Semi-implication: A Chapter in Universal Logic

Intuitionistic logic encompasses the principles of reasoning which were used by L. Brouwer in developing his intuitionistic mathematics, beginning in [Bro07]. Intuitionistic logic can be succinctly described as classical logic without the law ϕ ∨ ¬ϕ of excluded middle. Brouwer observed that this law was abstracted from finite situations and its application to statements about infinite collections is not justified. One of the consequences of the rejection of the law of excluded middle is that every intuitionistic proof of an existential sentence can be effectively transformed into an intuitionistic proof of an instance of that sentence. More precisely, if a formula of the form ∃xϕ(x) without free variables is provable in the intuitionistic predicate logic, then there is a term t without free variables such that ϕ(t) is provable. In particular, if ϕ ∨ ψ is provable, then either ϕ or ψ is provable. In that sense intuitionistic logic may provide a logical basis for constructive reasoning. A formal system of intuitionistic logic was proposed in [Hey30]. A relationship between the classical propositional logic PC and intuitionistic propositional logic INT was proved by Glivenko in [Gli29], namely a PC-formula ϕ is PC-provable if and only if ¬ ¬ϕ is INT-provable. Kripke semantics for intuitionistic logic was developed in [Kri65].

According to Beall and Restall’s logical pluralism, classical logic, relevant logic, and intuitionistic logic are all correct. On this version of logical pluralism, logic is considered to be normative, in the sense that someone who accepts the truth of the premises of a valid argument, is bound to accept the conclusion. So-called collapse arguments are designed to show the incompatibility of the simultaneous acceptance of logical pluralism and the normativity of logic. Caret, however, by proposing logical contextualism, and Blake-Turner and Russell by proposing telic pluralism, have sought to nullify the collapse problem. In the present article, after setting out these two approaches to the collapse problem, we argue that by using the concept of the ‘rationality of beliefs’ in order to frame the canonical purpose of logic, it can be demonstrated that if logical contextualism and telic pluralism are considered as philosophically significant logical pluralisms, a refined version of the collapse argument is still a threat for both of these kinds of logical pluralism.

This article aims to generate all theorems of a given size in the implicational fragment of propositional intuitionistic linear logic. It was presented at the 36th International Conference on Logic Programming (ICLP).

Full classical S5 in natural deduction with weak normalization

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.

The present article is a brief informal survey o$\textit {computability logic}$ (CoL). This relatively young and still evolving nonclassical logic can be characterized as a formal theory of computability in the same sense as classical logic is a formal theory of truth. In a broader sense, being conceived semantically rather than proof-theoretically, CoL is not just a particular theory but an ambitious and challenging long-term project for redeveloping logic. 
 
 In CoL, logical operators stand for operations on computational problems, formulas represent such problems, and their "truth" is seen as algorithmic solvability. In turn, computational problems – understood in their most general, interactive sense – are defined as games played by a machine against its environment, with "algorithmic solvability" meaning existence of a machine which wins the game against any possible behavior of the environment. With this semantics, CoL provides a systematic answer to the question "What can be computed?", just like classical logic is a systematic tool for telling what is true. Furthermore, as it happens, in positive cases "What can be computed" always allows itself to be replaced by "How can be computed", which makes CoL a problem-solving tool. 
 
 CoL is a conservative extension of classical first order logic but is otherwise much more expressive than the latter, opening a wide range of new application areas. It relates to intuitionistic and linear logics in a similar fashion, which allows us to say that CoL reconciles and unifies the three traditions of logical thought (and beyond) on the basis of its natural and "universal" game semantics. 

Here I show that the one-variable fragment of several first-order relevant logics corresponds to certain S5ish extensions of the underlying propositional relevant logic. In particular, given a fairly standard translation between modal and one-variable languages and a permuting propositional relevant logic L, a formula A\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {A}$$\\end{document} of the one-variable fragment is a theorem of LQ (QL) iff its translation is a theorem of L5 (L.5). The proof is model-theoretic. In one direction, semantics based on the Mares-Goldblatt [15] semantics for quantified L are transformed into ternary (plus two binary) relational semantics for S5-like extensions of L (for a general presentation, see Seki [26, 27]). In the other direction, a valuation is given for the full first-order relevant logic based on L into a model for a suitable S5 extension of L. I also discuss this work’s relation to finding a complete axiomatization of the constant domain, non-general frame ternary relational semantics for which RQ is incomplete [11].

The purpose of the research presented in this paper is to extend the author's results on sequent forms of Herbrand theorems for classical and intuitionistic logics onto classical and intuitionistic modal sequent logics. It was found that the technique reported at the SYNASC 2008 symposium and based on the original notions of admissibility and compatibility can satisfactorily be applied for proving Herbrand theorems for the logics under consideration in the sequent form.

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 $\mathbb{VA}$ .

The classical analytic tableau method has been extended successfully to modal logics (see e.g. Fitting 1983; Fitting 1993; Gore 1992) and also to relevant and paraconsistent logics Bloesch 1993a; Bloesch 1993b. The classical connection method has been extended to modal and intuitionistic logics Wallen 1990, and the purpose of this paper is to investigate whether a similar adaptation to relevant logic is possible. A hybrid method is developed for B+, with a specific solution to the “multiplicity problem”, as in the technique of modal semantic diagrams introduced in Hughes and Cresswell 1968. Proofs of soundness and completeness are also given.

Prawitz suggested expanding a natural deduction system for intuitionistic logic to include rules for classical logic constructors, allowing both intuitionistic and classical elements to coexist without losing their inherent characteristics. Looking at the added rules from the point of view of the Gödel-Gentzen translation, led us to propose a general method for the coexistent combination of two logics when a conservative translation exists from one logic (the source) to another (the host). Then we prove that the combined logic is a conservative extension of the original logics, thereby preserving the unique characteristics of each component logic. In this way there is no collapse of one logic into the other in the combination. We also demonstrate that a Gentzen calculus for the combined logic can be induced from a Gentzen calculus for the host logic by considering the translation. This approach applies to semantics as well. We then establish a general sufficient condition for ensuring that the combined logic is both sound and complete. We apply these principles by combining classical and intuitionistic logics capitalizing on the Gödel-Gentzen conservative translation, intuitionistic and S4 modal logics relying on the Gödel-McKinsey-Tarski conservative translation, and classical and Jaśkowski’s paraconsistent logics taking into account the existence of a conservative translation.

Intuitionistic logic has developed as a result of certain philosophical views on the foundation of mathematics, known as intuitionism. Intuitionism was originated by L. E. J. Brouwer in 1908. The first Hilbert style formalization of the intuitionistic logic, formulated as a proof system, is due to A. Heyting (1930). In this chapter we present a Hilbert style proof system I that is equivalent to the Heyting’s original formalization and discuss the relationship between intuitionistic and classical 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.

We extend the relational algebra of Chin and Tarski so that it is multisorted or, as we prefer, typed. Each type supports a local Boolean algebra outfitted with a converse operator. From Lyndon, we know that relation algebras cannot be represented as proper relation algebras where a proper relation algebra has binary relations as elements and the algebra is singly-typed. Here, the intensional conjunction, which was to represent relational composition in Chin and Tarski, spans three different local algebras, thus the term distributed in the title. Since we do not rely on proper relation algebras, we are free to re-express the algebras as typed. In doing so, we allow many different intensional conjunction operators.We construct a typed logic over these algebras, also known as heterogeneous algebras of Birkhoff and Lipson. The logic can be seen as a form of relevance logic with a classical negation connective where the Routley-Meyer star operator is reified as a converse connective in the logic. Relevance logic itself is not typed but our work shows how it can be made so. Some of the properties of classical relevance logic are weakened from Routley-Meyer’s version which is too strong for a logic over relation algebras.

The Tarskian classical relevant logic \(\mathsf {TR}\) arises from Tarski’s work on the foundations of the calculus of relations and on first-order logic restricted to finitely many variables, presented by Tarski and Givant their book, A Formalization of Set Theory without Variables, and summarized in the first nine sections. \(\mathsf {TR}\) is closely related to the well-known logic \(\mathsf {KR}\). Every formula of relevance logic has a corresponding sentence in Tarski’s extended first-order logic of binary relations with operators on the relation symbols. A formula is in \(\mathsf {TR}\) (by definition), or in \(\mathsf {KR}\) (by a theorem), if and only if its corresponding sentence can be proved in first-order logic, using at most four variables, from the assumptions that all binary relations are dense and, for \(\mathsf {TR}\), commute under composition, or, for \(\mathsf {KR}\), are symmetric. The vocabulary of \(\mathsf {TR}\) is the same as the classical relevant logic \(\mathsf {CR^*}\) proposed by Meyer and Routley but \(\mathsf {TR}\) properly contains \(\mathsf {CR^*}\). The frames characteristic for \(\mathsf {TR}\) are the ones that are characteristic for \(\mathsf {CR^*}\) and satisfy an extra frame condition. There are formulas in \(\mathsf {TR}\) (but not in \(\mathsf {CR^*}\)) that correspond to this frame condition and provide a counterexample to a theorem of T. Kowalski. The frames characteristic for \(\mathsf {TR}\), or \(\mathsf {KR}\), are the ones whose complex algebras are integral dense relation algebras that are commutative, or symmetric, respectively. For both classes, the number of isomorphism types grows like the number of isomorphism types of ternary relations. Asymptotic formulas are obtained for both classes. Similar results apply to a hierarchy of logics defined by the number of variables used in the first-order proofs of their corresponding sentences.