Michael Kremer defines fixed-point logics of truth based on Saul Kripke’s fixed point semantics for languages expressing their own truth concepts. Kremer axiomatizes the strong Kleene fixed-point logic of truth and the weak Kleene fixed-point logic of truth, but leaves the axiomatizability question open for the supervaluation fixed-point logic of truth and its variants. We show that the principal supervaluation fixed point logic of truth, when thought of as consequence relation, is highly complex: it is not even analytic. We also (...) consider variants, engendered by a stronger notion of ‘fixed point’, and by variant supervaluation schemes. A ‘logic’ is often thought of, not as a consequence relation, but as a set of sentences – the sentences true on each interpretation. We axiomatize the supervaluation fixed-point logics so conceived. (shrink)

I identify a notion of logical grounding, clarify it, and show how it can be used (i) to characterise various consequence relations, and (ii) to give a precise syntactic account of the notion of “groundedness” at work in the literature on the paradoxes of truth.

The sentences employed in semantic paradoxes display a wide range of semantic behaviours. However, the main theories of truth currently available either fail to provide a theory of paradox altogether, or can only account for some paradoxical phenomena by resorting to multiple interpretations of the language. In this paper, I explore the wide range of semantic behaviours displayed by paradoxical sentences, and I develop a unified theory of truth and paradox, that is a theory of truth that also provides a (...) unified account of paradoxical sentences. The theory I propose here yields a threefold classification of paradoxical sentences – liar-like sentences, truth-teller-like sentences, and revenge sentences. Unlike existing treatments of semantic paradox, the theory put forward in this paper yields a way of interpreting all three kinds of paradoxical sentences, as well as unparadoxical sentences, within a single model. (shrink)

In response to the liar’s paradox, Kripke developed the fixed-point semantics for languages expressing their own truth concepts. Kripke’s work suggests a number of related fixed-point theories of truth for such languages. Gupta and Belnap develop their revision theory of truth in contrast to the fixed-point theories. The current paper considers three natural ways to compare the various resulting theories of truth, and establishes the resulting relationships among these theories. The point is to get a sense of the lay of (...) the land amid a variety of options. Our results will also provide technical fodder for the methodological remarks of the companion paper to this one. (shrink)

In this article I present HYPER-REF, a model to determine the referent of any given expression in First-Order Logic. I also explain how this model can be used to determine the referent of a first-order theory such as First-Order Arithmetic. By reference or referent I mean the non-empty set of objects that the syntactical terms of a well-formed formula pick out given a particular interpretation of the language. To do so, I will first draw on previous work to make explicit (...) the notion of reference and its hyperintensional features. Then I present HYPER-REF and offer a heuristic method for determining the reference of any formula. Then I discuss some of the benefits and most salient features of HYPER-REF, including some remarks on the nature of self-reference in formal languages. (shrink)

A method of supervaluation for Kripke’s theory of truth is presented. It differs from Kripke’s own method in that it employs trees; results in a compositional semantics; assigns the intuitively correct truth values to the sentences of a particularly tricky example of Gupta’s; and – it is argued – is acceptable as an explication of the correspondence theory of truth.

In diesem Aufsatz wird ein neues Paradoxon vorgestellt, der Super-Lügner. Er ist stärker als alle bekannten Lügner-Sätze, nicht mehr eindeutig selbstreferentiell und läßt sich darüber hinaus in eindeutig in die Tarski-Hierarchie einordnen. Eine unendlich große Familie von Super-Lügnern auf Metaebenen ist konstruierbar. Schließlich widersetzt sich der Super-Lügner der Auflösung durch die neue vielversprechende Reflexionslogik LR von U. Blau.

In diesem Aufsatz wird ein neues Paradoxon vorgestellt, der Super-Lügner. Er ist stärker als alle bekannten Lügner-Sätze, nicht mehr eindeutig selbstreferentiell und läßt sich darüber hinaus in eindeutig in die Tarski-Hierarchie einordnen. Eine unendlich große Familie von Super-Lügnern auf Metaebenen ist konstruierbar. Schließlich widersetzt sich der Super-Lügner der Auflösung durch die neue vielversprechende Reflexionslogik LR von U. Blau.