Dual Tableaux for Some Logics Based on Intuitionism

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.

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.

Full classical S5 in natural deduction with weak normalization

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.

Prawitz proposed certain notions of proof-theoretic validity and conjectured that intuitionistic logic is complete for them [11, 12]. Considering propositional logic, we present a general framework of five abstract conditions which any proof-theoretic semantics should obey. Then we formulate several more specific conditions under which the intuitionistic propositional calculus (IPC) turns out to be semantically incomplete. Here a crucial role is played by the generalized disjunction principle. Turning to concrete semantics, we show that prominent proposals, including Prawitz’s, satisfy at least one of these conditions, thus rendering IPC semantically incomplete for them. Only for Goldfarb’s [1] proof-theoretic semantics, which deviates from standard approaches, IPC turns out to be complete. Overall, these results show that basic ideas of proof-theoretic semantics for propositional logic are not captured by IPC.

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 paper we present a neighborhood semantics for Intuitionistic Propositional Logic (IPL). We show that for each Kripke model of the logic there is a pointwise equivalent neighborhood model and vice versa. In this way, we establish soundness and completeness of IPL with respect to the neighborhood semantics. The relation between neighborhood and topological semantics are also investigated. Moreover, the notions of bisimulation and n-bisimulation between neighborhood models of IPL are defined naturally and some of their basic properties are proved. We also consider Basic Propositional Logic (BPL), a logic weaker than IPL introduced by Albert Visser, and introduce and study its neighborhood models in the same manner.

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).

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.

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.

In this paper, we present a propositional logic (called mixed logic) containing disjoint copies of minimal, intuitionistic and classical logics. We prove a completeness theorem for this logic with respect to a Kripke semantics. We establish some relations between mixed logic and minimal, intuitionistic and classical logics. We present at the end a sequent calculus version for this logic.

Intermediate logics are logics stronger than intuitionistic logic I but weaker than classical logic C. Most of them are motivated by their semantical characterization. In this chapter we see how the goal-directed approach can be extended to this area by analysing two case-studies. We have seen in the previous chapter that intuitionistic logic is complete with respect to the class of finite Kripke models. One can refine the completeness theorem and show that intuitionistic logic is complete with respect to Kripke models whose possible-worlds structure form a finite tree. Given a Kripke model M = (W, ≤, w 0, V), we can concentrate on the structure (W, ≤, w 0), which is a finite tree, and forget about the evaluation function V for atoms. We write λ(E) for the set of labels occurring in E. Changing the terminology a bit, we will speak about models based on a given finite tree (W, ≤, w 0), since varying V we will have several models based on it. The completeness result can then be re-phrased to assert that intuitionistic logic is complete with respect to the class of finite trees, that is to say, with respect to Kripke models based on finite trees. This change of terminology matters as we are naturally lead to consider subclasses of finite trees and ask what axioms we can add to obtain a characterization of valid formulas in these subclasses. For instance here are two natural subclasses: (1) for any n, the class of finite trees of height ≤ n; a finite tree T = (W, ≤, w 0) is in this class if there are not n + 1 different elements w 0, w 1, ... , w n , such that $${w_o} \leqslant {w_1} \leqslant ... \leqslant {w_n}hold.$$ This means that all chains are of length ≤ n. Valid formulas in these subclasses are axiomatized by the axioms BH n of the next section. (2) for any n, the class of finite trees of width ≤ n; a finite tree T = (W, ≤, w 0) is in this class if there are not n different elements which are pairwise incomparable. Valid formulas in these subclasses are axiomatized by taking the axiom schema $$V_{i = 1}^n({A_i} \to {V_{i \ne j}}{A_j})$$ for finite width trees of width ≤ n. Notice that for n = 2, we have the axiom $$({A_1} \to {A_2}) \vee ({A_2} \to {A_1})$$ which gives the well known logic LC introduced independently by Dummett [1959] and Gödel [1932] that we have already mentioned in the previous chapter. This axiom corresponds to the property of linearity: the models are finite ordered sequence of worlds (i.e. there are not two incomparable points). KeywordsInduction HypothesisClassical LogicProof SystemIntuitionistic LogicKripke ModelThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

"This book describes the aspects of mathematical logic related to computer sciences. The materials adopted in this book are intended to attend to both the peculiarities of logical systems and the requirements of computer science."

Only propositional logics are at issue here. Such a logic is contra-classical in a superficial sense if it is not a sublogic of classical logic, and in a deeper sense, if there is no way of translating its connectives, the result of which translation gives a sublogic of classical logic. After some motivating examples, we investigate the incidence of contra-classicality (in the deeper sense) in various logical frameworks. In Sections 3 and 4 we will encounter, originally as an example of what (in Section 2) we call a contra-classical modal logic, an unusual logic boasting a connective (“demi-negation”) whose double application is equivalent to a single application of the negation connective. Pondering the example points the way to a general characterization of contra-classicality (Theorems 3.3 and 4.6). In an Appendix (Section 5), we look at one alternative to classical logic as the target for such translational assimilation, intuitionistic logic, calling logics which resist the assimilation, in this case, contraintuitionistic. We will show that one such logic is classical logic itself, thereby strengthening a result of Wojcicki’s to the effect that the consequence relation of classical logic cannot be faithfully embedded by any connective-by-connective translation into that of intuitionistic logic. (What the “faithfully” means here is that not only is the translation of anything provable in the ‘source’ logic provable in the ‘target’ logic, but that also anything whose translation is provable in the target logic is provable in the source logic.)

The paper investigates interpretations of propositional and first-order logic in which validity is defined in terms of partial indices; sometimes called possibilities but here understood as non-empty subsets of a set W of possible worlds. Truth at a set of worlds is understood to be truth at every world in the set. If all subsets of W are permitted the logic so determined is classical first-order predicate logic. Restricting allowable subsets and then imposing certain closure conditions provides a modelling for intuitionistic predicate logic. The same semantic interpretation rules are used in both logics for all the operators.