Degrees of Interpretation

Abstract

What has been learned about logic by means of “uninterpreted” logistic systems can be supplemented by comparing the latter with systems which are more uninterpreted, as well as with others which are less uninterpreted than the well-known logistic systems. By somewhat extending the meaning of ‘uninterpreted’, I hope to establish certain claims about the nature of logistic systems and also to cast some light on the nature of “logic itself.”

My procedure involves looking at three major “degrees” of interpretation: first, systems uninterpreted both semantically and syntactically, second, systems uninterpreted semantically but not syntactically, and third, systems uninterpreted neither semantically nor syntactically. We shall be forced to limit ourselves to the truth-functional part of logic in this brief study. What are usually called uninterpreted systems can be seen on a continuum of “degrees” of interpretation, from ordinary reasoning at one extreme to a “thoroughly uninterpreted system” at the other.

“Logic itself” apparently lies nearer to the interpreted, deformalized end of the spectrum than to the uninterpreted, formalized end. Logic is not identical with any particular logistic system, but is that which the particular logistic systems aim to formalize or model or capture. I propose that it is that minimum set of logical, rather than syntactical, primitive terms, definitions, and rules which is needed to generate logically, rather than syntactically, the principles of ordinary reasoning as it is used by logicians in their metalinguistic discussions and informal proofs of metatheorems. This minimum set is somewhat larger than the primitive bases of most logistic systems; its truth-functional part is presented in the “Deformalized Logic” at degree twelve.