Finite additivity,another lottery paradox and conditionalisation

In this paper I argue that de Finetti provided compelling reasons for rejecting countable additivity. It is ironical therefore that the main argument advanced by Bayesians against following his recommendation is based on the consistency criterion, coherence, he himself developed. I will show that this argument is mistaken. Nevertheless, there remain some counter-intuitive consequences of rejecting countable additivity, and one in particular has all the appearances of a full-blown paradox. I will end by arguing that in fact it is no paradox, and that what it shows is that conditionalisation, often claimed to be integral to the Bayesian canon, has to be rejected as a general rule in a finitely additive environment.     

Finite Agents,Sublime Feelings: Response to Hanauer

Tom Hanauer's thoughtful discussion of my article “The Pleasures of Contra‐purposiveness: Kant, the Sublime, and Being Human” (2014) puts pressure on two important issues concerning the affective phenomenology of the sublime. My aim in that article was to present an analysis of the sublime that does not suffer from the problems identified by Jane Forsey in “Is a Theory of the Sublime Possible?” (2007). I argued that Kant's notion of reflective judgment can help with this task, because it allows us to capture the experience of failure that characterizes the sublime without committing us to ontologically transcendent items. In a significant departure from Kant, however, my account does not require references to our moral vocation to explain the pleasure we take in the sublime; the pleasure comes from getting the right measure of our agency. For Hanauer, trouble for my analysis comes both from the discursive presentation of the sublime, its focus on judgment, and from the removal of references to our moral vocation.     

Finite mathematics

A system of finite mathematics is proposed that has all of the power of classical mathematics. I believe that finite mathematics is not committed to any form of infinity, actual or potential, either within its theories or in the metalanguage employed to specify them. I show in detail that its commitments to the infinite are no stronger than those of primitive recursive arithmetic. The finite mathematics of sets is comprehensible and usable on its own terms, without appeal to any form of the infinite. That makes it possible to, without circularity, obtain the axioms of full Zermelo-Fraenkel Set Theory with the Axiom of Choice (ZFC) by extrapolating (in a precisely defined technical sense) from natural principles concerning finite sets, including indefinitely large ones. The existence of such a method of extrapolation makes it possible to give a comparatively direct account of how we obtain knowledge of the mathematical infinite. The starting point for finite mathematics is Mycielski's work on locally finite theories.I would like to thank Jeff Barrett, Akeel Bilgrami, Leigh Cauman, John Collins, William Craig, Gary Feinberg, Haim Gaifman, Yair Guttmann, Hidé Ishiguro, Isaac Levi, James Lewis, Vann McGee, Sidney Morgenbesser, George Shiber, Sarah Stebbins, Mark Steiner, and an anonymous referee for encouragement and various useful suggestions. The research described in this article and the preparation of the article were supported in part by the Columbia University Council for Research in the Humanities.     

Finite rational self-deceivers

I raise three puzzles concerning self-deception: (i) a conceptual paradox, (ii) a dilemma about how to understand human cognitive evolution, and (iii) a tension between the fact of self-deception and Davidson’s interpretive view. I advance solutions to the first two and lay a groundwork for addressing the third. The capacity for self-deception, I argue, is a spandrel, in Gould’s and Lewontin’s sense, of other mental traits, i.e., a structural byproduct. The irony is that the mental traits of which self-deception is a spandrel/byproduct are themselves rational.

The Problem of Obligation,the Finite Rational Will,and Kantian Value Realism

Abstract

Robert Stern's Understanding Moral Obligation is a remarkable achievement, representing an original reading of Kant's contribution to modern moral philosophy and the legacy he bequeathed to his later-eighteenth- and early-nineteenth-century successors in the German tradition. On Stern's interpretation, it was not the threat to autonomy posed by value realism, but the threat to autonomy posed by the obligatory nature of morality that led Kant to develop his critical moral theory grounded in the concept of the self-legislating moral agent. Accordingly, Stern contends that Kant was a moral realist of sorts, holding certain substantive views that are best characterized as realist commitments about value. In this paper, I raise two central objections to Stern's reading of Kant. The first objection concerns what Stern identifies as Kant's solution to the problem of moral obligation. Whereas Stern sees the distinction between the infinite will and the finite will as resolving the problem of moral obligation, I argue that this distinction merely explains why moral obligations necessarily take the form of imperatives for us imperfect human beings, but does not solve the deeper problem concerning the obligatory nature of morality—why we should take moral norms to be supremely authoritative laws that override all other norms based on our non-moral interests. The second objection addresses Stern's claim that Kantian autonomy is compatible with value realism. Although this is an idea with which many contemporary readers will be sympathetic, I suggest that the textual evidence actually weighs in favor of constructivism.     

Finite Reasons Without Foundations

This article develops a theory of reasons that has strong similarities to Peter Klein's infinitism. The view it develops, Framework Reasons, upholds Klein's principles of avoiding arbitrariness (PAA) and avoiding circularity (PAC) without requiring an infinite regress of reasons. A view of reasons that holds that the “reason for” relation is constrained by PAA and that PAC can avoid an infinite regress if the “reason for” relation is contextual. Moreover, such a view of reasons can maintain that skepticism is false by the maintaining that there is more to epistemic justification than can be expressed in any reasoning session. One crucial argument for Framework Reasons is that justification depends on a background of plausibility considerations. The final section of the article applies this view of reasons to Michael Bergmann's argument that any nonskeptical epistemology must embrace epistemic circularity.     

Finite Cardinals in Quasi-set Theory

Quasi-set theory is a ZFU-like axiomatic set theory, which deals with two kinds of ur-elements: M-atoms, objects like the atoms of ZFU, and m-atoms, items for which the usual identity relation is not defined. One of the motivations to advance such a theory is to deal properly with collections of items like particles in non-relativistic quantum mechanics when these are understood as being non-individuals in the sense that they may be indistinguishable although identity does not apply to them. According to some authors, this is the best way to understand quantum objects. The fact that identity is not defined for m-atoms raises a technical difficulty: it seems impossible to follow the usual procedures to define the cardinal of collections involving these items. In this paper we propose a definition of finite cardinals in quasi-set theory which works for collections involving m-atoms.     

Finite quasivarieties and self-referential conditions

In this paper, we concentrate on finite quasivarieties (i.e. classes of finite algebras defined by quasi-identities). We present a motivation for studying finite quasivarieties. We introduce a new type of conditions that is well suited for defining finite quasivarieties and compare these new conditions with quasi-identities.Special issue of Studia Logica: Algebraic Theory of Quasivarieties Presented by M. E. Adams, K. V. Adaricheva, W. Dziobiak, and A. V. Kravchenko     

Dimensional Characterization In Finite Quasi-Analysis

The method of Quasi-Analysis used by Carnap in his program of theconstitution of concepts from finite observations has the following twogoals: (1) Given unsharp observations in terms of similarity relations thetrue properties of the observed objects shall be obtained by a suitablelogical construction. (2) From a single relation on a finite domain,different dimensions of qualities shall be reconstructed and identified. Inthis article I show that with a slight modification Quasi-Analysis iscapable of fulfilling the first goal for a single observable dimension. Weobtain a partition of the so-called Quality Classes representing thepairwise disjoint and exhaustive extensions associated to the ``values' ofthe observable. On the other hand, an example demonstrates that the methodfails, as Goodman has pointed out, for a relation expressing similarity withregard to at least one out of many properties.Since it seems to be impossible in general to reconstruct more-dimensionalqualities from a single similarity relation, the constitution of at least asmany similarity relations as there are qualities have to be presumed. Thenit is possible to state adequate sufficient conditions for the dimension ofthe observable space, even if some of the similarity relations might dependon others. The concept of topological dimension cannot be used for thispurpose on finite sets of observations. We replace it by a set-algebraicalcondition on the Quality Classes.     

Annotation Theories over Finite Graphs

In the current paper we consider theories with vocabulary containing a number of binary and unary relation symbols. Binary relation symbols represent labeled edges of a graph and unary relations represent unique annotations of the graph’s nodes. Such theories, which we call annotation theories, can be used in many applications, including the formalization of argumentation, approximate reasoning, semantics of logic programs, graph coloring, etc. We address a number of problems related to annotation theories over finite models, including satisfiability, querying problem, specification of preferred models and model checking problem. We show that most of considered problems are NPTime- or co-NPTime-complete. In order to reduce the complexity for particular theories, we use second-order quantifier elimination. To our best knowledge none of existing methods works in the case of annotation theories. We then provide a new second-order quantifier elimination method for stratified theories, which is successful in the considered cases. The new result subsumes many other results, including those of [2, 28, 21].