Truth, deflationism and the ontology of expressions: an axiomatic study. - Nicolai, C (2014)
Philosophical enquiry on the notion of truth has traditionally involved the identification of a class of objects to which truth is ascribed. At the same time, formal investigations are often required when the notion of truth is at issue: semantic paradoxes force in fact philosophers to shape their arguments in a precise way. Objects of truth, in formal context, are always reduced to other, more manageable objects that mimic their structural properties such as numbers or sets. This form of reduction renders the distinction between linguistic or syntactic objects, to which truth is usually applied, and their mathematical counterparts opaque, at least from the point of view of the theory of truth. In informal metatheoretic discussion, in fact, they are clearly different entities. In this thesis we focus on an alternative way of constructing axiomatic theories of truth in which syntactic objects and mathematical objects belong to different universes. A brief introduction tries to situate the proposed theories in the context of different investigations on axiomatic truth. Chapter 2 is devoted to the discussion of historical and more theoretical motivations behind the proposed alternative. Chapter 3 will present the syntactic koinè spoken by our theories. Morphological categories of the object language and logical concepts concerning the object theory will be formalised in a recent axiomatisation of hereditarily finite sets. In Chapter 4 we finally introduce theories of truth with a built-in syntactic theory and examine some of their consequences. We briefly focus on disquotational truth, then consider compositional axioms for truth. Chapter 5 investigates a possible application of the setting just introduced: a realisation of the all-present interaction, in metamathematical practice, between informal metatheoretic claims and their (suitably chosen) coded counterparts. In the final chapter, after a brief characterisation of the key doctrines of the delflationary conception of truth, we evaluate the impact that the theories of truth studied in this work can have on the debate on the so-called conservativeness argument, which tries to match the alleged insubstantiality of the notion of truth, advocated by deflationists, with the deductive power of deflationary acceptable theories of truth.