Tarski's Undefinability of Truth Theorem comes in two versions: that no consistent theory which interprets Robinson's Arithmetic (Q) can prove all instances of the T-Scheme and hence define truth; and that no such theory, if sound, can even express truth. In this note, I prove corresponding limitative results for validity. While Peano Arithmetic already has the resources to define a predicate expressing logical validity, as Jeff Ketland has recently pointed out (2012, Validity as a primitive. Analysis 72: 421-30), no theory (...) which interprets Q closed under the standard structural rules can define nor express validity, on pain of triviality. The results put pressure on the widespread view that there is an asymmetry between truth and validity, viz. that while the former cannot be defined within the language, the latter can. I argue that Vann McGee's and Hartry Field's arguments for the asymmetry view are problematic. (shrink)

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 (...) principles cannot be consistently retained once the structural rule of contraction is restricted. I suggest that revisionary logicians have independent reasons for restricting such a rule. (shrink)

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 these principles cannot (...) be consistently retained once the structural rule of contraction is restricted. I suggest that revisionary logicians have independent reasons for restricting such a rule. (shrink)

In the literature on self-referential paradoxes one of the hardest and most challenging problems is that of revenge. This problem can take many shapes, but, typically, it besets non-classical accounts of some semantic notion, such as truth, that depend on a set of classically defined meta-theoretic concepts, like validity, consistency, and so on. A particularly troubling form of revenge that has received a lot of attention lately involves the concept of validity. The difficulty lies in that the non-classical logician cannot (...) accept her own definition of validity because it is given in a classical meta-theory. It is often suggested that this mismatch between the consequence relation of the account being espoused and the consequence relation of the meta-theory is a serious embarrassment. The main goal of the paper is to explore whether certain substructural accounts of the paradoxes can avoid this sort of embarrassment. Typically, these accounts are expressively incomplete, since they cannot assert in the object language that certain invalid arguments are in fact invalid. To overcome this difficulty I develop a novel type of hybrid proof-procedure, one that takes invalidities to be just as fundamental as validities. I prove that this proof-procedure enjoys a number of interesting properties and I analyze the prospects of applying it to languages capable of expressing self-referential statements. (shrink)

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. (shrink)

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. (...) Secondly, I show how to add a validity predicate to ST while avoiding the dilemma. (shrink)

Beall and Murzi :143–165, 2013) introduce an object-linguistic predicate for naïve validity, governed by intuitive principles that are inconsistent with the classical structural rules. As a consequence, they suggest that revisionary approaches to semantic paradox must be substructural. In response to Beall and Murzi, Field :1–19, 2017) has argued that naïve validity principles do not admit of a coherent reading and that, for this reason, a non-classical solution to the semantic paradoxes need not be substructural. The aim of this paper (...) is to respond to Field’s objections and to point to a coherent notion of validity which underwrites a coherent reading of Beall and Murzi’s principles: grounded validity. The notion, first introduced by Nicolai and Rossi, is a generalisation of Kripke’s notion of grounded truth, and yields an irreflexive logic. While we do not advocate the adoption of a substructural logic, we take the notion of naïve validity to be a legitimate semantic notion that points to genuine expressive limitations of fully structural revisionary approaches. (shrink)

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. (shrink)

This special issue collects together nine new essays on logical consequence :the relation obtaining between the premises and the conclusion of a logically valid argument. The present paper is a partial, and opinionated,introduction to the contemporary debate on the topic. We focus on two influential accounts of consequence, the model-theoretic and the proof-theoretic, and on the seeming platitude that valid arguments necessarilypreserve truth. We briefly discuss the main objections these accounts face, as well as Hartry Field’s contention that such objections (...) show consequenceto be a primitive, indefinable notion, and that we must reject the claim that valid arguments necessarily preserve truth. We suggest that the accountsin question have the resources to meet the objections standardly thought to herald their demise and make two main claims: (i) that consequence, as opposed to logical consequence, is the epistemologically significant relation philosophers should be mainly interested in; and (ii) that consequence is a paradoxical notion if truth is. (shrink)

Logical orthodoxy has it that classical first-order logic, or some extension thereof, provides the right extension of the logical consequence relation. However, together with naïve but intuitive principles about semantic notions such as truth, denotation, satisfaction, and possibly validity and other naïve logical properties, classical logic quickly leads to inconsistency, and indeed triviality. At least since the publication of Kripke’s Outline of a theory of truth , an increasingly popular diagnosis has been to restore consistency, or at least non-triviality, by (...) restricting some classical rules. Our modest aim in this note is to briefly introduce the main strands of the current debate on paradox and logical revision, and point to some of the potential challenges revisionary approaches might face, with reference to the nine contributions to the present volume.For a recent introduction to non-classical theories of truth and other semantic notions, see the excellent Beall a .. (shrink)

The perhaps most important criticism of the nontransitive approach to semantic paradoxes is that it cannot truthfully express exactly which metarules preserve validity. I argue that this criticism overlooks that the admissibility of metarules cannot be expressed in any logic that allows us to formulate validity-Curry sentences and that is formulated in a classical metalanguage. Hence, the criticism applies to all approaches that do their metatheory in classical logic. If we do the metatheory of nontransitive logics in a nontransitive logic, (...) however, there is no reason to think that the argument behind the criticism goes through. In general, asking a logic to express its own admissible metarules may not be a good idea. (shrink)

Several authors have argued that a version of Curry's paradox involving validity motivates rejecting the structural rule of contraction. This paper criticizes two recently suggested alternative responses to “validity Curry.” There are three salient stages in a validity Curry derivation. Rejecting contraction blocks the first, while the alternative responses focus on the second and third. I show that a distinguishing feature of validity Curry, as contrasted with more familiar forms of Curry's paradox, is that paradox arises already at the first (...) stage. Accordingly, blocking the second or third stages won't suffice for resolving the paradox. (shrink)

In this paper we explain our pretense account of truth-talk and apply it in a diagnosis and treatment of the Liar Paradox. We begin by assuming that some form of deflationism is the correct approach to the topic of truth. We then briefly motivate the idea that all T-deflationists should endorse a fictionalist view of truth-talk, and, after distinguishing pretense-involving fictionalism (PIF) from error- theoretic fictionalism (ETF), explain the merits of the former over the latter. After presenting the basic framework (...) of our PIF account of truth-talk, we demonstrate a few advantages it offers over T-deflationist accounts that do not explicitly acknowledge pretense at work in the discourse. In turning to the Liar Paradox, we explain how the quasi-anaphoric functioning that our account attributes to truth-talk provides a diagnosis of the Liar Paradox (and other instances of semantic pathology) as having no content—in the sense of not specifying any of what we call M-conditions. At the same time, however, we vindicate the intuition that we can understand liar sentences, thereby avoiding one standard objection to “meaningless strategy” responses to the Liar Paradox. With this diagnosis in place, we then, by way of treatment, introduce a new predicate, ‘semantically defective’, and show how the explanation we give for its application allows for a consistent, yet revenge-immune, (dis)solution of the Liar Paradox, and semantic pathology generally. (shrink)

Consider the following argument written on the board in room 227: 1 = 1. So, the argument on the board in room 227 is not valid. This argument generates a paradox. The aim of this paper is to present a resolution of this paradox and related paradoxes of validity, including a version of the Curry paradox. The proposal stresses the close connections between these validity paradoxes and paradoxes of truth and paradoxes of denotation. So a more general aim is to (...) provide a unified response to semantic paradox. The positive proposal is in part inspired by a brief, tantalizing suggestion of Gödel’s, that the paradoxes might appear “as something analogous to dividing by zero”—so that the concept of validity, for example, is everywhere applicable except for certain singular points or singularities. A second central claim is that ‘valid’ is a context-sensitive predicate. The key notions of this contextual-singularity theory are presented and applied to a variety of cases. Any purported solution to paradox must deal with the phenomenon of revenge, and a response to revenge is outlined. The paper closes with remarks about the accommodation of Tarski’s claim that natural languages are universal. (shrink)

Rejecting structural contraction has been proposed as a strategy for escaping semantic paradoxes. The challenge for its advocates has been to make intuitive sense of how contraction might fail. I offer a way of doing so, based on a “naive” interpretation of the relation between structure and logical vocabulary in a sequent proof system. The naive interpretation of structure motivates the most common way of blaming Curry-style paradoxes on illicit contraction. By contrast, the naive interpretation will not as easily motivate (...) one recent noncontractive approach to the Liar paradox. (shrink)

We discuss the principles for a primitive, object-linguistic notion of consequence proposed by ) that yield a version of Curry’s paradox. We propose and study several strategies to weaken these principles and overcome paradox: all these strategies are based on the intuition that the object-linguistic consequence predicate internalizes whichever meta-linguistic notion of consequence we accept in the first place. To these solutions will correspond different conceptions of consequence. In one possible reading of these principles, they give rise to a notion (...) of logical consequence: we study the corresponding theory of validity by showing that it is conservative over a wide range of base theories: this result is achieved via a well-behaved form of local reduction. The theory of logical consequence is based on a restriction of the introduction rule for the consequence predicate. To unrestrictedly maintain this principle, we develop a conception of object-linguistic consequence, which we call grounded consequence, that displays a restriction of the structural rule of reflexivity. This construction is obtained by generalizing Saul Kripke’s inductive theory of truth. Grounded validity will be shown to satisfy several desirable principles for a naïve, self-applicable notion of consequence. (shrink)

Semantic paradoxes pose a real threat to logics that attempt to be capable of expressing their own semantic concepts. Particularly, Curry paradoxes seem to show that many solutions must change our intuitive concepts of truth or validity or impose limits on certain inferences that are intuitively valid. In this way, the logic of a universal language would have serious problems. In this paper, we explore a different solution that tries to avoid both limitations as much as possible. Thus, we argue (...) that it is possible to capture the naive concepts of truth and validity without losing any of the valid inferences of classical logic. This approach is called the Buenos Aires plan. We present the logic of truth and validity, $$\mathsf {STTV}_{\omega }$$ based on the hierarchy of logics $$\textsf{ST}_{\omega }$$, whose validity predicate has the same semantic conditions as the material conditional. We argue that $$\mathsf {STTV}_{\omega }$$ is capable of blocking the problematic results while keeping the deductive power of classical logic as much as possible and offering an adequate semantic theory. On the other hand, one could object that it is not possible to reason with $$\mathsf {STTV}_{\omega }$$ because it is not closed under its logical principles. We respond to this objection and argue that the local characterization of validity shows how to make inferences using the logic $$\textsf{ST}_{\omega }$$. (shrink)

El objetivo de este artículo es investigar diversos resultados limitativos acerca del concepto de validez. En particular, argumento que ninguna teoría lógica de orden superior con semántica estándar puede tener recursos expresivos suficientes como para capturar su propio concepto de validez. Además, muestro que la lógica de la verdad transparente que Hartry Field desarrolló recientemente conduce a resultados limitativos similares.

We defend deflationism about truth against a pressing challenge, which is to explain how deflationists can understand the role that the _concept_ of truth appears to play in accounts of several other philosophically important concepts. We provide three strategies that deflationists can employ in response to the specific challenge regarding assertion that has been raised in several recent articles, viz., that the truth concept plays an ineliminable explanatory role in an account of assertion. We then show how to extend our (...) strategies to accounts of other central philosophical concepts, by applying them to accounts of belief, knowledge, and logical validity. The result is a set of recipes for deflationists about truth to employ in developing responses to worries that might be raised about the explanatory role of the truth concept. (shrink)

There have been recent arguments against the idea that substructural solutions are uniform. The claim is that even if the substructuralist solves the common semantic paradoxes uniformly by targeting Cut or Contraction, with additional machinery, we can construct higher-level paradoxes (e.g., a higher-level Liar, a higher-level Curry, and a meta-validity Curry). These higher-level paradoxes do not use metainferential Cut or Contraction, but rather, higher-level Cuts and higher-level Contractions. These kinds of paradoxes suggest that targeting Cut or Contraction is not enough (...) for solving semantic paradoxes; the substructuralist must target Cut of every level or Contraction of every level to solve the paradoxes. Hence, the substructuralists do not provide as uniform of a solution as they hoped they did. In response, we argue that the substructuralists need not admit these additional machineries. In fact, they are redundant in light of the validity predicate (i.e., there is no gain in terms of expressive power). The validity predicate is powerful enough to creep these paradoxes in the object level. The substructuralist does not need to ascend to metainferences to construct higher-level paradoxes. Moreover, there is a reading available to the substructuralist such that all the higher-level structural rules would collapse to instances of the object-level structural rules (e.g., meta \(_n\) Cut and meta \(_n\) Contraction would become instances of Cut and Contraction). We then address Barrio et al.’s worry that the validity predicate has its shortcomings; the substructuralist cannot internalize some of its metarules. We claim that the validity of metarules can be internalized without the need to strengthen the validity predicate. However, a problem raised by Barrio et al. is still present—the problem of internalizing unwanted instances of Cut in Cut-free approaches. We argue that this internalization problem is not unique to the validity predicate; the same problem is present with other problematic predicates, such as the truth predicate and the provability predicate. (shrink)