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 (...) there is more than one correct logic. The second challenge is to determine which form of logicalmonism is the correct one. One True Logic is the first monograph to explicitly articulate a version of logical monism and defend it against the first challenge. It provides a critical overview of the monism vs pluralism debate and argues for the former. It also responds to the second challenge by defending a particularmonism, based on a highly infinitary logic. It breaks new ground on a number of fronts and unifies disparate discussions in the philosophical and logical literature. In particular, it generalises the Tarski-Sher criterion of logicality, provides a novel defence of this generalisation, offers a clearnew argument for the logicality of infinitary logic and replies to recent pluralist arguments. (shrink)

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 →.

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 (...) first half traces the evolution of the Indispensability Argument from its origins in Quine and Putnam's works, taking in naturalism, confirmational holism, Field's program, and the use of idealisations in science along the way. Its second half examines the explanatory version of the Indispensability Argument, and focuses on several more recent versions of easy-road and hard-road fictionalism respectively. (shrink)

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, (...) our definition eschews metric assumptions, which are unrealistic in most applications of interest. (shrink)

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 (...) picture which builds on his work, but differs in a number of important respects. (shrink)

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.

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 (...) arithmetic. But why? Perhaps the best argument is that known instances of an arithmetical conjecture are almost always small: they appear at the start of the natural number sequence. Evidence of this kind consequently suffers from size bias. My essay shows that this sort of scepticism comes in many different flavours, raises some challenges for them all, and explores their respective responses. (shrink)

Thought: A Journal of Philosophy, Volume 10, Issue 3, Page 188-198, September 2021.

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 (...) is in good epistemic standing. (shrink)

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, (...) even more dramatically, was objects. Curiously, although Benacerraf’s article appeared in the heyday of Quine’s influence, it declined to engage the Quinean position squarely, even seemed to think it was not its business to do so. Despite that, in my experience, most philosophers believe that Benacerraf’s article put paid to the reductionist view that numbers are sets (though perhaps not the view that numbers are objects). My chapter will attempt to overturn this orthodoxy. (shrink)

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 (...) to mirror the inferential network of a natural language or some fragment of it. This paper takes some exploratory steps towards a quantitative account of the main ingredient in the goodness of a formalisation. We introduce and critically examine a mathematical model of how well a formalisation mirrors natural-language inferential relations. (shrink)

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 (...) that none of them fits the bill. In particular, atomized arguments do not ground the correctness of their unatomized counterparts; they are typically not credence-preserving; and they need not reveal the source of inferential disagreement. The moral this suggests is that complete rigour is not even a defeasible epistemic ideal. (shrink)

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 (...) datum that must be taken into account in constructing a theory of fairness. (shrink)

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 (...) either unsound or lacks a troubling conclusion. (shrink)

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 (...) more subtle and more interesting. (shrink)

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.

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.

A brief article introducing *One True Logic*. The book argues that there is one correct foundational logic and that it is highly infinitary.

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 (...) either unsound or lacks a troubling conclusion. (shrink)

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 →.

A focussed issue of The Reasoner on the topic of 'Infinitary Reasoning'. Owen Griffiths and A.C. Paseau were the guest editors.

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 (...) of page 13, ‘it’ should be ‘them’.Four lines from the end of section 12.3 on page 20, ‘premisses’ should be ‘premiss sets’, and .. (shrink)

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 (...) then compares the reconstructed programme with select historical sources: Euclid's Elements, Aristotle's Posterior Analytics, Descartes's Discourse on Method, Pascal's On the Geometric Mind and a twentieth-century account of axiomatisation. The second half of the book philosophically assesses the programme, exploring whether various areas of contemporary mathematics conform to it. The book concludes by outlining a replacement for the Euclidean Programme. (shrink)

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 (...) good epistemic standing. (shrink)

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.

The Laws of Belief: Ranking Theory & its Philosophical Applications, by SpohnWolfgang. New York: Oxford University Press, 2012. Pp. xv + 598.

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 (...) possible precisifications of the overgeneration argument and find them all unconvincing. (shrink)

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 (...) to the standard test for logicality. We deploy this test and show that it favors the monist. (shrink)

_Justin Clarke-Doane* * Morality and Mathematics. _ Oxford University Press, 2020. Pp. xx + 208. ISBN: 978-0-19-882366-7 ; 978-0-19-2556806.† †.

A traditional approach to teaching medical ethics aims to provide knowledge about ethics. This is in line with an epistemological view on ethics in which moral expertise is assumed to be located in theoretical knowledge and not in the moral experience of healthcare professionals. The aim of this paper is to present an alternative, contextual approach to teaching ethics, which is grounded in a pragmatic-hermeneutical and dialogical ethics. This approach is called moral case deliberation. Within moral case deliberation, healthcare professionals (...) bring in their actual moral questions during a structured dialogue. The ethicist facilitates the learning process by using various conversation methods in order to find answers to the case and to develop moral competencies. The case deliberations are not unique events, but are a structural part of the professional training on the work floor within healthcare institutions. This article presents the underlying theory on ethics and illustrates this approach with an example of a moral case deliberation project in a Dutch psychiatric hospital. The project was evaluated using the method of responsive evaluation. This method provided us with rich information about the implementation process and effects the research process itself also lent support to the process of implementation. (shrink)

In the light of increasing public mistrust, there is an urgent need to clarify the moral status of the medical profession and of the relationship of the clinician to his/her patients. In addressing this question, I first establish the coherence, within moral philosophy generally, of the concept of supererogation . I adopt the notion of an act of “unqualified” supererogation as one that is non-derivatively good, praiseworthy, and freely undertaken for others' benefit at the risk of some cost to the (...) agent. I then argue that committing oneself to the profession of clinical medicine is an act of this kind. This is the case, not because the aim of medicine is to help patients, but because of the open ended commitment of time and the vulnerability to the consequences of failure that the clinician must accept. (shrink)

In her groundbreaking book, Epistemic Injustice, renowned moral philosopher and social epistemologist Miranda Fricker coined the term epistemic injustice to draw attention to the pervasive impact of epistemic oppression on marginalized social groups. Fricker’s account spurred a flurry of scholarship regarding the discriminatory impact of epistemic injustice and gave birth to a domain of philosophical inquiry that has extended far beyond the disciplinary boundaries of philosophy. In this interview, Fricker responds to questions posed by A. C. Nikolaidis and Winston C. (...) Thompson that address the ways in which epistemic injustice intersects with education as a human endeavour and social institution. In doing so, Fricker reflects on her motivations in writing her book more than fifteen years ago and explicitly addresses some of the most significant contributions of the concept of epistemic injustice as a tool for analysing issues in education. She also offers insights on the purpose of education, outlines educational manifestations of epistemic injustice, and discusses the virtues that educators must exhibit and inculcate in their students in order for epistemic justice to obtain. (shrink)

This paper notes how Jim influenced my own use of Foucault and also focuses on two of James Marshall's New Zealand oriented texts. In the first, Discipline and Punishment in New Zealand Education he provides a Foucauldian genealogy of New Zealand approaches to both punishment and discipline, in particular corporal punishment. The second, his 1996 book co‐written with Michael Peters, Individualism and Community: Education and Social Policy in the Postmodern Condition, analyses political philosophy and social and educational policy as New (...) Zealand changed from being a welfare state since the 1930s to a neoliberal one since the mid 1980s. Foucauldian understandings about power, bio‐power, governmentality, autonomy and subjectivity are brought to bear in their analysis. (shrink)

In this paper I shall use terms such as “intrinsically good” which may be deemed old fashioned by many readers and which certainly to my own mind presuppose an objective non-naturalistic theory of ethics. I still hold such a theory and I have not mastered the new jargon by which a sort of higher synthesis between that and other theories is supposed to have been effected, but I do not think that such a view as mine of ethics in general (...) is necessarily presupposed if one is to understand or even agree with the contentions of my article. These relate to a specific problem as to certain ethical actions, which will arise on any view that admits the possibility of giving any sort of legitimate reasons for ethical judgments, as we all do in practice. After all a naturalist can easily translate “intrinsically good” into his own terms, say, valued for its own sake by most people who experience it, and there will still be a question as to what is intrinsically and what is merely instrumentally good and other questions as to what is the logical nature of certain arguments in ethics. (shrink)