Abstract
Let L be a first-order language, and let U be a structure for L. On the basis of the Tarskian definition of truth, the sentences of L are then dichotomized into those which are true in U and those which are false in U. Intuitively, it often seems possible to supplement this two-fold partition of the set S L of the sentences of L with an additional fine structure. For example, some of the truths in L are ‘better’ than others at least in the sense that they are more comprehensive, i.e., tell more about the structure U. Some false sentences of L are more mistaken than others — thus, the statement ‘3 is the smallest natural number’ can be said to be closer to the truth than the statement ‘5 is the smallest natural number’, and the claim that all prime numbers are odd is less erroneous than the claim that all prime numbers are even. Some false sentences of L may even contain (or entail) more true information about the structure than some of the true sentences of L (like tautologies). Similar remarks seem to apply not only to single sentences in L but to theories (i.e., deductively closed subsets of S L as well.
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