Abstract

A formal semantics for a theory T* is constructed which has the following properties: (a) T* embodies a modal logic (in fact, S5) and a so-called "free" logic which allow for terms such as "Pegasus" or "the round square" which may fail to denote actually existing individuals. (b) The language of T* contains a 1-place operation symbol whose intended interpretation is that of the operation of quotation, and a 2-place operation symbol whose intended interpretation is the operation of concatenation. Semantic principles are adopted in the meta-theory which ensure that these receive at least part of their intended interpretations. (c) The logical quantifiers receive what is often called "the substitution interpretation". Since variables may occur in formulas both in the place of terms (as they usually do) or in the place of formulas, this interpretation is more general than the one usually encountered. As a result of the treatment of quantification and the introduction of quotation and concatenation, the expressive power of T* is significantly greater than that of a standard first-order predicate calculus. (d) A truth predicate for T* is defined within T* itself and it is shown that this predicate satisfies criteria of adequacy which were adopted in earlier chapters. T* is classically 2-valued and Tarski's Convention is valid relative to the semantics provided. (e) Several familiar semantic paradoxes (including the Liar and the Grelling-Nelson) are formulated in the language of T*, and it is shown that no contradictions are generated within the theory. Preliminary sections contain precise but informal statements of several well-known semantic paradoxes as well as informal discussion of the concepts which play key roles in the development of the theory T*. In addition, one chapter is devoted to criticisms of other recent attempts to solve the semantic paradoxes. The final chapter contains criticisms of several unusual features of T* and suggestions are made regarding how these features might be eliminated.