Textual and historical subtleties aside, let's call the idea that numbers are properties (or classes) of equinumerous sets ‘the Fregean thesis.’ In a recent paper, Palle Yourgrau claims to have found a decisive refutation of this thesis. More surprising still, he claims in addition that the essence of this refutation is found in the Grundlagen itself – the very masterpiece in which Frege first proffered his thesis. My intention in this note is to evaluate these claims, and along the way to shed some light on relevant passages of the Grundlagen. I will argue that Yourgrau does not make his case.

The arguments with which we are concerned are found in the last three sections of Yourgrau's paper. A pervasive difficulty in these sections is that it is not clear exactly what Yourgrau is arguing against. The stated object of his attack is the Fregean thesis (d. 581-6), a thesis about what numbers are; however, instead of a frontal assault, his strategy is to embark on a foray into the ill-defined issue of what it is that numbers number (586ff.), where, roughly speaking, a number n numbers an object x just in case n can be legitimately assigned to x. The reason for this shift in emphasis appears to be rooted in a misconception. As we’ll see in more detail shortly, Yourgrau's argument against the Fregean thesis is based on an extension of a well known argument of Frege's found in §§22-3 of the Grundlagen, which Glenn Kessler has tagged the ‘relativity argument,’ (henceforth ‘RA’). According to Yourgrau, this is an argument ‘to the effect that what is literally numbered cannot simply be concrete objects’ (586). This is incorrect. Frege himself clarifies the point of the argument in §21 with the following preface:

To many atheists the existence of evil seems to provide a conclusive argument against the existence of God. God is by definition omnipotent and perfectly good; a perfectly good being will remove evil in so far as he can, an omnipotent being can remove any evil he chooses, so if there is a God there will be no evil, but there is evil, hence there is no God. Theists normally challenge this argument by challenging the premiss that a perfectly good being will remove evil in so far as he can. The theistic defence is usually put as the defence that many evils are logically necessary conditions of greater goods, and hence a perfectly good being may allow them to occur in order to bring about the greater good; so a perfectly good being may well allow some evils to occur.

Although there are a very few occasions on which Aristotle speaks of words, on the one hand, or mental concepts, on the other, as universals, he was no nominalist and no conceptualist. This negative thesis I have argued sufficiently, at least to my own satisfaction, in an earlier paper. He was, rather, a realist, but of a very tenuous sort. As I said in the earlier paper, he viewed universals as real entities but lacking numerical oneness; each is numerically many, and yet each is also one in some sense. The specific identity of numerically distinct particulars creates something like a class, and this is the universal.

This interpretation was not, in the earlier paper, defended against those who would attribute to Aristotle a much more robust form of realism, and it is that defense which this paper will undertake to provide. The evidence which can be marshalled for the more robust realism is impressive – more so, I would say, than that which can be brought forth for either the nominalist or the conceptualist interpretations. It rests on numerous passages which either explicitly or implicitly seem to tie together as identical, forms, substances, universals, and intelligible objects.

In Impartial Reason, Stephen Darwall presents an account of rational agency in which reasons to act are both motivational and normative in nature. On the one hand, they are facts about an action reflective awareness of which can genuinely influence preference and conduct. On the other hand, they are also capable of justifying action, of showing in an all-things-considered sense that a particular action is at least as choiceworthy as are alternatives to it. Furthermore, these two aspects of reasons to act are intimately related, since, ideally, at any rate, an agent's conduct is motivated by his awareness of himself as a particular subject of a given set of norms.

An ethical theory is axiological just in case it makes the permissibility of actions depend solely on considerations of goodness. Act utilitarianism is the paradigm axiological theory. An ethical theory is a pure rights theory just in case it judges an action permissible if and only if it violates no one’s rights. Libertarianism is a paradigm pure rights theory. I shall formulate and defend a type of axiological theory that, unlike act utilitarianism, is sensitive in a new and interesting way to whether rights are violated.

Let us start by briefly considering the strengths and weakness of act utilitarianism. Act utilitarianism judges an action permissible just in case it maximizes social (e.g., total or average) welfare. It has many attractive features. It is a clearly formulated theory that draws on the machinery of decision theory. It takes a tough minded approach to morality: always focusing on the outcomes of actions. And it takes human welfare very seriously. The axiological theory I shall defend will have all these features.

George Terzis makes several objections to claims and arguments I advanced in Impartial Reason. I cannot take them all up, but I would like to respond to some, which I shall group into three: (a) whether reasons depend on norms applying to all rational agents; (b) how the unity of agency relates to such norms; and (c) the self-support condition. Since the objections concerning (a) cut most deeply against the central thesis of Impartial Reason, I shall begin with them. Before I do that, however, I should make some preliminary remarks.

Impartial Reason offers an internalist theory of reasons, but one that is, I believe, more sensitive to the normative character of reasons than internalist theories usually are.1 A theory of reasons is internalist if it holds that something's being a reason depends somehow on its capacity to affect motivation. Unlike internalist theories that identify reasons with de facto motives, however, IR insists on the normative character of reasons as tending to justify conduct as rational. It does this by holding that a reason to act is something which motivates when appropriately (or rationally) considered. The normative or justificatory weight of reasons, then, is held to derive from a normative ideal of rational consideration. Reasons inherit as justificatory weight the motivational force they would come to have in an ideally rational process of practical reflection.

Jerome Bickenbach has provided a fair and sympathetic account of my argument in Trials and Punishments, and has clarified some of the book’s obscurities - for which I am very grateful: I will focus my response on his main objection to my account of punishment, since I am not persuaded that the objection holds.

Bickenbach argues that my ideal account of what punishment ought to be if it is to be adequately justified would actually show, if it succeeds, that criminal punishment (or at least punishment involving hard treatment) cannot be justified at all. ‘Criminal punishment, it would appear, is unjustifiable when it is needed, but justifiable only if it is no longer required’ (786). It would be justifiable if it was imposed on a criminal who shared the values embodied in the laws of the true community to which she belonged: but it would then be unnecessary, since in such cases ‘blame alone’ (or, one might add, a formal conviction or a purely symbolic punishment) would suffice for the communicative and persuasive purposes which punishment should ideally serve. It would be necessary if the criminal was a committed nonconformist who rejected the community’s values: but it would then be unjustifiable, since in such cases it could not serve the purposes which punishment should ideally serve.

In the chapter entitled ‘Philosophy and the Meaning of Life’ in Robert Nozick’s Philosophical Explanations there is an admonitory passage with many applications. ‘It is a puzzle,’ says Nozick, ‘how so many people, including intellectuals and academics devote enormous energy to work in which nothing of themselves or their important goals shines forth, not even in the way their work is presented. If they were struck down, their children upon growing up and examining their work would never know why they had done it, would never know who it was that did it. They work that way and sometimes live that way, too’ (1981, 578).

Towards the ends of their reviews, Annette Baier and Jean Hampton allow, if only momentarily, the real spectres to surface. Baier writes, ‘Gauthier rightly sees the dangers of exploitation and subjection inherent in a kin-based and affection-dependent morality, so purports to try for something totally different. Even if our moral natures cannot recognize themselves in Gauthier’s version of them, the problem that drives the attempt [for an individualist and unsentimental morality] is a real one, and so far, I think, an unsolved one; unsolved for morality as well as for moral theory.’ Hampton writes, ‘Gauthier will point out that anyone who insists that a human being has objective value must develop a theory that will not only justify that claim but also explain what reason one would ever have for respecting this value. I believe this is a challenge one has no choice but to accept, given what the moral facts are.’