On Constructive Connectives and Systems

There are two traditions concerning the definition and characterization of logical connectives. The better known one is the semantic tradition, which is based on the idea that an n-ary connective ⋄ is defined by the conditions which make a sentence of the form ⋄(ϕ 1 , . . . , ϕ n ) true. The other is the proof-theoretic tradition (originated from [9] -see e.g. [14] for discussions and references). This tradition implicitly divides the connectives into basic connectives and compound connectives, where the latter are defined in terms of the basic ones. The meaning of a basic connective, in turn, is determined by a set of derivation rules which are associated with it. Here one usually has in mind a natural deduction or a sequent system, in which every logical rule is an introduction rule (or perhaps an elimination rule, in the case of natural deduction) of some unique connective. However, it is well-known that not every set of rules can be taken as a definition of a basic connective. A minimal requirement is that whenever some sentence involving exactly one basic connective is provable, then it has a proof which involves no other connectives. In "normal" sequent systems, in which every rule except cut has the subformula property, this condition is guaranteed by a cut-elimination theorem. Therefore only sequent systems for which such a theorem obtains are considered as useful for defining connectives.

In [3] the semantic and the proof-theoretic traditions were shown to be equivalent for a a large family of what may be called semi-classical connectives (which includes all the classical connectives, as well as many others). In these papers multiple-conclusion canonical (= ‘ideal’) propositional rules and systems were defined in precise terms. A simple coherence criterion for the non-triviality of such a system was given, and it was shown that a canonical system is coherent if and only if it admits cut-elimination. Semi-classical connectives were characterized using canonical rules in coherent canonical systems. In addition, each of these connectives was given a semantic characterization. This characterization uses twovalued non-deterministic truth-tables -a natural generalization of the classical truth-tables. Moreover, it was shown there how to translate a semantic definition of a connective to a corresponding proof-theoretic one, and vice-versa. 1 In this paper we attempt to provide similar characterizations for the class of basic constructive connectives.

What exactly is a constructive connective? Several different answers to this question have been given in the literature, each adopting either of the traditions described above (but not both!). Thus in [12] McCullough gave a purely semantic characterization of constructive connectives, using a generalization of the Kripke-style semantics for intuitionistic logic. On the other hand Bowen suggested in [7] a quite natural proof-theoretic criterion for (basic) constructivity: an n-ary connective ⋄, defined by a set of sequent rules, is constructive if whenever a sequent of the form ⇒ ⋄(ϕ 1 , . . . , ϕ n ) is provable, then it has a proof ending by an application of one of right-introduction rules for ⋄.

In what follows we generalize and unify the syntactic and the semantic approaches by adapting the ideas and methods used in [3]. The crucial observation on which our theory is based is that every connective of a “normal” single-conclusion sequent system that admits cut-elimination is necessarily constructive according to Bowen’s criterion (because without using cuts, the only way to derive ⇒ ⋄(ϕ 1 , . . . , ϕ n ) in such a system is to prove first the premises of one of its right-introduction rules). This indicates that only single-conclusion sequent rules are useful for defining constructive connectives. In addition, for defining basic connectives, only canonical derivation rules (in a sense similar to that used in [3]) should be used. Therefore, our proof-theoretic characterization of basic constructive connectives is done using cut-free single-conclusion canonical systems. These systems are the natural constructive counterparts of the multiple-conclusion canonical systems of [3]. On the other hand, McCullough’s work suggests that an appropriate counterpart of the semantics of nondeterministic truth-tables should be given by a non-deterministic generalization of Kripkestyle semantics.

General single-conclusion canonical rules and systems were first introduced and investigated in [4]. A general non-deterministic Kripke-style semantics for such systems was also developed there, and a constructive necessary and sufficient coherence criterion for their non-triviality was provided. Moreover: it was shown that a system of this kind admits a strong form of cut-elimination iff it is coherent. However, [4] dealt only with strict single-conclusion systems, in which the succeedents of sequents contain exactly one formula. Unfortunately, in such a framework it is imposs

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