Olszewski claims that the Church-Turing thesis can be used in an argument against platonism in philosophy of mathematics. The key step of his argument employs an example of a supposedly effectively computable but not Turing-computable function. I argue that the process he describes is not an effective computation, and that the argument relies on the illegitimate conflation of effective computability with there being a way to find out . ‘Ah, but,’ you say, ‘what’s the use of its being right twice (...) a day, if I can’t tell when the time comes?’ Why, suppose the clock points to eight o’clock, don’t you see that the clock is right at eight o’clock? Consequently, when eight o’clock comes round your clock is right. Lewis Carroll. (shrink)

When we talk about the coherence of a story, we seem to think of how well its individual pieces fit together—how to explicate this notion formally, though? We develop a Bayesian network based coherence measure with implementation in _R_, which performs better than its purely probabilistic predecessors. The novelty is that by paying attention to the network structure, we avoid simply taking mean confirmation scores between all possible pairs of subsets of a narration. Moreover, we assign special importance to the (...) weakest links in a narration, to improve on the other measures’ results for logically inconsistent scenarios. We illustrate and investigate the performance of the measures in relation to a few philosophically motivated examples, and (more extensively) using the real-life example of the Sally Clark case. (shrink)

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)

While there is a general consensus around the role of religion in the abolition of the Slave Trade, historians continue to give little to no detail on exactly how Christian theology influenced the abolitionist movement. This article seeks to interrogate one major theological factor inherent in the spirituality that underpinned the activism of the British abolitionists, namely their notion of Divine Providence, and particularly its moral-emotive correlate: the fear of God’s wrath. These theological notions are discussed based mainly on the (...) analysis of the primary sources and within the theoretical framework of judicial providentialism, aptly captured by John Coffey among others. (shrink)

While there is a general consensus around the role of religion in the abolition of the Slave Trade, historians continue to give little to no detail on exactly how Christian theology influenced the abolitionist movement. This article seeks to interrogate one major theological factor inherent in the spirituality that underpinned the activism of the British abolitionists, namely their notion of Divine Providence, and particularly its moral-emotive correlate: the fear of God’s wrath. These theological notions are discussed based mainly on the (...) analysis of the primary sources and within the theoretical framework of judicial providentialism, aptly captured by John Coffey among others. (shrink)

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 ...

"Williams's masterful translation satisfies (at last!) a long-standing need. There are lots of good translations of Augustine's great work, but until now we have been forced to choose between those that strive to replicate in English something of the majesty and beauty of Augustine's Latin style and those that opt instead to convey the careful precision of his philosophical terminology and argumentation. Finally, Williams has succeeded in capturing both sides of Augustine's mind in a richly evocative, impeccably reliable, elegantly readable (...) presentation of one of the most impressive achievements in Western thought--Augustine's Confessions." --Scott MacDonald, Professor of Philosophy and Norma K. Regan Professor in Christian Studies, Cornell University. (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)

The Moral Nexus develops and defends a new interpretation of morality—namely, as a set of requirements that connect agents normatively to other persons in a nexus of moral relations. According to this relational interpretation, moral demands are directed to other individuals, who have claims that the agent comply with these demands. Interpersonal morality, so conceived, is the domain of what we owe to each other, insofar as we are each persons with equal moral standing. The book offers an interpretative argument (...) for the relational approach. Specifically, it highlights neglected advantages of this way of understanding the moral domain; explores important theoretical and practical presuppositions of relational moral duties; and considers the normative implications of understanding morality in relational terms. The book features a novel defense of the relational approach to morality, which emphasizes the special significance that moral requirements have, both for agents who are deliberating about what to do and for those who stand to be affected by their actions. The book argues that relational moral requirements can be understood to link us to all individuals whose interests render them vulnerable to our agency, regardless of whether they stand in any prior relationship to us. It also offers fresh accounts of some of the moral phenomena that have seemed to resist treatment in relational terms, showing that the relational interpretation is a viable framework for understanding our specific moral obligations to other people. (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)

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 (SO for short), 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)

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 Leśniewski, a Polish logician. After a brief survey of its origins, we show that the myth is highly dubious. First, no place in Leśniewski's published or unpublished work is known where the standard conditions are discussed. Second, Leśniewski's own logical theories allow for creative definitions. Third, Leśniewski'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 Łukasiewicz and Ajdukiewicz. (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)

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)

This article builds on my recent engagement with James Cone’s binary view of Africanness and Christianity which focused on his Western locus of enunciation and the criticism he received from his African American colleagues. I believe that analogical questions regarding Christian theology’s attitude towards Africanness in general and African religiosity in particular present themselves to us who live in and try to make sense of South African reality today, including white people like myself. I start by introducing a decolonial perspective (...) as it manifests itself through the recent #MustFall student movements. In this context, I offer three theses regarding the decolonial perspective on African religiosity, each of which constitutes a more or less direct critique of Cone’s ambivalent attitude towards Africanness, and African Traditional Religions in particular. The first thesis concerns the distinction between postcoloniality and decoloniality; the second thesis concerns engaging African religiosity as a requirement for decolonising Christian theology; and the third thesis concerns problematising the relationship between the categories of blackness and Africanness. (shrink)

According to the dialetheist argument from the inconsistency of informal mathematics, the informal version of the Gödelian argument leads us to a true contradiction. On one hand, the dialetheist argues, we can prove that there is a mathematical claim that is neither provable nor refutable in informal mathematics. On the other, the proof of its unprovability is given in informal mathematics and proves that very sentence. We argue that the argument fails, because it relies on the unjustified and unlikely assumption (...) that the informal Gödel sentence is informally provable. (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)

Classical logic of formal provability includes Löb’s theorem, but not reflection. In contrast, intuitions about the inferential behavior of informal provability (in informal mathematics) seem to invalidate Löb’s theorem and validate reflection (after all, the intuition is, whatever mathematicians prove holds!). 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)

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)

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)

In postmodern societies the symbolic vacuum, a result of the loss of a unified religious tradition, calls for substitutes in the form of fragmentary and isolated memories. By drawing from the reservoir of those memories in an arbitrary and subjective way, privatised religion creates a kind of symbolic bricolage. Can such a bricolage become more than a mere ‘counterfeit’ of collective meaning that religion once used to provide? Can religious tradition, based on a broken continuity of memory, still bring about (...) a matrix of the ways of expressing one’s faith? If so, how? This twofold study seeks to explore those and similar questions by means of showing, firstly, in what sense religion can be conceived of as memory which produces collective meanings and, secondly, what may happen when individualised and absolutised memories alienate themselves from a continuity of tradition, thus beginning to function as a sort of private religion. Being the second part of the study in question, this article aims at exploring the postmodern crisis of religious memory, which includes the pluralisation of the channels of the sacred and the differentiation of a total religious memory into a plurality of specialised circles of memory. Firstly, it examines the three main aspects of the current crisis of continuity at large, namely the affirmation of the autonomous individual, the advance of rationalisation, and the process of institutional differentiation. Secondly, the plurality of the channels of the sacred is discussed in light of religion’s apparently unique way of drawing legitimisation from its reference to tradition. This is followed by two illustrations of the reconstruction of religious memory. In the final section of the article, a theological reflection on possible directions that may be taken in the face of the postmodern crisis of religious memory is offered. (shrink)

In this article, we analyse the relation of philosophy and theology in the work of Jean‑Luc Marion in order to be able to see not only how the phenomenology of givenness can serve as a “new apologetics” for theology, but also how Marion’s phenomenology itself, in its historical development and in its core principle and method, is influenced and changed by theological phenomena. We present three ways of describing the relation, tension, mutual influence and separation of philosophy and theology: firstly, (...) in line with Pascal’s distinction between the orders of reason and of the heart; secondly, in phenomenology, in terms of indications to the effect that there can be a phenomenon of revelation in the mode of possibility that is distinguished from the phenomenon of Revelation in theology in the mode of historicity; and thirdly, by analogy with Christian apologetics. In particular, we analyse this third dimension, putting forward the thesis that Marion’s phenomenology itself has some characteristics of the Christian apologetics he describes. We try to demonstrate this interpretation of his phenomenology in its key dimensions, such as the counter-method and descriptions of the phenomena of love and revelation, which constitute the culmination of the phenomenology of givenness, although at the same time, as it were, its limit, crossing over into the theological order. (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)

Provability logics are, roughly speaking, modal logics meant to capture the formal principles of various provability operators or predicates.

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)

As a theologian coming from Europe, a ‘postcolonial import’ into South Africa, it is my white privilege in particular that continues to queer my understanding of a social revolution on which our future, as a people, may depend. In this article, I seek to turn my personal experience of grappling with my whiteness into the source of my reflection. Drawing inspiration from fallism – a recent student movement that inscribes itself into a larger decolonial ‘struggle against the globalised system of (...) racist capitalism’ – I ponder what it could mean, in the South African context, for whiteness to fall upward (Rohr). Here, the metaphor of ‘falling upward’ as a kenosis of whiteness is considered specifically with regard to a white theologian’s (my own) attempt to open spaces that could be filled with blackness. Contribution: This auto-ethnographic essay inscribes itself into a transdisciplinary study of theology and race from both socio-cultural and religiospiritual perspectives. The author’s personal reflections, inspired by his own engagement with the fallist narratives and his ever-evolving attitude towards the blackness–whiteness binary, as experienced in the South African social and academic contexts, are shared as a means to crack open the societal and theological (notably Christian) imagination, both of which appear to suffer from a serious crisis. (shrink)

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.

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.