283 research outputs found
A Burgessian critique of nominalistic tendencies in contemporary mathematics and its historiography
We analyze the developments in mathematical rigor from the viewpoint of a
Burgessian critique of nominalistic reconstructions. We apply such a critique
to the reconstruction of infinitesimal analysis accomplished through the
efforts of Cantor, Dedekind, and Weierstrass; to the reconstruction of Cauchy's
foundational work associated with the work of Boyer and Grabiner; and to
Bishop's constructivist reconstruction of classical analysis. We examine the
effects of a nominalist disposition on historiography, teaching, and research.Comment: 57 pages; 3 figures. Corrected misprint
Noncomparabilities & Non Standard Logics
Many normative theories set forth in the welfare economics, distributive justice and cognate literatures posit noncomparabilities or incommensurabilities between magnitudes of various kinds. In some cases these gaps are predicated on metaphysical claims, in others upon epistemic claims, and in still others upon political-moral claims. I show that in all such cases they are best given formal expression in nonstandard logics that reject bivalence, excluded middle, or both. I do so by reference to an illustrative case study: a contradiction known to beset John Rawls\u27s selection and characterization of primary goods as the proper distribuendum in any distributively just society. The contradiction is avoided only by reformulating Rawls\u27s claims in a nonstandard form, which form happens also to cohere quite attractively with Rawls\u27s intuitive argumentation on behalf of his claims
Normative Alethic Pluralism
Some philosophers have argued that truth is a norm of judgement and have provided a variety of formulations of this general thesis. In this paper, I shall side with these philosophers and assume that truth is a norm of judgement. What I am primarily interested in here are two core questions concerning the judgement-truth norm: (i) what are the normative relationships between truth and judgement? And (ii) do these relationships vary or are they constant? I argue for a pluralist pictureâwhat I call Normative Alethic Pluralism (NAP)âaccording to which (i) there is more than one correct judgement-truth norm and (ii) the normative relationships between truth and judgement vary in relation to the subject matter of the judgement. By means of a comparative analysis of disagreement in three areas of the evaluative domainârefined aesthetics, basic taste and moralityâI show that there is an important variability in the normative significance of disagreementâI call this the variability conjecture. By presenting a variation of Lynchâs scope problem for alethic monism, I argue that a monistic approach to the normative function of truth is unable to vindicate the conjecture. I then argue that normative alethic pluralism provides us with a promising model to account for it
Towards a Proof Theory of G\"odel Modal Logics
Analytic proof calculi are introduced for box and diamond fragments of basic
modal fuzzy logics that combine the Kripke semantics of modal logic K with the
many-valued semantics of G\"odel logic. The calculi are used to establish
completeness and complexity results for these fragments
Intuitionistic implication makes model checking hard
We investigate the complexity of the model checking problem for
intuitionistic and modal propositional logics over transitive Kripke models.
More specific, we consider intuitionistic logic IPC, basic propositional logic
BPL, formal propositional logic FPL, and Jankov's logic KC. We show that the
model checking problem is P-complete for the implicational fragments of all
these intuitionistic logics. For BPL and FPL we reach P-hardness even on the
implicational fragment with only one variable. The same hardness results are
obtained for the strictly implicational fragments of their modal companions.
Moreover, we investigate whether formulas with less variables and additional
connectives make model checking easier. Whereas for variable free formulas
outside of the implicational fragment, FPL model checking is shown to be in
LOGCFL, the problem remains P-complete for BPL.Comment: 29 pages, 10 figure
Bohrification of operator algebras and quantum logic
Following Birkhoff and von Neumann, quantum logic has traditionally been
based on the lattice of closed linear subspaces of some Hilbert space, or, more
generally, on the lattice of projections in a von Neumann algebra A.
Unfortunately, the logical interpretation of these lattices is impaired by
their nondistributivity and by various other problems. We show that a possible
resolution of these difficulties, suggested by the ideas of Bohr, emerges if
instead of single projections one considers elementary propositions to be
families of projections indexed by a partially ordered set C(A) of appropriate
commutative subalgebras of A. In fact, to achieve both maximal generality and
ease of use within topos theory, we assume that A is a so-called Rickart
C*-algebra and that C(A) consists of all unital commutative Rickart
C*-subalgebras of A. Such families of projections form a Heyting algebra in a
natural way, so that the associated propositional logic is intuitionistic:
distributivity is recovered at the expense of the law of the excluded middle.
Subsequently, generalizing an earlier computation for n-by-n matrices, we
prove that the Heyting algebra thus associated to A arises as a basis for the
internal Gelfand spectrum (in the sense of Banaschewski-Mulvey) of the
"Bohrification" of A, which is a commutative Rickart C*-algebra in the topos of
functors from C(A) to the category of sets. We explain the relationship of this
construction to partial Boolean algebras and Bruns-Lakser completions. Finally,
we establish a connection between probability measure on the lattice of
projections on a Hilbert space H and probability valuations on the internal
Gelfand spectrum of A for A = B(H).Comment: 31 page
What lies beneath: exploring links between asylum policy and hate crime in the UK
This paper explores the link between increasing incidents of hate crime and the asylum policy of successive British governments with its central emphasis on deterrence. The constant problematisation of asylum seekers in the media and political discourse ensures that 'anti-immigrant' prejudice becomes mainstr earned as a common-sense response. The victims are not only the asylum seekers hoping for a better life but democratic society itself with its inherent values of pluralism and tolerance debased and destabilised
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