Richard Kimberly Heck

It has been known for a few years that no more than Π 1 1 ∆ 1 3 comprehension is needed for the proof of “Frege’s Theorem.” One can at least imagine a view that would regard Pi-1-1 comprehension axioms as logical truths but deny that status to any that are more complex—a view that would, in particular, deny that full second-order logic deserves the name. Such a view would serve the purposes of neo-logicists. It is, in fact, no part of my view that, say, Delta-3-1 comprehension axioms are not logical truths. What I suggest, however, is that there is a special case to be made on behalf of Pi-1-1 comprehension. Making the case involves investigating extensions of first-order logic that do not rely upon the presence of second-order quantifiers. A formal system for so-called “ancestral logic” is developed, and it is then extended to yield what I call “Arché logic.”

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. This chapter argues, first, that the ontological issue is at best secondary: the crucial issue about second-order logic, at least for a neo-logicist, is epistemological. It is then argued 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, the chapter defends the author’s own earlier attempt to finesse this issue, in “A Logic for Frege’s Theorem,” from Hale’s criticisms.

Crispin Wright’s work on vagueness has been extremely influential. This piece provides an overview of his contributions and the ways his views have changed over the forty-plus years he has been writing about the topic. Heck argues that Wright’s discussion of the Tachometer paradox provides us with the resources to disarm the Sorites paradox. Sadly, that still leaves us remarkably far from any positive account of the semantics of vague concepts. In his most recent work, Wright has pursued an ‘intuitionistic’ account of vagueness whose virtues and vices are described.

The paper formulates and proves a strengthening of 'Frege's Theorem', which states that axioms for second-order arithmetic are derivable in second-order logic from Hume's Principle, which itself says that the number of Fs is the same as the number of Gs just in case the Fs and Gs are equinumerous. The improvement consists in restricting this claim to finite concepts, so that nothing is claimed about the circumstances under which infinite concepts have the same number. 'Finite Hume's Principle' also suffices for the derivation of axioms for arithmetic and, indeed, is equivalent to a version of them, in the presence of Frege's definitions of the primitive expressions of the language of arithmetic. The philosophical significance of this result is also discussed

Donald Davidson is generally supposed to have offered an argument for the principle of compositionality in “Theories of Meaning and Learnable Languages”. Peter Pagin, however, has argued that Davidson offers no such argument. Indeed, Pagin claims that Davidson explicitly rejects any demand that compositionality should be justified and, moreover, that the argument in question is invalid. I argue here that the first claim is mistaken and that the latter, though it offers a helpful corrective, does not undermine the interest of the argument Davidson gives. But there is a deep tension in Davidson's work here, to which Pagin may well be responding, and that tension is not resolvable.

As is well-known, the formal system in which Frege works in his Grundgesetze der Arithmetik is formally inconsistent, Russell's Paradox being derivable in it. This system is, except for minor differences, full second-order logic, augmented by a single non-logical axiom, Frege's Axiom V. It has been known for some time now that the first-order fragment of the theory is consistent. The present paper establishes that both the simple and the ramified predicative second-order fragments are consistent, and that Robinson arithmetic, Q, is relatively interpretable in the simple predicative fragment. The philosophical significance of the result is discussed.

In this exciting new collection, a distinguished international group of philosophers contribute new essays on central issues in philosophy of language and logic, in honor of Michael Dummett, one of the most influential philosophers of the late twentieth century. The essays are focused on areas particularly associated with Professor Dummett. Five are contributions to the philosophy of language, addressing in particular the nature of truth and meaning and the relation between language and thought. Two contributors discuss time, in particular the reality of the past. The last four essays focus on Frege and the philosophy of mathematics. The volume represents some of the best work in contemporary analytical philosophy.

So-called ‘facial’ cumshots, when a man ejaculates onto a woman’s face, are very common in pornography. While they are frequently said to be degrading and misogynistic, the fact that women are usually shown as enjoying this act should make us think again. Facials are instead rooted in male insecurity: of a fear that an aspect of how men orgasm—semen—is disgusting to women. By contrast, the fantasy, which pornography makes vivid, is that women might not just tolerate but celebrate and eroticize both ejaculation and its product. The way mainstream pornography presents facials may often be misogynistic, but it does not have to be, and it is not always. NOTE: This may not be a philosophy paper. My intention is to submit it to a sexuality studies journal.

Many people have sexual fantasies about being forced to have sex, or forcing someone to have sex. Several authors have argued that it is wrong to enjoy such fantasies: They lead to harm, or reinforce oppressive social structures, are liable to corrupt our character, or, mostly interestingly, are wrong in themselves, because they involve the eroticization of things that are wrong. I argue here that all such arguments fail properly to distinguish between fantasy and desire (despite authors' acknowledgement of that distinction), and between objects of desire and sources of arousal. The broader significance of this point is also discussed. This paper is intended, in part, as a defense of claims about sexual fantasy made at the end of "Does Pornography Presuppose Rape Myths?".

Cezary Cieśliński has proved a result shows that highlights `logical reflection': The principle that every logically provable sentence is true. He suggests further that this result has a good deal of philosophical significance, specifically for the so-called `conservativeness argument' against deflationism. This note discusses the question to what extent Cieśliński's result generalizes, and just how strong `logical reflection' is, and suggests that the answers to these questions call the philosophical (though not the technical) significance of Cieśliński's result into doubt.

This paper discusses the question to what extent Frege made serious use of semantical notions such as reference and truth. It focuses on his apparent uses of these notions in his apparently semantical discussions of his formal system in Grundgesetze der Arithmetik and defends the view that they are to be taken at face value. This paper is in some ways a companion to "Grundgesetze der Arithmetik I §§29-32", in which there is an extended, but mostly technical, discussion of Frege's attempt to prove that every well-formed expression in his formal language denotes: This paper contains more in the way of a discussion of the wider, interpretive significance of the technical interpretation given there.

This paper contains a close analysis of Frege's proofs of the axioms of arithmetic §§70-83 of Die Grundlagen, with special attention to the proof of the existence of successors in §§82-83. Reluctantly and hesitantly, we come to the conclusion that Frege was at least somewhat confused in those two sections and that he cannot be said to have outlined, or even to have intended, any correct proof there. The proof he sketches is in many ways similar to that given in Grundgesetze der Arithmetik, but fidelity to what Frege wrote in Die Grundlagen and in Grundgesetze requires us to reject the charitable suggestion that it was this (beautiful) proof that he had in mind in §§82-83.

Rae Langton and Caroline West argue that pornography silences women by presupposing misogynistic attitudes, such as that women enjoy being raped. More precisely, they claim that a somewhat infamous pictorial, ‘Dirty Pool’, makes such presuppositions, and that it is typical in this respect. I argue for four claims. (1) There are empirical reasons to doubt that women are silenced in the way that Langton and West claim they are. (2) There is no evidence that very much pornography makes the sorts of presuppositions that Langton and West's explanation of silencing requires it to make. (3) Even ‘Dirty Pool’, for all its other problems, does not make such presuppositions. (4) Langton and West misread ‘Dirty Pool’ because they do not take proper account of the fact that pornography often traffics in sexual fantasy. The broader lesson is that we need to read pornography more sensitively if we are to understand its capacity to shape socio‐sexual norms (for good or for ill).

Central to the liar paradox is the phenomenon of 'self-reference'. The paradox typically begins with a sentence like: (L): (L) is not true. Historically, doubts about the intelligibility of self-reference have been quite common. In some sense, though, these doubts were answered by Kurt Gödel's famous 'diagonal lemma'. This paper surveys some of the methods by which self-reference can be achieved, focusing first on purely syntactic methods before turning attention to the 'arithmetized' methods introduced by Gödel. It's primary lesson is that we need to be more careful than we usually have been about self-reference.

There are two main models for explaining Frege's notion of sense, both of which have their roots in the work of Sir Michael Dummett. One, nowadays most familiar from the work of David Chalmers, is broadly internalist and descriptivist in character. The other, most familiar from the work of Gareth Evans, is externalist and anti-descriptivist. I first consider the former project, arguing that Dummett anticipated Chalmers's version of the view, and that no version of this view is going to be defensible. The arguments are somewhat different from those familiar from the literature. I then consider Evans's view and argue that, while it is not vulnerable to many of the objections that have been made to it, it does not really succeed as an account of sense, because it forces us to abandon Frege's view that sense is an aspect of representational content.

Some years ago, Machery, Mallon, Nichols, and Stich reported the results of experiments that reveal, they claim, cross-cultural differences in speaker’s ‘intuitions’ about Kripke’s famous Gödel–Schmidt case. Several authors have suggested, however, that the question they asked their subjects is ambiguous between speaker’s reference and semantic reference. Machery and colleagues have since made a number of replies. It is argued here that these are ineffective. The larger lesson, however, concerns the role that first-order philosophy should, and more importantly should not, play in the design of such experiments and in the evaluation of their results.