Consequence and complexity in infinite-valued logic: a survey

In general, every logic L comes equipped with: (i) a syntax, consisting of a finite set 𝒜 of symbols, called the alphabet, and an inductive definition of which strings over 𝒜 are to be called formula of L; (ii) a semantics, telling the meaning of each formula, in particular telling when two formulae are equivalent; and (iii) an algorithmic procedure whereby, given a finite set F of formulae, one can in principle obtain all consequences of F. In certain fortunate cases - e.g. in classical logic - formulae up to equivalence form an interesting class of algebraic structures. The infinite-valued calculus of Łukasiewicz is such a fortunate case. Our aim in this paper is to review semantic-algorithmic issues for this logic, with particular reference to recent research