Internal and External Calculi: Ordering the Jungle without Being Lost in Translations

A Survey of Decidability Results for Modal, Tense and Intermediate Logics

Another possible title for this chapter might have been “some non-standard logics”, since its principal aim is to illustrate how the methods we introduced for the standard logics M, I and C are applicable in different settings as well.

We consider a general format for sequent rules for not necessarily normal modal logics based on classical or intuitionistic propositional logic and provide relatively simple local conditions ensuring cut elimination for such rule sets. The rule format encompasses e.g. rules for the boolean connectives and transitive modal logics such as S4 or its constructive version. We also adapt the method of constructing suitable rule sets by saturation to the intuitionistic setting and provide a criterium for translating axioms for intuitionistic modal logics into sequent rules. Examples include constructive modal logics and conditional logic $\mathbb{VA}$ .

In this thesis, we consider bi-intuitionistic logic and tense logic, as well as the combined bi-intuitionistic tense logic. Each of these logics contains a pair of dual connectives, for example, Rauszer’s bi-intuitionistic logic [100] contains intuitionistic implication and dual intuitionistic exclusion. The interaction between these dual connectives makes it non-trivial to develop a cut-free sequent calculus for these logics. In the first part of this thesis we develop a new extended sequent calculus for biintuitionistic logic using a framework of derivations and refutations. This is the first purely syntactic cut-free sequent calculus for bi-intuitionistic logic and thus solves an open problem. Our calculus is sound, semantically complete and allows terminating backward proof search, hence giving rise to a decision procedure for bi-intuitionistic logic. In the second part of this thesis we consider the broader problem of taming proof search in display calculi [12], using bi-intuitionistic logic and tense logic as case studies. While the generality of display calculi makes it an excellent framework for designing sequent calculi for logics where traditional sequent calculi fail, this generality also leads to a large degree of non-determinism, which is problematic for backward proof-search. We control this non-determinism in two ways: 1. First, we limit the structural connectives used in the calculi and consequently, the number of display postulates. Specifically, we work with nested structures which can be viewed as a tree of traditional Gentzen’s sequents, called nested sequents, which have been used previously by Kashima [73] and, independently, by Brunnler and Strasburger [17; 21; 20] and Poggiolesi [97] to present several modal and tense logics. 2. Second, since residuation rules are largely responsible for the difficulty in finding a proof search procedure for display-like calculi, we show how to eliminate these residuation rules using deep inference in nested sequents. Finally, we study the combined bi-intuitionistic tense logic, which contains the well-known intuitionistic modal logic as a sublogic. We give a nested sequent calculus for bi-intuitionistic tense logic that has cut-elimination, and a derived deep inference nested sequent calculus that is complete with respect to the first calculus and where contraction and residuation rules are admissible. We also show how our calculi can capture Simpson’s intuitionistic modal logic [104] and Ewald’s intuitionistic tense logic [39].

This chapter focuses on the development of Gerhard Gentzen's structural proof theory and its connections with intuitionism. The latter is important in proof theory for several reasons. First, the methods of Hilbert's old proof theory were limited to the “finitistic” ones. These methods proved to be insufficient, and they were extended by infinitistic principles that were still intuitionistically meaningful. It is a general tendency in proof theory to try to use weak principles. A second reason for the importance of intuitionism for proof theory is that the proof-theoretical study of intuitionistic logic has become a prominent part of logic, with applications in computer science.

The paper presents a survey of author's results on definable fixed points in modal, temporal, and intuitionistic propositional logics. The well-known Fixed Point Theorem considers the modalized case, but here we investigate the positive case. We give a classification of fixed point theorems, describe some classes of models with definable least fixed points of positive operators, special positive operators, and give some examples of undefinable least fixed points. Some other interesting phenomena are discovered – definability by formulas that do not preserve positivity of parameters and definability by finite sets of formulas. We also consider negative operators, graded modalities, construct undefinable inflationary fixed points, and put some problems.

We propose a modal linear logic to reformulate intuitionistic modal logic S4 (IS4) in terms of linear logic, establishing an S4-version of Girard translation from IS4 to it. While the Girard translation from intuitionistic logic to linear logic is well-known, its extension to modal logic is non-trivial since a naive combination of the S4 modality and the exponential modality causes an undesirable interaction between the two modalities. To solve the problem, we introduce an extension of intuitionistic multiplicative exponential linear logic with a modality combining the S4 modality and the exponential modality, and show that it admits a sound translation from IS4. Through the Curry-Howard correspondence we further obtain a Geometry of Interaction Machine semantics of the modal lambda-calculus by Pfenning and Davies for staged computation.

Substructural logics are traditionally obtained by dropping some or all of the structural rules from Gentzen's sequent calculi LK or LJ. It is well known that the usual logical connectives then split into more than one connective. Alternatively, one can start with the (intuitionistic) Lambek calculus, which contains these multiple connectives, and obtain numerous logics like: exponential-free linear logic, relevant logic, BCK logic, and intuitionistic logic, in an incremental way. Each of these logics also has a classical counterpart, and some also have a 'cyclic' counterpart. These logics have been studied extensively and are quite well understood. Generalising further, one can start with intuitionistic Bi-Lambek logic, which contains the dual of every connective from the Lambek calculus. The addition of the structural rules then gives Bi-linear, Bi-relevant, Bi-BCK and Bi-intuitionistic logic, again in an incremental way. Each of these logics also has a classical counterpart, and some even have a 'cyclic' counterpart. These (bi-intuitionistic and bi-classical) extensions of Bi-Lambek logic are not so well understood. Cut-elimination for Classical Bi-Lambek logic, for example, is not completely clear since some cut rules have side conditions requiring that certain constituents be empty or non-empty. The display logic of Nuel Belnap is a general Gentzen-style proof theoretical framework designed to capture many different logics in one uniform setting. The beauty of display logic is a general cut-elimination theorem, due to Belnap, which applies whenever the rules of the display calculus obey certain, easily checked, conditions. The original display logic, and its various incarnations, are not suitable for capturing bi-intuitionistic and bi-classical logics in a uniform way. We remedy this situation by giving a single (cut-free) Display calculus for the Bi-Lambek Calculus, from which all the well-known (bi-intuitionistic and bi-classical) extensions are obtained by the incremental addition of structural rules to a constant core of logical introduction rules. We highlight the inherent duality and symmetry within this framework obtaining 'four proofs for the price of one'. We give algebraic semantics for the Bi-Lambek logics and prove that our calculi are sound and complete with respect to these semantics. We show how to define an alternative display calculus for bi-classical substructural logics using negations, instead of implications, as primitives. Borrowing from other display calculi, we show how to extend our display calculus to handle bi-intuitionistic or bi-classical substructural logics containing the forward and backward modalities familiar from tense logic, the exponentials of linear logic, the converse operator familiar from relation algebra, four negations, and two unusual modalities corresponding to the non-classical analogues of Sheffer's 'dagger' and 'stroke', all in a modular way. Using the Gaggle Theory of Dunn we outline relational semantics for the binary and unary intensional connectives, but make no attempt to do so for the extensional connectives, or the exponentials. Finally, we flesh out a suggestion of Lambek to embed intuitionistic logic using two unusual 'exponentials', and show that these 'exponentials' are essentially tense logical modalities, quite at odds with the usual exponentials. Using a refinement of the display property, you can pick and choose from these possibilities to construct a display calculus for your needs. Keywords:display logic, substructural logics, linear logic, relevant logic, BCK-logic, bi-Heyting logic, intuitionistic substructural tense logics, intuitionistic modal logics, proof theory, gaggle theory

Fitch [8] first developed an intuitionistic modal logic based on Heyting’s intuitionistic logic. No special attention to this system was paid by those researchers working on intuitionism nor by those on modal logic. This is probably because Fitch gave only a proof theory, and because it was before the appearance of the so-called Kripke semantics for modal and intuitionistic logics. Recently, there has been a considerable interest in this topic as shown in the works of Ono [18], Gabbay [10], Ewald [7], Font [9], Božić and Došen [5], Došen [6].

Proof theory is a central area of mathematical logic of special interest to philosophy. It has its roots in the foundational debate of the 1920s, in particular, in Hilbert’s program in the philosophy of mathematics, which called for a formalization of mathematics, as well as for a proof, using philosophically unproblematic, “finitary” means, that these systems are free from contradiction. Structural proof theory investigates the structure and properties of proofs in different formal deductive systems, including axiomatic derivations, natural deduction, and the sequent calculus. Central results in structural proof theory are the normalization theorem for natural deduction, proved here for both intuitionistic and classical logic, and the cut-elimination theorem for the sequent calculus. In formal systems of number theory formulated in the sequent calculus, the induction rule plays a central role. It can be eliminated from proofs of sequents of a certain elementary form: every proof of an atomic sequent can be transformed into a “simple” proof. This is Hilbert’s central idea for giving finitary consistency proofs. The proof requires a measure of proof complexity called an ordinal notation. The branch of proof theory dealing with mathematical systems such as arithmetic thus has come to be called ordinal proof theory. The theory of ordinal notations is developed here in purely combinatorial terms, and the consistency proof for arithmetic presented in detail.

In this thesis we consider generic tools and techniques for the proof-theoretic investigation of not necessarily normal modal logics based on minimal, intuitionistic or classical propositional logic. The underlying framework is that of ordinary symmetric or asymmetric two-sided sequent calculi without additional structural connectives, and the point of interest are the logical rules in such a system. We introduce the format of a sequent rule with context restrictions and the slightly weaker format of a shallow rule. The format of a rule with context restrictions is expressive enough to capture most normal modal logics in the S5 cube, standard systems for minimal, intuitionistic and classical propositional logic and a wide variety of non-normal modal logics. For systems given by such rules we provide sufficient criteria for cut elimination and decidability together with generic complexity results. We also explore the expressivity of such systems with the cut rule in terms of axioms in a Hilbert-style system by exhibiting a corresponding syntactically defined class of axioms along with automatic translations between axioms and rules. This enables us to show a number of limitative results concerning amongst others the modal logic S5. As a step towards a generic construction of cut free and tractable sequent calculi we then introduce the notion of cut trees as representations of rules constructed by absorbing cuts. With certain limitations this allows the automatic construction of a cut free and tractable sequent system from a finite number of rules. For cases where such a system is to be constructed by hand we introduce a graphical representation of rules with context restrictions which simplifies this process. Finally, we apply the developed tools and techniques and construct new cut free sequent systems for a number of Lewis’ conditional logics extending the logic V. The systems yield purely syntactic decision procedures of optimal complexity and proofs of the Craig interpolation property for the logics at hand.

CHAPTER 16 - Temporal and Modal Logic

Modal and temporal logics are extensions to classical logic that have operators that deal with necessity and possibility (modal logics) and such as sometime, always and next (temporal logics). Models are sets of worlds that are connected by an accessibility relation. Restrictions imposed on this relation and the operators allowed give rise to different families of logic. This paper discusses an approach to theorem proving for temporal and modal logics based on clausal resolution. The main ideas are the translation to a normal form and the application of resolution rules that relate to the same world. This research initially focused on propositional linear time temporal logic but has been extended to computation tree logic, monodic first order temporal logic and normal modal logics. We describe the approach, explain the adaptations necessary for the logics mentioned and discuss the results of the provers developed for these logics.

This guide accompanies the following article(s): Meghan Sullivan, ‘Problems with Temporary Existence in Tense Logic’. Philosophy Compass 7/1 (2012): 43–57. doi: 10.1111/j.1747‐9991.2011.00457.x Author’s Introduction Over the past century, there has been considerable debate over whether and how anything changes with respect to existence. Most A‐theorists of time (presentists, growing block theorists, and branch theorists) think things come to exist or cease to exist. B‐theorists of time (four‐dimensionalists, in particular) think objects do not change with respect to existence. In my Compass article, I outline a serious difficulty that A‐theorists face in trying to reason about temporary existents. The most straightforward logics for time and existence entail that nothing exists merely temporarily. The problem arises from a set of theorems of the simplest temporal logic – the converse Barcan formulas. But attempts to fix the logic to get rid of the Barcan formulas pressure A‐theorists to abandon an intuitive and widespread assumption about existence. I survey the logical and metaphysical options for solving the problem. Author Recommends Burgess, John P. Philosophical Logic. Princeton: Princeton University Press, 2009. An introductory textbook in philosophical logic. Chapter 2 focuses on temporal logic and motivates a logic‐based response to problems with the temporal Barcan schemas. Prior, A. N. Past, Present, and Future . Oxford: Oxford University Press, 1967. The first attempt to rigorously formulate propositional and quantified tense logic. Chapter 8 especially provides philosophical insight into problems with change in existence. Prior uses Polish notation for his proofs and formalism, which requires a bit of background to translate. Sider, Theodore. Four‐Dimensionalism. Oxford: Oxford University Press, 2001. Provides a useful background on the debates in the philosophy of time. The first three chapters that precisely define the different theories are especially relevant. Sider, Theodore. Logic for Philosophy. Oxford: Oxford University Press, 2010. A useful guide to the semantics and proof theory of modal and temporal logics. van Inwagen, Peter. ‘Meta‐Ontology.’ Erkenntnis 48 (1998):233–50. Gives an explanation and defense of neo‐Quinean assumptions. Williamson, Timothy. ‘Bare Possibilia.’ Erkenntnis 48 (1998):257–73. Provides a logic‐based argument for necessary, permanent existence and gives an A‐theory‐friendly model for explaining change on such an ontology. Zimmerman, Dean W. ‘Temporary Intrinsics and Presentism.’ Metaphysics: The Big Questions. Eds. Peter van Inwagen and Dean W. Zimmerman. Oxford: Blackwell, 1998. Surveys a problem in formulating presentist theories of change and motivates the need for tense operators. Sample Syllabus: Here is a sample syllabus for a course on time in metaphysics and logic: Week I: Introduction: A‐Theories and B‐Theories We will consider precise ways of differentiating A‐theories of time and B‐theories of time, looking in particular at how A‐theorists and B‐theorists think of intrinsic properties. Reading: Chap 4.2., Lewis, David. On the Plurality of Worlds . Oxford: Blackwell, 1986. Zimmerman, Dean W. ‘Temporary Intrinsics and Presentism.’ Metaphysics: The Big Questions. Eds. Peter van Inwagen and Dean W. Zimmerman. Oxford: Blackwell, 1998. Chap 2, Sider, Theodore. Four‐Dimensionalism. Oxford: Oxford University Press, 2001. Week II: The Bug: Temporary Existence in Tense Logic We will consider why A‐theorists use tense logics to express their views, and we will look at the difficulties A‐theorists have expressing temporary existence in tense logic. Reading: Sullivan, Meghan. ‘Problems for Temporary Existence in Tense Logic.’ Philosophy Compass. Chap 8, Prior, A. N. Past, Present, and Future . Oxford: Oxford University Press, 1967. Week III: Option 1: Rewire Tense Logic? We will learn about Kripke’s solution to the parallel problem in modal logic, consider how it might be applied to tense logic, and then consider philosophical difficulties for the proposal. Reading: Chap 2, Burgess, John P. Philosophical Logic. Princeton: Princeton University Press. 2009. Kripke, Saul. ‘Semantical Considerations in Modal Logic.’ Reference and Modality. Ed. Bernard Linsky. Oxford: Oxford University Press, 1971. Optional: Chap 10, Sider, Theodore. Logic for Philosophy. Oxford: Oxford University Press, 2010. Week IV: Option 2: Believe in Permanent Existence?

A relatively new way of combining modal logics is to consider their products. The main application of these product logics lies in the description of parallel computing processes. Axiomatics and decidability of the validity problem have been rather extensively investigated and many logics behave well in these respects. In this paper we look at the product construction from a computational complexity point of view. We show that in many cases there is a drastic increase in complexity, e.g., all products containing the finite S5×S5 products as models have an nexptime-hard satisfaction problem. Products with a functional modality however do not lead to an increase in complexity. For the products K× S5 and S5× S5, we provide a matching upper bound. Combining (modal) logics is a very active area, witness e.g., [4] and the book [1]. A rather special way of combining two modal logics is to consider their products. This approach started with [20], and has recently been developed in great detail in [5]. In temporal logic, products of two logics have been used to describe the temporal logic of intervals (cf. the “product treatment” of the system HS from [8] in [15]: Chapter 4 and the references therein). Almost all products of temporal logics are undecidable, sometimes the validities are not even recursively enumerable [8, 22]. Products of modal (and modal and temporal) logics have applications in the theory of parallel computing [18]. Here we are concerned with the general mathematical theory of products of modal logics, in particular the complexity of several natural decision problems, like the validity problem. With respect to (Hilbert style) axiomatizability a lot of general results are obtained in [5], cf e.g., Theorem 5.7. That paper also contains decidability results for a large number of cases. The general trend for these results is that they are rather hard to prove, but become a lot easier if one of the logics is S5, though even then the filtration arguments are rather involved, and lead to models whose size is in general double exponential in the length of the formula which is to be satisfied. The upper bounds we obtain from these proofs are very bad, in the general case (when none of the logics is S5), the decision-algorithm is non-elementary, and when one of the logics is S5 we only obtain a non-deterministic double exponential time upper bound for the satisfaction problem. Questions concerning computational complexity, have hitherto not been addressed, and we will make a start here. The overall trend is that these logics have a very bad complexity for the satisfaction problem: in many simple cases it is nexptime-hard. Also, even if the satisfaction problem is decidable, the problem whether for a formula φ there exists a model The author is supported by UK EPSRC grant No. GR/K54946.