Along with offering an historically-oriented interpretive reconstruction of the syntax of PM ( rst ed.), I argue for a certain understanding of its use of propositional function abstracts formed by placing a circum ex on a variable. I argue that this notation is used in PM only when de nitions are stated schematically in the metalanguage, and in argument-position when higher-type variables are involved. My aim throughout is to explain how the usage of function abstracts as “terms” (loosely speaking) is (...) not inconsistent with a philosophy of types that does not think of propositional functions as mind- and language-independent objects, and adopts a nominalist/substitutional semantics instead. I contrast PM’s approach here both to function abstraction found in the typed λ-calculus, and also to Frege’s notation for functions of various levels that forgoes abstracts altogether, between which it is a kind of intermediary. (shrink)

The Institute Vienna Circle held a conference in Vienna in 2003, Cambridge and Vienna a?

In these articles, I describe Cantor’s power-class theorem, as well as a number of logical and philosophical paradoxes that stem from it, many of which were discovered or considered (implicitly or explicitly) in Bertrand Russell’s work. These include Russell’s paradox of the class of all classes not members of themselves, as well as others involving properties, propositions, descriptive senses, class-intensions, and equivalence classes of coextensional properties. Part I focuses on Cantor’s theorem, its proof, how it can be used to manufacture (...) paradoxes, Frege’s diagnosis of the core difficulty, and several broad categories of strategies for offering solutions to these paradoxes. (shrink)

Sequel to Part I. In these articles, I describe Cantor’s power-class theorem, as well as a number of logical and philosophical paradoxes that stem from it, many of which were discovered or considered (implicitly or explicitly) in Bertrand Russell’s work. These include Russell’s paradox of the class of all classes not members of themselves, as well as others involving properties, propositions, descriptive senses, class-intensions and equivalence classes of coextensional properties. Part II addresses Russell’s own various attempts to solve these paradoxes, (...) including strategies that he considered and rejected (limitation of size, the zigzag theory, etc.), as well as his own final views whereupon many purported entities that, if reified, lead to these contradictions, must not be genuine entities, but ‘logical fictions’ or ‘logical constructions’ instead. (shrink)

Aristotle’s words in the Metaphysics: “to say of what is that it is, or of what is not that it is not, is true” are often understood as indicating a correspondence view of truth: a statement is true if it corresponds to something in the world that makes it true. Aristotle’s words can also be interpreted in a deflationary, i.e., metaphysically less loaded, way. According to the latter view, the concept of truth is contained in platitudes like: ‘It is true (...) that snow is white iff snow is white’, ‘It is true that neutrinos have mass iff neutrinos have mass’, etc. Our understanding of the concept of truth is exhausted by these and similar equivalences. This is all there is to truth. In his book Truth (Second edition 1998), Paul Horwich develops minimalism, a special variant of the deflationary view. According to Horwich’s minimalism, truth is an indefinable property of propositions characterized by what he calls the minimal theory, i.e., all (nonparadoxical) propositions of the form: It is true that p if and only if p. Although the idea of minimalism is simple and straightforward, the proper formulation of Horwich’s theory is no simple matter. In this paper, I shall discuss some of the difficulties of a logical nature that arise. First, I discuss problems that arise when we try to give a rigorous characterization of the theory without presupposing a prior understanding of the notion of truth. Next I turn to Horwich’s treatment of the Liar paradox and a paradox about the totality of all propositions that was first formulated by Russell (1903). My conclusion is that Horwich’s minimal theory cannot deal with these difficulties in an adequate way, and that it has to be revised in fundamental ways in order to do so. Once such revisions have been carried out the theory may, however, have lost some of its appealing simplicity. (shrink)

Certain commentators on Russell's “no class” theory, in which apparent reference to classes or sets is eliminated using higher-order quantification, including W. V. Quine and (recently) Scott Soames, have doubted its success, noting the obscurity of Russell’s understanding of so-called “propositional functions”. These critics allege that realist readings of propositional functions fail to avoid commitment to classes or sets (or something equally problematic), and that nominalist readings fail to meet the demands placed on classes by mathematics. I show that Russell (...) did thoroughly explore these issues, and had good reasons for rejecting accounts of propositional functions as extra-linguistic entities. I argue in favor of a reading taking propositional functions to be nothing over and above open formulas which addresses many such worries, and in particular, does not interpret Russell as reducing classes to language. (shrink)

Metaphysically possible worlds have many uses. Epistemically possible worlds promise to be similarly useful, especially in connection with propositions and propositional attitudes. However, I argue that there is a serious threat to the natural accounts of epistemically possible worlds, from a version of Russell’s paradox. I contrast this threat with David Kaplan’s problem for metaphysical possible world semantics: Kaplan’s problem can be straightforwardly rebutted, the problems I raise cannot. I argue that although there may be coherent accounts of epistemically possible (...) worlds with fruitful applications, any such an account must fundamentally compromise the basic idea behind epistemic possibility. (shrink)

The period from 1900 to 1935 was particularly fruitful and important for the development of logic and logical metatheory. This survey is organized along eight "itineraries" concentrating on historically and conceptually linked strands in this development. Itinerary I deals with the evolution of conceptions of axiomatics. Itinerary II centers on the logical work of Bertrand Russell. Itinerary III presents the development of set theory from Zermelo onward. Itinerary IV discusses the contributions of the algebra of logic tradition, in particular, Löwenheim (...) and Skolem. Itinerary V surveys the work in logic connected to the Hilbert school, and itinerary V deals specifically with consistency proofs and metamathematics, including the incompleteness theorems. Itinerary VII traces the development of intuitionistic and many-valued logics. Itinerary VIII surveys the development of semantical notions from the early work on axiomatics up to Tarski's work on truth. (shrink)

The nature and topology of time remains an open question in philosophy, both tensed and tenseless concepts of time appear to have merit. A concept of time including both kinds of time evolution of physical systems in quantum mechanics subsumes the properties of both notions. The linear dynamics defines the universe probabilistically throughout space-time, and can be seen as the definition of a block universe. The collapse dynamics is the time evolution of the linear dynamics, and is thus of different (...) logical type to the linear dynamics. These two different kinds of time evolution are respectively tensed and tenseless. Ascribing tensed semantics to the collapse dynamics is problematic in the light of special relativity, but this difficulty does not apply to a relational quantum mechanics. In this context, while the linear dynamics is the time evolution of the universe objectively, the collapse dynamics is the time evolution of the universe subjectively, applying solely in the functional frame of reference of the observer. (shrink)

What is predicativity? While the term suggests that there is a single idea involved, what the history will show is that there are a number of ideas of predicativity which may lead to different logical analyses, and I shall uncover these only gradually. A central question will then be what, if anything, unifies them. Though early discussions are often muddy on the concepts and their employment, in a number of important respects they set the stage for the further developments, and (...) so I shall give them special attention. NB. Ahistorically, modern logical and set-theoretical notation will be used throughout, as long as it does not conflict with original intentions. (shrink)

Many philosophers still countenance senses or meanings in the broadly Fregean vein. However, it is difficult to posit the existence of senses without positing quite a lot of them, including at least one presenting every entity in existence. I discuss a number of Cantorian paradoxes that seem to result from an overly large metaphysics of senses, and various possible solutions. Certain more deflationary and nontraditional understanding of senses, and to what extent they fare better in solving the problems, are also (...) discussed. In the end, it is concluded that one must divide senses into various ramified-orders in order to avoid antinomy, but that the philosophical justification of such orders is, as yet, still somewhat problematic. (shrink)

In this paper, the authors show that there is a reading of St. Anselm's ontological argument in Proslogium II that is logically valid (the premises entail the conclusion). This reading takes Anselm's use of the definite description "that than which nothing greater can be conceived" seriously. Consider a first-order language and logic in which definite descriptions are genuine terms, and in which the quantified sentence "there is an x such that..." does not imply "x exists". Then, using an ordinary logic (...) of descriptions and a connected greater-than relation, God's existence logically follows from the claims: (a) there is a conceivable thing than which nothing greater is conceivable, and (b) if <em>x</em> doesn't exist, something greater than x can be conceived. To deny the conclusion, one must deny one of the premises. However, the argument involves no modal inferences and, interestingly, Descartes' ontological argument can be derived from it. (shrink)

Is the assumption of a fundamental distinction between particulars and universals another unsupported dogma of metaphysics? F. P. Ramsey famously rejected the particular – universal distinction but neglected to consider the many different conceptions of the distinction that have been advanced. As a contribution to the piecemeal investigation of this issue three interrelated conceptions of the particular – universal distinction are examined: universals, by contrast to particulars, are unigrade; particulars are related to universals by an asymmetric tie of exemplification; universals (...) are incomplete whereas particulars are complete. It is argued that these conceptions are wanting in several respects. Sometimes they fail to mark a significant division amongst entities. Sometimes they make substantial demands upon the shape of reality; once these demands are understood aright it is no longer obvious that the distinction merits our acceptance. The case is made via a discussion of the possibility of multigrade universals. (shrink)

This paper aims to outline an analysis and interpretation of the process that led to First-Order Logic and its consolidation as a core system of modern logic. We begin with an historical overview of landmarks along the road to modern logic, and proceed to a philosophical discussion casting doubt on the possibility of a purely rational justification of the actual delimitation of First-Order-Logic. On this basis, we advance the thesis that a certain historical tradition was essential to the emergence of (...) modern logic; this traditional context is analyzed as consisting in some guiding principles and, particularly, a set of exemplares (i.e., paradigmatic instances). Then, we proceed to interpret the historical course of development reviewed in section 1, which can broadly be described as a two-phased movement of expansion and then restriction of the scope of logical theory. We shall try to pinpoint ambivalencies in the process, and the main motives for subsequent changes. Among the latter, one may emphasize the spirit of modern axiomatics, the situation of foundational insecurity in the 1920s, the resulting desire to find systems well-behaved from a proof-theoretical point of view, and the metatheoretical results of the 1930s. Not surprisingly, the mathematical and, more specifically, the foundational context in which Firs-Order-Logic matured will be seen to have played a primary role in its shaping. (shrink)

All paradoxes of self-reference seem to share some structural features. Russell in 1908 and especially Priest nowadays have advanced structural descriptions that successfully identify necessary conditions for having a paradox of this kind. I examine in this paper Priest’s description of these paradoxes, the Inclosure Scheme (IS), and consider in what sense it may help us understand and solve the problems they pose. However, I also consider the limitations of this kind of structural descriptions and give arguments against Priest’s use (...) of IS in favour of dialetheism. IS fails to identify sufficient conditions for having a paradox of self-reference. That means that, even if we identified a problem common to any reasoning satisfying IS, that problem would not explain why some of those reasonings are paradoxical and some others are not. Therefore IS cannot justify by itself the claim that some particular theory offers the best solution to the paradoxes of self-reference. We still need to consider aspects concerning the content and context of occurrence of every paradox. (shrink)

While sets are combinatorial collections, defined by their elements, classes are logical collections, defined by their membership conditions. We develop, in a potentialist setting, a predicative approach to (logical) classes of (combinatorial) sets. Some reasons emerge to adopt a stricter form of potentialism, which insists, not only that each object is generated at some stage of an incompletable process, but also that each truth is “made true” at some such stage. The natural logic of this strict form of potentialism is (...) semi-intuitionistic: where each set-sized domain is classical, the domain of all sets or all classes is intuitionistic. (shrink)

Toward a theory of n-tuples of individuals and concepts as surrogates for Russellian singular propositions and singular concepts. Alonzo Church proposed a powerful and elegant theory of sequences of functions and their arguments as singular-concept surrogates. Church’s account accords with his Alternative (0), the strictest of his three competing criteria for strict synonymy. The currently popular objection to strict criteria like (0) on the basis of the Russell-Myhill paradox is misguided. Russell-Myhill is not a problem specifically for Alternative (0). Rather (...) it is a disproof of unrestricted concept comprehension. Unrestricted comprehension is also inconsistent with facts about sets of properties. It is demonstrated furthermore that the principal rival conception of propositions, as classes of possible worlds, is subject to a fatal philosophical collapse: It follows on that conception, given that each of us is fallible, that everyone believes everything. Although it is far superior to its principal rival conception, Church’s proposed theory is vulnerable under (0) to a version of Russell’s notorious Gray’s /Elegy/ objection. Some amendments to Church’s proposal are proffered that address Russell’s objection, including an amendment first proposed in the present author’s /Frege’s Puzzle/ (1986). Church’s response (personal correspondence) is considered. (shrink)

I propose a revision of Cantor’s account of set size that understands comparisons of set size fundamentally in terms of surjections rather than injections. This revised account is equivalent to Cantor's account if the Axiom of Choice is true, but its consequences differ from those of Cantor’s if the Axiom of Choice is false. I argue that the revised account is an intuitive generalization of Cantor’s account, blocks paradoxes—most notably, that a set can be partitioned into a set that is (...) bigger than it—that can arise from Cantor’s account if the Axiom of Choice is false, illuminates the debate over whether the Axiom of Choice is true, is a mathematically fruitful alternative to Cantor’s account, and sheds philosophical light on one of the oldest unsolved problems in set theory. (shrink)

What is the explanatory role of ‘status-truths’ such as essence-truths, necessity-truths and law-truths? A plausible principle, suggested by various authors, is Ground by Status, according to which status truths ground their prejacents. For instance, if it is essential to a that p, then this grounds the fact that p. But Ground by Status faces a forceful objection: it is inconsistent with widely accepted principles regarding the logic of grounding (Glazier in Philos Stud 174(11):2871–2889, 2017a, Synthese 174(198):1409–1424, 2017b; Kappes in Synthese (...) 199(1–2):2575–2595, 2020, Philos Stud 178(4):1267–1284, 2021). I defend Ground by Status against this objection. (shrink)

Typically, if I understand a sentence, then it expresses a proposition that I entertain. Nonsensical sentences don’t express propositions, but there are contexts in which we talk about understanding nonsensical sentences. For example, we accept various kinds of semantically defective sentences in fiction, philosophy, and everyday life. Furthermore, it is a standard assumption that if a sentence is nonsensical, then it makes no sense to say that it implies anything or is implied by other sentences. Semantically uninterpreted sentences don’t have (...) logical characteristics. Hence, the riddle of understanding nonsense arises. We seem to use nonsensical sentences in reasoning, thinking, judging, and drawing conclusions, but they convey no propositions, which are the vehicles of their semantic properties. In this article, I propose the pretence theory of understanding nonsense to explain the riddle of understanding nonsense, and discuss alternative frameworks that are insufficient to solve it. -/- . (shrink)

A historically popular response to the liar paradox (“this sentence is false”) is to say that the liar sentence is meaningless (or semantically defective, or malfunctions, or…). Unfortunately, like all other supposed solutions to the liar, this approach faces a revenge challenge. Consider the revenge liar sentence, “this sentence is either meaningless or false”. If it is true, then it is either meaningless or false, so not true. And if it is not true, then it can’t be either meaningless or (...) false, so it must be true. Either way, we are back in a paradox. This paper provides a detailed and exhaustive discussion of the options for responding to revenge on behalf of “meaningless” theories. Though I attempt to discuss all of the options fairly, I will ultimately opt for one specific response and discuss some of its challenges. Various technical and logical matters will be discussed throughout the paper, but my focus will be philosophical, throughout. My overall conclusion is that the “meaningless” strategy is at least as well off in the face of revenge as any other approach to the liar and related paradoxes. (shrink)

Higher-order logic, with its type-theoretic apparatus known as the simple theory of types (STT), has increasingly come to be employed in theorizing about properties, relations, and states of affairs—or ‘intensional entities’ for short. This paper argues against this employment of STT and offers an alternative: ordinal type theory (OTT). Very roughly, STT and OTT can be regarded as complementary simplifications of the ‘ramified theory of types’ outlined in the Introduction to Principia Mathematica (on a realist reading). While STT, understood as (...) a theory of intensional entities, retains the Fregean division of properties and relations into a multiplicity of categories according to their adicities and ‘input types’ and discards the division of intensional entities into different ‘orders’, OTT takes the opposite approach: it retains the hierarchy of orders (though with some modifications) and discards the categorisation of properties and relations according to their adicities and input types. In contrast to STT, this latter approach avoids intensional counterparts of the Epimenides and related paradoxes. Fundamental intensional entities lie at the base of the proposed hierarchy and are also given a prominent part to play in the individuation of non-fundamental intensional entities. (shrink)

The key to the measurement problem is the entity at the heart of Everett's formulation, the state of the memory, defined as the record of observations. In humans, the integrated synthesis defines the perceptual reality, a projective, three-dimensional representation of the world. This 'world hologram' is the conscious point of view, the mind in Lockwood's interpretation, the 'phenomenal perspective'. As Everett demonstrates, the collapse dynamics operates only judged by the state of the memory; the physical observer remains in a superposed (...) state. An operational mind-body property dualism is defined. This field of information is multiply instantiated in the many worlds of the unitary wave function. As these are superposed and coincident, the net result is a single world hologram. The physical frame of reference on this view is the superposition of the class of instantiating worlds, a second-logical-type phenomenon. By definition determinate only where defined by observations, this is the physical ontology of Everett's relative state. Collapse on observation is inherent. The evidence is the retrodiction of the holographic universe. This is the cosmology of the relative state. As it is determinate solely where defined by the world hologram that delineates the outer boundary, entropy is defined solely by this surface. The quasi-classical world is the cosmology in the absolute state. The incompatible dynamics of quantum mechanics operate in these two different types of physical frames of reference. The linear dynamics is the time evolution of the quasi-classical worlds. On observation, judged by the state of the world hologram, the class of instantiating worlds is redefined, becoming the class of worlds in which the observed outcome occurred. Collapse effectively operates. The holographic universe of the mind updates. The dynamics operate at different levels of logical type. The measurement problem is a category error. (shrink)

In the language of second-order logic, first- and second-order variables are distinguished syntactically and cannot be grammatically substituted. According to a prominent argument for the deployment of these languages, these substitution failures are necessary to block the derivation of paradoxes that result from attempts to generalize over predicate interpretations. I first examine previous approaches which interpret second-order sentences using expressions of natural language and argue that these approaches undermine these syntactic restrictions. I then examine Williamson’s primitivist approach according to which (...) second-order sentences are not offered readings in a previously understood language. I argue that the syntactic restrictions alone do not block the derivation of the paradox, unless they are backed by a principled reason that the language cannot be expanded to allow the grammatical substitution of first- and second- order variables. I argue that there is neither a syntactic nor a semantic principle that prohibits such an expansion. (shrink)

The austere view of nonsense says that the source of nonsense is not a violation of the rules of logical syntax, but nonsense is always due to a lack of meaning in one of the components of a sentence. In other words, the necessary and sufficient condition for nonsensicality is that no meaning has been assigned to a constituent in a sentence. The austere conception is the key ingredient of the resolute reading of Tractatus Logico-Philosophicus that presents a therapeutical interpretation (...) of the work, and rejects a possibility of conveying some ineffable truths by means of nonsensical sentences. In this paper, I defend the austere view against the most important objections. Firstly, I discuss textual sources for ascribing to Wittgenstein the austere view that have not been sufficiently thoroughly discussed yet. Next, I indicate that the apparent occurrence of words out of the context of a sentence on lists of verbs and in definitions is not in conflict with the version of the context principle recommended by austere theorists, since this argument does not sufficiently take into account the distinction between sign and symbol. Furthermore, I argue that the austere conception and the principle of compositionality are jointly tenable, for the former view does not exclude the existence of syntactic and semantic rules, but only shows that they are recognizable conditionally. Finally, I indicate that the austere view of nonsense should be seen as a viable option by a contemporary theorist of nonsense, for it helps us to avoid making controversial assumptions about the nature of meaning and content. (shrink)

This paper argues that we should solve paradoxes for propositions (such as the Russell–Myhill paradox) in essentially the same way that we solve Russellian paradoxes for sets. That is, the standard, iterative approach to sets is extended to include properties, and then the resulting hierarchy of sets and properties is used to construct propositions. Propositions on this account are structured in the sense of mirroring the sentences that express them, and they would seem to serve the needs of philosophers of (...) language and metaphysicians who rely on such propositions. The resulting account has limitations, comparable to those faced by the iterative approach to sets, but it is argued to be in important ways preferable to others that have been proposed. (shrink)

Some propositions are true, and it is true that some propositions are true. Each of these facts looks like an impeccable ground of the other. But they cannot both ground each other, since grounding is asymmetric. This paper explores two new diagnoses of this much discussed puzzle. The tools of higher-order logic are used to show how both diagnoses can be fleshed out into strong and consistent theories of grounding. These theories of grounding in turn demand new theories of the (...) granularity of propositions, properties, and relations. Even those who are uninterested in grounding should take seriously these pictures of reality’s logical structure, which are in many ways reminiscent of Russell’s and Wittgenstein’s logical atomism. (shrink)

Logical monism is the view that there is ‘One True Logic’. This is the default position, against which pluralists react. If there were not ‘One True Logic’, it is hard to see how there could be one true theory of anything. A theory is closed under a logic! But what is logical monism? In this article, I consider semantic, logical, modal, scientific, and metaphysical proposals. I argue that, on no ‘factualist’ analysis (according to which ‘there is One True Logic’ expresses (...) a factual claim, rather than an attitude like approval), does the doctrine have both metaphysical and methodological import. Metaphysically, logics abound. Methodologically, what to infer from what is not settled by the facts, even the normative ones. I conclude that the only interesting sense in which there could be One True Logic is noncognitive. The same may be true of monism about normative areas, like moral, epistemic, and prudential ones, generally. (shrink)

What explains the truth of a universal generalization? Two types of explanation can be distinguished. While an ‘instance-based explanation’ proceeds via some or all instances of the generalization, a ‘generic explanation’ is independent of the instances, relying instead on completely general facts about the properties or operations involved in the generalization. This intuitive distinction is analyzed by means of a truthmaker semantics, which also sheds light on the correct logic of quantification. On the most natural version of the semantics, this (...) analysis vindicates some claims made—without a proper defense—by Michael Dummett, Solomon Feferman, and others. Where instance-based explanations are freely available, classical logic is shown to be warranted. By contrast, intuitionistic logic remains warranted regardless of what explanations are available. (shrink)

Mi objetivo es discutir las principales dificultades que Frank P. Ramsey encontró en Principia Mathematica y la solución que, vía el Tractatus Logico-Philosophicus, propuso al respecto. Sostengo que las principales dificultades que Ramsey encontró en Principia Mathematica están, todas, relacionadas con que Russell y Whitehead desatendieron la forma lógica de las proposiciones matemáticas, las cuales, según Ramsey, deben ser tautológicas.

This book proposes a novel theory of truth and falsity. It argues that truth is a form of reference and falsity is a form of reference failure. -/- Most of the philosophical literature on truth concentrates on certain ontological and epistemic problems. This book focuses instead on language. By utilizing the Fregean idea that sentences are singular referring expressions, the author develops novel connections between the philosophical study of truth and falsity and the huge literature in in the philosophy of (...) language on the notion of reference. The first part of the book constructs the author’s theory and argues for it in length. Part II addresses the ways in which the theory relates to, and is different from, some of the basic theories of truth. Part III takes up how to account for the truth of sentences with logical operators and quantifiers. Finally, Part IV discusses the applications and implications of the theory for longstanding problems in philosophy of language, metaphysics, and epistemology. -/- A Referential Theory of Truth and Falsity will appeal to researchers and advanced students working in philosophy of language, epistemology, metaphysics, and linguistics. -/- . (shrink)

The Russell–Myhill theorem threatens a familiar structured conception of propositions according to which two sentences express the same proposition only if they share the same syntactic structure and their corresponding syntactic constituents share the same semantic value. Given the role of the principle of universal instantiation in the derivation of the theorem in simple type theory, one may hope to rehabilitate the core of the structured view of propositions in ramified type theory, where the principle is systematically restricted. We suggest (...) otherwise. The ramified core of the structured theory of propositions remains inconsistent in ramified type theory augmented with axioms of reducibility. This is significant because reducibility has been thought to be perfectly consistent with the ramified approach to the intensional antinomies. Nor is the addition of reducibility to ramified type theory sufficient to restore other intensional puzzles such as Prior’s paradox or Kripke’s puzzle about time and thought. (shrink)

Call a quantifier ‘unrestricted’ if it ranges over absolutely all objects. Arguably, unrestricted quantification is often presupposed in philosophical inquiry. However, developing a semantic theory that vindicates unrestricted quantification proves rather difficult, at least as long as we formulate our semantic theory within a classical first-order language. It has been argued that using a type theory as framework for our semantic theory provides a resolution of this problem, at least if a broadly Fregean interpretation of type theory is assumed. However, (...) the intelligibility of this interpretation has been questioned. In this paper I introduce a type-free theory of properties that can also be used to vindicate unrestricted quantification. This alternative emerges very naturally by reflecting on the features on which the type-theoretic solution of the problem of unrestricted quantification relies. Although this alternative theory is formulated in a non-classical logic, it preserves the deductive strength of classical strict type theory in a natural way. The ideas developed in this paper make crucial use of Russell’s notion of range of significance. (shrink)

Inspired by Cantor's Theorem (CT), orthodoxy takes infinities to come in different sizes. The orthodox view has had enormous influence in mathematics, philosophy, and science. We will defend the contrary view---Countablism---according to which, necessarily, every infinite collection (set or plurality) is countable. We first argue that the potentialist or modal strategy for treating Russell's Paradox, first proposed by Parsons (2000) and developed by Linnebo (2010, 2013) and Linnebo and Shapiro (2019), should also be applied to CT, in a way that (...) vindicates Countabilism. Our discussion dovetails with recent independently developed treatments of CT in Meadows (2015), Pruss (2020), and Scambler (2021), aimed at establishing the mathematical viability, and therefore epistemic possibility, of Countabilism. Unlike these authors, our goal isn't to vindicate the mathematical underpinnings of Countabilism. Rather, we aim to argue that, given that Countabilism is mathematically viable, Countabilism should moreover be regarded as true. After clarifying the modal content of Countabilism, we canvas some of Countabilism's many positive implications, including that Countabilism provides the best account of the pervasive independence phenomena in set theory, and that Countabilism has the power to defuse several persistent puzzles and paradoxes found in physics and metaphysics. We conclude that in light of its theoretical and explanatory advantages, Countabilism is more likely true than not. (shrink)

A resolution to the Russell Paradox is presented that is similar to Russell's “theory of types” method but is instead based on the definition of why a thing exists as described in previous work by this author. In that work, it was proposed that a thing exists if it is a grouping tying "stuff" together into a new unit whole. In tying stuff together, this grouping defines what is contained within the new existent entity. A corollary is that a thing, (...) such as a set, does not exist until after the stuff is tied together, or said another way, until what is contained within is completely defined. A second corollary is that after a grouping defining what is contained within is present and the thing exists, if one then alters what is tied together (e.g., alters what is contained within), the first existent entity is destroyed and a different existent entity is created. A third corollary is that a thing exists only where and when its grouping exists. Based on this, the Russell Paradox's set R of all sets that aren't members of themselves does not even exist until after the list of the elements it contains (e.g. the list of all sets that aren't members of themselves) is defined. Once this list of elements is completely defined, R then springs into existence. Therefore, because it doesn't exist until after its list of elements is defined, R obviously can't be in this list of elements and, thus, cannot be a member of itself; so, the paradox is resolved. This same type of reasoning is then applied to Godel's first Incompleteness Theorem. Briefly, while writing a Godel Sentence, one makes reference to a future, not yet completed and not yet existent sentence, G, that claims its unprovability. However, only once the sentence is finished does it become a new unit whole and existent entity called sentence G. If one then goes back in and replaces the reference to the future sentence with the future sentence itself, a totally different sentence, G1, is created. This new sentence G1 does not assert its unprovability. An objection might be that all the possibly infinite number of possible G-type sentences or their corresponding Godel numbers already exist somehow, so one doesn't have to worry about references to future sentences and springing into existence. But, if so, where do they exist? If they exist in a Platonic realm, where is this realm? If they exist pre-formed in the mind, this would seem to require a possibly infinite-sized brain to hold all these sentences. This is not the case. What does exist in the mind is the system for creating G-type sentences and their corresponding numbers. This mental system for making a G-type sentence is not the same as the G-type sentence itself just as an assembly line is not the same as a finished car. In conclusion, a new resolution of the Russell Paradox and some issues with proofs of Godel's First Incompleteness Theorem are described. (shrink)

A paradox about sets of properties is presented. The paradox, which invokes an impredicatively defined property, is formalized in a free third-order logic with lambda-abstraction, through a classically proof-theoretically valid deduction of a contradiction from a single premise to the effect that every property has a unit set. Something like a model is offered to establish that the premise is, although classically inconsistent, nevertheless consistent, so that the paradox discredits the logic employed. A resolution through the ramified theory of types (...) is considered. Finally, a general scheme that generates a family of analogous paradoxes and a generally applicable resolution are proposed. (shrink)

Hegel often says that his "logic" is meant to replace metaphysics. Since Hegel's Science of Logic is so different from a standard logic, most commentators have not treated the portion of that work devoted to logical forms as relevant to this claim. This paper argues that Hegel's discussion of logical forms of judgment and syllogism is meant to be the foundation of his reformation of metaphysics. Implicit in Hegel's discussion of the logical forms is the view that the metaphysical concepts (...) discussed in Books I and II of the Science of Logic supervene on the role of subject and predicate terms in the logical forms discussed in Book III. Hegel thus has an explanation for the nature and significance of metaphysical concepts that resembles Kant's "metaphysical deduction," according to which the categories can be derived from the table of judgments. Though Hegel's metaphysics is often supposed to be influenced by Kant, prevailing interpretations do not show how Hegel's fine-grained treatment of logical forms is relevant to his critical view of metaphysics. The present interpretation provides a model for Hegel's explanation of metaphysical concepts, as well as a new picture of the structure of his Science of Logic that emphasizes the priority of its Doctrine of the Concept. (shrink)

A survey of logical problems for the concept of omniscience.

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)

This dissertation examines several of the problems that Hilbert discovered in the foundations of mathematics, from a metalogical perspective. The problems manifest themselves in four different aspects of Hilbert’s views: (i) Hilbert’s axiomatic approach to the foundations of mathematics; (ii) His response to criticisms of set theory; (iii) His response to intuitionist criticisms of classical mathematics; (iv) Hilbert’s contribution to the specification of the role of logical inference in mathematical reasoning. This dissertation argues that Hilbert’s axiomatic approach was guided primarily (...) by model theoretical concerns. Accordingly, the ultimate aim of his consistency program was to prove the model-theoretical consistency of mathematical theories. It turns out that for the purpose of carrying out such consistency proofs, a suitable modification of the ordinary first-order logic is needed. To effect this modification, independence-friendly logic is needed as the appropriate conceptual framework. It is then shown how the model theoretical consistency of arithmetic can be proved by using IF logic as its basic logic. Hilbert’s other problems, manifesting themselves as aspects (ii), (iii), and (iv)—most notably the problem of the status of the axiom of choice, the problem of the role of the law of excluded middle, and the problem of giving an elementary account of quantification—can likewise be approached by using the resources of IF logic. It is shown that by means of IF logic one can carry out Hilbertian solutions to all these problems. The two major results concerning aspects (ii), (iii) and (iv) are the following: (a) The axiom of choice is a logical principle; (b) The law of excluded middle divides metamathematical methods into elementary and non-elementary ones. It is argued that these results show that IF logic helps to vindicate Hilbert’s nominalist philosophy of mathematics. On the basis of an elementary approach to logic, which enriches the expressive resources of ordinary first-order logic, this dissertation shows how the different problems that Hilbert discovered in the foundations of mathematics can be solved. (shrink)

In this dissertation, we shall investigate whether Tennant's criterion for paradoxicality(TCP) can be a correct criterion for genuine paradoxes and whether the requirement of a normal derivation(RND) can be a proof-theoretic solution to the paradoxes. Tennant’s criterion has two types of counterexamples. The one is a case which raises the problem of overgeneration that TCP makes a paradoxical derivation non-paradoxical. The other is one which generates the problem of undergeneration that TCP renders a non-paradoxical derivation paradoxical. Chapter 2 deals with (...) the problem of undergeneration and Chapter 3 concerns the problem of overgeneration. Chapter 2 discusses that Tenant’s diagnosis of the counterexample which applies CR−rule and causes the undergeneration problem is not correct and presents a solution to the problem of undergeneration. Chapter 3 argues that Tennant’s diagnosis of the counterexample raising the overgeneration problem is wrong and provides a solution to the problem. Finally, Chapter 4 addresses what should be explicated in order for RND to be a proof-theoretic solution to the paradoxes. (shrink)

Based on his Inclosure Schema and the Principle of Uniform Solution (PUS), Priest has argued that Curry’s paradox belongs to a different family of paradoxes than the Liar. Pleitz (2015, The Logica Yearbook 2014, pp. 233–248) argued that Curry’s paradox shares the same structure as the other paradoxes and proposed a scheme of which the Inclosure Schema is a particular case and he criticizes Priest’s position by pointing out that applying the PUS implies the use of a paraconsistent logic that (...) does not validate Contraction, but that this can hardly seen as uniform. In this paper, we will develop some further reasons to defend Pleitz’ thesis that Curry’s paradox belongs to the same family as the rest of the self-referential paradoxes & using the idea that conditionals are generalized negations. However, we will not follow Pleitz in considering doubtful that there is a uniform solution for the paradoxes in a paraconsistent spirit. We will argue that the paraconsistent strategies can be seen as special cases of the strategy of restricting Detachment and that the latter uniformly blocks all the connective-involving self-referential paradoxes, including Curry’s. (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)

Our approach to the liar paradox is based on the Wittgensteinian approach to semantic and logical paradoxes. The main aim of this article is to point out that the liar sentence is only seemingly intelligible, and that it has not been given any sense. First, we will present the traditional solutions of the paradox, especially those which we call modificational. Then we will determine what the defects of these solutions are. Our main objection is that the modificational approaches assume that (...) we can express in languages certain senses which are improper. Next, we will explain why we think that the liar sentence is a mere nonsense. This sentence does not have any role in any language game – it is completely useless. We will also respond to several objections to our approach. 1. That it is not consistent with the principle of compositionality of sense. 2. According to the Quineian philosophy of logic, paradoxical sentences can be conceived as false assumptions leading to crises of logical paradigms. 3. The liar sentence seems to be, contrary to our approach, intelligible. (shrink)

In two recent papers, Bob Hale has attempted to free second-order logic of the 'staggering existential assumptions' with which Quine famously attempted to saddle it. I argue, first, that the ontological issue is at best secondary: the crucial issue about second-order logic, at least for a neo-logicist, is epistemological. I then argue that neither Crispin Wright's attempt to characterize a `neutralist' conception of quantification that is wholly independent of existential commitment, nor Hale's attempt to characterize the second-order domain in terms (...) of definability, can serve a neo-logicist's purposes. The problem, in both cases, is similar: neither Wright nor Hale is sufficiently sensitive to the demands that impredicativity imposes. Finally, I defend my own earlier attempt to finesse this issue, in "A Logic for Frege's Theorem", from Hale's criticisms. (shrink)