Can we quantify over everything: absolutely, positively, definitely, totally, every thing? Some philosophers have claimed that we must be able to do so, since the doctrine that we cannot is self-stultifying. But this treats restrictivism as a positive doctrine. Restrictivism is much better viewed as a kind of militant quietism, which I call dadaism. Dadaists advance a hostile challenge, with the aim of silencing everyone who holds a positive position about ‘absolute generality’.

Call an explanation in which a non-mathematical fact is explained—in part or in whole—by mathematical facts: an extra-mathematical explanation. Such explanations have attracted a great deal of interest recently in arguments over mathematical realism. In this article, a theory of extra-mathematical explanation is developed. The theory is modelled on a deductive-nomological theory of scientific explanation. A basic DN account of extra-mathematical explanation is proposed and then redeveloped in the light of two difficulties that the basic theory faces. The final view (...) appeals to relevance logic and uses resources in information theory to understand the explanatory relationship between mathematical and physical facts. 1Introduction2Anchoring3The Basic Deductive-Mathematical Account4The Genuineness Problem5Irrelevance6Relevance and Information7Objections and Replies 7.1Against relevance logic7.2Too epistemic7.3Informational containment8Conclusion. (shrink)

In this paper, I introduce and defend a notion of analyticity for formal languages. I first uncover a crucial flaw in Timothy Williamson’s famous argument template against analyticity, when it is applied to sentences of formal mathematical languages. Williamson’s argument targets the popular idea that a necessary condition for analyticity is that whoever understands an analytic sentence assents to it. Williamson argues that for any given candidate analytic sentence, there can be people who understand that sentence and yet who fail (...) to assent to it. I argue that, on the most natural understanding of the notion of assent when it is applied to sentences of formal mathematical languages, Williamson’s argument fails. Formal analyticity is the notion of analyticity that is based on this natural understanding of assent. I go on to develop the notion of formal analyticity and defend the claim that there are formally analytic sentences and rules of inference. I conclude by showing the potential payoffs of recognizing formal analyticity. (shrink)

It is commonly held that the ascription of truth to a sentence is intersubstitutable with that very sentence. However, the simplest subclassical logics available to proponents of this view, namely K3 and LP, are hopelessly weak for many purposes. In this article, I argue that this is much more of a problem for proponents of LP than for proponents of K3. The strategies for recapturing classicality offered by proponents of LP are far less promising than those available to proponents of (...) K3. This undermines the ability of proponents LP to engage in public reasoning in classical domains. (shrink)

This paper proposes a new dialetheic logic, a Dialetheic Logic with Exclusive Assumptions and Conclusions ), including classical logic as a particular case. In \, exclusivity is expressed via the speech acts of assuming and concluding. In the paper we adopt the semantics of the logic of paradox extended with a generalized notion of model and we modify its proof theory by refining the notions of assumption and conclusion. The paper starts with an explanation of the adopted philosophical perspective, then (...) we propose our \ logic. Finally, we show how \ supports the dialetheic solution of the liar paradox. (shrink)

Many advocates of non-classical logic for reasons external to mathematics claim that their proposed revisions are consistent with the use of classical logic within pure mathematics. Doubts are raised about such claims, concerning the applicability of pure mathematics to natural and social science. -/- .

According to the iterative conception of set, each set is a collection of sets formed prior to it. The notion of priority here plays an essential role in explanations of why contradiction-inducing sets, such as the Russell set, do not exist. Consequently, these explanations are successful only to the extent that a satisfactory priority relation is made out. I argue that attempts to do this have fallen short: understanding priority in a straightforwardly constructivist sense threatens the coherence of the empty (...) set and raises serious epistemological concerns; but the leading realist interpretations---ontological and modal interpretations of priority---are deeply problematic as well. I conclude that the purported explanatory virtues of the iterative conception are, at present, unfounded. (shrink)

Restrictionism is a response to the Liar and other paradoxes concerning truth. Restrictionists—as I will call proponents of the strategy—respond to these paradoxes by giving up instances of the schema -/- <p> is true iff p. -/- My aim is to show that the current unpopularity of restrictionism is undeserved. I will argue that, whilst cautious versions of the strategy may face serious problems, a radical and previously overlooked version of restrictionism provides a strong and defensible response to the paradoxes.

According to Markus Gabriel, the world does not exist. This view—baptised metametaphysical nihilism—is exposited at length in his recent book Fields of Sense, which updates his earlier project of transcendental ontology. In this paper, I question whether meta-metaphysical nihilism is internally coherent, specifically whether the proposition ‘the world does not exist’ is expressible without performative contradiction on that view. Call this the inexpressibility objection. This is not an original objection—indeed it is anticipated in Gabriel’s book. However, I believe that his (...) response to it is inadequate and that I have something illuminating to say about this state of affairs. My claim is that we can distinguish between two senses of ‘the world’, one of which is benign and acceptable, the other not. The acceptable sense of ‘the world’ suffices to answer the inexpressibility objection—at a certain theoretical cost, of course. To explain what this cost is, I turn briefly to an examination of Martin Hägglund’s radical atheism. (shrink)

This paper concerns two paradoxes involving propositions. The first is Russell's paradox from Appendix B of The Principles of Mathematics, a version of which was later given by Myhill. The second is a paradox in the framework of possible worlds, given by Kaplan. This paper shows a number of things about these paradoxes. First, we will see that, though the Russell/myhill paradox and the Kaplan paradox might appear somewhat different, they are really just variants of the same phenomenon. Though they (...) do this in different ways, the core of each paradox is to use the notion of a proposition to construct a function, f, from the power set of some set into the set itself. Next we will see how this paradox fits into the Inclosure Schema. Finally, I will provide a model of the paradox in question, showing its results to be non‐trivial, though inconsistent. (shrink)

Religious believers react in one of four ways to apparent contradictions among their beliefs: Redirection, Resistance, Restraint, or Resolution. This paper evaluates positive mysterian Resistance, the view that believers may rationally believe and know apparently contradictory religious doctrines. After locating this theory by comparing and contrasting it with others, I explore the best developed version of it, that of James Anderson’s Paradox in Christian Theology. I argue that it faces steep epistemic problems, and is at best a temporarily reasonable but (...) ultimately unsustainable stance. (shrink)

Graham Priest proposed an argument for the conclusion that ‘nothing’ occurs as a singular term and not as a quantifier in a sentence like (1) ‘The cosmos came into existence out of nothing’. Priest's point is that, intuitively, (1) entails (C) ‘The cosmos came into existence at some time’, but this entailment relation is left unexplained if ‘nothing’ is treated as a quantifier. If Priest is right, the paradoxical notion of an object that is nothing plays a role in our (...) very understanding of reality. In this note, we argue that Priest's argument is unsound: the intuitive entailment relation between (1) and (C) does not offer convincing evidence that ‘nothing’ occurs as a term in (1). Moreover, we provide an explanation of why (1) is naturally taken to entail (C), which is both plausible and consistent with the standard, quantificational treatment of ‘nothing’. (shrink)

The paper proposes with Davidson that the talk of metaontology is literally meaningless, but with Wittgenstein that it is so in a way that grants a unique type of insight. More specifically, it argues both that Davidson’s arguments have a cogency that is hard to dismiss, and also that, since his own arguments are metaontological, they are self-referential, and consequently in turn undermine their own meaning as well. The paper argues further that metaontological statements cannot be avoided. Consequently, this kind (...) of statement constitutes an unavoidable self-referential paradox that means what it also excludes as capable of meaning. The result is a reinstatement of the meaning of ontological insight and in fact, the paper argues, a deep enrichment and also a particularly cogent justification of it. In addition, the logical peculiarity of the paradox involved has further useful consequences for the outcome of this justification, including a mutually illuminating commonality with some versions of metaethics. (shrink)

In this paper I reassess Floridi's solution to the Bar-Hillel-Carnap paradox (the information-yield of inconsistent propositions is maximal) by questioning the orthodox view that contradictions cannot be true. The main part of the paper is devoted to showing that the veridicality thesis (semantic information has to be true) is compatible with dialetheism (there are true contradictions), and that unless we accept the additional non-falsity thesis (information cannot be false) there is no reason to presuppose that there is no such thing (...) like contradictory information. (shrink)