Mathematical Truth

Edited by Mark Balaguer (California State University, Los Angeles)
Assistant editor: Sam Roberts (University of Sheffield)
About this topic
Summary

The topic of mathematical truth is importantly tied to the ontology of mathematics.  In particular, a central question is what kinds of objects we commit ourselves to when we endorse the truth of ordinary mathematical sentences, like ‘4 is even’ and ‘There are infinitely many prime numbers.’   But there are other important philosophical questions about mathematical truth as well.  For instance: Is there any plausible way to maintain that mathematical truths are analytic, i.e., true solely in virtue of meaning?  And given that most ordinary mathematical sentences (e.g., the two sentences listed above) follow from the axioms of our various mathematical theories (e.g., from sentences like ‘0 is a number’), how can we account for the truth of the axioms?  And how can we account for the objectivity of mathematics (i.e., for the fact that some mathematical sentences are objectively correct and others are objectively incorrect)?  Can we do this without endorsing the existence of mathematical objects?  Do mathematical objects even help?  And so on.

Key works

Some key works on these topics include the following: Carnap 1950; Benacerraf 1973; Putnam 1980; Field 1993; Field 1998; Wright & Hale 1992; Gödel 1964; Maddy 1988; and Maddy 1988.

Introductions

Introductory works include Shapiro 2000 and Colyvan 2012.

Related categories

245 found
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  1. The Benacerraf Problem of Mathematical Truth and Knowledge.Eileen S. Nutting - 2022 - Internet Encyclopedia of Philosophy.
    The Benacerraf Problem of Mathematical Truth and Knowledge Before philosophical theorizing, people tend to believe that most of the claims generally accepted in mathematics—claims like “2+3=5” and “there are infinitely many prime numbers”—are true, and that people know many of them. Even after philosophical theorizing, most people remain committed to mathematical truth and mathematical knowledge. … Continue reading The Benacerraf Problem of Mathematical Truth and Knowledge →.
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  2. In defense of Countabilism.David Builes & Jessica M. Wilson - 2022 - Philosophical Studies 179 (7):2199-2236.
    Inspired by Cantor's Theorem (CT), orthodoxy takes infinities to come in different sizes. The orthodox view has had enormous influence in mathematics, philosophy, and science. We will defend the contrary view---Countablism---according to which, necessarily, every infinite collection (set or plurality) is countable. We first argue that the potentialist or modal strategy for treating Russell's Paradox, first proposed by Parsons (2000) and developed by Linnebo (2010, 2013) and Linnebo and Shapiro (2019), should also be applied to CT, in a way that (...)
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  3. Calculus of Qualia: Introduction to Qualations 7 2 2022.Paul Merriam - manuscript
    The basic idea is to put qualia into equations (broadly understood) to get what might as well be called qualations. Qualations arguably have different truth behaviors than the analogous equations. Thus ‘black’ has a different behavior than ‘ █ ’. This is a step in the direction of a ‘calculus of qualia’. It might help clarify some issues.
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  4. What is the Point of Persistent Disputes? The Meta-Analytic Answer.Alexandre Billon & Philippe Vellozzo - forthcoming - Dialectica.
    Many philosophers regard the persistence of philosophical disputes as symptomatic of overly ambitious, ill-founded intellectual projects. There are indeed strong reasons to believe that persistent disputes in philosophy (and more generally in the discourse at large) are pointless. We call this the pessimistic view of the nature of philosophical disputes. In order to respond to the pessimistic view, we articulate the supporting reasons and provide a precise formulation in terms of the idea that the best explanation of persistent disputes entails (...)
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  5. Human Thought, Mathematics, and Physical Discovery.Gila Sher - forthcoming - In Yemima Ben Menahem & Carl Posy (eds.), Mathematical Knowledge, Objects and Applications: Essays in Memory of Mark Steiner. Berlin: Springer Nature.
    In this paper I discuss Mark Steiner's view of the contribution of mathematics to physics and take up some of the questions it raises. In particular, I take up the question of discovery and explore two aspects of this question ‒ a metaphysical aspect and a related epistemic aspect. The metaphysical aspect concerns the formal structure of the physical world. Does the physical world have mathematical or formal features or constituents, and what is the nature of these constituents? The related (...)
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  6. Beyond the “Formidable Circle”: Race and the Limits of Democratic Inclusion in Tocqueville's Democracy in America.Christine Dunn Henderson - 2022 - Journal of Political Philosophy 30 (1):94-115.
    Journal of Political Philosophy, Volume 30, Issue 1, Page 94-115, March 2022.
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  7. The Static Model of Inventory Management Without a Deficit with Neutrosophic Logic.Maissam Jdid, Rafif Alhabib & A. A. Salama - 2021 - International Journal of Neutrosophic Science 16 (1):42-48.
    In this paper, we present an expansion of one of the well-known classical inventory management models, which is the static model of inventory management without a deficit and for a single substance, based on the neutrosophic logic, where we provide through this study a basis for dealing with all data, whether specific or undefined in the field of inventory management, as it provides safe environment to manage inventory without running into deficit , and give us an approximate ideal volume of (...)
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  8. Reading Davis and Fanon: A Creolizing Approach to Race, Gender and Sexuality.Gamal Abdel-Shehid - 2021 - Philosophy and Global Affairs 1 (2):343-350.
    The paper uses insights from Jane Anna Gordon’s Creolizing Political Theory to come up with a different way to read the work of Frantz Fanon in general and his discussion of gender and sexuality in particular. The paper argues against a hermetic reading of Fanon, one which reads him outside of context and influences. Instead of this close, or primary reading of Fanon, I offer a “conversation” between Fanon and the early work of Angela Y. Davis. The paper shows that (...)
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  9. Creolizing Theory in Conversation with Theorizing Race in the Americas.Juliet Hooker - 2021 - Philosophy and Global Affairs 1 (2):277-281.
    This review essay situates Jane Anna Gordon’s in light of methodological debates about the nature and role of “comparison.” Gordon repurposes the concept of “creolization” as a means for political theory to grapple with heterogeneity and mixture, not as discrete sets of thinkers and traditions, but as co-constituting. Gordon’s use of creolizing is then read alongside Hooker’s concept of juxtaposition as an alternative to comparison.
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  10. Race, Ethnicity and a Post-Racial/Ethnic Future.Ovett Nwosimiri - 2021 - Filosofia Theoretica 10 (2):159-174.
    Ethnicity and racial identity formation are elements of our social world. In recent years, there has been numerous works on ethnicity and race. Both concepts are controversial in different disciplines. The controversies around these concepts have been heated up by scholars who have devoted their time to the discourse of ethnicity and race, and to understand the ascription of both concepts. Ethnicity and race have been causes of conflict, prejudice and discrimination among various ethnic and racial groups around the world. (...)
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  11. Infinite Reasoning.Jared Warren - 2021 - Philosophy and Phenomenological Research 103 (2):385-407.
    Our relationship to the infinite is controversial. But it is widely agreed that our powers of reasoning are finite. I disagree with this consensus; I think that we can, and perhaps do, engage in infinite reasoning. Many think it is just obvious that we can't reason infinitely. This is mistaken. Infinite reasoning does not require constructing infinitely long proofs, nor would it gift us with non-recursive mental powers. To reason infinitely we only need an ability to perform infinite inferences. I (...)
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  12. The Dream of Recapture.Carlo Nicolai - manuscript
    As a response to the semantic and logical paradoxes, theorists often reject some principles of classical logic. However, classical logic is entangled with mathematics, and giving up mathematics is too high a price to pay, even for nonclassical theorists. The so-called recapture theorems come to the rescue. When reasoning with concepts such as truth/class membership/property instantiation, if ones is interested in consequences of the theory that only contain mathematical vocabulary, nothing is lost by reasoning in the nonclassical framework. It is (...)
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  13. Mathematical Selves and the Shaping of Mathematical Modernism: Conflicting Epistemic Ideals in the Emergence of Enumerative Geometry.Nicolas Michel - 2021 - Isis 112 (1):68-92.
  14. How (and Why) the Conservation of a Circle is the Core (and Only) Dynamic in Nature.Ilexa Yardley - 2021 - Https://Medium.Com/the-Circular-Theory/.
    Solving Navier-Stokes and integrating it with Bose-Einstein. Moving beyond ‘mathematics’ and ‘physics.’ And, philosophy. Integrating 'point' 'line' 'circle.' (Euclid with 'reality.').
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  15. Notes on How-Possible Reasoning.Besim Karakadılar - manuscript
    A brief outline for some alternative lines of thought on the general structure of how-possible reasoning and its implications for formally and informally conceivable things, and the concept of mathematical truth.
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  16. Hermann Cohen’s Principle of the Infinitesimal Method: A Defense.Scott Edgar - 2020 - Hopos: The Journal of the International Society for the History of Philosophy of Science 10 (2):440-470.
    In Bertrand Russell's 1903 Principles of Mathematics, he offers an apparently devastating criticism of the neo-Kantian Hermann Cohen's Principle of the Infinitesimal Method and its History (PIM). Russell's criticism is motivated by his concern that Cohen's account of the foundations of calculus saddles mathematics with the paradoxes of the infinitesimal and continuum, and thus threatens the very idea of mathematical truth. This paper defends Cohen against that objection of Russell's, and argues that properly understood, Cohen's views of limits and infinitesimals (...)
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  17. Shadows of Syntax: Revitalizing Logical and Mathematical Conventionalism.Jared Warren - 2020 - New York, USA: Oxford University Press.
    What is the source of logical and mathematical truth? This book revitalizes conventionalism as an answer to this question. Conventionalism takes logical and mathematical truth to have their source in linguistic conventions. This was an extremely popular view in the early 20th century, but it was never worked out in detail and is now almost universally rejected in mainstream philosophical circles. Shadows of Syntax is the first book-length treatment and defense of a combined conventionalist theory of logic and mathematics. It (...)
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  18. How Can Necessary Facts Call for Explanation.Dan Baras - 2020 - Synthese 198 (12):11607-11624.
    While there has been much discussion about what makes some mathematical proofs more explanatory than others, and what are mathematical coincidences, in this article I explore the distinct phenomenon of mathematical facts that call for explanation. The existence of mathematical facts that call for explanation stands in tension with virtually all existing accounts of “calling for explanation”, which imply that necessary facts cannot call for explanation. In this paper I explore what theoretical revisions are needed in order to accommodate this (...)
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  19. The Residual Access Problem.Sharon Berry - manuscript
    A range of current truth-value realist philosophies of mathematics allow one to reduce the Benacerraf Problem to a problem concerning mathematicians' ability to recognize which conceptions of pure mathematical structures are coherent – in a sense which can be cashed out in terms of logical possibility. In this paper I will clarify what it takes to solve this `residual' access problem and then present a framework for solving it.
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  20. MANY 1 - A Transversal Imaginative Journey Across the Realm of Mathematics.Jean-Yves Beziau - 2017 - Journal of Indian Council of Philosophical Research 34 (2):259-287.
    We discuss the many aspects and qualities of the number one: the different ways it can be represented, the different things it may represent. We discuss the ordinal and cardinal natures of the one, its algebraic behaviour as a neutral element and finally its role as a truth-value in logic.
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  21. Structure and Categoricity: Determinacy of Reference and Truth Value in the Philosophy of Mathematics.Tim Button & Sean Walsh - 2016 - Philosophia Mathematica 24 (3):283-307.
    This article surveys recent literature by Parsons, McGee, Shapiro and others on the significance of categoricity arguments in the philosophy of mathematics. After discussing whether categoricity arguments are sufficient to secure reference to mathematical structures up to isomorphism, we assess what exactly is achieved by recent ‘internal’ renditions of the famous categoricity arguments for arithmetic and set theory.
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  22. Cyberpunk Entre Literatura E Matemática: Processos Comunicacionais da Literatura Massiva Na Crítica Científica da Realidade.Rafael Duarte Oliveira Venancio - 2013 - Conexão 12 (23).
    O presente artigo busca definir o movimento literário cyberpunk a partir da sua influência teórica vinda do campo da matemática. Utilizando a teorização interna ao movimento, centrada em Rudy Rucker, o objetivo aqui é entender como os campos da análise e dos fundamentos da matemática criam uma importante distinção entre os cyberpunks e as demais distopias literárias. Com isso, há a pressuposição de um movimento de uma crítica sociomatemática feita pelos cyberpunks cujos conceitos matemáticos tornam possível criticar o tempo presente, (...)
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  23. Regel und Witz. Wittgensteinsche Perspektiven auf Mathematik, Sprache und Moral. [REVIEW]Ulf Hlobil - 2010 - Zeitschrift für Philosophische Forschung 64 (3):416-419.
    Review of Timo-Peter Ertz's "Regel und Witz. Wittgensteinsche Perspektiven auf Mathematik, Sprache und Moral," Berlin & New York: de Gruyter, 2008.
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  24. Automated Theorem Proving and Its Prospects. [REVIEW]Desmond Fearnley-Sander - 1995 - PSYCHE: An Interdisciplinary Journal of Research On Consciousness 2.
    REVIEW OF: Automated Development of Fundamental Mathematical Theories by Art Quaife. (1992: Kluwer Academic Publishers) 271pp. Using the theorem prover OTTER Art Quaife has proved four hundred theorems of von Neumann-Bernays-Gödel set theory; twelve hundred theorems and definitions of elementary number theory; dozens of Euclidean geometry theorems; and Gödel's incompleteness theorems. It is an impressive achievement. To gauge its significance and to see what prospects it offers this review looks closely at the book and the proofs it presents.
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  25. What Mathematical Truth Could Not Be--1.Paul Benacerraf - 1998 - In Matthias Schirn (ed.), The Philosophy of Mathematics Today. Clarendon Press.
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  26. Introduction.Justin Clarke-Doane - 2012 - Review of Symbolic Logic 5 (3):379-379.
  27. Truth and Mathematics (Prawda a Matematyka).Lemanska Anna - 2010 - Studia Philosophiae Christianae 46 (1):37-54.
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  28. Axioms for Grounded Truth.Thomas Schindler - 2014 - Review of Symbolic Logic 7 (1):73-83.
    We axiomatize Leitgeb's (2005) theory of truth and show that this theory proves all arithmetical sentences of the system of ramified analysis up to $\epsilon_0$. We also give alternative axiomatizations of Kripke's (1975) theory of truth (Strong Kleene and supervaluational version) and show that they are at least as strong as the Kripke-Feferman system KF and Cantini's VF, respectively.
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  29. A Problem for Intuitionism: The Apparent Possibility of Performing Infinitely Many Tasks in a Finite Time.A. W. Moore - 1990 - Proceedings of the Aristotelian Society 90:17 - 34.
  30. On the Nature of Mathematical Truth.Carl G. Hempel - 1945 - In P. Benacerraf H. Putnam (ed.), Philosophy of Mathematics. Prentice-Hall. pp. 366--81.
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Analyticity in Mathematics
  1. Carnap and Beth on the Limits of Tolerance.Benjamin Marschall - 2021 - Canadian Journal of Philosophy 51 (4):282–300.
    Rudolf Carnap’s principle of tolerance states that there is no need to justify the adoption of a logic by philosophical means. Carnap uses the freedom provided by this principle in his philosophy of mathematics: he wants to capture the idea that mathematical truth is a matter of linguistic rules by relying on a strong metalanguage with infinitary inference rules. In this paper, I give a new interpretation of an argument by E. W. Beth, which shows that the principle of tolerance (...)
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  2. From Metasemantics to Analyticity.Zeynep Soysal - 2021 - Philosophy and Phenomenological Research 103 (1):57-76.
    In this paper, I argue from a metasemantic principle to the existence of analytic sentences. According to the metasemantic principle, an external feature is relevant to determining which concept one expresses with an expression only if one is disposed to treat this feature as relevant. This entails that if one isn’t disposed to treat external features as relevant to determining which concept one expresses, and one still expresses a given concept, then something other than external features must determine that one (...)
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  3. The Making of Peacocks Treatise on Algebra: A Case of Creative Indecision.Menachem Fisch - 1999 - Archive for History of Exact Sciences 54 (2):137-179.
    A study of the making of George Peacock's highly influential, yet disturbingly split, 1830 account of algebra as an entanglement of two separate undertakings: arithmetical and symbolical or formal.
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  4. Remarks on Wittgenstein, Gödel, Chaitin, Incompleteness, Impossiblity and the Psychological Basis of Science and Mathematics.Michael Richard Starks - 2019 - In Remarks on Impossibility, Incompleteness, Paraconsistency, Undecidability, Randomness, Computability, Paradox, Uncertainty and the Limits of Reason in Chaitin, Wittgenstein, Hofstadter, Wolpert, Doria, da Costa, Godel, Searle, Rodych, Berto, Floyd, Moyal. Las Vegas, NV USA: Reality Press. pp. 24-38.
    It is commonly thought that such topics as Impossibility, Incompleteness, Paraconsistency, Undecidability, Randomness, Computability, Paradox, Uncertainty and the Limits of Reason are disparate scientific physical or mathematical issues having little or nothing in common. I suggest that they are largely standard philosophical problems (i.e., language games) which were resolved by Wittgenstein over 80 years ago. -/- Wittgenstein also demonstrated the fatal error in regarding mathematics or language or our behavior in general as a unitary coherent logical ‘system,’ rather than as (...)
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  5. Is Geometry Analytic?David Mwakima - 2017 - Dianoia 1 (4):66 - 78.
    In this paper I present critical evaluations of Ayer and Putnam's views on the analyticity of geometry. By drawing on the historico-philosophical work of Michael Friedman on the relativized apriori; and Roberto Torretti on the foundations of geometry, I show how we can make sense of the assertion that pure geometry is analytic in Carnap's sense.
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  6. Is Hume’s Principle Analytic?Eamon Darnell & Aaron Thomas-Bolduc - forthcoming - Synthese 198 (1):169-185.
    The question of the analyticity of Hume’s Principle is central to the neo-logicist project. We take on this question with respect to Frege’s definition of analyticity, which entails that a sentence cannot be analytic if it can be consistently denied within the sphere of a special science. We show that HP can be denied within non-standard analysis and argue that if HP is taken to depend on Frege’s definition of number, it isn’t analytic, and if HP is taken to be (...)
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  7. Formal Analyticity.Zeynep Soysal - 2018 - Philosophical Studies 175 (11):2791-2811.
    In this paper, I introduce and defend a notion of analyticity for formal languages. I first uncover a crucial flaw in Timothy Williamson’s famous argument template against analyticity, when it is applied to sentences of formal mathematical languages. Williamson’s argument targets the popular idea that a necessary condition for analyticity is that whoever understands an analytic sentence assents to it. Williamson argues that for any given candidate analytic sentence, there can be people who understand that sentence and yet who fail (...)
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  8. Two Criticisms Against Mathematical Realism.Seungbae Park - 2017 - Diametros 52:96-106.
    Mathematical realism asserts that mathematical objects exist in the abstract world, and that a mathematical sentence is true or false, depending on whether the abstract world is as the mathematical sentence says it is. I raise two objections against mathematical realism. First, the abstract world is queer in that it allows for contradictory states of affairs. Second, mathematical realism does not have a theoretical resource to explain why a sentence about a tricle is true or false. A tricle is an (...)
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  9. A Case Study of Misconceptions Students in the Learning of Mathematics; The Concept Limit Function in High School.Widodo Winarso & Toheri Toheri - 2017 - Jurnal Riset Pendidikan Matematika 4 (1): 120-127.
    This study aims to find out how high the level and trends of student misconceptions experienced by high school students in Indonesia. The subject of research that is a class XI student of Natural Science (IPA) SMA Negeri 1 Anjatan with the subject matter limit function. Forms of research used in this study is a qualitative research, with a strategy that is descriptive qualitative research. The data analysis focused on the results of the students' answers on the test essay subject (...)
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  10. On Deductionism.Dan Bruiger - manuscript
    Deductionism assimilates nature to conceptual artifacts (models, equations), and tacitly holds that real physical systems are such artifacts. Some physical concepts represent properties of deductive systems rather than of nature. Properties of mathematical or deductive systems can thereby sometimes falsely be ascribed to natural systems.
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  11. Minimal Type Theory (MTT).Pete Olcott - manuscript
    Minimal Type Theory (MTT) is based on type theory in that it is agnostic about Predicate Logic level and expressly disallows the evaluation of incompatible types. It is called Minimal because it has the fewest possible number of fundamental types, and has all of its syntax expressed entirely as the connections in a directed acyclic graph.
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  12. Russell on Logicism and Coherence.Conor Mayo-Wilson - 2011 - Russell: The Journal of Bertrand Russell Studies 31 (1):89-106.
    According to Quine, Charles Parsons, Mark Steiner, and others, Russell's logicist project is important because, if successful, it would show that mathematical theorems possess desirable epistemic properties often attributed to logical theorems, such as a prioricity, necessity, and certainty. Unfortunately, Russell never attributed such importance to logicism, and such a thesis contradicts Russell's explicitly stated views on the relationship between logic and mathematics. This raises the question: what did Russell understand to be the philosophical importance of logicism? Building on recent (...)
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  13. Meaning, Presuppositions, Truth-Relevance, Gödel's Sentence and the Liar Paradox.X. Y. Newberry - manuscript
    Section 1 reviews Strawson’s logic of presuppositions. Strawson’s justification is critiqued and a new justification proposed. Section 2 extends the logic of presuppositions to cases when the subject class is necessarily empty, such as (x)((Px & ~Px) → Qx) . The strong similarity of the resulting logic with Richard Diaz’s truth-relevant logic is pointed out. Section 3 further extends the logic of presuppositions to sentences with many variables, and a certain valuation is proposed. It is noted that, given this valuation, (...)
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  14. In Carnap’s Defense: A Survey on the Concept of a Linguistic Framework in Carnap’s Philosophy.Parzhad Torfehnezhad - 2016 - Abstracta 9 (1):03-30.
    The main task in this paper is to detail and investigate Carnap’s conception of a “linguistic framework”. On this basis, we will see whether Carnap’s dichotomies, such as the analytic-synthetic distinction, are to be construed as absolute/fundamental dichotomies or merely as relative dichotomies. I argue for a novel interpretation of Carnap’s conception of a LF and, on that basis, will show that, according to Carnap, all the dichotomies to be discussed are relative dichotomies; they depend on conventional decisions concerning the (...)
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  15. Double Vision: Two Questions About the Neo-Fregean Program.John MacFarlane - 2009 - Synthese 170 (3):443-456.
    Much of The Reason’s Proper Study is devoted to defending the claim that simply by stipulating an abstraction principle for the “number-of” functor, we can simultaneously fix a meaning for this functor and acquire epistemic entitlement to the stipulated principle. In this paper, I argue that the semantic and epistemological principles Hale and Wright offer in defense of this claim may be too strong for their purposes. For if these principles are correct, it is hard to see why they do (...)
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  16. Against Mathematical Convenientism.Seungbae Park - 2016 - Axiomathes 26 (2):115-122.
    Indispensablists argue that when our belief system conflicts with our experiences, we can negate a mathematical belief but we do not because if we do, we would have to make an excessive revision of our belief system. Thus, we retain a mathematical belief not because we have good evidence for it but because it is convenient to do so. I call this view ‘ mathematical convenientism.’ I argue that mathematical convenientism commits the consequential fallacy and that it demolishes the Quine-Putnam (...)
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  17. New Perspectives on the Philosophy of Paul Benacerraf: Truth, Objects, Infinity (Fabrice Pataut, Editor).Fabrice Pataut Jody Azzouni, Paul Benacerraf Justin Clarke-Doane, Jacques Dubucs Sébastien Gandon, Brice Halimi Jon Perez Laraudogoitia, Mary Leng Ana Leon-Mejia, Antonio Leon-Sanchez Marco Panza, Fabrice Pataut Philippe de Rouilhan & Andrea Sereni Stuart Shapiro - forthcoming - Springer.
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  18. On the Quinean-Analyticity of Mathematical Propositions.Gregory Lavers - 2012 - Philosophical Studies 159 (2):299-319.
    This paper investigates the relation between Carnap and Quine’s views on analyticity on the one hand, and their views on philosophical analysis or explication on the other. I argue that the stance each takes on what constitutes a successful explication largely dictates the view they take on analyticity. I show that although acknowledged by neither party (in fact Quine frequently expressed his agreement with Carnap on this subject) their views on explication are substantially different. I argue that this difference not (...)
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  19. The Syntheticity of Time.Stephen R. Palmquist - 1989 - Philosophia Mathematica (2):233-235.
    In a recent article in this journal Phil. Math., II, v.4 (1989), n.2, pp.? ?] J. Fang argues that we must not be fooled by A.J. Ayer (God rest his soul!) and his cohorts into believing that mathematical knowledge has an analytic a priori status. Even computers, he reminds us, take some amount of time to perform their calculations. The simplicity of Kant's infamous example of a mathematical proposition (7+5=12) is "partly to blame" for "mislead[ing] scholars in the direction of (...)
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  20. Inwiefern Sind Die Mathematischen Sätze Analytisch?Gerhard Frey - 1972 - Philosophia Mathematica (2):145-157.
    A SUMMARY IN ENGLISH [by Editor]The problem is to find out whether mathematical propositions are analytical, and if so, or if not, to what extent.Kant defined the analyticity in terms of Cartesian res extensa, exemplified by “A body is extended”, while he considered, because of such examples, mathematical propositions to be synthetic. The recent studies in set theory by Gödel, P.J.Cohen, etc., indicate, however, that such a proposition as the continuum hypothesis is certainly not “analytic (tautological)” in the strict sense (...)
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