Putnam famously attempted to use model theory to draw metaphysical conclusions. His Skolemisation argument sought to show metaphysical realists that their favourite theories have countable models. His permutation argument sought to show that they have permuted models. His constructivisation argument sought to show that any empirical evidence is compatible with the Axiom of Constructibility. Here, I examine the metamathematics of all three model-theoretic arguments, and I argue against Bays (2001, 2007) that Putnam is largely immune to metamathematical challenges.

Massimo Dell'Utri (1990) provides a reconstruction of Hilary Putnam's argument (1981, chapter 1) to show that the hypothesis that we are brains in a vat is self-refuting. I will explain why the argument Dell'Utri offers us is, on the face of it, quite problematic. Then I will provide a way out of the difficulty.

Knowledge and its Limits presents a systematic new conception of knowledge as a kind of mental stage sensitive to the knower's environment. It makes a major contribution to the debate between externalist and internalist philosophies of mind, and breaks radically with the epistemological tradition of analyzing knowledge in terms of true belief. The theory casts new light on such philosophical problems as scepticism, evidence, probability and assertion, realism and anti-realism, and the limits of what can be known. The arguments are (...) illustrated by rigorous models based on epistemic logic and probability theory. The result is a new way of doing epistemology and a notable contribution to the philosophy of mind. (shrink)

I use some ideas of Keith DeRose's to develop an (invariantist!) account of why sceptical reasoning doesn't show that I don't know that I'm not a brain in a vat. I argue that knowledge is subject to the risk-of-error constraint: a true belief won’t have the status of knowledge if there is a substantial risk of the belief being in error that hasn’t been brought under control. When a substantial risk of error is present (i.e. beliefs in propositions that are (...) false in nearby worlds), satisfying the constraint requires bringing the risk under control. This is achieved either by sensitivity, i.e. you wouldn’t have the belief if it were false, or by identifying evidence for the proposition. However, when the risk of error is not substantial (i.e. beliefs in propositions that are not false in nearby worlds), the constraint is satisfied by default. My belief that I am not a brain in a vat is insensitive and I have no evidence for it, but since it is not false in nearby worlds, it satisfies the constraint by default. (shrink)

Crispin Wright; IV*—On Putnam's Proof that We are not Brains-in-a-Vat1, Proceedings of the Aristotelian Society, Volume 92, Issue 1, 1 June 1992, Pages 67–94, h.

This essay defends a view which is near enough to Putnam's characterization of metaphysical realism for it to be called by the same name. Indeterminacy of reference is conceded, in the sense that there may be multiple reference relations, but it is denied that this implied belief in unknowable noumena. It is enough for metaphysical realism as conceived here, that there be at least one reference relation. The essay also argues against defining truth epistemically. Even a Peircean ideal theory might (...) be false, in a contextual sense of might' stronger than that of the logically possible. (shrink)

Anyone who has pondered the limitlessness of space and time, or the endlessness of numbers, or the perfection of God will recognize the special fascination of this question. Adrian Moore's historical study of the infinite covers all its aspects, from the mathematical to the mystical.

This second edition includes a new Afterword by the author.

Anyone who has pondered the limitlessness of space and time, or the endlessness of numbers, or the perfection of God will recognize the special fascination of this question. Adrian Moore's historical study of the infinite covers all its aspects, from the mathematical to the mystical.

It is not my claim that the ‘L¨ owenheim-Skolem paradox’ is an antinomy in formal logic; but I shall argue that it is an antinomy, or something close to it, in philosophy of language. Moreover, I shall argue that the resolution of the antinomy—the only resolution that I myself can see as making sense—has profound implications for the great metaphysical dispute about realism which has always been the central dispute in the philosophy of language.