ABSTRACT In our response Field's ‘Properties, Propositions and Conditionals’, we explore the methodology of Field's program. We begin by contrasting it with a proof-theoretic approach and then commenting on some of the particular choices made in the development of Field's theory. Then, we look at issues of property identity in connection with different notions of equivalence. We close with some comments relating our discussion to Field's response to Restall’s [2010] ‘What Are We to Accept, and What Are We to Reject, (...) While Saving Truth from Paradox?’. (shrink)

This paper presents a range of new triviality proofs pertaining to naïve truth theory formulated in paraconsistent relevant logics. It is shown that excluded middle together with various permutation principles such as A → (B → C)⊩B → (A → C) trivialize naïve truth theory. The paper also provides some new triviality proofs which utilize the axioms ((A → B)∧ (B → C)) → (A → C) and (A → ¬A) → ¬A, the fusion connective and the Ackermann constant. An (...) overview over various ways to formulate Leibniz’s law in non-classical logics and two new triviality proofs for naïve set theory are also provided. (shrink)

We argue that distinct conditionals—conditionals that are governed by different logics—are needed to formalize the rules of Truth Introduction and Truth Elimination. We show that revision theory, when enriched with the new conditionals, yields an attractive theory of truth. We go on to compare this theory with one recently proposed by Hartry Field.

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)