1. Causal knowledge is an indispensable element in science. Causal assertions are embedded in both the results and the procedures of scientific investigation. 2. It is therefore worthwhile to investigate the meaning of causal statements and the ways in which we can arrive at causal knowledge.

Annotation In this book Mr. Mackie selects for critical discussion six related topic which are prominent in John Locke's Essay concerning Human Understanding: ...

A comprehensive one-year graduate (or advanced undergraduate) course in mathematical logic and foundations of mathematics. No previous knowledge of logic is required; the book is suitable for self-study. Many exercises (with hints) are included.

The idea of a 'logic of quantum mechanics' or quantum logic was originally suggested by Birkhoff and von Neumann in their pioneering paper [1936]. Since that time there has been much argument about whether, or in what sense, quantum 'logic' can be actually considered a true logic (see, e.g. Bell and Hallett [1982], Dummett [1976], Gardner [1971]) and, if so, how it is to be distinguished from classical logic. In this paper I put forward a simple and natural semantical framework (...) for quantum logic which reveals its difference from classical logic in a strikingly intuitive way, viz. through the fact that quantum logic admits (suitably formulated versions of) the characteristic quantum-mechanical notions of superposition and incompatibility of attributes. That is, precisely the features that distinguish quantum from classical physics also serve, within this framework, to distinguish quantum from classical logic. Some light is shed on the question of whether quantum logic is a genuine logical system by introducing a natural entailment relation for quantum-logical formulas with the implication symbol. The novelty is that, although implication behaves as it should (i.e. the 'deduction theorem' holds), the order of introduction of premises is significant. The fact that a reasonable entailment relation can be formulated for quantum logic supports the view that it is a genuine logical system and not merely an algebraic formalism. (shrink)

In this paper (a sequel to [4]) I put forward a "local" interpretation of mathematical concepts based on notions derived from category theory. The fundamental idea is to abandon the unique absolute universe of sets central to the orthodox set-theoretic account of the foundations of mathematics, replacing it by a plurality of local mathematical frameworks - elementary toposes - defined in category-theoretic terms.

This paper argues that, while there is a difference between personal and sub‐personal explanation, claims of autonomy should be treated with scepticism. It distinguishes between horizontal and vertical explanatory relations that might hold between facts at the personal and farts at the sub‐personal level. Noting that many philosophers are prepared to accept vertical explanatory relations between the two levels, I argue for the stronger claim that, in the case of at least three central personal level phenomena, the demands of explanatory (...) adequacy require postulating horizontal explanatory relations. (shrink)

Autobiographical memories typically give rise either to memory reports (“I remember going swimming”) or to first person past-tense judgements (“I went swimming”). This article focuses on first person past-tense judgements that are (epistemically) based on autobiographical memories. Some of these judgements have the IEM property of being immune to error through misidentification. This article offers an account of when and why first person past-tense judgements have the IEM property.

This paper is an introduction to topos theory which assumes no prior knowledge of category theory. It includes a discussion of internal logic in a topos, A characterization of the category of sets, And an investigation of the notions of topology and sheaf in a topos.

We analyze Zorn's Lemma and some of its consequences for Boolean algebras in a constructive setting. We show that Zorn's Lemma is persistent in the sense that, if it holds in the underlying set theory, in a properly stated form it continues to hold in all intuitionistic type theories of a certain natural kind. (Observe that the axiom of choice cannot be persistent in this sense since it implies the law of excluded middle.) We also establish the persistence of some (...) familiar results in the theory of (complete) Boolean algebras--notably, the proposition that every complete Boolean algebra is an absolute subretract. This (almost) resolves a question of Banaschewski and Bhutani as to whether the Sikorski extension theorem for Boolean algebras is persistent. (shrink)

Consider the following well-known result from the theory of normed linear spaces ([2], p. 80, 4(b)): (g) the unit ball of the (continuous) dual of a normed linear space over the reals has an extreme point. The standard proof of (~) uses the axiom of choice (AG); thus the implication AC~(w) can be proved in set theory. In this paper we show that this implication can be reversed, so that (*) is actually eq7I2valent to the axiom of choice. From this (...) we derive various corollaries, for example: the conjunction of the Boolean prime ideal theorem and the Krein-Milman theorem implies the axiom of choice, and the Krein-Milman theorem is not derivable from the Boolean prime ideal theorem. (shrink)

Tommie Shelby has offered an influential, carefully stated, and well-argued set of objections to any volitional analysis of racism (VAR) as consisting centrally in certain forms of race-based disregard. Here I hope to defend aspects of VAR by analyzing, evaluating, and sometimes countering several of his major contentions, which have stood unchallenged in the literature over more than two decades. First, I sketch and respond to his Methodological objection to VAR, which criticizes VAR's reliance on language and linguistic intuitions; then (...) to a Psychological objection, which suggests that VAR implausibly distances an agent's intentions from her beliefs; and, finally, to a Psychopathological objection, which holds that VAR requires treating as unproblematic some psychological states better seen as aberrant. The following section treats three subtle counterexamples Shelby uses to challenge VAR's core contentions about what's necessary and what's sufficient for racism. I close with some gestures toward what I see as a promising approach, very different from his, to studying these topics. (shrink)