Approximate truth and Łukasiewicz logic

Lukasiewicz' infinite-valued logic L has been used by a number of authors to study reasoning with vague, "fuzzy", or uncertain statements ([5], [2], [11], [12]). Scott and, more recently, Katz have also argued that L is appropriate for the logic of degrees of error or degrees of approximation to the truth ([6], p. 421 and [4], p. 773). The aim of this note is to show that L is not appropriate for this purpose, regardless of its possible application to vagueness, and to derive a few facts about alternative systems which are appropriate, or more nearly so. Both Scott and Katz indicate that a leading idea in their analysis of degree of error is the case of equations r = s, where some metric is used to measure the distance between r and s. If r and s are reals, then a convenient example is the metric |r — s|. In this case, r = s true iff |r — s\ = 0, and if r Φ s, \r — s| measures the degree of error of the equation r = s9 larger values indicating larger errors. Scott and Katz both use L to extend to compound formulas the measure of error that a metric like | r — s\ gives for equations. My contention that L does not do this correctly is based on the following claim: If we are to take at all seriously the idea that the "truth values" of our system measure the degree of error, we must insist that true statements have zero error. This aim is, of course, already met for the equation example just mentioned, but we ought to meet it for all statements whose "nearness" to the truth is to be assessed. In fact, it is just the property that true statements are zero distance from the truth —call it the "accuracy property"—that distinguishes an assessment of accuracy, at which Scott and Katz are clearly aiming, from an assessment of proximity to the comprehensive truth, which is the aim (or an aim) of the theories of so-called verisimilitude or truthlikeness and of theories of vagueness. A statement can be inaccurate only by stating something which is false. It may be less than the comprehensive truth, on the other hand, merely by sins of omission. Similarly, a statement might possess some degree of vagueness and still be true, but it could not possess any degree of inaccuracy and still be true.