Combining Possibilities and Negations

Combining non-classical (or sub-classical) logics is not easy, but it is very interesting. In this paper, we combine nonclassical logics of negation and possibility (in the presence of conjunction and disjunction), and then we combine the resulting systems with intuitionistic logic. We will find that Kracht's results on the undecidability of classical modal logics generalise to a non-classical setting. We will also see conditions under which intuitionistic logic can be combined with a non-intuitionistic negation without corrupting the intuitionistic fragment of the logic.     

Temporal and atemporal truth in intuitionistic mathematics

In section 1 we argue that the adoption of a tenseless notion of truth entails a realistic view of propositions and provability. This view, in turn, opens the way to the intelligibility of theclassical meaning of the logical constants, and consequently is incompatible with the antirealism of orthodox intuitionism. In section 2 we show how what we call the potential intuitionistic meaning of the logical constants can be defined, on the one hand, by means of the notion of atemporal provability and, on the other, by means of the operator K of epistemic logic. Intuitionistic logic, as reconstructed within this perspective, turns out to be a part of epistemic logic, so that it loses its traditional foundational role, antithetic to that of classical logic. In section 3 we uphold the view that certain consequences of the adoption of atemporal notion of truth, despite their apparent oddity, are quite acceptable from an antirealist point of view.     

Reflections on “difficult” embeddings

The main purpose of this note is to present difficult embeddings of minimal and full intuitionistic logic into classical linear logic, and to prove their soundness and faithfulness. Moreover, it is also pointed out that Girard's translation of intuitionistic logic into classical linear logic is provably equivalent to one of the translations considered in this paper.     

Intuitionistic Epistemic Logic,Kripke Models and Fitch’s Paradox

The present work is motivated by two questions. (1) What should an intuitionistic epistemic logic look like? (2) How should one interpret the knowledge operator in a Kripke-model for it? In what follows we outline an answer to (2) and give a model-theoretic definition of the operator K. This will shed some light also on (1), since it turns out that K, defined as we do, fulfills the properties of a necessity operator for a normal modal logic. The interest of our construction also lies in a better insight into the intuitionistic solution to Fitch’s paradox, which is discussed in the third section. In particular we examine, in the light of our definition, DeVidi and Solomon’s proposal of formulating the verification thesis as \(\phi \rightarrow \neg \neg K\phi\). We show, as our main result, that this definition excapes the paradox, though it is validated only under restrictive conditions on the models.     

The Pleasures of Anticipation: Enriching Intuitionistic Logic

We explore a relation we call anticipation between formulas, where A anticipates B (according to some logic) just in case B is a consequence (according to that logic, presumed to support some distinguished implicational connective ) of the formula AB. We are especially interested in the case in which the logic is intuitionistic (propositional) logic and are much concerned with an extension of that logic with a new connective, written as a, governed by rules which guarantee that for any formula B, aB is the (logically) strongest formula anticipating B. The investigation of this new logic, which we call ILa, will confront us on several occasions with some of the finer points in the theory of rules and with issues in the philosophy of logic arising from the proposed explication of the existence of a connective (with prescribed logical behaviour) in terms of the conservative extension of a favoured logic by the addition of such a connective. Other points of interest include the provision of a Kripke semantics with respect to which ILa is demonstrably sound, deployed to establish certain unprovability results as well as to forge connections with C. Rauszer's logic of dual intuitionistic negation and dual intuitionistic implication, and the isolation of two relations (between formulas), head-implication and head-linkage, which, though trivial in the setting of classical logic, are of considerable significance in the intuitionistic context.     

Quantized Linear Logic,Involutive Quantales and Strong Negation

A new logic, quantized intuitionistic linear logic (QILL), is introduced, and is closely related to the logic which corresponds to Mulvey and Pelletier's (commutative) involutive quantales. Some cut-free sequent calculi with a new property quantization principle and some complete semantics such as an involutive quantale model and a quantale model are obtained for QILL. The relationship between QILL and Wansing's extended intuitionistic linear logic with strong negation is also observed using such syntactical and semantical frameworks.     

Intuitionistic mathematics does not needex falso quodlibet

We define a system IR of first-order intuitionistic relevant logic. We show that intuitionistic mathematics (on the assumption that it is consistent) can be relevantized, by virtue of the following metatheorem: any intuitionistic proof of A from a setX of premisses can be converted into a proof in IR of eitherA or absurdity from some subset ofX. Thus IR establishes the same inconsistencies and theorems as intuitionistic logic, and allows one to prove every intuitionistic consequence of any consistent set of premisses.This paper grew out of discussion of a survey talk, on earlier work, that I gave to the 5th A.N.U. Paraconsistency Conference in January 1988. I am greatly indebted to the suggestion by Michael MacRobbie on that occasion that I investigate the so-called non-Ketonen form of the sequent rule for on the right. That suggestion inspired the correspondingly modified rule of Introduction in the system of natural deduction given above.     

Information-seeking dialogues: Some of their logical properties

The dialogical games introduced in Jaakko Hintikka, Information-Seeking Dialogues: A Model, (Erkenntnis, vol. 14, 1979) are studied here to answer the question as to what the natural logic or the logic of natural language is. In a natural language certain epistemic elements are not explicitly indicated, but they determine which inference rules are valid. By means of dialogical games, the question is answered: all classical first-order rules have to be modified in the same way in which some of them are modified in the transition to intuitionistic logic. (Furthermore, in some cases quantificational rules have to be modified further.) The rules that are left unmodified by intuitionists are applicable only to the output of certain game rules, but not to others. In. this sense, neither classical nor yet intuitionistic logic is the logic of natural language. We need a new type of nonclassical logic, justified by our information-seeking dialogues.     

Decidability of Quantified Propositional Intuitionistic Logic and S4 on Trees of Height and Arity ≤ω

Quantified propositional intuitionistic logic is obtained from propositional intuitionistic logic by adding quantifiers p, p, where the propositional variables range over upward-closed subsets of the set of worlds in a Kripke structure. If the permitted accessibility relations are arbitrary partial orders, the resulting logic is known to be recursively isomorphic to full second-order logic (Kremer, 1997). It is shown that if the Kripke structures are restricted to trees of at height and width at most , the resulting logics are decidable. This provides a partial answer to a question by Kremer. The result also transfers to modal S4 and some Gödel–Dummett logics with quantifiers over propositions.     

Constructing a continuum of predicate extensions of each intermediate propositional logic

Wajsberg and Jankov provided us with methods of constructing a continuum of logics. However, their methods are not suitable for super-intuitionistic and modal predicate logics. The aim of this paper is to present simple ways of modification of their methods appropriate for such logics. We give some concrete applications as generic examples. Among others, we show that there is a continuum of logics (1) between the intuitionistic predicate logic and the logic of constant domains, (2) between a predicate extension ofS4 andS4 with the Barcan formula. Furthermore, we prove that (3) there is a continuum of predicate logics with equality whose equality-free fragment is just the intuitionistic predicate logic.Dedicated to the memory of the late Professor S. MaeharaThis research was supported in part by Grant-in Aid for Encouragement of Young Scientists No. 06740140, Ministry of Education, Science and Culture, Japan.Presented byHiroakira Ono     

An application of Rieger-Nishimura formulas to the intuitionistic modal logics

The main results of the paper are the following: For each monadic prepositional formula which is classically true but not intuitionistically so, there is a continuum of intuitionistic monotone modal logics L such that L+ is inconsistent.There exists a consistent intuitionistic monotone modal logic L such that for any formula of the kind mentioned above the logic L+ is inconsistent.There exist at least countably many maximal intuitionistic monotone modal logics.The author appreciates very much referees' suggestions which helped to improve the exposition.     

Intuitionist logic — Subsystem of,extension of,or rival to,classical logic?

Strictly speaking, intuitionistic logic is not a modal logic. There are, after all, no modal operators in the language. It is a subsystem of classical logic, not [like modal logic] an extension of it. But... (thus Fitting, p. 437, trying to justify inclusion of a large chapter on intuitionist logic in a book that is largely about modal logics).     

Completeness of quantum logic

This paper is based on a semantic foundation of quantum logic which makes use of dialog-games. In the first part of the paper the dialogic method is introduced and under the conditions of quantum mechanical measurements the rules of a dialog-game about quantum mechanical propositions are established. In the second part of the paper the quantum mechanical dialog-game is replaced by a calculus of quantum logic. As the main part of the paper we show that the calculus of quantum logic is complete and consistent with respect to the dialogic semantics. Since the dialog-game does not involve the excluded middle the calculus represents a calculus of effective (intuitionistic) quantum logic.In a forthcoming paper it is shown that this calculus is equivalent to a calculus of sequents and more interestingly to a calculus of propositions. With the addition of the excluded middle the latter calculus is a model for the lattice of subspaces of a Hilbert space.On leave of absence from the Institut für Theoretische Physik der Universität zu Köln, W.-Germany.     

On Proof Terms and Embeddings of Classical Substructural Logics

There is an intimate connection between proofs of the natural deduction systems and typed lambda calculus. It is well-known that in simply typed lambda calculus, the notion of formulae-as-types makes it possible to find fine structure of the implicational fragment of intuitionistic logic, i.e., relevant logic, BCK-logic and linear logic. In this paper, we investigate three classical substructural logics (GL, GLc, GLw) of Gentzen's sequent calculus consisting of implication and negation, which contain some of the right structural rules. In terms of Parigot's -calculus with proper restrictions, we introduce a proof term assignment to these classical substructural logics. According to these notions, we can classify the -terms into four categories. It is proved that well-typed GLx--terms correspond to GLx proofs, and that a GLx--term has a principal type if stratified where x is nil, c, w or cw. Moreover, we investigate embeddings of classical substructural logics into the corresponding intuitionistic substructural logics. It is proved that the Gödel-style translations of GLx--terms are embeddings preserving substructural logics. As by-products, it is obtained that an inhabitation problem is decidable and well-typed GLx--terms are strongly normalizable.     

AN INTUITIONISTIC CHARACTERIZATION OF CLASSICAL LOGIC

By introducing the intensional mappings and their properties, we establish a new semantical approach of characterizing intermediate logics. First prove that this new approach provides a general method of characterizing and comparing logics without changing the semantical interpretation of implication connective. Then show that it is adequate to characterize all Kripke_complete intermediate logics by showing that each of these logics is sound and complete with respect to its (unique) ‘weakest characterization property’ of intensional mappings. In particular, we show that classical logic has the weakest characterization property , which is the strongest among all possible weakest characterization properties of intermediate logics. Finally, it follows from this result that a translation is an embedding of classical logic into intuitionistic logic, iff. its semantical counterpart has the property .      

On structural completeness of implicational logics

We consider the notion of structural completeness with respect to arbitrary (finitary and/or infinitary) inferential rules. Our main task is to characterize structurally complete intermediate logics. We prove that the structurally complete extension of any pure implicational in termediate logic C can be given as an extension of C with a certain family of schematically denned infinitary rules; the same rules are used for each C. The cardinality of the family is continuum and, in the case of (the pure implicational fragment of) intuitionistic logic, the family cannot be reduced to a countable one. It means that the structurally complete extension of the intuitionistic logic is not countably axiomatizable by schematic rules.This work was supported by the Polish Academy of Sciences, CPBP 08.15, Struktura logiczna rozumowa niesformalizowanych.     

Deduction theorems for weak implicational logics

The standard deduction theorem or introduction rule for implication, for classical logic is also valid for intuitionistic logic, but just as with predicate logic, other rules of inference have to be restricted if the theorem is to hold for weaker implicational logics.In this paper we look in detail at special cases of the Gentzen rule for and show that various subsets of these in effect constitute deduction theorems determining all the theorems of many well known as well as not well known implicational logics. In particular systems of rules are given which are equivalent to the relevance logics E,R, T, P-W and P-W-I.     

The Classification of Propositional Calculi

We discuss Smirnovs problem of finding a common background for classifying implicational logics. We formulate and solve the problem of extending, in an appropriate way, an implicational fragment H of the intuitionistic propositional logic to an implicational fragment TV of the classical propositional logic. As a result we obtain logical constructions having the form of Boolean lattices whose elements are implicational logics. In this way, whole classes of new logics can be obtained. We also consider the transition from implicational logics to full logics. On the base of the lattices constructed, we formulate the main classification principles for propositional logics.     

Gentzen-type formulation of the prepositional logic LQ

We give a Gentzen-type formulation GQ for the intermediate logic LQ and prove the cut-elimination theorem on it, where LQ is the propositional logic obtained from the intuitionistic propositional logic LI by adding the axioms of the form AV A.     

Connectification forn-contraction

In this paper, we introduce connectification operators for intuitionistic and classical linear algebras corresponding to linear logic and to some of its extensions withn-contraction. In particular,n-contraction (n2) is a version of the contraction rule, wheren+1 occurrences of a formula may be contracted ton occurrences. Since cut cannot be eliminated from the systems withn-contraction considered most of the standard proof-theoretic techniques to investigate meta-properties of those systems are useless. However, by means of connectification we establish the disjunction property for both intuitionistic and classical affine linear logics withn-contraction.Presented byHiroakira Ono