The paper begins with a more carefully stated version of ontologically neutral (ON) logic, originally introduced in (Hailperin, 1997). A non-infinitistic semantics which includes a definition of potential infinite validity follows. It is shown, without appeal to the actual infinite, that this notion provides a necessary and sufficient condition for provability in ON logic.

This article offers a limited defense of metaphysical “infinitism,” the view that there are, or might be, infinite chains of ontological dependence. According to a widespread presupposition, there must be an ultimate ground of being—most likely, a plurality of fundamental atoms. Contrary to this view, this article shows that metaphysical infinitism is internally coherent. In particular, a parallel with the debate concerning infinitism about epistemic justification is suggested, and an “emergence model” of being is put forward. (...) According to the emergence model, the being of any given entity gradually arises out of an infinite series of progressively less dependent entities—it is not wholly transmitted, as it were, from a basic, ungrounded level to all the dependent ones in a step-by-step fashion. Some objections are considered and rebutted. (shrink)

Infinitists argue that their view outshines foundationalism because infinitism can, whereas foundationalism cannot, explain two of epistemic justification’s crucial features: it comes in degrees and it can be complete. I present four different ways that foundationalists could make sense of those two features of justification, thereby undermining the case for infinitism.

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. Unfortunately, apophatic finitism cannot adopt that way without losing the ability to represent the positive ontological commitments of infinitism. (shrink)

In recent work, the interrelated questions of whether there is a fundamental level to reality, whether ontological dependence must have an ultimate ground, and whether the monist thesis should be endorsed that the whole universe is ontologically prior to its parts have been explored with renewed interest. Jonathan Schaffer has provided arguments in favour of 'priority monism' in a series of articles (2003, 2004, 2007a, 2007b, forthcoming). In this paper, these arguments are analysed, and it is claimed that they (...) are not compelling: in particular, the possibility that there is no ultimate level of basic entities that compose everything else is on a par with the possibility of infinite 'upward' complexity. The idea that we must, at any rate, postulate an ontologically fundamental level for methodological reasons ( Cameron 2008 ) is also discussed and found unconvincing: all things considered, there may be good reasons for endorsing 'metaphysical infinitism'. In any event, a higher degree of caution in formulating metaphysical claims than found in the extant literature appears advisable. (shrink)

In this chapter, a generic definition of fundamentality as an ontological minimality thesis is sought and its applicability examined. Most discussions of fundamentality are focused on a mereological understanding of the hierarchical structure of reality, which may be combined with an atomistic, object-oriented metaphysics. But recent work in structuralism, for instance, calls for an alternative understanding and it is not immediately clear that the conception of fundamentality at work in structuralism is commensurable with the mereological conception. However, it is (...) proposed that once we understand fundamentality as an ontological minimality thesis, these two as well as further conceptions of fundamentality can all be treated on a par, including metaphysical infinitism of the ‘boring’ type, where the same structure repeats infinitely. (shrink)

An elaboration in detail of the contention made in an earlier paper 1 that quantifier logic can be given an adequate formulation in which neither the notion of an individual nor that of a predicate appears. The logic is compatible with either an infinitistic or non-infinitistic completeness theorem.

The paper begins with a more carefully stated version of ontologically neutral (ON) logic, originally introduced in (Hailperin, 1997). A non-infinitistic semantics which includes a definition of potential infinite validity follows. It is shown, without appeal to the actual infinite, that this notion provides a necessary and sufficient condition for provability in ON logic.

I shall explain the notions of propositions and states of affairs as they are understood in the current ontological debate and I shall briefly relate them to similar notions in Aristotle and some Medieval authors. In contrast with the point of view of some philosophers who identify propositions and states of affairs, I shall argue that they need to be sharply distinguished. I shall then move on to a problem for propositions and, above all, states of affairs, known as (...) Bradley’s regress, and hint at discussions of this or analogous puzzles in the philosophical tradition before Bradley. Finally, I shall present my own approach to this issue, Fact Infinitism, focusing on why it would have been hardly acceptable before ontology came to have the Cantorian conception of the infinite at its disposal. (shrink)

The rational explanation of the Eucharist is at the centre of a revived debate in philosophical theology. After describing gunk-relational ontology, I show how it allows us to understand transubstantiation differently than other traditional and contemporary accounts, from which it draws a few points but combining them in a new way. In gunk-relational ontology every substance is its own relations, which constitute a gunky fundamental reality. The liturgical celebration of the Last Supper therefore creates a new relational situation for the (...) bread: the substance of the bread (its relations) profoundly changes its nature. (shrink)

Integrating cosmological and ontological lines of reasoning, I argue that there is a self-necessary being that (a) serves as the sufficient condition for everything, that (b) has the most perfect collection of whatever attributes of perfection there might be, and that (c) is an independent, eternal, unique, simple, indivisible, immutable, all-actual, all-free, all-present, all-powerful, all-knowing, all-good, personal creator of every expression of itself that everything is. My cosmo-ontological case for such a being, an everything-maker with the core features (...) ascribed to the God of classical theism, addresses the standard worries plaguing these lines of reasoning: (1) the richness required of such a being dissolves it into many beings; (2) the metaphysical possibility of such a being is assumed on insufficient grounds; (3) the features we ascribe to such a being are mere human-all-too-human projections. (shrink)

A review of U. Mäki (ed.). Fact and Fiction in Economics, Cambridge: Cambridge University Press, 2003. pp. xvi 384. ISBN 0521 00957. As people interested mainly in theory, methodologists and philos...

This paper provides a defence of the account of partial resemblances between properties according to which such resemblances are due to partial identities of constituent properties. It is argued, first of all, that the account is not only required by realists about universals à la Armstrong, but also useful (of course, in an appropriately re-formulated form) for those who prefer a nominalistic ontology for material objects. For this reason, the paper only briefly considers the problem of how to conceive of (...) the structural universals first posited by Armstrong in order to explain partial resemblances, and focuses instead on criticisms that have been levelled against the theory (by Pautz, Eddon, Denkel and Gibb) and that apply regardless of one’s preferred ontological framework. The partial identity account is defended from these objections and, in doing so, a hitherto quite neglected connection—between the debate about partial similarity as partial identity and that concerning ontological finitism versus infinitism—is looked at in some detail. (shrink)

In formal ontology, infinite regresses are generally considered a bad sign. One debate where such regresses come into play is the debate about fundamentality. Arguments in favour of some type of fundamentalism are many, but they generally share the idea that infinite chains of ontological dependence must be ruled out. Some motivations for this view are assessed in this article, with the conclusion that such infinite chains may not always be vicious. Indeed, there may even be room for a (...) type of fundamentalism combined with infinite descent as long as this descent is “boring,” that is, the same structure repeats ad infinitum. A start is made in the article towards a systematic account of this type of infinite descent. The philosophical prospects and scientific tenability of the account are briefly evaluated using an example from physics. (shrink)

The notion of fundamentality, as it is used in metaphysics, aims to capture the idea that there is something basic or primitive in the world. This metaphysical notion is related to the vernacular use of “fundamental”, but philosophers have also put forward various technical definitions of the notion. Among the most influential of these is the definition of absolute fundamentality in terms of ontological independence or ungroundedness. Accordingly, the notion of fundamentality is often associated with these two other technical (...) notions. (shrink)

This paper is about metaphysical ‘infinitism’, the view that there are, or could be, infinite chains of ontological dependence. Its main aim is to show that, contrary to widespread opinion, metaphysical infinitism is a coherent position. On the basis of this, it is then additionally argued that metaphysical infinitism need not fare worse than the more canonical ‘foundationalist’ alternatives when it comes to formulating metaphysical explanations. In the course of the discussion, a rather unexplored parallel with (...) the debate concerning infinitism about justification is suggested. (shrink)

According to Madhyamaka Buddhist philosophers, everything depends for its existence on something else. But what would a world devoid of fundamentalia look like? In this paper, I argue that the anti-foundationalist “neither-one-nor-many argument” of the Indian Mādhyamika Śrīgupta commits him to a position I call “metaphysical indefinitism.” I demonstrate how this view follows from Śrīgupta’s rejection of mereological simples and ontologically independent being, when understood in light of his account of conventional reality. Contra recent claims in the secondary literature, I (...) clarify how the Madhyamaka metaphysical dependence structure is not a straightforward infinitism since it does not honor strict asymmetry or transitivity. Instead, its dependence relations are irreflexive and extendable, admitting of dependence chains of indefinite (though not actually infinite) length and dependence loops of non-zero length. Yet, the flexible ontology of Śrīgupta's Madhyamaka can accommodate a contextualist account of asymmetry and support a revisable theory of conventional truth, delivering significant payoffs for the view, including the capacity to accommodate developments in scientific explanation. (shrink)

Modal metaphysics consumed much of the philosophical discussion at the turn of the century, yielding a number of epistemological insights. Modal analyses were applied within epistemology, yielding sensitivity and safety theories of knowledge as well as counterfactual accounts of the basing relation. The contemporary conversation has now turned to a new metaphysical notion – grounding – opening the way to a fresh wave of insights by bringing grounding into epistemology. In this paper, I attempt one such application, making sense of (...) the epistemic regress problem in terms of grounding. I argue that the relation that generates the epistemic regress is a grounding relation, showing that grounding can make sense of proposals by epistemic foundationalists and charting the course for similar applications to epistemic coherentism and epistemic infinitism. If it is right that grounding is involved in the epistemic regress, this points the way forward both for epistemologists and metaphysicians, revealing the prospects of solutions to the epistemic regress problem while providing grounding advocates with yet another example of grounding with which to theorize. (shrink)

A prominent Thomistic cosmological argument maintains that an infinite regress of causes, which exhibits a certain pattern of ontological dependence among its members, would be vicious and so must terminate in a first member. Interestingly, Jonathan Schaffer offers a similar argument in the contemporary grounding literature for the view called metaphysical foundationalism. I consider the striking similarities between both arguments and conclude that both are unsuccessful for the same reason. I argue this negative result gives us indirect reason to (...) consider metaphysical infinitism as a genuine possibility, the view that chains of ontological dependence or ground can descend indefinitely. (shrink)

A version of Bradley's regress can be endorsed in an effort to address the problem of the unity of states of affairs or facts, thereby arriving at a doctrine that I have called fact infinitism . A consequence of it is the denial of the thesis, WF, that all chains of ontological dependence are well-founded or grounded. Cameron has recently rejected fact infinitism by arguing that WF, albeit not necessarily true, is however contingently true. Here fact (...) class='Hi'>infinitism is supported by showing that Cameron's argument for the contingent truth of WF is unsuccessful. (shrink)

A version of Bradley's regress can be endorsed in an effort to address the problem of the unity of states of affairs or facts, thereby arriving at a doctrine that I have called fact infinitism. A consequence of it is the denial of the thesis, WF, that all chains of ontological dependence are well‐founded or grounded. Cameron has recently rejected fact infinitism by arguing that WF, albeit not necessarily true, is however contingently true. Here fact infinitism (...) is supported by showing that Cameron's argument for the contingent truth of WF is unsuccessful. (shrink)

This is the first part of a two-tier overview article on fundamentality in metaphysics and the philosophy of physics. It provides an introduction to the notion of fundamentality in metaphysics, as well as to several related concepts. The key issues in the contemporary debate on the topic are summarised, making systematic reference to the most relevant literature. In particular, various ways in which the fundamental entities and the fundamental structure of reality may be conceived are illustrated and discussed. A final (...) brief section looks at the methodological issue of naturalism, paving the way for the survey of fundamentality in the philosophy of physics, which is carried out in the second part. (shrink)

This paper examines the challenge that the argument known as "Bradley's regress" poses to the friends of states of affairs (facts), in its requesting an explanation of the existence of a fact as a unitary whole in addition to its constituents. All the main theoretical options, short of denying that there are facts, are considered. It is argued that only two of them are viable, namely a "Brute fact approach", according to which the existence of a fact cannot be explained (...) with the typical tools of ontologists, and "Fact infinitism", according to which, for any given fact, one should posit for explanatory purposes an infinite number of distinct states of affairs involving exemplification relations of higher and higher levels. Fact infinitism is defended against various objections and it is suggested that it should be preferred to the brute fact approach by those who are not too shy in vindicating for ontology a role, alongside with the empirical sciences, in the task of postulating entities in order to achieve explanations. (shrink)

This paper examines two objections to the infinitist theory of epistemic justification, namely “the finite mind objection” and “the distinction objection.” It criticizes Peter Klein’s response to the distinction objection and offers a more plausible response. It is then argued that this response is incompatible with Klein’s response to the finite mind objection. Infinitists, it would seem, cannot handle both objections when taken together.

Klein’s account of epistemic justification, infinitism, supplies a novel solution to the regress problem. We argue that concentrating on the normative aspect of justification exposes a number of unpalatable consequences for infinitism, all of which warrant rejecting the position. As an intermediary step, we develop a stronger version of the ‘finite minds’ objection.

In a recent paper, “Infinitism and Epistemic Normativity,” we have problematized the relationship between infinitism and epistemic normativity. Responding to our criticisms, John Turri has offered a defense of infinitism. In this paper, we argue that Turri’s defense fails, leaving infinitism vulnerable to the originally raised objections.

This paper expands upon a way in which we might rationally doubt that there are multiple sizes of infinity. The argument draws its inspiration from recent work in the philosophy of truth and philosophy of set theory. More specifically, elements of contextualist theories of truth and multiverse accounts of set theory are brought together in an effort to make sense of Cantor’s troubling theorem. The resultant theory provides an alternative philosophical perspective on the transfinite, but has limited impact on everyday (...) mathematical practice. (shrink)

Modest Infinitism -/- Jeremy Fantl -/- Abstract -/- Infinitism, a theory of justification most recently developed and defended by Peter Klein, is the view that justification is a matter of having an infinite series of non-repeating reasons for a proposition. I argue that infinitism is preferable to other theories (like foundationalism) in that only infinitism can plausibly account for two important features of justification: 1) that it admits of degrees and 2) that a concept of complete (...) justification makes sense. (shrink)

The article discusses infinitism, the view that a belief is justified for a subject only if she possesses an infinite chain of available reasons for that belief. In its most recent and sophisticated version, the view allegedly escapes the problems that trouble its main competitors, foundationalism and coherentism, while avoiding the traditional objections which relegated it to a marginal place. The article argues that despite these improvements, sophisticated versions of infinitism face a pressing problem, viz. they are unable (...) to appropriately articulate propositional and doxastic justification. (shrink)

According to infinitism, beliefs can be justified by an infinite chain of reasons. So far, infinitism has rarely been taken seriously and often even dismissed as inconsistent. However, Peijnenburg and Atkinson have recently argued that beliefs can indeed be justified by an infinite chain of reasons, if justification is understood probabilistically. In the following, I will discuss the formal result that has led to this conclusion. I will then introduce three probabilistic explications of justification and examine to which (...) extent they support Peijnenburg’s and Atkinson’s claim. (shrink)

Consider the following process of epistemic justification: proposition $E_{0}$ is made probable by $E_{1}$ which in turn is made probable by $E_{2}$ , which is made probable by $E_{3}$ , and so on. Can this process go on indefinitely? Foundationalists, coherentists, and sceptics claim that it cannot. I argue that it can: there are many infinite regresses of probabilistic reasoning that can be completed. This leads to a new form of epistemic infinitism.

I evaluate two new objections to an infinitist account of epistemic justification, and conclude that they fail to raise any new problems for infinitism. The new objections are a refined version of the finite-mind objection, which says infinitism demands more than finite minds can muster, and the normativity objection, which says infinitism entails that we are epistemically blameless in holding all our beliefs. I show how resources deployed in response to the most popular objection to infinitism, (...) the original finite-mind objection, can be redeployed to address the two new objections. (shrink)