` ´? : G ª À ª ´?µ : Ê ª Ë ª ´` µ : d ª e~ ª ´d µµ
ÌCÊ :
:-E -E
~
¡ ` ´:: : G : Ê : d µ ÌCÊ
::-E
~
¡ ` ´ : G ª ´ µ : Ê ª ´` µ : d ª ´d µµ
ÌCÊ ::
::-E -E
~
¡; ´ : ; : ; : ;µ ` ´ : G : Ê : dµ ÌCÊ
!-I
~
¡ ` ´ ! : G ª ´ ! µ : Ê ª ´` µ : d ª ´d µµ
ÌCÊ !
!-I -I
~
¡ ` ´ : G : Ê : dµ ¡ ` ´ ! : À : Ë : e~µ ÌCÊ ÌCÊ and
!-E
~
¡ ` ´ : G ª À ª ´µ : Ê ª Ë ª ´` µ : d ª e~ ª ´d µµ
ÌCÊ ! !-E -E
Figure 1: The TL Consequence Relation
5 We then assume the two agents agree to construct discussion of the implications of adopting or not adopt-
6
a logical language Ä which adopts all inference rules ing the contested rule or the proposition . Suppose
B
d 6= d
in the union of their two respective sets of rules instead then that ½ . In this circumstance, al-
k
Ä `
(i.e. contains all those rules which either agent has though B’s logic does not include k , B may be willing
5 A
¡ `
adopted). We next assume that two databases, to accept k some of the time. For instance, if the labels
B
Ä D
and ¡ , of 4-tuples are constructed from as outlined in had a probabilistic interpretation, B may agree to
A
¡ `
above, with containing agent A’s assignments of use k a proportion of the times it is asked to do so,
dictionary labels in the fourth place of each tuple, while analogously with statistical confidence values. Alterna- B
¡ contains B’s assignments. Thus, the elements of the tively, B may accept the use of contested rules on the two databases may potentially only differ in the fourth basis of the label assigned to them being above some
places of the tuples each contains, since Ä uses all in- threshold value; such thresholds may differ according ference rules of both agents. One can readily imagine to the identity of the requesting agent, A, for example, cases where such differences may arise. For example, with contested rules being accepted more readily from
we noted in the previous section that the TL disjunction trusted agents than from others. _ introduction rules, _-I1 and -I2, permit the assertion Our approach so far has assumed that A is seeking to
of a LEM. If one agent does not agree with the use of persuade B to adopt a proposition , and hence an infer- `
this rule in this way they may assign it an AF value of ence rule k . However, if the two agents are engaged d
½ . As mentioned, this assignment can be context- in some joint task, for instance agreeing common inten- d
specific, i.e. an agent could assign the value ½ only tions or prioritizations, both A and B may be simultane- when either of these rules is used to assert LEM,and not ously seeking to persuade each other to adopt proposi-
when they are used for two unrelated propositions and tions and thus inference rules. In these circumstances, it
. Likewise with the double negation elimination rule may behoove the two agents to agree common accept-
::-E, which may be considered appropriate for some ability labels for contested inference rules, as a means
propositions and not others. Similarly, agents may as- of ranking or prioritizing propositions. How might this A
sign differential dictionary values to the use of inference be done? Suppose, as above, that database ¡ contains
B
~
A
: G : Ê : d µ ¡
rules which are derived from those in Figure 1, such as the tuple ´ , while contains the tuple ~
the two distributive laws mentioned in Section 1 in rela- B
: G : Ê : d µ ´ . We can readily construct a common
tion to Birkhoff and von Neumann’s logic for quantum ~
´ : G : Ê : dµ
database ¡ of tuples , where the labels
~ ~
A B
mechanics. ~
d d d are defined from and by some agreed method.
As in Section 2, assume thereis a claim which A as- For instance, A and B may agree to define each element
A B
serts but which B contests since its proof uses an infer- ~
d d d = ÑiÒfd ; d g i
i of by .
i i
ence rule which B has not adopted, nor which is deriv- It would also be straightforward to define a func- £
able from, nor admissible in, B’s logic. For simplicity, ~ d tion which maps a sequence d to a single value ,
we first assume there is only one such rule and that it is to provide some form of summary assessment of a
deployed only once in A’s proof of . Suppose the tuple £ =
chain of inferences. For instance, the mapping d
~
A
´ : G : Ê : d µ
which contains A’s proof of is , and
ÑiÒ fd g
=½;:::;Ò i
i would be equivalent in this context to ` that the contested rule is k , for some k. B’s assignment saying that “A chain is only as strong as its weakest
of labels to the inference rules used in the proof of is link.” If AF dictionary values were real numbers be-
~
B
´ : G : Ê : d µ
the fourth element of the tuple . Since tween 0 and 1 (e.g. statistical Ô-values), then an ap-
É
Ò
£
= ½ ´½ d µ the k-th rule is contested by B, we should expect the d
propriate mapping may be i .
i=½
A B
~ ~
A B
d 6= d d
k-th elements of d and to differ, i.e. that . With such a mapping agreed, the two agents could then
k k
B
= d d readily define a rank order of propositions. For in-
If ½ , then B has assigned the contested rule
k
£
d = ÑiÒ fd g i
a label which indicates its use is completely unaccept- stance, if the weakest-link mapping i
~
´ : G : Ê : dµ
able to B. This would eliminate any possibility of com- was used, and ¡ contains the tuples
: À : Ë : e~µ promise between the two agents over the use of the rule. and ´ , then we could define to be ranked The dialogue could proceed only by the second of the 6Agent A could seek to contest the assignment by B of the label
two approachesoutlined in Section 2, i.e. by means of a d ½ , an approach we do not pursue here. As Heathcote has demon- strated [11], to justify an assertion that the rule represented an invalid 5We assume for simplicity that the axioms of the logics of the two form of argument B may ultimately require some form of abduction, agents are not inconsistent. which thus provides the possibility of continuing contestation by A.
6
ÑiÒ fe g < ÑiÒ fd g
j i i higher than whenever j . This Acknowledgments may be of value if the propositions represent, for ex- ample, alternative joint intentions, or competing allo- This work was partially funded by the UK EPSRC un- cations of resources. Recent work in AI has explored der grant GR/L84117 and a studentship. We are also methods for combining preferences of different agents grateful for comments from Trevor Bench-Capon, Mark in argumentation systems [1]. Colyvan, Susan Haack, Vladimir Rybakov & Bart Ver- heij, and from the anonymous reviewers.
Note also that the AF labels and the summary map- £
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