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.

2. The Contingency and A posteriority Constraint: A formulation of the thesis must make physicalism come out contingent and a posteriori. First, physicalism is a contingent truth, if it is a truth. This means that physicalism could have been false, i.e. there are counterfactual worlds in which physicalism is false, for example, counterfactual worlds in which there are miracle -performing angels.[9] Moreover, if physicalism is true, our knowledge of its truth is a posteriori. This is to say that there are (...) ways the world could turn out to be such that physicalism is false. For example, if there are miracle -performing angels, then physicalism is false. So there are worlds considered as actual in which physicalism is false.[10] For short, call this ‘the a posteriority constraint’.[11]. (shrink)

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)

I first sketch an account of humility as a character trait in which we are unimpressed with our good, envied, or admired features, achievements, etc., where these lack significant salience for our image of ourselves, because of the greater prominence of our limitations and flaws. I situate this view among several other recent conceptions of humility (also called modesty), dividing them between the inward-directed and outward-directed, distinguish mine from them, pose problems for each alternative account, and show how my understanding (...) of humility captures truths present but exaggerated in several of them. Responding to some problems for my view, including what I call “Driver’s Paradox”(i.e., the strangeness of someone’s proclaiming ‘I’m humble!’), I suggest that some over-ambitious claims about our moral responsibilities may indicate a lack of proper humility. I discuss the relationship of the character trait of humility both to what humiliates and to what humbles, concluding with consideration of the background assumptions against which, and the circumstances in which, humility may reasonably be classified as a moral virtue. (shrink)

One strategy for blocking Chalmers's overall case against physicalism has been to deny his claim that showing that phenomenal properties are in some sense physical requires an a priori entailment of the phenomenal truths from the physical ones. Here I avoid this well-trodden ground and argue instead that an a priori entailment of the phenomenal truths from the physical ones does not require an analysis in the Jackson/Chalmers sense. This is to sever the dualist's link between conceptual analysis and a (...) priori entailment by showing that the lack of the former does not imply the absence of the latter. Moreover, given the role of the argument from conceptual analysis in Chalmers's overall case for dualism, undermining that argument effectively undermines that case as a whole in a way that, I'll argue, undermining the conceivability arguments as stand-alone arguments does not. (shrink)

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.

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)

One of the aims of this paper is to discuss the different senses of the term 'existence' as used by the nyaya philosophers. this discussion leads us to a discussion on absence or negation and its role in logic. a discussion on empty terms has also been introduced in this context. according to the nyaya, existence, knowability and nameability are considered as universal properties. the distinction between these universal properties has been discussed in this context. i have also discussed the (...) question whether the nyaya has used redundant terms in designating the same imposed property by using three different terms. a distinction between different senses of the term 'property' has also been discussed in this context. (shrink)

This essay rebuts Gary Seay's efforts to show that committing euthanasia need not conflict with a physician's professional duties. First, I try to show how his misunderstanding of the correlativity of rights and duties and his discussion of the foundation of moral rights undermine his case. Second, I show aspects of physicians' professional duties that clash with euthanasia, and that attempts to avoid this clash lead to absurdities. For professional duties are best understood as deriving from professional virtues and the (...) commitments and purposes with which the professional as such ought to act, and there is no plausible way in which her death can be seen as advancing the patient's medical welfare. Third, I argue against Prof. Seay's assumption that apparent conflicts among professional duties must be resolved through "balancing" and argue that, while the physician's duty to extend life is continuous with her duty to protect health, any duty to relieve pain is subordinate to these. Finally, I show that what is morally determinative here, as throughout the moral life, is the agent's intention and that Prof. Seay's implicitly preferred consequentialism threatens not only to distort moral thinking but would altogether undermine the medical (and any other) profession and its internal ethics. (shrink)

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)

On s'occupe de la logique quantique depuis 1935 et pendant cette période beaucoup de résultats ont été obtenus, encore me suis-je abstenu de mentionner les résultats obtenus ces dernières années par les logiciens dans ce domaine. Certains de ces résultats ont un intérêt direct pour les théories physiques. En particulier on voit de quelles conditions impératives on doit tenir compte quand on effectue un changement de théorie pour passer à une théorie meilleure. L'exemple de la théorie fonctionnelle des corpuscules est (...) significatif. (shrink)

The essay contrasts the thesis that deserved punishment is punishment which, as deserved, is obligatory with the weaker thesis that it is punishment which, as deserved, is permissible. The author first outlines an account of the meaning of desert-claims which entails only the weaker thesis and then defends this account against criticisms levied in a recent article that it is ambiguous, cannot explain the moral significance of desert, justifies letting people profit from their crimes, and permits unequal treatment. The essay (...) proceeds to a critique of George Sher's view of deserved punishment, faulting Sher for: (1) his reliance on an implausible understanding of benefits, (2) his inability to justify the punishment of crime-victims for their own crimes, and (3) the inadequacy of his defense of mercy. Finally, the author sketches a role-centered conception of morality within which it becomes clearer how deserved punishment can be justified as the victim's ties to the criminal, and the role-responsibilities derivative therefrom, are vitiated by the latter's misdeeds. (shrink)

'Ideologies of Discrimination' considers the implications of the new genetics for understandings of personhood and for understandings of the relationship between people in groups. In particular, the essay delineates and examines the emerging notion of a 'genetic group' and considers the social implications of redefining families, racial groups and ethnic groups through express, and often exclusive, reference to a shared genome. One consequence of such redefinition has been the justification and elaboration of stigmatizing images of and discrimination against such groups-especially (...) those, such as Jews and African-Americans, that have long been identified with specific somatic characteristics. A few worrisome trends begin to emerge in the response of the American legal system to the notion of genetic groups. Among these is a shift in the locus of privacy and identity from the autonomous individual to the genetic group. This shift challenges (and threatens) long-standing Western values (including equality and liberty) that depend upon the ideological centrality of autonomous individuality. (shrink)

This article won the Mind prize essay competition announced last year, for an essay defending the academic pursuit of philosophy. The author is at present a free-lance philosopher; at the time of writing the paper he was a part time lecturer at the University of Newcastle upon Tyne, in the department that has since closed. His collection of aphorisms, Sentences and Slogans has recently been published privately (for details, see 'Books Received').