On the definition of the classical connectives and quantifiers

Classical logic is embedded into constructive logic, through a definition of the classical connectives and quantifiers in terms of the constructive ones. The history of the notion of constructivity started with a dispute on the deduction rules that one should or should not use to prove a theorem. Depending on the rules accepted by the ones and the others, the proposition P ∨ ¬P , for instance, had a proof or not. A less controversial situation was reached with a classification of proofs, and it became possible to agree that this proposition had a classical proof but no constructive proof. An alternative is to use the idea of Hilbert and Poincaré that axioms and deduction rules define the meaning of the symbols of the language and it is then possible to explain that some judge the proposition P ∨ ¬P true and others do not because they do not assign the same meaning to the symbols ∨, ¬, etc. The need to distinguish several meanings of a common word is usual in mathematics. For instance the proposition “there exists a number x such that 2x = 1” is true of false depending on whether the word “number” means “natural number” or “real number”. Even for logical connectives, the word “or” has to be disambiguated into inclusive and exclusive. Taking this idea seriously, we should not say that the proposition P ∨ ¬P has a classical proof but no constructive proof, but we should say that the proposition P ∨ ¬P has a proof and the proposition P ∨¬P does not, that is we should introduce two symbols for each connective and quantifier, for instance a symbol ∨ for the constructive disjunction and a symbol ∨ for the classical one, instead of introducing two judgments: “has a classical proof” and “has a constructive proof”. We should also be able to address the question of the provability of mixed propositions and, for instance, express that the proposition (¬(P ∧Q))⇒ (¬P ∨ ¬Q) has a proof. The idea that the meaning of connectives and quantifiers is expressed by the deduction rules leads to propose a logic containing all the constructive and ∗INRIA, 23 avenue d’Italie, CS 81321, 75214 Paris Cedex 13, France. [email protected]. classical connectives and quantifiers and deduction rules such that a proposition containing only constructive connectives and quantifiers has a proof in this logic if and only if it has a proof in constructive logic and a proposition containing only classical connectives and quantifiers has a proof in this logic if and only if it has a proof in classical logic. Such a logic containing classical, constructive, and also linear, connectives and quantifiers has been proposed by J.-Y. Girard [3]. This logic is a sequent calculus with unified sequents that contain a linear zone and a classical zone and rules treating differently propositions depending on the zone they belong. Our goal in this paper is slightly different, as we want to define the meaning of a small set of primitive connectives and quantifiers with deduction rules and define the others explicitly, in the same way the exclusive or is explicitly defined in terms of conjunction, disjunction and negation: A⊕ B = (A ∧ ¬B) ∨ (¬A ∧ B). A first step in this direction has been made by Gödel [4] who defined a translation of constructive logic into classical logic, and Kolmogorov [7], Gödel [5], and Gentzen [2] who defined a translation of classical logic into constructive logic. As the first translation requires a modal operator, we shall focus on the second. This leads to consider constructive connectives and quantifiers as primitive and search for definitions of the classical ones. Thus, we want to define classical connectives and quantifiers >, ⊥, ¬, ∧, ∨, ⇒, ∀, and ∃ and embed classical propositions into constructive logic with a function ‖ ‖ defined as follows. x