Computers and relevant logic : a project in computing matrix model structures for propositional logics

Abstract

I present and discuss four classes of algorithm designed as solutions to the problem of generating matrix representations of model structures for some non-classical propositional logics. I then go on to survey the output from implementations of these algorithms and finally exhibit some logical investigations suggested by that output. All four algorithms traverse a search tree depthfirst. In the case of the first and fourth methods the tree is fixed by imposing a lexicographic order on possible matrices, while the second and third create their search tree dynamically as the job progresses. The first algorithm is a simple "backtrack" with some pruning of the tree in response to refutations of possible matrices. The fourth, the most efficient we have for time, maximises the amount of pruning while keeping the same basic form. The second, which uses a large number of special properties of the logics in question, and so requires some logical and algebraic knowledge on the part of the programmer, finds the matrices at the tips of branches only, while the third, due to P.A. Pritchard, is far easier to program and tests a matrix at every node of the search tree. The logics with which I am concerned are in the "relevant" group first seriously investigated by A.R. Anderson and N.D. Belnap (see their Entailment: the logic of relevance and necessity, 1975). The most surprising observation in my preliminary survey of the numbers of matrices validating such systems is that the typical models are not much like the models normally taken as canonical for the logics. In particular the proportion of inconsistent models (validating some cases of the scheme 'A & ~A') is much higher than might have been expected. Among the logical investigations already suggested by the quasi-empirical data now available in the form of matrices are some work on the system R-W, including my theorem, proved in chapter 2.3, that with the law of excluded middle it suffices to trivialise naive set theory, and the little-noticed subject of Ackermann constants (sentential constants) in these logics. The formula which collapses naive set theory in R-W plus A v ~A is the most damaging set-theoretic antinomy known. The theorem that there are at least 3088 Ackermann constants in the logic R (chapter 2.4) could not reasonably have been proved without the aid of a computer. My major conclusion is that this work on applications of computers in logical research has reached a point where we are able not only to relieve logicians of some drudgery, but to suggest theorems and insights of new and possibly important kinds.