Roy T. Cook

A number of formal constraints on acceptable abstraction principles have been proposed, including conservativeness and irenicity. Hume’s Principle, of course, satisfies these constraints. Here, variants of Hume’s Principle that allow us to count concepts instead of objects are examined. It is argued that, prima facie, these principles ought to be no more problematic than HP itself. But, as is shown here, these principles only enjoy the formal properties that have been suggested as indicative of acceptability if certain constraints on the size of the continuum hold. As a result, whether or not these higher-order versions of Hume’s Principle are acceptable seems to be independent of standard (ZFC) set theory. This places the abstractionist in an uncomfortable dilemma: Either there is some inherent difference between counting objects and counting concepts, or new criteria for acceptability will need to be found. It is argued that neither horn looks promising.

One of the main reasons for providing formal semantics for languages is that the mathematical precision afforded by such semantics allows us to study and manipulate the formalization much more easily than if we were to study the relevant natural languages directly. Michael Tye and R. M. Sainsbury have argued that traditional set-theoretic semantics for vague languages are all but useless, however, since this mathematical precision eliminates the very phenomenon (vagueness) that we are trying to capture. Here we meet this objection by viewing formalization as a process of building models, not providing descriptions. When we are constructing models, as opposed to accurate descriptions, we often include in the model extra ‘machinery’ of some sort in order to facilitate our manipulation of the model. In other words, while some parts of a model accurately represent actual aspects of the phenomenon being modelled, other parts might be merely artefacts of the particular model. With this distinction in place, the criticisms of Sainsbury and Tye are easily dealt with—the precision of the semantics is artefactual and does not represent any real precision in vague discourse. Although this solution to this problem is independent of any particular semantics a detailed account of how we would distinguish between representor and artefact within Dorothy Edgington's degree-theoretic semantics is presented.

The Liar paradox is the directly self-referential Liar statement: This statement is false.or : " Λ: ∼ T 1" The argument that proceeds from the Liar statement and the relevant instance of the T-schema: " T ↔ Λ" to a contradiction is familiar. In recent years, a number of variations on the Liar paradox have arisen in the literature on semantic paradox. The two that will concern us here are the Curry paradox, 2 and the Yablo paradox. 3The Curry paradox demonstrates that neither negation nor a falsity predicate is required in order to generate semantic paradoxes. Given any statement Φ whatsoever, we need merely consider the statement: If this statement is true, then Φor: " Ξ: T → Φ" Here, via familiar reasoning, one can ‘prove’ Φ merely through consideration of statement Ξ and the Ξ-instance of the T-schema.Interestingly, the Liar paradox can be viewed as nothing more than a special case of the Curry paradox. If we define negation in terms of the conditional and a primitive absurdity constant ‘⊥’: 4" ∼ Ψ = df Ψ → ⊥" then the Liar paradox is simply the instance of the Curry paradox obtained by substituting ‘⊥’ for Φ.The Yablo paradox demonstrates that circularity is also not required in order to generate semantic paradox. 5 The paradox proceeds by considering an infinite ω-sequence of statements of the form: S 1: ) S 2: ) S 3: ) : : : : : S i: ) : : : : : :– that is, the set of …

We begin with a prepositional languageLpcontaining conjunction (Λ), a class of sentence names {Sα}αϵA, and a falsity predicateF. We (only) allow unrestricted infinite conjunctions, i.e., given any non-empty class of sentence names {Sβ}βϵB,is a well-formed formula (we will useWFFto denote the set of well-formed formulae).The language, as it stands, is unproblematic. Whether various paradoxes are produced depends on which names are assigned to which sentences. What is needed is a denotation function:For example, theLPsentence “F(S1)” (i.e.,Λ{F(S1)}), combined with a denotation functionδsuch thatδ(S1)“F(S1)”, provides the (or, in this context, a)Liar Paradox.To give a more interesting example,Yablo's Paradox[4] can be reconstructed within this framework.Yablo's Paradoxconsists of an ω-sequence of sentences {Sk}kϵωwhere, for eachnϵω:WithinLPan equivalent construction can be obtained using infinite conjunction in place of universal quantification - the sentence names are {Si}iϵωand the denotation function is given by:We can express this in more familiar terms as:etc.

Logical pluralism is the view that there is more than one correct logic. In this article, I explore what logical pluralism is, and what it entails, by: (i) distinguishing clearly between relativism about a particular domain and pluralism about that domain; (ii) distinguishing between a number of forms logical pluralism might take; (iii) attempting to distinguish between those versions of pluralism that are clearly true and those that are might be controversial; and (iv) surveying three prominent attempts to argue for logical pluralism and evaluating them along the criteria provided by (ii) and (iii).

Gottlob Frege defined cardinal numbers in terms of value-ranges governed by the inconsistent Basic Law V. Neo-logicists have revived something like Frege's original project by introducing cardinal numbers as primitive objects, governed by Hume's Principle. A neo-logicist foundation for set theory, however, requires a consistent theory of value-ranges of some sort. Thus, it is natural to ask whether we can reconstruct the cardinal numbers by retaining Frege's definition and adopting an alternative consistent principle governing value-ranges. Given some natural assumptions regarding what an acceptable neo-logicistic theory of value-ranges might look like, successfully implementing this alternative approach is impossible

On the Dummettian understanding, anti-realism regarding a particular discourse amounts to (or at the very least, involves) a refusal to accept the determinacy of the subject matter of that discourse and a corresponding refusal to assert at least some instances of excluded middle (which can be understood as expressing this determinacy of subject matter). In short: one is an anti-realist about a discourse if and only if one accepts intuitionistic logic as correct for that discourse. On careful examination, the strongest Dummettian arguments for anti-realism of this sort fail to secure intuitionistic logic as the single, correct logic for anti-realist discourses. Instead, antirealists are placed in a situation where they fail to be justified in asserting monism (that intuitionistic logic is the unique correct logic). Thus, antirealists seem forced either to accept pluralism (i.e. one or more intermediate logic is at least as `correct’ as intuitionistic logic–an option I take to be unattractive from the anti-realist perspective), or they must be anti-realists about the realism/anti-realism debate (and, in particular, must refuse to assert the instance of excluded middle equivalent to logical monism or logical pluralism)

Paradoxes are arguments that lead from apparently true premises, via apparently uncontroversial reasoning, to a false or even contradictory conclusion. Paradoxes threaten our basic understanding of central concepts such as space, time, motion, infinity, truth, knowledge, and belief. In this volume Roy T Cook provides a sophisticated, yet accessible and entertaining, introduction to the study of paradoxes, one that includes a detailed examination of a wide variety of paradoxes. The book is organized around four important types of paradox: the semantic paradoxes involving truth, the set-theoretic paradoxes involving arbitrary collections of objects, the Soritical paradoxes involving vague concepts, and the epistemic paradoxes involving knowledge and belief. In each of these cases, Cook frames the discussion in terms of four different approaches one might take towards solving such paradoxes. Each chapter concludes with a number of exercises that illustrate the philosophical arguments and logical concepts involved in the paradoxes. _Paradoxes_ is the ideal introduction to the topic and will be a valuable resource for scholars and students in a wide variety of disciplines who wish to understand the important role that paradoxes have played, and continue to play, in contemporary philosophy.

In this paper, we present a formal recipe that Frege followed in his magnum opus “Grundgesetze der Arithmetik” when formulating his definitions. This recipe is not explicitly mentioned as such by Frege, but we will offer strong reasons to believe that Frege applied it in developing the formal material of Grundgesetze. We then show that a version of Basic Law V plays a fundamental role in Frege’s recipe and, in what follows, we will explicate what exactly this role is and explain how it differs from the role played by extensions in his earlier book “Die Grundlagen der Arithmetik”. Lastly, we will demonstrate that this hitherto neglected yet foundational aspect of Frege’s use of Basic Law V helps to resolve a number of important interpretative challenges in recent Frege scholarship, while also shedding light on some important differences between Frege’s logicism and recent neo-logicist approaches to the foundations of mathematics.

In this paper I examine the prospects for a successful neo–logicist reconstruction of the real numbers, focusing on Bob Hale's use of a cut-abstraction principle. There is a serious problem plaguing Hale's project. Natural generalizations of this principle imply that there are far more objects than one would expect from a position that stresses its epistemological conservativeness. In other words, the sort of abstraction needed to obtain a theory of the reals is rampantly inflationary. I also indicate briefly why this problem is likely to reappear in any neo–logicist reconstruction of real analysis.

One of the main problems plaguing neo-logicism is the Bad Company challenge: the need for a well-motivated account of which abstraction principles provide legitimate definitions of mathematical concepts. In this article a solution to the Bad Company challenge is provided, based on the idea that definitions ought to be conservative. Although the standard formulation of conservativeness is not sufficient for acceptability, since there are conservative but pairwise incompatible abstraction principles, a stronger conservativeness condition is sufficient: that the class of acceptable abstraction principles be strictly logically symmetrically class conservative . The article concludes with an examination of which classes of abstraction principles meet this criteria