Some structure results for propositional calculi

Abstract

1. Introduction. Since this paper is a written version of one presented as a survey lecture at a meeting of the Association for Symbolic Logic, its form and content have been essentially determined by the form and content of that lecture, and these latter were themselves considerably influenced by a deliberate attempt to avoid the lecture consisting essentially either of a catalogue of results in some fairly wide field or of the presentation of detailed proofs of a few particular results in a very limited field, with, in either case, a consequential serious limitation of the time available for discussion of the motivation for the results and of the connections between them. There is thus no attempt to cover, even in outline, the full range of topics which could reasonably fall within the scope of its title.The paper consists mainly of a general discussion of some results concerned with finite models of axiomatically defined calculi of propositional form, with particular reference to the occurrence of such models in proofs that certain propositional calculi are decidable. In addition, there is an account of some results concerned with the existence of undecidable propositional calculi and of others connected with the decidability or undecidability of certain structure problems which relate either to sets of propositional calculi or to different presentations of the same calculus.

Algorithms for finding matrix models of propositional calculi

“…Given is a propositional language L. A matrix M is an ordered triple (M, O, D), where M is a set, D is a proper subset of M comprising the designated elements, and O a set of operations on M in 1-1 arity-preserving correspondence with the connectives of L. An operation of arity n is a function from the cross product of n copies of M into M. A formulafis valid in M iffevery mapping from the variables off into M, when extended to a homomorphism h from the set of all formulas over the variables off into M, gives h(f) ~ D. M is said to satisfy an axiomatically characterized logic L on L (and vice-versa) iff all theorems of L are valid in M. Additionally requiring the rules of L to preserve designation gives the strong models of L (see [5]), which we henceforth assume as our subject.…”

“…The first step towards this goal was taken by Brady [3], who noted that the so-called (by Harrop [5]) strong matrix models are recursively enumerable, and can therefore be discovered in principle by fully-automatic means. He presented some basic structural results which reduce the sizes of the search-spaces involved, but as discovered by Meyer (and reported in [16, pp.…”

Varieties of algebraic systems and propositional calculuses

“…In exactly the same way as for the intuitionistic and modal calculuses, and for each propositional calculus (defined in the usual way [5]), it is easy to find the corresponding simple signature and foundation.…”

“…5]). Such a calculus, which coincides In one special case wlth the equational calculus, may be a propositional calculus in another special case (for example, Intuitlonlstlc or modal [5,6,7]). The corresponding generalizations are carried out in Sec.…”

A Single‐Axiom Impligational Calculus of Given Unsolvability

“…By substitution into C, we get both where X = ( ( A z 3 (B, 3 (B, 3 (d 3 A , ) ) ) ) 2 (B, 3 (d 2 A , ) ) ) .…”