Legal probabilism is the view that juridical fact-finding should be modeled using Bayesian methods. One of the alternatives to it is the narration view, according to which instead we should conceptualize the process in terms of competing narrations of what happened. The goal of this paper is to develop a reconciliatory account, on which the narration view is construed from the Bayesian perspective within the framework of formal Bayesian epistemology.

With material on his early philosophical views, his contributions to set theory and his work on nominalism and higher-order quantification, this book offers a uniquely expansive critical commentary on one of analytical philosophy’s great ...

We investigate what happens when ‘truth’ is replaced with ‘provability’ in Yablo’s paradox. By diagonalization, appropriate sequences of sentences can be constructed. Such sequences contain no sentence decided by the background consistent and sufficiently strong arithmetical theory. If the provability predicate satisfies the derivability conditions, each such sentence is provably equivalent to the consistency statement and to the Gödel sentence. Thus each two such sentences are provably equivalent to each other. The same holds for the arithmetization of the existential Yablo (...) paradox. We also look at a formulation which employs Rosser’s provability predicate. (shrink)

One of the standard views on plural quantification is that its use commits one to the existence of abstract objects–sets. On this view claims like ‘some logicians admire only each other’ involve ineliminable quantification over subsets of a salient domain. The main motivation for this view is that plural quantification has to be given some sort of semantics, and among the two main candidates—substitutional and set-theoretic—only the latter can provide the language of plurals with the desired expressive power (given that (...) the nominalist seems committed to the assumption that there can be at most countably many names). To counter this approach I develop a modal-substitutional semantics of plural quantification (on which plural variables, roughly speaking, range over ways names could be) and argue for its nominalistic acceptability. (shrink)

The goal is to sketch a nominalist approach to mathematics which just like neologicism employs abstraction principles, but unlike neologicism is not committed to the idea that mathematical objects exist and does not insist that abstraction principles establish the reference of abstract terms. It is well-known that neologicism runs into certain philosophical problems and faces the technical difficulty of finding appropriate acceptability criteria for abstraction principles. I will argue that a modal and iterative nominalist approach to abstraction principles circumvents those (...) difficulties while still being able to put abstraction principles to a foundational use. (shrink)

Brown (The laboratory of the mind. Thought experiments in the natural science, 1991a , 1991b ; Contemporary debates in philosophy of science, 2004 ; Thought experiments, 2008 ) argues that thought experiments (TE) in science cannot be arguments and cannot even be represented by arguments. He rest his case on examples of TEs which proceed through a contradiction to reach a positive resolution (Brown calls such TEs “platonic”). This, supposedly, makes it impossible to represent them as arguments for logical reasons: (...) there is no logic that can adequately model such phenomena. (Brown further argues that this being the case, “platonic” TEs provide us with irreducible insight into the abstract realm of laws of nature). I argue against this approach by describing how “platonic” TEs can be modeled within the logical framework of adaptive proofs for prioritized consequence operations. To show how this mundane apparatus works, I use it to reconstruct one of the key examples used by Brown, Galileo’s TE involving falling bodies. (shrink)

It is a commonplace remark that the identity relation, even though not expressible in a first-order language without identity with classical set-theoretic semantics, can be defined in a language without identity, as soon as we admit second-order, set-theoretically interpreted quantifiers binding predicate variables that range over all subsets of the domain. However, there are fairly simple and intuitive higher-order languages with set-theoretic semantics in which the identity relation is not definable. The point is that the definability of identity in higher-order (...) languages not only depends on what variables range over, but also is sensitive to how predication is construed. This paper is a follow-up to , where it has been proven that no actual axiomatization of Leśniewski’s Ontology determines the standard semantics for the epsilon connective. (shrink)

Sobocinski in his paper on Leśniewski's solution to Russell's paradox (1949b) argued that Leśniewski has succeeded in explaining it away. The general strategy of this alleged explanation is presented. The key element of this attempt is the distinction between the collective (mereological) and the distributive (set-theoretic) understanding of the set. The mereological part of the solution, although correct, is likely to fall short of providing foundations of mathematics. I argue that the remaining part of the solution which suggests a specific (...) reading of the distributive interpretation is unacceptable. It follows from it that every individual is an element of every individual. Finally, another Leśniewskian-style approach which uses so-called higher-order epsilon connectives is used and its weakness is indicated. (shrink)

A theory of definitions which places the eliminability and conservativeness requirements on definitions is usually called the standard theory. We examine a persistent myth which credits this theory to Le?niewski, a Polish logician. After a brief survey of its origins, we show that the myth is highly dubious. First, no place in Le?niewski's published or unpublished work is known where the standard conditions are discussed. Second, Le?niewski's own logical theories allow for creative definitions. Third, Le?niewski's celebrated ?rules of definition? lay (...) merely syntactical restrictions on the form of definitions: they do not provide definitions with such meta-theoretical requirements as eliminability or conservativeness. On the positive side, we point out that among the Polish logicians, in the 1920s and 1930s, a study of these meta-theoretical conditions is more readily found in the works of ?ukasiewicz and Ajdukiewicz. (shrink)

The main focus of this paper is to develop an adaptive formal apparatus capable of capturing (certain types of) reasoning conducted within the framework of the so-called dynamic conceptual frames. I first explain one of the most recent theories of concepts developed by cognitivists, in which a crucial part is played by the notion of a dynamic frame. Next, I describe how a dynamic frame may be captured by a finite set of first-order formulas and how a formalized adaptive framework (...) for reasoning within a dynamic frame can be developed. (shrink)

BAT is a logic built to capture the inferential behavior of informal provability. Ultimately, the logic is meant to be used in an arithmetical setting. To reach this stage it has to be extended to a first-order version. In this paper we provide such an extension. We do so by constructing non-deterministic three-valued models that interpret quantifiers as some sorts of infinite disjunctions and conjunctions. We also elaborate on the semantical properties of the first-order system and consider a couple of (...) its strengthenings. It turns out that obtaining a sensible strengthening is not straightforward. We prove that most strategies commonly used for strengthening non-deterministic logics fail in our case. Nevertheless, we identify one method of extending the system which does not. (shrink)

Salmon and Soames argue against nominalism about numbers and sentence types. They employ (respectively) higher-order and first-order logic to model certain natural language inferences and claim that the natural language conclusions carry commitment to abstract objects, partially because their renderings in those formal systems seem to do that. I argue that this strategy fails because the nominalist can accept those natural language consequences, provide them with plausible and non-committing truth conditions and account for the inferences made without committing themselves to (...) abstract objects. I sketch a modal account of higher-order quantification, on which instead of ranging over sets, higher order quantifiers are used to make (logical) possibility claims about which predicate tokens can be introduced. This approach provides an alternative account of truth conditions for natural language sentences which seem to employ higher-order quantification, thus allowing the nominalist to evade Salmon’s argument. I also show how the nominalist can account for the occurrence of apparently singular abstract terms in certain true statements. I argue that the nominalist can achieve this by, first, dividing singular terms into real singular terms (referring to concrete objects) and only apparent singular terms (called onomatoids), introduced for the sake of brevity and simplicity, and then providing an account of nominalistically acceptable truth conditions of sentences containing onomatoids. I develop such an account in terms of modally interpreted abstraction principles and argue that applying this strategy to Soames’s argument allows the nominalists to defend themselves. (shrink)

Richard Swinburne (Swinburne and Shoemaker 1984; Swinburne 1986) argues that human beings currently alive have non{bodily immaterial parts called souls. In his main argument in support of this conclusion (modal argument), roughly speaking, from the assumption that it is logically possible that a human being survives the destruction of their body and a few additional premises, he infers the actual existence of souls. After a brief presentation of the argument we describe the main known objection to it, called the substitution (...) objection (SO for short), which is raised by Alston and Smythe (1994), Zimmerman (1991) and Stump and Kretzmann (1996). We then explain Swinburne's response to it (1996). This constitutes a background for the discussion that follows. First, we formalize Swinburne's argument in a quantified propositional modal language so that it is logically valid and contains no tacit assumptions, clearing up some notational issues as we go. Having done that, we explain why we find Swinburne's response unsatisfactory. Next, we indicate that even though SO is quite compelling (albeit for a slightly different reason than the one given previously in the literature), a weakening of one of the premises yields a valid argument for the same conclusion and yet immune to SO. Even this version of the argument, we argue, is epistemically circular. (shrink)

Classical logic of formal provability includes Löb’s theorem, but not reflection. In contrast, intuitions about the inferential behavior of informal provability seem to invalidate Löb’s theorem and validate reflection. We employ a non-deterministic many-valued semantics and develop a modal logic T-BAT of an informal provability operator, which indeed does validate reflection and invalidates Löb’s theorem. We study its properties and its relation to known provability-related paradoxical arguments. We also argue that T-BAT is a fairly sensible candidate for a formal logic (...) of informal provability. (shrink)

In the debate about the legal value of naked statistical evidence, Di Bello argues that (1) the likelihood ratio of such evidence is unknown, (2) the decision-theoretic considerations indicate that a conviction based on such evidence is unacceptable when expected utility maximization is combined with fairness constraints, and (3) the risk of mistaken conviction based on such evidence cannot be evaluated and is potentially too high. We argue that Di Bello’s argument for (1) works in a rather narrow context, and (...) that (1) is not exactly in line with the way expert witnesses are required to formulate their opinions. Consequently, Di Bello’s argument for (2), which assumes (1), does not apply uniformly to all convictions based on naked evidence. Moreover, if Di Bello’s analysis is correct, it applies also to eyewitness testimony, given empirical results about its quality, and so the distinctions drawn by DiBello cut across the distinction between naked statistical evidence and other types of evidence. Finally, if we weaken the rather strong requirement of precise measurability of the risk of mistaken conviction, to the availability of reasonable but imprecise and fallible estimates, many field and empirical studies show that often the risk of mistaken conviction based on naked statistical evidence can be estimated to a similar extent as the risk of mistaken conviction based on any other sort of evidence. (shrink)

David Lewis has formulated a well-known challenge to his Best System account of lawhood: the content of any system whatever can be formulated very simply if one allows for perverse choices of primitive vocabulary. We show that the challenge is not that dangerous, and that to account for it one need not invoke natural properties or relativized versions of the Best System account. This way, we help to move towards an even better Best System account. We discuss extensions of our (...) strategy to the discussions about the indexicality of the notion of laws of nature, and to another trivialization argument. (shrink)

The first installment of a paper comparing the standard slingshot argument with the doxastic version.

This is the third installment of a paper which deals with comparison and evaluation of the standard slingshot argument (for the claim that all true sentences, if they refer, refer to the same object) with the doxastic formulation.

The goal is to sketch a nominalist approach to mathematics which just like neologicism employs abstraction principles, but unlike neologicism is not committed to the idea that mathematical objects exist and does not insist that abstraction principles establish the reference of abstract terms. It is well-known that neologicism runs into certain philosophical problems and faces the technical difficulty of finding appropriate acceptability criteria for abstraction principles. I will argue that a modal and iterative nominalist approach to abstraction principles circumvents those (...) difficulties while still being able to put abstraction principles to a foundational use. (shrink)

I illustrate with three classical examples the mistakes arising from using a modal operator admitting multiple interpretations in the same argument; the flaws arise especially easily if no attention is paid to the range of propositional variables. Premisses taken separately might seem convincing and a substitution for a propositional variable in a modal context might seem legitimate. But there is no single interpretation of the modal operators involved under which all the premisses are plausible and the substitution successful.

In some cases one is provided with inconsistent information and has to reason about various consistent scenarios contained within that information. Our goal is to argue that filtered paraconsistent logics are not the right tool to handle such cases and that the problems generalise to a large class of paraconsistent logics. A wide class of paraconsistent logics is obtained by filtration: adding conditions to the classical consequence operation . We start by surveying the most promising candidates and comparing their strengths. (...) Then we discuss the mainstream views on how non-classical logics should be chosen for an application and argue that none of these allows us to choose any of the filtered logics for action-guiding reasoning with inconsistent information, because such reasoning has to start with the selection of possible scenarios – and such a.. (shrink)

One of the streams in the early development of set theory was an attempt to use mereology, a formal theory of parthood, as a foundational tool. The first such attempt is due to a Polish logician, Stanisław Leśniewski . The attempt failed, but there is another, prima facie more promising attempt by Jerzy Słupecki , who employed his generalized mereology to build mereological foundations for type theory. In this paper I situate Leśniewski's attempt in the development of set theory, describe (...) and evaluate Leśniewski's approach, describe Słupecki's strategy without unnecessary technical details, and evaluate it with a rather negative outcome. The issues discussed go beyond merely historical interests due to the current popularity of mereology and because they are related to nominalistic attempts to understand mathematics in general. The introduction describes very briefly the situation in which mereology entered the scene of foundations of mathematics — it can be safely sk.. (shrink)

This book features mathematical and formal philosophers’ efforts to understand philosophical questions using mathematical techniques. It offers a collection of works from leading researchers in the area, who discuss some of the most fascinating ways formal methods are now being applied. It covers topics such as: the uses of probable and statistical reasoning, rational choice theory, reasoning in the environmental sciences, reasoning about laws and changes of rules, and reasoning about collective decision procedures as well as about action. Utilizing mathematical (...) techniques has been very fruitful in the traditional domains of formal philosophy – logic, philosophy of mathematics and metaphysics – while formal philosophy is simultaneously branching out into other areas in philosophy and the social sciences. These areas particularly include ethics, political science, and the methodology of the natural and social sciences. Reasoning about legal rules, collective decision-making procedures, and rational choices are of interest to all those engaged in legal theory, political science and economics. Statistical reasoning is also of interest to political scientists and economists. (shrink)

In this paper we argue that an existing theory of concepts called dynamic frame theory, although not developed with that purpose in mind, allows for the precise formulation of a number of problems associated with induction from a single instance. A key role is played by the distinction we introduce between complete and incomplete dynamic frames, for incomplete frames seem to be very elegant candidates for the format of the background knowledge used in induction from a single instance. Furthermore, we (...) show how dynamic frame theory provides the terminology to discuss the justification and the fallibility of incomplete frames. In the Appendix, we give a formal account of incomplete frames and the way these lead to induction from a single instance. (shrink)

This paper evaluates Richard Swinburne’s modal argument for the existence of souls. After a brief presentation of the argument, wedescribe the main known objection to it, which is called the substitution objection, and explain Swinburne’s response to that objection. With this as background, we formalize Swinburne’s argument in a quantified propositional modal language, modifying it so that it is logically valid and contains no tacit assumptions, and we explain why we find Swinburne’s response to SO unsatisfactory. Next, we indicate that, (...) even though SO is quite compelling, a weakening of one of the premises yields a valid argument for the same conclusion which is immune to SO. This version of the argument, however, is epistemically circular. (shrink)