VIII—Truth in a Structure

Abstract

When Alfred Tarski died in 1983 at the age of 82, obituaries very justly compared him with Aristotle and Frege. His vast output (over three hundred papers, abstracts and books) contained fundamental advances in nearly all branches of mathematical logic. Unlike Aristotle and Frege, Tarski was not a philosopher. In his published works he never discussed philosophical issues, except where he could handle them mathematically, and he was cautious even in this. He made sceptical remarks about applying logical notions to sentences of natural languages-it 'leads inevitably to confusions and contradictions' ([1935] p. 267), he said. But before you dismiss him as a mere theorem prover, you should ask yourself what your grandsons and granddaughters are likely to study when they settle down to their 'Logic for computing' class at 9.30 after school assembly. Will it be syllogisms? Just possibly it could be the difference between saturated objects and unsaturated concepts, though I doubt it. I put my money on Tarski's definition of truth for formalised languages. It has already reached the university textbooks of logic programming, and another ten years should see it safely into the sixth forms. This is a measure of how far Tarski has influenced the whole framework of logic. My topic today is Tarski's definition of truth. But first let me mention some historical background. Since the end of last century, algebraists and geometers have often found themselves studying certain objects which we now call structures, or more loosely models. For the first fifty years of this century they were usually known as Systeme von Dingen, or more briefly as systems. Roughly speaking, a structure is a collection of elements together with certain labelled relations which are defined on those elements. A typical example is what