Thomas Hofweber

Although there is a prima facie strong case for a close connection between the meaning and inferential role of certain expressions, this connection seems seriously threatened by the semantic and logical paradoxes which rely on these inferential roles. Some philosophers have drawn radical conclusions from the paradoxes for the theory of meaning in general, and for which sentences in our language are true. I criticize these overreactions, and instead propose to distinguish two conceptions of inferential role. This distinction is closely tied to two conceptions of deductive logic, and it is the key, I argue, for understanding first the connection between meaning and inferential role, and second what the paradoxes show more generally.

forthcoming in Meanings and other Things: essays on Stephen Schiffer Gary Ostertag (ed.) MIT Press 2007. Schiffer substantially changed his view about propositions and that-clauses somewhere between his two most recent books: Remnants of Meaning and The Things We Mean. I look at what problems his earlier view had, and what reason Schiffer gives for giving it up in favor of his more recent view. I argue that Schiffer’s reasons are not very good reasons, and that instead the problems for Remnants can be solved, contrary to the ones Things faces. I outline how a view in the spirit of the one Schiffer held in Remnants can be formulated and defended against the problems that his version faces. In the end we should go back to a view like the one he held in Remnants.

The relevant alternatives approach in epistemology1 arose some years ago partly out of the hope to be able to reconcile our ordinary claims of knowledge with our inability to answer the skeptic. It was supposed to give rise to an account of knowledge according to which our ordinary claims of knowledge are true, even though the claims about our lack of knowledge that the skeptics make in one of their more persuasive moments are also true. To know, according to such an account, is to have evidence sufficient to rule out all the relevant alternatives. In ordinary life few alternatives are relevant. For example, whether or not we are brains in a vat is not a relevant alternative that we have to be able to rule out. In the debate with the skeptic it may become relevant, and accordingly we might not know something any more then, even though we have the same evidence as in ordinary life. The skeptics cleverly make more and more alternatives relevant, and that is how they succeed. But their success in the philosophy seminar is no threat to our ordinary claims of knowledge, or so the theory goes.

The problem of change plays a central role in the metaphysics of time and material objects, and whoever does best in solving this problem has a leg up when it comes to choosing a metaphysics of time and material objects. But whether this central role of the problem of change in metaphysics is legitimate is not at all clear. This is so in part since it is not clear what the problem of change is, and why it is a problem in metaphysics. We will investigate what metaphysical problem the problem of change might be, and how it relates to various other problems related to change that are studied in the empirical sciences. The problem of change can thus be a case study of what makes a problem a metaphysical one and how metaphysics relates to other parts of inquiry. We conclude that the central role the problem of change is given in the contemporary metaphysics of time is not justified.

Hilbert’s program in the philosophy of mathematics comes in two parts. One part is a technical part. To carry out this part of the program one has to prove a certain technical result. The other part of the program is a philosophical part. It is concerned with philosophical questions that are the real aim of the program. To carry out this part one, basically, has to show why the technical part answers the philosophical questions one wanted to have answered. Hilbert probably thought that he had completed the philosophical part of his program, maybe up to a few details. What was left to do was the technical part. To carry it out one, roughly, had to give a precise axiomatization of mathematics and show that it is consistent on purely finitistic grounds. This would come down to giving a relative consistency proof of mathematics in finitist mathematics, or to give a proof-theoretic reduction of mathematics on to finitist mathematics (we will look at these notions in more detail soon). It is widely believed that Gödel’s theorems showed that the technical part of Hilbert’s program could not be carried out. Gödel’s theorems show that the consistency of arithmetic can not even be proven in arithmetic, not to speak of by finitistic means alone. So, the technical part of Hilbert’s program is hopeless, and since Hilbert’s program essentially relied on both the technical and the philosophical part, Hilbert’s program as a whole is hopeless. Justified as this attitude is, it is a bit unfortunate. It is unfortunate because it takes away too much attention from the philosophical part of Hilbert’s program. And this is unfortunate for two reasons.

In his groundbreaking Grundlagen, Frege (1884) pointed out that number words like ‘four’ occur in ordinary language in two quite different ways and that this gives rise to a philosophical puzzle. On the one hand ‘four’ occurs as an adjective, which is to say that it occurs grammatically in sentences in a position that is commonly occupied by adjectives. Frege’s example was (1) Jupiter has four moons, where the occurrence of ‘four’ seems to be just like that of ‘green’ in (2) Jupiter has green moons. On the other hand, ‘four’ occurs as a singular term, which is to say that it occurs in a position that is commonly occupied by paradigmatic cases of singular terms, like proper names: (3) The number of moons of Jupiter is four. Here ‘four’ seems to be just like ‘Wagner’ in (4) The composer of Tannhäuser is Wagner, and both of these statements seem to be identity statements, ones with which we claim that what two singular terms stand for is identical. But that number words can occur both as singular terms and as adjectives is puzzling. Usually adjectives cannot occur in a position occupied by a singular term, and the other way round, without resulting in ungrammaticality and nonsense. To give just one example, it would be ungrammatical to replace ‘four’ with ‘the number of moons of Jupiter’ in (1): (5) *Jupiter has the number of moons of Jupiter moons. This ungrammaticality results even though ‘four’ and ‘the number of moons of Jupiter’ are both apparently singular terms standing for the same object in (3). So, how can it be that number words can occur both as singular terms and as adjectives, while other adjectives cannot occur as singular terms, and other singular terms cannot occur as adjectives?

It is natural to think that questions in the metaphysics of chance are independent of the mathematical representation of chance in probability theory. After all, chance is a feature of events that comes in degrees and the mathematical representation of chance concerns these degrees but leaves the nature of chance open. The mathematical representation of chance could thus, un-controversially, be taken to be what it is commonly taken to be: a probability measure satisfying Kolmogorov’s axioms. The metaphysical questions about chance seem to be left open by all this. I argue that this is a mistake. The employment of real numbers as measures of chance in standard probability theory brings with it commitments in the metaphysics of (objective) chance that are not only substantial but also mistaken. To measure chance properly we need to employ extensions of the real numbers that contain infinitesimals: positive numbers that are infinitely small. But simply using infinitesimals alone is not enough, as a number of arguments show. Instead we need to put three ideas together: infinitesimals, the non-locality of chance and flexibility in measurement. Only those three together give us a coherent picture of chance and its mathematical representation.