We consider two categorical syllogisms, valid or invalid, to be equivalent if they can be transformed into each other by certain transformations, going back to Aristotle, that preserve validity. It is shown that two syllogisms are equivalent if and only if they have the same models. Counts are obtained for the number of syllogisms in each equivalence class. For a more natural development, using group-theoretic methods, the space of syllogisms is enlarged to include nonstandard syllogisms, and various groups of transformations (...) on that space are studied. (shrink)

Gleason's theorem for ������³ says that if f is a nonnegative function on the unit sphere with the property that f(x) + f(y) + f(z) is a fixed constant for each triple x, y, z of mutually orthogonal unit vectors, then f is a quadratic form. We examine the issues raised by discussions in this journal regarding the possibility of a constructive proof of Gleason's theorem in light of the recent publication of such a proof.

We consider notions of boundedness of subsets of the natural numbers ℕ that occur when doing mathematics in the context of intuitionistic logic. We obtain a new characterization of the notion of a pseudobounded subset and we formulate the closely related notion of a detachably finite subset. We establish metric equivalents for a subset of ℕ to be detachably finite and to satisfy the ascending chain condition. Following Ishihara, we spell out the relationship between detachable finiteness and sequential continuity. Most (...) of the results do not require countable choice. (shrink)

The modern theory of computability is based on the works of Church, Markov and Turing who, starting from quite different models of computation, arrived at the same class of computable functions. The purpose of this paper is the show how the main results of the Church-Markov-Turing theory of computable functions may quickly be derived and understood without recourse to the largely irrelevant theories of recursive functions, Markov algorithms, or Turing machines. We do this by ignoring the problem of what constitutes (...) a computable function and concentrating on the central feature of the Church-Markov-Turing theory: that the set of computable partial functions can be effectively enumerated. In this manner we are led directly to the heart of the theory of computability without having to fuss about what a computable function is.The spirit of this approach is similar to that of [RGRS]. A major difference is that we operate in the context of constructive mathematics in the sense of Bishop [BSH1], so all functions are computable by definition, and the phrase “you can find” implies “by a finite calculation.” In particular ifPis some property, then the statement “for eachmthere isnsuch thatP” means that we can construct a functionθsuch thatP)for allm. Church's thesis has a different flavor in an environment like this where the notion of a computable function is primitive.One point of such a treatment of Church's thesis is to make available to Bishopstyle constructivists the Markovian counterexamples of Russian constructivism and recursive function theory. The lack of serious candidates for computable functions other than recursive functions makes it quite implausible that a Bishopstyle constructivist could refute Church's thesis, or any consequence of Church's thesis. Hence counterexamples such as Specker's bounded increasing sequence of rational numbers that is eventually bounded away from any given real number [SPEC] may be used, as Brouwerian counterexamples are, as evidence of the unprovability of certain assertions. (shrink)

A very weak omniscience principle is formulated, related omniscience principlesare considered, and the theorem that a function of bounded variation is the difference of two increasing functions is shown to be equivalent to the omniscience principle WLPO. It is a so shown that an arbitrary function with located variation on an interval is the difference of two increasing functions.

A form of Kripke's schema turns out to be equivalent to each of the following two statements from metric topology: every open subspace of a separable metric space is separable; every open subset of a separable metric space is a countable union of open balls. Thus Kripke's schema serves as a point of reference for classifying theorems of classical mathematics within Bishop-style constructive reverse mathematics.

A notion of completeness and completion suitable for use in the absence of countable choice is developed. This encompasses the construction of the real numbers as well as the completion of an arbitrary metric space. The real numbers are characterized as a complete Archimedean Heyting field, a terminal object in the category of Archimedean Heyting fields.

The notions of linear and metric independence are investigated in relation to the property: if U is a set of n+1 independent vectors, and X is a set of n independent vectors, then adjoining some vector in U to X results in a set of n+1 independent vectors. It is shown that this property holds in any normed linear space. A related property – that finite-dimensional subspaces are proximinal – is established for strictly convex normed spaces over the real or (...) complex numbers. It follows that metric independence and linear independence are equivalent in such spaces. Proofs are carried out in the context of intuitionistic logic without the axiom of countable choice. (shrink)

The concept of lying is generally assumed to be closely related to the concept of assertion. However, the literature on lying has focused almost exclusively on lies expressed by unqualified assertions. Sometimes a speaker chooses to qualify her assertion by hedging, making her utterance a hedged declarative. This paper defends the thesis that lies can be expressed by untruthful hedged declaratives, and explores the implications of this thesis for the definition of lying. Many standard approaches to the definition of lying (...) allow lies to be expressed by untruthful hedged declaratives, which may explain why these cases have been ignored in the literature. However, untruthful hedged declaratives present a problem for a recently-proposed theory that analyzes lying in terms of the Knowledge Norm of Assertion. According to this theory, a necessary condition on lying is that in asserting that p, the speaker represents herself as knowing that p; however, in making an untruthful hedged declarative a speaker represents herself as having an epistemic attitude weaker than knowledge. The fact that such a theory fails to adequately explain untruthful hedged declaratives provides a reason to prefer alternative theories of lying. The discussion draws out some important observations about the concepts of lying, assertion, belief, and the relations between them. (shrink)

The most popular concepts of happiness among psychologists and philosophers nowadays are concepts of happiness according to which happiness is defined as " satisfaction with life as a whole ". Such concepts are " Whole Life Satisfaction " concepts of happiness. I show that there are hundreds of non-equivalent ways in which a WLS conception of happiness can be developed. However, every precise conception either requires actual satisfaction with life as a whole or requires hypothetical satisfaction with life as a (...) whole. I show that a person can be "happy" at a time even though he is not actually satisfied with his life as a whole at that time. I also show that a person can be "happy" at a time even though it is not correct to say that if he were to think about his life at that time, he would be satisfied with it as a whole. My thesis is that if you think that happiness is the Good, you should avoid defining happiness as whole life satisfaction. (shrink)

Explores some steps toward non-assimilative encounters in the "global village.".

A Critical Introduction to Fictionalism provides a clear and comprehensive understanding of an important alternative to realism. Drawing on questions from ethics, the philosophy of religion, art, mathematics, logic and science, this is a complete exploration of how fictionalism contrasts with other non-realist doctrines and motivates influential fictionalist treatments across a range of philosophical issues. Defending and criticizing influential as well as emerging fictionalist approaches, this accessible overview discuses physical objects, universals, God, moral properties, numbers and other fictional entities. Where (...) possible it draws general lessons about the conditions under which a fictionalist treatment of a class of items is plausible. Distinguishing fictionalism from other views about the existence of items, it explains the central features of this key metaphysical topic. Featuring an historical survey, definitions of key terms, characterisations of important subdivisions, objections and problems for fictionalism, and contemporary fictionalist treatments of several issues, A Critical Introduction to Fictionalism is a valuable resource for students of metaphysics as well as students of philosophical methodology. It is the only book of its kind. (shrink)

The Profound Limitations of Knowledge explores the limitations of knowledge and argues that neither reasoning nor direct or indirect observations can be trusted. We cannot even assign probabilities to claims of what we can know. Furthermore, for any set of data, there are an infinite number of possible interpretations. Evidence suggests that we live in a participatory universe--that is, our observations shape reality.

The fragility of democracy and the political aesthetics of public art -- Voices and places: the space of public art and Wodiczko's the homeless projection -- Democracy's "empty place": Rawls's political liberalism and Derrida's democracy to come -- Public art's "plain tablet": the political aesthetics of contemporary art -- Democracy and public art: Badiou and Ranciere -- The political aesthetics of Chicago's Millennium Park -- The political aesthetics of New York's National 9/11 Memorial -- Public art as an act of (...) citizenship -- Badiou on "being and the void.". (shrink)

Endorses the pursuit of paradigm shifts in our understandings of faith, truth, and nature to remedy the "underside" of modernity and thus to inaugurate a post-modern (but not anti-modern) and post-secular (but not anti-secular) view of the world.

It is incorrect to construe Hume as a Pyrrhonian sceptic. Or so I have argued elsewhere. To the contrary, Hume in fact offers a detailed defence of the thesis that the norms of scientific inference, that is, the “rules by which to judge of causes and effects”, arereasonablerules to follow in forming our beliefs. Conforming to these rules in its formation of causal beliefs is astrategythe understanding employs in order to satisfy the end of curiosity (T271). Science is reasonable because, (...) so far as we can, within our fallible limits, discover, it is the strategy of belief formation that is the best means of achieving our end of truth, insofar as the latter is, within our fallible limits, attainable. (shrink)

We suggest that there is confusion between why consciousness developed and what additional functions, through continued evolution, it has co-opted. Consider episodic memory. If we believe that episodic memory evolved solely to accurately represent past events, it seems like a terrible system—prone to forgetting and false memories. However, if we believe that episodic memory developed to flexibly and creatively combine and rearrange memories of prior events in order to plan for the future, then it is quite a good system. We (...) argue that consciousness originally developed as part of the episodic memory system—quite likely the part needed to accomplish that flexible recombining of information. We posit further that consciousness was subsequently co-opted to produce other functions that are not directly relevant to memory per se, such as problem-solving, abstract thinking, and language. We suggest that this theory is compatible with many phenomena, such as the slow speed and the after-the-fact order of consciousness, that cannot be explained well by other theories. We believe that our theory may have profound implications for understanding intentional action and consciousness in general. Moreover, we suggest that episodic memory and its associated memory systems of sensory, working, and semantic memory as a whole ought to be considered together as the conscious memory system in that they, together, give rise to the phenomenon of consciousness. Lastly, we suggest that the cerebral cortex is the part of the brain that makes consciousness possible, and that every cortical region contributes to this conscious memory system. (shrink)

A message from the stars -- A dispute over buoyancy -- Inertia, Empiricism, and spots on the sun -- Science and religion -- Troubles in Rome: 1615-1616 -- Mathematics and the book of nature -- Showdown -- Matter and motion.

Ethicists, researchers and policy makers have paid increasing attention to the ethical conduct of research, especially research involving human beings. Research performed with and by undergraduates poses a specific set of ethical challenges. These challenges are often overlooked by the research community because it is assumed that undergraduate student researchers do not have a significant impact on the research community and that their projects are not host to research posing important ethical issues. This paper identifies several features characteristic of research (...) in undergraduate environments. The paper pays special attention to selective small liberal arts colleges, an important segment of higher education in the US, and advocates integrating instruction in research ethics wherever undergraduates are engaged in research. (shrink)

This chapter contains sections titled: Information and Meaning Information and Knowledge Acknowledgments References.