ABSTRACT
One of the most basic intuitive ideas behind the notion of following-from is that a conclusion follows from some premises iff it is true whenever all the premises are; hence, that following-from is, in essence, truth-preservation. Thus, a promising approach to following-from, and an alternative to its calculization, might be one that takes this at face value, and it seems that this is one of the things that became possible thanks to Tarski. We can semantically characterize a language by specifying its set of acceptable truth-valuations, that is, mappings of its sentences on 1 and 0. Given this, we have the relation of consequence (a conclusion is a consequence of some premises if it is mapped on 1 by every acceptable truth-valuation that maps all the premises on 1) and the set of tautologies (sentences that are mapped on 1 by every acceptable truth-valuation). Needless to say that for the calculus of classical propositional logic there exists a matching truth-functional semantics (such that the inference generated by the former coincides with the consequence as generated by the latter). From the viewpoint of specification of acceptable truth-valuations, truth-functional semantics can be seen as a generalization of calculi, for while the latter can be seen as restricted to prescriptions that a sentence must be true if some other sentences are true, the former may also work with falsity.
