Expressing Validity: Towards a Self-Sufficient Inferentialism

For semantic inferentialists, the basic semantic concept is validity. An inferentialist theory of meaning should offer an account of the meaning of "valid." If one tries to add a validity predicate to one's object language, however, one runs into problems like the v-Curry paradox. In previous work, I presented a validity predicate for a non-transitive logic that can adequately capture its own meta-inferences. Unfortunately, in that system, one cannot show of any inference that it is invalid. Here I extend the system so that it can capture invalidities

Dissolving the Paradoxicality Paradox

Non-classical solutions to semantic paradox can be associated with conceptions of paradoxicality understood in terms of entailment facts. In a K3-based theory of truth, for example, it is prima facie natural to say that a sentence φ is paradoxical iff φ ∨ ¬φ entails an absurdity. In a recent paper, Julien Murzi and Lorenzo Rossi exploit this idea to introduce revenge paradoxes for a number of non-classical approaches, including K3. In this paper, I show that on no understanding of ‘is paradoxical’ (for K3) should both rules needed for their paradox be expected to hold unrestrictedly. Just which rule fails, however, depends on various factors, including whether the derivability relation of a target system of reasoning is arithmetically definable

Non-Representational Mathematical Realism

This paper is an attempt to convince anti-realists that their correct intuitions against the metaphysical inflationism derived from some versions of mathematical realism do not force them to embrace non-standard, epistemic approaches to truth and existence. It is also an attempt to convince mathematical realists that they do not need to implement their perfectly sound and judicious intuitions with the anti-intuitive developments that render full-blown mathematical realism into a view which even Gödel considered objectionable (Gödel 1995, p. 150).I will argue for the following two theses: (i) that realism, in its standard characterization, is our default position, a position in agreement with our pre-theoretical intuitions and with the results of our best semantic theories, and (ii) that most of the metaphysical qualms usually related to it depends on a poor understanding of truth and existence as higher-order concepts.

Dissolving the paradoxicality paradox

Non-classical solutions to semantic paradox can be associated with conceptions of paradoxicality understood in terms of entailment facts. In a K3-based theory of truth, for example, it is prima facie natural to say that a sentence φ is paradoxical iff φ ∨ ¬φ entails an absurdity. In a recent paper, Julien Murzi and Lorenzo Rossi exploit this idea to introduce revenge paradoxes for a number of non-classical approaches, including K3. In this paper, I show that on no understanding of ‘is paradoxical’ (for K3) should both rules needed for their paradox be expected to hold unrestrictedly. Just which rule fails, however, depends on various factors, including whether the derivability relation of a target system of reasoning is arithmetically definable

Capturing naive validity in the Cut-free approach

Rejecting the Cut rule has been proposed as a strategy to avoid both the usual semantic paradoxes and the so-called v-Curry paradox. In this paper we consider if a Cut-free theory is capable of accurately representing its own notion of validity. We claim that the standard rules governing the validity predicate are too weak for this purpose and we show that although it is possible to strengthen these rules, the most obvious way of doing so brings with it a serious problem: an internalized version of Cut can be proved for a Curry-like sentence. We also evaluate a number of possible ways of escaping this difficulty.Fil: Barrio, Eduardo Alejandro. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad de Buenos Aires; ArgentinaFil: Rosenblatt, Lucas Daniel. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad de Buenos Aires; ArgentinaFil: Tajer, Diego. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad de Buenos Aires; Argentin