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  1.  64
    One true logic: a monist manifesto.A. C. Paseau & Owen Griffiths - 2022 - Oxford: Oxford University Press. Edited by A. C. Paseau.
    Logical monism is the claim that there is a single correct logic, the 'one true logic' of our title. The view has evident appeal, as it reflects assumptions made in ordinary reasoning as well as in mathematics, the sciences, and the law. In all these spheres, we tend to believe that there aredeterminate facts about the validity of arguments. Despite its evident appeal, however, logical monism must meet two challenges. The first is the challenge from logical pluralism, according to which (...)
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  2.  27
    Indispensability.A. C. Paseau & Alan Baker - 2023 - Cambridge University Press.
    Our best scientific theories explain a wide range of empirical phenomena, make accurate predictions, and are widely believed. Since many of these theories make ample use of mathematics, it is natural to see them as confirming its truth. Perhaps the use of mathematics in science even gives us reason to believe in the existence of abstract mathematical objects such as numbers and sets. These issues lie at the heart of the Indispensability Argument, to which this Element is devoted. The Element's (...)
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  3.  86
    Non-metric Propositional Similarity.A. C. Paseau - 2022 - Erkenntnis 87 (5):2307-2328.
    The idea that sentences can be closer or further apart in meaning is highly intuitive. Not only that, it is also a pillar of logic, semantic theory and the philosophy of science, and follows from other commitments about similarity. The present paper proposes a novel way of comparing the ‘distance’ between two pairs of propositions. We define ‘\ is closer in meaning to \ than \ is to \’ and thereby give a precise account of comparative propositional similarity facts. Notably, (...)
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  4.  33
    Arithmetic, enumerative induction and size bias.A. C. Paseau - 2021 - Synthese 199 (3-4):9161-9184.
    Number theory abounds with conjectures asserting that every natural number has some arithmetic property. An example is Goldbach’s Conjecture, which states that every even number greater than 2 is the sum of two primes. Enumerative inductive evidence for such conjectures usually consists of small cases. In the absence of supporting reasons, mathematicians mistrust such evidence for arithmetical generalisations, more so than most other forms of non-deductive evidence. Some philosophers have also expressed scepticism about the value of enumerative inductive evidence in (...)
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  5. Lakatos and the Euclidean Programme.A. C. Paseau & Wesley Wrigley - forthcoming - In Roman Frigg, Jason Alexander, Laurenz Hudetz, Miklos Rédei, Lewis Ross & John Worrall (eds.), The Continuing Influence of Imre Lakatos's Philosophy: a Celebration of the Centenary of his Birth. Springer.
    Euclid’s Elements inspired a number of foundationalist accounts of mathematics, which dominated the epistemology of the discipline for many centuries in the West. Yet surprisingly little has been written by recent philosophers about this conception of mathematical knowledge. The great exception is Imre Lakatos, whose characterisation of the Euclidean Programme in the philosophy of mathematics counts as one of his central contributions. In this essay, we examine Lakatos’s account of the Euclidean Programme with a critical eye, and suggest an alternative (...)
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  6.  46
    Is English consequence compact?A. C. Paseau & Owen Griffiths - 2021 - Thought: A Journal of Philosophy 10 (3):188-198.
    Thought: A Journal of Philosophy, Volume 10, Issue 3, Page 188-198, September 2021.
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  7.  76
    Did Frege commit a cardinal sin?A. C. Paseau - 2015 - Analysis 75 (3):379-386.
    Frege’s _Basic Law V_ is inconsistent. The reason often given is that it posits the existence of an injection from the larger collection of first-order concepts to the smaller collection of objects. This article explains what is right and what is wrong with this diagnosis.
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  8. What’s the Point of Complete Rigour?A. C. Paseau - 2016 - Mind 125 (497):177-207.
    Complete inferential rigour is achieved by breaking down arguments into steps that are as small as possible: inferential ‘atoms’. For example, a mathematical or philosophical argument may be made completely inferentially rigorous by decomposing its inferential steps into the type of step found in a natural deduction system. It is commonly thought that atomization, paradigmatically in mathematics but also more generally, is pro tanto epistemically valuable. The paper considers some plausible candidates for the epistemic value arising from atomization and finds (...)
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  9. Non-deductive justification in mathematics.A. C. Paseau - 2023 - Handbook of the History and Philosophy of Mathematical Practice.
    In mathematics, the deductive method reigns. Without proof, a claim remains unsolved, a mere conjecture, not something that can be simply assumed; when a proof is found, the problem is solved, it turns into a “result,” something that can be relied on. So mathematicians think. But is there more to mathematical justification than proof? -/- The answer is an emphatic yes, as I explain in this article. I argue that non-deductive justification is in fact pervasive in mathematics, and that it (...)
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  10.  97
    A measure of inferential-role preservation.A. C. Paseau - 2019 - Synthese 196 (7):2621-2642.
    The point of formalisation is to model various aspects of natural language. Perhaps the main use to which formalisation is put is to model and explain inferential relations between different sentences. Judged solely by this objective, a formalisation is successful in modelling the inferential network of natural language sentences to the extent that it mirrors this network. There is surprisingly little literature on the criteria of good formalisation, and even less on the question of what it is for a formalisation (...)
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  11. Reducing Arithmetic to Set Theory.A. C. Paseau - 2009 - In Øystein Linnebo & Otavio Bueno (eds.), New Waves in Philosophy of Mathematics. Palgrave Macmillan. pp. 35-55.
    The revival of the philosophy of mathematics in the 60s following its post-1931 slump left us with two conflicting positions on arithmetic’s ontological relationship to set theory. W.V. Quine’s view, presented in 'Word and Object' (1960), was that numbers are sets. The opposing view was advanced in another milestone of twentieth-century philosophy of mathematics, Paul Benacerraf’s 'What Numbers Could Not Be' (1965): one of the things numbers could not be, it explained, was sets; the other thing numbers could not be, (...)
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  12.  58
    Propositionalism.A. C. Paseau - 2021 - Journal of Philosophy 118 (8):430-449.
    Propositionalism is the claim that all logical relations can be captured by propositional logic. It is usually regarded as obviously false, because propositional logic seems too weak to capture the rich logical structure of language. I show that there is a clear sense in which propositional logic can match first-order logic, by producing formalizations that are valid iff their first-order counterparts are, and also respect grammatical form as the propositionalist construes it. I explain the real reason propositionalism fails, which is (...)
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  13.  81
    Isomorphism invariance and overgeneration.Owen Griffiths & A. C. Paseau - 2016 - Bulletin of Symbolic Logic 22 (4):482-503.
    The isomorphism invariance criterion of logical nature has much to commend it. It can be philosophically motivated by the thought that logic is distinctively general or topic neutral. It is capable of precise set-theoretic formulation. And it delivers an extension of ‘logical constant’ which respects the intuitively clear cases. Despite its attractions, the criterion has recently come under attack. Critics such as Feferman, MacFarlane and Bonnay argue that the criterion overgenerates by incorrectly judging mathematical notions as logical. We consider five (...)
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  14. An exact measure of paradox.A. C. Paseau - 2013 - Analysis 73 (1):17-26.
    We take seriously the idea that paradoxes come in quantifiable degree by offering an exact measure of paradox. We consider three factors relevant to the degree of paradox, which are a function of the degree of belief in each of the individual propositions in the paradox set and the degree of belief in the set as a whole. We illustrate the proposal with a particular measure, and conclude the discussion with some critical remarks.
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  15.  85
    Fairness and Aggregation.A. C. Paseau & Ben Saunders - 2015 - Utilitas 27 (4):460-469.
    Sometimes, two unfair distributions cancel out in aggregate. Paradoxically, two distributions each of which is fair in isolation may give rise to aggregate unfairness. When assessing the fairness of distributions, it therefore matters whether we assess transactions piecemeal or focus only on the overall result. This piece illustrates these difficulties for two leading theories of fairness before offering a formal proof that no non-trivial theory guarantees aggregativity. This is not intended as a criticism of any particular theory, but as a (...)
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  16.  72
    The overgeneration argument(s): A succinct refutation.A. C. Paseau - 2014 - Analysis 74 (1):ant097.
    The overgeneration argument attempts to show that accepting second-order validity as a sound formal counterpart of logical truth has the unacceptable consequence that the Continuum Hypothesis is either a logical truth or a logical falsehood. The argument was presented and vigorously defended in John Etchemendy’s The Concept of Logical Consequence and it has many proponents to this day. Yet it is nothing but a seductive fallacy. I demonstrate this by considering five versions of the argument; as I show, each is (...)
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  17.  81
    The overgeneration argument(s): A succinct refutation.A. C. Paseau - 2014 - Analysis 74 (1):40-47.
    The overgeneration argument attempts to show that accepting second-order validity as a sound formal counterpart of logical truth has the unacceptable consequence that the Continuum Hypothesis is either a logical truth or a logical falsehood. The argument was presented and vigorously defended in John Etchemendy’s The Concept of Logical Consequence and it has many proponents to this day. Yet it is nothing but a seductive fallacy. I demonstrate this by considering five versions of the argument; as I show, each is (...)
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  18. Ways of Being and Logicality.Owen Griffiths & A. C. Paseau - 2023 - Journal of Philosophy 120 (2):94-116.
    Ontological monists hold that there is only one way of being, while ontological pluralists hold that there are many; for example, concrete objects like tables and chairs exist in a different way from abstract objects like numbers and sets. Correspondingly, the monist will want the familiar existential quantifier as a primitive logical constant, whereas the pluralist will want distinct ones, such as for abstract and concrete existence. In this paper, we consider how the debate between the monist and pluralist relates (...)
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  19. Logos, Logic and Maximal Infinity.A. C. Paseau - 2022 - Religious Studies 58:420-435.
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  20.  46
    One Logic, Or Many?Owen Griffiths & A. C. Paseau - 2023 - Philosophy Now 154:8-9.
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  21.  57
    Ancestral Links.A. C. Paseau - 2022 - The Reasoner 16 (7):55-56.
    This short article discusses the fact that the word ‘ancestor’ features in certain arguments that a) are apparently logically valid, b) contain infinitely many premises, and c) are such that none of their finite sub-arguments are logically valid. The article's aim is to motivate, within its brief compass, the study of infinitary logics.
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  22.  47
    Dissemination Corner: One True Logic.A. C. Paseau & Owen Griffiths - 2022 - The Reasoner 16 (1):3-4.
    A brief article introducing *One True Logic*. The book argues that there is one correct foundational logic and that it is highly infinitary.
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  23.  25
    Compactness.A. C. Paseau, and & Robert Leek - 2023 - Internet Encyclopedia of Philosophy.
    The Compactness Theorem The compactness theorem is a fundamental theorem for the model theory of classical propositional and first-order logic. As well as having importance in several areas of mathematics, such as algebra and combinatorics, it also helps to pinpoint the strength of these logics, which are the standard ones used in mathematics and arguably … Continue reading Compactness →.
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  24.  78
    Justin Clarke-Doane* Morality and Mathematics.Michael Bevan & A. C. Paseau - 2020 - Philosophia Mathematica 28 (3):442-446.
    _Justin Clarke-Doane* * Morality and Mathematics. _ Oxford University Press, 2020. Pp. xx + 208. ISBN: 978-0-19-882366-7 ; 978-0-19-2556806.† †.
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  25.  41
    Compactness Theorem.A. C. Paseau & Robert Leek - 2022 - Internet Encyclopedia of Philosophy.
    The Compactness Theorem The compactness theorem is a fundamental theorem for the model theory of classical propositional and first-order logic. As well as having importance in several areas of mathematics, such as algebra and combinatorics, it also helps to pinpoint the strength of these logics, which are the standard ones used in mathematics and arguably … Continue reading Compactness Theorem →.
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  26.  62
    Justin Clarke-Doane*Morality and Mathematics.Michael Bevan & A. C. Paseau - forthcoming - Philosophia Mathematica.
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  27.  47
    Isomorphism invariance and overgeneration – corrigendum.O. Griffiths & A. C. Paseau - 2017 - Bulletin of Symbolic Logic 23 (4):546-546.
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  28.  40
    Erratum to: A measure of inferential-role preservation.A. C. Paseau - 2017 - Synthese 194 (4):1425-1425.
    Erratum to: Synthese DOI 10.1007/s11229-015-0705-5In line 3 of footnote 8 on page 4, ‘allow’ should be ‘disallow’.In line 8 of page 5, \ should be \ and \ should be \. Similarly for lines 1, 2, 3, 7, 8, 13 and 14 of page 6.The entry in row 20 column 6 of the table on page 5 should be 1 rather than 0.The entry \ in row 30 column 5 of the table on page 5 should be \.In line 27 (...)
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  29.  84
    Focussed Issue of The Reasoner on Infinitary Reasoning.A. C. Paseau & Owen Griffiths (eds.) - 2022
    A focussed issue of The Reasoner on the topic of 'Infinitary Reasoning'. Owen Griffiths and A.C. Paseau were the guest editors.
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  30.  61
    JOHN P. BURGESS Rigor and Structure.A. C. Paseau - 2016 - British Journal for the Philosophy of Science 67 (4):1185-1187.
  31.  1
    Non-deductive Justification in Mathematics.A. C. Paseau - 2024 - In Bharath Sriraman (ed.), Handbook of the History and Philosophy of Mathematical Practice. Cham: Springer. pp. 2401-2416.
    In mathematics, the deductive method reigns. Without proof, a claim remains unsolved, a mere conjecture, not something that can be simply assumed; when a proof is found, the problem is solved, it turns into a “result,” something that can be relied on. So mathematicians think. But is there more to mathematical justification than proof?The answer is an emphatic yes, as I explain in this chapter. I argue that non-deductive justification is in fact pervasive in mathematics, and that it is in (...)
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  32. Propositional logics of logical truth.A. C. Paseau & Owen Griffiths - 2021 - In Gil Sagi & Jack Woods (eds.), The Semantic Conception of Logic : Essays on Consequence, Invariance, and Meaning. New York, NY: Cambridge University Press.
  33.  81
    Philosophy of the Matrix.A. C. Paseau - 2017 - Philosophia Mathematica 25 (2):246-267.
    A mathematical matrix is usually defined as a two-dimensional array of scalars. And yet, as I explain, matrices are not in fact two-dimensional arrays. So are we to conclude that matrices do not exist? I show how to resolve the puzzle, for both contemporary and older mathematics. The solution generalises to the interpretation of all mathematical discourse. The paper as a whole attempts to reinforce mathematical structuralism by reflecting on how best to interpret mathematics.
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  34.  28
    The Euclidean Programme.A. C. Paseau & Wesley Wrigley - 2024 - Cambridge, UK: Cambridge University Press.
    The Euclidean Programme embodies a traditional sort of epistemological foundationalism, according to which knowledge – especially mathematical knowledge – is obtained by deduction from self-evident axioms or first principles. Epistemologists have examined foundationalism extensively, but neglected its historically dominant Euclidean form. By contrast, this book offers a detailed examination of Euclidean foundationalism, which, following Lakatos, the authors call the Euclidean Programme. The book rationally reconstructs the programme's key principles, showing it to be an epistemological interpretation of the axiomatic method. It (...)
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  35.  46
    The Laws of Belief: Ranking Theory & its Philosophical Applications, by Wolfgang Spohn.A. C. Paseau - 2017 - Mind 126 (501):273-278.
    The Laws of Belief: Ranking Theory & its Philosophical Applications, by SpohnWolfgang. New York: Oxford University Press, 2012. Pp. xv + 598.
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  36. David Papineau. Philosophical Devices: Proofs, Probabilities, Possibilities, and Sets. Oxford: Oxford University Press, 2012. ISBN 978-0-19965173-3. Pp. xix + 224. [REVIEW]A. C. Paseau - 2013 - Philosophia Mathematica (1):nkt006.
  37.  49
    James Robert Brown. Platonism, naturalism, and mathematical knowledge. New York and London: Routledge, 2012. Isbn 978-0-415-87266-9. Pp. X + 182. [REVIEW]A. C. Paseau - 2012 - Philosophia Mathematica 20 (3):359-364.