This paper develops the idea that valid arguments are equivalent to true conditionals by combining Kripke’s theory of truth with the evidential account of conditionals offered by Crupi and Iacona. As will be shown, in a first-order language that contains a naïve truth predicate and a suitable conditional, one can define a validity predicate in accordance with the thesis that the inference from a conjunction of premises to a conclusion is valid when the corresponding conditional is true. The validity predicate (...) so defined significantly increases our expressive resources and provides a coherent formal treatment of paradoxical arguments. (shrink)

ABSTRACT In ‘Properties, Propositions, and Conditionals’ Field [2021] advances further on our understanding of the logic and meaning of naive theories – theories that maintain, in the face of paradox, basic assumptions about properties and propositions. His work follows in a tradition going back over 40 years now, of using Kripke fixed-point model constructions to show how naive schemas can be (Post) consistent, as long as one embeds in a non-classical logic. A main issue in all this research is the (...) question of finding a suitable conditional for the naive schemas, an implication connective that is well-behaved enough to be useful but weak enough to maintain coherence when there are paradoxes in the surrounding aether. Field takes several cues from the exemplary work of Ross Brady. Brady's work is predominantly concerned with models for naive theories in relevant logic. Field's considered view is that ‘relevant conditionals are just the wrong tool for naive theories’ [ibid.: 140]. In this note, I reply that a relevant conditional is an indispensable tool for at least one very important job: giving identity conditions for properties. (shrink)

ABSTRACT In this brief note we explore a couple of features of the semantics for indicative conditionals provided by Field. Those features strikingly resemble some controversial principles in connexive logic. We will show that although Field’s semantics has the technical means to stand to the mentioned features, more work is needed to make some of its outcomes less unintuitive.

Nonclassical theories of truth that take truth to be transparent have some obvious advantages over any classical theory of truth. But several authors have recently argued that there’s also a big disadvantage of nonclassical theories as compared to their “external” classical counterparts: proof-theoretic strength. While conceding the relevance of this, the paper argues that there is a natural way to beef up extant internal theories so as to remove their proof-theoretic disadvantage. It is suggested that the resulting internal theories are (...) preferable to their external counterparts. (shrink)

This paper is a step toward showing what is achievable using non-classical metatheory—particularly, a substructural paraconsistent framework. What standard results, or analogues thereof, from the classical metatheory of first order logic can be obtained? We reconstruct some of the originals proofs for Completeness, Löwenheim-Skolem and Compactness theorems in the context of a substructural logic with the naive comprehension schema. The main result is that paraconsistent metatheory can ‘re-capture’ versions of standard theorems, given suitable restrictions and background assumptions; but the shift (...) to non-classical logic may recast the meanings of these apparently ‘absolute’ theorems. (shrink)