Solutions to semantic paradoxes often involve restrictions of classical logic for semantic vocabulary. In the paper we investigate the costs of these restrictions in a model case. In particular, we fix two systems of truth capturing the same conception of truth: (a variant) of the system KF of [Feferman 1991] formulated in classical logic, and (a variant of) the system PKF of [Halbach & Horsten 2006], formulated in basic De Morgan logic. The classical system is known to be much stronger than the nonclassical one. We assess the reasons for this asymmetry by showing that the truth theoretic principles of PKF cannot be blamed: PKF with induction restricted to non-semantic vocabulary coincides in fact with what the restricted version of KF proves true.
We study the relationships between two clusters of axiomatizations of Kripke's fixed-point models for languages containing a self-applicable truth predicate. The first cluster is represented by what we will call 'PKF-like' theories, originating in recent work Halbach and Horsten, whose axioms and rules (in Basic De Morgan Logic) are all valid in fixed-point models; the second by 'KF-like' theories first introduced by Solomon Feferman, that lose this property but reflect the classicality of the metatheory in which Kripke's construction is carried out. We show that to any natural system in one cluster -corresponding to natural variations on induction schemata -there is a corresponding system in the other proving the same sentences true, addressing a problem left open by Halbach and Horsten and accomplishing a suitably modified version of the project sketched by Reinhardt aiming at an instrumental reading of classical theories of self-applicable truth.
We discuss the principles for a primitive, object-linguistic notion of consequence proposed by (Beall and Murzi, Journal of Philosophy, 3 pp. 143-65 (2013)) that yield a version of Curry's paradox. We propose and study several strategies to weaken these principles and overcome paradox: all these strategies are based on the intuition that the object-linguistic consequence predicate internalizes whichever meta-linguistic notion of consequence we accept in the first place. To these solutions will correspond different conceptions of consequence. In one possible reading of these principles, they give rise to a notion of logical consequence: we study the corresponding theory of validity (and some of its variants) by showing that it is conservative over a wide range of base theories: this result is achieved via a well-behaved form of local reduction. The theory of logical consequence is based on a restriction of the introduction rule for the consequence predicate. To unrestrictedly maintain this principle, we develop a conception of object-linguistic consequence, which we call grounded consequence, that displays a restriction of the structural rule of reflexivity. (strong Kleene version). Grounded validity will be shown to satisfy several desirable principles for a naïve, self-applicable notion of consequence.
Iterated reflection principles have been employed extensively to unfold epistemic commitments that are incurred by accepting a mathematical theory. Recently this has been applied to theories of truth. The idea is to start with a collection of Tarski-biconditionals and arrive by finitely iterated reflection at strong compositional truth theories. In the context of classical logic it is incoherent to adopt an initial truth theory in which A and 'A is true' are inter-derivable. In this article we show how in the context of a weaker logic, which we call Basic De Morgan Logic, we can coherently start with such a fully disquotational truth theory and arrive at a strong compositional truth theory by applying a natural uniform reflection principle a finite number of times.
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