This paper criticizes Soames’s main argument against a variant of two-dimensionalism that he calls strong two-dimensionalism. The idea of Soames’s argument is to show that the strong two-dimensionalist’s semantics for belief ascriptions delivers wrong semantic verdicts about certain complex modal sentences that contain both such ascriptions and claims about the truth of the ascribed beliefs. A closer look at the formal semantics underlying strong two-dimensionalism reveals that there are two feasible ways of specifying the truth conditions for claims of the (...) latter sort. Only one of the two yields the problematic semantic verdicts, so strong two-dimensionalists can avoid Soames’s argument by settling for the other way. (shrink)

This paper argues against Logical Realism, in particular against the view that there are facts of matters of logic that obtain independently of us, our linguistic conventions and inferential practices. The paper challenges logical realists to provide a non-intuition based epistemology, one which would be compatible with the empiricist and naturalist convictions motivating much recent anti-realist philosophy of mathematics.

This paper defends three theses: that presentism is either trivial or untenable; that the debate between tensed and tenseless theories of time is not about the status of presentism; and that there is no temporal analogue of the modal thesis of actualism.

This article surveys some of the key issues that arise when one tries to use tense logic as a metaphysical theory of the nature of time. Topics discussed include basic tense logic, tense logic and verb tense, the structure of the time series, instants of time, quantified tense logic, and the expressive resources of tense logic.

Some theories of time entail that the present can change before or after it has happened. Examples include views on which time-travelers can change the past, the glowing block theory, Peter Geach’s mutable future view, and the moving spotlight theory. This paper argues that such ante factum or posthumous change requires a heterodox “split time” view on which earlier-than is not the converse of later-than.

To define new property terms, we combine already familiar ones by means of certain logical operations. Given suitable constraints, these operations may presumably include the resources of first-order logic: truth-functional sentence connectives and quantification over objects. What is far less clear is whether we can also use modal operators for this purpose. This paper clarifies what is involved in this question, and argues in favor of modal property definitions.

I present and defend an “indexical” version of the Fine-Tuning Argument. I begin by outlining the dialectic between the Fine-Tuning Argument, the Multiverse Objection, and the This-Universe Reply. Next, I sketch an indexical fine-tuning argument and defend it from two new objections. Then, I show that such an argument is immune to the Multiverse Objection. I explain how a further augmentation to the argument allows it to avoid an objection I call the “Indifference Objection.” I conclude that my indexical version (...) of the Fine-Tuning Argument is no less cogent than the standard version, and yet it is immune to the Multiverse Objection. (shrink)

In earlier work we proposed an account of information grounded in counterfactual conditionals rather than probabilities, and argued that it might serve philosophical needs that more familiar probabilistic alternatives do not. Demir [2008] and Scarantino [2008] criticize the counterfactual approach by contending that its alleged advantages are illusory and that it fails to secure attractive desiderata. In this paper we defend the counterfactual account from these criticisms, and suggest that it remains a useful account of information.

According to one account, vagueness is "metaphysical." The friend of metaphysical vagueness believes that, for some object and some property, there can be no determinate fact of the matter whether that object exemplifies that property. A second account maintains that vagueness is due only to ignorance. According to the epistemic account, vagueness is explained completely by and is nothing over and above our not knowing some relevant fact or facts. These are the minority views. The dominant position maintains that there (...) is a third possible variety of vagueness, linguistic vagueness. And, it goes on to insist, all vagueness is of this third variety. I shall argue, however, that linguistic vagueness is not a third variety of vagueness. Either it is a species of metaphysical vagueness or a kind of ignorance. And this, I argue, makes trouble for the claim that all vagueness is linguistic. (shrink)

‘Restricted composition’ says that there are some composite objects. And it says that some objects jointly compose nothing at all. The main threat to restricted composition is the influential and widely defended Vagueness Argument. We shall see that the Vagueness Argument fails. In seeing how this argument fails, we shall discover a new focus for the debate over composition's extent.

In this paper, an objective conception of contexts based loosely upon situation theory is developed and formalized. Unlike subjective conceptions, which take contexts to be something like sets of beliefs, contexts on the objective conception are taken to be complex, structured pieces of the world that (in general) contain individuals, other contexts, and propositions about them. An extended first-order language for this account is developed. The language contains complex terms for propositions, and the standard predicate "ist" that expresses the relation (...) that holds between a context and a proposition just in case the latter is true in the former. The logic for the objective conception features a global classical predicate calculus, a local logic for reasoning within contexts, and axioms for propositions. The specter of paradox is banished from the logic by allowing "ist" to be nonbivalent in problematic cases: it is not in general the case, for any context c and proposition p, that either ist(c,p) or ist(c, ¬p). An important representational capability of the logic is illustrated by proving an appropriately modified version of an illustrative theorem from McCarthy's classic Blocks World example. (shrink)

The fundamental principle of the theory of possible worlds is that a proposition p is possible if and only if there is a possible world at which p is true. In this paper we present a valid derivation of this principle from a more general theory in which possible worlds are defined rather than taken as primitive. The general theory uses a primitive modality and axiomatizes abstract objects, properties, and propositions. We then show that this general theory has very small (...) models and hence that its ontological commitments—and, therefore, those of the fundamental principle of world theory—are minimal. (shrink)

In Modal Logic as Metaphysics, Timothy Williamson claims that the possibilism-actualism (P-A) distinction is badly muddled. In its place, he introduces a necessitism-contingentism (N-C) distinction that he claims is free of the confusions that purportedly plague the P-A distinction. In this paper I argue first that the P-A distinction, properly understood, is historically well-grounded and entirely coherent. I then look at the two arguments Williamson levels at the P-A distinction and find them wanting and show, moreover, that, when the N-C (...) distinction is broadened (as per Williamson himself) so as to enable necessitists to fend off contingentist objections, the P-A distinction can be faithfully reconstructed in terms of the N-C distinction. However, Williamson’s critique does point to a genuine shortcoming in the common formulation of the P-A distinction. I propose a new definition of the distinction in terms of essential properties that avoids this shortcoming. (shrink)

The relation between Kant’s conception of modalities in the Postulates of Empirical Thought and Hegel’s conception in the Logic of Essence has not been addressed in the current scholarship. I argue that there is in fact a close connection that becomes visible if the desideratum which is implied by Kant’s conceptions is understood. Thus, after an analysis of the Kantian modal postulates, the article shows that they are sufficient to characterize the necessity of Kant’s Grundsätze and, hence, a specific form (...) of synthetic propositions a priori. However, I will prove that they are insufficient to characterize Kant’s synthetic propositions a priori about transcendental philosophy. This insufficiency in the characterization of transcendental philosophy itself brings into the arena Hegel, who supplements the Kantian modalities with a self-reflexive form of transcendental modality. (shrink)

The paper looks at the semantics and ontology of dispositions in the light of recent work on the subject. Objections to the simple conditionals apparently entailed by disposition statements are met by replacing them with so-called 'reduction sentences' and some implications of this are explored. The usual distinction between categorical and dispositional properties is criticised and the relation between dispositions and their bases examined. Applying this discussion to two typical cases leads to the conclusion that fragility is not a real (...) property and that, while both temperature and its bases are, this does not generate any problem of overdetermination. (shrink)

1. The problemLewis identifies the monadic property being F with the set of all actual and possible Fs; the dyadic relation R is identified with the set of actual and possible pairs of things that are related by R; and so on . 1 Egan has argued that the fact that some properties have some of their properties contingently leads to trouble: " Let @ be the actual world, in which being green is [someone's] favourite property, and let w be (...) a world in which being green is [nobody's] favourite property. Since being green is somebody's favourite property in @, it must be a member of … being somebody's favourite property. Since being green is not anybody's favourite property in w, it must not be a member of being somebody's favourite property. Contradiction. " Egan suggests identifying properties with functions from worlds to extensions. However, as Lewis and others have complained, this has the drawback of turning monadic properties into relations . 2 In this article, we shall show that the Lewisian has the resources to block the above argument while simultaneously respecting the adicity of the relevant property and the contingency of the predication without changing his theory of properties. However, our analysis reveals a hidden ambiguity in the Lewisian conception of properties, and on either disambiguation the resulting account of properties seems inadequate.2. On ‘being someone's favourite property’The phrase ‘ being someone's favourite property’ is ambiguous. Given Lewis's ontology of possibilia, where the phrase picks out a monadic property it is not one that is possessed contingently; where it picks out a property that is possessed contingently it is not one that is monadic. But the argument works only if being someone's favourite property is both monadic and possessed contingently.Why should …. (shrink)

In this paper I pour a little cold water on claims of global supervenience, not by arguing that they are false, and not by arguing that they possess no philosophical utility whatsoever, but by building a case for the following conditional conclusion: if you expect claims of global supervenience to play a certain role in a certain metaphysical project, then you will be disappointed, since they cannot play such a role. The metaphysical project is to give an illuminating and suitably (...) physicalist account of the relations between properties instantiated at different levels of reality. The paper makes a prima facie case for holding that claims of global supervenience can provide no illumination in such an account, a case that emerges when the metaphysics of global supervenience is probed a little more deeply than is customary. (shrink)

The principle of Summation, which is a technically sharpened version of the familiar claim that a whole is a sum of its parts, is presented by Peter van Inwagen as a trivial truth. I argue to the contrary, that it is incompatible with the natural assumption that a whole may gain or lose parts non-instantaneously. For, as I show, the latter assumption implies that something can be determinately a whole without being determinately a sum of parts, and this, in turn, (...) indicates the falsity of Summation. I point out that the tension between Summation and the possibility of non-instantaneous gain or loss of parts compels us to rethink the relations between the concepts of whole and sum, and may have far reaching consequences for the mereology of physical objects. (shrink)

This paper aims to motivate a new beginning in metaphysical thinking about persistence by drawing attention to the disappearance of change in current accounts of persistence. I defend the claim that the debate is stuck in a dilemma which results from neglecting the constructive role of change for persistence. Neither of the two main competing views, perdurantism and endurantism, captures the idea of persistence as an identity through time. I identify the fundamental ontological reasons for this, namely the shared commitment (...) to what I call ‘thing ontology’: an ontology that gives the ontological priority to static things. I conclude by briefly indicating how switching to a process ontological framework that takes process and change to be ontologically primary may allow for overcoming the dilemma of persistence. (shrink)

The paper argues that four-dimensionalism is incompatible with the existence of additively cumulative properties, including mass, volume, and electrical charge. These properties add up over disjoint objects: for example, the mass of a whole composed of two disjoint objects is a sum of the individual masses of the objects. The difficulty with such properties for four-dimensionalism stems from the way this theory makes persistence depend on the existence of disjoint objects at disjoint times. I consider various possible responses to this (...) difficulty and conclude that they all fail. (shrink)

In this paper, I relate key features of Adolf Reinach’s abundant ontology of propositional states of affairs of his to Armstrong’s—or an Armstrongian—state of affairs ontology, with special regard to finding out how sparse or abundant the latter is with respect to negative states of affairs. After introducing the issue, I clarify the notion of a propositional state of affairs, paying special attention to the notion of abstract versus concrete. I show how Reinach’s states of affairs are propositional, and how (...) they compare with Chisholm’s well-known propositional states of affairs. In the next section, I outline Reinach’s five roles for states of affairs and show that only one of them is relevant to Armstrongian state of affairs ontology. In the following section, I utilise this role to create a ranking of state of affairs ontologies according to how abundant they are. It is, however, unclear which level Armstrongian state of affairs ontology is at, since it is unclear if, like Reinach’s ontology, it includes negative states of affairs. In the final section, I argue that the answer is a qualified ‘yes’, i.e. it does not occupy the sparsest level. (shrink)

What I call Mellor’s Question is the problem of whether determinables are properties of their determinates or properties of the particulars that possess these determinates. One can distinguish two basic competing theories of determinables that address the issue, implicitly if not explicitly. On the second-order theory, determinables are second-order properties of determinate properties; on the second-level theory, determinables are first-order properties of the particulars with these determinate properties. Higher-order properties are prima facie ontologically uneconomical, and in line with my general (...) view that ontological parsimony is vital to metaphysics, I consider it highly important which of the two theories is true. Firstly, I argue that the second-level theory offers the best explanation of the explananda (though the race is close), including the important but neglected phenomenon of ‘intermediate determinables’. Secondly, by paying attention to intermediate determinables and instantiation of higher-order properties, I argue that the second-level theory also is more ontologically economical. For these two reasons, this theory is preferable. (shrink)

The semantical framework is fundamentally intensional: neither possible worlds nor sets as basic entities, but rather, besides individuals, propositions, properties and relations (in intension). Logical truth is defined in terms of logical form (without mentioning this notion) without employing sets of models and the concept of truth in a model. Truth itself is explicitly defined (without recursion); the truth-conditions for the logical constants of the object-language become theorems derivable from the axioms for "to intend"--the basic semantical relation.

I formulate and defend a version of the many universes (or multiverse) reply to the atheistic argument from evil. Specifically, I argue that (i) if we know that any argument from evil (be it a logical or evidential argument) is sound, then we know that God would be (or at least probably would be) unjustified in actualizing our universe. I then argue that (ii) there might be a multiverse and (iii) if so, then we do not know that God would (...) be (or at least probably would be) unjustified in actualizing our universe. It follows that we cannot know that the atheistic argument from evil is sound, in which case we cannot be certain that the argument succeeds, and so it is rational to refuse to reject theism because of such arguments. (shrink)

In The Philosophical Criticism, Alois Riehl developed a realistic interpretation of Kant’s transcendental idealism based on his theory of space and time. In doing so, more than 100 years ago, he formulated an interpretation of the relation between the thing in itself and appearances that is discussed in current research as the metaphysical „dual aspect“ interpretation, although it is rarely attributed to Riehl. To reconstruct Riehl’s position, the research results of comparative studies on Moritz Schlick are systematically extended and applied (...) to the current debate on Kant’s transcendental idealism. (shrink)

A basic way of evaluating metaphysical theories is to ask whether they give satisfying answers to the questions they set out to resolve. I propose an account of “third-order” virtue that tells us what it takes for certain kinds of metaphysical theories to do so. We should think of these theories as recipes. I identify three good-making features of recipes and show that they translate to third-order theoretical virtues. I apply the view to two theories—mereological universalism and plenitudinous platonism—and draw (...) out their third-order virtues and vices. One lesson is that there is an important difference between essentially and non-essentially third-order vicious theories. I also argue that if a theory is essentially third-order vicious, it cannot be assessed for more standard “second-order” theoretical virtues and vices, like parsimony. This motivates the idea that third-order virtues are distinct from second-order ones. Finally, I suggest that the relationship between truth, progress, and third-order virtue is more complex than it seems. (shrink)

Logical realism is the view that there is logical structure in the world. I argue that, if logical realism is true, then we are deeply ignorant of that logical structure: either we can’t know which of our logical concepts accurately capture it, or none of our logical concepts accurately capture it at all. I don’t suggest abandoning logical realism, but instead discuss how realists should adjust their methodology in the face of this ignorance.

Ethical non-naturalists often charge that their naturalist competitors cannot adequately explain the distinctive normativity of moral or more broadly practical concepts. I argue that the force of the charge is mitigated, because non-naturalism is ultimately committed to a kind of mysterianism about the metaphysics of practical norms that possesses limited explanatory power. I then show that focusing on comparative judgments about the explanatory power of various metaethical theories raises additional problems for the non-naturalist, and suggest grounds for optimism that a (...) naturalistic realist about practical normativity will ultimately be able to explain the distinctive normativity of practical norms. I then show that radical pluralism or particularism about the structure of normative ethics would complicate the naturalistic strategy that I defend. This suggests a perhaps surprising way in which the resolution of the debate between ethical naturalists and non-naturalists may rest in part on the answers to substantive normative questions. (shrink)

Ethical non-naturalists often charge that their naturalist competitors cannot adequately explain the distinctive normativity of moral or more broadly practical concepts. I argue that the force of the charge is mitigated, because non-naturalism is ultimately committed to a kind of mysterianism about the metaphysics of practical norms that possesses limited explanatory power. I then show that focusing on comparative judgments about the explanatory power of various metaethical theories raises additional problems for the non-naturalist, and suggest grounds for optimism that a (...) naturalistic realist about practical normativity will ultimately be able to explain the distinctive normativity of practical norms. I then show that radical pluralism or particularism about the structure of normative ethics would complicate the naturalistic strategy that I defend. This suggests a perhaps surprising way in which the resolution of the debate between ethical naturalists and non-naturalists may rest in part on the answers to substantive normative questions. (shrink)

Assuming minimal fine-individuation--that there are some necessarily equivalent intensional objects (e.g. propositions) that are nonetheless distinct objects, on standard actualist frameworks, the answer to our title question is "No". First I specify a fully cognitively accessible, purely qualitative maximal consistent state of affairs (MCS). (That there is an MCS that is either fully graspable or purely qualitative is in itself quite contrary to conventional dogma.) Then I identify another MCS, one necessarily equivalent to the first. It follows that there could (...) have been more than one obtaining MCS. I then argue that there is more than one obtaining MCS. So there is nothing answering to "the actual world", and the set of worlds is not the set of MCSs. I explore various patch-ups. Finally, I compare the actualist and modal realist notions of worlds, and I argue that even if the realist were right about concrete world ensembles, our reflections indicate that necessary truth is not truth in all worlds anyway, which undercuts the rationale for the realist's program. (shrink)

Gareth Evans has argued influentially against vague identities. David Lewis and Theodore Sider have argued against vague parthood. Much of the distaste among philosophers for metaphysical vagueness is sourced in these arguments. I argue that even if the considerations adduced by Evans, Lewis and Sider are conclusive, metaphysical boundary vagueness remains possible.

Ontic structural realism is a thesis of fundamentality metaphysics: the thesis that structure, not objects, has fundamental status. Claimed as the metaphysic most befitting of modern physics, OSR first emerged as an entreaty to eliminate objects from the metaphysics of fundamental physics. Such elimination was urged by Steven French and James Ladyman on the grounds that only it could resolve the ‘underdetermination of metaphysics by physics’ that they claimed reduced any putative objectual commitment to a merely ‘ersatz’ form of realism. (...) Few, however, have joined French and Ladyman either in acknowledging that such underdetermination exists or in attributing to it such drastic consequences. However, an alternative view that physics does sanction objects, albeit merely as ontologically secondary entities, represents a different and seemingly less extreme route to the same conclusion regarding the fundamentality of structure. But since what it means to be ‘ontologically prior’ is itself a vexed philosophical question, a stance must be taken as to how we are to understand priority before its prospects may be evaluated. In an earlier paper, I outlined how Fine’s notion of ontological dependence might be utilized to defend the priority-based approach to structuralism. Since then, however, I have become convinced that that ontological dependence is not a relation of priority after all. As a result, the arguments outlined in that paper stand in need of reassessment. In this work, I consider the prospects for priority-based structuralism when expressed in the idiom of determination. My conclusion will be that it has yet to be vindicated by our best physical theories, owing to the failure of symmetry structures to determine the world’s inventory of fundamental kinds. Nevertheless, the same symmetry considerations point towards there being renewed prospects for eliminativism—an eliminativism, moreover, of more naturalistic appeal than that hitherto associated with OSR. 1Introduction 2Structuralist Strategies 3Defining Ontological Priority: Dependence or Determination? 4Structuralism in the Idiom of Determination 4.1Determining plurality 4.2Determining kind properties 5A Reinvigorated Eliminativism. (shrink)

Presentists standardly conform to the eternalist’s paradigm of treating all cases of property-exemplification as involving a single relation of instantiation. This, we argue, results in a much less parsimonious and philosophically explanatory picture than is possible if other alternatives are considered. We argue that by committing to primitive past and future tensed instantiation ties, presentists can make gains in both economy and explanatory power. We show how this metaphysical picture plays out in cases where an individual exists to partake in (...) facts about its past and future, and also in cases where that individual no longer exists, and proxies (or surrogates) for that thing must be found. (shrink)

An account of validity that makes what is invalid conditional on how many individuals there are is what I call a conditional account of validity. Here I defend conditional accounts against a criticism derived from Etchemendy’s well-known criticism of the model-theoretic analysis of validity. The criticism is essentially that knowledge of the size of the universe is non-logical and so by making knowledge of the extension of validity depend on knowledge of how many individuals there are, conditional accounts fail to (...) reflect that the former knowledge is basic, i.e., independent of knowledge derived from other sciences. Appealing to Russell’s pre-Principia logic, I defend conditional accounts against this criticism by sketching a rationale for thinking that there are infinitely many logical objects. (shrink)

Deflationists about truth embrace the positive thesis that the notion of truth is useful as a logical device, for such purposes as blanket endorsement, and the negative thesis that the notion doesn’t have any legitimate applications beyond its logical uses, so it cannot play a significant theoretical role in scientific inquiry or causal explanation. Focusing on Christopher Hill as exemplary deflationist, the present paper takes issue with the negative thesis, arguing that, without making use of the notion of truth conditions, (...) we have little hope for a scientific understanding of human speech, thought, and action. For the reference relation, the situation is different. Inscrutability arguments give reason to think that a more-than-deflationary theory of reference is unattainable. With respect to reference, deflationism is the only game in town. (shrink)

I argue that a solution to puzzles concerning the relationship ofobjects and their properties – a version of the `bundle' theory ofparticulars according to which ordinary objects are mereologicalfusions of monadic and relational tropes – is also a solution topuzzles of material constitution involving the allegedco-location of material objects. Additionally, two argumentsthat have played a prominent role in shaping the current debate,Mark Heller's argument for Four Dimensionalism and Peter vanInwagen's argument against Mereological Universalism, are shownto be unsound given this version (...) of the bundle theory. (shrink)

The pressure to individuate propositions more finely than intensionally—that is, hyper-intensionally—has two distinct sources. One source is the philosophy of mind: one can believe a proposition without believing an intensionally equivalent proposition. The second source is metaphysics: there are intensionally equivalent propositions, such that one proposition is true in virtue of the other but not vice versa. I focus on what our theory of propositions should look like when it's guided by metaphysical concerns about what is true in virtue of (...) what. In this paper I articulate and defend a metaphysical theory of the individuation of propositions, according to which two propositions are identical just in case they occupy the same nodes in a network of invirtuation relations. Invirtuation is here taken to be a primitive relation of metaphysical explanation exemplified by propositions that, in conjunction with truth, defines the notion of true in virtue of. After formulating the theory, I compare it with a view.. (shrink)

If _presentism_ is true, then no wholly non-present events exist. If _absence orthodoxy_ is true, then no absences exist. I discuss a well-known causal argument against presentism, and develop a very similar argument against absence orthodoxy. I argue that solutions to the argument against absence orthodoxy can be adopted by the presentist as solutions to the argument against presentism. The upshot is that if the argument against absence orthodoxy fails, then so does the argument against presentism.

Possibilism—the view that there are non-actual, merely possible entities—is a surprisingly resilient doctrine.1 One particularly hardy strand of possibilism—the modal realism championed by David Lewis—continues to attract both foes who seek to demonstrate its falsity (or at least stare its advocates into apostasy) and friends who hope to defend modal realism (or, when necessary, modify modal realism so as to avoid problematic objections).2 Although I am neither a foe nor friend of modal realism (but some of my best friends are!), (...) like many I continue to be fascinated by the doctrine. (shrink)

© The Author 2017. Published by Oxford University Press on behalf of The Analysis Trust. All rights reserved. For Permissions, please email: [email protected] wonderful collection of most of van Inwagen’s recent essays on topics in fundamental ontology is certainly to be welcomed.1 Many of the essays are focused on articulating and arguing for van Inwagen’s preferred meta-ontology, which he calls neo-Quineanism. In addition to these essays, Existence also contains essays on the eliminability of variables, the status of fictional entities, the (...) nature of ontological categories, the nature of properties, mereology and change over time, the comparative merits of constituent and relational ontologies, a probing discussion of metaphysical issues in the philosophy of mind, as well as an interview with van Inwagen that describes his intellectual development and his assessment of his philosophical contributions. Each essay is written in van Inwagen’s characteristically charming and impeccable style. I... (shrink)

I argue that extended simples are possible. The argument given here parallels an argument given elsewhere for the claim that the shape properties of material objects are extrinsic, not intrinsic as is commonly supposed. In the final section of the paper, I show that if the shape properties of material objects are extrinsic, the most popular argument against extended simples fails.

This paper argues that a counterpart-theoretic treatment of events, combined with a counterfactual theory of causation, can help resolve three puzzles from the causation literature. First, CCT traces the apparent contextual shifts in our causal attributions to shifts in the counterpart relation which obtains in those contexts. Second, being sensitive to shifts in the counterpart relation can help diagnose what goes wrong in certain prominent examples where the transitivity of causation appears to fail. Third, CCT can help us resurrect the (...) much-maligned fragility response to the problems of late pre-emption by understanding fragility in counterpart-theoretic terms. Some reasons to prefer this CCT approach to rivals are discussed. (shrink)

Given Lewis’s views about recombination and spatial relations, there are possible worlds in which space is discrete and yet the Pythagorean theorem is true – contrary to the so-called Weyl-Tile argument that concluded that the Pythagorean theorem must fail if space is discrete.