While knowledge of mere possibilities is difficult to understand, knowledge of possibilities that are actual seems unproblematic (as far as we know the actual world). The principle that what is actual is possible has been near-universally accepted. After summarizing some sporadic dissent, I present a proposal for how the validity of the principle might be restricted. While the principle certainly holds for sufficiently inclusive objective and epistemic possibilities, it may not hold when the accessibility of possibilities is contextually restricted.

My aim in this paper is to sketch, with a broad brush and in bare outlines, an approach to modal epistemology that is characterized by three distinctive features. First, the approach is agency-based: it locates the roots of our modal thought and knowledge in our experience of our own agency. Second, the approach is ambitious in that it takes the experience of certain modal properties in agency to be the sole distinctive feature of specifically modal thought and knowledge; everything that (...) we know about modality beyond the experience of agency is a matter of applying standard methods of inquiry such as deduction, induction and abductive methods for choosing between theories. Third, the account holds that modal thought and knowledge is, first and foremost, about modal properties that are sufficiently like those encountered in agency. (shrink)

The motivating question of this paper is: ‘How are our beliefs in the theorems of mathematics justified?’ This is distinguished from the question ‘How are our mathematical beliefs reliably true?’ We examine an influential answer, outlined by Russell, championed by Gödel, and developed by those searching for new axioms to settle undecidables, that our mathematical beliefs are justified by ‘intuitions’, as our scientific beliefs are justified by observations. On this view, axioms are analogous to laws of nature. They are postulated (...) to best systematize the data to be explained. We argue that there is a decisive difference between the cases. There is agreement on the data to be systematized in the scientific case that has no analog in the mathematical one. There is virtual consensus over observations, but conspicuous dispute over intuitions. In this respect, mathematics more closely resembles stereotypical philosophy. We conclude by distinguishing two ideas that have long been associated -- realism (the idea that there is an independent reality) and objectivity (the idea that in a disagreement, only one of us can be right). We argue that, while realism is true of mathematics and philosophy, these domains fail to be objective. One upshot of the discussion is that even questions of fundamental physics may fail to be objective in roughly the sense that the question, ‘Is the Parallel Postulate is true?’, understood as one of pure mathematics, fails to be. Another is a kind of pragmatism. Factual questions in mathematics, modality, logic, and evaluative areas go proxy for non-factual practical ones. -/- . (shrink)

Some have argued for a division of epistemic labor in which mathematicians supply truths and philosophers supply their necessity. We argue that this is wrong: mathematics is committed to its own necessity. Counterfactuals play a starring role.

Some have argued for a division of epistemic labor in which mathematicians supply truths and philosophers supply their necessity. We argue that this is wrong: mathematics is committed to its own necessity. Counterfactuals play a starring role.

[This paper is available as open access from the publisher.]The conventional wisdom in ethics is that pure moral laws are at least metaphysically necessary. By contrast, Moral Contingentism holds that pure moral laws are metaphysically contingent. This paper raises a normative objection to Moral Contingentism: it is worse equipped than Moral Necessitarianism to account for the normative standing or authority of the pure moral laws to govern the lives of the agents to whom they apply. Since morality is widely taken (...) to have such a standing, failing to account for it would be a significant problem. The objection also shows that the debate about the modal status of moral principles isn’t a debate solely within modal metaphysics, but has implications for topics in moral philosophy. (shrink)

The consensus among spacetime substantivalists is to respond to Leibniz's classic shift arguments, and their contemporary incarnation in the form of the hole argument, by pruning the allegedly problematic metaphysical possibilities that generate these arguments. Some substantivalists do so by directly appealing to a modal doctrine akin to anti-haecceitism. Other substantivalists do so by appealing to an underlying hyperintensional doctrine that implies some such modal doctrine. My first aim in this paper is to pose a challenge for all extant forms (...) of this consensus position. My second aim is to show what form substantivalism must take in order to uphold the consensus while addressing this challenge. The result is a novel "plenitudinous" substantivalist view, which predicts that certain modal facts about spacetime are vague or indeterminate. I then argue against this view on independent grounds, concluding that substantivalists should reject the consensus position. The paper also discusses the way forward for substantivalists in light of this conclusion. (shrink)

Are the categorical laws of ontology metaphysically contingent? I do not intend to give a full answer to this question in this paper. But I shall give a partial answer to it. In particular, Gideon Rosen, in his article “The Limits of Contingency”, has distinguished a certain conception of metaphysical necessity, which he calls the Non-Standard conception, which, together with the assumption that all natures or essences are Kantian, is supposed to entail that many laws of ontology are metaphysically contingent. (...) Now, the argument Rosen gives supports the conclusion that all categorical laws of ontology are contingent. I shall argue that the Non-Standard conception and the thesis that all natures are Kantian are incompatible with each other and that, if the Non-Standard conception is true, there must be at least one metaphysically necessary categorical law of ontology, and I shall identify such a law. Thus my contribution to the question of the title of the paper will be that not all categorical ontological laws can be contingent if the Non-Standard conception is true. (shrink)

We explore a ‘Neo-Cartesian’ account of divine ultimacy that raises the concept of God to its ultimate level of abstraction so that we can do away with even the question of his existence. Our starting point is God’s relation to the logical and metaphysical order of reality and the views of Descartes and Leibniz on this topic. While Descartes held the seemingly bizarre view that the eternal truths are freely created by God, Leibniz stands for the mainstream view that the (...) eternal truths are grounded in God’s nature. We argue that the implausibility of Descartes’ doctrine stems mainly from the assumption that there is a non-epistemic notion of absolute necessity that constitutes the ultimate court of appeal for all modal questions and that this assumption is questionable. We also question the assumption that God’s ultimacy merely requires that all reality be grounded in God in the sense of mere explanation, so that it suffices if the necessary truths are grounded in God’s nature but not in God’s will. This will lead us to a reassessment of Descartes’ position. In the final and main part of the paper, we push Descartes’ doctrine of the creation of the eternal truths to its ‘logical’ conclusion with the aim of getting to a novel conception of ‘God.’. (shrink)

In this article, I show that (i) from what I call a “Kripkean” account of the relations between conceivability and metaphysical necessities, (ii) an apparently plausible principle relating conceivability and epistemic modality, and (iii) the duality of epistemic modalities, one can show the utterly anti-Kripkean result that every metaphysical necessity is an epistemic necessity. My aim is to present and diagnose the problem and evaluate the costs of some possible Kripkean reactions. In particular, I will evaluate the consequences and theoretical (...) costs of rejecting the main ingredients of the argument, namely that we cannot genuinely conceive the negations of metaphysical necessities, that there is no postulated relation between conceivability and epistemic possibility (actually, between unconceivability and epistemic impossibility), and that epistemic possibility and necessity are not dualities. (shrink)

The philosophical problem that stems from the successful application of mathematics in the empirical sciences has recently attracted growing interest within philosophers of mathematics and philosophers of science. Nevertheless, little attention has been devoted to the converse applicability issue of how physical considerations find successful application in mathematics. In this article, focusing on some case studies, I address the latter issue and argue that some successful applications of physics to mathematics essentially depend on the use of conservation principles. I conclude (...) with a short discussion of how the successful interplay between conservation principles and mathematics can be accounted for. (shrink)

In this paper I present two cases, taken from the history of science, in which mathematics and physics successfully interplay. These cases provide, respectively, an example of the successful application of mathematics in astronomy and an example of the successful application of mechanics in mathematics. I claim that an illustration of these cases has a twofold value in the context of the applicability debate. First, it enriches the debate with an historical perspective which is largely omitted in the contemporary discussion. (...) Second, it reveals a component of the applicability problem that has received little attention. This component concerns the successful application of physical principles in mathematical practice. With the help of the two examples, in the final part of the paper I address the following question: are successful applications of mathematics to physics and successful applications of physics to mathematics two sides of the same problem? (shrink)

Talk of metaphysical modality as “absolute” is ambiguous, as it appears to convey multiple ideas. Metaphysical possibility is supposedly completely unrestricted or unqualified; metaphysical necessity is unconditional and exceptionless. Moreover, metaphysical modality is thought to be absolute in the sense that it’s real or genuine and the most objective modality: metaphysical possibility and necessity capture ways things could and must have really been. As we disentangle these ideas, certain talk of metaphysical modality qua “absolute” turns out to be misguided. Metaphysical (...) possibility isn't completely unrestricted or most inclusive compared to the other modalities; metaphysical necessity, like all kinds of necessities, is relative to or conditional upon a specific framework of reference. Still, metaphysical modality captures how things could and must have really been most generally because it deals with reality and the nature of things or their essence. That’s the chief interest of metaphysics. Arguments against the alleged absoluteness of metaphysical modality may not thereby undermine its philosophical significance. (shrink)

I propose that we approach the epistemology of modality by putting modal metaphysics first and, specifically, by investigating the metaphysics of essence. Following a prominent Neo-Aristotelian view, I hold that metaphysical necessity depends on the nature of things, namely their essences. I further clarify that essences are core properties having distinctive superexplanatory powers. In the case of natural kinds, which is my focus in the paper, superexplanatoriness is due to the fact that the essence of a kind is what causes (...) all the many properties and behaviors that are typically shared by all the instances of the kind. Accordingly, we know what is necessarily true of kinds by knowing what is essential to them in the sense of actually playing such causal-explanatory roles. Modal reasoning aimed at discovering metaphysical necessity thus proceeds via essentialist deduction: we move from essentialist truths to reach necessary truths. (shrink)

This is the second part of a two-part series on the logic of hyperlogic, a formal system for regimenting metalogical claims in the object language (even within embedded environments). Part A provided a minimal logic for hyperlogic that is sound and complete over the class of all models. In this part, we extend these completeness results to stronger logics that are sound and complete over restricted classes of models. We also investigate the logic of hyperlogic when the language is enriched (...) with hyperintensional operators such as counterfactual conditionals and belief operators. (shrink)

Scanlon’s Being Realistic about Reasons (BRR) is a beautiful book – sleek, sophisticated, and programmatic. One of its key aims is to demystify knowledge of normative and mathematical truths. In this article, I develop an epistemological problem that Scanlon fails to explicitly address. I argue that his “metaphysical pluralism” can be understood as a response to that problem. However, it resolves the problem only if it undercuts the objectivity of normative and mathematical inquiry.

Recent epistemology of modality has seen a growing trend towards metaphysics-first approaches. Contrastingly, this paper offers a more philosophically modest account of justifying modal claims, focusing on the practices of scientific modal inferences. Two ways of making such inferences are identified and analyzed: actualist-manipulationist modality and relative modality. In AM, what is observed to be or not to be the case in actuality or under manipulations, allows us to make modal inferences. AM-based inferences are fallible, but the same holds for (...) practically all empirical inquiry. In RM, modal inferences are evaluated relative to what is kept fixed in a system, like a theory or a model. RM-based inferences are more certain but framework-dependent. While elements from both AM and RM can be found in some existing accounts of modality, it is worth highlighting them in their own right and isolating their features for closer scrutiny. This helps to establish their relevant epistemologies that are free from some strong philosophical assumptions often attached to them in the literature. We close by showing how combining these two routes amounts to a view that accounts for a rich variety of modal inferences in science. (shrink)

What is the relation between metaphysical necessity and essence? This paper defends the view that the relation is one of identity: metaphysical necessity is a special case of essence. My argument consists in showing that the best joint theory of essence and metaphysical necessity is one in which metaphysical necessity is just a special case of essence. The argument is made against the backdrop of a novel, higher-order logic of essence, whose core features are introduced in the first part of (...) the paper. The second part investigates the relation between metaphysical necessity and essence in the context of HLE. Reductive hypotheses are among the most natural hypotheses to be explored in the context of HLE. But they also have to be weighed against their non-reductive rivals. I investigate three different reductive hypotheses and argue that two of them fare better than their non-reductive rivals: they are simpler, more natural, and more systematic. Specifically, I argue that one candidate reduction, according to which metaphysical necessity is truth in virtue of the nature of all propositions, is superior to the others, including one proposed by Kit Fine, according to which metaphysical necessity is truth in virtue of the nature of all objects. The paper concludes by offering some reasons to think that the best joint theory of essence and metaphysical necessity is one in which the logic of metaphysical necessity includes S4, but not S5. (shrink)

Modal rationalism is the claim that for all p, if it is ideally conceivable that p, then there is a metaphysically possible world, W, in which p is true. This will be true just if there are no strong a posteriori necessities, where a strong necessity (for short) is a proposition that is conceivably false, but which is true in all metaphysically possible worlds. But are there any strong necessities? Various alleged examples have been proposed and argued over in the (...) literature, but there is no consensus on whether any is genuine. In this paper, I aim to move the debate forwards by proving that there are in fact no strong necessities. I argue that there are no strong necessities because they are impossible; they are impossible because the very notion of a strong necessity is – despite prima facie appearances – ultimately incoherent. Thus, I argue, it is an a priori truth that there are no strong necessities, and that modal rationalism itself is true. (shrink)

Gideon Rosen, Brian Leiter, and Catarina Dutilh Novaes raise deep questions about the arguments in Morality and Mathematics (M&M). Their objections bear on practical deliberation, the formulation of mathematical pluralism, the problem of universals, the argument from moral disagreement, moral ‘perception’, the contingency of our mathematical practices, and the purpose of proof. In this response, I address their objections, and the broader issues that they raise.

In this article, we discuss a simple argument that modal metaphysics is misconceived, and responses to it. Unlike Quine's, this argument begins with the simple observation that there are different candidate interpretations of the predicate ‘could have been the case’. This is analogous to the observation that there are different candidate interpretations of the predicate ‘is a member of’. The argument then infers that the search for metaphysical necessities is misguided in much the way the ‘set-theoretic pluralist’ claims that the (...) search for the true axioms of set theory is. We show that the obvious responses to this argument fail. However, a new response has emerged that purports to prove, from higher-order logical principles, that metaphysical possibility is the broadest kind of possibility applying to propositions, and is to that extent special. We distill two lines of reasoning from the literature, and argue that their import depends on premises that any ‘modal pluralist’ should deny. Both presuppose that there is a unique typed hierarchy, which is what the modal pluralist, in the context of higher-order logic, should disavow. In other words, both presupposes that there is a unique candidate for what higher-order claims could mean. We consider the worry that, in a higher-order setting, modal pluralism faces an insuperable problem of articulation, collapses into modal monism, is vulnerable to the Russell-Myhill paradox, or even contravenes the truism that there is a unique actual world, and argue that these worries are misplaced. We also sketch the bearing of the resulting ‘Higher-Order Pluralism’ on the theory of content. One theme of the discussion is that, if Higher-Order Pluralism is correct, then there is no fixed metatheory from which to characterize higher-order reality. (shrink)

It is widely alleged that metaphysical possibility is “absolute” possibility Conceivability and possibility, Clarendon, Oxford, 2002, p 16; Stalnaker, in: Stalnaker Ways a world might be: metaphysical and anti-metaphysical essays, Oxford University Press, Oxford, 2003, pp 201–215; Williamson in Can J Philos 46:453–492, 2016). Kripke calls metaphysical necessity “necessity in the highest degree”. Van Inwagen claims that if P is metaphysically possible, then it is possible “tout court. Possible simpliciter. Possible period…. possib without qualification.” And Stalnaker writes, “we can agree (...) with Frank Jackson, David Chalmers, Saul Kripke, David Lewis, and most others who allow themselves to talk about possible worlds at all, that metaphysical necessity is necessity in the widest sense.” What exactly does the thesis that metaphysical possibility is absolute amount to? Is it true? In this article, I argue that, assuming that the thesis is not merely terminological, and lacking in any metaphysical interest, it is an article of faith. I conclude with the suggestion that metaphysical possibility may lack the metaphysical significance that is widely attributed to it. (shrink)

Platonists affirm the existence of abstract mathematical objects, and Nominalists deny the existence of abstract mathematical objects. While there are standard arguments in favor of Nominalism, these arguments fail to account for the necessity of Nominalism. Furthermore, these arguments do nothing to explain why Nominalism is true. They only point to certain theoretical vices that might befall the Platonist. The goal of this paper is to formulate and defend a simple, valid argument for the necessity of Nominalism that seeks to (...) precisify the widespread intuition that mathematical objects are somehow ‘spooky’ or ‘mysterious’. (shrink)

In many different ontological debates, anti-arbitrariness considerations push one towards two opposing extremes. For example, in debates about mereology, one may be pushed towards a maximal ontology (mereological universalism) or a minimal ontology (mereological nihilism), because any intermediate view seems objectionably arbitrary. However, it is usually thought that anti-arbitrariness considerations on their own cannot decide between these maximal or minimal views. I will argue that this is a mistake. Anti-arbitrariness arguments may be used to motivate a certain popular thesis in (...) the philosophy of mathematics that rules out the maximalist view in many different ontological debates. (shrink)

Inspired by Cantor's Theorem (CT), orthodoxy takes infinities to come in different sizes. The orthodox view has had enormous influence in mathematics, philosophy, and science. We will defend the contrary view---Countablism---according to which, necessarily, every infinite collection (set or plurality) is countable. We first argue that the potentialist or modal strategy for treating Russell's Paradox, first proposed by Parsons (2000) and developed by Linnebo (2010, 2013) and Linnebo and Shapiro (2019), should also be applied to CT, in a way that (...) vindicates Countabilism. Our discussion dovetails with recent independently developed treatments of CT in Meadows (2015), Pruss (2020), and Scambler (2021), aimed at establishing the mathematical viability, and therefore epistemic possibility, of Countabilism. Unlike these authors, our goal isn't to vindicate the mathematical underpinnings of Countabilism. Rather, we aim to argue that, given that Countabilism is mathematically viable, Countabilism should moreover be regarded as true. After clarifying the modal content of Countabilism, we canvas some of Countabilism's many positive implications, including that Countabilism provides the best account of the pervasive independence phenomena in set theory, and that Countabilism has the power to defuse several persistent puzzles and paradoxes found in physics and metaphysics. We conclude that in light of its theoretical and explanatory advantages, Countabilism is more likely true than not. (shrink)

There is an old meta-philosophical worry: very roughly, metaphysical theories have no observational consequences and so the study of metaphysics has no value. The worry has been around in some form since the rise of logical positivism in the early twentieth century but has seen a bit of a renaissance recently. In this paper, I provide an apology for metaphysics in the face of this kind of concern. The core of the argument is this: pure mathematics detaches from science in (...) much the same manner as metaphysics and yet it is valuable nonetheless. The source of value enjoyed by pure mathematics extends to metaphysics as well. Accordingly, if one denies that metaphysics has value, then one is forced to deny that pure mathematics has value. The argument places an added burden on the sceptic of metaphysics. If one truly believes that metaphysics is worthless (as some philosophers do), then one must give up on pure mathematics as well. (shrink)

We develop a theory of necessity operators within a version of higher-order logic that is neutral about how fine-grained reality is. The theory is axiomatized in terms of the primitive of *being a necessity*, and we show how the central notions in the philosophy of modality can be recovered from it. Various questions are formulated and settled within the framework, including questions about the ordering of necessities under strength, the existence of broadest necessities satisfying various logical conditions, and questions about (...) their logical behaviour. We also wield the framework to probe the conditions under which a logicist account of necessities is possible, in which the theory is completely reducible to logic. (shrink)