The present research is a logico-philosophical analysis of the issues concerning the semantics for plural logic, with particular attention to the recent work by A. Oliver and T. Smiley, Plural Logic (OUP).
The first chapter introduces into the model-theoretic semantics for second-order languages. Three versions are presented: standard, Henkin and multi-sorted. All three differ in the definition of the assignment function to the second-order variables. The second chapter analyzes the relationship between model-theoretic semantics and ontology, in particular realism and nominalism. On the one hand, realism relies on the so called referential (or objectual) semantics; on the other hand, nominalism must rely on the so called substitutional semantics, for what concerns second-order variables, in order to avoid any ontological commitment with respect to such variables.
The third chapter introduces plural semantics. W.O. Quine (in 1970), argued that second-order logic is a ‘set theory in sheep’s clothing’ and, so, not a pure logic. Quine’s approach was strongly criticized by G. Boolos, in a series of articles in the 70s and 80s of the last century. He proposed a new sort of referential semantics for (monadic) secon-order variables and quantifiers, the so called plural interpretation based on a (one-many) ralation assignment.
The fourth chapter presents an outline of the more ideas of the work by Oliver and Smiley. In particular, their system 1) recasts predication in terms of plural predication and 2) attempts to capture plural denotation phenomena. In order to extend the category of terms to the plural case, Autors propose a theory of definite descriptions that contrasts with the Russellian one. Plural functional terms, obtained by means of the descriptive apparatus, denote so called multivalued functions – to be added to the usual functions, now singlevalued. In the fifth chapter it is provided an analysis of such function with respect to mathematics and logic. Multivalued functions play also a key role in the semantics of plural logic, modeling the assignment function for plural variables. It is also considered some semantic consequence due to their assumption.
The final chapter concludes the analysis of plural logic. Ø. Linnebo (2003) presented a criterion of logicality. From the application of such a criterion, it emerges that there are no compelling reasons not to define plural logic a pure logic. The only main point against such plural logics is the modal rigidity of the notion of plurality. Such a rigidity reveals that the alleged formalization of some typical features of that fragment of natural language that is related to plural phenomena is not fully adequate in these sort of plural logics.