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2018-04-23 Hypersequent Calculi for Modal Logics

Burns, Samara Elizabeth

Burns, S. A. (2018). Hypersequent Calculi for Modal Logics (Unpublished master's thesis). University of Calgary, Calgary, AB. doi:10.11575/PRISM/31825 http://hdl.handle.net/1880/106539 master thesis

University of Calgary graduate students retain copyright ownership and moral rights for their thesis. You may use this material in any way that is permitted by the Copyright Act or through licensing that has been assigned to the document. For uses that are not allowable under copyright legislation or licensing, you are required to seek permission. Downloaded from PRISM: https://prism.ucalgary.ca UNIVERSITY OF CALGARY

Hypersequent Calculi for Modal Logics

by

Samara Elizabeth Burns

A THESIS

SUBMITTED TO THE FACULTY OF GRADUATE STUDIES

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE

DEGREE OF MASTER OF ARTS

GRADUATE PROGRAM IN PHILOSOPHY

CALGARY, ALBERTA

April, 2018

© Samara Elizabeth Burns 2018 Abstract

This thesis surveys and examines hypersequent approaches to the proof theory of modal logics. Traditional sequent calculi for modal logics often fail to have many of the desirable properties that we expect of a sequent calculus. Cut cannot be eliminated from the system for S5, the axioms of each logic are not straightforwardly related to the sequent rules, and vari- ation between modal sequent calculi occurs in the presence and absence of logical rules, rather than structural rules, which violates Došen’s princi- ple. The hypersequent framework is beneficial as we can provide Cut-free complete treatments of many modal logics. However, hypersequent ap- proaches often lack generality, or do not conform to Došen’s principle. A recent development in the proof theory of modal logics, called relational hypersequents, appears to overcome many of these issues. Relational hy- persequents provide a unified proof theory for many modal logics, where the logical rules are held constant between modal systems. This thesis pro- vides some preliminary results for relational hypersequents by providing a

Cut-free completeness proof for the modal logic K.

ii Acknowledgements

I would like to thank everyone in the Department of Philosophy at the Uni- versity of Calgary for their support and kindness throughout my time in the department. In particular, I would like to thank my thesis supervisor

Richard Zach. Your endless encouragement, pep talks and wealth of knowl- edge have been invaluable to me. This thesis would not have been possible without you.

I would also like to thank my partner, Mike, who has been the greatest support for me throughout the writing process. Thank you for believing in me, and for everything that you do. Finally, thank you to my parents, who have always encouraged me to do my best.

iii Table of Contents

Abstract ...... ii

Acknowledgements ...... iii

Table of Contents ...... iv

List of Tables ...... vi

1 Introduction ...... 1

2 Gentzen Systems for Modal Logics ...... 7

2.1 Syntax and Semantics ...... 7

2.2 The Sequent Calculus ...... 8

2.3 Natural Deduction ...... 16

2.4 Modal Sequent Calculi ...... 19

2.5 Modal Natural Deduction ...... 24

2.6 Conclusion ...... 27

3 Hypersequent Systems ...... 28

3.1 Introduction ...... 28

3.2 Hypersequent Calculi ...... 29

3.3 Hypersequents and Modal Logic ...... 30

3.4 The Development of Hypersequent Calculi ...... 32

3.5 Hypersequent Rules from Frame Properties ...... 39

3.6 Related Sequent Systems ...... 44

3.7 Conclusion ...... 53

4 Relational Hypersequents ...... 55

4.1 Introduction ...... 55

iv Table of Contents

4.2 Relational Hypersequents ...... 56

4.3 Soundness ...... 61

4.4 Completeness for RK ...... 70

5 Conclusion ...... 85

Bibliography ...... 88

v List of Tables

2.1 Sequent Rules for the System LK ...... 11

2.2 Natural Deduction Rules for the System NK ...... 17

2.3 Modal Sequent Calculus Rules ...... 20

2.4 Modal Natural Deduction Rules ...... 25

3.1 Hypersequent Rules for the System HLK ...... 31

3.2 Mints’ Sequent Rules ...... 35

3.3 Simple Frame Properties and their Normal Descriptions ...... 40

3.4 Tree Hypersequent Rules for the Base Calculus THSK ...... 46

3.5 Modal Tree Hypersequent Rules ...... 46

3.6 Structural Tree Hypersequent Rules ...... 46

3.7 Modal 2-Sequent Calculus Rules ...... 49

3.8 Linear Nested Sequent Rules for the Base Calculus LNSK ...... 50

3.9 Modal Linear Nested Sequent Rules ...... 51

3.10 Basic Rules for the Labelled System G3K ...... 51

3.11 Modal Labelled Sequent Rules ...... 52

4.1 Relational Hypersequent Rules for the system RK ...... 57

4.2 External Structural Rules for Relational Hypersequents ...... 58

5.1 Modal Proof Systems and their Important Properties ...... 86

vi Chapter 1

Introduction

Gerhard Gentzen introduced the sequent calculus and natural deduction systems in his “Untersuchungen über das logische Schließen I-II" [Investi- gations into Logical Deduction I-II] (1935a/1935b). Gentzen’s natural deduc- tion systems NJ and NK (for intuitionistic logic and classical logic, respec- tively) were the centerpiece of his investigation. He developed these calculi as a way of formally mimicking the structure of mathematicians’ “natural” reasoning. Gentzen’s natural deduction was a departure from the axiomatic deduction systems that were used at the time. Axiomatic proofs proceed top-down from a set of axioms with a limited set of rules (often just modus ponens), whereas natural deduction allows one to make and discharge as- sumptions within a formal derivation. In addition, there are two rules that govern each logical operator, and no axioms. Since mathematicians do not usually reason from axioms alone, natural deduction appears to be a better analogue to actual mathematical reasoning. At the same time as Gentzen’s calculi NJ and NK were developed, Stanisław Jaskowski´ presented similar ideas in his paper “On the Rules of Suppositions in Formal Logic” (1934),

1 1. Introduction which was inspired by lectures from Jan Lukasiewicz (Szabo, 1969, 4). A history of natural deduction can be found in Pelletier and Hazen (2012).

The sequent calculus systems LJ and LK do not have the same natural structure as Gentzen’s natural deduction systems. Gentzen developed the sequent calculus as a meta-calculus for his natural deduction systems. At the time, Gentzen theorized that every natural deduction derivation could be converted into a determinate normal form. He thought that no formula need enter a derivation that is not a component of the end-formula. In other words, every natural deduction derivation has a presentation that is

“not roundabout" (Gentzen, 1935a, 289). This technical result would come to be known as the normalization theorem. This theorem was not proved directly for classical logic until much later. Gentzen did prove normalization directly for intuitionistic logic, but did not publish the result (von Plato,

2008). Instead, Gentzen proved an analogous result in the sequent calculus.

In the context of the sequent calculus, Gentzen was able to prove the

Hauptsatz (or main theorem), also known as the cut elimination theorem.

The cut rule is a unique structural rule of the sequent calculus, as it is the only rule where a formula can be “removed” from a derivation.

,'',⇤ ⇥ Cut ) ,⇤ ,⇥ ) ) Gentzen showed that any sequent calculus derivation can be algorith- mically converted into a derivation with the same end-formula but without using the Cut rule. Due to the correspondence between natural deduc- tion and the sequent calculus, the Cut elimination theorem for the sequent calculus implies that every proof in the corresponding natural deduction

2 1. Introduction system has a normal form. The Cut elimination theorem also has important philosophical and technical consequences. Any sequent calculus where Cut can be eliminated, and where the formulas in the premise of all other rules are subformulas of the formulas in the conclusion, is analytic: derivations can be constructed from the bottom-up simply by breaking formulas down into their subformulas. In some cases, the Cut elimination theorem can also be used to prove consistency and decidability. In computer science, the Cut elimination theorem theorem makes derivation systems amenable to proof search.

Perhaps another important feature of Gentzen’s calculi, and one that is especially pertinent given the variety of logics we encounter in a contem- porary context, is its usefulness and applicability. Avron (1996) uses the sequent calculus as a paradigm example of a successful logical framework. It can handle a great many logics, and in fact, it is almost expected that a logic will have a well-behaved sequent calculus. It says something fundamental about the logic: that it is useful, or should be investigated further.

It is surprising, then, to find that some important non-classical logics lack well-behaved Gentzen-type calculi. Modal logics in particular are an interesting case. Although modal logics are philosophically interesting and useful, and though they have a robust semantics, they do not have a uniform proof theory. Axiomatic methods and tree methods are the preferred mode of derivation in modal contexts, and it has been difficult to develop a unified method for sequent calculus and natural deduction proofs for modal logics.

Although K, and many of its normal extensions (such as T, D, B, S4 and S5) all have sound and complete sequent calculi, S5 is not complete without the Cut rule. It follows that the sequent calculus for S5 is not analytic—a

3 1. Introduction property that we will show has some significant philosophical and technical importance. The inability of the sequent calculus to uniformly accommo- date all of the normal extensions of K is puzzling. Insofar as we view modal logics as a cohesive group of logics, we should expect a proof system to deal with them all in an elegant manner.

The situation is similar when considering natural deduction for modal logics. Although Lemmon-style and Fitch-style natural deduction systems have been given for some modal logics, Gentzen-style natural deduction systems have been largely neglected. Dag Prawitz developed modal natural deduction systems for S4 and S5 in his dissertation “Natural Deduction: A

Proof-Theoretical Study” (1965), where he gave a normalization procedure for these calculi. Prawitz’s work with S4 and S5 has recently been expanded upon by Medeiros (2006) and Martins and Martins (2008). Unfortunately, there is still no Gentzen-style natural deduction system for K and its other extensions. Although we may translate Fitch-style natural deduction sys- tems into Gentzen-style, modal logics still lack the correspondence between the sequent calculus and natural deduction.

This thesis outlines some of the attempts at overcoming these issues, and presents some preliminary results for a recently developed sequent system we call relational hypersequents. The structure of the thesis is as follows. In the next chapter we introduce the sequent calculus system LK and natural deduction system NK for classical logic, and discuss some of their important features. In particular, we consider the Cut elimination theorem and its philosophical and technical upshots. We then introduce extensions of these calculi that have been developed for modal logics. We show how many of the modal sequent systems do not have the desirable properties that the

4 1. Introduction system LK has. In particular, we show that the Cut elimination theorem fails for S5. In addition, we explain how the relationship between natural deduction and sequent calculi has been lost in a modal context due to the lack of appropriate modal rules.

The third chapter discusses some extensions of Gentzen’s sequent frame- work. Various hypersequent frameworks have been developed, and with varying degrees of success, for modal logics. We detail the development of these systems and discuss the extent to which they meet the philosophical and technical desiderata discussed in the second chapter. We also intro- duce some other variations on the sequent framework, namely 2-sequents, nested sequents, and labelled sequents. Although not strictly hypersequent frameworks, these systems have connections to the relational hypersequent systems introduced in the final chapter.

The fourth chapter will focus on an extension of Gentzen’s sequent cal- culus we call relational hypersequents. These sequents display features of hypersequents, nested sequents and labelled sequents. These hybrid features make relational hypersequents a powerful framework: K and its normal extensions can be handled in a uniform way, unlike other hyper- sequent systems. In addition, these systems retain their connection to the notion of natural deduction. These systems are fairly new, and we provide some preliminary results: a Cut-free completeness proof for K.

This thesis makes three main contributions to the literature on modal proof theory. First, we provide criteria for comparing and evaluating ap- proaches to the proof theory of modal logics. Specifically, we consider generality (how many logics the proof system accommodates), Cut elimina- tion, modularity and Došen’s principle. We also provide a new, systematic

5 1. Introduction survey of hypersequent systems for modal logics, which includes the new re- lational hypersequent framework. Finally, we give the first proof of Cut-free completeness for the relational hypersequent system RK.

6 Chapter 2

Gentzen Systems for Modal Logics

2.1 Syntax and Semantics

The calculi presented in this thesis are for the propositional fragment of modal logic. We define our language LÉ as follows. Let p1,p2,p3,... be a { } denumerable set of sentence letters. We include the propositional connec- tives , , and . We use ', ,✓ ,... as variables ranging over arbitrary ¬ _ ^ ! formulas of our language. We also add the logical symbol É, for necessity.

The possibility operator Ü can be defined as É '. ¬ ¬

Definition 1 (Atomic Formula). Atomic formulas of LÉ are defined as fol- lows:

1. All sentence letters are atomic formulas.

Definition 2 (Formula). The set of formulas of LÉ is defined as:

1. All atomic formulas are formulas.

2. If ' is a formula, then ' is also a formula. ¬

7 2. Gentzen Systems for Modal Logics

3. If ' and are formulas, then (' ) is also a formula. _ 4. If ' and are formulas, then (' ) is also a formula. ^ 5. If ' and are formulas, then (' ) is also a formula. !

6. If ' is a formula, then É' is also a formula.

Definition 3 (Frame). A frame F = W,R is an ordered pair where W is a h i non-empty set of worlds and R is a 2-place relation between worlds, called the accessibility relation. A model is defined as an ordered pair, M = F, v h i where F is a frame and v is a valuation function. The valuation function takes as argument an ordered pair w,' where w W and ' LÉ and h i 2 2 maps it to the set T, F . { } Definition 4 (Satisfiability). We define the satisfiability of the well-formed formulas inductively as follows:

M, w è ' iff v (w,')=T .

M, w è ' iff M, w è '. ¬ 6 M, w è ' iff M è ' or M, w è . _ M, w è ' iff M, w è ' and M, w è . ^ M, w è ' iff M, w è ' or M, w è . ! 6 M, w èÉ' iff for all u W such that wRu,M, u è '. 2

2.2 The Sequent Calculus

As the basis for our modal sequent systems, we use the propositional frag- ment of Gentzen’s sequent calculus for classical logic, LK.1

1This thesis is concerned with classical modal logic and so we do not consider any extensions of Gentzen’s intuitionistic calculus LJ.

8 2. Gentzen Systems for Modal Logics

Definition 5 (Sequent). A sequent is any expression of the form ) where and are sequences of formulas of LÉ.

We call the sequent arrow, which separates the left hand and right ) hand side of the sequent. These may alternatively be referred to as the antecedent and succeedent of the sequent, respectively. The symbol is ) purely syntactic. For clarity, we will occasionally place brackets around in- dividual sequents. A sequent may be thought of intuitively as a conjunction of the formulas on the left hand side, and a disjunction of the formulas on the right hand side, where the sequent arrow represents implication. More formally, the sequent can be thought of as:

! ^ _ Note that we have defined a sequent as a pair of sequences of formulas.

Sequents can be defined alternatively as multisets or sets of formulas. When defined as sets, the order and quantity of each formula on each side of the sequent does not need to be taken into consideration. The structural rules of contraction and exchange are not required in these cases. When defined as multisets, the number of appearances of a formula on each side of the sequent must be taken into consideration, although the order of the formulas is not important. When considered as sequences, both the order and number of appearances of the formulas must be considered.

Derivations in the sequent calculus are trees of sequents. Each sequent in the tree is either an axiom or stands below another sequent in the tree, separated by a line representing the application of an inference rule. Each line is labeled according to the rule applied. Inference rules are spit into

9 2. Gentzen Systems for Modal Logics two main groups: structural rules, and logical rules. Structural rules affect the order and quantity of a formula in a sequent. Logical rules manipulate formulas according to the main operator. There are two rules for each logical operator: a left rule and a right rule, which are applied to the left- and right-hand side of the sequent, respectively.

The axioms and rules of the sequent calculus can be found in table 2.1.

Note that all of the rules of Gentzen’s sequent calculus, excluding Cut, have the subformula property: the formulas in the premises of the rule are subformulas of the formulas occurring in the conclusion of the rule. Any rule that obeys the subformula property is analytic. It can be guaranteed that any sequent calculus where Cut can be eliminated, and where the other rules obey the subformula property, is itself analytic.

We use the term principal formula to refer to the formula that is intro- duced in either the antecedent or succeedent of the conclusion of a logical inference rule. The formulas that are explicitly shown in the premises of the inference are side formulas. For example, in the inference rule R, the ^ formula ' is the principal formula, while ' and are side formulas. All ^ other formulas in an inference are called parametric formulas.

Cut Elimination

The Cut elimination theorem (or Hauptsatz) is the central theorem of Gentzen’s dissertation. He provides a list of transformation steps to show how any proof using the Cut rule can be transformed into one without the use of Cut.

In order to prove this theorem, he introduces an equivalent rule called a

Mix.

10 2. Gentzen Systems for Modal Logics

Axioms

' ' )

Structural Rules

TL TR ', ) ) ,' ) ) ',', ,',' CL CR ', ) ) ,' ) ) ,', , 0 ,', ,0 ) EL ) ER ,', 0 , ,',0 ) ) ,'',⇤ ⇥ Cut ) ,⇤ ,⇥ ) ) We call ' the Cut formula of the above infer- ence.

Logical Rules

,' ', L R ', ) ) , ' ¬ ¬ ¬ ) ) ¬ ', , ,' L R1 ) ) ) , ' , _ ' _ _ ) ) _ , ', ) R2 ) L1 ,' _ ' , ^ ) _ ^ ) , ,' , ) L2 ) ) R ' , ^ ,' ^ ^ ) ) ^ ,' , ', , ) ) L ) R ' , ! ,' ! ! ) ) ! Table 2.1: Sequent Rules for the System LK

11 2. Gentzen Systems for Modal Logics

1 1 2 2 ) ) Mix 1,2⇤ ⇤1,2 )

In the above inference figure, there is some formula ' that occurs at least once in 2 and 1, and 2⇤, ⇤1 are the result of removing all copies of

' from 2, 1 with the application of the Mix rule. We call this formula the Mix formula. Every application of Cut can be converted into a Mix through exchange and weakening on the Mix formula after applying the Mix rule. Similarly, an application of Mix can be converted into a series of Cuts through exchange and contraction on the Cut formula before applying the

Cut rule. Gentzen shows that any derivation whose lowest inference is a Mix can be converted into one without a Mix inference. We use a superscript

cf to denote that is derivable in the Cut-free portion of the ` ) ) calculus.

cf cf Theorem 6 (Cut Elimination). If LK ( ,',...,') and LK (',...,',⌃ ` ) ` ) cf ⇤) then LK ( ,⌃ ,⇤) ` ) Gentzen’s proof proceeds through double induction on the rank and grade of the Mix formula.

Definition 7 (Rank).

1. The left rank is the greatest number of sequents on the left-hand

branch such that the Mix formula occurs in the succeedent the se-

quent and the last sequent in the branch is the left-hand premise of

the Mix.

12 2. Gentzen Systems for Modal Logics

2. The right rank is the greatest number of sequents on the right-hand

branch such that the Mix formula occurs in the antecedent of the

sequent and the last sequent in the branch is the right-hand premise

of the Mix.

3. The rank of a derivation is the sum of the left and right ranks.

Definition 8 (Grade). The grade of a formula ' is defined as the number of

logical symbols that occur in '.

Through a series of transformations, the Mix can be moved through the

derivation to act on a formula that is of lower grade or a lower rank. Consider

the following example, with the left and right rank = 1...... ,' , ', 1 1,' ',2 2 1 1 1 1 2 2 † Mix ) ) R ) L ) , , ) 1 1,' ^ ' ,2 2 ^ 1 2⇤ ⇤1 2 ) ^ ^ ) Mix ) 1,2 1,2 1,2 1,2 ) )

The Mix is moved up the derivation to act on the constituent formula ',

which is of a lower grade. Eventually, the Mix will be acting on axioms and

can be eliminated from the proof altogether.

A semantic Cut elimination proof may be given in place of an algorithmic

Cut elimination procedure. This type of proof usually shows that the Cut-

free fragment of the calculus is complete: if a sequent is valid, then it is

provable in the calculus without the use of Cut. This type of proof, however,

does not show how to remove a Cut inference from any given proof, just that

a Cut-free proof is possible. We refer to such systems as Cut-free complete.If

a system has a full Cut-elimination proof, we say that the system is Cut-free

or Cut-admissible.

13 2. Gentzen Systems for Modal Logics

The notion of Cut elimination has several important consequences, but of particular interest is the notion of analyticity.

Definition 9 (Analytic proof). A sequent calculus system is analytic just in case

1. The system is Cut-admissible or Cut-free complete and,

2. All other rules have the subformula property.

Analytic proofs (whether a formal derivation in a proof system, or an informal demonstration of a concept or proposition) are built from the bottom-up. The object of demonstration is taken as the starting point, and is broken down into smaller parts until the proof reaches axioms. A synthetic proof, on the other hand, is built from the top-down. Synthetic proofs begin with axioms, and the desired proposition or concept is derived top-down. If a sequent calculus system has the properties listed above, then it is analytic: every provable sequent has a derivation that is analytic. This is not the case in a sequent calculus where either of the above conditions do not hold.

Due to the difference in structure between analytic and synthetic proofs, synthetic proofs may contain information that is not strictly related to, or necessary for, the demonstration of the conclusion. When constructing a proof in a synthetic way, we might wind down an unnecessary path before reaching our conclusion. Analytic proofs ensure that no excess information enters into the proof itself. This is not always desirable: because analytic proofs are focused on the structure of the formulas rather than the structure of the proof itself, analytic proofs can be long and complex. Synthetic proofs are often shorter and more elegant.2 2George Boolos gives an example of a proof in his paper “Don’t Eliminate Cut” that is

14 2. Gentzen Systems for Modal Logics

The distinction between analytic and synthetic proofs has a long histor- ical background. The notion began in the Western world at least as early as Ancient Greece, with evidence that Plato and Aristotle endorsed some conception of analytic methods (Poggiolesi, 2010, 12). Other important proponents of analytic proof were Bernard Bolzano and René Descartes.

For Bolzano, analytic proofs were a way of providing an objective ground for mathematical truths. By taking a proposition and reducing it to its smallest components, we give reasons for the proposition to be true (Poggiolesi, 2010,

13).

Similarly to Bolzano, Descartes cited epistemic reasons for preferring an- alytic proofs. Although analytic proofs are often longer and harder to follow,

Descartes privileged the use of analytic methods over synthetic methods. In a response to an objection to his Meditations, Descartes claims that analytic proofs would convince an attentive reader of the truth of the proposition

“as if he had himself discovered it”; a synthetic proof, on the other hand, may convince a less precocious reader, but lacks the formal strength of an analytic proof (Descartes, 1984, 110).

Beyond historical and philosophical considerations, the Cut elimination theorem has several important technical consequences. The consistency of the system LK (likewise LJ) can be proven using the fact that the system is Cut-free. The empty sequent is derivable in LK iff the system ) is inconsistent. However, without the Cut rule, there is no possible way of deriving the empty sequent. If were derivable, then it would ) be derivable without the use of Cut. But since Cut-free proofs have the mere pages long in a tree system, whereas the corresponding proof in natural deduction “contains more symbols than there are nanoseconds between Big Bangs" (Boolos, 1984, 373).

15 2. Gentzen Systems for Modal Logics subformula property, this is impossible.

In addition, Gentzen used the subformula property in order to give a decision procedure for the propositional fragments of LK and LJ (Gentzen,

1935b, 204). The proof proceeds as follows. Consider a sequent S. Let S 0 be reduct of S where no formula ' occurs more than three times in the succeedent or the antecedent of the sequent. Consider now the set ⇧ of all sequents that contain only subformulas of the formulas in S 0. This set is finite. Now consider all of the axioms in ⇧, and examine the remaining sequents. Find all sequents that are possible conclusions of an inference applied to axioms in ⇧. The lower sequent is provable. Now see if there is a sequent in ⇧ that is possibly the conclusion of an inference rule applied to one of the provable sequents in ⇧. This procedure will continue until either a proof of S 0 is found or the process terminates without finding a proof. The sequent calculus also has an important connection to natural de- duction.

2.3 Natural Deduction

Gentzen-style natural deduction is meant to represent “natural" mathemat- ical reasoning. Unlike sequent calculus proofs, natural deduction systems allow us to derive conclusions based on assumptions. NK derivations are trees whose leaves are the assumptions that the conclusion depends on.

Certain assumptions may be discharged when a rule is applied, signaling that the conclusion no longer depends on the discharged formula. Assump- tions are labelled, and when an assumption is discharged by a rule, the index is written beside the inference line. The rules for classical natural de-

16 2. Gentzen Systems for Modal Logics

n ['] . ' ' . ¬ Elim ¬ Intro ? ?' ¬ ¬

' Intro1 Intro2 ' _ ' _ _ _ n m ['] [ ] . . ' . . Intro ' ✓ ✓ ' ^ Elim ^ _ ✓ _

' Elim Elim2 '^ 1 ^ ^ ^ n ['] . . '' ! Elim

Intro ! ' ! !n [ '] ¬ . .

'? ? Table 2.2: Natural Deduction Rules for the System NK duction are given in table 2.2. Gentzen includes in his language the symbol for falsity, . ?

Normalization

Prawitz (1965) presents a discussion of the relationship between the se- quent calculus and natural deduction systems. The sequent calculus can be thought of as representing the deducibility relation in the corresponding

17 2. Gentzen Systems for Modal Logics

natural deduction system: If a sequent ( '1,...,'n ) is derivable in LK, ) then '1,...,'n 1 NK 'n (and similarly for the intuitionistic calculi). It [{ }` follows that if the Cut elimination theorem holds in the sequent calculus

(i.e., no formula enters the derivation that is not a component of the end- sequent), then there is a normal derivation of every theorem in the natural deduction system.

There are different kinds of normalization: strong normalization, weak normalization, and the normal form theorem. The first states that every sequence of reductions of a derivation results in a normal proof; the second that there is a sequence of reductions that yields a normal proof; and the normal form theorem states that a normal derivation exists (Stålmarck,

1991, 130). The normal form theorem follows from both strong and weak normalization. It is also equivalent to the Cut elimination theorem: if Cut can be eliminated in a sequent calculus system, then the corresponding natural deduction system has the normal form property. Although a stronger normalization result is desirable, the Cut elimination proof still tells us something important about the corresponding natural deduction system.

Direct normalization for first-order classical logic was first proved by

Prawitz (1965) for a logic without and . There he defined a series of _ 9 reduction steps that, when applied to a non-normal proof, removes all

“detours” and results in an equivalent, normalized proof. The reduction steps involve identifying detours and removing them, as below.

18 2. Gentzen Systems for Modal Logics

1 2 . . . 1 . . ' . Intro ' ' ^ † '^ Elim . ^ . . . 3 3

Several attempts have been made to give a normalization proof for full classical logic, but it was not until recently that the result was actually proven

(von Plato and Siders, 2012). Unfortunately, we lack a normalization result for many modal logics.

2.4 Modal Sequent Calculi

In order to accommodate modal logics, the sequent calculus LK is expanded by adding rules governing the É operator. The modal rules change, however, depending upon the modal logic under consideration. The various modal rules and the corresponding frame properties are summarized in table 2.3.

By convention, É is the sequence with all members prefixed with the É operator, i.e., if = '1,...,'n then É = É'1,...,É'n . A history of modal sequent calculi can be found in Wansing (2002). It has been shown that the sequent systems in table 2.3 are sound and complete with respect to the frames they represent. Although Cut can be eliminated from the sequent systems for K, T, B, D and S4, Ohnishi and Matsumoto

(1957) show that the sequent calculus for S5 is not Cut-free complete. When

Cut is removed from the calculus for S5 it results in a weaker system: the

19 2. Gentzen Systems for Modal Logics

Sequent System Rule Frame Restrictions

' K ) k É É' ) ', Tk Reflexive + ) t É', ) ,' BkÉ Symmetric + ) b É ,É' ) Dk+ d Serial É ) ) É ' S4 k + t + 4 Transitive and Re- ) ' É É flexive ) ,' S5 k t É É Universal + + ) 5 É É,É' ) Table 2.3: Modal Sequent Calculus Rules sequent É É',' is not derivable. ) ¬ É' É' ) ', ' R ' ' É É ¬5 t ) É',ɬ É' É' ) ' Cut ) ¬ É É',' ) ) ¬

Braüner (2000) provides a solution to the problem of Cut elimination in

S5 by exploiting the connection between S5 and monadic predicate logic.

It is well known that S5 can be embedded in monadic predicate logic (see

Mints 1992a, 40, for example). Braüner’s calculus requires only two modal rules. ', ,' ) ÉL ) ÉR É', ,É' ) ) Where no formula or ✓ in the application of the É R rule 2 2 depends on the occurrence of ' in the premise. In order to define the notion of dependency between formulas, we first define the notion of a

20 2. Gentzen Systems for Modal Logics connection between formula occurrences in a proof, and the notion of a modally closed formula. We say that the occurence of a parametric formula

' in the conclusion of an inference rule is inherited from the occurrence of

' in the premise.

Definition 10 (Immediate connection). In a derivation , two formulas ' and are said to be immediately connected iff one of the following hold:

1. ' is the principal formula of some inference rule, and is a side

formula of that inference rule, or vice versa,

2. ' and are both principal formulas in an axiom,

3. ' and are side formulas in a Cut inference, or

4. ' and are parametric formulas in an inference, where is inherited

from ', or vice versa.

Definition 11 (Connection). A sequence of formulas '1,...,'n in a deriva- tion is a connection between '1 and 'n iff for all i , where 1 i n 1, 'i   is immediately connected to 'i +1.

Definition 12 (Modally Closed Formula). A modally closed formula is a formula in which each occurrence of a propositional letter is within the scope of a É connective.

Definition 13 (Dependent). Two formulas ' and in a derivation are dependent iff there is a connection between ' and that does not contain a modally closed formula.

21 2. Gentzen Systems for Modal Logics

Soundness and completeness follows based on the connections with monadic predicate logic, and Cut elimination can be proved algorithmically

(Braüner, 2000, 636). The sequent É É',' is derivable in the calculus. ) ¬ Proof. ' ' L É' ) ' É ) ',' R É ¬ R )ɬ É',' É ) ¬

Although this solution provides us with a Cut-free system for S5 it is in some ways unsatisfactory: so far there is no unified way of accommodating the modal logics in a sequent framework.

Došen’s Principle and Modularity

Another issue with the logic S5 is that its usual sequent calculus presentation is not modular. Generally speaking, a sequent calculus system is said to be modular just in case “a single axiom schema is captured by a single sequent rule (or a finite set of such rules)" (Wansing, 2002, 8). S5, then, is not modular as the system is not complete without the Cut rule. Wansing claims that one way of ensuring that a calculus is modular is by varying calculi only in terms of structural rules — not logical rules. He calls this

Došen’s Principle, based on Došen (1988).

Definition 14 (Došen’sPrinciple). Variation between sequent systems should occur only by varying structural rules, while preserving the logical rules be- tween systems.

22 2. Gentzen Systems for Modal Logics

Došen’s principle is not necessarily concerned with the properties of a “good” sequent calculus. Rather, the principle is concerned with the ways in which we obtain new sequent calculi from existing ones (Poggiolesi,

2010, 32). Došen’s principle gives us an idea of how to expand the proof structure in order to accommodate different axiom rules. Ideally, certain parameters of the sequent calculus should be systematically modifiable in order to obtain rules characteristic of particular modal systems (Wansing,

1994, 128). Sequent systems have so far been obtained for only a small collection of modal logics, and it is unclear how to systematically expand on the traditional modal sequent framework in order to accommodate other axiom systems. Specifically, given the structural limitations of the sequent calculus, it is unclear how to modify the structural rules in order to accommodate specific axioms.

Wansing equates Došen’s principle with the notion of modularity. How- ever, this may be a mistake: modularity seems to be concerned mainly with the relationship between a Hilbert system and the sequent calculus. If a

Hilbert-type system contains an axiom H , modularity tells us how H should be represented in the sequent calculus. This is a relationship between axioms and sequent calculi, not a relationship between sequent calculi

(Poggiolesi, 2010, 34). While the properties often go hand in hand, they may also be separate.

While it might seem like a sequent calculus framework for modal logics should have both of these properties, it may not always result in the best possible calculus. A sequent calculi that is modular may not have a corre- sponding natural deduction system if there are more than two rules for the modal operator, or if the modal rules act on both the left- and right-hand

23 2. Gentzen Systems for Modal Logics side of the sequent. For example, one might obtain a modal sequent system from another by adding a rule that manipulates the modal operator (a logi- cal rule). The resultant system might be modular, but it does not respect

Došen’s principle. It might be the case, then, that we end up with more than two modal rules (as is is the case with S4), and so the sequent systems lose their property as a meta-calculus for natural deduction.

2.5 Modal Natural Deduction

Natural deduction for classical modal logics has been developed mainly for the logics S4 and S5. Prawitz (1965) was the first to develop natural de- duction systems for these logics. In addition to specifying the modal rules, he gave a normalization procedure for his calculi. These logics, however, were restricted: reductio ad absurdum can only be applied to atomic formu- las in Prawitz’s systems, and the language excluded , , and Ü. However, _ 9 Medeiros (2006) has shown that the reduction procedure used for S4 is incor- rect: some instantiations of Prawitz’s reduction steps violate the conditions on É Intro. Medeiros (2006) provides a logically equivalent formulation of S4 and provides a normalization procedure for it. Prawitz’ procedure for

S5 did not have this flaw, but excluded and . Martins and Martins (2008) 9 _ expanded on Prawitz’ system by providing a normalization procedure for full S5.

These systems are unique in the landscape of modal natural deduction, as they are the only representations of Gentzen-style natural deduction.

As in the case of the sequent calculus, modal natural deduction systems are given by adding a rule governing the modal operator to the above cal-

24 2. Gentzen Systems for Modal Logics

Natural Intro Rule Elim Rule Deduction System

i1 in [É'1] ,...,[É'n ] . ' NS4 . É ' É Elim4 É'1 ... É'n É Intro4 É

Where assumptions i1,...,in are all dis- charged, and does not depend on any other assumptions.

. ' NS5 . É Elim ' ' É 5 É Intro5 É' Where ' is modally independent of ev- ery formula 2

Table 2.4: Modal Natural Deduction Rules culus. These additions are summarized in table 2.4. As can be seen, we lack rules corresponding to K, T, B, and D. The rule for S4 is from Medeiros

(2006), and the rule for S5 is from Martins and Martins (2008).

As with quantifiers in a classical natural deduction system, there are restrictions on the application of the É introduction rules. In most cases, when applying a rule that discharges an assumption, you have the option to discharge a single occurrence of the assumption, more than one copy of the assumption, or to not discharge the assumption at all. In the case of the S4 rule, you must discharge all assumptions. In addition, must not depend on any assumptions other than the ones listed in the rule.

25 2. Gentzen Systems for Modal Logics

The restriction on the S5 rule is more complex. This restriction is similar to that of Braüner (2000), and we retain the concept of an essentially modal formula. The notion of a connection is redefined for the natural deduction context. We call a formula a minor premise of an inference rule iff it is a premise that does not contain the operator being acted upon. Otherwise, a premise is considered a major premise.

Definition 15 (Side-connected). Let ⇧ be a subtree of a derivation ⇥, and suppose that ⇧ has endformula '. Let ⇧1,...,⇧n be prooftrees with endfor- mulas 1,..., n that stand immediately above '. Graphically, ⇧ has the following structure:

1 ... n '

We say that i is side-connected with j for all i , j n.  Definition 16 (Connection).

A connection between two formulas '1 and 'n in a derivation is a sequence of formulas '1,...,'n such that for i where 1 i n, one of the   following holds:

1. 'i is not the major premise of an application of Elim, and either _ 'i +1 stands immediately below 'i or vice versa,

2. 'i is a premise of an application of Elim or Elim and 'i +1 is side- ! ¬ connected with 'i ,

3. 'i is the major premise of an application of Elim, and 'i +1 is an _ assumption discharged by the rule, or vice versa,

26 2. Gentzen Systems for Modal Logics

4. 'i is a consequence of an application of Intro, Intro, or , and ! ¬ ? 'i +1 is an assumption discharged by this application, or vice versa.

Definition 17 (Modal independence). Two formulas ' and in a derivation are modally independent iff every connection between ' and contains at least one occurrence of an essentially modal formula.

The É Intro5 rule of NS5 may only be used in contexts where ' is modally independent of every formula . 2 Thus far, we lack normalizable natural deduction systems for K and many of its extensions.

2.6 Conclusion

Gentzen’s proof systems are both philosophically interesting and technically useful. Unfortunately, modal logics lack the robust proof theory that classi- cal logic has. Many philosophical and technical attributes of the sequent calculus do not translate when LK is extended to the modal logic S5. There are also obvious gaps to be filled in modal natural deduction. Gentzen-type natural deduction systems are missing for K, T, B and D. For this reason, we also lack the connection between the Cut elimination theorem and normal- ization theorem in the context of modal logics.

In the next section, we introduce an extension of the sequent calculus, the hypersequent calculus, and show how this system can be used to give a

Cut-admissible system for S5.

27 Chapter 3

Hypersequent Systems

3.1 Introduction

Many generalizations of the sequent calculus have been developed. La- belled sequents, nested sequents (or tree hypersequents), display sequents, and hypersequents are all extensions of Gentzen’s original framework. Al- though modal logics can be dealt with using any of these systems, we focus on the hypersequent framework. Such a framework has been used very successfully in the study of non-classical logics. The transition from a tra- ditional sequent framework to hypersequents is motivated by the failure of traditional sequents to handle certain non-classical logics. As has been discussed, one such case is the modal logic S5: although the logic has a sound and complete sequent calculus, the system is not Cut-free complete.

This issue can be overcome by switching to a hypersequent framework.

In this chapter we survey the development of hypersequent calculi for modal logics and discuss their relation to other extensions of the sequent calcu- lus. Cut-free hypersequent systems for S5 were developed independently

28 3. Hypersequent Systems by Pottinger (1983) and Avron (1996). However, these systems are closely related to Kripke (1959)’s tableau system for S5, and Mints (1992b)’s hybrid systems. We also present the systems of Lahav (2013), who gives a general method for developing Cut-free hypersequent calculi from sets of frame properties. Although the hypersequent framework is a powerful tool, the hypersequent systems developed thus far do not respect Došen’s principle.

Finally, we provide an overview of other extensions of the sequent cal- culus, namely nested sequents, 2-sequents and labelled sequents. In the next chapter, we will give a detailed overview of Parisi (2016), who provides a modified version of the hypersequent calculus for K, T, D, B, S4 and S5.

These systems blend features of hypersequents, linear sequents and labelled sequents in order to create a unified proof theoretic framework that respects

Došen’s principle.

3.2 Hypersequent Calculi

We now turn to the hypersequent calculus HLK. This system is an extension of Gentzen’s original sequent calculus for classical logic, LK, and is used as the base calculus for each of the hypersequent systems introduced below.

HLK is extended to modal logics by adding rules that govern the modal operator. The rules for HLK are given in 3.1.

Definition 18 (Hypersequent). A hypersequent is any expression of the form

1 1 ... n n ) | | ) where i i are sequents. )

29 3. Hypersequent Systems

The symbol “ " is used syntactically to separate the sequents. A hyperse- | quent can be thought of as disjunction of sequents, which we refer to as the component sequents. In a modal context, a hypersequent H is valid just in case for every model M, at least one of its component sequents is true at a world w W . Throughout this thesis we will use G and H to stand in for 2 arbitrary hypersequents (sometimes called side-hypersequents, or external contexts).

As in the traditional sequent calculus, rules for the hypersequent calculus are separated into structural and logical rules. All derivations begin with an axiom.

Hypersequents provide more structural flexibility than traditional se- quent systems. The structure of the hypersequent allows one to apply rules that act on multiple sequents at a time. External structural rules can either change the structure of the hypersequent by adding, removing or exchang- ing sequents in the hypersequent. However, they may also move formulas between sequents, as in Avron’s modal “splitting" rule below.

3.3 Hypersequents and Modal Logic

The development of hypersequent systems for modal logics may be traced back to Kripke (1959)’s semantic tableaux for S5 (Avron, 1996, 22). Following this, Mints (1992b) developed a Gentzen-type calculus with implicit hyper- sequent structure based on Kripke’s method. The first explicit hypersequent calculus was introduced by Pottinger (1983), whose system is equivalent to that of Mints. Pottinger’s was also the first Cut-free hypersequent calculus for S5.

30 3. Hypersequent Systems

Axioms

' ' )

Internal Structural Rules

G H H H TL TR G | ', ) | H G | ) ,' | H | ) | | ) | G ',', H H ,',' H CL CR G | ', ) | H G | ) ,' | H | ) | | ) | G ,', , 0 H H ,', ,0 H | ) | EL | ) | ER G , ,', 0 H G , ,',0 H | ) | | ) | G ,' HG',⇤ ⇥ H Cut | ) G | ,⇤ ,⇥| H ) | | ) | External Structural Rules

G H G H EW EC G | H | G) | ) H | |)| | ) | G ⇥ ⇤ H EE G | ⇥) ⇤ | ) | H | ) | ) | Logical Rules

G ' H G ', H | ) | L | ) | R G ', H ¬ G , ' H ¬ |¬ ) | | ) ¬ | G ,' HG , H G ,' H 0 0 L R | G )G | ,' | H )H | G | ) ,' | H 1 0 0 _ _ | | _ ) | | | ) _ | G , H G ', H | ) | R2 | ) | L1 G ,' H _ G ' , H ^ | ) _ | | ^ ) | G , , H G ,' HG , H | ) | L2 | ) | | ) | R G ' , H ^ G ,' H ^ | ^ ) | | ) ^ | G ,' HG , H G ', , H | ) | | ) | L | ) | R G ' , H ! G ,' H ! | ! ) | | ) ! | Table 3.1: Hypersequent Rules for the System HLK

31 3. Hypersequent Systems

3.4 The Development of Hypersequent Calculi

Kripke (1959)’s tableaux method for S5 can be seen as the first step to- wards a hypersequent calculus for modal logics. This method is based on the semantic tableaux method introduced by Beth (1955). Although not a hypersequent calculus, it is evident that Kripke’s system has particular hyper-properties: instead of operating within a single tableau, the rules for the modal operators introduce auxiliary (side) tableaux. Modal derivations, then, consist of a set of tableaux rather than a single tableau.

The semantic tableaux method is used to test whether some sentence

is entailed by a conjunction of sentences '1,...,'n . When n = 0, is universally valid. We write '1,...,'n in the left hand column of the tableau, and in the right hand column. Intuitively, this represents a state of affairs where is false and '1,...,'n is true.

An individual tableau is closed if and only if a formula occurs in both its left- and right-hand columns. A set of tableaux is said to be closed if at least one of its members is closed. If the tableaux closes, then this state of affairs is not possible, thus showing '1,...,'n è .

We begin a tableaux by placing '1,...,'n in the left hand column and in the right hand column. We then apply the following rules. Some of the rules refer to alternative tableaux as well as auxiliary tableaux. Alternative tableaux are opened when there are two possible truth value assignments that make a formula true, as in the case of a true disjunction. Alternative tableaux can be thought of as a new possible world that is accessible from the tableau it opened fom.

1. If ' appears in the left-hand column of a tableau, write ' in the ¬ 32 3. Hypersequent Systems

right-hand column.

2. If ' appears in the right-hand column of a tableau, write ' in the ¬ left-hand column.

3. If ' appears in the left-hand column of a tableau, then the tableau _ splits into two alternative tableaux: one with ' in the left-hand col-

umn, and one with in the left-hand column.

4. If ' appears in the right-hand column of a tableau, write ' and _ in the right-hand column.

5. If ' appears in the left-hand column of a tableau, write ' and ^ in the left-hand column.

6. If ' appears in the right-hand column of a tableau, then the tableau ^ splits into two alternative tableaux: one with ' in the right-hand

column, and one with in the right-hand column.

7. If ' appears in the left-hand column of a tableau, then the tableau ! splits into two alternative tableaux: one with ' in the right-hand

column, and one with in the left-hand column.

8. If ' appears in the right-hand column of a tableau, write ' in ! the left-hand column, and in the right-hand column.

9. If É' appears in the left column of a tableau, then write ' in all the left-hand column of all tableau in the set.

10. If É' appears in the right-hand column of the tableau, then create a new auxiliary tableau with ' in the right-hand column.

33 3. Hypersequent Systems

We show that É ' èÉ É ' in S5. For clarity, we place checkmarks ¬ ¬ ¬ ¬ beside formulas when a rule has been applied to them, except for any É formulas in the lefthand column. In addition, we use to denote that an ) auxiliary tableaux has been opened.

(1) É ' ÿ (2) É É ' ÿ (4) É ' ÿ 1 ¬ ¬ (3) ɬ'¬ÿ 1.1 (7) É ' ¬ ¬ (9) ' ¬ ) (8) '¬ ¬ ¬

+

(5) ' ÿ (6) ' ¬ 1.2 (10) ' ¬ ⇥

We see that the É rule requires us to open a new auxiliary tableau. Al- though these tableaux create a more branch-like structure than a traditional hypersequent, the similarities are apparent. Unlike the traditional hyperse- quent framework, the separate tableaux are not disjunctive. Rather, they are connected by an implicit accessibility relation. As such, this system is similar in structure to the relational hypersequent calculi introduced in the next chapter. By altering the calculus slightly to accommodate other accessibility relations, one can obtain a tableaux method for other modal logics.

The influence of this system is apparent in Mints (1992b). Mints recon-

figures Kripke’s tableaux system for S5 by converting each tableau into a sequent. He reconfigures the sequent by removing the sequent arrow and

34 3. Hypersequent Systems

Axioms

',' h i ? Structural Rules

S, , S, ,',' Contraction (external) Contraction (internal) Sh, ih i Sh, ,' i h i h i S, S, ,' T, ,' Weakening Cut S, h,'i h Si,T , h i h i h i Logical Rules

S, ,', S, ,' T, , h i R h i h i L S, ,' ! S,T, ,' ! h ! i h ! i S, , ' S, ,' , ⌃ h i h i É R h i h i É L1 S, ,É' S, , ⌃,É' h i h i h i S, ,' h i É L2 S, ,É' h i Table 3.2: Mints’ Sequent Rules considering instead a single multiset of formulas. We write ' instead of ' to denote that a formula is in the antecedent of the sequent. If is a multiset of formulas, then is a tableau. Each tableau represents a world. Mints h i calls a sequent an arbitrary list of tableaux.

The rules for Mints’ system are summarized in table 3.2. In this context we let S and T stand in for arbitrary sequents. The system includes rules for contraction and weakening. Notably, this sytem has rules corresponding to both the external contraction rule and the internal contraction rule. In addition, Cut can be eliminated (Mints, 1992b, 235).

The interpretation ⌧ of a sequent is defined as follows.

35 3. Hypersequent Systems

1. If '1,...,'n , 1,..., m is a tableau, then ⌧( '1,...,'n , 1,..., m )= h n m i h i É( 'i k ) i =1 ! k=1 V W n 2. If S1,...,Sn is a sequent, then ⌧(S1,...,Sn )= Si i =1 W Pottinger’s (1983) hypersequent is the first explicit hypersequent system.

In his system, the hypersequent is interpreted as a sequence of sequents.

Basic axioms and propositional rules are supplemented by the following

two modal rules:

G 1,2 H ' É'_G É',',1,2 É'_H | ) | | ) É R | ) | É L G 1,É',2 H G 1,É',2 H | ) | | ) | The rules require some explanation. Pottinger uses general notation,

and by varying the interpretation, the rules can be sound and complete for

the modal logics T, S4 and S5.

1. T: É'_G = G , and is the result of deleting all formulas not of the

form É' from , and then deleting the first occurrence of É from each of the remaining formulas.

2. S4: É'_G = G and is the result of deleting all formulas not of the

form É' from .

3. S5:IfG = 1 1 ... n n then É'_G = É',1 ) | | ) ) 1 ... É',n n , and is the result of deleting all formulas | | ) not of the form É' from .

The rules for Pottinger’s systems were only ever published in an abstract.

Although he claims that the systems are Cut-free, it is unclear if this was a

full Cut elimination proof or a proof of Cut-free completeness.

36 3. Hypersequent Systems

Avron (1996)’s system GS5 for S5 is a proposed simplification of Pot- tinger’s original hypersequent method. His system is interesting insofar as it is related to the more general hypersequent framework by using a modified splitting rule. Splitting rules are external structural rules that break the components of a single sequent into two sequents. These rules are used in other hypersequent contexts, like calculi for intermediate and substructural logics. Avron’s calculus is thus nicely integrated into the larger framework of hypersequent calculi. To the rules of the calculus HLK we add the following modal rules, as well as a modalized splitting rule.

G ', H G É ' H | ) | É L | ) | É R G É', H G É É' H | ) | | ) |

G É1,2 É1,2 H | ) | MS G É1 É1 2 2 H | ) | ) |

Unfortunately, Avron does not provide rules for other modal logics, nor does he suggest a way to expand upon this system in order to do so. However,

Avron presents a full Cut elimination proof for this system. We provide a case here for clarity.

Theorem 19 (Cut-Elimination). Cut can be eliminated in GS5.

The theorem is proved by induction on the length of a proof.

cf If GS5 H1 = G1 1 1,',...,' ... n n ,',...,' and ` | ) | | ) cf GS5 H2 = G2 ',...,',⌃1 ⇤1 ... ',...,', ⌃k ⇤k then: ` | ) | | )

cf 1. GS5 H = G1 G2 1,...,n ,⌃1,...⌃k 1,...,n ,⇤1,...,⇤k , or ` | | )

37 3. Hypersequent Systems

cf 2. If ' = ÉB then GS5 G1 G2 1 1 ... n n ⌃1 ` | | ) | | ) | ) ⇤1 ... ⌃k ⇤k | | )

Note that Avron’s induction hypothesis is different than Gentzen’s as presented in section 1. In hypersequent calculi, the EC rule presents a problem for Cut elimination. Clause (1) in the induction hypothesis allows us to reduce several cuts in parallel, which avoids the problems caused by EC. Clause (2) is required in cases where the Cut formula is a modal formula. There are modal cases where (1) is not sufficient to derive the desired conclusion, and so in modal contexts the second clause is used. It is easy to see that (1) follows from (2) through a series of external contractions and internal weakening.

Consider the following case.

1,É 1,É,É' ) MS 1 1 É É,É' É',2 2 ) | ) ) Cut 1 1 É ,2 É,2 ) | )

Using (2) above, the Cut can be moved upwards to one of a lower rank.

1,É 1,É,É' É',2 2 ) ) Cut 1,É 1,É 2 2 ) | ) MS 1 1 É É 2 2 ) | ) | ) WL, WR 1 1 É ,2 É,2 É ,2 É,2 ) | ) | ) EC 1 1 É ,2 É,2 ) | )

Avron’s system is a straightforward and elegant way of solving the prob- lem of Cut-elimination for S5. This system, however, has not been adapted to other modal logics. In general, this issue arises with all the above hy- persequent treatments for modal logics: they are not unified. Although

38 3. Hypersequent Systems

Cut-free systems have been developed for S5, T and S4, these systems do not give a general way of obtaining hypersequent calculi. Below, we intro- duce Lahav (2013)’s method for obtaining hypersequent calculi from frame properties. Unlike the above systems, Lahav’s method gives us a uniform way of developing hypersequent calculi.

3.5 Hypersequent Rules from Frame Properties

In this section we outline the general method given by Lahav (2013) for obtaining Cut-admissible hypersequent calculi from simple frame proper- ties. The properties of seriality, reflexivity, and universality, to name a few, can be converted into n-simple formulas which are then translated into hypersequent rules. The main result is that any calculus obtained by adding a set of rules R to a base calculus HK is sound and Cut-free complete.

We begin our overview of Lahav (2013)’s method by defining several key concepts. In order to define simple frame properties, we use a language that includes the universal and existential quantifiers ( , ), with their usual 8 9 interpretation, a denumerable set of variables, and the predicates R and =.

Definition 20 (n-Simple Formula). Let u, w1, w2, w3,... be variables in our language. A formula is called n-simple if it is of the form w1,...wn u⇥ 8 9 where ⇥ is a series of conjunctions and disjunctions joining atomic formulas of the form wm Ru or wm = u where 1 m n.   There are potentially several ways in which a frame property can be represented as an n-simple formula, and a method for determining the n-simple formula based on the frame property is desired. One way of doing

39 3. Hypersequent Systems

Seriality w1 u(w1Ru) 1 , 8 9 {h{ } ;i} Reflexivity w1 u(w1Ru w1 = w2) 1 , 1 8 9 ^ {h{ } { }i} Universality w1, w2 u(w1Ru w2 = u) 1 , 2 8 9 ^ {h{ } { }i}

Table 3.3: Simple Frame Properties and their Normal Descriptions this is by first determining the normal description of the frame property, and then converting that into the n-simple formula.

Definition 21 (Normal Description). For any n-simple formula ✓ : w1,...wn u⇥, 8 9 a normal description of ✓ is a finite, non-empty set S of ordered pairs SR ,S= h i where SR ,S 1,...n 1,...n , and ⇥ is equal to wi Ru = SR ,S= S ( i SR h i✓{ }⇥{ } h i2 2 ^ wi u . We denote the normal description of someW n-simpleW formulas i S= = ) 2 W✓ by S(⇥).

Table 3.3 lists several simple frame properties, along with their corre- sponding n-simple formula and normal descriptions, from Lahav (2013,

410).

Note that the properties of transitivity and symmetry cannot be captured using n-simple formulas. These logics are accommodated by modifying the base calculus, as shown below. n-Simple Modal Logics n-Simple modal logics use the calculus HK as their base, which is obtained by adding the following É rule to the basic hypersequent calculus HLK. This system is sound and complete for the modal logic K. Note that in the case

40 3. Hypersequent Systems of HK, the set of n-simple formulas is empty.

G ' H | ) | É G É É' H | ) | Given any normal description S(⇥) of some n-simple formula ✓ , there is corresponding hypersequent rule of the form:

G i , i : SR ,S S i S= i SR i0 i S= = { | 2 2 ) 2 h i2 } Ind G 1,É10 1 ... n ,Én0 n S| S) | S| ) We say that any instance of the above rule based on some n-simple formula

✓ is the hypersequent rule induced by ✓ . Any instance of the above rule is a representation of the the corresponding frame property.

In the following subsections we give example derivations in several sys- tems. A complete list of induced hypersequent rules can be found in Lahav

(2013).

We show that the axiom É(' ) (É' É ) is derivable in HK. ! ! ! ' ' ) ) L ',' ! ! ) É R É',É(' ) É ! ) R É(' ) É' É ! ! ) ! R É(' ) É' É ! ) ! ! ! The modal logic T is characterized by reflexive frames. The following rule is added to the above calculus to capture reflexivity:

G 1,10 1 H | ) | Ref G 1,É10 1 H | ) | The axiom É' ' is derivable in HKT . ! ' ' Ref É' ) ' R ) É' ' ) ! !

41 3. Hypersequent Systems

Serial frames are characteristic of the modal logic D. To obtain a corre- sponding calculus, the following rule is added to HK :

G 10 H | )| Ser G 1,É10, 1 H | ) | The axiom É' É ' is derivable in HKD . !¬ ¬ ' ' ', ' ) L ¬ ', ¬' ) Ser É É R ɬ' ) É ' ¬ R )ɬ'¬ É ' ) !¬ ¬ ! Transitive Modal Logics

As mentioned, the above method works for simple frame properties, but not all frame properties are of this type: the properties of transitivity and symmetry cannot be captured using the above schema. However, these frame properties can be represented by changing the modal rule for HK.In the case of transitive modal logics, the É rule is replaced with the modal rule É4 in order to obtain the calculus HK4.

G ,É ' H | ) | É4 G É , É' H | ) | The schemata for converting n-simple frame properties into hyperse- quent rules is also adjusted for transitive logics. Any set of rules R induced according to the schema Ind4 can then be added to HK4.

G i , i : SR ,S S i S= i SR i0 É i0 i S= = { | 2 2 [ ) 2 h i2 } Ind4 G 1,É10 1 ... n ,Én0 n S | S ) | |S ) The basic calculus for transitive modal logics is characteristic of K4, the logic transitive frames. We show that the axiom É' ÉÉ' is derivable in )

42 3. Hypersequent Systems

HK4. ' ' WL ',É' ) ' É4 É' ) É' WL ',É' ) É' É4 É' ) ÉÉ' R ) É' ÉÉ' ) ! ! The modal logic S4 is characterized by transitive and reflexive frames.

The calculus HK4 is augmented with a rule for reflexivity to obtain HKS4:

G 1,10,É10 H | ) | Ref4 G 1,É10, H | ) | The axiom É' ' is derivable using the Ref4 rule. ! ' ' WL É',' ) ' Ref4 É' ) ' ) Symmetric Modal Logics

Like transitivity, the property of symmetry is not captured by any n-simple formula. Symmetric modal logics can be accommodated by again altering the basic calculus HK. We replace the É4 rule with the following:

G ',É H | ) | ÉB G É É', H | ) | The scheme for introducing new hypersequent rules from n-simple frame properties is also updated accordingly.

G i , i , : SR ,S S i S= i SR i0 i S= i SR É 0i = { | 2 2 ) 2 2 h i2 } G 1,É10 1,01 ... n ,Én0 n ,0n S | S ) S| | S ) It is important to note that the calculi for symmetric modal logics are not

Cut-free complete. They do, however, still enjoy the subformula property, as only cuts on formulas that are subformulas of the end formulas are necessary.

43 3. Hypersequent Systems

However, no Cut-free calculus is available. This is not simply a shortcoming of this method: no Cut-free hypersequent calculus has been constructed for the logic B (Lahav, 2013, 7).

Also note that Lahav’s hypersequent systems are not entirely modular.

Although the calculi for n-simple modal logics are modular, transitive and symmetric modal logics are not. This is because the axioms for K and transitivity (likewise for symmetry) are captured by a single rule: they cannot be separated. It is therefore not the case that each individual axiom is captured by a set of rules. In addition, the systems do not satisfy Došen’s principle, as variation between calculi occurs in the presence/absence of logical rules, not just structural rules.

Of course Lahav’s method has one marked advantage over the other hy- persequent methods: its generality. Due to the general conversion method, any modal logic whose axioms can be captured using n-simple formulas may be handled using this technique. Avron and Pottinger’s hypersequent systems do not have this generality, and it is unclear how one could expand on their methods to make them more general.

3.6 Related Sequent Systems

It is also pertinent to consider other extensions of the sequent calculus: tree hypersequents, linear nested sequents, and labelled sequents. These systems have a special relationship to the systems considered in chapter 4.

44 3. Hypersequent Systems

Tree Hypersequents

Tree hypersequents (also called nested sequents or deep sequents) are hy- persequents that admit of a branching or tree-like structure. In this way, tree hypersequents incorporate the structure of Kripke frames directly into the proof system itself. This system was developed by Brünnler (2009) for modal logics, who uses the same notation as Kashima (1994)’s systems for temporal logic. Further work on tree hypersequents can be found in Poggiolesi (2010).

Tree hypersequents begin with the usual definition of a sequent, , ) where and are multisets of formulas. A sequent is taken to determine a state of affairs at a corresponding possible world. Two meta-linguistic symbols “;" and “/", denote different relations between sequents (or worlds).

The symbol “/", used between two sequents, S1/S2, intuitively means that

S2 is accessible from S1. The hypersequent S1/S2;S3 means that both S2 and

S3 are accessible from S1, but S2 and S3 cannot see each other.

Definition 22 (Tree Hypersequent). The set of tree hypersequents THS is defined by:

1. If is a sequent, then THS ) ) 2

2. If is a sequent, and G1,...,Gn THS, then /G1;...;Gn THS. ) 2 2 The interpretation ⌧ of a tree hypersequent is defined by:

1. ⌧[ ]= ] ) ! V W 2. ⌧[ /G1,...,Gn ]=t [ ] É⌧[G1] ... É⌧[Gn ] _ _ _ The basic modal rules for the tree hypersequent system, THSK, are given in table 3.4. In displaying the rules we use the notation G to denote { ) } 45 3. Hypersequent Systems

G / ' G / (',⌃ ⇧ / H ) { ) ) } É R { ) ) } É L G ,É' G É', / (⌃ ⇧ / H ) { ) } { ) ) }

Table 3.4: Tree Hypersequent Rules for the Base Calculus THSK

G ', G / (É',⌃ ⇤ / H ) { ) } Ét { ) ) } É4 G É', G É', / (⌃ ⇤ / H ) { ) } { ) ) }

G ', / (⌃ ⇤ / H ) G /' { ) ) } Éb { ) )}Éd G / (É',⌃ ⇤ / H ) G É', { ) ) } { ) }

G É', / (⌃ ⇤ / H ) { ) ) } É5 G / (É',⌃ ⇤ / H ) { ) ) }

Table 3.5: Modal Tree Hypersequent Rules

G / (⌃ ⇤ / H ) G / (⌃ ⇤ / H ) t¯ 4¯ {G ) ,⌃ ),⇤ / H } G { )/ ( /)⌃ ⇤ /}H ) { ) } { ) ) ) } G / ⌃ ⇤ / ⇥ ⇧ / H ;H G / ( ( ) 0) ¯ ¯ { ) ) ) } b { ) )}d G ,⇥ ,⇧ / (⌃ ⇤ / H 0);H G { ) ) } { ) }

G / (⌃ ⇤ / (⇥ ⇧ / H );H 0) { ) ) ) 5¯ G / (⇥ ⇧ / H );(⌃ ⇤ / H 0) { ) ) ) }

Table 3.6: Structural Tree Hypersequent Rules

that a rule is applied at a specific point in the tree hypersequent. In other

words, G is a tree hypersequent, and the sequents within the braces are

the specific point in the tree hypersequent where the rule is being applied.

In addition, H in this context represents a multiset of tree hypersequents.

46 3. Hypersequent Systems

These rules, together with the axiom p p, structural rules (internal and ) external contraction and weakening), and the usual logical rules, are sound and complete for the modal logic K.

In order to obtain the extensions of K, pairs of rules are added to THSK.

For each axiom, a structural rule and a modal rule are added to the calculus.

The modal rules are summarized in table 3.5, the corresponding structural rules in table 3.6. Note that these systems are not fully modular. In order to obtain the calculus THSS5 for S5, one adds the pairs of rules for 4, b , ¯ and 5. The axiom 5, however, is not entirely captured by the rules É5 and 5, ¯ as the system that results when one adds É5 and 5 to THSK is not Cut free (Poggiolesi, 2010, 126). Tree hypersequents also do not conform to Došen’s principle.

2-Sequents

Linear nested sequents are a subgroup of tree hypersequents that permit only linear constructions rather than tree-like constructions. Normal and non-normal modal logics have been successfully accommodated within the context of linear nested sequents (Lellmann 2015; Lellmann and Pimentel

2016). Linear nested sequents share similar properties to the 2-sequent system developed by Masini (1992). Although Masini’s 2-sequent system was developed in particular for the modal logic D, linear nested sequents expand beyond D to give systems for K and its extensions.

2-Sequents

Masini’s 2-sequents resemble vertical hypersequents. We can think of

2-sequents as vertical successions of sequences of formulas, separated by

47 3. Hypersequent Systems the symbol , as in the example below. `

', ✓(1)

(2) ` ⇡ (3)

The “level" of each formula in the 2-sequent is represented by the brack- eted number at the end of each line. More formally, we can define a 2- sequent as follows.

Definition 23 (2-sequence). A 2-sequence ⌦ = si 1 i ! is an infinite vertical { }   succession of sequences si of LÉ formuas, where k 1 such that j k, sj 9 8 is the empty sequence.

Definition 24 (2-sequent). A 2-sequent is an any expression where ` and are 2-sequences.

Definition 25 (Depth). The depth of a 2-sequence , written #( ) is defined by min i : i 0, j > i , j is empty . { 8 } The depth of a 2-sequent , written #( ), is defined as max # ,# . ` ` { } We call a formula maximal iff the depth of the formula is equal to the depth of the 2-sequent, and a formula is a maximum iff it is the unique maximal formula. In the above example, the depth of the 2-sequent is 3, and ⇡ is a maximum formula.

The modal rules for the 2-sequent calculus are given in table 3.7. There is a restriction on the É rule: the formula ' must be a maximum for- ` mula. This restriction mimics the eigenvariable restriction on the universal quantifier in the first-order sequent calculus (Masini, 1992, 234).

48 3. Hypersequent Systems

si +1 ` si ,' si ` ' 0 É É ` ` si +1,É' si ,É' ` ` si Where ' is a maximum formula 0 Table 3.7: Modal 2-Sequent Calculus Rules

The interpretation of a 2-sequent is defined as:

If s and k are 2-sequences, then

1. ⌧[s k]= s k. ` ! skV W 2. ⌧[ ]=( s k) É(⌧[ ]). ! _ ` ` V W As mentioned, 2-sequents were never widely adopted, and it is unclear how to expand this method to modal logics other than D. Linear nested sequents are a related framework that has overcome this issue.

Linear Nested Sequents

Like 2-sequents, linear nested sequents are interpreted under nested modal operators. Since the empty part of a 2-sequent is not interpreted, we may truncate the 2-sequent to a finite length. As we will see, the result is simply a linear nested sequent. This framework is important as it can be expanded to deal with the normal extensions of K, as well as many non-normal modal logics. They are also modular: in order to accommodate each different logic,

49 3. Hypersequent Systems

G ' G ',⌃ ⇧ H ) ) É R ) ) É L G ,É' G É', ⌃ ⇧ H ) ) ) G ', , H G ,' HG , H ) L ) ) R G ' , H ^ G ,' H ^ ^ ) ) ^ Table 3.8: Linear Nested Sequent Rules for the Base Calculus LNSK

a new rule can be added to the base calculus LNSK that captures a specific frame property.

Definition 26 (Linear Nested Hypersequent). The set LNS of linear nested sequents is defined by:

1. If is a sequent, then LNS. ) ) 2 2. If is a sequent, and G LNS, then G LNS ) 2 ) 2 The modal interpretation of a linear nested sequent (⌧É) is given by:

1. If is a sequent, then ⌧É( )= ) ) ! V W 2. ⌧É( G )= É⌧É(G ) ) ) _ V W As above, we begin with the usual definition of a sequent , where ) the sequent is interpreted as a multiset of formulas. The set LNS of linear nested hypersequents is defined below. The modal rules and rules for con- junction of the base calculus are given in table 3.8, and rules for the modal extensions of LNSK are given in table 3.9. In addition to these rules, the internal structural rules of weakening and contraction are added, and the usual axioms.

50 3. Hypersequent Systems

G ', H G ' ) Ét ) ) Éd G É', H G É', ) ) G É',⌃ ⇧ H G ⌃ ⇧,É' H ) ) É4 ) ) É5 G É', ⌃ ⇧ H G ,É' ⌃ ⇧ H ) ) ) ) Table 3.9: Modal Linear Nested Sequent Rules

y : ', x : É', xRy, xRy, , y : ' ) É L ) É R x : É', xRy, , x : É' ) ) Where y does not occur in the con- clusion of the inference.

Table 3.10: Basic Rules for the Labelled System G3K

Unlike tree hypersequents, linear nested sequents only require the ad- dition of a single modal rule in order to capture an axiom. In addition, there is currently no linear nested sequent system that captures the axiom

B (Lellmann, 2015, 15).

Labelled Sequents

Another attempt at accommodating modal logics is through labelled se- quents. Although there are several formulations of labelled sequents, the basic strategy incorporates the semantics into the sequent itself through labels representing worlds and the accessibility relation. Such systems can be found in Negri (2005) and Gabbay (1994). We give an overview of Negri’s systems here.

The modal rules for the system G3K are summarized in table 3.10. The

51 3. Hypersequent Systems

xRy, xRx, d t ) ) ) ) (y ,) 62 yRx, xRy, xRz, xRy, yRz, b 4 xRy, ) xRy, yRz, ) ) ) yRz, xRy, xRz, 5 xRy, xRz, ) ) Table 3.11: Modal Labelled Sequent Rules other rules of the system are labelled variants of Gentzen’s rules for LK. This system is sound and complete for the modal logic K. To our language, we add a set of variables x , y,...ranging over a set of possible worlds W , the symbol

“:" which stands in for èM and the two-place predicate R which represents the accessibility relation. We call any instance of xRy a relational atom.

The base calculus can be expanded to the other extensions of K by con- verting the frame properties to “structural" rules. These rules are summa- rized in table 3.11. In order to obtain a new modal systems from the base calculus, one adds the rule corresponding to the desired axiom plus a rule that satisfies the closure condition. Negri’s calculus does not contain a con- traction rule. However, it is possible that instantiating the free variables in the premise of the rule results in two copies of a single atom iRj. The closure condition requires that the two identical atoms be contracted to- gether. If 1,...,,,...,n are instatiated relational atoms in the premise of a rule, the closure condition requires that the two copies of be contracted together.

52 3. Hypersequent Systems

Consider, for instance, the closure condition on the rule 5:

iRj, iRj, jRj, iRj, jRj, 5 5c iRj, iRj, ) † iRj, ) ) ) Notably, all of Negri’s labelled calculi are Cut-free (Negri, 2005, 523).

However, although Cut can be eliminated, the labelled calculi do not have the subformula property. Negri notes that the calculi do have what she calls a subterm property: all terms in a derivation are either eigenvariables or terms in the conclusion (Negri, 2005, 533). The calculi are also not modular: some rules end up being redundant (Poggiolesi, 2010, 96). For instance, the rule 5c above is redundant when added to the base calculus to obtain the system KT5. However, in cases without the rule t , 5c is necessary in order to capture the axiom 5.

Like tree hypersequents, labelled calculi internalize the structure of a Kripke model. However, tree hypersequents are less explicit about the semantic parameters than semantic proof methods like labelled calculi

Poggiolesi (2010, 119). In tree hypersequents, the relational structure is in- corporated into the structure of the derivations themselves, whereas labelled calculi make the relations explicit through relational atoms and labelled formulas. In Negri’s calculus, for example, the sequent does not have a straightforward formula interpretation, whereas both the sequent and the tree hypersequent in Poggiolesi’s calculus have a formula interpretation.

3.7 Conclusion

There are many extensions of the sequent calculus, and they have varying degrees of success when applied to modal logic. Perhaps most striking is the

53 3. Hypersequent Systems fact that, besides labelled sequents, none of the systems obey Došen’s prin- ciple. In the next chapter we introduce a recently developed hypersequent system that solves this problem.

54 Chapter 4

Relational Hypersequents

4.1 Introduction

In this section we introduce and discuss a recent variation on the traditional hypersequent framework, which we call relational hypersequents. In the previous section, hypersequents were interpreted as disjunctions of indi- vidual sequents. A relational hypersequent instead represents a branch of possible worlds. Each sequent in the branch represents a state of affairs corresponding to a possible world, which are separated by an accessibility relation. In order to represent this change in interpretation, and to make the relational properties clear, we replace the symbol “ " with “ ”. | i A relational hypersequent system for S5 was originally developed by

Restall (2009). This system was adapted for K and its other extensions by

Parisi (2016). The relational hypersequent framework is advantageous as the modal logics K, T, D, B, S4 and S5 can be uniformly accommodated within the framework, and so variation between relational sequent systems occurs only in the structural rules of the calculi. The rules governing the modal

55 4. Relational Hypersequents operators are held constant between each system. Although it is not clear if relational hypersequents can be adapted to a larger set of frame properties

(as in Lahav 2013), this method gives us a unified way of accommodating many of the most important modal logics.

A proof of Cut-free completeness for S5 can be found in Restall (2009) and Parisi (2016). We provide a novel Cut-free completeness result for the base calculus RK.

4.2 Relational Hypersequents

We begin with the usual definition of a sequent, , interpreted as )

, where and are sequences of formulas of LÉ. ! V W Definition 27 (Relational Hypersequent). The set RHS of relational hyper- sequents is defined by:

1. If is a sequent, then RHS. ) ) 2 2. If is a sequent, and G RHS, then G RHS. ) 2 ) i 2

The rules for relational hypersequents are given in table 4.1 below. This base calculus, RK, is sound and complete for the modal logic K. Unlike tra- ditional hypersequents, the base calculus does not contain typical external structural rules of weakening, contraction, ad exchange. Relational calculi for extensions of K are obtained by adding additional structural rules. These additions are summarized in table 4.2.

With relational hypersequents, special attention must be paid to side sequents. Although side sequents are still arbitrary, the order of the side

56 4. Relational Hypersequents

Axioms p p )

Internal Structural Rules

G H G H TL TR G i', ) i H G i ) ,' i H i ) i i ) i G ',', H G ,',' H CL CR G i ', ) i H G i ) ,' i H i ) i i ) i G ,', , 0 H G ,', ,0 H i ) i EL i ) i ER G , ,', 0 H G , ,',0 H i ) i i ) i G ,' HG',⇤ ⇥ H Cut i ) G i,⇤ ,⇥i H ) i i ) i External Structural Rules

G G EWR EWL G G i) )i Logical Rules

G ,' H G ', H i ) i L i ) i R G ', H ¬ G , ' H ¬ i¬ ) i i ) ¬ i G ,' HG , H G ,' H i ) i i ) i L i ) i R1 G ,' H _ G ,' H _ i _ ) i i ) _ i G , H G ', H i ) i R2 i ) i L1 G ,' H _ G ' , H ^ i ) _ i i ^ ) i G , H G ,' HG , H i ) i L2 i ) i i ) i R G ' , H ^ G ,' H ^ i ^ ) i i ) ^ i G ,' HG , H G ', , H i ) i i ) i L i ) i R G ' , H ! G ,' H ! i ! ) i i ) ! i G ', ⇤ ⇥ H ' H i ) i ) i É L ) i ) i É R G É',⇤ ⇥ H ,É' H i ) i ) i ) i Table 4.1: Relational Hypersequent Rules for the system RK

57 4. Relational Hypersequents

Logic External Structural Rules Derivability Relation

G H T i ) i ) i EC RT G H ` i ) i

G D )i Drop RD G `

... B 1 1 n n ) i i ) Sym RB n n ... 1 1 ` ) i i )

G H S4 EC i G) i )H i i ) i G H i EW RS4 G H ` i)i

G H S5 EC i G) i )H i i ) i G H i EW RS5 G H ` i)i

G 1 1 2 2 H i ) i ) i EE G 2 2 1 2 H i ) i ) i Table 4.2: External Structural Rules for Relational Hypersequents sequents matter as we do not have external exchange, contraction and weakening in the base calculus. For this reason, when applying rules with multiple premises, we must ensure that the side sequents of each premise are identical.

The following definitions will be needed in the rest of this chapter. We use the notation M, w è to mean that if = '1,...,'n , then for all 1 k  

58 4. Relational Hypersequents

n,M, w è 'k . 6

Definition 28 (Branch of worlds). Let F be a frame and w1,...,wn be worlds.

The sequence w1,...,wn forms a branch of worlds in F if for each 1 i   n, wi Rwi 1

Definition 29 (Countermodel). A model M is a countermodel to a sequent

at a world w iff M, w è and M, w è . ) Definition 30 (Counter-example). A model M is a counter-example to a hypersequent 1 1 ... n n iff there is a branch of worlds w1,...,wn ) i i ) such that M is a countermodel to each sequent i i at wi for all 1 i n. )   Definition 31 (Valid hypersequent). We say that a hypersequent H is valid in a class of frames F just in case there is no counter-example to it that is in

F. Otherwise, we say that the hypersequent is invalid.

Note that the interpretation of the relational hypersequent is equivalent to that of the linear nested sequent.

Theorem 32. For any sequents 1 1,...,n n , 1 1 ... n ) ) ) i i ) n is valid iff 1 1 ... n n is valid. ) ) Proof. We prove this fact by induction on the length of a hypersequent.

1. Base Case: Suppose that there is a counter-example to the relational

hypersequent 1 1. So there is a M, w1 such that M, w1 è 1, and ) M, w1 è 1. It follows that the linear nested sequent 1 1 is not ) valid.

59 4. Relational Hypersequents

2. Suppose that there is a counter-example to the relational hyperse-

quent

1 1 ... n 1 n 1 n n ) i i ) i )

This is the case iff there is a branch of words w1,...,wn such that

for all 1 i n, wi Rwi 1, and each M, wi forms a countermodel to   i i . It follows that M, w1,...,wn 1 forms a counter-example to ) the sequent 1 1 ... n 1 n 1. By our inductive hypothesis, ) i i ) there is a counter-example to (1 1 ... n 1 n 1) iff (1 1 ) i i ) ) ... n 1 n 1) is invalid. It follows that (1 1 ... n 1 n 1) ) ) ) is invalid. This is the case iff M è 1 1 É(... É( n 1 6 ! _ _ ! n 1)...). We know from the definitionV ofW the counter-exampleV that

Wwn Rwn 1, and that M, wn is a countermodel to n n . It follows ) that there is a counter-example to 1 1 ... n 1 n 1 n n , ) ) ) as 1 1 É(...É( n 1 n 1 É( n n ))...) is false ! _ ! _ ! whenV wn RwW n 1 and thereV is a countermodelW Vn Wn at wn . )

Although the interpretation of the two systems is the same, relational hypersequent calculi have a few advantages over the linear nested sequent framework. First, relational hypersequents obey Došen’sprinciple: variation between the calculi occurs only in the presence/absence of structural rules. In addition, there are only two left- and right-hand rules for the modal operator. In the context of relational hypersequents, we do not need to add new rules that govern the modal operator when switching between calculi.

60 4. Relational Hypersequents

Linear nested sequents, however, have the advantage of being a purely syntactic method: both the sequent and the hypersequent have a straightfor- ward formula interpretation. In the relational case, the component sequents retain their usual interpretation, but the hypersequent itself does not have a straightforward formula interpretation. For this reason, relational hyper- sequent calculi might be seen as an intermediate between labelled calculi and linear nested sequents.

It is also important to note that the calculus RS5 is not modular, as the rule EE represents both symmetry and transitivity.

4.3 Soundness

Soundness proofs for the relational calculi can be found in Parisi (2016).

Most of the cases are routine. We omit many of the propositional cases and present the cases for , É, EWL and EWR. !

Theorem 33 (Soundness). If RK H , then there is no counter-example to H . ` The proof proceeds by induction on the length of a derivation.

Definition 34 (Length of a Derivation). The length of a derivation , written l () is defined inductively as follows.

1. If is an instance of an axiom, then l ()=1.

2. If is derived from premises 1,...n , l ()=(⌃1 i n l (i )) + 1.  

Proof. Let be the last inference of . We prove the above theorem by induction in the length of the derivation .

61 4. Relational Hypersequents

1. Base Case: is an instance of an axiom, (p p). There is no model M ) and possible world wi such that M, wi è p and M, wi è p. It follows 6 that there is no counter-example to the axiom.

2. is an instance of the L rule. So the derivation runs as follows. ! . . . . G i i ,' H G ,i i H i ) i i ) i L G ' ,i i H ! i ! ) i By our inductive hypothesis, the proof is correct up until the appli-

cation of the L rule. Consider an arbitrary model M. We suppose ! for contradiction that M, w1,...,wn forms a counter-example to the

conclusion.

G ' ,i i H i ! ) i

So w1,...,wn forms a branch of worlds such that wk Rwk 1 and M

is a countermodel to each sequent k k at wk for all 1 k n. )   Since M must be a countermodel to (' ,i i ) at wi , M, wi è ! ) i ' , and M, wi è i . It follows that M, wi è ' or M, wi è . [{ ! } 6

Suppose that M, wi è '. Since i and i remain unchanged in the 6 derivation, this means that M is a countermodel to i i ,' at wi , ! which contradicts our hypothesis.

On the other hand, suppose that M, wi è . Again, since i and

i remain unchanged in the derivation step, it follows that M is a

countermodel to ,i i at wi , which contradicts our hypothesis. ) In either case, the hypothesis has been violated. We can infer that

the conclusion of the inference is valid.

62 4. Relational Hypersequents

3. is an instance of the R rule. So the derivation runs as follows. ! . . G ',i i , H i ) i R G i i ,' H ! i ) ! i By our inductive hypothesis, the proof is correct up until the appli-

cation of the R rule. Consider an arbitrary model M and suppose ! for contradiction that M, w1,...,wn forms a counter-example to the

conclusion of the inference.

G i i ,' H i ) ! i

So w1,...,wn forms a branch of worlds such that wk Rwk 1 and M

is a countermodel to each sequent k k at wk for all 1 k n. M is )   a countermodel to (i i ,' ) at wi .SoM, wi è i , and M, wi è ) ! i ' . Since M, wi è ' , it follows that M, wi è ' and [{ ! } 6 ! M, wi è . But since i and i remain unchanged in the derivation, it 6 follows that M is a counter-example to the sequent (',i i , ) at ) wi . This contradicts our hypothesis. So the conclusion of the inference

step must be valid.

4. is an instance of É L. So the end of the derivation runs as follows. . . G ',i 1 i 1 i i H i ) i ) i É L G i 1 i 1 É',i i H i ) i ) i Suppose that the derivation is sound up until the application of

the É L rule. Consider an arbitrary model M. We suppose for reductio

that M, w1,...,wn forms a counter-example to the conclusion.

G i 1 i 1 É',i i H i ) i ) i 63 4. Relational Hypersequents

It follows that w1,...,wn forms a branch of worlds such that wk Rwk 1

and M is a countermodel to each sequent k k at wk for all 1 )  k n. M must be a countermodel to É',i i at wi . It follows that  ) wi èÉ'. So, for any possible world v such that wi Rv, M, v è '.In

particular, we know that wi Rwi 1. Since M, w1,...,wn is a counter- example to the lower hypersequent, M, wi 1 è i 1 and M, wi 1 è i 1.

So M is also a countermodel to the sequent (',i 1 i 1) at wi 1. ) So M is a counter-example to the premise

G ',i 1 i 1 É',i i H i ) i ) i

which contradicts our supposition.

5. is an instance of É R. So the end of the derivation runs as follows. . . ' 1 1 H ) i ) i É R 1 1,É' H ) i By our inductive hypothesis, the derivation is sound up until the

application of the ÉR rule. We consider an arbitrary model M, and

suppose that M, w1,...,wn forms a counter-example to the conclu-

sion.

1 1,É' H ) i

This means that w1,...,wn forms a branch of worlds such that

wk Rwk 1 and M is a countermodel to k k at wk for all 1 k n. )   M must be a counter-example to 1 1,É' at w1. This means that ) M, w1 èÉ'. So, there is some possible world v such that w1Rv and 6 M, v è '. But this means that M is also a countermodel to the sequent 6 64 4. Relational Hypersequents

( ') at w0, since w1Rw0.SoM is a counter-example to the premise )

' 1 1 H ) i ) i

which contradicts our assumption. So the conclusion of the infer-

ence must be valid.

6. is an instance of EWR. So the end of the derivation runs as follows. . . G EWR G i) By our inductive hypothesis, the derivation is sound up until the

application of EWR. Consider a model M, and suppose by way of

contradiction that M, w1,...,wn forms a counter-example to the con-

clusion of the inference.

G i)

This means that w1,...,wn forms a branch of worlds such that

wk Rwk 1 and M is a countermodel to k k at wk for all 1 k n. )   It follows that M, w1,...,wn 1 must also be a counter-example to G , which is a contradiction.

7. is an instance of EWL. The proof is symmetrical to the EWR case.

Adding the following rule to RK results in the calculus RT, sound for the class of reflexive frames.

G i i i i H i ) i ) i EC G i i H i ) i 65 4. Relational Hypersequents

Theorem 35. If RT H , then there is no reflexive counter-example to H . ` Proof. We simply show that the EC rule is sound for reflexive frames. Sup- pose that is an instance of EC. So the derivation ends in the following inference. . . G i i i i H i ) i ) i EC G i i H By inductive hypothesis, wei know) thati the derivation is sound up un- til the application of the EC rule. Consider an arbitrary model M with a reflexive frame, and let M be a counter-example to the conclusion of the inference. So there is a branch of worlds w1,...,wn , where wk Rwk 1 for all 1 k n. If the model is a counter-example to the conclusion, then it is   a countermodel to i i at wi for all 1 i n. However, the frame is )   reflexive, so wi Rwi .Sow1,...,wi , wi ,...,wn is a branch of worlds where wk is a counter-example to k k for all 1 k n. But this means that M )   is a counter-example to (G i i i i H ), which contradicts our i ) i ) i hypothesis. So the conclusion of the inference must be valid.

The calculus RB, sound for symmetric frames, is the result of adding the following rule to RK.

1 1 ... n n ) i i ) Sym n n ... 1 1 ) i i ) Theorem 36. If RB H , then there is no symmetric counter-example to H . ` Proof. We show that the Sym rule is sound for symmetric frames. Suppose that is an instance of the Sym rule. So the end of the derivation runs as follows. . . 1 1 ... n n ) i i ) Sym n n ... 1 1 ) i i ) 66 4. Relational Hypersequents

By our inductive hypothesis, we know that the premise of the inference is sound. Let M be an arbitrary model with a symmetric frame. Suppose toward a contradiction that M, w1,...,wn forms a counter-example to the conclusion.

n n ... 1 1 ) i i )

This means that wn ,...,w1 forms a branch of worlds such that wi 1Rwi and M is a countermodel to (i i ) at wi for all 1 i n. Since the frame )   is symmetric, it follows that wi Rwi 1.Sow1,...,wn also forms a branch of worlds that is a countermodel to each i i at wi for all 1 i n.SoM is )   also a counter-example to the premise

1 1 ... n n ) i i ) which contradicts our hypothesis. So the conclusion of the inference must be valid.

The calculus RD, sound for serial frames, is the result of adding the rule

Drop to RK. G Drop )iG

Theorem 37. If RD H , then there is no serial counter-example to H . ` Proof. We show that the Drop rule is sound for symmetric frames. Sup- pose that is an instance of Drop. So the derivation ends in the following inference. . . G Drop )iG By our inductive hypothesis, we know that the premise of the inference is valid. Consider an arbitrary model M with a serial frame. Suppose toward a

67 4. Relational Hypersequents

contradiction that M, w1,...,wn forms a counter-example to the conclusion of the inference. This means that there is a branch of worlds w1,...,wn such that wi Rwi 1 and M is a countermodel to each i i at wi for all 1 i n. )   Since the frame is serial, it must be the case that there is some world w0 such that w1Rw0. M is a countermodel to the empty sequent at w0. It follows that M is a counter-example to the hypersequent ( G ). )i The calculus RS4 is the result of adding the rule EW to RT. RS4 is sound for the class of reflexive and transitive frames.

G H EW G i H i)i

Theorem 38. If RS4 H , then there is no transitive counter-example to H . ` Proof. It suffices to show that the EW rule is sound for transitive frames. Let

be an instance of EW. . . G H EW G i H i)i By our inductive hypothesis, the derivation is sound up until the applica- tion of the EW rule. Consider an arbitrary model M with a transitive frame.

Let M be a counter-example to the conclusion

G i i i +1 i +1 H i ) i)i ) i where G = 1 1, ... i 1 i 1 and H = i +2 i +2 ... n n . ) i i ) ) i i ) So there is a branch of worlds w1,...,wi , w, wi +1,...,wn such that w j Rwj 1 for all 1 j n, wi +1Rw, wRwi and M is a countermodel to each j j   ) at w j for all 1 j n. Since wi +1Rw, wRwi , and the frame is transitive, it   follows that wi +1Rwi .Sow1,...,wi , wi +1,...,wn is also a branch of worlds

68 4. Relational Hypersequents

and M is a countermodel to each j j at w j for all 1 j n. But this )   means that M is also a counter-example to the premise.

If either G or H is empty, then this is an application of EWL or EWL, respectively, which we have shown are sound.

The calculus RS5 is the result of adding rule EE to RS4. The resultant calculus is sound for the class of reflexive, transitive and symmetric frames.

G S1 S2 H i i i EE G S2 S1 H i i i Note that the sequent É É',' is derivable in RS5 without the use ) ¬ of Cut.

Proof. p p L p ) p É É EE )p i )p É R )i)p p É ¬ R )¬p,Éi)Ép É ) ¬

Theorem 39. If RS5 H , then there is no reflexive, transitive and symmetric ` counter-example to H .

Proof. It suffices to show that the EE rule is sound for transitive, symmetric frames. Suppose that is an application of the EE rule. So the end of the derivation runs as follows. . .

G i i i +1 i +1 H i ) i ) i EE G i +1 i +1 i i H i ) i ) i

69 4. Relational Hypersequents

By inductive hypothesis, we know that the derivation is sound up until the application of the EE rule. Consider an arbitrary model M with a sym- metric, transitive frame. Suppose for reductio that M, w1,...,wn forms a counter-example to the conclusion of the inference.

G i +1 i +1 i i H i ) i ) i

So there is a branch of worlds w1,...,wi 1, wi +1, wi , wi +2,...,wn such that M is a countermodel to k k at wk for all 1 k n. We know that )   wi +1Rwi 1, wi Rwi +1 and wi +2Rwi . Since the frame is transitive, wi Rwi 1 and wi +2Rw1+1. Since the frame is also symmetric, wi +1Rwi . It follows that w1,...,wi 1, wi , wi +1, wi +2,...,wn is a branch of worlds such that M is a countermodel to j j at w j for 1 j n. It follows that M is is a )   counter-example to the premise

G i i i +1 i +1 H i ) i ) i which contradicts our hypothesis. So the conclusion must be valid. The above also holds with G ,H empty.

4.4 Completeness for RK

Theorem 40 (RK Completeness). If there is no counter-example to H , then

RK H. `

Given a hypersequent H , we give all formulas ' H a unique index 'k 2 beginning with the leftmost formula and index 1. In addition we give each sequent a label j corresponding to a possible world. When new sequents

70 4. Relational Hypersequents are added during the reduction process, new labels are given according to the reduction step.

Definition 41 (Partial reduction). Given any hypersequent H , the partial reduction tree R(H ), with H as the root hypersequent, is the result of the following reduction steps applied to H .

Reduction steps are applied to the formula with the lowest index. When a reduction has been applied to a formula 'k we remove the index from that formula. When a new formula is added to a sequent, if labels 1,...,n are in use, we give the new formula the label n + 1.

Definition 42 (Closed Leaf). A leaf H 0 of R(H ) is closed iff some sequent

i = i of H 0 is such that i i = . If a leaf is not closed, then it is open. ) \ 6 ;

Let 'k be the formula with the lowest index.

1. If 'k is a propositional variable, then do nothing.

2. a) 'k is .If'k i , then expand the branch as follows: ¬ 2 j G i = i , l G 0 i )j i G i = i G 0 i ) i b) If 'k i , then expand the branch as follows: 2 j G l ,i = i G 0 i )j i G i = i G 0 i ) i 3. a) 'k is ✓ .If'k i , then do the following: _ 2

j j G l ,i = i G 0 G ✓l +1,i = i G 0 i ) i j i ) i G i = i G 0 i ) i 71 4. Relational Hypersequents

b) If 'k i , then expand the branch as follows: 2 j G i = i , l ,✓l +1 G 0 i ) j i G i = i G 0 i ) i 4. a) 'k is ✓ .If'k i , then expand the branch: ^ 2 j G l ,✓l +1,i = i G 0 i )j i G i = i G 0 i ) i b) If 'k i , then expand the branch as follows: 2

j j G i = i , l G 0 G i = i ,✓l +1 G 0 i ) i j i ) i G i = i G 0 i ) i

5. a) 'k is ✓ .If'k i , then expand the branch: ! 2

j j G i = i , l G 0 G ✓l +1,i = i G 0 i ) i j i ) i G i = i G 0 i ) i

b) If 'k i , then expand the branch as follows: 2 j G l ,i = i ,✓l +1 G 0 i )j i G i = i G 0 i ) i 6. 'k is É .If'k i , then expand the branch as follows: 2

k j G l ,i 1 = i 1 i = i G 0 i )k i )j i G i 1 = i 1 i = i G 0 i ) i ) i If at any point all leaves of R(H ) close, then we stop. A derivation of the hypersequent H may be found through a series of external and internal weakenings.

72 4. Relational Hypersequents

Lemma 43. For any hypersequent H 0, if RK H 0, then the application of a 6` reduction step ↵ to H 0 results in at least one unprovable premise.

Proof. We prove the above lemma by induction on the length of the reduc- tion. We consider only reductions on formulas of the form ✓ and É . ! The cases for the other propositional connectives are similar.

a. ↵ is a L reduction. By hypothesis, we know that the conclusion of ! the reduction step,

j G i = i G i ) i

is not derivable in RK. Suppose for reductio that both premises of the

reduction are provable. So we have the following derivations:

⇡ . . . . j j G , G i = i 0 G ✓ ,i = i G 0 i ) i i ) i From these we obtain the following proof,

⇡ . . . . j j G i = i , G G ✓ ,i = i G 0 0 L i ) i j i ) i G ✓ ,i = i G ! i ! j ) i G i = i G i ) i where the double line indicates a series of internal structural rules.

This derivation contradicts our initial hypothesis. So the premises

must be unprovable in RK.

73 4. Relational Hypersequents b. ↵ is a R reduction. By our inductive hypothesis, the conclusion of ! the inference step,

j G i = i G 0 i ) i

is unprovable in RK. Suppose for reductio that the premise of the

reduction rule is provable. So we have a proof of the following.

. . j G ,i = i ,✓ G 0 i ) i From this we can derive the conclusion, which contradicts our

hypothesis.

. . j G ,i = i ,✓ G 0 R i )j i ! G i = i , ✓ G 0 i )j ! i G i = i G 0 i ) i c. ↵ is a É L reduction. By our inductive hypothesis, the conclusion of the inference rule

k j G i 1 = i 1 i = i G 0 i ) i ) i

is not derivable in RK. Suppose for reductio that we have a derivation

of the following form.

. . k j G ,i 1 = i 1 i = i G 0 i ) i ) i From this we can derive the conclusion, which contradicts our

hypothesis.

74 4. Relational Hypersequents

. . k j G ,i 1 = i 1 i = i G 0 R i k ) i )j i É G i 1 = i 1 É ,i = i G 0 i ) i ) i k j G i 1 = i 1 i = i G 0 i ) i ) i

1 m Lemma 44. Let H 0 =(1 = 1 ... n = n ) be a node in the reduction tree

)1 i i )m for R(H ), and let H 00 =(10 = 01 ... n0 = 0n ) be the result of a reduction

) i i )k k step ↵ on H 0. We show that for each j = j , and j0 = 0j , j j0 and ) ) ✓ j 0j . ✓ Proof. We prove the above lemma by induction on the length of the reduc- tion tree R(H ).

1. ↵ is a L reduction on the formula . By the reduction rule (2a), ¬ ¬ has been added to the succeedent of the sequent, so j = 0j . \{¬ } It follows that j 0j . Since j remains unchanged, it is trivial that ✓ j j0. All other sequents are unchanged. ✓ 2. ↵ is a R reduction on the formula . By the reduction rule (2b), ¬ ¬ j = j0 .Soj j0. Since j is unchanged in the reduction, it is \{¬ } ✓ trivial that j 0j . ✓ 3. ↵ is a L reduction on the formula ✓ . We distinguish cases. _ _ In the first case, consider the leftmost branch. By the reduction

rule (3a), has been added to the antecedent of the sequent. So

j = j0 , and j j0. Since is unchanged in the reduction, it is \{ } ✓ trivial that j 0j . ✓

75 4. Relational Hypersequents

In the rightmost branch ✓ has been added to the antecedent of

the sequent. So j = j0 ✓ and 0. Since j = 0j , j 0j .In \{ } ✓ ✓ both cases, all other sequents are unchanged.

4. ↵ is a R reduction on ✓ .By(3b), and ✓ have been added to the _ _ succeedent of the sequent. So j = 0j ,✓ , and therefore j 0j . \{ } ✓ Since j is unchanged in the reduction, it is trivial that j j0. All other ✓ sequents are left the same.

5. ↵ is a L reduction on ✓ . By the reduction rule (4a), and ✓ have ^ ^ been added to the antecedent of the sequent, so j = j0 ,✓ .It \{ } follows that j j0. Since j is unchanged, it is trivial that j 0j . ✓ ✓ All other sequents remain the same.

6. ↵ is a R reduction on some formula ✓ . We distinguish cases. ^ ^

In the leftmost branch, by (4b), j = 0j .Soj 0j . Since \{ } ✓ j is unchanged in the reduction, it is trivial that j j0. ✓

In the rightmost branch, by the reduction rule, j = j0.Soj 0. ✓ It is also the case that j = 0j ✓ , so j 0j . All other sequents \{ } ✓ remain the same in both cases.

7. ↵ is a L reduction on some formula ✓ . We distinguish cases. ! !

In the first case, consider the leftmost branch. By (5a), j = 0j \ .Soj 0j . Since j is unchanged in the reduction, it is trivial { } ✓ that j j0. ✓ In the second case, consider the rightmost branch. By the reduc-

tion rule, j = j0 ✓ .Soj j0. It is also the case that j = 0j , so \{ } ✓ j 0j . All other sequents remain unchanged. ✓ 76 4. Relational Hypersequents

8. ↵ is a R reduction on a formula ✓ . By the reduction rule (5b), ! ! j = j0 .Soj j0. Also by the reduction rule, j = 0j , and \{ } ✓ \{ } so j 0j . All other sequents are unchanged. ✓

9. ↵ is a É L reduction on a formula É . By the reduction rule (6), j = j0

and j = 0j .Soj j0 and j 0j . Also by the reduction rule, j 1 = ✓ ✓ j0 1 É , and j 1 = 0j 1. It follows that j 1 j0 1 and j 1 = 0j 1. \ ✓

It follows that = , as if a formula ever occurs in () then it occurs \ ; 0 (0) in every successive hypersequent. It is therefore the case that the formula never occurs in ( ) at any point, as if it did, the branch would be closed.

Lemma 45. All partial reduction trees for RK terminate

Proof. Let r ('i ) denote the grade of a labelled formula 'i , and let r (H ) be the sum of all r ('i ) such that 'i H . We call r (H ) the grade of the 2 hypersequent H.

Let G 0 be the result of a reduction step applied to G . If a formula i is being reduced, we remove all labels from it during the reduction step and add one or more new labelled formulas to the hypersequent. We show that r (G 0) < r (G ). We consider only reductions of É in the antecedent, and . ! The other cases are similar.

1. is a É reduction in the antecedent on some formula É'i . During

the reduction step, the formula 'k has been added to G 0, with grade

r (É'i ) 1. The index has been removed from É' in G 0, since the

77 4. Relational Hypersequents

formula has been reduced. It follows that r (G 0)=r (G ) 1, so r (G 0) < r (G ).

2. is a reduction in the antecedent on some formula ' ✓i , where ! ! r (' ✓ )=r (')+r (✓ )+1. Consider the leftmost branch. The label has ! been removed from ' ✓ in G 0 since the formula has been reduced. ! However, ' has been added to G 0. But we know that r (') < r (' ✓ ), ! and it follows that r (G 0) < r (G ). Similarly for the rightmost branch.

3. is a reduction in the antecedent on some formula ' ✓i , where ! ! r (' ✓ )=r (')+r (✓ )+1. The index has been removed from ' ! ! ✓ G 0 since the formula has been reduced. However, ' and ✓ have 2 been added to G 0. But we know that r (')+r (✓ )=r (' ✓ ) 1, and ! so it follows that r (G 0) < r (G ).

Either the tree terminates when all leaves G ⇤ are such that r (G ⇤)=0, or there remain only unreduced formulas of the form É' to which no reduction step applies. In either case, the reduction tree is finite.

1 m If R(H ) has not closed, then pick an open leaf H ⇤ = 1 = 1 ... n = j ) i i )j n . For each sequent i = i in H ⇤, where 1 i n, let Hi = i =

m )   ) i ... n = n . i i )

Definition 46 (Successor). Given the hypersequent Hi , if É 1,...,É n i 2 and É'1,...,É'm i , then for each modal formula É l , the hypersequent j2.l j Hil = '1,...,'m = l i = i G is a successor of H ⇤. ) i ) i

The reduction process is then repeated on each Hil in order to reduce any formulas uncovered during the successor step.

78 4. Relational Hypersequents

Lemma 47. If H 0 is an unprovable hypersequent, then all of its successors are also unprovable.

Proof.

1. By hypothesis, we know that H 0 is an unprovable hypersequent. Sup- j.l pose by way of contradiction that some successor of the form ( '1,...,'m = j ) l i = i G 0) is provable in RK. So we have a proof of the following. i ) i . . j.l j '1,...,'m = l i = i (É l ) G 0 ) i ) i And from this we can derive,

. . j.l j '1,...,'m = l i = i G 0 Lm )j.l i )j i É = l '1,..., 'n , i = i G É É 0 R ) i )j i É É'1,...,É'n , i = i ,É l G 0 )j i i = i G 0 ) i Where is several internal exchanges and a contractions on each

j É'k and É l . This contradicts our initial assumption that (i = ) i G 0) is unprovable. i

For any hypersequent H we define the reduction set for H , ⇡(H ), as follows.

Definition 48 (Reduction Set).

1. H ⇤ ⇡(H ) 2 79 4. Relational Hypersequents

2. If G ⇡(H ), and G 0 is a successor of G , then (G 0)⇤ ⇡(H ) 2 2 Lemma 49. If G ,G 0 ⇡(H ) and = G , 0 = 0 G 0, then = 0 and 2 ) 2 ) 2 = 0.

Proof. We prove the above by induction.

1. The sequent H is such that each component sequent = is given ) a unique label . No new labels are added in the partial reduction

process. It follows that no two sequents in H ⇤ have the same label.

2. Suppose that G ⇡(H ) and G 0 is a successor of G . (G 0)⇤ is the result 2 of the partial reduction process applied to G 0. Suppose that = is ) a member of G , and 0 = 0 is an element of (G 0)⇤. = j.l , as this ) 6 is a new world added to the hypersequent, and thus not present in

l l G 0. For all other sequents k = k in G that such that k0 = 0k is an ) ) element (G )⇤, k = k0 and k = 0k , as these sequents are unchanged in the successor step. Labels have been removed from all formulas

except for unreduced É formulas, the reduction of which only affects the new world that is introduced in the reduction step. It follows that

= 0 and = 0.

Definition 50. Suppose that ⇡(H ) is the reduction set for an unprovable hypersequent H . Define MH = W,R, v as follows: { } 1. W = : = ⇡(H ) . { ) 2 }

2. R0 iff = n1,...,nk and either 0 = n1,...,nk 1 or 0 = n1,...nk ,nk.nk 1 .

80 4. Relational Hypersequents

3. v (p)= : p . { 2 } Where is such that = ⇡(H ). By lemma 49, is unique, and ) 2 so our definition is well defined.

Lemma 51. and have the following properties, based on the definition of a partial reduction:

1. If , then . ¬ 2 2

2. If , then . ¬ 2 2

3. If ✓ , then or ✓ . _ 2 2 2

4. If ✓ , then ,✓ . _ 2 2

5. If ✓ , then ,✓ . ^ 2 2

6. If ✓ , then or ✓ . ^ 2 2 2

7. If ✓ , then or ✓ . ! 2 2 2

8. If ✓ , then and ✓ . ! 2 2 2

Proof. The above follows directly from the properties of the reduction steps

2a — 5b.

Lemma 52. The following also hold for ,:

9. If É n , then either n 1 and for all n.k W , n.k or there 2 2 2 2

is no n 1 or n.k ..

10. If É n , then there is some n.i such that n Rn.i , and n.i . 2 2

81 4. Relational Hypersequents

Proof.

9. Suppose that É n . We distinguish between two cases: either 2

some É✓1,...,É✓m n , or not. 2

Suppose that É✓1,...,É✓m n . Then, for each 1 k m there is

2 n.k n   some successor of the form ( ,..., i = ✓k = G 0). It follows

) i ) in that n.k for each 1 k m. In addition, if = is not the 2   )

leftmost sequent in H , then the rule 6 has been applied, so n 1 . 2 n Suppose that there is no É✓ n , and that = is not the 2 )

leftmost sequent in H . Reduction step 6 has been applied, so n 1 . 2 n In the case where there is no É✓ n , and = is the leftmost 2 ) sequent in H , then there is no n0 such that n Rn0 .

10. Suppose that É n . The successor step has been applied. It 2 follows that there is some n.k such that n.k , and n Rn.k . 2

Theorem 53. For any formula ', if ' then MK , è ', and if ' 2 2 then MK , è '. 6 Proof. We now show that the model determined by the reduction tree is

a counter-example to the root hypersequent, i.e., for any sets j and j , and formula ', if ' j then MK ,j è ', and if ' j , then MK ,j è '. 2 2 6 We proceed by induction on the complexity of the formula '.

1. ' is a propositional formula p.Ifp j , then MK ,j è p by con- 2

struction. If p j , then p j by the fact that the tree is open. It 2 62 follows that MK ,j è p by the definition of v . 6 82 4. Relational Hypersequents

2. ' is , where j . By lemma 51 j .SoMK ,j è . But ¬ ¬ 2 2 6 this means that MK ,j è . ¬

3. ' is , where j . By lemma 51 j .SoMK ,j è . But ¬ ¬ 2 2 this means that MK ,j è . 6 ¬

4. ' is ✓ , where ✓ j . By lemma 51, either or ✓ are in j .By _ _ 2 our inductive hypothesis, MK ,j è or MK ,j è ✓ . In either case,

MK ,j è ✓ . _

5. ' is ✓ , where ✓ j . By lemma 51, and ✓ are in j . By our _ _ 2 inductive hypothesis, MK ,j è , and MK ,j è ✓ . But this means 6 6 that MK ,j è ✓ . 6 _

6. ' is ✓ , where ✓ j . By lemma 51 and ✓ are in j .So ^ ^ 2 MK ,j è , and MK ,j è ✓ . But this means that MK ,j è ✓ . ^

7. ' is ✓ , where ✓ j . By lemma 51, either or ✓ are in j .By ^ ^ 2 our inductive hypothesis, this means that MK ,j è or MK ,j è ✓ . 6 6 In either case, MK ,j è ✓ . 6 ^

8. ' is ✓ , where ' j . By lemma 51, we know that either j , ! 2 2

or ✓ j . By our inductive hypothesis, Mk ,j è or Mk ,j è ✓ .In 2 6 either case, Mk ,j è ✓ . !

9. ' is ✓ , where ' j . By lemma 51, j and ✓ j .By ! 2 2 2 our inductive hypothesis, this means that Mk ,j è and Mk ,j è ✓ . 6 But this means that Mk ,j è ✓ . 6 !

10. ' is É , where É j . We distinguish between two cases. In the 2

first case, j 1 and for all j.i W , j.i , by lemma 52. By the 2 2 2 83 4. Relational Hypersequents

inductive hypothesis, we know that MK , j 1 è and MK , j.i è , for all i . It follows that MK ,j èÉ', since by the construction of the

model, there is no other world 0 such that j R0. In the other case,

j is the leftmost sequent in every hypersequent in ⇡(H ). It follows

that there is no 0 such that j R0. So we know that É' is vacuously

true in j , and so MK ,j èÉ'.

11. Suppose that ' is É , such that É j . By lemma 52 there is some 2

j.i such that j.i . By our inductive hypothesis, MK ,j.i è . 2 6 Since by construction, j Rj.i , we know that MK ,j èÉ . 6

cf Corollary 54. If RK H then RK H. ` `

84 Chapter 5

Conclusion

In this thesis we have addressed many of the issues that arise in the search for Gentzen-type systems for modal logics. Sequent calculus systems for modal logics are not unified, and many of the calculi for S5 are not Cut-free.

Natural deduction systems for modal logics are in a worse state. Gentzen- style natural deduction has been developed for S4 and S5, but not for K and its other extensions.

Extensions of Gentzen’s sequent calculus have been developed in or- der to overcome these issues. The systems of Pottinger (1983) and Avron

(1996) were the first to introduce a Cut-free calculus for S5 using hyperse- quents. However, these systems lack general properties that are desirable in a sequent calculus: there is no obvious way to extend their systems to incorporate the most important extensions of K.

Lahav (2013) gave a general way of creating hypersequent rules from frame properties. However, his systems do not satisfy Došen’s principle, as logical rules are varied between calculi, not structural rules. For this reason,

Gentzen’s original reason for developing the sequent calculus—to be used

85 5. Conclusion

Proof System Došen’s Modularity Generality Cut Principle Elimination/ Cut-free Completeness

Mints N/A Yes No Yes Pottinger No No No Yes, but no available proof

Avron N/A Yes No Yes Lahav No Non- Most general Yes, except for symmetric, symmetric non-transitive logics logics Tree No Yes, except S5 Yes Yes Hypersequents Linear Nested No Yes Yes, but no Yes Sequents calculus for B Labelled Se- Yes No Yes Yes quents Relational Yes Yes, except S5 Yes Proofs for K and Hypersequents S5

Table 5.1: Modal Proof Systems and their Important Properties

as a meta-calculus for natural deduction—is lost in these contexts. Other

successful extensions of the sequent calculus, like tree hypersequents and

linear nested sequents, also do not respect Došen’s principle. These proof

systems and their important properties are summarized in table 5.1.

We have presented some preliminary results that suggest that relational

hypersequents might be able to overcome some of the main issues in the

proof theory of modal logics. Relational hypersequent systems blend fea-

tures of hypersequents, linear nested sequents, and labelled sequents. Al-

though this is not a purely syntactic method, it has advantages over other

86 5. Conclusion extensions of the sequent calculus insofar as it respects Došen’s principle.

We have shown that the base calculus RK is Cut-free complete. It remains to be shown that the other extensions, specifically RT, RS4, RD and RB are also Cut-free complete. Given that the calculi respect Došen’s principle, it is possible that relational hypersequents may give us an idea of how to construct a so-called hyper-natural deduction system for modal logics, the development of which will be a step towards a unified Gentzen-style proof theory for modal logics.

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