A Constructive Logic with Classical Proofs and Refutations

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].

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.

The fact that classical mathematical proofs of simply existential statements\ncan be read as programs was established by Goedel and Kreisel half a century\nago. But the possibility of extracting useful computational content from\nclassical proofs was taken seriously only from the 1990s on when it was\ndiscovered that proof interpretations based on Goedel's and Kreisel's ideas can\nprovide new nontrivial algorithms and numerical results, and the Curry-Howard\ncorrespondence can be extended to classical logic via programming concepts such\nas continuations and control operators.\n The workshop series "Classical Logic and Computation" aims to support a\nfruitful exchange of ideas between the various lines of research on\ncomputational aspects of classical logic. This volume contains the abstracts of\nthe invited lectures and the accepted contributed papers of the third CL&C\nworkshop which was held jointly with the workshop "Program Extraction and\nConstructive Mathematics" at the University of Brno in August 21-22, 2010, as a\nsatellite of CSL and MFCS. The workshops were held in honour of Helmut\nSchwichtenberg who became "professor emeritus" in September 2010.\n The topics of the papers include the foundations, optimizations and\napplications of proof interpretations such as Hilbert's epsilon substitution\nmethod, Goedel's functional interpretation, learning based realizability and\nnegative translations as well as special calculi and theories capturing\ncomputational and complexity-theoretic aspects of classical logic such as the\nlambda-mu-calculus, applicative theories, sequent-calculi, resolution and\ncut-elimination\n

Full classical S5 in natural deduction with weak normalization

Debates concerning philosophical grounds for the validity of classical and intuitionistic logics often have the very nature of proofs as a point of controversy. The intuitionist advocates for a strictly constructive notion of proof, while the classical logician advocates for a notion which allows the use of non-constructive principles such as reductio ad absurdum. In this paper we show how to coherently combine logical ecumenism and proof-theoretic semantics ( $$\boldsymbol{\mathrm{PtS}}$$ ) by providing not only a medium in which classical and intuitionistic logics coexist, but also one in which their respective notions of proof coexist. Intuitionistic proofs receive the standard treatment of $$\boldsymbol{\mathrm{PtS}}$$ , whereas classical proofs are given a semantics based on ideas by David Hilbert. Furthermore, we advance the state of the art in $$\boldsymbol{\mathrm{PtS}}$$ by introducing a key contribution: treating the absurdity constant $$\bot$$ as an atomic proposition and requiring all bases to be consistent. This treatment is essential for the obtainment of some ecumenical results, and it can also be used in standard intuitionistic $$\boldsymbol{\mathrm{PtS}}$$ . Additionally, we employ normalization techniques to demonstrate the consistency of simulation bases. These innovations provide fresh technical and conceptual insights into the study of bases in $$\boldsymbol{\mathrm{PtS}}$$ .

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. 

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.

Classical logic is symmetric with respect to True and False but intuitionistic logic is not. We introduce and study a conservative extension of first-order intuitionistic logic that is symmetric with respect to True and False.

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.

A formulation of lattice theory as a system of rules added to sequent calculus is given. The analysis of proofs for the contraction-free calculus of classical predicate logic known as G3c extends to derivations with the mathematical rules of lattice theory. It is shown that minimal derivations of quantifier-free sequents enjoy a subterm property: all terms in such derivations are terms in the endsequent.An alternative formulation of lattice theory as a system of rules in natural deduction style is given, both with explicit meet and join constructions and as a relational theory with existence axioms. A subterm property for the latter extends the standard decidable classes of quantificational formulas of pure predicate calculus to lattice theory.

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 $\mathbb{VA}$ .

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.

We consider some advanced topics in logic including fuzzy logic, temporal logic, intuitionist logic, undefined values in logic, logic and AI and theorem provers. Fuzzy logic is an extension of classical logic that acts as a mathematical model for vagueness. Temporal logic is concerned with the expression of properties that have time dependencies. Brouwer and others developed intuitionist logic as the logical foundation for intuitionism, which was a controversial theory of the foundations of mathematics based on a rejection of the law of the excluded middle and an insistence on constructive existence. We discuss several approaches that have been applied to dealing with undefined values that arise with partial functions including the logic of partial functions; Dijkstra’s approach with his cand and cor operators; and Parnas’s approach which preserves a classical two-valued 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.