We show the admissibility for BCI of a rule form of the characteristic implicational axiom of abelian logic, this rule taking us from →β to α. This is done in Section 8, with surrounding sections exploring the admissibility and derivability of various related rules in several extensions of BCI.

After some generalities about connections between functions and relations in Sections 1 and 2 recalls the possibility of taking the semantic values of ‐ary Boolean connectives as ‐ary relations among truth‐values rather than as ‐ary truth functions. Section 3, the bulk of the paper, looks at correlates of these truth‐value relations as applied to formulas, and explores in a preliminary way how their properties are related to the properties of “logical relations” among formulas such as equivalence, implication (entailment) and contrariety (...) (logical incompatibility), concentrating for illustrative purposes on binary logical relations such as those just listed. To avoid an excess of footnotes, some points have been deferred to an Appendix as “Longer Notes”. (shrink)

In his 1964 note, ‘Two Additions to Positive Implication’, A. N. Prior showed that standard axioms governing conjunction yield a nonconservative extension of the pure implicational intermediate logic OIC of R. A. Bull. Here, after reviewing the situation with the aid of an adapted form of the Kripke semantics for intuitionistic and intermediate logics, we proceed to illuminate this example by transposing it to the setting of modal logic, and then relate it to the propositional logic of what have been (...) called ‘Hilbert algebras with infimum’. (shrink)

In a paper on the logical work of the Jains, Graham Priest considers a consequence relation, semantically characterized, which has a natural analogue in modal logic. Here we give a syntactic/axiomatic description of the modal formulas which are consequences of the empty set by this relation, which is to say: those formulas which are, for every model, true at some point in that model.

Accounts of logical independence which coincide when applied in the case of classical logic diverge elsewhere, raising the question of what a satisfactory all-purpose account of logical independence might look like. ‘All-purpose’ here means: working satisfactorily as applied across different logics, taken as consequence relations. Principal candidate characterizations of independence relative to a consequence relation are that there the consequence relation concerned is determined by only by classes of valuations providing for all possible truth-value combinations for the formulas whose independence (...) is at issue, and that the consequence relation ‘says’ nothing special about how those formulas are related that it does not say about arbitrary formulas. Each of these proposals returns counterintuitive verdicts in certain cases—the truth-value inspired approach classifying certain cases one would like to describe as involving failures of independence as being cases of independence, and the de Jongh approach counting some intuitively independent pairs of formulas as not being independent after all. In final section, a modification of the latter approach is tentatively sketched to correct for these misclassifications. The attention is on conceptual clarification throughout, rather than the provision of technical results. Proofs, as well as further elaborations, are lodged in the ‘longer notes’ in a final Appendix. (shrink)

A term td is called a ternary deductive term for a variety of algebras V if the identity td≈r holds in V and ∈θ yields td≈td for any A∈V and any principal congruence θ on A. A connective f is called td-distributive if td)≈ f,…,td). If L is a propositional logic and V is a corresponding variety that has a TD term td, then any admissible in L rule, the premises of which contain only td-distributive operations, is derivable, and the (...) substitution r↦td is a projective L-unifier for any formula containing only td-distributive connectives. The above substitution is a generalization of the substitution introduced by T. Prucnal to prove structural completeness of the implication fragment of intuitionistic propositional logic. (shrink)