This article is about the ontological dispute between finitists, who claim that only finitely many numbers exist, and infinitists, who claim that infinitely many numbers exist. Van Bendegem set out to solve the ‘general problem’ for finitism: how can one recast finite fragments of classical mathematics in finitist terms? To solve this problem Van Bendegem comes up with a new brand of finitism, namely so-called ‘apophatic finitism’. In this article it will be argued that apophatic finitism is unable to represent the negative ontological commitments of infinitism or, in other words, that which does not exist according to infinitism. However, there is a brand of infinitism, so-called ‘apophatic infinitism’, that is able to represent both the positive and the negative ontological commitments of apophatic finitism.

If one is interested in reasoning counterfactually within a physical theory, one cannot adequately use the standard possible world semantics. As developed by Lewis and others, this semantics depends on entertaining possible worlds with miracles, worlds in which laws of nature, as described by physical theory, are violated. Van Fraassen suggested instead to use the models of a theory as worlds, but gave up on determining the needed comparative similarity relation for the semantics objectively. I present a third way, in which this similarity relation is determined from properties of the models contextually relevant to the truth of the counterfactual under evaluation. After illustrating this with a simple example from thermodynamics, I draw some implications for future work, including a renewed possibility for a viable deflationary account of laws of nature.

This paper argues that the theory of structured propositions is not undermined by the Russell-Myhill paradox. I develop a theory of structured propositions in which the Russell- Myhill paradox doesn’t arise: the theory does not involve ramification or compromises to the underlying logic, but rather rejects common assumptions, encoded in the notation of the λ-calculus, about what properties and relations can be built out of others. I argue that the structuralist had independent reasons to reject these underlying assumptions. The theory is given both a diagrammatic representation, and a logical representation in a special purpose language.

We show that there is a mathematical obstruction to complete Turing computability of intelligence. This obstruction can be circumvented only if human reasoning is fundamentally unsound, with the latter formally interpreted here as certain stable soundness. To this end, we first develop in a specific setting a certain analogue of a Gödel statement, which has universality with respect to a certain class of Turing machines / formal systems. As a partial consequence of this universality, this Gödel statement, or Gödel string G as we call it in the language of Turing machines, does not require soundness but only stable soundness. Moreover, this G is constructed explicitly, given the general form of our class of Turing machines.

Schoen eld has constructed examples of proper inaccuracy measures that value verisimilitude (in a certain sense) in spaces of worlds equipped with a particular variety of verisimilitude metric. However, Schoen eld left it as an open question whether `for every space of worlds, there is a proper inaccuracy measure that values verisimilitude.' Here we answer this question in the armative.

We present a new frame semantics for positive relevant and substructural propositional logics. This frame semantics is both a generalization of Routley–Meyer ternary frames and a simplification of them. The key innovation of this semantics is the use of a single accessibility relation to relate collections of points to points.

Neo-Fregeans in the philosophy of mathematics hold that the key to a correct understanding of mathematics is the implicit definition of mathematical terms. In this paper, I discuss and advocate the rejection of abstractionism, the putative constraint (latent within the recent neo-Fregean tradition) according to which all acceptable implicit definitions take the form of abstraction principles. I argue that there is reason to think that neo-Fregean aims would be better served by construing the axioms of mathematical theories themselves as implicit definitions, and consider and respond to several lines of objection to this thought.

If a conditional is to be relevant—if A → B is to be true only when there is a genuine connection between the antecedent A and the consequent B—any ‘worlds’ semantics for that conditional must look rather unlike the well-known modal semantics for strict conditionals, counterfactuals and other non-classical conditional connectives. If I wish to evaluate the conditional A → B at some ‘world’ x, it will never suffice to find some class of worlds, related to x (whether that choice depends on A, or on B, or on anything else) and then check of those worlds where A is true, whether B is true at those selected worlds, too. For then, the identity conditional A → A (in which the consequent is identical to the antecedent) is guaranteed to be true at absolutely any world whatsoever.

It is known from the work of Specker [3] that Quine’s NF is consistent iff the theory TZZT of Typed Set Theory with types indexed by Z remains consistent when we add the scheme of biconditionals φ ←→ φ , where φ is the result of raising all type indices in φ by 1. Since evidently TZZT |= φ iff TZZT |= φ+ it looks as if there should be realizers for the corresponding biconditionals φ ←→ φ and thereby a proof of consistency for INF (the constructive fragment of NF) that is not at the same time a reason to believe in the consistency of the full classical theory. There seems to be a connection here with Visser’s Logic BPC in [4].

This paper aims to shed light on the relation between Boltzmannian statistical mechanics and Gibbsian statistical mechanics by studying the Mechanical Averaging Principle, which says that, under certain conditions, Boltzmannian equilibrium values and Gibbsian phase averages are approximately equal. What are these conditions? We identify three conditions each of which is individually sufficient (but not necessary) for Boltzmannian equilibrium values to be approximately equal to Gibbsian phase averages: the Khinchin condition, and two conditions that result from two new theorems, the Average Equivalence Theorem and the Cancelling Out Theorem. These conditions are not trivially satisfied, and there are core models of statistical mechanics, the six-vertex model and the Ising model, in which they fail.

In Richard Bradley's book, Decision Theory with a Human Face (2017), we have selected two themes for discussion. The first is the Bolker-Jeffrey (BJ) theory of decision, which the book uses throughout as a tool to reorganize the whole field of decision theory, and in particular to evaluate the extent to which expected utility (EU) theories may be normatively too demanding. The second theme is the redefinition strategy that can be used to defend EU theories against the Allais and Ellsberg paradoxes, a strategy that the book by and large endorses, and even develops in an original way concerning the Ellsberg paradox. We argue that the BJ theory is too specific to fulfil Bradley’s foundational project and that the redefinition strategy fails in both the Allais and Ellsberg cases. Although we share Bradley’s conclusion that EU theories do not state universal rationality requirements, we reach it not by a comparison with BJ theory, but by a comparison with the non-EU theories that the paradoxes have heuristically suggested.

This paper proposes a way of doing type theory informally, assuming a cubical style of reasoning. It can thus be viewed as a first step toward a cubical alternative to the program of informalization of type theory carried out in the homotopy type theory book for dependent type theory augmented with axioms for univalence and higher inductive types. We adopt a cartesian cubical type theory proposed by Angiuli, Brunerie, Coquand, Favonia, Harper and Licata as the implicit foundation, confining our presentation to elementary results such as function extensionality, the derivation of weak connections and path induction, the groupoid structure of types and the Eckmman–Hilton duality.

The “four-color” theorem seems to be generalizable as follows. The four-letter alphabet is sufficient to encode unambiguously any set of well-orderings including a geographical map or the “map” of any logic and thus that of all logics or the DNA (RNA) plan(s) of any (all) alive being(s). Then the corresponding maximally generalizing conjecture would state: anything in the universe or mind can be encoded unambiguously by four letters.

One alternative to Tarski’s hierarchy of metalanguages is to consider paraconsistent logics for theories of truth, to deal with paradoxical sentences. In the face of the possibility of inconsistency, critics and proponents of paraconsistency alike have then sought ‘consistency operators’, to characterize non-paradoxical sentences. For strong forms of paraconsistency—dialetheism—this is called the ‘just true’ problem. In this paper we consider various options for treatments of the issue, and follow the ‘just true’ problem to a stark divide. If a paraconsistentist uses a classical metatheory, then they can have a ‘just true’ operator, but only by accepting a paracomplete logic, and in fact ruling out any truth value gluts. If a paraconsistentist uses a paraconsistent metatheory, then the ‘just true’ problem is easily resolved, albeit not in a way that would be satisfying to a non-paraconsistentist.

We present an elementary system of axioms for the geometry of Minkowski spacetime. It strikes a balance between a simple and streamlined set of axioms and the attempt to give a direct formalization in first-order logic of the standard account of Minkowski spacetime in [Maudlin 2012] and [Malament, unpublished]. It is intended for future use in the formalization of physical theories in Minkowski spacetime. The choice of primitives is in the spirit of [Tarski 1959]: a predicate of betwenness and a four place predicate to compare the square of the relativistic intervals. Minkowski spacetime is described as a four dimensional ‘vector space’ that can be decomposed everywhere into a spacelike hyperplane - which obeys the Euclidean axioms in [Tarski and Givant, 1999] - and an orthogonal timelike line. The length of other ‘vectors’ are calculated according to Pythagora’s theorem. We conclude with a Representation Theorem relating models M of our system M that satisfy second order continuity to the mathematical structure xR , ηaby, called ‘Minkowski spacetime’ in physics textbooks.

Careful formulations of the Bayesian norm of Conditionalization acknowledge that it governs how you should plan to update your credences, or how you should be disposed to update them. It does not govern how you should in fact update, or at least not directly. That is, Conditionalization does not govern the relationship between your prior credences and your posterior credences, but rather the relationship between your prior credences and your plans or dispositions for updating those priors. In particular, it governs those plans or dispositions you have for updating your credences in response to a certain sort of learning situation, namely, one in which you take yourself to learn a proposition with certainty.

We develop a logic-based framework for formal specification and algorithmic verification of homogeneous and dynamic concurrent multi-agent transition systems. Homogeneity means that all agents have the same available actions at any given state and the actions have the same effects regardless of which agents perform them. The state transitions are therefore determined only by the vector of numbers of agents performing each action and are specified symbolically, by means of conditions on these numbers definable in Pres-burger arithmetic. The agents are divided into controllable (by the system supervisor/controller) and uncontrollable, representing the environment or adversary. Dynamicity means that the numbers of controllable and uncontrollable agents may vary throughout the system evolution, possibly at every transition. As a language for formal specification we use a suitably extended version of Alternating-time Temporal Logic, where one can specify properties of the type “a coalition of (at least) n controllable agents can ensure against (at most) m uncontrollable agents that any possible evolution of the system satisfies a given objective ? ″, where ? is specified again as a formula of that language and each of n and m is either a fixed number or a variable that can be quantified over. We provide formal semantics to our logic LHDMAS and define normal form of its formulae. We then prove that every formula in LHDMAS is equivalent in the finite to one in a normal form and develop an algorithm for global model checking of formulae in normal form in finite HDMAS models, which invokes model checking truth of Presburger formulae. We establish worst case complexity estimates for the model checking algorithm and illustrate it on a running example.

We study pure coordination games where in every outcome, all players have identical payoffs, ‘win’ or ‘lose’. We identify and discuss a range of ‘purely rational principles’ guiding the reasoning of rational players in such games and compare the classes of coordination games that can be solved by such players with no preplay communication or conventions. We observe that it is highly nontrivial to delineate a boundary between purely rational principles and other decision methods, such as conventions, for solving such coordination games.

The received view of computation is methodologically bifurcated: it offers different accounts of computation in the mathematical and physical cases. But little in the way of argument has been given for this approach. This paper rectifies the situation by arguing that the alternative, a unified account, is untenable. Furthermore, once these issues are brought into sharper relief we can see that work remains to be done to illuminate the relationship between physical and mathematical computation.

This paper aims to answer the question of whether or not Frege’s solution limited to value-ranges and truth-values proposed to resolve the “problem of indeterminacy of reference” in section 10 of Grundgesetze is a violation of his principle of complete determination, which states that a predicate must be defined to apply for all objects in general. Closely related to this doubt is the common allegation that Frege was unable to solve a persistent version of the Caesar problem for value-ranges. It is argued that, in Frege’s standards of reducing arithmetic to logic, his solution to the indeterminacy does not give rise to any sort of Caesar problem in the book.

The contraposing conditional ‘If A then C’ is defined by the conjunction of A > C and ¬C > ¬A, where > is a conditional of the kind studied by Stalnaker, Lewis and others. This idea has recently been explored, under the name ‘evidential conditional’, in a sequence of papers by Crupi and Iacona and Raidl, and it has been found of independent interest by Booth and Chandler. I discuss various properties of these conditionals and compare them to the ‘difference-making conditionals’ studied by Rott, which are defined by the conjunction of A > C and not ¬A > C. I raise some doubts about Crupi and Iacona’s claim that contraposition captures the idea of evidence or support.

David Builes presents a paradox concerning how confident you should be that any given member of an infinite collection of fair coins landed heads, conditional on the information that they were all flipped and only finitely many of them landed heads. We argue that if you should have any conditional credence at all, it should be ₂ .

A principle, according to which any scientific theory can be mathematized, is investigated. That theory is presupposed to be a consistent text, which can be exhaustedly represented by a certain mathematical structure constructively.

In this article, I survey some philosophical attitudes to talk concerning ‘the’ universe of sets. I separate out four different strands of the debate, namely: (i) Universism, (ii) Multiversism, (iii) Potentialism, and (iv) Pluralism. I discuss standard arguments and counterarguments concerning the positions and some of the natural mathematical programmes that are suggested by the various views.

Let H be a finite-dimensional complex Hilbert space and D the set of density matrices on H , i.e., the positive operators with trace 1. Our goal in this note is to identify a probability measure u on D that can be regarded as the uniform distribution over D. We propose a measure on D, argue that it can be so regarded, discuss its properties, and compute the joint distribution of the eigenvalues of a random density matrix distributed according to this measure.

One of the most expected properties of a logical system is that it can be algebraizable, in the sense that an algebraic counterpart of the deductive machinery could be found. When this happens, a lot of logical problems can be faithfully and conservatively translated into some given algebra, and then algebraic tools can be used to tackle them. This happens so naturally with the brotherhood between classical logic and Boolean algebra, that a similar relationship is expected to hold for non-standard logics as well. And indeed it holds for some, but not for all logics. In any case, the task of finding such an algebraic counterpart is far from trivial. The intuitive idea behind the search for algebraization for a given logic system, generalizing the pioneering proposal of Lindenbaum and Tarski, usually starts by trying to find a congruence on the set of formulas that could be used to produce a quotient algebra, defined over the algebra of formulas of the logic.

We critically review two extant paradigms for understanding the systematic interaction between modality and tense, as well as their respective modifications designed to do justice to the contingency of time’s structure and composition. We show that on either type of theory, as well as their respective modifications, some principles prove logically valid whose truth might sensibly be questioned on metaphysical grounds. These considerations lead us to devise a more general logical framework that allows accommodation of those metaphysical views that its predecessors rule out by fiat.

There are at least three vaguely atomistic principles that have come up in the literature, two explicitly and one implicitly. First, standard atomism is the claim that everything is composed of atoms, and is very often how atomism is characterized in the literature. Second, superatomism is the claim that parthood is well-founded, which implies that every proper parthood chain terminates, and has been discussed as a stronger alternative to standard atomism (Cotnoir 2013). Third, there is a principle that lies between these two theses in terms of its relative strength: strong atomism, the claim that every maximal proper parthood chain terminates. Although strong atomism is equivalent to superatomism in classical extensional mereology, it is strictly weaker than it in strictly weaker systems in which parthood is a partial order. And it is strictly stronger than standard atomism in classical extensional mereology and, given the axiom of choice, in such strictly weaker systems as well. Though strong atomism has not, to my knowledge, been explicitly identified, Shiver (2015) appears to have it in mind, though it is unclear whether he recognizes that it is not equivalent to standard atomism in each of the mereologies he considers. I prove these logical relationships which hold amongst these three atomistic principles, and argue that, whether one adopts classical extensional mereology or a system strictly weaker than it in which parthood is a partial order, standard atomism is a more defensible addition to one’s mereology than either of the other two principles, and it should be regarded as the best formulation of the atomistic thesis.