Probabilistic Logic and Induction

We give a probabilistic interpretation of first-order formulas based on Valiants model of pac-learning. We study the resulting notion of probabilistic or approximate truth and take some first steps in developing its model theory. In particular we show that every fixed error parameter determining the precision of universal quantification gives rise to a different class of tautologies. Finally we study the inductive inference of first-order formulas from atomic truths.