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)

Substructural approaches to paradoxes have attracted much attention from the philosophical community in the last decade. In this paper we focus on two substructural logics, named ST and TS, along with two structural cousins, LP and K3. It is well known that LP and K3 are duals in the sense that an inference is valid in one logic just in case the contrapositive is valid in the other logic. As a consequence of this duality, theories based on either logic are (...) tightly connected since many of the arguments for and objections against one theory reappear in the other theory in dual form. The target of the paper is making explicit in exactly what way, if any, ST and TS are dual to one another. The connection will allow us to gain a more fine-grained understanding of these logics and of the theories based on them. In particular, we will obtain new insights on two questions concerning ST which are being intensively discussed in the current literature: whether ST preserves classical logic and whether it is LP in sheep’s clothing. Explaining in what way ST and TS are duals requires comparing these logics at a metainferential level. We provide to this end a uniform proof theory to decide on valid metainferences for each of the four logics. This proof procedure allows us to show in a very simple way how different properties of inferences (unsatisfiability, supersatisfiability and antivalidity) that behave in very different ways for each logic can be captured in terms of the validity of a metainference. (shrink)

Ultralogic as Universal? is a seminal text in non-classcial logic. Richard Routley presents a hugely ambitious program: to use an 'ultramodal' logic as a universal key, which opens, if rightly operated, all locks. It provides a canon for reasoning in every situation, including illogical, inconsistent and paradoxical ones, realized or not, possible or not. A universal logic, Routley argues, enables us to go where no other logic—especially not classical logic—can. Routley provides an expansive and singular vision of how a universal (...) logic might one day solve major problems in set theory, arithmetic, linguistics, physics, and more. It circulated in typescript in the late 1970s before appearing as the Appendix to Exploring Meinong's Jungle and Beyond. With engaging, forceful prose, unsparing criticism of entrenched institutions, and many tantalizing proof sketches, Ultralogic? has had a major influence on the development of paraconsistent and relevant logic. This new edition makes this work available for a modern audience, newly typeset and corrected, along with extensive notes, and new commentary essays. (shrink)

Standard reasoning about Kripke semantics for modal logic is almost always based on a background framework of classical logic. Can proofs for familiar definability theorems be carried out using anonclassical substructural logicas the metatheory? This article presents a semantics for positive substructural modal logic and studies the connection between frame conditions and formulas, via definability theorems. The novelty is that all the proofs are carried out with anoncontractive logicin the background. This sheds light on which modal principles are invariant under (...) changes of metalogic, and provides (further) evidence for the general viability of nonclassical mathematics. (shrink)

An important question for proponents of non-contractive approaches to paradox is why contraction fails. Zardini offers an answer, namely that paradoxical sentences exhibit a kind of instability. I elaborate this idea using revision theory, and I argue that while instability does motivate failures of contraction, it equally motivates failure of many principles that non-contractive theorists want to maintain.

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)

I provide an interpretation of Wittgenstein's much criticized remarks on Gödel's First Incompleteness Theorem in the light of paraconsistent arithmetics: in taking Gödel's proof as a paradoxical derivation, Wittgenstein was right, given his deliberate rejection of the standard distinction between theory and metatheory. The reasoning behind the proof of the truth of the Gödel sentence is then performed within the formal system itself, which turns out to be inconsistent. I show that the models of paraconsistent arithmetics (obtained via the Meyer-Mortensen (...) collapsing filter on the standard model) match with many intuitions underlying Wittgensteins philosophy of mathematics, such as its strict finitism and the insistence on the decidability of any meaningful mathematical question. (shrink)

Dialetheists believe some sentences are both true and false. Objectors have argued that this makes it unclear how people can disagree with each other because, given the dialetheist’s commitments, if I make a claim and you tell me my claim is false, we might both be correct. Graham Priest (2006a) thinks that people disagree by rejecting or denying what is said rather than ascribing falsehood to it. We build on the work of Julien Murzi and Massimiliano Carrara (2015) and show (...) that Priest’s approach cannot succeed: given the same dialetheist’s commitments you may be correct to reject a claim that I correctly believe. We argue further that any attempt to solve the problem by identifying a new attitude of disagreement will also fail. The culprit, we claim, is the attempt to find a pair of attitudes that satisfy ‘exclusivity’—that is, attitudes such that both cannot be simultaneously correct. Instead of identifying disagreement by the kinds of attitudes involved, we propose dialetheists focus on the normative landscape and identify it in part by whether parties have reasons to change their attitudes. We offer our own normative theory of disagreement to help dialetheists with this challenge. (shrink)

Nonclassical theories of truth have in common that they reject principles of classical logic to accommodate an unrestricted truth predicate. However, different nonclassical strategies give up different classical principles. The paper discusses one criterion we might use in theory choice when considering nonclassical rivals: the maxim of minimal mutilation.

Some theorists have developed formal approaches to truth that depend on counterexamples to the structural rules of contraction. Here, we study such approaches, with an eye to helping them respond to a certain kind of objection. We define a contractive relative of each noncontractive relation, for use in responding to the objection in question, and we explore one example: the contractive relative of multiplicative-additive affine logic with transparent truth, or MAALT. -/- .

We observe that removing contraction from a standard sequent calculus for first-order predicate logic preserves completeness for the modal fragment.

This paper begins an axiomatic development of naive set theoryin a paraconsistent logic. Results divide into two sorts. There is classical recapture, where the main theorems of ordinal and Peano arithmetic are proved, showing that naive set theory can provide a foundation for standard mathematics. Then there are major extensions, including proofs of the famous paradoxes and the axiom of choice (in the form of the well-ordering principle). At the end I indicate how later developments of cardinal numbers will lead (...) to Cantor’s theorem, the existence of large cardinals, and a counterexample to the continuum hypothesis. (shrink)

I propose an approach to liar and Curry paradoxes inspired by the work of Roger Swyneshed in his treatise on insolubles (1330-1335). The keystone of the account is the idea that liar sentences and their ilk are false (and only false) and that the so-called ''capture'' direction of the T-schema should be restricted. The proposed account retains what I take to be the attractive features of Swyneshed's approach without leading to some worrying consequences Swyneshed accepts. The approach and the resulting (...) logic (called ''Swynish Logic'') are non-classical, but are consistent and compatible with many elements of the classical picture including modus ponens, modus tollens, and double-negation elimination and introduction. It is also compatible with bivalence and contravalence. My approach to these paradoxes is also immune to an important kind of revenge challenge that plagues some of its rivals. (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)

In recent years, substructural approaches to paradoxes have become quite popular. But whatever restrictions on structural rules we may want to enforce, it is highly desirable that such restrictions be accompanied by independent philosophical motivation, not directly related to paradoxes. Indeed, while these recent developments have shed new light on a number of issues pertaining to paradoxes, it seems that we now have even more open questions than before, in particular two very pressing ones: what (independent) motivations do we have (...) (if any) for restrictions on structural rules, and what to make of the plurality of new logics emerging from these restrictions, i.e. how to ‘choose’ among the different options. In this paper, we address these two questions from the perspective of a dialogical conception of logic that we've been advocating in recent years. We will argue that dialogical interpretations of structural rules, that is, as rules determining specific properties of the dialogues in question, provide a conveniently neutral framework to adjudicate between the different substructural proposals that have been made in the literature on paradoxes. (shrink)

Closure is the idea that what is true about a theory of truth should be true in it. Commitment to closure under truth motivates non-classical logic; commitment to closure under validity leads to substructural logic. These moves can be thought of as responses to revenge problems. With a focus on truth in mathematics, I will consider whether a noncontractive approach faces a similar revenge problem with respect to closure under provability, and argue that if a noncontractive theory is to be (...) genuinely closed, then it must be free of all contraction, even in the metatheory. (shrink)

The naive set theory problem is to begin with a full comprehension axiom, and to find a logic strong enough to prove theorems, but weak enough not to prove everything. This paper considers the sub-problem of expressing extensional identity and the subset relation in paraconsistent, relevant solutions, in light of a recent proposal from Beall, Brady, Hazen, Priest and Restall [4]. The main result is that the proposal, in the context of an independently motivated formalization of naive set theory, leads (...) to triviality. (shrink)

In this paper, I consider the connection between consequence relations and closure operations. I argue that one familiar connection makes good sense of some usual applications of consequence relations, and that a largeish family of familiar noncontractive consequence relations cannot respect this familiar connection.

Substructural logics and their application to logical and semantic paradoxes have been extensively studied. In the paper, we study theories of naïve consequence and truth based on a non-reflexive logic. We start by investigating the semantics and the proof-theory of a system based on schematic rules for object-linguistic consequence. We then develop a fully compositional theory of truth and consequence in our non-reflexive framework.

$$\mathbf {Z}$$ is a new type of non-finitist inference, i.e., an inference that involves treating some infinite collection as completed, designed for contraction free logic with unrestricted abstraction. It has been introduced in Petersen (Studia Logica 64:365–403, 2000) and shown to be consistent within a system $$\mathbf {{}L^iD{}}{}$$ $$_{\uplambda }$$ of contraction free logic with unrestricted abstraction. In Petersen (Arch Math Log 42(7):665–694, 2003) it was established that adding $$ \mathbf {Z}$$ to $$\mathbf {{}L^iD{}}{}$$ $$_{\uplambda }$$ is sufficient to prove (...) the totality of primitive recursive functions but it was also indicated that this would not extend to 2-recursive functions such as the Ackermann–Péter function, for instance. The purpose of the present paper is to expand the underlying idea in the construction of $$\mathbf {Z}$$ to gain a stronger notion, conveniently labeled $$\mathbf {Z}_2$$, which is sufficient to prove a form of nested double induction and thereby the totality of 2-recursive functions. (shrink)

There is significant interest in type-free systems that allow flexible self-application. Such systems are of interest in property theory, natural language semantics, the theory of truth, theoretical computer science, the theory of classes, and category theory. While there are a variety of proposed type-free systems, there is a particularly natural type-free system that we believe is prototypical: the logic of recursive algorithms. Algorithmic logic is the study of basic statements concerning algorithms and the algorithmic rules of inference between such statements. (...) As shown in [1], the threat of paradoxes, such as the Curry paradox, requires care in implementing rules of inference in this context. As in any type-free logic, some traditional rules will fail. The first part of the paper develops a rich collection of inference rules that do not lead to paradox. The second part identifies traditional rules of logic that are paradoxical in algorithmic logic, and so should be viewed with suspicion in type-free logic generally. (shrink)

In a recent series of papers, I and others have advanced new logical approaches to familiar paradoxes. The key to these approaches is to accept full classical logic, and to accept the principles that cause paradox, while preventing trouble by allowing a certain sort ofnontransitivity. Earlier papers have treated paradoxes of truth and vagueness. The present paper will begin to extend the approach to deal with the familiar paradoxes arising in naive set theory, pointing out some of the promises and (...) pitfalls of such an approach. (shrink)

After an introduction to set the stage, we consider some variations on the reasoning behind Curry's Paradox arising against the background of classical propositional logic and of BCI logic and one of its extensions, in the latter case treating the "paradoxicality" as a matter of nonconservative extension rather than outright inconsistency. A question about the relation of this extension and a differently described (though possibly identical) logic intermediate between BCI and BCK is raised in a final section, which closes with (...) a handful of questions left unanswered by our discussion. (shrink)

In this paper I propose a new approach to the foundation of mathematics: non-monotonic set theory. I present two completely different methods to develop set theories based on adaptive logics. For both theories there is a finitistic non-triviality proof and both theories contain (a subtle version of) the comprehension axiom schema. The first theory contains only a maximal selection of instances of the comprehension schema that do not lead to inconsistencies. The second allows for all the instances, also the inconsistent (...) ones, but restricts the conclusions one can draw from them in order to avoid triviality. The theories have enough expressive power to form a justification/explication for most of the established results of classical mathematics. They are therefore not limited by Gödel’s incompleteness theorems. This remarkable result is possible because of the non-recursive character of the final proofs of theorems of non-monotonic theories. I shall argue that, precisely because of the computational complexity of these final proofs, we cannot claim that non-monotonic theories are ideal foundations for mathematics. Nevertheless, thanks to their strength, first order language and the recursive dynamic (defeasible) proofs of theorems of the theory, the non-monotonic theories form (what I call) interesting pragmatic foundations. (shrink)

Z\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {Z}$$\end{document} is a new type of non-finitist inference, i.e., an inference that involves treating some infinite collection as completed, designed for contraction free logic with unrestricted abstraction. It has been introduced in Petersen and shown to be consistent within a system LiD\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {{}L^iD{}}{}$$\end{document}λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{\uplambda }$$\end{document} of contraction free logic with unrestricted abstraction. In Petersen (...) :665–694, 2003) it was established that adding Z\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathbf {Z}$$\end{document} to LiD\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {{}L^iD{}}{}$$\end{document}λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{\uplambda }$$\end{document} is sufficient to prove the totality of primitive recursive functions but it was also indicated that this would not extend to 2-recursive functions such as the Ackermann–Péter function, for instance. The purpose of the present paper is to expand the underlying idea in the construction of Z\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {Z}$$\end{document} to gain a stronger notion, conveniently labeled Z2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {Z}_2$$\end{document}, which is sufficient to prove a form of nested double induction and thereby the totality of 2-recursive functions. (shrink)

This paper is a sequel to my [7]. It focuses on the notion of natural number as introduced in section 11 of that paper with regard to forms of induction and recursive definitions. One point is that this notion of natural number is somewhat weaker than the classical one in so far as it is defined in terms of a weak implication. The other point is the lack of even a weak form of extensionality. As a main result of the (...) present paper it will turn out that the means provided in [7] are sufficient to account for an interpretation of primitive recursive arithmetic. (shrink)

This note shows that the permutation instructions presented by Zardini (2011) for eliminating cuts on universally quantified formulas in the sequent calculus for the noncontractive theory of truth IKTωare inadequate. To that purpose the note presents a derivation in the sequent calculus for IKTωending with an application of cut on a universally quantified formula which the permutation instructions cannot deal with. The counterexample is of the kind that leaves open the question whether cut can be shown to be eliminable in (...) the sequent calculus for IKTωwith an alternative strategy. (shrink)