Intuitionistic relevant logic and perfect validity

Abstract

In [3] Neil Tennant shows that the provable sequents of his system of Classical Relevant Logic (without a conditional) are exactly the substitution instances of classically perfectly valid sequents. A sequent is perfectly valid, relative to a notion of validity, if it is valid but ceases to be so on removal of any sentence that occurs in it as either a premiss or as conclusion. In [4] he conjectures that the analogous property holds of his Intuitionistic Relevant Logic ([4], p. 255). Here I shall refute that conjecture, suggest an amendment that circumvents the refutation, and indicate a further difficulty. In its sequent formulation Intuitionistic Relevant Logic (IR) is like standard intuitionist propositional logic save that the structural rules of Cut and Thinning (Dilution) are forsworn and left v-introduction is slightly strengthened. All sequents are of the form X : Y where X is a possibly empty set of sentences and Y is either empty or contains a single sentence. Below we indicate sets by listing their members, i.e. we omit set braces. The differences between IR in its natural deduction formulation and intuitionist propositional logic are more marked: the introduction and elimination rules, other than v-elimination, are those of intuitionist propositional logic without the rule of absurdity; to make up for that omission the rule of v-elimination is strengthened so as to allow derivation of disjunctive syllogism (modus tollendo ponens); all proofs must be in normal form and there can be no vacuous discharge of assumptions in -introduction and v-elimination. Tennant has shown that the sequent and natural deduction formulations are equivalent. (For further details see [4], pp. 185-200, 253-65.) I shall now derive in IR a sequent that is intuitionistically valid but neither itself intuitionistically perfectly valid nor a substitution instance of an intuitionistically perfectly valid sequent: