Logical Pluralism Research Papers - Academia.edu (original) (raw)

The thesis is an investigation into the logical pluralism debate, aiming to understand how the philosophical commitments sustaining each side to the debate connects to more general issues connected to the foundations of logic. My... more

The thesis is an investigation into the logical pluralism debate, aiming to understand how the philosophical commitments sustaining each side to the debate connects to more general issues connected to the foundations of logic. My investigation centers on the following three notions: (1) Epistemic justification, (2) The metaphysical "ground" for logical truth, and (3) Normativity. Chapter 1 traces the monistic and pluralistic conception of logic back to its philosophical/mathematical roots, which we find in the writings of Rudolf Carnap and Gottlob Frege. I argue that logical pluralism - in its more plausible, epistemic (rather than ontic) form - was enabled by the semantic shift which Carnap seems to have anticipated and that, from a conventionalist perspective, his 'Principle of Tolerance' follows as a consequence of that shift. Chapter 2 concerns the issues ensuing from Willard V. O Quine’s critique of Carnap's conventionalism, which had a devastating effect for his foundationalist project. The aim is in particular to address the issue of meaning-variance, a crucial assumption for the conventionalist approach to pluralism. In chapter 3, I present another framework for pluralism, due to Stewart Shapiro’s [2014] ‘modelling’ conception of logic, according which logic is conceived as a mathematical model of natural language. Shapiro argues that our concept of logical consequence is vague and in need of a sharpening to attain a fixed meaning. Pluralism follows from there being two or more equally "correct" such sharpenings; i.e., relative to our theoretical aims. I argue that the modelling-conception is the best way to approach a justification of basic logical laws. However, since that conception also grounds Timothy Williamson’s [2017] argument for monism, I argue that the conception ultimately fails to establish logical pluralism. Since both Williamson and Shapiro take a pragmatic approach to justification, I conclude that the question of pluralism does not turn on epistemological commitments (i.e., on (1)), and suggest instead that it is a matter of (2), i.e., of one's conception of the "ground" for logical truth.

Logical pluralism is the view that there is more than one correct logic. This very general characterization gives rise to a whole family of positions. I argue that not all of them are stable. The main argument in the paper is inspired by... more

Logical pluralism is the view that there is more than one correct logic. This very general characterization gives rise to a whole family of positions. I argue that not all of them are stable. The main argument in the paper is inspired by considerations in Keefe (2014), Priest (2006a), Read (2006), and Williamson (1988), and it aims at the most popular form of logical pluralism advocated by Beall and Restall (2000, 2006). I argue that there is a more general argument available that challenges all variants of logical pluralism that meet the following three conditions: (i) that there are at least two correct logical systems characterized in terms of different consequence relations, (ii) that there is some sort of rivalry among the correct logics, and (iii) that logical consequence is normative. The hypothesis I argue for amounts to what Caret (2016) calls a ‘collapse problem’ in form the of a conditional claim: If a position satisfies all these conditions, then that position is unstable in the sense that it collapses into competing positions.

In this article, we discuss a simple argument that modal metaphysics is misconceived, and responses to it. Unlike Quine’s, this argument begins with the banal observation that there are different candidate interpretations of the... more

In this article, we discuss a simple argument that modal metaphysics is misconceived, and responses to it. Unlike Quine’s, this argument begins with the banal observation that there are different candidate interpretations of the predicate ‘could have been the case’. This is analogous to the observation that there are different candidate interpretations of the predicate ‘is a member of’. The argument then infers that the search for metaphysical necessities is misguided in much the way the ‘set-theoretic pluralist’ (Clarke-Doane & Hamkins [2017]) claims that the search for the true axioms of set theory is. We show that the obvious responses to this argument fail. However, a new response has emerged that purports to prove, from higher order logical principles, that metaphysical possibility is the broadest kind of possibility applying to propositions, and is to that extent special. We distill two lines of reasoning from the literature, and argue that their import depends on premises that a ‘modal pluralist’ should deny. Both presuppose that there is a unique typed hierarchy, which is what the modal pluralist, in the context of higher-order logic, should disavow. In other words, both presupposes that there is a unique candidate for what higher-order claims could mean. We consider the worry that, in a higher-order setting, modal pluralism faces an insuperable problem of articulation, collapses into modal monism, is vulnerable to the Russell-Myhill paradox, or even contravenes the truism that there is a unique actual world, and argue that these worries are misplaced. We also sketch the bearing of the resulting ‘Higher Order Pluralism’ on the theory of content. An upshot is that, if Higher Order Pluralism is true, then there is no stable metatheory from which to characterize higher order reality.

I propose a comprehensive account of negation as a modal operator, vindicating a moderate logical pluralism. Negation is taken as a quantifier on worlds, restricted by an accessibility relation encoding the basic concept of compatibility.... more

I propose a comprehensive account of negation as a modal operator, vindicating a moderate logical pluralism. Negation is taken as a quantifier on worlds, restricted by an accessibility relation encoding the basic concept of compatibility. This latter captures the core meaning of the operator. While some candidate negations are then ruled out as violating plausible constraints on compatibility, different specifications of the notion of world support different logical conducts for (the admissible) negations. The approach unifies in a philosophically motivated picture the following results: nothing can be called a negation properly if it does not satisfy (Minimal) Contraposition and Double Negation Introduction; the pair consisting of two split or Galois negations encodes a distinction without a difference; some paraconsistent negations also fail to count as real negations, but others may; intuitionistic negation qualifies; and classical Boolean negation does as well, to the extent that constructivist and paraconsistent doubts on it do not turn on the basic concept of compatibility, but on the interpretation of worlds.

This article examines the traditional and modern doctrines of categorical propositions and argues that both doctrines have serious problems. While the doctrines disagree about existential imports of categorical propositions, they share... more

This article examines the traditional and modern doctrines of categorical
propositions and argues that both doctrines have serious problems.
While the doctrines disagree about existential imports of categorical
propositions, they share a fundamental assumption: matching categorical propositions of the same kind (e.g., universal affirmatives, particular negatives) are logically equivalent (the logical
homogeneity thesis). The article shows that this common thesis is false, and
argues that one cannot deal with the logical diversity of categorical propositions with minor modifications of the traditional or modern
doctrine. It also gives a sketch of an account that pays close attention
to the logical diversity of categorical propositions. In doing so, the article examines differences
among various versions of the traditional doctrine and discusses categorical propositions
involving plural constructions and non-distributive predicates.

In this paper I investigate how differences in approach to truth and logic (in particular, a deflationist vs. a substantivist approach to these fields) affect philosophers’ views concerning pluralism and normativity in these fields. My... more

In this paper I investigate how differences in approach to truth and logic (in particular, a deflationist vs. a substantivist approach to these fields) affect philosophers’ views concerning pluralism and normativity in these fields. My perspective on truth and logic is largely epistemic, focusing on the role of truth in knowledge (rather than on the use of the words “true” and “truth” in natural language), and my reference group includes Carnap (1934), Harman (1986), Horwich (1990), Wright (1992), Beall and Restall (2006), Field (2009), Lynch (2009), and Sher (2016a).1 Whenever possible, I focus on positive rather than negative views on the issues involved, although in some cases this is not possible.

Proof-theoretic semantics is a well-established inferentialist theory of meaning that develops ideas proposed by Prawitz and Dummett. The main aim of this theory is to find a foundation of logic based on some aspects of the linguistic use... more

Proof-theoretic semantics is a well-established inferentialist theory of meaning that develops ideas proposed by Prawitz and Dummett. The main aim of this theory is to find a foundation of logic based on some aspects of the linguistic use of the logical terms, as opposed to the regular foundation offered by a model-theoretic approach à la Tarski, in which the denotation of non-linguistic entities is central. Traditionally, intuitionistic logic is considered justified in proof-theoretic semantics (although some doubts are sometimes raised regarding ex falso quodlibet). Even though this approach to semantics has greatly progressed in the last decades, it remains nonetheless controversial the existence of a justification of classical logic that suits its restraints. In this thesis I examine various proposals that try to give such a justification and propose a new one greatly inspired by one of Peter Milne’s papers. The conclusion is, to some extent, open since a reformulation of some notions of proof-theoretic semantics is needed in order to justify classical logic. I conclude the thesis with a general defence of logical pluralism and a description of the kind of pluralism that can be applied to our reformulation of proof-theoretic semantics.

This is a survey paper in German covering a number of central themes in the philosophy of logic, including the debates surrounding the notion of logical consequence, the problem of demarcating logic, logical pluralism vs. logical monism... more

This is a survey paper in German covering a number of central themes in the philosophy of logic, including the debates surrounding the notion of logical consequence, the problem of demarcating logic, logical pluralism vs. logical monism and the question of the normativity of logic.

Italian translation of "On Logical Relativity" (2002)

In this paper I argue against the view that logical consequence is a truth-preserving relation and, in general, that it is a reflexive and transitive relation. I expound a number of non-reflexive or non-transitive notions of logical... more

In this paper I argue against the view that logical consequence is a truth-preserving relation and, in general, that it is a reflexive and transitive relation. I expound a number of non-reflexive or non-transitive notions of logical consequence and argue that they satisfy the requirements to be considered part of the so-called “core tradition” of logic, and that they thus give rise to logics bona fide.

This paper explores an apparent tension between two widely held views about logic: that logic is normative and that there are multiple equally legitimate logics. The tension is this. If logic is normative, it tells us something about how... more

This paper explores an apparent tension between two widely held views about logic: that logic is normative and that there are multiple equally legitimate logics. The tension is this. If logic is normative, it tells us something about how we ought to reason. If, as the pluralist would have it, there are several correct logics, those logics make incompatible recommendations as to how we ought to reason. But then which of these logics should we look to for normative guidance? I argue that inasmuch as pluralism draws its motivation from its ability to defuse logical disputes—that is, disputes between advocates of rival logics—it is unable to provide an answer: pluralism collapses into monism with respect to either the strongest or the weakest admissible logic.

I discuss Benacerraf's epistemological challenge for realism about areas like mathematics, metalogic, and modality, and describe the pluralist response to it. I explain why normative pluralism is peculiarly unsatisfactory, and use this... more

I discuss Benacerraf's epistemological challenge for realism about areas like mathematics, metalogic, and modality, and describe the pluralist response to it. I explain why normative pluralism is peculiarly unsatisfactory, and use this explanation to formulate a radicalization of Moore's Open Question Argument. According to the argument, the facts -- even the normative facts -- fail to settle the practical questions at the center of our normative lives. One lesson is that the concepts of realism and objectivity, which are widely identified, are actually in tension.

According to the logical pluralism of Beall and Restall, there are several distinct relations of logical consequence. Some critics argue that logical pluralism suffers from what I call the collapse problem: that despite its intention to... more

According to the logical pluralism of Beall and Restall, there are several distinct relations of logical consequence. Some critics argue that logical pluralism suffers from what I call the collapse problem: that despite its intention to articulate a radically pluralistic doctrine about logic, the view unintentionally collapses into logical monism. In this paper, I propose a contextualist resolution of the collapse problem. This clarifies the mechanism responsible for a plurality of logics and handles the motivating data better than the original view. It is a major improvement that should be embraced by all logical pluralists.

Up to now theories of semantic information have implicitly relied on logical monism, or the view that there is one true logic. The latter position has been explicitly challenged by logical pluralists. Adopting an unbiased attitude in the... more

Up to now theories of semantic information have implicitly relied on logical monism, or the view that there is one true logic. The latter position has been explicitly challenged by logical pluralists. Adopting an unbiased attitude in the philosophy of information, we take a suggestion from Beall and Restall at heart and exploit logical pluralism to recognise another kind of pluralism. The latter is called informational pluralism, a thesis whose implications for a theory of semantic information we explore.

According to the form of logical pluralism elaborated by Beall and Restall there is more than one relation of logical consequence. Since they take the relation of logical consequence to reside at the very heart of a logical system,... more

According to the form of logical pluralism elaborated by Beall and Restall there is more than one relation of logical consequence. Since they take the relation of logical consequence to reside at the very heart of a logical system, different relations of logical consequence yield different logics. In this paper, we are especially interested in understanding what are the consequences of endorsing Beall and Restall’s version of logical pluralism vis-à-vis the normative guidance that logic is taken to provide to reasoners. In particular, the aim of this paper is threefold. First, in sections 2 and 3, we offer an exegesis of Beall and Restall’s logical pluralism as a thesis of semantic indeterminacy of our concept of logical consequence—i.e. understood as indeterminacy logical pluralism. Second, in sections 4 and 5, we elaborate and critically scrutinise three models of semantic indeterminacy that we think are fit to capture Beall and Restall’s indeterminacy logical pluralism. Third, in section 6, following Beall and Restall’s assumption that the notion of logical consequence has normative significance for deductive reasoning, we raise a series of normative problems for indeterminacy logical pluralism. The overall conclusion that we aim to establish is that Beall and Restall’s indeterminate logical pluralism cannot offer an adequate account of the normative guidance that logic is taken to provide us with in ordinary contexts of reasoning.

In this paper, I try to find a way of dissolving the tension existing between logical pluralism and the normative status of logic. To do this, on the one hand, regarding the normative status of logic, I defend an externalist perspective... more

In this paper, I try to find a way of dissolving the tension existing between logical pluralism and the normative status of logic. To do this, on the one hand, regarding the normative status of logic, I defend an externalist perspective of normativity as long as logic has a normative role for social practices of rational interaction. On the other hand, regarding the nature of logic, I argue that logic is the result of a process of reflective equilibrium; logical laws are obtained from a process of mutual adjustment and adaptation with argumentative practices. Therefore, different processes of reflective equilibrium give rise to different logical systems. I show that this externalist characterization of the normativity of logic in conjunction with the thesis of reflective equilibrium enables to defend a version of logical pluralism that doesn't enter into conflict with the normativity.

The answers to the question(s) in the title depend on which pluralisms one is talking about. There are a lot of them, and they have little in common. We will focus here on our own views. For truth pluralism, we consider the view... more

The answers to the question(s) in the title depend on which pluralisms one is talking about. There are a lot of them, and they have little in common. We will focus here on our own views. For truth pluralism, we consider the view articulated in Michael Lynch’s Truth as one and many (2009) and Crispin Wright’s Truth and objectivity (1992). For logical pluralism, it is Stewart Shapiro’s Varieties of logic (2014).

When discussing Logical Pluralism several critics argue that such an open-minded position is untenable. The key to this conclusion is that, given a number of widely accepted assumptions, the pluralist view collapses into Logical Monism.... more

When discussing Logical Pluralism several critics argue that such an open-minded position is untenable. The key to this conclusion is that, given a number of widely accepted assumptions, the pluralist view collapses into Logical Monism. In this paper we show that the arguments usually employed to arrive at this conclusion do not work. The main reason for this is the existence of certain substructural logics which have the same set of valid inferences as Classical Logic—although they are, in a clear sense, non-identical to it. We argue that this phenomenon can be generalized, given the existence of logics which coincide with Classical Logic regarding a number of metainfer-ential levels—although they are, again, clearly different systems. We claim this highlights the need to arrive at a more refined version of the Collapse Argument, which we discuss at the end of the paper.

L'obiettivo di questo elaborato è affrontare la posizione filosofica del pluralismo logico, analizzando gli sviluppi più recenti di questa teoria (principalmente quelli dovuti a J.C. Beall e Greg Restall) e illustrarne le questioni... more

L'obiettivo di questo elaborato è affrontare la posizione filosofica del pluralismo logico, analizzando gli sviluppi più recenti di questa teoria (principalmente quelli dovuti a J.C. Beall e Greg Restall) e illustrarne le questioni lasciate aperte. Vengono distinti due modi di caratterizzare la conseguenza logica - quello realista orientato ai modelli e quello antirealista orientato alla pratica inferenziale - e si analizza il modo in cui il pluralismo può essere argomentato in entrambe queste cornici teoriche.
In entrambi questi ambienti si mostra che è possibile ricostruire la logica classica, quella intuizionista e una variante di quella rilevante. Viene poi mostrato che in nessuna di queste prospettive siamo costretti ad accettare la tesi della variazione del significato sostenuta da Quine, secondo cui uno stesso connettivo otterrebbe significati diversi nelle varie logiche. Si argomenta anzi sostenendo che la prospettiva di Beall e Restall permette di ricostruire differenti logiche mantenendo fisso il significato adottato dalle costanti logiche, e che questa caratteristica distingue il pluralismo sviluppato da questi due autori da quelli tradizionali, adottati da Rudolf Carnap e Susan Haack.
In conclusione vengono confrontati i pluralismi realista e antirealista ottenuti, evidenziandone similitudini e differenze.

The starting point of this paper concerns the apparent difference between what we might call absolute truth and truth in a model, following Donald Davidson. The notion of absolute truth is the one familiar from Tarski’s T-schema: ‘Snow is... more

主张逻辑的多元性，是一个极具争议的逻辑哲学立场。后承多元论是近来最为显著的逻辑多元论学说，它主张逻辑的多元性来自逻辑后承的多元性。本文首先考虑并排除了两类典型的针对逻辑多元论的整体性反驳; 其次阐述了直觉性的、非形式的后承概念的未确定性，以及毕奥和雷斯托尔以此为动机所提出的后承多元论; 然后 考虑了这个理论的一个直接但饱受质疑的结果，即存在两个不同的经典逻辑后承，尤其说明这个结果在处理逻辑哲 学难题时的用处; 最后再次回到后承概念的未确定性，探讨了对它的两种理解:... more

主张逻辑的多元性，是一个极具争议的逻辑哲学立场。后承多元论是近来最为显著的逻辑多元论学说，它主张逻辑的多元性来自逻辑后承的多元性。本文首先考虑并排除了两类典型的针对逻辑多元论的整体性反驳; 其次阐述了直觉性的、非形式的后承概念的未确定性，以及毕奥和雷斯托尔以此为动机所提出的后承多元论; 然后 考虑了这个理论的一个直接但饱受质疑的结果，即存在两个不同的经典逻辑后承，尤其说明这个结果在处理逻辑哲 学难题时的用处; 最后再次回到后承概念的未确定性，探讨了对它的两种理解: 含混性和歧义性。It is a controversial standpoint in the philosophy of logic to claim logical pluralism. Pluralism about logical consequence, which claims that the plurality of logic results from the plurality of logical consequence, is the most prominent theory in recent years. In this paper, we first consider and exclude two kinds of typical holistic refutations against logical pluralism. Secondly, we expound the unsettledness of the informal and pretheoretical notion of consequence, as well as the pluralism proposed and defended by JC Beall and Greg Restall. Then, we consider a direct but questionable result of this theory, that is, there are two different classical consequences. In particular, we explain the use of this result in dealing with some puzzling problems in the philosophy of logic. Finally, we go back to the unsettled conception of consequence, and its two sorts of understanding: vagueness and ambiguity.

Logical pluralism is the view that there is more than one correct logic. Most logical pluralists think that logic is normative in the sense that you make a mistake if you accept the premisses of a valid argument but reject its conclusion.... more

Logical pluralism is the view that there is more than one correct logic. Most logical pluralists think that logic is normative in the sense that you make a mistake if you accept the premisses of a valid argument but reject its conclusion. Some authors have argued that this combination is self-undermining: Suppose that L 1 and L 2 are correct logics that coincide except for the argument from Γ to φ, which is valid in L 1 but invalid in L 2. If you accept all sentences in Γ, then, by normativity, you make a mistake if you reject φ. In order to avoid mistakes, you should accept φ or suspend judgment about φ. Both options are problematic for pluralism. Can pluralists avoid this worry by rejecting the normativity of logic? I argue that they cannot. All else being equal, the argument goes through even if logic is not normative.

This paper examines two syllogistic arguments contrasted in an ancient Chinese book, the Mozi, which expounds doctrines of the Mohist school of philosophers. While the arguments seem to have the same form, one of them (the one-horse... more

This paper examines two syllogistic arguments contrasted in an ancient Chinese book, the Mozi, which expounds doctrines of the Mohist school of philosophers. While the arguments seem to have the same form, one of them (the one-horse argument) is valid but the other (the two-horse argument) is not. To explain this difference, the paper uses English plural constructions to formulate the arguments. Then it shows that the one-horse argument is valid because it is an instance of the plural cousin of a standard form of valid categorical syllogisms (Plural Barbara), and argues that the two-horse argument involves equivocal uses of a key predicate (the Chinese counterpart of ‘have four feet’) that has the distributive/non-distributive ambiguity. In doing so, the paper discusses linguistic differences between Chinese and English and explains why the logic of plural constructions is applicable to Chinese arguments that involve no plural constructions.

In western philosophy and beyond, a tension between pluralism and monism has sparked many developments and debates. Pluralism of norms, of forms of knowledge, of aesthetic and moral values, of interests etc. has often been pitted against... more

In western philosophy and beyond, a tension between pluralism and monism has sparked many developments and debates. Pluralism of norms, of forms of knowledge, of aesthetic and moral values, of interests etc. has often been pitted against monism. Monism usually implies a hierarchical order of such norms etc. After having traced the origin of this tension between pluralism and monism in ancient tragedy and philosophy, I’m asking in this article whether a rejection of monism and embrace of pluralism necessarily raises the specter of inconsistency and contradiction.
The threat of inconsistency need not be associated with pluralism as even with regard to logic an argument can be made for a ‘multiplicity of logics’, as van Lambalgen and Stenning argue in several places. They refer to the varieties in reasoning that can be observed in humans and which are partly due to there being ‘dual systems’ of reasoning: System 1 processes information fast, automatized, and emotional, while System 2 is rather slow, more deliberative, and more rational. In contrast to a widely held view, the authors argue that System 1 processing is not without logic, even though it is a different form of logic from System 2. In addition, the multiplicity of logics they discuss is related to the multiple semantics required for distinct domains of reasoning. This perspective raises the subsequent question how this pluralism can be available while maintaining consistency, how should the available options be configured or related to each other?
In the remainer of this chapter I’m addressing several such options for configurating pluralism like: non-foundational pluralism and foundational pluralism, moderate or temporary pluralism, antagonistic pluralism, incompatible pluralism, incommensurable pluralism, ‘Anything goes’ pluralism, complementary pluralism, integrative pluralism, and interactive pluralism. The discussion and examples of these options show that irrespective of its domain, the encounter with different configurations of pluralism shows how some forms of pluralism are likely to be productive, whereas others are less so - which is an important lesson given the prominent pluralism of pluralisms in philosophy and beyond.

Substructural pluralism about the meaning of logical connectives is best understood as the view that natural language connectives have all (and only) the properties conferred by classical logic, but that particular occurrences of these... more

Substructural pluralism about the meaning of logical connectives is best understood as the view that natural language connectives have all (and only) the properties conferred by classical logic, but that particular occurrences of these connectives cannot simultaneously exhibit all these properties. This is just a more sophisticated way of saying that while natural language connectives are ambiguous, they are not so in the way classical logic intends them to be. Since this view is usually framed as a means to resolve paradoxes, little attention is paid to the logical properties of the am- biguous connectives themselves. The present paper sets out to fill this gap. First, I argue that substructural logicians should care about these connectives; next, I describe a consequence relation between a set of ambiguous premises and an ambiguous conclusion, and review the logical properties of ambiguous connectives; finally, I highlight how ambiguous connectives might explain our intuitions about logical rivalry.

This article examines two syllogistic arguments contrasted in an ancient Chinese book, the Mozi, which expounds doctrines of the Mohist school of philosophers. While the arguments seem to have the same form, one of them (the one-horse... more

This article examines two syllogistic arguments contrasted in an ancient Chinese book, the Mozi, which expounds doctrines of the Mohist school of philosophers. While the arguments seem to have the same form, one of them (the one-horse argument) is valid but the other (the two-horse argument) is not. To explain this difference, the article uses English plural constructions to formulate the arguments. Then it shows that the one-horse argument is valid because it has a valid argument form, the plural cousin of a standard form of valid categorical syllogisms (Plural Barbara), and argues that the two-horse argument involves equivocal uses of a key predicate (the Chinese counterpart of ‘have four feet’) that has the distributive/nondistributive ambiguity. In doing so, the article discusses linguistic differences between Chinese and English and explains why the logic of plural constructions is applicable to Chinese arguments that involve no plural constructions.

In this paper, I defend Rudolf Carnap's Principle of Tolerance from an accusation, due to Michael Friedman, that it is self-defeating by prejudicing any debate towards the logically stronger theory. In particular, Friedman attempts to... more

In this paper, I defend Rudolf Carnap's Principle of Tolerance from an accusation, due to Michael Friedman, that it is self-defeating by prejudicing any debate towards the logically stronger theory. In particular, Friedman attempts to show that Carnap's reconstruction of the debate between classicists and intuitionists over the foundations of mathematics in his book The Logical Syntax of Language, is biased towards the classical standpoint since the metalanguage he constructs to adjudicate between the rival positions is fully classical. I argue that this criticism is mistaken on two counts: (1) it fails to fully appreciate the freedom with regard to the construction of linguistic frameworks that Carnap intended his Principle to embody, and (2) Friedman's objection underestimates the extent to which the evaluation of a framework is task-relative. I conclude that Tolerance is not self-undermining in the way that Friedman claims it is. While this is a restricted conclusion - and is not a vindication of Carnap's views on logic and mathematics tout court - it nonetheless suggests that his tolerant perspective has been dismissed too quickly, even by his supporters.