A simple theory containing its own truth predicate

Tarski's indefinability theorem shows us that truth is not definable in arithmetic. The requirement to define truth for a language in a stronger language (if contradiction is to be avoided) lapses for particularly weak languages. A weaker language, however, is not necessary for that lapse. It also lapses for an adequately weak theory. It turns out that the set of G{\"o}del numbers of sentences true in arithmetic modulo $n$ is definable in arithmetic modulo $n$

The Vacuity of Postmodernist Methodology

Many of the philosophical doctrines purveyed by postmodernists have been roundly refuted, yet people continue to be taken in by the dishonest devices used in proselytizing for postmodernism. I exhibit, name, and analyse five favourite rhetorical manoeuvres: Troll's Truisms, Motte and Bailey Doctrines, Equivocating Fulcra, the Postmodernist Fox Trot, and Rankly Relativising Fields. Anyone familiar with postmodernist writing will recognise their pervasive hold on the dialectic of postmodernism and come to judge that dialectic as it ought to be judged

The Nothing from infinity paradox versus plenitudinous indeterminism

The Nothing from Infinity paradox arises when the combination of two infinitudes of point particles meet in a supertask and disappear. Corral-Villate claims that my arguments for disappearance fail and concedes that this failure also produces an extreme kind of indeterminism, which I have called plenitudinous. So my supertask at least poses a dilemma of extreme indeterminism within Newtonian point particle mechanics. Plenitudinous indeterminism might be trivial, although easy attempts to prove it so seem to fail in the face of plausible continuity principles. However, the question of its triviality is here moot, since I show that, except in one case, Corral-Villate’s disproofs fail, and with a correction, the original arguments are unrefuted. Consequently, of the two contenders for the outcome of my supertask, the Nothing from Infinity paradox has won out

Shutting dretske’s door

Dretske proposes a theory of knowledge in terms of a theory of information, but wishes to deny that empirical knowledge settles the large question of scepticism. This leads him to deny the closure of knowledge under known entailment. In a recent paper Jäger argues that Dretske’s theory of information entails closure for knowledge, ‘at least for the kind of propositions here at issue’ (Jäger 2004:194). If Jäger is right, Dretske is seriously embarrassed and must give something up. In this paper I show that there are two flaws in Jäger’s argument. The principle of informational closure considered by Jäger is incompatible with Dretske’s theory of information, and Jäger’s argument that Dretske is committed to a certain kind of substitution instance of that principle of informational closure is invalid. I propose adequacy conditions on signalled information and use them to motivate a formulation of a general closure principle for signalled information. I show that Dretske’s account of information satisfies the adequacy conditions, but in a way which commits him to an instance of the general closure principle. I argue that Dretske is consequently committed to closure for some cases of knowledge for which he wishes to deny closure. Finally, I sketch how, on the basis of the closure principle to which Dretske is committed, Jäger’s broader argument may yet go through

Paradoxes of Probability

We call something a paradox if it strikes us as peculiar in a certain way, if it strikes us as something that is not simply nonsense, and yet it poses some difficulty in seeing how it could make sense. When we examine paradoxes more closely, we find that for some the peculiarity is relieved and for others it intensifies. Some are peculiar because they jar with how we expect things to go, but the jarring is to do with imprecision and misunderstandings in our thought, failures to appreciate the breadth of possibility consistent with our beliefs. Other paradoxes, however, pose deep problems. Closer examination does not explain them away. Instead, they challenge the coherence of certain conceptual resources and hence challenge the significance of beliefs which deploy those resources. I shall call the former kind weak paradoxes and the latter, strong paradoxes. Whether a particular paradox is weak or strong is sometimes a matter of controversy—sometimes it has been realised that what was thought strong is in fact weak, and vice versa,— but the distinction between the two kinds is generally thought to be worth drawing. In this Cchapter, I shall cover both weak and strong probabilistic paradoxes

Uncertainty Phobia and Epistemic Forbearance in a Pandemic

In this chapter I show how challenges to our ability to tame the uncertainty of a pandemic leaves us vulnerable to uncertainty phobia. This is because not all the uncertainty that matters can be tamed by our knowledge of the relevant probabilities, contrary to what many believe. We are vulnerable because unrelievable wild uncertainty is a hard burden to bear, especially so when we must act in the face of it. The source of unrelievable wild uncertainty is that the nature of probability distributions matters for whether knowledge of them tames uncertainty. It matters because a warrant for the taming is provided by two theorems, but this warrant applies only to some kinds of probability distribution. Essentially, this is because the theorems are about what happens at a mathematical limit but real life never reaches the limit. Consequently, the warrant depends on how quickly the random processes producing the uncertainty converge towards their limit. If they are governed by one class of probability distributions, they converge quickly enough to possess the warrant. If they are governed by another class of probability distributions, they converge towards their limit too slowly and so do not possess that warrant. The random processes of pandemics involve the slow kind . Faced with such a burden, as we are in a pandemic, we are tempted to retreat into uncertainty phobia, leading to fixed definite opinions, precisely when the exercise of sound judgement to determine our responses requires our opinions to be hedged and mobile. Coping with a pandemic requires us to bear the burden of unrelievable wild uncertainty rather than give in to the temptation of uncertainty phobia. Pandemics require the virtue of epistemic forbearance

A problem for the unity of normativity

A prevalent assumption is that normativity is a unity. In this paper I argue against this assumption by demonstrating the problems it poses to a well known answer to a well known problem for taking rationality to be normative. John Broome's normative requirement relation does indeed avoid that problem, but insofar as the relation is supposed to offer a general characterisation of the normativity of rationality, it fails. It fails because it cannot capture an important aspect of the normativity of rationality, that it is available to guide us. I show that if we distinguish two kinds of normativity it need not fail in this way