Variable Sharing in Substructural Logics: an Algebraic Characterization

Abstract

<jats:p>We characterize the non-trivial substructural logics having the variable sharing property as well as its strong version. To this end, we find the algebraic counterparts over varieties of these logical properties.</jats:p>

The (Greatest) Fragment of Classical Logic that Respects the Variable-Sharing Principle (in the FMLA-FMLA Framework)

“…We would like to thank Shawn Standefer for urging us to clarify this issue. As suggested by an anonymous reviewer, for references on this issue see [22] and [2].…”

“…In this vein, although it has been pointed out that the satisfaction of this criterion is only necessary but not sufficient to establish the relevance of a target notion of implication, it could be interesting to consider its satisfaction as an appropriate filter on a previously given independent notion of implication-thus rendering an (at least weakly) relevant subsystem thereof. 2 In this vein we could conceive, for example, filtering Classical Logic (CL, hereafter). Then, although truth-preservation in CL is an unacceptable guide to implication (due to its permeability to irrelevancies in the form of the paradoxes of material and strict implication), it might well be the case that the simultaneous satisfaction of truth-preservation and the Variable-Sharing Principle is an acceptable criterion.…”

“…It is also sometimes customary to think about such a set of valid firstdegree entailments as a logical system on its own right. This can be easily 2 That the Variable-Sharing Principle can be seen as necessary but not sufficient is salient by noticing that there have been many relevant logicians (Anderson and Belnap among them) who rejected implications that are valid in Classical Logic and comply with the Variable-Sharing Principle-e.g. those going from ¬ϕ ∧ (ϕ ∨ ψ) to ψ, or from ϕ ∧ ¬ϕ to ϕ ∧ ψ.…”

“…3 That we choose to denote this fragment by CL VSP can be explained by noting that, in general, we may denote with the Fmla-Fmla fragment of L VSP the subsystem of a given logic L whose valid inferences are only those valid inferences of L that satisfy the Variable-Sharing Principle. 4 That is to say:…”

“…The Fmla-Fmla fragment of a logic L is the restriction of L to what is called, e.g., in[25, p. 108] the Fmla-Fmla framework. That is to say, set of inferences that are valid in such a logic which have exactly one formula as a premise and exactly one formula as a conclusion 4. As an anonymous reviewer points out, this constitutes a slightly different variant of the VSP-a deductive version of the VSP, one may claim.…”