Intuitionistic Autoepistemic Logic

In this paper we address the problem of combining a logic with nonmonotonic modal logic. In particular we study the intuitionistic case. We start from a formal analysis of the notion of intuitionistic consistency via the sequent calculus. The epistemic operator M is interpreted as the consistency operator of intuitionistic logic by introducing intuitionistic stable sets. On the basis of a bimodal structure we also provide a semantics for intuitionistic stable sets.     

Quantum Logic in Intuitionistic Perspective

In their seminal paper Birkhoff and von Neumann revealed the following dilemma:[ ] whereas for logicians the orthocomplementation properties of negation were the ones least able to withstand a critical analysis, the study of mechanics points to the distributive identities as the weakest link in the algebra of logic.In this paper we eliminate this dilemma, providing a way for maintaining both. Via the introduction of the "missing" disjunctions in the lattice of properties of a physical system while inheriting the meet as a conjunction we obtain a complete Heyting algebra of propositions on physical properties. In particular there is a bijective correspondence between property lattices and propositional lattices equipped with a so called operational resolution, an operation that exposes the properties on the level of the propositions. If the property lattice goes equipped with an orthocomplementation, then this bijective correspondence can be refined to one with propositional lattices equipped with an operational complementation, as such establishing the claim made above. Formally one rediscovers via physical and logical considerations as such respectively a specification and a refinement of the purely mathematical result by Bruns and Lakser (1970) on injective hulls of meet-semilattices. From our representation we can derive a truly intuitionistic functional implication on property lattices, as such confronting claims made in previous writings on the matter. We also make a detailed analysis of disjunctivity vs. distributivity and finitary vs. infinitary conjunctivity, we briefly review the Bruns-Lakser construction and indicate some questions which are left open.     

Basic Intuitionistic Conditional Logic

Journal of Philosophical Logic - Conditional logics have traditionally been intended to formalize various intuitively correct modes of reasoning involving (counterfactual) conditional expressions...     

Weakly Intuitionistic Quantum Logic

In this article von Neumann’s proposal that in quantum mechanics projections can be seen as propositions is followed. However, the quantum logic derived by Birkhoff and von Neumann is rejected due to the failure of the law of distributivity. The options for constructing a distributive logic while adhering to von Neumann’s proposal are investigated. This is done by rejecting the converse of the proposal, namely, that propositions can always be seen as projections. The result is a weakly Heyting algebra for describing the language of quantum mechanics.     

On an Intuitionistic Modal Logic

In this paper we consider an intuitionistic variant of the modal logic S4 (which we call IS4). The novelty of this paper is that we place particular importance on the natural deduction formulation of IS4— our formulation has several important metatheoretic properties. In addition, we study models of IS4— not in the framework of Kirpke semantics, but in the more general framework of category theory. This allows not only a more abstract definition of a whole class of models but also a means of modelling proofs as well as provability.     

Truth-Maker Semantics for Intuitionistic Logic

I propose a new semantics for intuitionistic logic, which is a cross between the construction-oriented semantics of Brouwer-Heyting-Kolmogorov and the condition-oriented semantics of Kripke. The new semantics shows how there might be a common semantical underpinning for intuitionistic and classical logic and how intuitionistic logic might thereby be tied to a realist conception of the relationship between language and the world.     

On Sahlqvist Formulas in Relevant Logic

This paper defines a Sahlqvist fragment for relevant logic and establishes that each class of frames in the Routley-Meyer semantics which is definable by a Sahlqvist formula is also elementary, that is, it coincides with the class of structures satisfying a given first order property calculable by a Sahlqvist-van Benthem algorithm. Furthermore, we show that some classes of Routley-Meyer frames definable by a relevant formula are not elementary.     

Expressing Second-order Sentences in Intuitionistic Dependence Logic

Intuitionistic dependence logic was introduced by Abramsky and Väänänen [1] as a variant of dependence logic under a general construction of Hodges’ (trump) team semantics. It was proven that there is a translation from intuitionistic dependence logic sentences into second order logic sentences. In this paper, we prove that the other direction is also true, therefore intuitionistic dependence logic is equivalent to second order logic on the level of sentences.     

Undecidability of First-Order Modal and Intuitionistic Logics with Two Variables and One Monadic Predicate Letter

Studia Logica - We prove that the positive fragment of first-order intuitionistic logic in the language with two individual variables and a single monadic predicate letter, without functional...     

Lambda Calculus and Intuitionistic Linear Logic

The introduction of Linear Logic extends the Curry-Howard Isomorphism to intensional aspects of the typed functional programming. In particular, every formula of Linear Logic tells whether the term it is a type for, can be either erased/duplicated or not, during a computation. So, Linear Logic can be seen as a model of a computational environment with an explicit control about the management of resources.This paper introduces a typed functional language ! and a categorical model for it.The terms of ! encode a version of natural deduction for Intuitionistic Linear Logic such that linear and non linear assumptions are managed multiplicatively and additively, respectively. Correspondingly, the terms of ! are built out of two disjoint sets of variables. Moreover, the -abstractions of ! bind variables and patterns. The use of two different kinds of variables and the patterns allow a very compact definition of the one-step operational semantics of !, unlike all other extensions of Curry-Howard Isomorphism to Intuitionistic Linear Logic. The language ! is Church-Rosser and enjoys both Strong Normalizability and Subject Reduction.The categorical model induces operational equivalences like, for example, a set of extensional equivalences.The paper presents also an untyped version of ! and a type assignment for it, using formulas of Linear Logic as types. The type assignment inherits from ! all the good computational properties and enjoys also the Principal-Type Property.     

A Translation of Intuitionistic Predicate Logic into Basic Predicate Logic

Basic Predicate Logic, BQC, is a proper subsystem of Intuitionistic Predicate Logic, IQC. For every formula in the language {, , , , , , }, we associate two sequences of formulas 0,1,... and 0,1,... in the same language. We prove that for every sequent , there are natural numbers m, n, such that IQC , iff BQC n m. Some applications of this translation are mentioned.     

Absolute Probability Functions for Intuitionistic Propositional Logic

Provided here is a characterisation of absolute probability functions for intuitionistic (propositional) logic L, i.e. a set of constraints on the unary functions P from the statements of L to the reals, which insures that (i) if a statement A of L is provable in L, then P(A) = 1 for every P, L's axiomatisation being thus sound in the probabilistic sense, and (ii) if P(A) = 1 for every P, then A is provable in L, L's axiomatisation being thus complete in the probabilistic sense. As there are theorems of classical (propositional) logic that are not intuitionistic ones, there are unary probability functions for intuitionistic logic that are not classical ones. Provided here because of this is a means of singling out the classical probability functions from among the intuitionistic ones.     

Inverse Images of Box Formulas in Modal Logic

We investigate, for several modal logics but concentrating on KT, KD45, S4 and S5, the set of formulas B for which ${\square B}$ is provably equivalent to ${\square A}$ for a selected formula A (such as p, a sentence letter). In the exceptional case in which a modal logic is closed under the (‘cancellation’) rule taking us from ${\square C \leftrightarrow \square D}$ to ${C \leftrightarrow D}$ , there is only one formula B, to within equivalence, in this inverse image, as we shall call it, of ${\square A}$ (relative to the logic concerned); for logics for which the intended reading of “ ${\square}$ ” is epistemic or doxastic, failure to be closed under this rule indicates that from the proposition expressed by a knowledge- or belief-attribution, the propositional object of the attitude in question cannot be recovered: arguably, a somewhat disconcerting situation. More generally, the inverse image of ${\square A}$ may comprise a range of non-equivalent formulas, all those provably implied by one fixed formula and provably implying another—though we shall see that for several choices of logic and of the formula A, there is not even such an ‘interval characterization’ of the inverse image (of ${\square A}$ ) to be found.     

The Lambek Calculus Extended with Intuitionistic Propositional Logic

We present sound and complete semantics and a sequent calculus for the Lambek calculus extended with intuitionistic propositional logic.     

Intuitionistic Propositional Logic without 'Contraction' but with 'Reductio'

Routley-Meyer type relational complete semantics are constructed for intuitionistic contractionless logic with reductio. Different negation completions of positive intuitionistic logic without contraction are treated in a systematical, unified and semantically complete setting.     

An Intuitionistic Completeness Theorem for Classical Predicate Logic

This paper presents an intuitionistic proof of a statement which under a classical reading is logically equivalent to Gödel’s completeness theorem for classical predicate logic.     

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.