On approximate truth

We propose a definition for the relation: structure U approximates structure S. A first order sentence is then defined to be approximately true in a structure just in case it is true (standardly) in an approximating structure. The deductive and inductive logic of approximate truth in this sense is discussed. Regarding deduction, we consider a modal language where ▪ θ is true in a structure S just in case θ is approximately true in S , and show that the set of theorems of this language is not recursively enumerable. Regarding induction, we define a paradigm of probably approximately correct truth detection, and show that successful induction is possible in this paradigm with respect to a wide class of sentences.