This study looks to the work of Tarski's mentors Stanislaw Lesniewski and Tadeusz Kotarbinski, and reconsiders all of the major issues in Tarski scholarship in light of the conception of Intuitionistic Formalism developed: semantics, truth, paradox, logical consequence.

Ontological pluralism is the view that there are different ways to exist. It is a position with deep roots in the history of philosophy, and in which there has been a recent resurgence of interest. In contemporary presentations, it is stated in terms of fundamental languages: as the view that such languages contain more than one quantifier. For example, one ranging over abstract objects, and another over concrete ones. A natural worry, however, is that the languages proposed by the pluralist (...) are mere notational variants of those proposed by the monist, in which case the debate between the two positions would not seem to be substantive. Jason Turner has given an ingenious response to this worry, in terms of a principle that he calls ‘logical realism’. This paper offers a counter-response on behalf of the ‘notationalist’. I argue that, properly applied, the principle of logical realism is no threat to the claim that the languages in question are notational variants. Indeed, there seems to be every reason to think that they are. (shrink)

Although the invariance criterion of logicality first emerged as a criterion of a purely mathematical interest, it has developed into a criterion of considerable linguistic and philosophical interest. In this paper I compare two different perspectives on this criterion. The first is the perspective of natural language. Here, the invariance criterion is measured by its success in capturing our linguistic intuitions about logicality and explaining our logical behavior in natural-linguistic settings. The second perspective is more theoretical. Here, the invariance criterion (...) is used as a tool for developing a theoretical foundation of logic, focused on a critical examination, explanation, and justification of its veridicality and modal force. (shrink)

I attempt an explication of what it means for an operation across domains to be the same on all domains, an issue that ) took to be central for a successful delimitation of the logical operations. Some properties that seem strongly related to sameness are examined, notably isomorphism invariance, and sameness under extensions of the domain. The conclusion is that although no precise criterion can satisfy all intuitions about sameness, combining the two properties just mentioned yields a reasonably robust and (...) useful explication of sameness across domains. (shrink)

This paper aims to provide hyperintensional foundations for mathematical platonism. I examine Hale and Wright's (2009) objections to the merits and need, in the defense of mathematical platonism and its epistemology, of the thesis of Necessitism. In response to Hale and Wright's objections to the role of epistemic and metaphysical modalities in providing justification for both the truth of abstraction principles and the success of mathematical predicate reference, I examine the Necessitist commitments of the abundant conception of properties endorsed by (...) Hale and Wright and examined in Hale (2013), and demonstrate how a two-dimensional approach to the epistemology of mathematics is consistent with Hale and Wright's notion of there being non-evidential epistemic entitlement rationally to trust that abstraction principles are true. A choice point that I flag is that between availing of intensional or hyperintensional semantics. The hyperintensional semantic approach that I advance is a topic-sensitive epistemic two-dimensional truthmaker semantics. I countenance a hyperintensional semantics for novel epistemic abstractionist modalities. I suggest that observational type theory can be applied to first-order abstraction principles in order to make abstraction principles recursively enumerable. Epistemic and metaphysical states and possibilities may thus be shown to play a constitutive role in vindicating the reality of mathematical objects and truth, and in providing a conceivability-based route to the truth of abstraction principles as well as other axioms and propositions in mathematics. (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)

Plural expressions found in natural languages allow us to talk about many objects simultaneously. Plural logic — a logical system that takes plurals at face value — has seen a surge of interest in recent years. This book explores its broader significance for philosophy, logic, and linguistics. What can plural logic do for us? Are the bold claims made on its behalf correct? After introducing plural logic and its main applications, the book provides a systematic analysis of the relation between (...) this logic and other theoretical frameworks such as set theory, mereology, higher-order logic, and modal logic. The applications of plural logic rely on two assumptions, namely that this logic is ontologically innocent and has great expressive power. These assumptions are shown to be problematic. The result is a more nuanced picture of plural logic's applications than has been given thus far. Questions about the correct logic of plurals play a central role in the final chapters, where traditional plural logic is rejected in favor of a "critical" alternative. The most striking feature of this alternative is that there is no universal plurality. This leads to a novel approach to the relation between the many and the one. In particular, critical plural logic paves the way for an account of sets capable of solving the set-theoretic paradoxes. (shrink)

This dissertation concerns the foundations of epistemic modality. I examine the nature of epistemic modality, when the modal operator is interpreted as concerning both apriority and conceivability, as well as states of knowledge and belief. The dissertation demonstrates how phenomenal consciousness and gradational possible-worlds models in Bayesian perceptual psychology relate to epistemic modal space. The dissertation demonstrates, then, how epistemic modality relates to the computational theory of mind; metaphysical modality; deontic modality; logical modality; the types of mathematical modality; to the (...) epistemic status of undecidable propositions and abstraction principles in the philosophy of mathematics; to the apriori-aposteriori distinction; to the modal profile of rational propositional intuition; and to the types of intention, when the latter is interpreted as a modal mental state. Examining the nature of epistemic logic itself, I develop a novel approach to conditions of self-knowledge in the setting of the modal μ-calculus, as well as novel epistemicist solutions to Curry's, the liar, and the knowability paradoxes. Solutions to previously intransigent issues concerning the first-person concept; the distinction between fundamental and derivative truths; and the unity of intention and its role in decision theory, are developed along the way. (shrink)

Resumo: Pedro Hispano define os sincategoremas como expressões que revelam de que maneira os sujeitos e os predicados estão de fato relacionados nas proposições, contribuindo assim para o estabelecer o que elas significam e fixar as condições de verdade e as formas lógicas correspondentes. Entre as expressões que ele julga serem sincategoremáticas, ‘não’, ‘e’, ‘ou’, ‘se’, ‘todo’ e ‘necessário’ se destacam atualmente como constantes lógicas. Todavia, opondo-se a grande parte dos lógicos contemporâneos para quem tais expressões possuem um significado fixo (...) na medida em que integram as formas lógicas das respectivas proposições, Pedro Hispano admite que seu significado pode ser modificado quando a expressões do mesmo tipo credita a capacidade de atuarem em determinados contextos como categoremas. Além disso, ao defender que as expressões ora em questão explicitam as formas lógicas das proposições mediante a articulação dos categoremas a elas associados, ele também demonstra divergir dos principais critérios de demarcação das constantes lógicas atualmente vigentes que não preveem a atribuição de tal comportamento a expressões desse tipo. Portanto, não resta dúvida de que a teoria de Pedro Hispano sobre os sincategoremas enriquece a nossa compreensão ainda insuficiente da natureza das constantes lógicas e contribui para a resolução do problema da demarcação de tais expressões.: According to Petrus Hispanus, syncategoremata are expressions that determine how subjects and predicates are actually related in propositions. They contribute to establishing what the categoremata mean and to specifying the truth conditions of the corresponding logical forms. Among the expressions Peter of Spain thinks of as syncategorematic, ‘not’, ‘and’, ‘or’, ‘if’, ‘all, and ‘necessary’ are nowadays considered to be logical operators. But unlike the contemporary logicians who argue that these expressions have fixed meanings because they belong to the logical forms of the corresponding propositions, Peter of Spain allows that their meanings can be modified, attributing to them the ability to act in certain contexts as categoremata. In addition, he argues that these expressions make explicit the logical forms through the articulation of corresponding propositions’ categoremata; thus he also diverges from the main contemporary criteria of demarcation of logical constants, which does not allow that these expressions have such behavior. Therefore, there is no doubt that Hispanus’ theory of the syncategoremata enriches our still insufficient understanding of the logical constants and contributes to the resolution of the problem of the demarcation of such expressions. (shrink)

The notion of a proposition is central to philosophy. But it is subject to paradoxes. A natural response is a hierarchical account and, ever since Russell proposed his theory of types in 1908, this has been the strategy of choice. But in this paper I raise a problem for such accounts. While this does not seem to have been recognized before, it would seem to render existing such accounts inadequate. The main purpose of the paper, however, is to provide a (...) new hierarchical account that solves the problem. (shrink)

The threat of ontological deflationism (the view that disagreement about what there is can be non‐substantive) is averted by appealing to realism about fundamental structure—or so tells us Ted Sider. In this paper, the notion of structural indeterminacy is introduced as a particular case of metaphysical indeterminacy; then it is argued that structural indeterminacy is not only compatible with a metaphysics of fundamental structure, but it can even safeguard it from a crucial objection; finally, it is shown that, if there (...) are instances of structural indeterminacy, a hitherto unacknowledged variety of ontological deflationism will arise. Unless structure is shown to be determinate, ontological deflationism remains a live option. Furthermore, I will consider whether structural indeterminacy could be challenged by adopting a naturalistic epistemology of structure; the question is answered in the negative on the basis of a formal result concerning theory choice. Finally, I submit a new way of articulating the epistemology of structure, which hinges on the very possibility of structural indeterminacy. (shrink)

On one influential view, metaphysical fundamentality can be understood in terms of joint‐carving. Ted Sider has recently argued that (i) some first order quantifier is joint‐carving, and (ii) modal notions are not joint‐carving. After vindicating the theoretical indispensability of quantification against recent criticism, I will defend a logical result due to Arnold Koslow which implies that (i) and (ii) are incompatible. I will therefore consider an alternative understanding of Sider's metaphysics to the effect that (i) some first order quantifier is (...) joint‐carving, and (iii) intensional notions are not joint‐carving. Another result due to Koslow entails that (i) and (iii) are also incompatible. I will argue that this second result is inconclusive. Nevertheless, (iii) is incompatible with another tenet of Sider's metaphysics, namely that (iv) ‘being joint‐carving’ is itself joint‐carving. In order to resolve the inconsistency, I will tentatively argue that condition (iv) should be renounced. (shrink)

What we call first‐order logic over fixed domain was initiated, in a certain guise, by Peirce around 1885 and championed, albeit in idiosyncratic form, by Zermelo in papers from the 1930s. We characterise such logics model‐ and proof‐theoretically and argue that they constitute exploration of a clearly circumscribed conception of domain‐dependent generality. Whereas a logic, or family of such, can be of interest for any of a variety of reasons, we suggest that one of those reasons might be that said (...) logic fosters some clarification regarding just what qualifies as a logical concept, a logical operation, or a logical law. (shrink)

In a recent paper, “The Concept of Logical Consequence,” W. H. Hanson criticizes a formal-structural characterization of logical consequence in Tarski and Sher. Hanson accepts many principles of the formal-structural view. Relating to Sher 1991 and 1996a, he says.

This paper offers a response to William’s Hanson’s criticism of Sher’s formal-structural conception of logical consequence and logical constants.

The construction of a systematic philosophical foundation for logic is a notoriously difficult problem. In Part One I suggest that the problem is in large part methodological, having to do with the common philosophical conception of “providing a foundation”. I offer an alternative to the common methodology which combines a strong foundational requirement with the use of non-traditional, holistic tools to achieve this result. In Part Two I delineate an outline of a foundation for logic, employing the new methodology. The (...) outline is based on an investigation of why logic requires a veridical justification, i.e., a justification which involves the world and not just the mind, and what features or aspect of the world logic is grounded in. Logic, the investigation suggests, is grounded in the formal aspect of reality, and the outline proposes an account of this aspect, the way it both constrains and enables logic, logic's role in our overall system of knowledge, the relation between logic and mathematics, the normativity of logic, the characteristic traits of logic, and error and revision in logic. (shrink)

Many philosophers are baffled by necessity. Humeans, in particular, are deeply disturbed by the idea of necessary laws of nature. In this paper I offer a systematic yet down to earth explanation of necessity and laws in terms of invariance. The type of invariance I employ for this purpose generalizes an invariance used in meta-logic. The main idea is that properties and relations in general have certain degrees of invariance, and some properties/relations have a stronger degree of invariance than others. (...) The degrees of invariance of highly-invariant properties are associated with high degrees of necessity of laws governing/describing these properties, and this explains the necessity of such laws both in logic and in science. This non-mysterious explanation has rich ramifications for both fields, including the formality of logic and mathematics, the apparent conflict between the contingency of science and the necessity of its laws, the difference between logical-mathematical, physical, and biological laws/principles, the abstract character of laws, the applicability of logic and mathematics to science, scientific realism, and logical-mathematical realism. (shrink)

Alfred Tarski a joué un rôle déterminant dans la déconstruction du clivage classique de l’analytique et du synthétique alors même que les ouvrages consacrés à ce chapitre de la philosophie restent, en général, relativement discrets au sujet de son rôle au profit de W. V. O. Quine. La critique de Tarski, qui s’articule dès 1930, s’organise le long de deux axes principaux. Le premier a trait à la difficulté de définir objectivement la notion de logicité ; le second, qui s’appuie (...) sur une conception holiste de la science, admet la possibilité d’une rectification des principes logiques au nom de l’expérience. Sur ce point, les objections ponctuelles de Tarski anticipent la critique plus systématique que Quine adressera à la notion de l’analyticité. L’analyse des principales étapes de la critique de Tarski révèle toutefois que sa pensée n’est pas toujours exempte de tensions internes. Alfred Tarski has played a decisive role in the refutation of the classic cleavage between analytic and synthetic truths, though the studies which deal with this chapter of philosophy generally stress the role of W. V. O. Quine in this question and omit to mention Tarski’s name. Tarski’s critique, which has been expressing itself since 1930, is developed along two main lines. The first concerns the difficulty of giving an objective definition of logical notions ; the second, based on a holistic conception of science, maintains the possibility of the refutation of logical principles by experience. In this question the selective objections of Tarski anticipate the more systematic critique of analyticity by Quine. However, the analysis of the different steps in Tarski’s critique shows that his way of thinking is not free of inner tensions. (shrink)

The essay discusses a recurrent criticism of the isomorphism-invariance criterion for logical terms, according to which the criterion pertains only to the extension of logical terms, and neglects the meaning, or the way the extension is fixed. A term, so claim the critics, can be invariant under isomorphisms and yet involve a contingent or a posteriori component in its meaning, thus compromising the necessity or apriority of logical truth and logical consequence. This essay shows that the arguments underlying the criticism (...) are flawed since they rely on an invalid inference from the modal or epistemic status of statements in the metalanguage to that of statements in the object-language. The essay focuses on McCarthy’s version of the argument, but refers to Hanson and McGee’s versions as well. (shrink)

In standard model-theoretic semantics, the meaning of logical terms is said to be fixed in the system while that of nonlogical terms remains variable. Much effort has been devoted to characterizing logical terms, those terms that should be fixed, but little has been said on their role in logical systems: on what fixing their meaning precisely amounts to. My proposal is that when a term is considered logical in model theory, what gets fixed is its intension rather than its extension. (...) I provide a rigorous way of spelling out this idea, and show that it leads to a graded account of logicality: the less structure a term requires in order for its intension to be fixed, the more logical it is. Finally, I focus on the class of terms that are invariant under isomorphisms, as they render themselves more easily to mathematical treatment. I propose a mathematical measure for the logicality of such terms based on their associated Löwenheim numbers. (shrink)

Invariance criteria are widely accepted as a means to demarcate the logical vocabulary of a language. In previous work, I proposed a framework of “semantic constraints” for model theoretic consequence which does not rely on a strict distinction between logical and nonlogical terms, but rather on a range of constraints on models restricting the interpretations of terms in the language in different ways. In this paper I show how invariance criteria can be generalized so as to apply to semantic constraints (...) on models. Some obviously unpalatable semantic constraints turn out to be invariant under isomorphisms. I shall connect our discussion to known counterexamples to invariance criteria for logical terms, and so the generalization will also shed light on the current existing debate on logicality. I analyse the failure of invariance to fulfil its role as a criterion for logicality, and argue that invariance conditions should best be thought of as merely methodological meta-constraints restricting the ways the model-theoretic apparatus should be used. (shrink)

Tarski characterized logical notions as invariant under permutations of the domain. The outcome, according to Tarski, is that our logic, which is commonly said to be a logic of extension rather than intension, is not even a logic of extension—it is a logic of cardinality. In this paper, I make this idea precise. We look at a scale inspired by Ruth Barcan Marcus of various levels of meaning: extensions, intensions and hyperintensions. On this scale, the lower the level of meaning, (...) the more coarse-grained and less “intensional” it is. I propose to extend this scale to accommodate a level of meaning appropriate for logic. Thus, below the level of extension, we will have a more coarse-grained level of form. I employ a semantic conception of form, adopted from Sher, where forms are features of things “in the world”. Each expression in the language embodies a form, and by the definition we give, forms will be invariant under permutations and thus Tarskian logical notions. I then define the logical terms of a language as those terms whose extension can be determined by their form. Logicality will be shown to be a lower level analogue of rigidity. Using Barcan Marcus’s principles of explicit and implicit extensionality, we are able to characterize purely logical languages as “sub-extensional”, namely, as concerned only with form, and we thus obtain a wider perspective on both logicality and extensionality. (shrink)

The goal of this paper is to develop a theory of content for vague language. My proposal is based on the following three theses: (1) language-mastery is not rulebased— it involves a certain kind of decision-making; (2) a theory of content is to be thought of instrumentally—it is a tool for making sense of our linguistic practice; and (3) linguistic contents are only locally defined—they are only defined relative to suitably constrained sets of possibilities. CiteULike Connotea Del.icio.us What's this?

Forthcoming in Philosophical Compass. I explain why plural quantifiers and predicates have been thought to be philosophically significant.

The logicality of language is the hypothesis that the language system has access to a ‘natural’ logic that can identify and filter out as unacceptable expressions that have trivial meanings—that is, that are true/false in all possible worlds or situations in which they are defined. This hypothesis helps explain otherwise puzzling patterns concerning the distribution of various functional terms and phrases. Despite its promise, logicality vastly over-generates unacceptability assignments. Most solutions to this problem rest on specific stipulations about the properties (...) of logical form—roughly, the level of linguistic representation which feeds into the interpretation procedures—and have substantial implications for traditional philosophical disputes about the nature of language. Specifically, contextualism and semantic minimalism, construed as competing hypotheses about the nature and degree of context-sensitivity at the level of logical form, suggest different approaches to the over-generation problem. In this paper, I explore the implications of pairing logicality with various forms of contextualism and semantic minimalism. I argue that to adequately solve the over-generation problem, logicality should be implemented in a constrained contextualist framework. (shrink)

I consider different versions of a structuralist view of mathematical objects, according to which characteristic mathematical objects have no more of a 'nature' than is given by the basic relations of a structure in which they reside. My own version of such a view is non-eliminative in the sense that it does not lead to a programme for eliminating reference to mathematical objects. I reply to criticisms of non-eliminative structuralism recently advanced by Keränen and Hellman. In replying to the former, (...) I rely on a distinction between 'basic' and 'constructed' structures. A conclusion is that ideas from the metaphysical tradition can be misleading when applied to the objects of modern mathematics. (shrink)

Permutation invariance is often presented as the correct criterion for logicality. The basic idea is that one can demarcate the realm of logic by isolating specific entities—logical notions or constants—and that permutation invariance would provide a philosophically motivated and technically sophisticated criterion for what counts as a logical notion. The thesis of permutation invariance as a criterion for logicality has received considerable attention in the literature in recent decades, and much of the debate is developed against the background of ideas (...) put forth by Tarski in a 1966 lecture (Tarski 1966/1986). But as noted by Tarski himself in the lecture, the permutation invariance criterion yields a class of putative ‘logical constants’ that are essentially only sensitive to the number of elements in classes of individuals. Thus, to hold the permutation invariance thesis essentially amounts to limiting the scope of logic to quantificational phenomena, which is controversial at best and possibly simply wrong. In this paper, I argue that permutation invariance is a misguided approach to the nature of logic because it is not an adequate formal explanans for the informal notion of the generality of logic. In particular, I discuss some cases of undergeneration of the criterion, i.e. the fact that it excludes from the realm of logic operators that we have good reason to regard as logical, especially some modal operators. (shrink)

On a widespread naturalist view, the meanings of mathematical terms are determined, and can only be determined, by the way we use mathematical language—in particular, by the basic mathematical principles we’re disposed to accept. But it’s mysterious how this can be so, since, as is well known, minimally strong first-order theories are non-categorical and so are compatible with countless non-isomorphic interpretations. As for second-order theories: though they typically enjoy categoricity results—for instance, Dedekind’s categoricity theorem for second-order PA and Zermelo’s quasi-categoricity (...) theorem for second-order ZFC—these results require full second-order logic. So appealing to these results seems only to push the problem back, since the principles of second-order logic are themselves non-categorical: those principles are compatible with restricted interpretations of the second-order quantifiers on which Dedekind’s and Zermelo’s results are no longer available. In this paper, we provide a naturalist-friendly, non-revisionary solution to an analogous but seemingly more basic problem—Carnap’s Categoricity Problem for propositional and first-order logic—and show that our solution generalizes, giving us full second-order logic and thereby securing the categoricity or quasi-categoricity of second-order mathematical theories. Briefly, the first-order quantifiers have their intended interpretation, we claim, because we’re disposed to follow the quantifier rules in an open-ended way. As we show, given this open-endedness, the interpretation of the quantifiers must be permutation-invariant and so, by a theorem recently proved by Bonnay and Westerståhl, must be the standard interpretation. Analogously for the second-order case: we prove, by generalizing Bonnay and Westerståhl’s theorem, that the permutation invariance of the interpretation of the second-order quantifiers, guaranteed once again by the open-endedness of our inferential dispositions, suffices to yield full second-order logic. (shrink)

Logic is usually thought to concern itself only with features that sentences and arguments possess in virtue of their logical structures or forms. The logical form of a sentence or argument is determined by its syntactic or semantic structure and by the placement of certain expressions called “logical constants.”[1] Thus, for example, the sentences Every boy loves some girl. and Some boy loves every girl. are thought to differ in logical form, even though they share a common syntactic and semantic (...) structure, because they differ in the placement of the logical constants “every” and “some”. By contrast, the sentences Every girl loves some boy. and Every boy loves some girl. are thought to have the same logical form, because “girl” and “boy” are not logical constants. Thus, in order to settle questions about logical form, and ultimately about which arguments are logically valid and which sentences logically true, we must distinguish the “logical constants” of a language from its nonlogical expressions. (shrink)

Let me start with a well-known story. Kant held that logic and conceptual analysis alone cannot account for our knowledge of arithmetic: “however we might turn and twist our concepts, we could never, by the mere analysis of them, and without the aid of intuition, discover what is the sum [7+5]” (KrV, B16). Frege took himself to have shown that Kant was wrong about this. According to Frege’s logicist thesis, every arithmetical concept can be defined in purely logical terms, and (...) every theorem of arithmetic can be proved using only the basic laws of logic. Hence, Kant was wrong to think that our grasp of arithmetical concepts and our knowledge of arithmetical truth depend on an extralogical source—the pure intuition of time (Frege 1884, §89, §109). Arithmetic, properly understood, is just a part of logic. (shrink)

This paper develops a novel theory of abstraction—what we call collective abstraction. The theory solves a notorious problem for noneliminative structuralism. The noneliminative structuralist holds that in addition to various isomorphic systems there is a pure structure that can be abstracted from each of these systems; but existing accounts of abstraction fail for nonrigid systems like the complex numbers. The problem with the existing accounts is that they attempt to define a unique abstraction operation. The theory of collective abstraction instead (...) simultaneously defines a collection of distinct abstraction operations, each of which maps a system to its corresponding pure structure. The theory is precisely formulated in an essentialist language. This allows us to throw new light on the question to what extent structuralists are committed to symmetric dependence. Finally, we apply the theory of collective abstraction to solve a problem about converse relations. (shrink)

Russell’s paradox is purely logical in the following sense: a contradiction can be formally deduced from the proposition that there is a set of all non-self-membered sets, in pure first-order logic—the first-order logical form of this proposition is inconsistent. This explains why Russell’s paradox is portable—why versions of the paradox arise in contexts unrelated to set theory, from propositions with the same logical form as the claim that there is a set of all non-self-membered sets. Burali-Forti’s paradox, like Russell’s paradox, (...) is portable. I offer the following explanation for this fact: Burali-Forti’s paradox, like Russell’s, is purely logical. Concretely, I show that if we enrich the language \ of first-order logic with a well-foundedness quantifier W and adopt certain minimal inference rules for this quantifier, then a contradiction can be formally deduced from the proposition that there is a greatest ordinal. Moreover, a proposition with the same logical form as the claim that there is a greatest ordinal can be found at the heart of several other paradoxes that resemble Burali-Forti’s. The reductio of Burali-Forti can be repeated verbatim to establish the inconsistency of these other propositions. Hence, the portability of the Burali-Forti’s paradox is explained in the same way as the portability of Russell’s: both paradoxes involve an inconsistent logical form—Russell’s involves an inconsistent form expressible in \ and Burali-Forti’s involves an inconsistent form expressible in \. (shrink)

The need to distinguish between logical and extra-logical varieties of inference, entailment, validity, and consistency has played a prominent role in meta-ethical debates between expressivists and descriptivists. But, to date, the importance that matters of logical form play in these distinctions has been overlooked. That’s a mistake given the foundational place that logical form plays in our understanding of the difference between the logical and the extra-logical. This essay argues that descriptivists are better positioned than their expressivist rivals to provide (...) the needed account of logical form, and so better able to capture the needed distinctions. This finding is significant for several reasons: First, it provides a new argument against expressivism. Second, it reveals that descriptivists can make use of this new argument only if they are willing to take a controversial—but plausible—stand on claims about the nature and foundations of logic. (shrink)

Informally, structural properties of mathematical objects are usually characterized in one of two ways: either as properties expressible purely in terms of the primitive relations of mathematical theories, or as the properties that hold of all structurally similar mathematical objects. We present two formal explications corresponding to these two informal characterizations of structural properties. Based on this, we discuss the relation between the two explications. As will be shown, the two characterizations do not determine the same class of mathematical properties. (...) From this observation we draw some philosophical conclusions about the possibility of a ‘correct’ analysis of structural properties. (shrink)

A number of recent works consider treating validity as a primitive notion rather than one defined in some standard manner. There seem to have been three motivations. First, to understand how truth and validity interact in potentially paradoxical settings. Second, to argue that validity is in fact afflicted with paradoxes analogous to the semantic paradoxes. Third, to develop a ‘deflationary’ conception of validity or consequence. This article treats the notion of validity as a primitive notion and shows how to provide (...) a consistent theory of classical validity (and self-applicative truth), conservative over Peano arithmetic. (shrink)

We ask, when is a property of a model a logical property? According to the so-called Tarski–Sher criterion this is the case when the property is preserved by isomorphisms. We relate this to model-theoretic characteristics of abstract logics in which the model class is definable. This results in a graded concept of logicality in the terminology of Sagi [46]. We investigate which characteristics of logics, such as variants of the Löwenheim–Skolem theorem, Completeness theorem, and absoluteness, are relevant from the logicality (...) point of view, continuing earlier work by Bonnay, Feferman, and Sagi. We suggest that a logic is the more logical the closer it is to first order logic. We also offer a refinement of the result of McGee that logical properties of models can be expressed in $L_{\infty \infty }$ if the expression is allowed to depend on the cardinality of the model, based on replacing $L_{\infty \infty }$ by a “tamer” logic. (shrink)

In the paper the following questions are discussed: What is logical consequence? What are logical constants? What is a logical system? What is logical pluralism? What is logic? In the conclusion, the main tendencies of development of modern logic are pointed out.

In his groundbreaking Grundlagen, Frege (1884) pointed out that number words like ‘four’ occur in ordinary language in two quite different ways and that this gives rise to a philosophical puzzle. On the one hand ‘four’ occurs as an adjective, which is to say that it occurs grammatically in sentences in a position that is commonly occupied by adjectives. Frege’s example was (1) Jupiter has four moons, where the occurrence of ‘four’ seems to be just like that of ‘green’ in (...) (2) Jupiter has green moons. On the other hand, ‘four’ occurs as a singular term, which is to say that it occurs in a position that is commonly occupied by paradigmatic cases of singular terms, like proper names: (3) The number of moons of Jupiter is four. Here ‘four’ seems to be just like ‘Wagner’ in (4) The composer of Tannhäuser is Wagner, and both of these statements seem to be identity statements, ones with which we claim that what two singular terms stand for is identical. But that number words can occur both as singular terms and as adjectives is puzzling. Usually adjectives cannot occur in a position occupied by a singular term, and the other way round, without resulting in ungrammaticality and nonsense. To give just one example, it would be ungrammatical to replace ‘four’ with ‘the number of moons of Jupiter’ in (1): (5) *Jupiter has the number of moons of Jupiter moons. This ungrammaticality results even though ‘four’ and ‘the number of moons of Jupiter’ are both apparently singular terms standing for the same object in (3). So, how can it be that number words can occur both as singular terms and as adjectives, while other adjectives cannot occur as singular terms, and other singular terms cannot occur as adjectives? (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)

According to the interpretational theory of logical validity (IR), logical validity is preservation of truth in all interpretations compatible with the intended meaning of logical expressions. IR suffers from a seemingly defeating objection, the so-called cardinality problem: any instance of the statement ‘There are n things’ is true under all interpretations, since it can be written down using only logical expressions that are not to be reinterpreted; yet ‘There are n things’ is not logically true. I argue that the cardinality (...) problem is indeed a serious problem for IR, when understood in terms of ‘asymmetry of information’. I then argue that IR can be rehabilitated by making quantifiers context-sensitive: what we do not reinterpret is the Kaplanian character of a quantifier, rather than its content. ‘There are n things’ is false in a context where fewer than n things are relevant, so it is not logically true in IR. I finally discuss some objections and ramifications of my account: I discuss how to make space for the possibility of an explicitly absolutely general quantifier in my framework, how terms can be logical even though context-sensitive, and how to recapture classical logic within my framework. (shrink)

The isomorphism invariance criterion of logical nature has much to commend it. It can be philosophically motivated by the thought that logic is distinctively general or topic neutral. It is capable of precise set-theoretic formulation. And it delivers an extension of ‘logical constant’ which respects the intuitively clear cases. Despite its attractions, the criterion has recently come under attack. Critics such as Feferman, MacFarlane and Bonnay argue that the criterion overgenerates by incorrectly judging mathematical notions as logical. We consider five (...) possible precisifications of the overgeneration argument and find them all unconvincing. (shrink)

There have been several different and even opposed conceptions of the problem of logical constants, i.e. of the requirements that a good theory of logical constants ought to satisfy. This paper is in the first place a survey of these conceptions and a critique of the theories they have given rise to. A second aim of the paper is to sketch some ideas about what a good theory would look like. A third aim is to draw from these ideas and (...) from the preceding survey the conclusion that most conceptions of the problem of logical constants involve requirements of a philosophically demanding nature which are probably not satisfiable by any minimally adequate theory. (shrink)

Deflationism about truth describes truth as a logical notion. In the present paper, I explore the implication of the alleged logicality of truth from the perspective of axiomatic theories of truth, and argue that the deflationist doctrine of the logicality of truth gives rise to two types of self-undermining arguments against deflationism, which I call the conservativeness argument from logicality and the topic-neutrality argument.

This is a survey of work on set-theoretical invariance criteria for logicality. It begins with a review of the Tarski-Sher thesis in terms, first, of permutation invariance over a given domain and then of isomorphism invariance across domains, both characterized by McGee in terms of definability in the language L∞,∞. It continues with a review of critiques of the Tarski-Sher thesis, and a proposal in response to one of those critiques via homomorphism invariance. That has quite divergent characterization results depending (...) on its formulation, one in terms of FOL, the other by Bonnay in terms of L∞,∞, both without equality. From that we move on to a survey of Bonnay’s work on similarity relations between structures and his results that single out invariance with respect to potential isomorphism among all such. Turning to the critique that calls for sameness of meaning of a logical operation across domains, the paper continues with a result showing that the isomorphism invariant operations that are absolutely definable with respect to KPU−Inf are exactly those definable in full FOL; this makes use of an old theorem of Manders. The concluding section is devoted to a critical discussion of the arguments for set-theoretical criteria for logicality. (shrink)

Tarski's general conception of logic placed it at the center of all rational thought, and he took its aim to be the creation of a unified conceptual apparatus. In pursuit of this conviction, from his base at the University of California in Berkeley in the post-war years he campaigned vigorously on behalf of logic, locally, nationally and internationally. Though Tarski was ecumenical in his efforts to establish the importance of logic in these various ways, in his own work—even that part (...) of it which is considered to be philosophically significant—he was first and foremost a mathematician, and, even more so, a set-theoretical one. The paper traces this particular conception of logic back to the influence of his teachers in set theory and topology in Warsaw, and then details how it affected his explication of the notions of satisfaction, truth, logical consequence and logical term. (shrink)

The paper starts with an examination and critique of Tarski’s wellknown proposed explication of the notion of logical operation in the type structure over a given domain of individuals as one which is invariant with respect to arbitrary permutations of the domain. The class of such operations has been characterized by McGee as exactly those definable in the language L∞,∞. Also characterized similarly is a natural generalization of Tarski’s thesis, due to Sher, in terms of bijections between domains. My main (...) objections are that on the one hand, the Tarski-Sher thesis thus assimilates logic to mathematics, and on the other hand fails to explain the notion of same logical operation across domains of different sizes. A new notion of homomorphism invariant operation over functional type structures (with domains M0 of individuals and {T, F} at their base) is introduced to accomplish the latter. The main result is that an operation is definable from.. (shrink)

This article improves two existing theorems of interest to neologicist philosophers of mathematics. The first is a classification theorem due to Fine for equivalence relations between concepts definable in a well-behaved second-order logic. The improved theorem states that if an equivalence relation E is defined without nonlogical vocabulary, then the bicardinal slice of any equivalence class—those equinumerous elements of the equivalence class with equinumerous complements—can have one of only three profiles. The improvements to Fine’s theorem allow for an analysis of (...) the well-behaved models had by an abstraction principle, and this in turn leads to an improvement of Walsh and Ebels-Duggan’s relative categoricity theorem. (shrink)