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A truthmaker for a proposition P is exact if it contains nothing irrelevant to P . What are the exact truthmakers for necessitated propositions? This paper makes progress on this issue by showing how to extend Fine’s truthmaker semantics for intuitionistic logic to an exact truthmaker semantics for intuitionistic modal logic. The project is of interest also to the classical logician: while all distinctively classical theorems may be true, they differ from the intuitionistic ones in how they are made true. This sheds new light on the status of the T and B axioms.

With help of a compact Prolog-based theorem prover for Intuitionistic Propositional Logic, we synthesize minimal assumptions under which a given formula formula becomes a theorem. After applying our synthesis algorithm to cover basic abductive reasoning mechanisms, we synthesize conjunctions of literals that mimic rows of truth tables in classical or intermediate logics and we abduce conditional hypotheses that turn the theorems of classical or intermediate logics into theorems in intuitionistic logic. One step further, we generalize our abductive reasoning mechanism to synthesize more expressive sequent premises using a minimal set of canonical formulas, to which arbitrary formulas in the calculus can be reduced while preserving their provability. Organized as a self-contained literate Prolog program, the paper supports interactive exploration of its content and ensures full replicability of our results.

Motivated by the definition of semi-Nelson algebras, a propositional calculus called semi-intuitionistic logic with strong negation is introduced and proved to be complete with respect to that class of algebras. An axiomatic extension is proved to have as algebraic semantics the class of Nelson algebras.

Bi-intuitionistic logic is the result of adding the dual of intuitionistic implication to intuitionistic logic. In this note, we characterize the expressive power of this logic by showing that the first order formulas equivalent to translations of bi-intuitionistic propositional formulas are exactly those preserved under bi-intuitionistic directed bisimulations. The proof technique is originally due to Lindström and, in contrast to the most common proofs of this kind of result, it does not use the machinery of neither saturated models nor elementary chains.

Semi-intuitionistic logic is the logic counterpart to semi-Heyting algebras, which were defined by H. P. Sankappanavar as a generalization of Hey ting algebras. We present a new, more streamlined set of axioms for semi-intuitionistic logic, which we prove translationally equivalent to the original one. We then study some formulas that define a semi-Heyting implication, and specialize this study to the case in which the formulas use only the lattice operators and the intuitionistic implication. We prove then that all the logics thus obtained are equivalent to intuitionistic logic, and give their Kripke semantics.

This paper considers logics which are formally dual to intuitionistic logic in order to investigate a co-constructive logic for proofs and refutations. This is philosophically motivated by a set of problems regarding the nature of constructive truth, and its relation to falsity. It is well known both that intuitionism can not deal constructively with negative information, and that defining falsity by means of intuitionistic negation leads, under widely-held assumptions, to a justification of bivalence. For example, we do not want to equate falsity with the non-existence of a proof since this would render a statement such as “pi is transcendental” false prior to 1882. In addition, the intuitionist account of negation as shorthand for the derivation of absurdity is inadequate, particularly outside of purely mathematical contexts. To deal with these issues, I investigate the dual of intuitionistic logic, co-intuitionistic logic, as a logic of refutation, alongside intuitionistic logic of proofs. Direct proof and refutation are dual to each other, and are constructive, whilst there also exist syntactic, weak, negations within both logics. In this respect, the logic of refutation is weakly paraconsistent in the sense that it allows for statements for which, neither they, nor their negation, are refuted. I provide a proof theory for the co-constructive logic, a formal dualizing map between the logics, and a Kripke-style semantics. This is given an intuitive philosophical rendering in a re-interpretation of Kolmogorov's logic of problems.

The variety Sℋ of semi-Heyting algebras was introduced by H. P. Sankappanavar (in: Proceedings of the 9th \"Dr. Antonio A. R. Monteiro\" Congress, Universidad Nacional del Sur, Bahía Blanca, 2008) [13] as an abstraction of the variety of Heyting algebras. Semi-Heyting algebras are the algebraic models for a logic HsH, known as semi-intuitionistic logic, which is equivalent to the one defined by a Hilbert style calculus in Cornejo (Studia Logica 98(1-2): 9-25, 2011) [6]. In this article we introduce a Gentzen style sequent calculus GsH for the semi-intuitionistic logic whose associated logic GsH is the same as HsH. The advantage of this presentation of the logic is that we can prove a cutelimination theorem for GsH that allows us to prove the decidability of the logic. As a direct consequence, we also obtain the decidability of the equational theory of semi-Heyting algebras.

Gödel logic (alias Dummett logic) is the extension of intuitionistic logic by the linearity axiom. Symmetric Gödel logic is a logical system, the language of which is an enrichment of the language of Gödel logic with their dual logical connectives. Symmetric Gödel logic is the extension of symmetric intuitionistic logic (L. Esakia, C. Rauszer). The proof-intuitionistic calculus, the language of which is an enrichment of the language of intuitionistic logic by modal operator was investigated by Kuznetsov and Muravitsky. Bimodal symmetric Gödel logic is a logical system, the language of which is an enrichment of the language of Gödel logic with their dual logical connectives and two modal operators. Bimodal symmetric Gödel logic is embedded into an extension of (bimodal) Gödel–Löb logic (provability logic), the language of which contains disjunction, conjunction, negation and two (conjugate) modal operators. The variety of bimodal symmetric Gödel algebras, that represent the algebraic counterparts of bimodal symmetric Gödel logic, is investigated. Description of free algebras and characterization of projective algebras in the variety of bimodal symmetric Gödel algebras is given. All finitely generated projective bimodal symmetric Gödel algebras are infinite, while finitely generated projective symmetric Gödel algebras are finite.

Ewald (J Symbolic Logic 51(1):166–179, 1986 ) considered tense operators G , H , F and P on intuitionistic propositional calculus and constructed an intuitionistic tense logic system called IKt. The aim of this paper is to give an algebraic axiomatization of the IKt system. We will also show that the algebraic axiomatization given by Chajda (Cent Eur J Math 9(5):1185–1191, 2011 ) of the tense operators P and F in intuitionistic logic is not in accordance with the Halmos definition of existential quantifiers. In this paper, we will study the IKt variety of IKt-algebras. First, we will introduce some examples and we will prove some properties. Next, we will prove that the IKt system has IKt-algebras as algebraic counterpart. We will also describe a discrete duality for IKt-algebras bearing in mind the results indicated by Orłowska and Rewitzky (Fundam Inform 81(1–3):275–295, 2007 ) for Heyting algebras. We will also get a general construction of tense operators on a complete Heyting algebra, which is a power lattice via the so-called Heyting frame. Finally, we will introduce the notion of tense deductive system which allowed us both to determine the congruence lattice in an IKt-algebra and to characterize simple and subdirectly irreducible algebras of the IKt variety.

We investigate monadic fragments of Intuitionistic Control Logic (ICL), which is obtained from Intuitionistic Propositional Logic (IPL) by extending language of IPL by a constant distinct from intuitionistic constants. In particular we present the complete description of purely negational fragment and show that most of monadic fragments are finite.