Citations of:
Selfreference and the languages of arithmetic
Philosophia Mathematica 15 (1):129 (2007)
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In this second and last paper of the two part investigation on "Modality and Axiomatic Theories of Truth" we apply a general strategy for constructing modal theories over axiomatic theories of truth to the theory KripkeFeferman. This general strategy was developed in the first part of our investigation. Applying the strategy to KripkeFeferman leads to the theory Modal KripkeFeferman which we discuss from the three perspectives that we had already considered in the first paper, where we discussed the theory Modal (...) 

A reply to two responses to an earlier paper, "A Liar Paradox". 

The purpose of this note is to present a strong form of the liar paradox. It is strong because the logical resources needed to generate the paradox are weak, in each of two senses. First, few expressive resources required: conjunction, negation, and identity. In particular, this form of the liar does not need to make any use of the conditional. Second, few inferential resources are required. These are: (i) conjunction introduction; (ii) substitution of identicals; and (iii) the inference: From ¬(p (...) 

Selfreference has played a prominent role in the development of metamathematics in the past century, starting with Gödel’s first incompleteness theorem. Given the nature of this and other results in the area, the informal understanding of selfreference in arithmetic has sufficed so far. Recently, however, it has been argued that for other related issues in metamathematics and philosophical logic a precise notion of selfreference and, more generally, reference is actually required. These notions have been so far elusive and are surrounded (...) 

This paper concerns the relationship between transitivity of entailment, omegainconsistency and nonstandard models of arithmetic. First, it provides a cutfree sequent calculus for nontransitive logic of truth STT based on Robinson Arithmetic and shows that this logic is omegainconsistent. It then identifies the conditions in McGee for an omegainconsistent logic as quantified standard deontic logic, presents a cutfree labelled sequent calculus for quantified standard deontic logic based on Robinson Arithmetic where the deontic modality is treated as a predicate, proves omegainconsistency (...) 

Nontransitive responses to the validity Curry paradox face a dilemma that was recently formulated by Barrio, Rosenblatt and Tajer. It seems that, in the nontransitive logic ST enriched with a validity predicate, either you cannot prove that all derivable metarules preserve validity, or you can prove that instances of Cut that are not admissible in the logic preserve validity. I respond on behalf of the nontransitive approach. The paper argues, first, that we should reject the detachment principle for naive validity. (...) 





In Heck, Richard Heck presents variants on the familiar liar paradox, intended to reveal limitations of theories of transparent truth. But all existing theories of transparent truth can respond to Heck's variants in just the same way they respond to the liar. These new variants thus put no new pressure on theories of transparent truth. 

Richard Heck has recently drawn attention on a new version of the Liar Paradox, one which relies on logical resources that are so weak as to suggest that it may not admit of any “truly satisfying, consistent solution”. I argue that this conclusion is too strong. Heck's Liar reduces to absurdity principles that are already rejected by consistent paracomplete theories of truth, such as Kripke's and Field's. Moreover, the new Liar gives us no reasons to think that (versions of) these (...) 