Informativnost matematických či analyticky pravdivých tvrzení a paradox inference

Abstract

In 1934 Cohen & Nagel formulated the famous paradox of inference: "If in an inference the conclusion is not contained in the premises, it cannot be valid; and if the conclusion is not different from the premises, it is useless; but the conclusion cannot be contained in the premises and also possess novelty; hence inferences cannot be both valid and useful." This paradox arises because of the tension between (a) the legitimacy (validity) of an inference, and (b) the utility of an inference. One can perhaps reformulate the question posed by the 'paradox': How can logic function as a useful epistemological tool? We formulate two answers to the question posed by the paradox, namely a negative and a positive one, by distinguishing between empirical information about the state of the world and the analytical content of a statement.

First we analyze the way in which "the conclusion is contained in the premises". We recapitulate the notion of semantic (empirical) information and also how we may compare statements regarding the semantic information they contribute to us. Further we define a logically valid argument and a logically valid statement by making use of a logical form literally corresponding to a statement. The analytical content of a statement S is defined as the respective construction expressed by S. By way of conclusion some criteria that make it possible to compare statements with regard to their analytical informativeness are formulated.