The simple substitution property provides a systematic and easy method for proving a theorem by an axiomatic way. The notion of the property was introduced in Hosoi [4] but without a definite name and he showed three examples of the axioms with the property. Later, the property was given it's name as above in Sasaki [7].Our main result here is that the necessary and sufficient condition for a logicL on a finite slice to have the simple substitution property is thatL (...) is finite. Here the necessity part is essentially new, for the sufficiency part has been proved in Hosoi and Sasaki [5]. Also the proof of sufficiency part is improved here. (shrink)

Section 1 recalls a point noted by A. N. Prior forty years ago: that a certain formula in the language of a purely implicational intermediate logic investigated by R. A. Bull is unprovable in that logic but provable in the extension of the logic by the usual axioms for conjunction, once this connective is added to the language. Section 2 reminds us that every formula is interdeducible with (i.e. added to intuitionistic logic, yields the same intermediate logic as) some conjunction-free (...) formula. Thus it would seem that any detour going via formulas with conjunction can be avoided, which raises a puzzle: how is this consistent with the point from Section 1? Sections 3 and 4 raise and discuss this puzzle. In fact, the puzzle turns out on closer inspection not to be so puzzling after all, but it does serve as a convenient centrepiece around which to organize a discussion of the phenomenon illustrated by the Bull–Prior example. Section 5 notes that Prior's observation can be extended to the case of the result of adding disjunction to Bull's logic, while Section 6 includes some further remarks aimed at diagnosing one source of possible residual puzzlement. A subtext of our discussion—spanning several of the notes—is that this work by Bull and Prior has been overlooked, their results having to be rediscovered, by many algebraists and logicians in more recent years. (shrink)

We give a Gentzen-type formulation GQ for the intermediate logic LQ and prove the cut-elimination theorem on it, where LQ is the propositional logic obtained from the intuitionistic propositional logic LI by adding the axioms of the form AV A.

A sequent calculus with the subformula property has long been recognised as a highly favourable starting point for the proof theoretic investigation of a logic. However, most logics of interest cannot be presented using a sequent calculus with the subformula property. In response, many formalisms more intricate than the sequent calculus have been formulated. In this work we identify an alternative: retain the sequent calculus but generalise the subformula property to permit specific axiom substitutions and their subformulas. Our investigation leads (...) to a classification of generalised subformula properties and is applied to infinitely many substructural, intermediate, and modal logics. We also develop a complementary perspective on the generalised subformula properties in terms of logical embeddings. This yields new complexity upper bounds for contractive-mingle substructural logics and situates isolated results on the so-called simple substitution property within a general theory. (shrink)