Valueless Measures on Pointless Spaces

Journal of Philosophical Logic · 2022

• Mathematics
• Pure mathematics
• Discrete mathematics

Evidence, ignorance, and symmetry

Philosophical Perspectives · 2021 · 2 citations

• Political Science
• Epistemology
• Philosophy

Abstract The Principle of Indifference (POI) is a rule for rationally assigning precise degrees of confidence to possibilities among which we have no reason to discriminate. Despite criticism of the principle stemming from Bertrand's paradox, many have recently come to the defense of POI or adopted some restricted version of that principle, especially in discussions of self‐locating belief. I argue that POI in both unrestricted and restricted forms is untenable, and that arguments for the more restricted principles are hostage to problems similar to those that bedevil arguments for traditional POI.

The content of indexical belief

Philosophy and Phenomenological Research · 2021

• Computer Science
• Epistemology
• Philosophy

Abstract I argue that a recent variation on the Sleeping Beauty case due to Titelbaum [2012a] puts significant pressure on Lewis's theory of doxastic content—the theory of content that Titelbaum and others presuppose. In particular, that theory cannot make sense of the rational constraints on credences imposed by the Principal Principle, as ordinarily understood. I then argue more generally that any theory on which contents of credences are sets of centered worlds cannot adequately represent all of the rational constraints on credences.

A Calculus of Regions Respecting Both Measure and Topology

Journal of Philosophical Logic · 2019-01-14 · 15 citations

Topology and measure in logics for region-based theories of space

Annals of Pure and Applied Logic · 2018-01-03 · 1 citations

Closure and Epistemic Modals

Philosophy and Phenomenological Research · 2017-04-17 · 29 citations

According to a popular closure principle for epistemic justification, if one is justified in believing each of the premises in set Φ and one comes to believe that ψ on the basis of competently deducing ψ from Φ—while retaining justified beliefs in the premises—then one is justified in believing that ψ . This principle is prima facie compelling; it seems to capture the sense in which competent deduction is an epistemically secure means to extend belief. However, even the single‐premise version of this closure principle is in conflict with certain seemingly good inferences involving the epistemic possibility modal ♢. According to other compelling principles concerning competent deduction and epistemic justification, one can competently infer ¬♢ φ from ¬ φ in deliberation even though there are cases in which one can justifiably believe ¬ φ but would be unjustified in believing ¬♢ φ . Thus, as we argue, philosophers must choose between unrestricted closure for justification and the validity of these other principles.

LOGICS ABOVE<i>S</i>4 AND THE LEBESGUE MEASURE ALGEBRA

The Review of Symbolic Logic · 2016-10-28

Abstract We study the measure semantics for propositional modal logics, in which formulas are interpreted in the Lebesgue measure algebra ${\cal M}$ , or algebra of Borel subsets of the real interval [0,1] modulo sets of measure zero. It was shown in Lando (2012) and Fernández-Duque (2010) that the propositional modal logic S 4 is complete for the Lebesgue measure algebra. The main result of the present paper is that every logic L above S 4 is complete for some subalgebra of ${\cal M}$ . Indeed, there is a single model over a subalgebra of ${\cal M}$ in which all nontheorems of L are refuted. This work builds on recent work by Bezhanishvili, Gabelaia, &amp; Lucero-Bryan (2015) on the topological semantics for logics above S 4. In Bezhanishvili et al. , (2015), it is shown that there are logics above that are not the logic of any subalgebra of the interior algebra over the real line, ${\cal B}$ (ℝ), but that every logic above is the logic of some subalgebra of the interior algebra over the rationals, ${\cal B}$ (ℚ), and the interior algebra over Cantor space, ${\cal B}\left( {\cal C} \right)$ .

Coincidence and Common Cause

Noûs · 2016-09-26 · 19 citations

Abstract According to the traditional view of the causal structure of a coincidence, the several parts of a coincidence are produced by independent causes. I argue that the traditional view is mistaken; even the several parts of a coincidence may have a common cause. This has important implications for how we think about the relationship between causation and causal explanation—and in particular, for why coincidences cannot be explained.

The stretch-length tradeoff in geometric networks: average case and worst case study

Mathematical Proceedings of the Cambridge Philosophical Society · 2015-05-05

Abstract Consider a network linking the points of a rate-1 Poisson point process on the plane. Write Ψ ave ( s ) for the minimum possible mean length per unit area of such a network, subject to the constraint that the route-length between every pair of points is at most s times the Euclidean distance. We give upper and lower bounds on the function Ψ ave ( s ), and on the analogous “worst-case” function Ψ worst ( s ) where the point configuration is arbitrary subject to average density one per unit area. Our bounds are numerically crude, but raise the question of whether there is an exponent α such that each function has Ψ( s ) ≍ ( s − 1) −α as s ↓ 1.

Conclusive Reasons and Epistemic Luck

Australasian Journal of Philosophy · 2015-08-25 · 20 citations

What is it to have conclusive reasons to believe a proposition P? According to a view famously defended by Dretske, a reason R is conclusive for P just in case [R would not be the case unless P were the case]. I argue that, while knowing that P is plausibly related to having conclusive reasons to believe that P, having such reasons cannot be understood in terms of the truth of this counterfactual condition. Simple examples show that it is possible to believe P on the basis of reasons that satisfy the counterfactual, and still get things right about P only as a matter of luck. Seeing where this account of conclusive reasons goes wrong points to an important distinction between having conclusive reasons and relying on reasons that are in point of fact conclusive. It also has wider consequences for whether modal principles like sensitivity and safety can rule out the pernicious kind of epistemic luck, or the kind of luck that interferes with knowledge.