Truthmakers

Edited by Jamin Asay (University of Hong Kong)
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  1. Truth-Maker Theory and the Stopped Clock: Why Heathcote Fails to Solve the Gettier Problem.Qilin Li - manuscript
    Adrian Heathcote has proposed a truth-making account of knowledge that combines traditional conditions of justified true belief with the truth-making condition, which would jointly provide us with the sufficient condition of knowledge, and this truth-maker account of knowledge in turn explains why a gettiered justified true belief fails to be regarded as a genuine instance of knowledge. In this paper, by the comparison of two different casual models that are illustrated by the thermometer and the clock respectively, however, it will (...)
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  2. A Quantificational Analysis of the Liar Paradox.Matheus Silva - manuscript
    It seems that the most common strategy to solve the liar paradox is to argue that liar sentences are meaningless and, consequently, truth-valueless. The other main option that has grown in recent years is the dialetheist view that treats liar sentences as meaningful, truth-apt and true. In this paper I will offer a new approach that does not belong in either camp. I hope to show that liar sentences can be interpreted as meaningful, truth-apt and false, but without engendering any (...)
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  3. Dependence Asymmetry Between Truth and Reality.Jiamin Yu - manuscript
    The thesis aims to account for the pre-theoretic intuition of asymmetric dependence between truth and reality, namely, the intuition that truth depends on reality, but reality doesn’t depend on truth. I start by delineating a framework to answer this question: according to this framework, a proper theory of dependence asymmetry should explain why truth depends on reality (truth-reality dependence) and why reality doesn’t depend on truth (asymmetry). There are two readings of the second requirement: a strong reading and a weak (...)
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  4. Against Representational Levels.Nicholas K. Jones - forthcoming - Philosophical Perspectives.
    Some views articulate reality's hierarchical structure using relations from the fundamental to representations of reality. Other views instead use relations from the fundamental to constituents of non-representational reality. This paper argues against the first kind of view.
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  5. Replies to De Rizzo & Schnieder, Tegtmeier and Vallicella.Bo R. Meinertsen - forthcoming - Philosophia:1-11.
    In this piece I respond to commentaries by Julio De Rizzo & Benjamin Schnieder, Erwin Tegtmeier and William Vallicella on my book Metaphysics of States of Affairs (Springer, 2018).
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  6. Pretence Fictionalism About the Non-Present.Kristie Miller - forthcoming - Inquiry: An Interdisciplinary Journal of Philosophy.
    Presentists hold that only present things exist. But we all, presentists included, utter sentences that appear to involve quantification over non-present objects, and so we all, presentists included, seem to commit ourselves to such objects. Equally, we all, presentists included, take utterances of many past-tensed (and some future-tensed) sentences to be true. But if no past or future things exist, it’s hard to see how there can be anything that those utterances are about, which makes them true. This paper presents (...)
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  7. Possibility Precedes Actuality.Tuomas E. Tahko - forthcoming - Erkenntnis:1-21.
    This paper is inspired by and develops on E. J. Lowe’s work, who writes in his book The Possibility of Metaphysics that ‘metaphysical possibility is an inescapable determinant of actuality’ (1998: 9). Metaphysics deals with possibilities – metaphysical possibilities – but is not able to determine what is actual without the help of empirical research. Accordingly, a delimitation of the space of possibilities is required. The resulting – controversial – picture is that we generally need to know whether something is (...)
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  8. Presentism Without Truth-Makers.Barry Ward - forthcoming - Chronos.
    We construct a presentist semantics on which there are no truth-makers for past and future tensed statements. The semantics is not an expressivist or projectivist one, and is not susceptible to the semantical difficulties that confront such theories. We discuss how the approach handles some standard concerns with presentism.
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  9. Relevance Without Minimality.Stephen Yablo - forthcoming - In Andy Egan & Dirk Kindermann (eds.), Unstructured Content. Oxford: Oxford University Press.
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  10. Non-Realist Cognitivism, Truthmaking, and Ontological Cheating.Farbod Akhlaghi - 2022 - Ethics 132 (2):291-321.
    Derek Parfit defended Non-Realist Cognitivism. It is an open secret that this metaethical theory is often thought at best puzzling and at worst objectionably unclear. Employing truthmaker theory, I provide an account of Non-Realist Cognitivism that dispels charges of objectionable unclarity, clarifies how to assess it, and explains why, if plausible, it would be an attractive theory. I develop concerns that the theory involves cheating into an objection that ultimately reveals Non-Realist Cognitivism faces a dilemma. Whether it can escape demands (...)
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  11. Ecumenical Truthmaking: A Précis of A Theory of Truthmaking.Jamin Asay - 2022 - Asian Journal of Philosophy 1 (1(3)):1-5.
    The theory of truthmaking has long aroused skepticism from philosophers who believe it to be tangled up in contentious ontological commitments and unnecessary theoretical baggage. I argue in A Theory of Truthmaking that this suspicion is unfounded. Philosophers across the spectrum can take advantage of truthmaking, and use it to better understand the ontological implications of topics that arise all over the philosophical landscape. Challenging the current orthodoxy that truthmaking's fundamental purpose is to be a tool for explaining why truths (...)
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  12. Metafísica para Juristas.Samuele Chilovi - 2022 - In D. Lagier & G. Lariguet (eds.), Filosofía. Una Introducción para Juristas. Madrid: Trotta.
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  13. Multi-Descriptional Physicalism, Level(s) of Being, and the Mind-Body Problem.Savvas Ioannou - 2022 - Dissertation, University of St. Andrews
    The main idea of this thesis is multi-descriptional physicalism. According to it, only physical entities are elements of our ontology, and there are different ways to describe them. Higher-level vocabularies (e.g., mental, neurological, biological) truly describe reality. Sentences about higher-level entities are made true by physical entities. Every chapter will develop multi-descriptional physicalism or defend it from objections. In chapter 1, I will propose a new conceptual reductive account that conceptually reduces higher-level entities to physical entities. This conceptual reductive account (...)
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  14. Generality.Nils Kürbis - 2022 - In Nils Kürbis, Jonathan Nassim & Bahram Assadian (eds.), Knowledge, Number and Reality. Encounters with the Work of Keith Hossack. London: Bloomsbury Academic. pp. 161-176.
    Hossack's 'The Metaphysics of Knowledge' develops a theory of facts, entities in which universals are combined with universals or particulars, as the foundation of his metaphysics. While Hossack argues at length that there must be negative facts, facts in which the universal 'negation' is combined with universals or particulars, his conclusion that there are also general facts, facts in which the universal 'generality' is combined with universals, is reached rather more swiftly. In this paper I present Hossack with three arguments (...)
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  15. The best thing about the deflationary theory of truth.Jamin Asay - 2021 - Philosophical Studies 179 (1):109-131.
    I argue that deflationary theories of truth reveal an important lesson for the broader theory of truth: although the notion of truthmaking has played an essential role in many traditional theories of truth, it can be separated from and survive the rejection of substantive theories of truth. I argue that many of the traditional substantive theories of truth are unified in defining truth in terms of the ontological grounds that are needed to account for truth. Deflationists reject the idea that (...)
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  16. An Argument for Completely General Facts.Landon D. C. Elkind - 2021 - Journal for the History of Analytical Philosophy 9 (7).
    In his 1918 logical atomism lectures, Russell argued that there are no molecular facts. But he posed a problem for anyone wanting to avoid molecular facts: we need truth-makers for generalizations of molecular formulas, but such truth-makers seem to be both unavoidable and to have an abominably molecular character. Call this the problem of generalized molecular formulas. I clarify the problem here by distinguishing two kinds of generalized molecular formula: incompletely generalized molecular formulas and completely generalized molecular formulas. I next (...)
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  17. On Metaphysics’ Independence From Truthmaking.Aldo Filomeno - 2021 - Erkenntnis 1:1-14.
    This paper aims to support the claim that analytic metaphysics should be more cautious regarding the constraints that truthmaking considerations impose on metaphysical theories. To this end, I reply to Briggs and Forbes (2017), whoargue that certain truthmaking commitments are incurred by a Humean metaphysics and by the Growing-Block theory. First, I argue that Humean Supervenience does not need to endorse a standard version of truthmaker maximalism. This undermines Briggs and Forbes’s conclusion that Humean Supervenience and the Growing-Block theory are (...)
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  18. Which Fitch?Mark Jago - 2021 - Analysis 81 (3):436-439.
    Jago uses a Fitch-style argument in an attempt to demonstrate that every truth has a truthmaker. But Trueman shows that there is a parallel argument, this time to the conclusion that no truth has a truthmaker. Since we cannot accept both, we must ditch at least one Fitch. But which? Keywords: Truth, truthmaking, truthmaker maximalism, Fitch paradox, Robert Trueman.
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  19. Backing as Truthmaking.Alexios Stamatiadis-Bréhier - 2021 - Canadian Journal of Philosophy 51 (5):367–383.
    Separatists about grounding take explanations to be separate from their corresponding grounding-facts. Grounding-facts are supposed to underlie, or back, such explanations. However, the backing relation hasn’t received much attention in the literature. The aim of this paper is to provide an informative definition of backing. First, I examine two prominent proposals: backing as explaining (Kovacs 2017; 2019a) and backing as grounding (see Sjölin Wirling 2020). Finally, I put forward my own proposal. I argue that under plausible assumptions about the role (...)
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  20. Purely Theoretical Explanations.Giacomo Andreoletti, Jonathan Tallant & Giuliano Torrengo - 2020 - Philosophia 49 (1):133-154.
    This paper introduces a new kind of explanation that we describe as ‘purely theoretical’. We first present an example, E, of what we take to be a case of purely theoretical explanation. We then show that the explanation we have in mind does not fit neatly into any of the existing categories of explanation. We take this to give us prima facie motivation for thinking that purely theoretical explanation is a distinctive kind of explanation. We then argue that it can (...)
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  21. Truthmakers Against Correspondence.Jamin Asay - 2020 - Grazer Philosophische Studien 97 (2):271-293.
    Many philosophers think truthmaker theory offers a correspondence theory of truth. Despite the similarities, however, this identification cannot be correct. Truthmaker theory offers no theory of truth, nor can it be employed to offer an acceptable substantive theory of truth. Instead, truthmaker theory takes truth for granted. Though truthmaker theory is not a correspondence theory, it shares with it the same motivational basis—that truth is worldly—and better accounts for what is pre-theoretically compelling about correspondence theories. As a result, those at (...)
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  22. Deflating Deflationary Truthmaking.Jamin Asay & Sam Baron - 2020 - Philosophical Quarterly 70 (278):1-21.
    In this paper we confront a challenge to truthmaker theory that is analogous to the objections raised by deflationists against substantive theories of truth. Several critics of truthmaker theory espouse a ‘deflationary’ attitude about truthmaking, though it has not been clearly presented as such. Our goal is to articulate and then object to the underlying rationale behind deflationary truthmaking. We begin by developing the analogy between deflationary truth and deflationary truthmaking, and then show how the latter can be found in (...)
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  23. David Armstrong on the Metaphysics of Mathematics.Thomas Donaldson - 2020 - Dialectica 74 (4):113-136.
    This paper has two components. The first, longer component (sec. 1-6) is a critical exposition of Armstrong’s views about the metaphysics of mathematics, as they are presented in Truth and Truthmakers and Sketch for a Systematic Metaphysics. In particular, I discuss Armstrong’s views about the nature of the cardinal numbers, and his account of how modal truths are made true. In the second component of the paper (sec. 7), which is shorter and more tentative, I sketch an alternative account of (...)
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  24. Truthmakers for 1st Order Sentences - a Proposal.Friedrich Wilhelm Grafe - 2020 - Archive.Org.
    The purpose of this paper is to communicate - as a proposal - a general method of assigning a 'truthmaker' to any 1st order sentence in each of its models. The respective construct is derived from the standard model theoretic (recursive) satisfaction definition for 1st order languages and is a conservative extension thereof. The heuristics of the proposal (which has been somewhat idiosyncratic from the current point of view) and some more technical detail of the construction may be found in (...)
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  25. Starting Rational Reconstruction of Spinoza's Metaphysics by "a Formal Analogy to Elements of 'de Deo' (E1)".Friedrich Wilhelm Grafe - 2020 - Archive.Org.
    We aim to compile some means for a rational reconstruction of a named part of the start-over of Baruch (Benedictus) de Spinoza's metaphysics in 'de deo' (which is 'pars prima' of the 'ethica, ordine geometrico demonstrata' ) in terms of 1st order model theory. In so far, as our approach will be judged successful, it may, besides providing some help in understanding Spinoza, also contribute to the discussion of some or other philosophical evergreen, e.g. 'ontological commitment'. For this text we (...)
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  26. A Short Argument for Truthmaker Maximalism.Mark Jago - 2020 - Analysis 80 (1):40-44.
    Each truth has a truthmaker: an entity in virtue of whose existence that truth is true. So say truthmaker maximalists. Arguments for maximalism are hard to find, whereas those against are legion. Most accept that maximalism comes at a significant cost, which many judge to be too high. The scales would seem to be balanced against maximalism. Yet, as I show here, maximalism can be derived from an acceptable premise which many will pre-theoretically accept.
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  27. Is Truth Made, and If So, What Do We Mean by That? Redefining Truthmaker Realism.Catherine Legg - 2020 - Philosophia 48 (2):587-606.
    Philosophical discussion of truthmaking has flourished in recent times, but what exactly does it mean to ‘make’ a truth-bearer true? I argue that ‘making’ is a concept with modal force, and this renders it a problematic deployment for truthmaker theorists with nominalist sympathies, which characterises most current theories. I sketch the outlines of what I argue is a more genuinely realist truthmaker theory, which is capable of answering the explanatory question: In virtue of what does each particular truthmaker make its (...)
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  28. Against Disjunctive Properties: Four Armstrongian Arguments.Bo R. Meinertsen - 2020 - Philosophia 49 (1):95-106.
    This paper defends the case against (sparse) disjunctive properties by means of four Armstrongian arguments. The first of these is a logical atomist argument from truthmaking, which is, broadly speaking, ‘Armstrongian’ (Armstrong 1997). This argument is strong – although it stands or falls with the relevant notion of truthmaking, as it were. However, three arguments, which are prima facie independent of truthmaking, can be found explicitly early in Armstrong’s middle period. Two of these early arguments face a serious objection put (...)
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  29. Truthmaker Semantics for Natural Language: Attitude Verbs, Modals, and Intensional Transitive Verbs.Friederike Moltmann - 2020 - Theoretical Linguistics 3:159-200.
    This paper gives an outline of truthmaker semantics for natural language against the background of standard possible-worlds semantics. It develops a truthmaker semantics for attitude reports and deontic modals based on an ontology of attitudinal and modal objects and on a semantic function of clauses as predicates of such objects. It also présents new motivations for 'object-based truthmaker semantics' from intensional transitive verbs such as ‘need’, ‘look for’, ‘own’, and ‘buy’ and gives an outline of their semantics. This paper is (...)
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  30. An account of truthmaking.Noël Blas Saenz - 2020 - Synthese 197 (8):3413-3435.
    In this paper, I both propose and discuss a novel account of truthmaking. I begin by showing what truthmaking is not: it is not grounding and it is not correspondence. I then show what truthmaking is by offering an account that appeals both to grounding and what I call ‘deep correspondence’. After I present the account and show that it is an account that unifies, I put it to work by showing how it can overcome an objection to truthmaking, how (...)
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  31. A Defence of Lucretian Presentism.Jonathan Tallant & David Ingram - 2020 - Australasian Journal of Philosophy 98 (4):675-690.
    In this paper, we defend Lucretian Presentism. Although the view faces many objections and has proven unpopular with presentists, we rehabilitate Lucretianism and argue that none of the objections stick.
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  32. Truthmaking.Kelly Trogdon - 2020 - In Michael J. Raven (ed.), Routledge Handbook of Metaphysical Grounding. Routledge. pp. 396-407.
    Discussion of grounding-theoretic accounts of truthmaking in terms of the theoretical role of “catching cheaters”.
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  33. Bo R. Meinertsen: Metaphysics of States of Affairs: Truthmaking, Universals, and a Farewell to Bradley’s Regress. [REVIEW]William F. Vallicella - 2020 - Metaphysica 21 (1):167-177.
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  34. Much Ado About Aboutness.Sam Baron, Reginald Mary Chua, Kristie Miller & James Norton - 2019 - Inquiry: An Interdisciplinary Journal of Philosophy (3).
    Strong non-maximalism holds that some truths require no ontological ground of any sort. Strong non-maximalism allows one to accept that some propositions are true without being forced to endorse any corresponding ontological commitments. We show that there is a version of truthmaker theory available—anti-aboutness truthmaking—that enjoys the dialectical benefits of the strong non-maximalist’s position. According to anti-aboutness truthmaking, all truths require grounds, but a proposition need not be grounded in the very thing(s) that the proposition is about. We argue that (...)
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  35. On Weak Truthmaking.Nicola Guarino, Daniele Porello & Giancarlo Guizzardi - 2019 - In Adrien Barton, Selja Seppälä & Daniele Porello (eds.), Proceedings of the Joint Ontology Workshops 2019. CEUR Workshop Proceedings.
    Informally speaking, a truthmaker is something in the world in virtue of which the sentences of a language can be made true. This fundamental philosophical notion plays a central role in applied ontology. In particular, a recent nonorthodox formulation of this notion proposed by the philosopher Josh Parsons, which we labelled weak truthamking, has been shown to be extremely useful in addressing a number of classical problems in the area of Conceptual Modeling. In this paper, after revisiting the classical notion (...)
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  36. Presentism and Cross-Time Relations.Rognvaldur Ingthorsson - 2019 - In Patrick Blackburn, Per Hasle & Peter Ohrstrom (eds.), Logic and Philosophy of Time: Further Themes from Prior, Vol. 2. Aalborg, Denmark: pp. 53–72.
    This paper is a partial defence of presentism against the argument from cross-time relations. It is argued, first, that the Aristotelian view of causation and persistence does not really depict these phenomena in terms of relations between entities existing at different times, and indeed excludes the possibility of such cross-time relations obtaining. Second, it is argued that to reject the existence of the past—and thereby be unable to ground the truth of claims about the past—does not lead to any absurd (...)
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  37. Review of Douglas Edwards, The Metaphysics of Truth. [REVIEW]Mark Jago - 2019 - Mind 128 (511):970–976.
    The Metaphysics of Truth, by EdwardsDouglas. Oxford: Oxford University Press, 2018. Pp. 208.
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  38. On Armstrong’s Difficulties with Adequate Truthmaking Restrictions.Brannon McDaniel - 2019 - Journal of Philosophical Research 44:145-156.
    D. M. Armstrong rejects various ontologies that posit truths without truthmakers. But, lest proponents of such questionable ontologies postulate suspicious truthmakers in a bid to regain ontological respectability, Armstrong requires a plausible restriction on truthmaking that eliminates such ontologies. I discuss three different candidate restrictions: categorical, natural, and intrinsic difference-making. While the categorical and natural restrictions eliminate the questionable ontologies, they also eliminate Armstrong’s own ontology. The intrinsic difference-making restriction, on the other hand, fails to eliminate any of them. Thus (...)
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  39. Nominals and Event Structure.Friederike Moltmann - 2019 - In Robert Truswell (ed.), Oxford Handbook of Event Structure. Oxford: Oxford UP.
    This paper discusses three approaches to the semantics of event nominalizations and adverbial modification: the Davidsonian account, the Kimian account, and the truthmaker account. It argues that a combination of all three accounts is needed for the semantics of the full range of event, trope, and state nominalizations in English.
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  40. O dilema presentista.Matheus Diesel Werberich - 2019 - Em Curso 6 (1):105-114.
    The aim of the present paper is to show that presentism cannot answer the truthmaker objection in a satisfactory manner. For such, two main categories of presentist solutions were studied: the first kind states that truthmakers can’t provide any objection to presentism, while the second type tries to ground past truths by postulating new ontological categories, which were rejected, mainly because of their use of ad hoc ontologies. Also, I showed how the presentist should adopt antirealism about the past in (...)
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  41. The Role of Truth in Psychological Science.Jamin Asay - 2018 - Theory and Psychology 28 (3):382-397.
    In a recent paper, Haig and Borsboom explore the relevance of the theory of truth for psychological science. Although they conclude that correspondence theories of truth are best suited to offer the resources for making sense of scientific practice, they leave open the possibility that other theories might accomplish those same ends. I argue that deflationary theories of truth, which deny that there is any substantive property that unifies the class of truths, makes equally good sense of scientific practice as (...)
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  42. We Don’T Need No Explanation.Jamin Asay - 2018 - Philosophical Studies 175 (4):903-921.
    Explanation has played myriad roles in truthmaker theory. The notion of explanation is sometimes thought to give content to the very idea of truthmaking, and is sometimes used as a weapon to undermine the entire point of truthmaker theory. I argue that the notion of explanation is dialectically useless in truthmaker theory: while it’s true that truthmaking offers a form of explanation, this claim is theoretically unilluminating, and leaves truthmaker theorists vulnerable to various kinds of attack. I advocate an alternative (...)
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  43. Putting Pluralism in its Place.Jamin Asay - 2018 - Philosophy and Phenomenological Research 96 (1):175–191.
    Pluralism about truth is the view that there are many properties, not just one, in virtue of which things are true. Pluralists hope to dodge the objections that face traditional monistic substantive views of truth, as well as those facing deflationary theories of truth. More specifically, pluralists hope to advance an explanatorily potent understanding of truth that can capture the subtleties of various realist and anti-realist domains of discourse, all while avoiding the scope problem. I offer a new objection to (...)
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  44. The Problem of Universals and the Asymmetry of Instantiation.Donald L. M. Baxter - 2018 - American Philosophical Quarterly 55 (2):189-202.
    Oliver's and Rodriguez-Pereyra's important interpretation of the problem of universals as one concerning truthmakers neglects something crucial: that there is a numerical identity between numerically distinct particulars. The problem of universals is rather how to resolve the apparent contradiction that the same things are both numerically distinct and numerically identical. Baxter's account of instantiation as partial identity resolves the apparent contradiction. A seeming objection to this account is that it appears to make instantiation symmetric, since partial identity is symmetric. Armstrong's (...)
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  45. God as Ultimate Truthmaker.Paul Clavier - 2018 - European Journal for Philosophy of Religion 10 (1):67-80.
    Theories of truthmaking have been introduced quite recently in epistemology. Having little to do with truth serums, or truths drugs, their concern is to define truth in terms of a certain relation between truthbearers and truthmakers. Those theories make an attempt to remedy what is supposed to be lacking in classical theories of truth, especially in Alfred Tarski’s semantic theory.
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  46. Three Problems with Metaethical Minimalism.Raff Donelson - 2018 - Southwest Philosophy Review 34 (1):125-131.
    Metaethical minimalism. sometimes called quietism, is the view that first-order moral judgments can be true but nothing makes them true. This article raises three worries for that view. First, minimalists have no good reason to insist that moral judgments can be true. Second, minimalism, in abandoning the requirement that true judgments need to have truthmakers, leads to a problematic proliferation of truths. Third, most versions of minimalism entail a disjointed and therefore unacceptable theory of language and thought.
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  47. Thisnesses, Propositions, and Truth.David Ingram - 2018 - Pacific Philosophical Quarterly 99 (3):442-463.
    Presentists, who believe that only present objects exist, should accept a thisness ontology, since it can do considerable work in defence of presentism. In this paper, I propose a version of presentism that involves thisnesses of past and present entities and I argue this view solves important problems facing standard versions of presentism.
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  48. Molnar on Truthmakers for Negative Truths.Nils Kürbis - 2018 - Metaphysica 19 (2):251-257.
    Molnar argues that the problem of truthmakers for negative truths arises because we tend to accept four metaphysical principles that entail that all negative truths have positive truthmakers. This conclusion, however, already follows from only three of Molnar´s metaphysical principles. One purpose of this note is to set the record straight. I provide an alternative reading of two of Molnar´s principles on which they are all needed to derive the desired conclusion. Furthermore, according to Molnar, the four principles may be (...)
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  49. Metaphysics of States of Affairs: Truthmaking, Universals, and a Farewell to Bradley’s Regress.Bo R. Meinertsen - 2018 - Springer Singapore.
    This book addresses the metaphysics of Armstrongian states of affairs, i.e. instantiations of naturalist universals by particulars. The author argues that states of affairs are the best candidate for truthmakers and, in the spirit of logical atomism, that we need no molecular truthmakers for positive truths. In the book's context, this has the pleasing result that there are no molecular states of affairs. Following this account of truthmaking, the author first shows that the particulars in (first-order) states of affairs are (...)
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  50. Essential Truths and Their Truth-Grounds.Robert Michels - 2018 - Ergo: An Open Access Journal of Philosophy 5:790-815.
    This paper motivates and defends a principle which captures a systematic connection between essence, truth, and grounding. It says that if a proposition expresses an essential truth, i.e., if it is true in virtue of the nature of some objects, then there are grounds for its truth which involve these objects. Together with the assumption that a fact can only be grounded in facts which are relevant to it, this principle is then applied in an argument against the monotonicity of (...)
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