The idea that some objects are metaphysically “cheap” has wide appeal. An influential version of the idea builds on abstractionist views in the philosophy of mathematics, on which numbers and other mathematical objects are abstracted from other phenomena. For example, Hume's Principle states that two collections have the same number just in case they are equinumerous, in the sense that they can be correlated one‐to‐one:. The principal aim of this article is to use the notion of grounding to develop this (...) sort of abstractionism. The appeal to grounding enables a unified response to the two main challenges that confront abstractionism. First, we must explicate the metaphor of metaphysical “cheapness.” Second, we must rebut the “bad company” objection, which rejects abstraction principles like (HP) as tarnished by their similarity to inconsistent principles like Frege's Basic Law V. By enforcing a simple requirement that all abstraction be properly grounded, we propose a unified solution to these two hard, and prima facie unrelated, problems. On our view, grounded abstraction simultaneously ensures “cheap” abstracta and permissible abstraction. (shrink)

I provide a theory of the metaphysical foundations of identity: an account what grounds facts of the form a=b. In particular, I defend the claim that indiscernibility grounds identity. This is typically rejected because it is viciously circular; plausible assumptions about the logic of ground entail that the fact that a=b partially grounds itself. The theory I defend is immune to this circularity.

This paper develops a novel theory of abstraction—what we call collective abstraction. The theory solves a notorious problem for noneliminative structuralism. The noneliminative structuralist holds that in addition to various isomorphic systems there is a pure structure that can be abstracted from each of these systems; but existing accounts of abstraction fail for nonrigid systems like the complex numbers. The problem with the existing accounts is that they attempt to define a unique abstraction operation. The theory of collective abstraction instead (...) simultaneously defines a collection of distinct abstraction operations, each of which maps a system to its corresponding pure structure. The theory is precisely formulated in an essentialist language. This allows us to throw new light on the question to what extent structuralists are committed to symmetric dependence. Finally, we apply the theory of collective abstraction to solve a problem about converse relations. (shrink)

I systematically defend a novel account of the grounds for identity and distinctness facts: they are all uniquely zero‐grounded. First, this Null Account is shown to avoid a range of problems facing other accounts: a relation satisfying the Null Account would be an excellent candidate for being the identity relation. Second, a plenitudinist view of relations suggests that there is such a relation. To flesh out this plenitudinist view I sketch a novel framework for expressing real definitions, use this framework (...) to give a definition of identity, and show how the central features of the identity relation can be deduced from this definition. (shrink)

We discuss abstraction principles in the context of modal and temporal logic. It is argued that abstractionism conflicts with both serious presentism and serious actualism.

A brief review of Øystein Linnebo's Thin Objects. The review ends with a brief discussion of cardinal number and metaphysical ground.

Though the divide between reason-based and causal-explanatory approaches in psychiatry and psychopathology is old and deeply rooted, current trends involving multi-factorial explanatory models and evidence-based approaches to interpersonal psychotherapy, show that it has already been implicitly bridged. These trends require a philosophical reconsideration of how reasons can be causes. This paper contributes to that trajectory by arguing that Donald Davidson’s classic paradigm of 1963 is still a valid option.

I explore proposals for stating identity criteria in terms of ground. I also address considerations for and against taking identity and distinctness facts to be ungrounded.

Let NNG be the claim that necessitation is necessary for grounding, and let NSG be the claim that necessitation is sufficient for grounding. The consensus view is that grounding cannot be reduced to necessitation, and this is due to the (approximately) universally-accepted claim that NSG is false. Among deniers of NSG: grounding contingentists think NNG is also false, but they are in the minority compared to grounding necessitarians who uphold NNG. For one who would defend the claim that grounding is (...) reducible to necessitation, the task is formidable: she must defend NSG and NNG. I consider two prominent objections against NSG, and two more against NNG before developing a reductive account of grounding as minimal necessitation that avoids not only all four of the previously mentioned objections, but also an additional objection that targets minimal necessitation accounts in particular. If my arguments are compelling, then, insofar as we thereby have a strong prima facie case for thinking that grounding can be reduced to (minimal) necessitation, we have a strong prima facie case for thinking the consensus view is mistaken. (shrink)

Contingent negative existentials give rise to a notorious paradox. I formulate a version in terms of metaphysical grounding: nonexistence can't be fundamental, but nothing can ground it. I then argue for a new kind of solution, expanding on work by Kit Fine. The key idea is that negative existentials are contingently zero-grounded – that is to say, they are grounded, but not by anything, and only in the right conditions. If this is correct, it follows that grounding cannot be an (...) internal relation, and that no complete account of reality can be purely fundamental. (shrink)

This paper offers a metaphysical explanation of the identity and distinctness of concrete objects. It is tempting to try to distinguish concrete objects on the basis of their possessing different qualitative features, where qualitative features are ones that do not involve identity. Yet, this criterion for object identity faces counterexamples: distinct objects can share all of their qualitative features. This paper suggests that in order to distinguish concrete objects we need to look not only at which properties and relations objects (...) instantiate but also how they instantiate these properties and relations. I propose that objects are identical when they stand in certain qualitative relations in virtue of their existence. And concrete objects are distinct when they do not stand in the same kinds of relations to one another in virtue of their existence. (shrink)

Discussion of atomistic and monistic theses about abstract reality.

This is part two of a two-part paper in which we develop an axiomatic theory of the relation of partial ground. The main novelty of the paper is the of use of a binary ground predicate rather than an operator to formalize ground. In this part of the paper, we extend the base theory of the first part of the paper with hierarchically typed truth-predicates and principles about the interaction of partial ground and truth. We show that our theory is (...) a proof-theoretically conservative extension of the ramified theory of positive truth up to. (shrink)

Sometimes a fact can play a role in a grounding explanation, but the particular content of that fact make no difference to the explanation—any fact would do in its place. I call these facts vacuous grounds. I show that applying the distinction between-vacuous grounds allows us to give a principled solution to Kit Fine and Stephen Kramer’s paradox of ground. This paradox shows that on minimal assumptions about grounding and minimal assumptions about logic, we can show that grounding is reflexive, (...) contra the intuitive character of grounds. I argue that we should never have accepted that grounding is irreflexive in the first place; the intuitions that support the irreflexive intuition plausibly only require that grounding be non-vacuously irreflexive. Fine and Kramer’s paradox relies, essentially, on a case of vacuous grounding and is thus no problem for this account. (shrink)

This book concerns the foundations of epistemic modality and hyperintensionality and their applications to the philosophy of mathematics. David Elohim examines the nature of epistemic modality, when the modal operator is interpreted as concerning both apriority and conceivability, as well as states of knowledge and belief. The book demonstrates how epistemic modality and hyperintensionality relate to the computational theory of mind; metaphysical modality and hyperintensionality; the types of mathematical modality and hyperintensionality; to the epistemic status of large cardinal axioms, undecidable (...) propositions, and abstraction principles in the philosophy of mathematics; to the modal and hyperintensional profiles of the logic of rational intuition; and to the types of intention, when the latter is interpreted as a hyperintensional mental state. Chapter \textbf{2} argues for a novel type of expressivism based on the duality between the categories of coalgebras and algebras, and argues that the duality permits of the reconciliation between modal and hyperintensional cognitivism and modal and hyperintensional expressivism. Elohim develops a novel, topic-sensitive truthmaker semantics for dynamic epistemic logic, and develops a novel, dynamic two-dimensional semantics grounded in two-dimensional hyperintensional Turing machines. Chapter \textbf{3} provides an abstraction principle for two-dimensional (hyper-)intensions. Chapter \textbf{4} advances a topic-sensitive two-dimensional truthmaker semantics, and provides three novel interpretations of the framework along with the epistemic and metasemantic. Chapter \textbf{5} applies the fixed points of the modal $\mu$-calculus in order to account for the iteration of epistemic states in a single agent, by contrast to availing of modal axiom 4 (i.e. the KK principle). The fixed point operators in the modal $\mu$-calculus are rendered hyperintensional, which yields the first hyperintensional construal of the modal $\mu$-calculus in the literature and the first application of the calculus to the iteration of epistemic states in a single agent instead of the common knowledge of a group of agents. Chapter \textbf{6} advances a solution to the Julius Caesar problem based on Fine's `criterial' identity conditions which incorporate conditions on essentiality and grounding. Chapter \textbf{7} provides a ground-theoretic regimentation of the proposals in the metaphysics of consciousness and examines its bearing on the two-dimensional conceivability argument against physicalism. The topic-sensitive epistemic two-dimensional truthmaker semantics developed in chapters \textbf{2} and \textbf{4} is availed of in order for epistemic states to be a guide to metaphysical states in the hyperintensional setting. -/- Chapters \textbf{8-12} provide cases demonstrating how the two-dimensional hyperintensions of hyperintensional, i.e. topic-sensitive epistemic two-dimensional truthmaker, semantics, solve the access problem in the epistemology of mathematics. Chapter \textbf{8} examines the interaction between Elohim's hyperintensional semantics and the axioms of epistemic set theory, large cardinal axioms, the Epistemic Church-Turing Thesis, the modal axioms governing the modal profile of $\Omega$-logic, Orey sentences such as the Generalized Continuum Hypothesis, and absolute decidability. These results yield inter alia the first hyperintensional Epistemic Church-Turing Thesis and hyperintensional epistemic set theories in the literature. Chapter \textbf{9} examines the modal and hyperintensional commitments of abstractionism, in particular necessitism, and epistemic hyperintensionality, epistemic utility theory, and the epistemology of abstraction. Elohim countenances a hyperintensional semantics for novel epistemic abstractionist modalities. Elohim suggests, too, that higher observational type theory can be applied to first-order abstraction principles in order to make first-order abstraction principles recursively enumerable, i.e. Turing machine computable, and that the truth of the first-order abstraction principle for two-dimensional hyperintensions is grounded in its being possibly recursively enumerable and the machine being physically implementable. Chapter \textbf{10} examines the philosophical significance of hyperintensional $\Omega$-logic in set theory and discusses the hyperintensionality of metamathematics. Chapter \textbf{11} provides a modal logic for rational intuition and provides a hyperintensional semantics. Chapter \textbf{12} avails of modal coalgebras to interpret the defining properties of indefinite extensibility, and avails of hyperintensional epistemic two-dimensional semantics in order to account for the interaction between interpretational and objective modalities and the truthmakers thereof. This yields the first hyperintensional category theory in the literature. Elohim invents a new mathematical trick in which first-order structures are treated as categories, and Vopenka's principle can be satisfied because of the elementary embeddings between the categories and generate Vopenka cardinals in the category of Set in category theory. Chapter \textbf{13} examines modal responses to the alethic paradoxes. Elohim provides a counter-example to epistemic closure for logical deduction. Chapter \textbf{14} examines, finally, the modal and hyperintensional semantics for the different types of intention and the relation of the latter to evidential decision theory. (shrink)

This book concerns the foundations of epistemic modality and hyperintensionality and their applications to the philosophy of mathematics. David Elohim examines the nature of epistemic modality, when the modal operator is interpreted as concerning both apriority and conceivability, as well as states of knowledge and belief. The book demonstrates how epistemic modality and hyperintensionality relate to the computational theory of mind; metaphysical modality and hyperintensionality; the types of mathematical modality and hyperintensionality; to the epistemic status of large cardinal axioms, undecidable (...) propositions, and abstraction principles in the philosophy of mathematics; to the modal and hyperintensional profiles of the logic of rational intuition; and to the types of intention, when the latter is interpreted as a hyperintensional mental state. Chapter \textbf{2} argues for a novel type of expressivism based on the duality between the categories of coalgebras and algebras, and argues that the duality permits of the reconciliation between modal and hyperintensional cognitivism and modal and hyperintensional expressivism. Elohim develops a novel, topic-sensitive truthmaker semantics for dynamic epistemic logic, and develops a novel, dynamic two-dimensional semantics grounded in two-dimensional hyperintensional Turing machines. Chapter \textbf{3} provides an abstraction principle for two-dimensional (hyper-)intensions. Chapter \textbf{4} advances a topic-sensitive two-dimensional truthmaker semantics, and provides three novel interpretations of the framework along with the epistemic and metasemantic. Chapter \textbf{5} applies the fixed points of the modal $\mu$-calculus in order to account for the iteration of epistemic states in a single agent, by contrast to availing of modal axiom 4 (i.e. the KK principle). The fixed point operators in the modal $\mu$-calculus are rendered hyperintensional, which yields the first hyperintensional construal of the modal $\mu$-calculus in the literature and the first application of the calculus to the iteration of epistemic states in a single agent instead of the common knowledge of a group of agents. Chapter \textbf{6} advances a solution to the Julius Caesar problem based on Fine's `criterial' identity conditions which incorporate conditions on essentiality and grounding. Chapter \textbf{7} provides a ground-theoretic regimentation of the proposals in the metaphysics of consciousness and examines its bearing on the two-dimensional conceivability argument against physicalism. The topic-sensitive epistemic two-dimensional truthmaker semantics developed in chapters \textbf{2} and \textbf{4} is availed of in order for epistemic states to be a guide to metaphysical states in the hyperintensional setting. -/- Chapters \textbf{8-12} provide cases demonstrating how the two-dimensional hyperintensions of hyperintensional, i.e. topic-sensitive epistemic two-dimensional truthmaker, semantics, solve the access problem in the epistemology of mathematics. Chapter \textbf{8} examines the interaction between Elohim's hyperintensional semantics and the axioms of epistemic set theory, large cardinal axioms, the Epistemic Church-Turing Thesis, the modal axioms governing the modal profile of $\Omega$-logic, Orey sentences such as the Generalized Continuum Hypothesis, and absolute decidability. These results yield inter alia the first hyperintensional Epistemic Church-Turing Thesis and hyperintensional epistemic set theories in the literature. Chapter \textbf{9} examines the modal and hyperintensional commitments of abstractionism, in particular necessitism, and epistemic hyperintensionality, epistemic utility theory, and the epistemology of abstraction. Elohim countenances a hyperintensional semantics for novel epistemic abstractionist modalities. Elohim suggests, too, that higher observational type theory can be applied to first-order abstraction principles in order to make first-order abstraction principles recursively enumerable, i.e. Turing machine computable, and that the truth of the first-order abstraction principle for two-dimensional hyperintensions is grounded in its being possibly recursively enumerable and the machine being physically implementable. Chapter \textbf{10} examines the philosophical significance of hyperintensional $\Omega$-logic in set theory and discusses the hyperintensionality of metamathematics. Chapter \textbf{11} provides a modal logic for rational intuition and provides a hyperintensional semantics. Chapter \textbf{12} avails of modal coalgebras to interpret the defining properties of indefinite extensibility, and avails of hyperintensional epistemic two-dimensional semantics in order to account for the interaction between interpretational and objective modalities and the truthmakers thereof. This yields the first hyperintensional category theory in the literature. Elohim invents a new mathematical trick in which first-order structures are treated as categories, and Vopenka's principle can be satisfied because of the elementary embeddings between the categories and generate Vopenka cardinals in the category of Set in category theory. Chapter \textbf{13} examines modal responses to the alethic paradoxes. Elohim provides a counter-example to epistemic closure for logical deduction. Chapter \textbf{14} examines, finally, the modal and hyperintensional semantics for the different types of intention and the relation of the latter to evidential decision theory. (shrink)

Identity and distinctness facts are ones like “The Eiffel Tower is identical to the Eiffel Tower,” and “The Eiffel Tower is distinct from the Louvre.” This paper concerns one question in the metaphysics of identity: Are identity and distinctness facts metaphysically fundamental or are they nonfundamental? I provide an overview of answers to this question.

A fundamental entity is an entity that is ‘ontologically independent’; it does not depend on anything else for its existence or essence. It seems to follow that a fundamental entity is ‘modally free’ in some sense. This assumption, that fundamentality entails modal freedom (or ‘FEMF’ as I shall label the thesis), is used in the service of other arguments in metaphysics. But as I will argue, the road from fundamentality to modal freedom is not so straightforward. The defender of FEMF (...) should provide positive reasons for believing it, especially in light of recent views that are incompatible with it. I examine both direct and indirect routes to FEMF. (shrink)

In this paper, we try to establish that some mathematical theories, like K-theory, homology, cohomology, homotopy theories, spectral sequences, modern Galois theory (in its various applications), representation theory and character theory, etc., should be thought of as (abstract) machines in the same way that there are (concrete) machines in the natural sciences. If this is correct, then many epistemological and ontological issues in the philosophy of mathematics are seen in a different light. We concentrate on one problem which immediately follows (...) the recognition of the particular status of these theories: the demarcation problem between ‘natural kinds’ and ‘artefacts’. (shrink)

Thin Objects has two overarching ambitions. The first is to clarify and defend the idea that some objects are ‘thin’, in the sense that their existence does not make a substantive demand on reality. The second is to develop a systematic and well-motivated account of permissible abstraction, thereby solving the so-called ‘bad company problem’. Here I synthesise the book by briefly commenting on what I regard as its central themes.

Minimalism and Trivialism are two recent forms of lightweight Platonism in the philosophy of mathematics: Minimalism is the view that mathematical objects arethinin the sense that “very little is required for their existence”, whereas Trivialism is the view that mathematical statements have trivial truth‐conditions, that is, that “nothing is required of the world in order for those conditions to be satisfied”. In order to clarify the relation between the mathematical and the non‐mathematical domain that these views envisage, it has recently (...) been proposed that both Linnebo's notion of sufficiency and Rayo's “just is”‐operator can, or even should, be interpreted in terms of metaphysicalgrounding. This interpretation makes Minimalism and Trivialism akin to Aristotelianism in the philosophy of mathematics, according to which mathematical entities depend for their existence and their properties on non‐mathematical ones. In this paper we raise a general objection to this interpretation. We highlight a tension – a “Big Picture” Problem – between the metaphysical picture underlying both Minimalism and Trivialism, on the one side, and the metaphysics of Platonism and Aristotelianism on the other. We then consider various ways in which Linnebo and Rayo could respond. We finally argue that Minimalism and Trivialism are closer to Aristotelianism than to Platonism; however, even though these positions are not forms of traditional Platonism, they are not standard forms of Aristotelianism either. (shrink)

Seemingly natural principles about the logic of ground generate cycles of ground; how can this be if ground is asymmetric? The goal of the theory of decycling is to find systematic and principled ways of getting rid of such cycles of ground. In this paper—drawing on graph-theoretic and topological ideas—I develop a general framework in which various theories of decycling can be compared. This allows us to improve on proposals made earlier by Fine and Litland. However, it turns out that (...) there is no unique method of decycling. An important upshot is that the notion of asymmetric ground may be indeterminate. (shrink)

This essay aims to redress the contention that epistemic possibility cannot be a guide to the principles of modal metaphysics. I introduce a novel epistemic two-dimensional truthmaker semantics. I argue that the interaction between the two-dimensional framework and the mereological parthood relation, which is super-rigid, enables epistemic possibilities and truthmakers with regard to parthood to be a guide to its metaphysical profile. I specify, further, a two-dimensional formula encoding the relation between the epistemic possibility and verification of essential properties obtaining (...) and their metaphysical possibility or verification. I then generalize the approach to haecceitistic properties, and examine the Julius Caesar problem as a test case. (shrink)

One conception of the Principle of Sufficient Reason maintains that every fact is metaphysically explained. There are different ways to challenge this version of the PSR; one type of challenge involves pinpointing a specific set of facts that resist metaphysical explanation. Certain identity and distinctness facts seem to constitute such a set. For example, we can imagine a scenario in which we have two qualitatively identical spheres, Castor and Pollux. Castor is distinct from Pollux but it is unclear what could (...) metaphysically explain this distinctness fact. In this paper, I argue that we should not treat identity and distinctness facts as metaphysically fundamental. As such, identity and distinctness facts do not challenge the PSR. We can metaphysically explain them. (shrink)

Abstraction principles and grounding can be combined in a natural way Modality: metaphysics, logic, and epistemology, Oxford University Press, Oxford, pp 109–136, 2010; Schwartzkopff in Grazer philosophische studien 82:353–373, 2011). However, some ground-theoretic abstraction principles entail that there are circles of partial ground :775–801, 2017). I call this problem auto-abstraction. In this paper I sketch a solution. Sections 1 and 2 are introductory. In Sect. 3 I start comparing different solutions to the problem. In Sect. 4 I contend that the (...) thesis that the right-hand side of an abstraction principle is prior to its left-hand side motivates an independence constraint, and that this constraint leads to predicative restrictions on the acceptable instances of ground-theoretic abstraction principles. In Sect. 5 I argue that auto-abstraction is acceptable unless the left-hand side is essentially grounded by the right-hand side. In Sect. 6 I highlight several parallelisms between auto-abstraction and the puzzles of ground. I finally compare my solution with the strategies listed in Sect. 3. (shrink)

There is a puzzle concerning the essences of fundamental entities that arises from considerations about essence, on one hand, and fundamentality, on the other. The Essence-Dependence Link (EDL) says that if x figures in the essence of y, then y is dependent upon x. EDL is prima facie plausible in many cases, especially those involving derivative entities. But consider the property negative charge. A negatively charged object exhibits certain behaviors that a positively charged object does not: it moves away from (...) other negatively charged objects, towards positively charged objects, etc. It is commonly thought that negative charge disposes its bearer to move away from other negatively charged objects, towards positively charged objects, etc. But if negative charge is fundamental, then no other entities—including the property positive charge—can figure in its essence. We thus have a prima facie puzzle: How can we say anything interesting about the essences of fundamental entities without running afoul of EDL? In this paper, I present and discuss the consequences of EDL for the debate between causal essentialists and quidditists about properties, and propose solutions to the puzzle. (shrink)

An argument going back to Russell shows that the view that propositions are structured is inconsistent in standard type theories. Here, it is shown that such type theories may nevertheless provide entities which can serve as proxies for structured propositions. As an illustration, such proxies are applied to the case of grounding, as standard views of grounding require a degree of propositional structure which suffices for a version of Russell’s argument. While this application solves some of the problems grounding faces, (...) it introduces problematic limitations: it becomes impossible to quantify unrestrictedly over the relata of ground. The proposed proxies may thus not save grounding, but they shed light on what exactly Russell’s argument does and does not show. (shrink)

The existence of mereological sums can be derived from an abstraction principle in a way analogous to numbers. I draw lessons for the thesis that “composition is innocent” from neo-Fregeanism in the philosophy of mathematics.

Platonism about mathematics (or mathematical platonism) isthe metaphysical view that there are abstract mathematical objectswhose existence is independent of us and our language, thought, andpractices. Just as electrons and planets exist independently of us, sodo numbers and sets. And just as statements about electrons and planetsare made true or false by the objects with which they are concerned andthese objects' perfectly objective properties, so are statements aboutnumbers and sets. Mathematical truths are therefore discovered, notinvented., Existence. There are mathematical objects.

Hume’s Principle states that the cardinal number of the concept F is identical with the cardinal number of G if and only if F and G can be put into one-to-one correspondence. The Schwartzkopff-Rosen Principle is a modification of HP in terms of metaphysical grounding: it states that if the number of F is identical with the number of G, then this identity is grounded by the fact that F and G can be paired one-to-one, 353–373, 2011, 362). HP is (...) central to the neo-logicist program in the philosophy of mathematics ; in this paper we submit that, even if the neo-logicists wish to venture into the metaphysics of grounding, they can avoid the SR Principle. In Section 1 we introduce neo-logicism. In Sections 2 and 3 we examine the SR Principle. We then formulate an account of arithmetical facts which does not rest on the SR Principle; we finally argue that the neo-logicists should avoid the SR Principle in favour of this alternative proposal. (shrink)

In Thin objects: an abstractionist account (Oxford University Press, 2018), Øystein Linnebo claims that ‘mathematical objects are thin in the sense that very little is required for their existence’. Linnebo articulates his view in an abstractionist manner: according to Linnebo, the truth of the right‐hand side of a Fregean abstraction principle, which states that two items stand in a given equivalence relation, is sufficient for the truth of its left‐hand side, which states that the same abstract object is associated to (...) both items. This special issue contains nine articles discussing different aspects of Linnebo's book. In this introduction, we provide an opinionated introduction to the Special Issue and an overview of the contributions. (shrink)

This paper sets out a new framework for discussing a long-standing problem in the philosophy of mathematics, namely the connection between the physical world and a mathematical domain when the mathematics is applied in science. I argue that considering counterfactual situations raises some interesting challenges for some approaches to applications, and consider an approach that avoids these challenges.