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)

What is the source of logical and mathematical truth? This book revitalizes conventionalism as an answer to this question. Conventionalism takes logical and mathematical truth to have their source in linguistic conventions. This was an extremely popular view in the early 20th century, but it was never worked out in detail and is now almost universally rejected in mainstream philosophical circles. Shadows of Syntax is the first book-length treatment and defense of a combined conventionalist theory of logic and mathematics. It (...) argues that our conventions, in the form of syntactic rules of language use, are perfectly suited to explain the truth, necessity, and a priority of logical and mathematical claims, as well as our logical and mathematical knowledge. (shrink)

Some have argued for a division of epistemic labor in which mathematicians supply truths and philosophers supply their necessity. We argue that this is wrong: mathematics is committed to its own necessity. Counterfactuals play a starring role.

Some have argued for a division of epistemic labor in which mathematicians supply truths and philosophers supply their necessity. We argue that this is wrong: mathematics is committed to its own necessity. Counterfactuals play a starring role.

This piece continues the tradition of arguments by John Lucas, Roger Penrose and others to the effect that the human mind is not a machine. Kurt Gödel thought that the intensional paradoxes stand in the way of proving that the mind is not a machine. According to Gödel, a successful proof that the mind is not a machine would require a solution to the intensional paradoxes. We provide what might seem to be a partial vindication of Gödel and show that (...) if a particular solution to the intensional paradoxes is adopted, one can indeed give an argument to the effect that the mind is not a machine. (shrink)

Since Montague’s work it is well known that treating a single modality as a predicate may lead to paradox. In their paper “No Future”, Horsten and Leitgeb show that if the two temporal modalities are treated as predicates paradox might arise as well. In our paper we investigate whether paradoxes of multiple modalities, such as the No Future paradox, are genuinely new paradoxes or whether they “reduce” to the paradoxes of single modalities. In order to address this question we develop (...) a notion of reducibility based on a version of Smoryński Diagonalized Operator Logic. We show that there are reducible multimodal paradoxes as well as irreducible paradoxes of interaction. In particular, we show the No Future paradox to be an irreducible paradox according to our notion of reducibility. (shrink)

Justified belief is a core concept in epistemology and there has been an increasing interest in its logic over the last years. While many logical investigations consider justified belief as an operator, in this paper, we propose a logic for justified belief in which the relevant notion is treated as a predicate instead. Although this gives rise to the possibility of liar-like paradoxes, a predicate treatment allows for a rich and highly expressive framework, which lives up to the universal ambitions (...) of investigating epistemological concepts. We start with a base theory for justified belief, and then systematically present putative additional axioms for justified belief. We provide an overview of consistency results when the additional principles are added to the base theory, and discuss their philosophical plausibility. (shrink)

Classical logic of formal provability includes Löb’s theorem, but not reflection. In contrast, intuitions about the inferential behavior of informal provability (in informal mathematics) seem to invalidate Löb’s theorem and validate reflection (after all, the intuition is, whatever mathematicians prove holds!). We employ a non-deterministic many-valued semantics and develop a modal logic T-BAT of an informal provability operator, which indeed does validate reflection and invalidates Löb’s theorem. We study its properties and its relation to known provability-related paradoxical arguments. We also (...) argue that T-BAT is a fairly sensible candidate for a formal logic of informal provability. (shrink)

According to the dialetheist argument from the inconsistency of informal mathematics, the informal version of the Gödelian argument leads us to a true contradiction. On one hand, the dialetheist argues, we can prove that there is a mathematical claim that is neither provable nor refutable in informal mathematics. On the other, the proof of its unprovability is given in informal mathematics and proves that very sentence. We argue that the argument fails, because it relies on the unjustified and unlikely assumption (...) that the informal Gödel sentence is informally provable. (shrink)

BAT is a logic built to capture the inferential behavior of informal provability. Ultimately, the logic is meant to be used in an arithmetical setting. To reach this stage it has to be extended to a first-order version. In this paper we provide such an extension. We do so by constructing non-deterministic three-valued models that interpret quantifiers as some sorts of infinite disjunctions and conjunctions. We also elaborate on the semantical properties of the first-order system and consider a couple of (...) its strengthenings. It turns out that obtaining a sensible strengthening is not straightforward. We prove that most strategies commonly used for strengthening non-deterministic logics fail in our case. Nevertheless, we identify one method of extending the system which does not. (shrink)

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)

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)

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)

It is often argued that fully structural theories of truth and related notions are incapable of expressing a nonstratified notion of defectiveness. We argue that recently much-discussed non-contractive theories suffer from the same expressive limitation, provided they identify the defective sentences with the sentences that yield triviality if they are assumed to satisfy structural contraction.

The aim of this paper is to provide epistemic reasons for investigating the notions of informal rigour and informal provability. I argue that the standard view of mathematical proof and rigour yields an implausible account of mathematical knowledge, and falls short of explaining the success of mathematical practice. I conclude that careful consideration of mathematical practice urges us to pursue a theory of informal provability.

An epistemic formalization of arithmetic is constructed in which certain non-trivial metatheoretical inferences about the system itself can be made. These inferences involve the notion of provability in principle, and cannot be made in any consistent extensions of Stewart Shapiro's system of epistemic arithmetic. The system constructed in the paper can be given a modal-structural interpretation.

New epistemic principles are formulated in the language of Shapiro's system of Epistemic Arithmetic. It is argued that some plausibility can be attributed to these principles. The relations between these principles and variants of controversial constructivistic principles are investigated. Special attention is given to variants of the intuitionistic version of Church's thesis and to variants of Markov's principle.

The present paper discusses a proposal which says,roughly and with several qualifications, that thecollection of mathematical truths is identical withthe set of theorems of ZFC. It is argued that thisproposal is not as easily dismissed as outright falseor philosophically incoherent as one might think. Some morals of this are drawn for the concept ofmathematical knowledge.

The topic of this article is the closure of a priori knowability under a priori knowable material implication: if a material conditional is a priori knowable and if the antecedent is a priori knowable, then the consequent is a priori knowable as well. This principle is arguably correct under certain conditions, but there is at least one counterexample when completely unrestricted. To deal with this, Anderson proposes to restrict the closure principle to necessary truths and Horsten suggests to restrict it (...) to formulas that belong to less expressive languages. In this article it is argued that Horsten’s restriction strategy fails, because one can deduce that knowable ignorance entails necessary ignorance from the closure principle and some modest background assumptions, even if the expressive resources do not go beyond those needed to formulate the closure principle itself. It is also argued that it is hard to find a justification for Anderson’s restricted closure principle, because one cannot deduce it even if one assumes very strong modal and epistemic background principles. In addition, there is an independently plausible alternative closure principle that avoids all the problems without the need for restriction. (shrink)

The deviation of mathematical proof—proof in mathematical practice—from the ideal of formal proof—proof in formal logic—has led many philosophers of mathematics to reconsider the commonly accepted view according to which the notion of formal proof provides an accurate descriptive account of mathematical proof. This, in turn, has motivated a search for alternative accounts of mathematical proof purporting to be more faithful to the reality of mathematical practice. Yet, in order to develop and evaluate such alternative accounts, it appears as a (...) necessary prerequisite to first possess a clear picture of what the deviation of mathematical proof from formal proof consists in. The present work aims to contribute building such a picture by investigating the relation between the elementary steps of deduction constituting the two types of proofs—mathematical inference and logical inference. Many claims have been made in the literature regarding the relation between mathematical inference and logical inference, most of them stating that the former is lacking properties that are constitutive of the latter. Such differentiating claims are, however, usually put forward without a clear conception of the properties occurring in them, and are generally considered to be immediately justified by our direct acquaintance, or phenomenological experience, with the two types of inferences. The present study purports to advance our understanding of the relation between mathematical inference and logical inference by developing a detailed philosophical analysis of the differentiating claims, that is, an analysis of the meaning of the differentiating claims—through the properties that occur in them—as well as the reasons that support them. To this end, we provide at the outset a representative list of the different properties of logical inference that have occurred in the differentiating claims, and we notice that they all boil down to the three properties of formality, generality, and mechanicality. For each one of these properties, our analysis proceeds in two steps: we first provide precise conceptual characterizations of the different ways logical inference has been said to be formal, general, and mechanical, in the philosophical and logical literature on formal proof; we then examine why mathematical inference does not appear to be formal, general, and mechanical, for the different variations of these notions identified. Our study results in a precise conceptual apparatus for expressing and discussing the properties differentiating mathematical inference from logical inference, and provides a first inventory of the various reasons supporting the observations of those differences. The differentiating claims constitute thus a set of data that any philosophical account of mathematical inference and proof purporting to be more faithful to mathematical practice ought to be able to accommodate and explain. (shrink)

In a series of articles dating from 1903 to 1906, Frege criticizes Hilbert’s methodology of proving the independence and consistency of various fragments of Euclidean geometry in his Foundations of Geometry. In the final part of the last article, Frege makes his own proposal as to how the independence of genuine axioms should be proved. Frege contends that independence proofs require the development of a ‘new science’ with its own basic truths. This paper aims to provide a reconstruction of this (...) New Science that meets modern standards and to examine possible problems surrounding Frege’s original proposal. The paper is organized as follows: the first two sections summarize the main points of the Frege–Hilbert controversy and discuss some issues surrounding the problem of independence proofs. Section 3 contains an informal presentation of Frege’s proposal. In section 4 a more detailed reconstruction of Frege’s New Science is set out while section 5 examines what is left out. The concluding section is devoted to a discussion of Frege’s strategy and its significance from a broader perspective. (shrink)

The knower paradox states that the statement ‘We know that this statement is false’ leads to inconsistency. This article presents a fresh look at this paradox and some well-known solutions from the literature. Paul Égré discusses three possible solutions that modal provability logic provides for the paradox by surveying and comparing three different provability interpretations of modality, originally described by Skyrms, Anderson, and Solovay. In this article, some background is explained to clarify Égré’s solutions, all three of which hinge on (...) intricacies of provability logic and its arithmetical interpretations. To check whether Égré’s solutions are satisfactory, we use the criteria for solutions to paradoxes defined by Susan Haack and we propose some refinements of them. This article aims to describe to what extent the knower paradox can be solved using provability logic and to what extent the solutions proposed in the literature satisfy Haack’s criteria. Finally, the article offers some reflections on the relation between knowledge, proof, and provability, as inspired by the knower paradox and its solutions. (shrink)

The Paradox of the Knower was originally presented by Kaplan and Montague [26] as a puzzle about the everyday notion of knowledge in the face of self-reference. The paradox shows that any theory extending Robinson arithmetic with a predicate K satisfying the factivity axiom K → A as well as a few other epistemically plausible principles is inconsistent. After surveying the background of the paradox, we will focus on a recent debate about the role of epistemic closure principles in the (...) Knower. We will suggest this debate sheds new light on the concept of knowledge which is at issue in the paradox – i.e. is it a “thin” notion divorced from concepts such as evidence or justification, or is it a “thick” notion more closely resembling mathematical provability? We will argue that a number of features of the paradox suggest that the latter option is more plausible. Along the way, we will provide a reconstruction of the paradox using a quantified extension of Artemovʼs [2] Logic of Proofs, as well as a series of results linking the original formulation of the paradox to reflection principles for formal arithmetic. On this basis, we will argue that while the Knower can be understood to motivate a distinction between levels of knowledge, it does not provide a rationale for recognizing a uniform hierarchy of knowledge predicates in the manner suggested by Anderson. (shrink)

The goal of this paper is to explore the significance of Montague’s paradox—that is, any arithmetical theory $T\supseteq Q$ over a language containing a predicate $P$ satisfying $P\rightarrow \varphi $ and $T\vdash \varphi \,\therefore\,T\vdash P$ is inconsistent—as a limitative result pertaining to the notions of formal, informal, and constructive provability, in their respective historical contexts. To this end, the paradox is reconstructed in a quantified extension $\mathcal {QLP}$ of Artemov’s logic of proofs. $\mathcal {QLP}$ contains both explicit modalities $t:\varphi $ (...) and also proof quantifiers $x:\varphi $. In this system, the basis for the rule NEC is decomposed into a number of distinct principles governing how various modes of reasoning about proofs and provability can be internalized within the system itself. A conceptually motivated resolution to the paradox is proposed in the form of an argument for rejecting the unrestricted rule NEC on the basis of its subsumption of an intuitively invalid principle pertaining to the interaction of proof quantifiers and the proof-theorem relation expressed by explicit modalities. (shrink)

After a brief discussion of Kreisel’s notion of informal rigour and Myhill’s notion of absolute proof, Gödel’s analysis of the subject is presented. It is shown how Gödel avoids the notion of informal proof because such a use would contradict one of the senses of “formal” that Gödel wants to preserve. This Gödelian notion of “formal” is directly tied to his notion of absolute proof and to the question of the general applicability of concepts, in a way that overcomes both (...) Kreisel and Myhill’s conceptions. This paper aims to contribute to the present-day debate on informal and epistemic mathematics, focusing on what appears necessary for a better understanding of the issues at stake. (shrink)

Since the time of Aristotle's students, interpreters have considered Prior Analytics to be a treatise about deductive reasoning, more generally, about methods of determining the validity and invalidity of premise-conclusion arguments. People studied Prior Analytics in order to learn more about deductive reasoning and to improve their own reasoning skills. These interpreters understood Aristotle to be focusing on two epistemic processes: first, the process of establishing knowledge that a conclusion follows necessarily from a set of premises (that is, on the (...) epistemic process of extracting information implicit in explicitly given information) and, second, the process of establishing knowledge that a conclusion does not follow. Despite the overwhelming tendency to interpret the syllogistic as formal epistemology, it was not until the early 1970s that it occurred to anyone to think that Aristotle may have developed a theory of deductive reasoning with a well worked-out system of deductions comparable in rigor and precision with systems such as propositional logic or equational logic familiar from mathematical logic. When modern logicians in the 1920s and 1930s first turned their attention to the problem of understanding Aristotle's contribution to logic in modern terms, they were guided both by the Frege-Russell conception of logic as formal ontology and at the same time by a desire to protect Aristotle from possible charges of psychologism. They thought they saw Aristotle applying the informal axiomatic method to formal ontology, not as making the first steps into formal epistemology. They did not notice Aristotle's description of deductive reasoning. Ironically, the formal axiomatic method (in which one explicitly presents not merely the substantive axioms but also the deductive processes used to derive theorems from the axioms) is incipient in Aristotle's presentation. Partly in opposition to the axiomatic, ontically-oriented approach to Aristotle's logic and partly as a result of attempting to increase the degree of fit between interpretation and text, logicians in the 1970s working independently came to remarkably similar conclusions to the effect that Aristotle indeed had produced the first system of formal deductions. They concluded that Aristotle had analyzed the process of deduction and that his achievement included a semantically complete system of natural deductions including both direct and indirect deductions. Where the interpretations of the 1920s and 1930s attribute to Aristotle a system of propositions organized deductively, the interpretations of the 1970s attribute to Aristotle a system of deductions, or extended deductive discourses, organized epistemically. The logicians of the 1920s and 1930s take Aristotle to be deducing laws of logic from axiomatic origins; the logicians of the 1970s take Aristotle to be describing the process of deduction and in particular to be describing deductions themselves, both those deductions that are proofs based on axiomatic premises and those deductions that, though deductively cogent, do not establish the truth of the conclusion but only that the conclusion is implied by the premise-set. Thus, two very different and opposed interpretations had emerged, interestingly both products of modern logicians equipped with the theoretical apparatus of mathematical logic. The issue at stake between these two interpretations is the historical question of Aristotle's place in the history of logic and of his orientation in philosophy of logic. This paper affirms Aristotle's place as the founder of logic taken as formal epistemology, including the study of deductive reasoning. A by-product of this study of Aristotle's accomplishments in logic is a clarification of a distinction implicit in discourses among logicians--that between logic as formal ontology and logic as formal epistemology. (shrink)

This is a paper for a special issue of Semiotic Studies devoted to Stanislaw Krajewski’s paper. This paper gives some supplementary notes to Krajewski’s on the Anti-Mechanist Arguments based on Gödel’s incompleteness theorem. In Section 3, we give some additional explanations to Section 4–6 in Krajewski’s and classify some misunderstandings of Gödel’s incompleteness theorem related to AntiMechanist Arguments. In Section 4 and 5, we give a more detailed discussion of Gödel’s Disjunctive Thesis, Gödel’s Undemonstrability of Consistency Thesis and the definability (...) of natural numbers as in Section 7–8 in Krajewski’s, describing how recent advances bear on these issues. (shrink)

The implicit commitment thesis is the claim that believing in a mathematical theory S carries an implicit commitment to further sentences not deductively entailed by the theory, such as the consistency sentence Con(S). I provide a new argument for this thesis based on the notion of mathematical certainty. I also reply to a recent argument by Walter Dean against the implicit commitment thesis, showing that my formulation of the thesis avoids the difficulties he raises.

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)

In 1933 Godel introduced a calculus of provability (also known as modal logic S4) and left open the question of its exact intended semantics. In this paper we give a solution to this problem. We find the logic LP of propositions and proofs and show that Godel's provability calculus is nothing but the forgetful projection of LP. This also achieves Godel's objective of defining intuitionistic propositional logic Int via classical proofs and provides a Brouwer-Heyting-Kolmogorov style provability semantics for Int which (...) resisted formalization since the early 1930s. LP may be regarded as a unified underlying structure for intuitionistic, modal logics, typed combinatory logic and λ-calculus. (shrink)

We analyse Kreisel’s notion of human-effective computability. Like Kreisel, we relate this notion to a concept of informal provability, but we disagree with Kreisel about the precise way in which this is best done. The resulting two different ways of analysing human-effective computability give rise to two different variants of Church’s thesis. These are both investigated by relating them to transfinite progressions of formal theories in the sense of Feferman.

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)