W. V. Quine famously defended two theses that have fallen rather dramatically out of fashion. The first is that intensions are “creatures of darkness” that ultimately have no place in respectable philosophical circles, owing primarily to their lack of rigorous identity conditions. However, although he was thoroughly familiar with Carnap’s foundational studies in what would become known as possible world semantics, it likely wouldn’t yet have been apparent to Quine that he was fighting a losing battle against intensions, due in (...) large measure to developments stemming from Carnap’s studies and culminating in the work of Kripke, Hintikka, and Bayart. These developments undermined Quine’s crusade against intensions on two fronts. First, in the context of possible world semantics, intensions could after all be given rigorous identity conditions by defining them (in the simplest case) as functions from worlds to appropriate extensions, a fact exploited to powerful and influential effect in logic and linguistics by the likes of Kaplan, Montague, Lewis, and Cresswell. Second, the rise of possible world semantics fueled a strong resurgence of metaphysics in contemporary analytic philosophy that saw properties and propositions widely, fruitfully, and unabashedly adopted as ontological primitives in their own right. This resurgence — happily, in my view — continues into the present day. -/- For a time, at any rate, Quine experienced somewhat better success with his second thesis: that higher-order logic is, at worst, confused and, at best, a quirky notational alternative to standard first-order logic. However, Quine notwithstanding, a great deal of recent work in formal metaphysics transpires in a higher-order logical framework in which properties and propositions fall into an infinite hierarchy of types of (at least) every finite order. Initially, the most philosophically compelling reason for embracing such a framework since Russell first proposed his simple theory of types was simply that it provides a relatively natural explanation of the paradoxes. However, since the seminal work of Prior there has been a growing trend to consider higher-order logic to be the most philosophically natural framework for metaphysical inquiry, many of the contributors to this volume being among the most important and influential advocates of this view. Indeed, this is now quite arguably the dominant view among formal metaphysicians. -/- In this paper, and against the current tide, I will argue in §1 that there are still good reasons to think that Quine’s second battle is not yet lost and that the correct framework for logic is first-order and type-free — properties and propositions, logically speaking, are just individuals among others in a single domain of quantification — and that it arises naturally out of our most basic logical and semantical intuitions. The data I will draw upon are not new and are well-known to contemporary higher-order metaphysicians. However, I will try to defend my thesis in what I believe is a novel way by suggesting that these basic intuitions ground a reasonable distinction between “pure” logic and non-logical theory, and that Russell-style semantic paradoxes of truth and exemplification arise only when we move beyond the purely logical and, hence, do not of themselves provide any strong objection to a type-free conception of properties and propositions. -/- Most of my arguments in §1 are largely independent of any specific account of the nature of properties and propositions beyond their type-freedom. However, I will in addition argue that there are good reasons to take propositions, at least, to be very fine-grained. My arguments are thus bolstered significantly if it can be shown that there are in fact well-defined examples of logics that are not only type-free but which comport with such a conception of propositions. It is the purpose of §2 to lay out a logic of this sort in some detail, drawing especially upon work by George Bealer and related work of my own. With the logic in place, it will be possible to generalize the line of argument noted above regarding Russell-style paradoxes and, in §3, apply it to two propositional paradoxes — the Prior-Kaplan paradox and the Russell-Myhill paradox — that are often taken to threaten the sort of account developed here. (shrink)

Although higher-order metaphysics seems prima facie to be a promising new approach to metaphysics, it is nonetheless based on a mistake. This mistake is tied to a misuse of formal languages in metaphysics in general, not just to the use of higher-order rather than lower-order languages. I hope to highlight the mistake by discussing a popular recent example of higher- order metaphysics: the argument that reality is not structured using reasoning inspired by (...) the Russell-Myhill paradox. A key issue will be the relationship between higher-order quantification in formal languages and quantification in English, in particular the questions whether such quantifiers are already to be found in English, and whether they could simply be added to English if not in order to gain expressive strength. I will conclude that little of metaphysical significance follows from this Russell-Myhill inspired argument against structure. The reason why the results of higher-order logic show little of metaphysical significance generalizes to other uses of higher-order logic in metaphysical theorizing, and with it supports that higher-order metaphysics is a flawed research program. I end with some reflections on what positive role formal tools like higher-order logic can have in metaphysics, and what the fundamental mistake is on which higher-order metaphysics is based. (shrink)

This paper argues that the theory of structured propositions is not undermined by the Russell-Myhill paradox. I develop a theory of structured propositions in which the Russell-Myhill paradox doesn't arise: the theory does not involve ramification or compromises to the underlying logic, but rather rejects common assumptions, encoded in the notation of the $\lambda$-calculus, about what properties and relations can be built. I argue that the structuralist had independent reasons to reject these (...) underlying assumptions. The theory is given both a diagrammatic representation, and a logical representation in a novel language. In the latter half of the paper I turn to some technical questions concerning the treatment of quantification, and demonstrate various equivalences between the diagrammatic and logical representations, and a fragment of the $\lambda$-calculus. (shrink)

Although higher-order metaphysics seems prima facie to be a promising new approach to metaphysics, it is nonetheless based on a mistake. This mistake is tied to a misuse of formal languages in metaphysics in general, not just to the use of higher-order rather than lower-order languages. I hope to highlight the mistake by discussing a popular recent example of higher- order metaphysics: the argument that reality is not structured using reasoning inspired by (...) the Russell-Myhill paradox. A key issue will be the relationship between higher-order quantification in formal languages and quantification in English, in particular the questions whether such quantifiers are already to be found in English, and whether they could simply be added to English if not in order to gain expressive strength. I will conclude that little of metaphysical significance follows from this Russell-Myhill inspired argument against structure. The reason why the results of higher-order logic show little of metaphysical significance generalizes to other uses of higher-order logic in metaphysical theorizing, and with it supports that higher-order metaphysics is a flawed research program. I end with some reflections on what positive role formal tools like higher-order logic can have in metaphysics, and what the fundamental mistake is on which higher-order metaphysics is based. (shrink)

This paper sets out a predicative response to the Russell-Myhill paradox of propositions within the framework of Church’s intensional logic. A predicative response places restrictions on the full comprehension schema, which asserts that every formula determines a higher-order entity. In addition to motivating the restriction on the comprehension schema from intuitions about the stability of reference, this paper contains a consistency proof for the predicative response to the Russell-Myhill paradox. The models used (...) to establish this consistency also model other axioms of Church’s intensional logic that have been criticized by Parsons and Klement: this, it turns out, is due to resources which also permit an interpretation of a fragment of Gallin’s intensional logic. Finally, the relation between the predicative response to the Russell-Myhill paradox of propositions and the Russell paradox of sets is discussed, and it is shown that the predicative conception of set induced by this predicative intensional logic allows one to respond to the Wehmeier problem of many non-extensions. (shrink)

This chapter explores the metaphysical views about higher-order logic held by two individuals responsible for introducing it to philosophy: Gottlob Frege (1848–1925) and Bertrand Russell (1872–1970). Frege understood a function at first as the remainder of the content of a proposition when one component was taken out or seen as replaceable by others, and later as a mapping between objects. His logic employed second-order quantifiers ranging over such functions, and he saw a deep division in nature (...) between objects and functions. Russell understood propositional functions as what is obtained when constituents of propositions are replaced by variables, but eventually denied that they were entities in their own right. Both encountered contradictions when supposing there to exist as many objects as functions, and both adopted views about the meaningfulness of higher-order discourse that were difficult to state from within their own strictures. (shrink)

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

The Russell-Myhill paradox puts pressure on the Russellian structured view of propositions by showing that it conflicts with certain prima facie attractive ontological and logical principles. I describe several versions of RMP and argue that structurists can appeal to natural assumptions about metaphysical grounding to provide independent reasons for rejecting the ontological principles used in these paradoxes. It remains a task for future work to extend this grounding-based approach to all variants of RMP.

According to relationism, for Alice to believe that some rabbits can speak is for Alice to stand in a relation to a further entity, some rabbits can speak. But what could this further entity possibly be? Higher-order metaphysics seems to offer a simple, natural answer. On this view (roughly put), expressions in different syntactic categories (for instance: names, predicates, sentences) in general denote entities in correspondingly different ontological categories. Alice's belief can thus be understood to relate her to (...) a sui generis entity denoted by "some rabbits can speak", belonging to a different ontological category than Alice herself. This straightforward account of the attitudes has historically been deemed so attractive that it was seen as providing an important motivation for higher-order metaphysics itself (Prior [1971]). But I argue that it is not as straightforward as it might seem, and in fact that propositional attitudes present a foundational challenge for higher-order metaphysics. (shrink)

This chapter provides an introduction to higher-order metaphysics as well as to the contributions to this volume. We discuss five topics, corresponding to the five parts of this volume, and summarize the contributions to each part. First, we motivate the usefulness of higher-order quantification in metaphysics using a number of examples, and discuss the question of how such quantifiers should be interpreted. We provide a brief introduction to the most common forms of higher-order logics (...) used in metaphysics, and indicate a number of questions which can be raised in such systems using logical vocabulary alone. Using a further example, we return to applications of higher-order logics in metaphysics. We also mention key developments in the history of higher-order logic as it pertains to metaphysics. Finally, we mention certain arguments which have been raised against the use of higher-order logic, and some ways of responding to them. (shrink)

This is the first comprehensive textbook on higher order logic that is written specifically to introduce the subject matter to graduate students in philosophy. The book covers both the formal aspects of higher-order languages -- their model theory and proof theory, the theory of λ-abstraction and its generalizations -- and their philosophical applications, especially to the topics of modality and propositional granularity. The book has a strong focus on non-extensional higher-order logics, making it more (...) appropriate for foundational metaphysics than other introductions to the subject from computer science, mathematics, and linguistics. A Philosophical Introduction to Higher Order Logics assumes only that readers have a basic knowledge of first-order logic. With an emphasis on exercises, it can be used as a textbook though is also ideal for self-study. Author Andrew Bacon organizes the book's 18 chapters around four main parts: I. Typed Language II. Higher Order Languages III. General Higher-Order Languages IV. Higher-Order Model Theory In addition, two appendices cover the Curry-Howard isomorphism and its applications for modeling propositional structure. Each chapter includes exercises that move from easier to more difficult, strategically placed throughout the chapter, and concludes with an annotated suggested reading list providing graduate students with most valuable additional resources. Key Features Is the first comprehensive introduction to higher-order logic as a grounding for addressing problems in metaphysics Introduces the basic formal tools that are needed to theorize in, and model, higher order languages Offers an abundance of: - Simple exercises throughout the book, serving as comprehension checks on basic concepts and definitions - More difficult exercises designed to facilitate long-term learning Contains annotated sections on further reading, pointing the reader to related literature, learning resources, and historical context. (shrink)

The Russell-Myhill Antinomy, also known as the Principles of Mathematics Appendix B Paradox, is a contradiction that arises in the logical treatment of classes and "propositions", where "propositions" are understood as mind-independent and language-independent logical objects. If propositions are treated as objectively existing objects, then they can be members of classes. But propositions can also be about classes, including classes of propositions. Indeed, for each class of propositions, there is a proposition stating that all propositions in that (...) class are true. Propositions of this form are said to "assert the logical product" of their associated classes. Some such propositions are themselves in the class whose logical product they assert. For example, the proposition asserting that all-propositions-in-the- class-of-all-propositions-are-true is itself a proposition, and therefore it itself is in the class whose logical product it asserts. However, the proposition stating that all-propositions-in-the-null-class-are-true is not itself in the null class. Now consider the class w, consisting of all propositions that state the logical product of some class m in which they are not included. This w is itself a class of propositions, and so there is a proposition r, stating its logical product. The contradiction arises from asking the question of whether r is in the class w. It seems that r is in w just in case it is not. This antinomy was discovered by Bertrand Russell in 1902, a year after discovering a simpler paradox usually called Russell's paradox ". It was discussed informally in Appendix B of his 1903 Principles of Mathematics . In 1958, the antinomy was independently rediscovered by John Myhill, who found it to plague the "Logic of Sense and Denotation" developed by Alonzo Church. (shrink)

We defend hylomorphism against Maegan Fairchild’s purported proof of its inconsistency. We provide a deduction of a contradiction from SH+, which is the combination of “simple hylomorphism” and an innocuous premise. We show that the deduction, reminiscent of Russell’s Paradox, is proof-theoretically valid in classical higher-order logic and invokes an impredicatively defined property. We provide a proof that SH+ is nevertheless consistent in a free higher-order logic. It is shown that the unrestricted comprehension principle (...) of property abstraction on which the purported proof of inconsistency relies is analogous to naïve unrestricted set-theoretic comprehension. We conclude that logic imposes a restriction on property comprehension, a restriction that is satisfied by the ramified theory of types. By extension, our observations constitute defenses of theories that are structurally similar to SH+, such as the theory of singular propositions, against similar purported disproofs. (shrink)

Subverting a once widely held Quinean paradigm, there is a growing consensus among philosophers of logic that higher-order quantifiers (which bind variables in the syntactic position of predicates and sentences) are a perfectly legitimate and useful instrument in the logico-philosophical toolbox, while neither being reducible to nor fully explicable in terms of first-order quantifiers (which bind variables in singular term position). This article discusses the impact of this quantificational paradigm shift on metaphysics, focussing on theories of properties, (...) propositions, and identity, as well as on the metaphysics of modality. (shrink)

Higher-order logic augments first-order logic with devices that let us generalize into grammatical positions other than that of a singular term. Some recent metaphysicians have advocated for using these devices to raise and answer questions that bear on many traditional issues in philosophy. In contrast to these 'higher-order metaphysicians', traditional metaphysics has often focused on parallel, but importantly different, questions concerning special sorts of abstract objects: propositions, properties and relations. The answers to the higher- (...) class='Hi'>order and the property-theoretic questions may coincide sometimes but will often come apart. I argue that when they do, the higher-order questions are closer to the metaphysical action and so it would be better for these debates to proceed in higher-order terms. (shrink)

Deleuze’s “transcendental empiricism” and the “empirical side” of Whitehead’s metaphysics are paradoxical unless placed in the context of their unorthodox readings of empiricism. I explore this context focusing on their engagements with Hume. Both subvert presumptions of a categorical gap between external nature and internal human experience and open possibilities for a speculative empiricism that is non-reductive while still affirming experience as source for philosophical thinking. Deleuze and Whitehead follow Hume in beginning with events of sensation as primary but do (...) not presume the logic of (human) subjects and objects (of nature) as necessary structuring polarities for their interpretation. This challenges a basic distinction (between inner and outer or between self and world) that seems inherent in the ordinary concept of experience, thus earning the moniker speculative. The speculative empiricist studies how these abstractions arise from events of experience prior to their consolidation in representation. This includes a critical component: to what extent do unexamined assumptions about conceptual abstraction hinder, block, or prefigure experiential attention? This critical component has existential implications for how we attend to the affective, intuitive, and preconceptual. (shrink)

This book articulates and defends Fregean realism, a theory of properties based on Frege's insight that properties are not objects, but rather the satisfaction conditions of predicates. Robert Trueman argues that this approach is the key not only to dissolving a host of longstanding metaphysical puzzles, such as Bradley's Regress and the Problem of Universals, but also to understanding the relationship between states of affairs, propositions, and the truth conditions of sentences. Fregean realism, Trueman suggests, ultimately leads to a version (...) of the identity theory of truth, the theory that true propositions are identical to obtaining states of affairs. In other words, the identity theory collapses the gap between mind and world. This book will be of interest to anyone working in logic, metaphysics, the philosophy of language or the philosophy of mind. (shrink)

This chapter offers an opinionated introduction to higher-order formal languages with an eye towards their applications in metaphysics. A simply relationally typed higher-order language is introduced in four stages: starting with first-order logic, adding first-order predicate abstraction, generalizing to higher-order predicate abstraction, and finally adding higher-order quantification. It is argued that both β-conversion and Universal Instantiation are valid on the intended interpretation of this language. Given these two principles, it is (...) then shown how we can use pure higher-order logic to ask, and begin to answer, metaphysical questions with non-trivial implications. In particular, while we must reject the popular idea that structural differences between sentences correspond to parallel distinctions in the logical structure of extra-linguistic reality, it may still be possible to give a purely logical characterization of objectual aboutness and related notions. (shrink)

The present work outlines a logical and philosophical conception of propositions in relation to a group of puzzles that arise by quantifying over them: the Russell-Myhill paradox, the Prior-Kaplan paradox, and Prior's Theorem. I begin by motivating an interpretation of Russell-Myhill as depending on aboutness, which constrains the notion of propositional identity. I discuss two formalizations of of the paradox, showing that it does not depend on the syntax of propositional variables. I then (...) extend to propositions a modal predicative response to the paradoxes articulated by an abstraction principle for propositions. On this conception, propositions are “shadows” of the sentences that express them. Modal operators are used to uncover the implicit relation of dependence that characterizes propositions that are about propositions. The benefits of this approach are shown by application to other intensional puzzles. The resulting view is an alternative to the plenitudinous metaphysics of impredicative comprehension principles. (shrink)

Martin-Löf's constructive type theory forms the basis of this paper. His central notions of category and set, and their relations with Russell's type theories, are discussed. It is shown that addition of an axiom - treating the category of propositions as a set and thereby enabling higher order quantification - leads to inconsistency. This theorem is a variant of Girard's paradox, which is a translation into type theory of Mirimanoff's paradox (concerning the set of all (...) well-founded sets). The occurrence of the contradiction is explained in set theoretical terms. Crucial here is the way a proof-object of an existential proposition is understood. It is shown that also Russell's paradox can be translated into type theory. The type theory extended with the axiom mentioned above contains constructive higher order logic, but even if one only adds constructive second order logic to type theory the contradictions arise. (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)

This is a review of "Properties and Propositions: The Metaphysics of Higher-Order Logic" by Robert Trueman. Following an overview of the main themes of the book, I discuss the metaphysical presuppositions of Trueman's Fregean notation for predicate abstraction and evaluate his argument for strict typing.

In this report I motivate and develop a type-free logic with predicate quantifiers within the general ontological framework of properties, relations, and propositions. In Part I, I present the major ideas of the system informally and discuss its philosophical significance, especially with regard to Russell's paradox. In Part II, I prove the soundness, consistency, and completeness of the logic.

It is fairly well known that Wittgenstein's criticisms of Russell's multiple‐relation theory of judgement had a devastating effect on the latter's philosophical enterprise. The exact nature of those criticisms however, and the explanation for the severity of their consequences, has been a source of confusion and disagreement amongst both Russell and Wittgenstein scholars. In this paper, I offer an interpretation of those criticisms which shows them to be consonant with Wittgenstein's general critique of Russell's conception of logic (...) and which serves to elucidate some of the notoriously enigmatic passages of the Tractatus. In particular, I seek to show the continuity of Wittgenstein's criticisms of the theory of judgement with his remarks on Russell's paradox and the theory of types. In addition, I place these issues in the context of Russell's own philosophical ambitions in order to reveal the deep divisions between the two over the nature of logical form and the analysis of propositional content. (shrink)

I take some initial steps toward a theory of real definition, drawing upon recent developments in higher-order logic. The resulting account allows for extremely fine- grained distinctions (i.e., it can distinguish between any relata that differ in their syntactic structure, while avoiding the Russell-Myhill problem). It is the first account that can consistently embrace three desirable logical principles that initially appear to be incompatible: the Identification Hypothesis (if F is, by definition, G then F is the (...) same as G), Irreflexivity (there are no reflexive definitions) and Leibniz’s Law. Additionally, it possesses the resources to resolve the paradox of analysis. (shrink)

Could space consist entirely of extended regions, without any regions shaped like points, lines, or surfaces? Peter Forrest and Frank Arntzenius have independently raised a paradox of size for space like this, drawing on a construction of Cantor’s. I present a new version of this argument and explore possible lines of response.

We define the notion of a model of higher-order modal logic in an arbitrary elementary topos E. In contrast to the well-known interpretation of higher-order logic, the type of propositions is not interpreted by the subobject classifier ΩE, but rather by a suitable complete Heyting algebra H. The canonical map relating H and ΩE both serves to interpret equality and provides a modal operator on H in the form of a comonad. Examples of such structures arise (...) from surjective geometric morphisms f : F → E, where H = f∗ΩF. The logic differs from non-modal higher-order logic in that the principles of functional and propositional extensionality are not longer valid but may be replaced by modalized versions. The usual Kripke, neighborhood, and sheaf semantics for propositional and first-order modal logic are subsumed by this notion. (shrink)

A comunidade de filósofos da Metafísica Analítica Contemporânea não consegue entender-se com respeito ao estatuto ontológico das “Propriedades”. Uma parte defende que as propriedades existem, mas, têm grandes dificuldades em explicar a sua existência. Outra parte acredita que as propriedades não existem, mas, não conseguem justificar o problema ontológico da sua predicação. Esta divisão é o tema central deste livro que vos apresentamos, intitulado, “Properties and Propositions: The Metaphysics of Higher-Order Logic” da autoria de Robert Trueman e publicado (...) pela Cambridge University Press. (shrink)

An increasing amount of twenty-first century metaphysics is couched in explicitly hyperintensional terms. A prerequisite of hyperintensional metaphysics is that reality itself be hyperintensional: at the metaphysical level, propositions, properties, operators, and other elements of the type hierarchy, must be more fine-grained than functions from possible worlds to extensions. In this paper I develop, in the setting of type theory, a general framework for reasoning about the granularity of propositions and properties. The theory takes as primitive the notion of a (...) substitution on a proposition (property, etc.) and, among other things, uses this idea to elucidate a number of theoretically important distinctions. A class of structures are identified which can be used to model a wide range of positions about the granularity of reality; certain of these structures are seen to receive a natural treatment in the category of M-sets. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:russell: the Journal of Bertrand Russell Studies n.s. 33 (winter 2013–14): 173–90 The Bertrand Russell Research Centre, McMaster U. issn 0036–01631; online 1913–8032 c:\users\kenneth\documents\type3302\rj 33,2 114 red.docx 2014-01-31 8:29 PM oeviews MODES AND LEVELS OF PERPLEXITY I. Grattan-Guinness Middlesex U. Business School Hendon, London nw4 4bt, uk [email protected] John Ongley and Rosalind Carey. Russell: a Guide for the Perplexed. London: Bloomsbury, 2013. Pp. ix, 212. (...) isbn: 978-0-8264-9753-6. £45 (hb), £14.99 (pb); us$80 (hb), us$24.95 (pb). i his book is the 41st in a series of guides to the work of various philosophers and philosopher groups, written at a level of readability for students. The authors survey the course of Bertrand Russell’s philosophical career, starting out with his work on logic and mathematics. The main philosophical tools are presented: in logic, they are term, proposition, proposition function, relation, paradox, and first- vis-à-vis higher-order logic. Then, moving on to epistemology, the principal philosophical notions are presented, such as sense-data, facts related to language, Russell’s overriding preference for positivistic philosophies and suspicions of metaphysics, knowledge by acquaintance and by description, theories of truth, logical atomism, percept, events, meaning, and inference in the context of scientific theories. It is nice to have many of the principal components of Russell’s philosophy exhibited together, including the later works An Inquiry into Meaning and Truth and Human Knowledge. The perplexed reader would like to be given precise references for Russell’s assertions on various points, especially when he was so prolific. A cited short item does not need more precision; but the two main chapters that cover the logic and epistemology story often seem to invoke The Principles of Mathematics or Principia Mathematica and yet contain only one precise citation of a longer text in their 64 pages (p. 85). The other chapters of the book are somewhat better in this respect. The index is excellent, though W. V. Quine was missed at pp. 29–30. The table of contents would have benefited from including the titles of the sections. q= 174 Reviews c:\users\kenneth\documents\type3302\rj 33,2 113 red.docx 2014-01-15 10:04 ii Much ground is covered; however, the account is mitigated by the failure to provide a biographical factual survey that should have constituted the opening chapter. Little or no description or explanation is given of, among others, Cambridge University, its mathematical tripos, Russell’s research fellowship in 1895, the philosophical journals Mind and The Monist, Russell’s conversion from neo-Hegelian to positivistic philosophy following G. E. Moore around 1899, the start of personal contacts with Giuseppe Peano in 1900, the gradual involvement of A. N. Whitehead from around 1902, the financial rescue of Principia Mathematica by the grant from the Royal Society in 1909, the serious mistake in Volume 2 detected by Whitehead in 1911 while it was in press, the initial contact with Ludwig Wittgenstein from 1911, Russell’s switch from logic to epistemology in the early 1910s, Whitehead’s intentions for the fourth volume of Principia that was abandoned in 1918, the circumstances of the second edition published in the mid-1920s, Russell’s partial return to serious philosophical writings from the mid-1930s onwards, or the Trinity fellowship from 1944. Presumably the authors have made most of these omissions intentionally ; but the target audience is the loser. The opening “introduction” does mention some of these events and publications ; but its purpose is to provide a preliminary look at some of the main features of Russell’s logic, including the comprehension axiom “that every predicate defines a class” (p. 3), type theory, definite descriptions, “the noclass theory of classes” (p. 10), and the status and role of “analysis”. Why do these features need prologues when the others do not? When they are re-discussed in the later chapters on logic and epistemology, several near-repetitions occur. Furthermore, quantification does not receive sufficient consideration in the book; and the logical connectives and rules of inference none at... (shrink)

This paper deals with higher-order vagueness in Williamson's 'logic of clarity'. Its aim is to prove that for 'fixed margin models' (W,d,α ,[ ]) the notion of higher-order vagueness collapses to second-order vagueness. First, it is shown that fixed margin models can be reformulated in terms of similarity structures (W,~). The relation ~ is assumed to be reflexive and symmetric, but not necessarily transitive. Then, it is shown that the structures (W,~) come along with naturally (...) defined maps h and s that define a Galois connection on the power set PW of W. These maps can be used to define two distinct boundary operators bd and BD on W. The main theorem of the paper states that higher-order vagueness with respect to bd collapses to second-order vagueness. This does not hold for BD, the iterations of which behave in quite an erratic way. In contrast, the operator bd defines a variety of tolerance principles that do not fall prey to the sorites paradox and, moreover, do not always satisfy the principles of positive and negative introspection. (shrink)

As to the misconceptions: In the first place, the existence of "undecidable propositions" or "unsolvable problems" has only remote connections with the failure of excluded middle. More precisely, from the fact that a certain problem is unsolvable, one cannot infer that the affirmative and negative answers to that problem are both incorrect. Both Gödel's and Church's theorems were originally proved for systems with the excluded middle, i.e. for systems in which 'p or not p' is provable for every proposition 'p'; (...) though in the light of Gödel's theorem there are propositions 'p' such that neither 'p' nor 'not p' is provable in those systems. Careful distinction should be made between the following locutions. (shrink)

We explore the view that Frege's puzzle is a source of straightforward counterexamples to Leibniz's law. Taking this seriously requires us to revise the classical logic of quantifiers and identity; we work out the options, in the context of higher-order logic. The logics we arrive at provide the resources for a straightforward semantics of attitude reports that is consistent with the Millian thesis that the meaning of a name is just the thing it stands for. We provide models (...) to show that some of these logics are non-degenerate. (shrink)

In their correspondence in 1902 and 1903, after discussing the Russell paradox, Russell and Frege discussed the paradox of propositions considered informally in Appendix B of Russell’s Principles of Mathematics. It seems that the proposition, p, stating the logical product of the class w, namely, the class of all propositions stating the logical product of a class they are not in, is in w if and only if it is not. Frege believed that this (...) class='Hi'>paradox was avoided within his philosophy due to his distinction between sense (Sinn) and reference (Bedeutung). However, I show that while the paradox as Russell formulates it is ill-formed with Frege’s extant logical system, if Frege’s system is expanded to contain the commitments of his philosophy of language, an analogue of this paradox is formulable. This and other concerns in Fregean intensional logic are discussed, and it is discovered that Frege’s logical system, even without its naive class theory embodied in its infamous Basic Law V, leads to inconsistencies when the theory of sense and reference is axiomatized therein. (shrink)

Higher-order realists about properties express their view that there are properties with the help of higher-order rather than first-order quantifiers. They claim two types of advantages for this way of formulating property realism. First, certain gridlocked debates about the nature of properties, such as the immanentism versus transcendentalism dispute, are taken to be dissolved. Second, a further such debate, the tropes versus universals dispute, is taken to be resolved. In this paper I first argue that (...) higher-order realism does not in fact resolve the tropes versus universals dispute. In a constructive spirit, I then develop higher-order realism in a way that leads to a dissolution, rather than a resolution, of this dispute too. (shrink)

An increasing amount of contemporary philosophy of mathematics posits, and theorizes in terms of special kinds of mathematical modality. The goal of this paper is to bring recent work on higher-order metaphysics to bear on the investigation of these modalities. The main focus of the paper will be views that posit mathematical contingency or indeterminacy about statements that concern the ‘width’ of the set theoretic universe, such as Cantor’s continuum hypothesis. Within a higher-order framework I show (...) that contingency about the width of the set-theoretic universe refutes two orthodoxies concerning the structure of modal reality: the view that the broadest necessity has a logic of S5, and the ‘Leibniz biconditionals’ stating that what is possible, in the broadest sense of _possible_, is what is true in some possible world. Nonetheless, I suggest that the underlying picture of modal set-theory is coherent and has attractions. (shrink)

This paper is a study of higher-order contingentism – the view, roughly, that it is contingent what properties and propositions there are. We explore the motivations for this view and various ways in which it might be developed, synthesizing and expanding on work by Kit Fine, Robert Stalnaker, and Timothy Williamson. Special attention is paid to the question of whether the view makes sense by its own lights, or whether articulating the view requires drawing distinctions among possibilities that, (...) according to the view itself, do not exist to be drawn. The paper begins with a non-technical exposition of the main ideas and technical results, which can be read on its own. This exposition is followed by a formal investigation of higher-order contingentism, in which the tools of variable-domain intensional model theory are used to articulate various versions of the view, understood as theories formulated in a higher-order modal language. Our overall assessment is mixed: higher-order contingentism can be fleshed out into an elegant systematic theory, but perhaps only at the cost of abandoning some of its original motivations. (shrink)

In explaining the notion of a fundamental property or relation, metaphysicians will often draw an analogy with languages. The fundamental properties and relations stand to reality as the primitive predicates and relations stand to a language: the smallest set of vocabulary God would need in order to write the “book of the world.” This paper attempts to make good on this metaphor. To that end, a modality is introduced that, put informally, stands to propositions as logical truth stands to (...) sentences. The resulting theory, formulated in higher-order logic, also vindicates the Humean idea that fundamental properties and relations are freely recombinable and a variant of the structural idea that propositions can be decomposed into their fundamental constituents via logical operations. Indeed, it is seen that, although these ideas are seemingly distinct, they are not independent, and fall out of a natural and general theory about the granularity of reality. (shrink)

Slaman and Wehner have constructed structures which distinguish the computable Turing degree 0 from the noncomputable degrees, in the sense that the spectrum of each structure consists precisely of the noncomputable degrees. Downey has asked if this can be done for an ordinary type of structure such as a linear order. We show that there exists a linear order whose spectrum includes every noncomputable Δ 0 2 degree, but not 0. Since our argument requires the technique of permitting (...) below a Δ 0 2 set, we include a detailed explanation of the mechanics and intuition behind this type of permitting. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Scientific Method in PhilosophyAuthor's note: Thanks to Gregory Landini for helpful clarifications.Gregory Landini. Repairing Bertrand Russell's 1913 Theory of Knowledge. (History of Analytic Philosophy.) London: Palgrave Macmillan, 2022. Pp. x, 397. isbn: 978-3-030-66355-1, us$139 (hb); 978-3-030-66356-8, us$109 (ebook).The title of this book might suggest a rather narrow study of a problem with Russell's Theory of Knowledge and a proposed solution. But as with Landini's first book, (...) class='Hi'>Russell's Hidden Substitutional Theory, the title does not capture the full scope of the work. That work did not just point out the lingering substitution theory in Principia, but gave a sweeping view of Russell's overall development from Principles to Principia and one of the most detailed accounts of the formal system in Principia in any work. This work gives a sweeping overview of Russell's philosophy up to 1918, including not just Russell's multiple-relation theory of judgment, but the whole project of what Russell called scientific method in philosophy, which was the subtitle to his 1914 Our Knowledge of the ExternalWorld. Landini thinks this is the height of Russell's philosophy and proposes a fix to the problems which led Russell to abandon the project.Landini does not just present Russell's views during this period, but vigorously defends them, and not just the multiple-relation theory of judgment mostly associated with Theory of Knowledge, but also the views that logic should be understood as synthetic a priori, Russell's doctrine of acquaintance, including the acquaintance with universals as the grounding of synthetic a priori knowledge, and Russell's view that logic is the essence of philosophy. The book contains a wide range of discussions, including issues in philosophy of mind, theories of representation, theories of universals, evolution in philosophy, and the metaphysics of time and space.The book opens with an overview of some main themes, including Landini's view of the three main phases of Russell's post-idealist philosophy and his discussion of what he calls the revolutions in logic and mathematics. The three phases Landini has in mind are the Principles phase, the Principia phase [End Page 81] (which includes The Problems of Philosophy and Theory of Knowledge) and the neutral monist phase, which Landini sees as a mistake Russell made when he didn't see how to repair his 1913 work. According to Landini, the Principles era ends with the failure of the substitution theory from the po /ao paradox, the Principia era ends by 1918 with the abandonment of the attempt to finish the project of the 1913 Theory of Knowledge, and the neutral monist era begins in 1919 and continues through 1948. Landini wants to concentrate on repairing the view in the Principia era. Any remarks of Russell's from later rejecting, for example, his view of the subject, of acquaintance, of neutral monism, and remarks concerning logic as consisting of tautologies are to be rejected. Landini firmly believes that after abandoning the 1913 project, Russell took a wrong turn. Landini's discussion of the multiple-relation theory of judgment is therefore quite different from the other discussion in the literature, in that most commentators have thought Russell was correct in abandoning the theory and most think there is something to Wittgenstein's criticism. Landini thinks Wittgenstein's criticism was based on his new vision of logic, and that Russell should have rejected it and proceeded with the project of Theory of Knowledge.Given the controversies involved in these claims, I think the best way forward is an explication of Landini's new emphasis on the revolutions in logic and mathematics and some of the key claims Landini makes concerning the Principia era. Then we can look at the details of Landini's repairing of the project.the two revolutionsWhat is striking about Landini's more recent work is his emphasis on these two revolutions. One is the revolution within logic, which originates with Frege; the other a revolution in mathematics which originates with Weierstrass, von Staudt, Pieri and Cantor. While most historians of this period are aware of the important shifts... (shrink)

This paper consists of an investigation of three debates concerning propositional identity: the tension between structured propositions and higher-order logic, the principle Only Logical Circles, and Kaplan’s Paradox. The literature at large has mistaken the consequences of each of these debates. Structuralists are not committed to the claim that identical properties have different extensions; rather, they are committed to existence monism. Only Logical Circles does not preclude the identification of green in terms of grue; some further (...) (and, as of yet, undisclosed) principle is required for that result. And Kaplan’s Paradox does not count against the possible-worlds conception of propositions; it applies to every account of propositions whatsoever. (shrink)

In Plato’s Parmenides 132a-133b, the widely known Third Man Paradox is stated, which has special interest for the history of logical reasoning. It is important for philosophers because it is often thought to be a devastating argument to Plato’s theory of Forms. Some philosophers have even viewed Aristotle’s theory of predication and the categories as inspired by reflection on it [Owen 1966]. For the historians of logic it is attractive, because of the phenomenon of self-reference that involves. Bocheński denies (...) any possibility of correct logical reasoning before Aristotle. In particular, he flatly declares of Plato that “correct logic we find none in his work” [1951, 15]. In this line, many papers have been written that call attention to the violation of a metalogical principle – the type rules – because of the Third Man Paradox. The problem of interpretation of the paradox raised many discussions, since the 50’s, and the literature devoted to this topic is today enormous. Nevertheless, many points in Plato’s reasoning remain obscure. Hitherto, it is commonly believed that the paradox was simply stated by Plato in his "Parmenides", but not actually solved. Even more, it is believed that it is hardly possible for Plato to have suggested a solution, because he confused the categories of substance and attribute. B. Russell has also argued that Plato violates in his arguments the restrictions imposed on language by the theory of logical types. This view is encouraged by the linguistic difficulties, which Plato has faced in his attempt to formulate an ontology of abstract entities, i.e. that in Greek language abstract and concrete terms are formally indistinguishable [Kneales 1984]. In this paper, we analyse the logical structure of the argument in an attempt to give a systematic consistent reconstruction of the text appealing to methods and concepts of modern logic and semantics. Already in 1954 G. Vlastos had noted in his seminal paper “The ‘Third Man’ Argument in the Parmenides” that “if any progress … is to be made at this juncture it must come from some advance in understanding the logical structure of the Argument”. Accordingly, our analysis is primarily focused on the line of logical reasoning, rather than the metaphysical underpinnings and philosophical implications of the paradox for Plato’s theory of Forms. We divide the Platonic text into three parts, presenting apparent thematic coherence: a) Formulation of the Third Man Paradox (132a-132b). The central issue of this part is a step-like generation of Forms that can continue ad infinitum. b) Discussion of the paradoxical situation (132b-c), by passing to the use of terminology echoing Eleatic philosophy. The concept of “thought” and the underlying semantics of Eleatic origin are central in this passage. c) Solution of the paradox (132d-133a). In our view, this part contains not only a resemblance regress, as most interpretators con-sider, but also the solution of the paradox by the introduction of a sound definition of the concept of “similarity”. We should note that the name ‘Third Man Paradox’ never occurs in Plato, who, strictly speaking, formulates a ‘Third Large Paradox’. The name ‘Third Man Paradox’ appears in the works of Aristotle and the commentators of the Peripatetic tradition. Later commentators have identified Aristotle’s Third Man with Plato’s Third Large Argument. In our paper, we also examine other testimonies about the Third Man Paradox found in the works of commentators in order to illuminate the logical structure of the argument. Although the vocabulary used in these versions is different, they do not affect its logical structure, but rather reveal different understandings by the ancient authors. These testimonies are divided into two main categories: those met in Neo-Platonic authors, notably in Proclus’ "Commentary on Plato’s Parmenides", and the versions of the Third Man Paradox found in texts of the Peripatetic tradition (Eudemus, Aristotle, Alexander). From our discussion, it becomes clear that the approaches to the paradox by the various scholars of antiquity are different, depending also on their participation in the one or the other campus of philosophical thought. We show that the Peripatetic authors are aware of the source of the paradox. However, the first scholar of antiquity who explicitly ascribes a solution to Plato seems to be Proclus. (shrink)

The paradox of mass voting is not, generally speaking, matched by a paradoxical mass attempt to be politically well informed. As Converse underscored, most people are grossly politically ignorant—just as they would be if, as rational‐ignorance theory holds, they realized that their votes don't matter. Yet many millions of them contradict the theory by voting. This contradiction, and the illogical reasons people offer for voting, suggest that the logic of collective action does not come naturally to people. To equate (...) public ignorance with “rational ignorance,” then, attributes too much theoretical self‐consciousness to people. Even if their ignorance is not based on a grasp of the collective‐action problem, however, people's political ignorance is “rational”: but it is less a matter of calculating the low benefit likely to flow from one vote in a sea of millions, than of encountering the much higher cost of being well informed as compared to voting. As the high cost of becoming well informed appears likely to grow, it may be advisable to move off of the political agenda issues that have no clear continuum toward which a mostly ignorant median voter may gravitate. (shrink)

Slaman and Wehner have constructed structures which distinguish the computable Turing degree 0 from the noncomputable degrees, in the sense that the spectrum of each structure consists precisely of the noncomputable degrees. Downey has asked if this can be done for an ordinary type of structure such as a linear order. We show that there exists a linear order whose spectrum includes every noncomputable $\Delta^0_2$ degree, but not 0. Since our argument requires the technique of permitting below a (...) $\Delta^0_2$ set, we include a detailed explanation of the mechanics and intuition behind this type of permitting. (shrink)

Relativized Metaphysical Modality (RMM: Murray and Wilson, 'Relativized metaphysical modality', Oxford Studies in Metaphysics, 2012; Murray, Perspectives on Modal Metaphysics, 2017) exploits 'two-dimensionalist' resources to metaphysical, rather than epistemological, ends: the second dimension offers perspective-dependence without contingency, diverting attacks on 'Classical' analyses of modals (in effect, analyses validating S5 and the Barcan Formulae). Here, we extend the RMM program in two directions. First, we harvest resources for RMM from Lewis's 1980 'Context--Index' (CI) framework: (a) the ban in CI on binding (...) into context-arguments (akin to Kaplan's 'monstrosity' ban) projects a bright line between perspective-dependence and contingency; and (b) CI-postulated connections among meaning, content, truth, argument-structure, context, and modality collectively generate a 'Generalized Humphrey Problem' for any non-Classical analysis (examples covered include appeals to accessibility, contingent domains, and counterpart relations). Second, we sharpen the tools of RMM-based metaphysical analysis, and extend their domain of coverage across familiar anomalies for Classical modals: we revisit earlier RMM-based bulwarks for S5 (against 'Chisholm's Paradox' for moderate flexibility of essence, and nomological necessitarianism); and we now similarly shore up the Barcan Formulae (against the apparent contingency of existence and nonexistence). (shrink)

While it has perhaps always accompanied philosophical thought – one immediately thinks of Plato’s Dialogues – the problem of the communication of that thought, and therefore of its capacity to be taught, has acquired a new insistence in the work of post-Kantian thinkers. As evidence of this one could cite Fichte’s repeated efforts to formulate a definitive version of his Wissenschaftslehre, the model of the Bildungsroman that Hegel adopts for his Phenomenology of Spirit, Kierkegaard’s pseudonymous works, Nietzsche’s Thus Spoke Zarathustra, (...) and Heidegger’s uncompleted Being and Time. Certainly each of these thinkers, as well as their projects, are quite distinct from the others, but their works are also signposts. It is not a question here of reducing the puzzles and difficulties of these works to some single element, some trace of a philosophical zeitgeist. Rather, it is enough to indicate this preoccupation with the problem of the communicability of thinking in order to raise the question of the modality of such a communication. How has thinking come to be so fundamentally troubled by its own expression? That Kant’s Critical philosophy has precipitated this preoccupation ought to come as no surprise, for it was Kant, most famously in the Critique of Pure Reason, who sought to definitively set the limits of philosophical thinking. In doing so, however, Kant is unable to remain utterly silent about what he himself sets outside the bounds of meaningful thought and speech: things-in-themselves. The mere fact that this meaningless element is an essential component of Kant’s philosophical exposition is an indication of the paradoxical logic of the limit: to postulate a limit is simultaneously to set out its obverse side. In Kant’s case, then, the demarcation of what can be thought carries with it the implicit thinking of what is ruled out of bounds for thought. Beginning with Fichte, the philosophical project of Kant’s successors and inheritors is then largely determined by an effort to incorporate or at least accommodate this alien element within thinking. The initial phase of this effort is marked by a tragic motif, a conjoining of thought with the experience of loss and suffering that is characteristic of Romanticism, that culminates in the dialectic of Hegel’s Absolute Knowing. In Absolute Knowing, thought has not vanquished its limitations, it has instead discovered the resources with which it can perpetually surmount them. This first phase is followed by a reaction or response to this seeming completion and overcoming of the Kantian limitation. Now, the tragic movement of thought comes to be understood as itself limited, and necessarily so according to the logic of the limit that imposes itself upon every gesture of closure. Tragedy thereby gives way to comedy insofar as the limitation that is incorporated as an obstacle to be overcome by tragic thought is shown to be the external condition of that very thought. Finally, if the comic is the fate of the limitations of tragedy, the thought of comedy itself demands a third mode of thinking, one capable of something more than the mere reiteration of the limit between comedy and tragedy that troubles them both. This third form of expression is parody, an expression appropriate to a thinking that repeats and completes the threefold movement of tragic thought while, in this very repetition, creating a limitless domain of thoughtful expression. (shrink)