A logical approach to Bell’s Inequalities of quantum mechanics has been introduced by Abramsky and Hardy (Abramsky & Hardy, 2012). We point out that the logical Bell’s Inequalities of Abramsky & Hardy (2012) are provable in the probability logic of Fagin, Halpern and Megiddo (Fagin et al., 1990). Since it is now considered empirically established that quantum mechanics violates Bell’s Inequalities, we introduce a modified probability logic, that we call quantum team logic, in which Bell’s Inequalities are not provable, and prove a Completeness theorem for this logic. For this end we generalise the team semantics of dependence logic (Väänänen, 2007) first to probabilistic team semantics, and then to what we call quantum team semantics.

The collapse models of arithmetic are inconsistent, nontrivial models obtained from ℕ and set out in the Logic of Paradox (LP). They are given a general treatment by Priest (Priest, 2000). Finite collapse models are decidable, and thus axiomatizable, because finite. LP, however, is ill-suited to normal axiomatic reasoning, as it invalidates Modus Ponens, and almost all other usual conditional inferences. I set out a logic, A3, first given by Avron (Avron, 1991), and give a first order axiom system for the finite collapse models. I present some standard arithmetical axioms in addition to a cyclic axiom and prove that these axioms are sound and complete for the cyclic models, reporting a similar result for the heap models. The state of the situation for the each of the kinds of infinite collapse model is, however, left an open question.

In this paper I present a precise version of Stalnaker’s thesis and show that it is both consistent and predicts our intuitive judgments about the probabilities of conditionals. The thesis states that someone whose total evidence is E should have the same credence in the proposition expressed by ‘if A then B’ in a context where E is salient as they have conditional credence in the proposition B expresses given the proposition A expresses in that context. The thesis is formalised rigorously and two models are provided that demonstrate that the new thesis is indeed tenable within a standard possible world semantics based on selection functions. Unlike the Stalnaker–Lewis semantics the selection functions cannot be understood in terms of similarity. A probabilistic account of selection is defended in its place.

I end the paper by suggesting that this approach overcomes some of the objections often levelled at accounts of indicatives based on the notion of similarity.

The big question at the end of Feferman (2013) is: Is it possible to find a foundation for unlimited category theory? I show that the answer is no by showing that unlimited category theory is inconsistent.

Bertrand Russell offered an influential paradox of propositions in Appendix B of The Principles of Mathematics, but there is little agreement as to what to conclude from it. We suggest that Russell’s paradox is best regarded as a limitative result on propositional granularity. Some propositions are, on pain of contradiction, unable to discriminate between classes with different members: whatever they predicate of one, they predicate of the other. When accepted, this remarkable fact should cast some doubt upon some of the uses to which modern descendants of Russell’s paradox of propositions have been put in recent literature.

This paper tackles an open problem posed by Åqvist. It is the problem of whether his dyadic deontic systems E and F are complete with respect to their intended Hanssonian preference-based semantics. It is known that there are two different ways of interpreting what it means for a world to be best or top-ranked among alternatives. This can be understood as saying that it is optimal among them, or maximal among them. First, it is established that, under either the maximality rule or the optimality rule, E is sound and complete with respect to the class of all preference models, the class of those in which the betterness relation is reflexive, and the class of those in which it is total. Next, an analogous result is shown to hold for F. That is, it is established that, under either rule, F is sound and complete with respect to the class of preference models in which the betterness relation is limited, the class of those in which it is limited and reflexive, and the class of those in which it is limited and total.

The rules for Core Logic are stated, and various important results about the system are summarized. We describe its relationship to other systems, such as Classical Logic, Intuitionistic Logic, Minimal Logic, and the Anderson–Belnap relevance logic R. A precise, positive explication is offered of what it is for the premises of a proof to connect relevantly with its conclusion. This characterization exploits the notion of positive and negative occurrences of atoms in sentences. It is shown that all Core proofs are relevant in this precisely defined sense. We survey extant results about variable-sharing in rival systems of relevance logic, and find that the variable-sharing conditions established for them are weaker than the one established here for Core Logic (and for its classical extension). Proponents of other systems of relevance logic (such as R and its subsystems) are challenged to formulate a stronger variable-sharing condition, and to prove that R or any of its subsystems satisfies it, but that Core Logic does not. We give reasons for pessimism about the prospects for meeting this challenge.

There are two perspectives from which formal theories can be viewed. On the one hand, one can take a theory to be about some privileged models. On the other hand, one can take all models of a theory to be on a par. In contrast with what is usually done in philosophical debates, we adopt the latter viewpoint. Suppose that from this perspective we want to add an adequate truth predicate to a background theory. Then on the one hand the truth theory ought to be semantically conservative over the background theory. At the same time, it is generally recognised that the central function of a truth predicate is an expressive one. A truth predicate ought to allow us to express propositions that we could not express before. In this article we argue that there are indeed natural truth theories which satisfy both the demand of semantical conservativeness and the demand of adequately extending the expressive power of our language.

It is a classic result (McKinsey & Tarski, 1944; Rasiowa & Sikorski, 1963) that if we interpret modal diamond as topological closure, then the modal logic of any dense-in-itself metric space is the well-known modal system S4. In this paper, as a natural follow-up, we study the modal logic of an arbitrary metric space. Our main result establishes that modal logics arising from metric spaces form the following chain which is order-isomorphic (with respect to the ⊃ relation) to the ordinal ω + 3:

$S4.Gr{z_1} \supset S4.Gr{z_2} \supset S4.Gr{z_3} \supset \cdots \,S4.Grz \supset S4.1 \supset S4.$

It follows that the modal logic of an arbitrary metric space is finitely axiomatizable, has the finite model property, and hence is decidable.

In a recent series of papers, I and others have advanced new logical approaches to familiar paradoxes. The key to these approaches is to accept full classical logic, and to accept the principles that cause paradox, while preventing trouble by allowing a certain sort of nontransitivity. Earlier papers have treated paradoxes of truth and vagueness. The present paper will begin to extend the approach to deal with the familiar paradoxes arising in naive set theory, pointing out some of the promises and pitfalls of such an approach.

Is evidential support transitive? The answer is negative when evidential support is understood as confirmation so that X evidentially supports Y if and only if p(Y | X) > p(Y). I call evidential support so understood “support” (for short) and set out three alternative ways of understanding evidential support: support-t (support plus a sufficiently high probability), support-t* (support plus a substantial degree of support), and support-tt* (support plus both a sufficiently high probability and a substantial degree of support). I also set out two screening-off conditions (under which support is transitive): SOC1 and SOC2. It has already been shown that support-t is non-transitive in the general case (where it is not required that SOC1 holds and it is not required that SOC2 holds), in the special case where SOC1 holds, and in the special case where SOC2 holds. I introduce two rather weak adequacy conditions on support measures and argue that on any support measure meeting those conditions it follows that neither support-t* nor support-tt* is transitive in the general case, in the special case where SOC1 holds, or in the special case where SOC2 holds. I then relate some of the results to Douven’s evidential support theory of conditionals along with a few rival theories.

We introduce a new understanding of deontic modals that we call obligations as weakest permissions. We argue for its philosophical plausibility, study its expressive power in neighborhood models, provide a complete Hilbert-style axiom system for it and show that it can be extended and applied to practical norms in decision and game theory.

In a recent article (Mancosu, 2009), I have explored the historical, mathematical, and philosophical issues related to the new theory of numerosities. The theory of numerosities provides a context in which to assign ‘sizes’ to infinite sets of natural numbers in such a way as to preserve the part-whole principle, namely if a set A is properly included in B then the numerosity of A is strictly less than the numerosity of B. Numerosity assignments differ from the standard assignment of size provided by Cantor’s cardinality assignments. In this paper I generalize some worries, raised by Richard Heck, emerging from the theory of numerosities to a line of thought resulting in what I call a ‘good company’ objection to Hume’s Principle (HP). The paper is centered around five main parts. The first (§3) takes a historical look at nineteenth-century attributions of equality of numbers in terms of one-one correlation and argues that there was no agreement as to how to extend such determinations to infinite sets of objects. This leads to the second part (§4) where I show that there are countably-infinite many abstraction principles that are ‘good’, in the sense that they share the same virtues of HP (or so I claim) and from which we can derive the axioms of second-order arithmetic. All the principles I present agree with HP in the assignment of numbers to finite concepts but diverge from it in the assignment of numbers to infinite concepts. The third part (§5) connects this material to a debate on Finite Hume’s Principle between Heck and MacBride. The fourth part (§6) states the ‘good company’ objection as a generalization of Heck’s objection to the analyticity of HP based on the theory of numerosities. In the same section I offer a taxonomy of possible neo-logicist responses to the ‘good company’ objection. Finally, §7 makes a foray into the relevance of this material for the issue of cross-sortal identifications for abstractions.

Many recent writers in the philosophy of mathematics have put great weight on the relative categoricity of the traditional axiomatizations of our foundational theories of arithmetic and set theory (Parsons, 1990; Parsons, 2008, sec. 49; McGee, 1997; Lavine, 1999; Väänänen & Wang, 2014). Another great enterprise in contemporary philosophy of mathematics has been Wright’s and Hale’s project of founding mathematics on abstraction principles (Hale & Wright, 2001; Cook, 2007). In Walsh (2012), it was noted that one traditional abstraction principle, namely Hume’s Principle, had a certain relative categoricity property, which here we term natural relative categoricity. In this paper, we show that most other abstraction principles are not naturally relatively categorical, so that there is in fact a large amount of incompatibility between these two recent trends in contemporary philosophy of mathematics. To better understand the precise demands of relative categoricity in the context of abstraction principles, we compare and contrast these constraints to (i) stability-like acceptability criteria on abstraction principles (cf. Cook, 2012), (ii) the Tarski-Sher logicality requirements on abstraction principles studied by Antonelli (2010b) and Fine (2002), and (iii) supervaluational ideas coming out of the work of Hodes (1984, 1990, 1991).

We generalize the construction of lattice-valued models of set theory due to Takeuti, Titani, Kozawa and Ozawa to a wider class of algebras and show that this yields a model of a paraconsistent logic that validates all axioms of the negation-free fragment of Zermelo-Fraenkel set theory.