Until fairly recently a common way of doing history of science was to pick up an important strand of contemporary scientific thought and to trace its origin back to the philosophical tangle of the scientific revolution. This approach conveniently by-passed the breakdowns of once useful and pervasive theories, and neglected the long intellectual journeys along devious routes. History of science read like a success story; the pioneers who failed were neither dismissed nor excused; they were simply ignored. The historian knew (...) what he was hunting for and he was careful to limit his search to areas where his quarry was sure to be found. This method, which has been dubbed the precursor-view, stands in contrast—albeit, not in opposition—to the contextual method, which aims at a better understanding of the actual thought-processes of the early scientists. On this second view, history of science must not only account for present theories in the light of past developments, it must also assess old theories in terms of the scientist's conceptual framework, and judge them against the background of the world picture of his age. This may lead the historian down the blind alleys of the past, chasing spurious attempts at explaining the nature of physical reality, but it can clear the ground for a less anachronistic interpretation of the emergence of modern science and the actual process of discovery. History of science acts as a winnowing fork, but we cannot suppose that the discoverer himself always separated the wheat from the chaff, and we must be ever wary of equating the dream with the task. (shrink)

Explanation in mathematics has recently attracted increased attention from philosophers. The central issue is taken to be how to distinguish between two types of mathematical proofs: those that explain why what they prove is true and those that merely prove theorems without explaining why they are true. This way of framing the issue neglects the possibility of mathematical explanations that are not proofs at all. This paper addresses what it would take for a non-proof to (...) explain. The paper focuses on a particular example of an explanatory non-proof: an argument that mathematicians regard as explaining why a given theorem holds regarding the derivative of an infinite sum of differentiable functions. The paper contrasts this explanatory non-proof with various non-explanatory proofs of the same theorem. The paper offers an account of what makes the given non-proof explanatory. This account is motivated by investigating the difficulties that arise when we try to extend Mark Steiner’s influential account of explanatory proofs to cover this explanatory non-proof. (shrink)

This paper uses famous problems from philosophy of science and philosophical psychology—underdetermination of theory by evidence, Nelson Goodman’s new riddle of induction, theory-ladenness of observation, and “Kripkenstein’s” rule-following paradox—to show that it is empirically impossible to reliably interpret which functions a large language model (LLM) AI has learned, and thus, that reliably aligning LLM behavior with human values is provably impossible. Sections 2 and 3 show that because of how complex LLMs are, researchers must interpret their learned (...) functions largely in terms of empirical observations of their outputs and network behavior. Sections 4–7 then show that for every “aligned” function that might appear to be confirmed by empirical observation, there is always an infinitely larger number of “misaligned”, arbitrarily time-limited functions equally consistent with the same data. Section 8 shows that, from an empirical perspective, we can thus never reliably infer that an LLM or subcomponent of one has learned any particular function at all before any of an uncountably large number of unpredictable future conditions obtain. Finally, Section 9 concludes that the probability of LLM “misalignment” is—at every point in time, given any arbitrarily large body of empirical evidence—always vastly greater than the probability of “alignment.”. (shrink)

This paper concerns the relation between a proof’s beauty and its explanatory power – that is, its capacity to go beyond proving a given theorem to explaining why that theorem holds. Explanatory power and beauty are among the many virtues that mathematicians value and seek in various proofs, and it is important to come to a better understanding of the relations among these virtues. Mathematical practice has long recognized that certain proofs but not others have (...) explanatory power, and this paper offers an account of what makes a proof explanatory. This account is motivated by a wide range of examples drawn from mathematical practice, and the account proposed here is compared to other accounts in the literature. The concept of a proof that explains is closely intertwined with other important concepts, such as a brute force proof, a mathematical coincidence, unification in mathematics, and natural properties. Ultimately, this paper concludes that the features of a proof that would contribute to its explanatory power would also contribute to its beauty, but that these two virtues are not the same; a beautiful proof need not be explanatory. (shrink)

Starting in 2005, general logical metatheorems have been developed that guarantee the extractability of uniform effective bounds from large classes of proofs of theorems that involve abstract metric structures X. In this paper we adapt this to the class of CAT\)-spaces X for \ and establish a new metatheorem that explains specific bound extractions that recently have been achieved in this context as instances of a general logical phenomenon.

The use of figurate numbers (e. g. in the context of elementary number theory) can be considered a heuristic in the field of problem solving or proving. In this paper, we want to discuss this heuristic from the perspectives of the semiotic theory of Peirce (“diagrammatic reasoning” and “collateral knowledge”) and cognitive psychology (“schema theory” and “Gestalt psychology”). We will make use of several results taken from our research to illustrate first-year students’ problems when dealing with figurate numbers in the (...) context of proving. The considerations taken from both theoretical perspectives will help to partly explain such phenomena. It will be shown that the use of figurate numbers must not be considered to be any kind of help for learners or some way of ‘easy’ mathematics. Working in this representational system has to be learned and practiced as another kind of knowledge is necessary for working with figurate numbers. The named findings also touch upon the concept of ‘proofs that explain’. Finally, we will highlight some implications for teaching and point to a number of demands for future research. (shrink)

On a traditional view, the primary role of a mathematical proof is to warrant the truth of the resulting theorem. This view fails to explain why it is very often the case that a new proof of a theorem is deemed important. Three case studies from elementary arithmetic show, informally, that there are many criteria by which ordinary proofs are valued. I argue that at least some of these criteria depend on the methods of (...) inference the proofs employ, and that standard models of formal deduction are not well-equipped to support such evaluations. I discuss a model of proof that is used in the automated deduction community, and show that this model does better in that respect. (shrink)

Over the past half-century, formal, machine-executable proofs have been developed for an impressive range of mathematical theorems. Formalists argue that such proofs should be seen as providing the fully worked out proofs of which mathematicians’ proofs are sketches. Nonformalists argue that this conception of the relationship of formal to informal proofs cannot explain the fact that formal proofs lack essential virtues enjoyed by mathematicians’ proofs, the fact, for example, that formal proofs are not convincing and (...) lack the explanatory power of their informal counterparts. Formal proofs do not yield mathematical insight or understanding, and they are not fruitful in the way that informal proofs can be, for instance, in suggesting novel approaches to old problems. And yet, there is a clear sense in which the formalist is right: Formal proofs do seem to make explicit what is otherwise taken for granted in the mathematician’s reasoning. Very recent work uncovers the source of the dilemma by showing that rigor does not entail formalism insofar as rigor, unlike formalism, is compatible with contentfulness. Mathematical proofs, were they to be recast in a Leibnizian language, at once a characteristica and a calculus, would be, or could be made to be, completely gap-free and hence machine checkable. Because as so recast they would be grounded in mathematical ideas, they would not be machine executable. It is on just this basis that a compelling account can be given of the fact that formal proofs are mostly irrelevant to mathematicians in their practice. (shrink)

Proof-theoretic semantics is an alternative to model-theoretic semantics. It aims at explaining the meaning of the logical constants in terms of the inference rules that govern their behaviour in proofs. We argue that this must be construed as the task of explaining these meanings relative to a logic, i.e., to a consequence relation. Alas, there is no agreed set of properties that a relation must have in order to qualify as a consequence relation. Moreover, the association (...) of a consequence relation to a logical calculus is not as straightforward as it may seem. We show that these facts are problematic for the proof-theoretic project but the problems can be solved. Our thesis is that the consequence relation relevant for proof-theoretic semantics is the one given by the sequent-to-sequent derivability relation in Gentzen systems. (shrink)

Many theorists hold that outright verdicts based on bare statistical evidence are unwarranted. Bare statistical evidence may support high credence, on these views, but does not support outright belief or legal verdicts of culpability. The vignettes that constitute the lottery paradox and the proof paradox are marshalled to support this claim. Some theorists argue, furthermore, that examples of profiling also indicate that bare statistical evidence is insufficient for warranting outright verdicts.I examine Pritchard's and Buchak's treatments (...) of these three kinds of case. Pritchard argues that his safety condition explains the insufficiency of bare statistical evidence for outright verdicts in each of the three cases, while Buchak argues that her treatment of the distinction between credence and belief explains this. In these discussions the three kinds of cases – lottery, proof paradox, and profiling – are treated alike. The cases are taken to exhibit the same epistemic features. I identity significant overlooked epistemic differences amongst these three cases; these differences cast doubt on Pritchard's explanation of the insufficiency of bare statistical evidence for outright verdicts. Finally, I raise the question of whether we should aim for a unified explanation of the three paradoxes. (shrink)

The aim of this paper is to provide epistemic reasons for investigating the notions of informal rigour and informal provability. I argue that the standard view of mathematical proof and rigour yields an implausible account of mathematical knowledge, and falls short of explaining the success of mathematical practice. I conclude that careful consideration of mathematical practice urges us to pursue a theory of informal provability.

Proof and Consequence is a rigorous, elegant introduction to classical first-order natural deductive logic; it provides an accurate and accessible first course in the study of formal systems. The text covers all the topics necessary for learning logic at the beginner and intermediate levels: this includes propositional and quantificational logic (using Suppes-style proofs) and extensive metatheory, as well as over 800 exercises. Proof and Consequence provides exclusive access to the software application Simon, an easily downloadable program designed to (...) facilitate an intuitive understanding of classical logic through the generation and analysis of proofs. It also aids with the representation of natural language sentences in the formal language. Equipped with nearly all the exercises found in the text, Simon helps students work efficiently and effectively by detecting and explaining errors in solutions as they proceed. Students can also submit assignments, view their own records, and check their standing in the class. The complete logic package includes: * The logic textbook, Proof and Consequence *A very helpful study guide to the textbook, containing extra exercises, Simple Simon *Access, through Simon, to the grading software, Simon Says, that allows students to submit assignments and track their grades. (shrink)

Philosophers who regard some mathematical proofs as explaining why theorems hold, and others as merely proving that they do hold, disagree sharply about the explanatory value of proofs by mathematical induction. I offer an argument that aims to resolve this conflict of intuitions without making any controversial presuppositions about what mathematical explanations would be.

In this chapter, Talbott explains how the Proof paradigm, a model of top-down reasoning, has led to a serious misunderstanding of how moral judgments are epistemically justified. Talbott develops an alternative equilibrium model of moral reasoning based on the work of Mill, Rawls, and Habermas and uses it to show how bottom-up reasoning could have led to the discovery of human rights. Talbott uses the U.S. Constitution to illustrate the idea that guarantees of basic human rights are components (...) of a self-improving self-regulating system for promoting justice. The system does not have to begin with self-evident or even true principles of justice. Bottom-up reasoning can lead to changes that make it more just over time. (shrink)

"Proof" has been and remains one of the concepts which characterises mathematics. Covering basic propositional and predicate logic as well as discussing axiom systems and formal proofs, the book seeks to explain what mathematicians understand by proofs and how they are communicated. The authors explore the principle techniques of direct and indirect proof including induction, existence and uniqueness proofs, proof by contradiction, constructive and non-constructive proofs, etc. Many examples from analysis and modern algebra are included. The (...) exceptionally clear style and presentation ensures that the book will be useful and enjoyable to those studying and interested in the notion of mathematical "proof.". (shrink)

The central debate of natural theology among medieval Muslims and Jews concerned whether or not the world was eternal. Opinions divided sharply on this issue because the outcome bore directly on God's relationship with the world: eternity implies a deity bereft of will, while a world with a beginning leads to the contrasting picture of a deity possessed of will. In this exhaustive study of medieval Islamic and Jewish arguments for eternity, creation, and the existence of God, Herbert Davidson provides (...) a systematic classification of the proofs, analyzes and explains them, and traces their sources in Greek philosophy. Throughout the study, Davidson tries to take into account every argument of a philosophical character, disregarding only those arguments that rest entirely on religious faith or which fall below a minimal level of plausibility. (shrink)

People are habitual explanation generators. At its most mundane, our propensity to explain allows us to infer that we should not drink milk that smells sour; at the other extreme, it allows us to establish facts (e.g., theorems in mathematical logic) whose truth was not even known prior to the existence of the explanation (proof). What do the cognitive operations underlying the inference that the milk is sour have in common with the proof (...) class='Hi'>that, say, the square root of two is irrational? Our ability to generate explanations bears striking similarities to our ability to make analogies. Both reflect a capacity to generate inferences and generalizations that go beyond the featural similarities between a novel problem and familiar problems in terms of which the novel problem may be understood. However, a notable difference between analogy-making and explanation-generation is that the former is a process in which a single source situation is used to reason about a single target, whereas the latter often requires the reasoner to integrate multiple sources of knowledge. This seemingly small difference poses a challenge to the task of marshaling our understanding of analogical reasoning to understanding explanation. We describe a model of explanation, derived from a model of analogy, adapted to permit systematic violations of this one-to-one mapping constraint. Simulation results demonstrate that the resulting model can generate explanations for novel explananda and that, like the explanations generated by human reasoners, these explanations vary in their coherence. (shrink)

Mathematicians sometimes judge a mathematical proof to be beautiful and in doing so seem to be making a judgement of the same kind as aesthetic judgements of works of visual art, music or literature. Mathematical proofs are also appraised for explanatoriness: some proofs merely establish their conclusions as true, while others also show why their conclusions are true. This paper will focus on the prima facie plausible assumption that, for mathematical proofs, beauty and explanatoriness tend to go together. (...) To make headway we need to have some grip on what it is for a proof to be beautiful, and for that we need some account of judgements of beauty in general. That is the concern of the first section. The second section faces the problem that it is far from obvious how abstract entities, such as mathematical proofs, can be beautiful, strictly and literally speaking. Reasons are given for the view that they can be. The third section introduces the distinction between proofs which explain their conclusions and proofs which do not. Finally, the question whether, for mathematical proofs, the beautiful and the explanatory tend to coincide is addressed. It is argued that we have reason to doubt that explanatory proofs tend to be beautiful, and insufficient reason to believe or disbelieve that beautiful proofs tend to be explanatory. (shrink)

The aim I am pursuing here is to describe some general aspects of mathematical proofs. In my view, a mathematical proof is a warrant to assert a non-tautological statement which claims that certain objects (possibly a certain object) enjoy a certain property. Because it is proved, such a statement is a mathematical theorem. In my view, in order to understand the nature of a mathematical proof it is necessary to understand the nature of mathematical objects. If we (...) understand them as external entities whose 'existence' is independent of us and if we think that their enjoying certain properties is a fact, then we should argue that a theorem is a statement that claims that this fact occurs. If we also maintain that a mathematical proof is internal to a mathematical theory, then it becomes very difficult indeed to explain how a proof can be a warrant for such a statement. This is the essential content of a dilemma set forth by P. Benacerraf (cf. Benacerraf 1973). Such a dilemma, however, is dissolved if we understand mathematical objects as internal constructions of mathematical theories and think that they enjoy certain properties just because a mathematical theorem claims that they enjoy them. (shrink)

This paper urges the importance of including conditional proof as an inference rule in the teaching of elementary symbolic logic. The paper explains how to make clear to students that conditional proof is valid. This is done by a little proof that shows that hypothetical syllogism (or the chain rule) is both intuitively valid yet redundant. Teaching conditional proof not only aids in a deeper understanding of the meaning of “if” but also provides (...) a strong reminder to the student that they have not proved that the conclusion is true but instead have shown that the conclusion follows from the premises. (shrink)

The notion of “the burden of proof” plays an important role in real-world argumentation contexts, in particular in law. It has also been given a central role in normative accounts of argumentation, and has been used to explain a range of classic argumentation fallacies. We argue that in law the goal is to make practical decisions whereas in critical discussion the goal is frequently simply to increase or decrease degree of belief in a proposition. In the latter (...) case, it is not necessarily important whether that degree of belief exceeds a particular threshold (e.g., ‘reasonable doubt’). We explore the consequences of this distinction for the role that the “burden of proof” has played in argumentation and in theories of fallacy. (shrink)

This paper concerns the role of intuitions in mathematics, where intuitions are meant in the Kantian sense, i.e. the “seeing” of mathematical ideas by means of pictures, diagrams, thought experiments, etc.. The main problem discussed here is whether Platonistic argumentation, according to which some pictures can be considered as proofs (or parts of proofs) of some mathematical facts, is convincing and consistent. As a starting point, I discuss James Robert Brown’s recent book Philosophy of Mathematics, in particular, his primarily examples (...) and analogies. I then consider some alternatives and counterarguments, namely John Norton’s opposite view, that intuitions are just pictorially represented logical arguments and are superfluous; and the Kantian transcendental theory of construction in imagination, as it is developed in the works of Marcus Giaquinto and Michael Friedman. Although I support the claim that some intuitions are essential in mathematical justification, I argue that a Platonistic approach to intuitions is partial and one should go further than a Platonist in explaining how some intuitions can deliver new mathematical knowledge. (shrink)

Moore's proof of an external world is a piece of reasoning whose premises, in context, are true and warranted and whose conclusion is perfectly acceptable, and yet immediately seems flawed. I argue that neither Wright's nor Pryor's readings of the proof can explain this paradox. Rather, one must take the proof as responding to a sceptical challenge to our right to claim to have warrant for our ordinary empirical beliefs, either for any particular empirical belief (...) we might have, or for belief in the existence of an external world itself. I show how Wright's and Pryor's positions are of interest when taken in connection with Humean scepticism, but that it is only linking it with Cartesian scepticism which can explain why the proof strikes us as an obvious failure. (shrink)

The Article addresses three main questions. First: Why do some scholars and decision-makers take evidence assessment criteria as standards of proof and vice versa? The answer comes from the fact that some legal systems are more concerned with assessment criteria and others with standards; therefore jurists educated in different contexts tend to emphasize what they are more familiar with, and to assimilate to it what they are less familiar with. Second: Why do systems differ in those respects? Here (...) the answer stems from the historical, institutional and procedural differences that explain why some systems are more concerned with assessment criteria and others with standards of proof. And third, assuming that both criteria and standards are necessary to legal decision-making about facts: How can a system work if it neglects one of these things? Here the Article argues that there is a functional connection between criteria and standards. The functional connection account is distinguished from a functional equivalence account, and some systems and jurisdictions are referred to in greater detail to support the functional connection claim. (shrink)

The view that a mathematical proof is a sketch of or recipe for a formal derivation requires the proof to function as an argument that there is a suitable derivation. This is a mathematical conclusion, and to avoid a regress we require some other account of how the proof can establish it.

“Proof of concept” is a phrase frequently used in descriptions of research sought in program announcements, in experimental studies, and in the marketing of new technologies. It is often coupled with either a short definition or none at all, its meaning assumed to be fully understood. This is problematic. As a phrase with potential implications for research and technology, its assumed meaning requires some analysis to avoid it becoming a descriptive category that refers to all things scientifically exciting. (...) I provide a short analysis of proof of concept research and offer an example of it within synthetic biology. I suggest that not only are there activities that circumscribe new epistemological categories but there are also associated normative ethical categories or principles linked to the research. I examine these and provide an outline for an alternative ethical account to describe these activities that I refer to as “extended agency ethics”. This view is used to explain how the type of research described as proof of concept also provides an attendant proof of principle that is the result of decision-making that extends across practitioners, their tools, techniques, and the problem solving activities of other research groups. (shrink)

In a recent paper, the mathematician Harold Edwards claimed that Euler’s alleged proof, that Fermat’s last theorem is true for the case n = 3, is flawed. Fermat’s last theorem is the conjecture that there are no positive integers x, y, z, or n, such that n is greater than two and such that xn + yn = zn. In this paper we shall first briefly explain the specific flaw to which Edwards called (...) attention. After that we briefly explain the nature of mathematical proofs and with reference to such proofs explain the nature of gaps in proofs. Then we critically discuss an alternative view concerning the nature of proofs in mathematics and discuss the alternative view critically. Specifically we argue that advocates of this alternative view, which we call mathematical postulationism, have not provided a satisfactory account of the nature of gaps in proofs. The unsatisfactoriness of postulationist accounts of gaps in proofs is revealed through reflection concerning appropriate repairs to proofs with gaps. (shrink)

I explain why model theory is unsatisfactory as a semantic theory and has drawbacks as a tool for proofs on logic systems. I then motivate and develop an alternative, the truth-valuational substitutional approach (TVS), and prove with it the soundness and completeness of the first order Predicate Calculus with identity and of Modal Propositional Calculus. Modal logic is developed without recourse to possible worlds. Along the way I answer a variety of difficulties that have been raised against TVS (...) and show that, as applied to several central questions, model-theoretic semantics can be considered TVS in disguise. The conclusion is that the truth-valuational substitutional approach is an adequate tool for many of our logic inquiries, conceptually preferable over model-theoretic semantics. Another conclusion is that formal logic is independent of semantics, apart from its use of the notion of truth, but that even with respect to it its assumptions are minimal. (shrink)

In order to perform certain actions – such as incarcerating a person or revoking parental rights – the state must establish certain facts to a particular standard of proof. These standards – such as preponderance of evidence and beyond reasonable doubt – are often interpreted as likelihoods or epistemic confidences. Many theorists construe them numerically; beyond reasonable doubt, for example, is often construed as 90 to 95% confidence in the guilt of the defendant. -/- A family of influential cases (...) suggests standards of proof should not be interpreted numerically. These ‘proof paradoxes’ illustrate that purely statistical evidence can warrant high credence in a disputed fact without satisfying the relevant legal standard. In this essay I evaluate three influential attempts to explain why merely statistical evidence cannot satisfy legal standards. (shrink)

The main issue André Porto raises in his paper concerns the use of dot notation to indicate an infinite set of hypotheses. Whereas I agree that one cannot extract a unique infinite expansion from a finite initial segment, in my response I argue that this holds for finite expansions as well. I further explain how my remarks on infinite proof structures are neither motivated by the impact of Gödel’s incompleteness theorems on Hilbert’s program, nor by a (...) negative view of strict finitism.O problema central que André Porto discute em seu artigo diz respeito ao uso da notação de pontos para indicar um conjunto infinito de hipóteses. Mesmo estando de acordo não ser possível extrair uma expansão infinita a partir de um segmento inicial finito, em minha réplica argumento que isto vale igualmente para expansões finitas. Explico também que minhas observações sobre estruturas de prova infinitas não são motivadas pelo impacto dos teoremas de incompletude de Gödel no programa de Hilbert, e tampouco por uma visão negativa do finitismo estrito. (shrink)

A proof P of a theorem T is transferable when a typical expert can become convinced of T solely on the basis of their prior knowledge and the information contained in P. Easwaran has argued that transferability is a constraint on acceptable proof. Meanwhile, a proof P is fixable when it’s possible for other experts to correct any mistakes P contains without having to develop significant new mathematics. Habgood-Coote and Tanswell have observed that some acceptable (...) proofs are both fixable and in need of fixing, in the sense that they contain nontrivial mistakes. The claim that acceptable proofs must be transferable seems quite plausible. The claim that some acceptable proofs need fixing seems plausible too. Unfortunately, these attractive suggestions stand in tension with one another. I argue that the transferability requirement is the problem. Acceptable proofs need only satisfy a weaker requirement I call “corrigibility”. I explain why, despite appearances, the corrigibility standard is preferable to stricter alternatives. (shrink)

I propose and defend a proof-based account of legal exceptions. The basic thought is that the characteristic behaviour of exceptions is to be explained in terms of the distinction, relative to some given decision-type C in some decision-making context, between two classes of relevant facts: those that may, and those that may not, remain uncertain if a token decision C is to count as correctly made. The former is the class of exceptions. A fact F is (...) an exception relative to some decision-type C, I claim, if (i) the ascertainment of F is sufficient for a token decision C not to count as correctly made, and (ii) the ascertainment of the negation of F is not necessary for a token decision C to count as correctly made. I also recuperate, reconstruct and discuss some of HLA Hart’s early views on defeasible judicial decisions. These two projects are closely connected: the latter is a vehicle for the former. (shrink)

In this article, I will discuss Mulla Sadrā's proof of ideas together with his evaluations of Fārābī, Ibn Sīnā and Suhrawardī's views. The aim of the article is to try to provide a certain opinion about the proof that Sadrā developed. It is seen that Sadrā generally exhibits a dual attitude about ideas. Sadrā's first attitude is to match the theologians' teaching of names, Suhrawardi's view of the master of genres, the sufists' a'yan al-sābita theory and (...) the Peripatetics’ concept of universal with ideas. His second attitude is that he is alone at the point of understanding Plato and reaching the clearness of the issue. In other words, he states that there has been no one who have understood and explained the issue as it should be. Sadrā attributes his own success on the subject both to discovery and inspiration, and to his intellectual research on existence. The aim of Sadrā is to try to unite between Plato and Aristotle. Therefore, he also follows Plotinos and Fārābī, who had this purpose before him. Sadrā, who criticizes Avicenna on the subject, sees Suhrawardī as the philosopher who comes closest to the truth, but states that he is also inadequate in some points. (shrink)

Following a brief preface, the second section of this paper discusses Mill's early reflections on the problem of how deductive inference can be illuminating. In the third section it is suggested that in his Logic Mill misconstrued the feature that the premises of a logically valid argument contain the conclusion as the ground of a charge that deductive proof is question-begging. The fourth section discusses the nature of the traditional petitio objection to syllogism, and the fifth (...) shows that Mill had a theory capable of answering it. The final section cites the confusion described in section 3 to explain why Mill nevertheless continued to think that syllogistic proofs beg the question. (shrink)

Time was when the proofs of the existence of God formed an essential part of any self-respecting system of Philosophy. But for many years now this has ceased to be the case. It may be due to the gradual increase of the influence of Kant that the idea seems to have become accepted, tacitly, in the main, but none the less very widely, that proof or disproof of a belief such as this was hardly a fit subject (...) for philosophical discussion at all. At any rate, it is noteworthy how rarely the question is faced directly in the philosophical discussions of the last forty years or so. What we have had, much more frequently, is discussion about how the God of religion might be thought of and what sort of place He might have in reality as conceived in this or that philosophical system. And certainly discussions on these lines have done much to clarify our conception of the Divine, even if they have sometimes seemed to explain it only by explaining it away altogether, or at any rate to present us with the idea of something which, whatever it is, is certainly not the God of any religion. But of recent years I seem to detect a growing readiness to face the question as squarely as the old theologians, whopropounded the first proofs of God's existence, did. (shrink)

Mathematicians only use deductive proofs to establish that mathematical claims are true. They never use inductive evidence, such as probabilistic proofs, for this task. Don Fallis (1997 and 2002) has argued that mathematicians do not have good epistemic grounds for this complete rejection of probabilistic proofs. But Kenny Easwaran (2009) points out that there is a gap in this argument. Fallis only considered how mathematical proofs serve the epistemic goals of individual mathematicians. Easwaran suggests that deductive (...) proofs might be epistemically superior to probabilistic proofs because they are transferable. That is, one mathematician can give such a proof to another mathematician who can then verify for herself that the mathematical claim in question is true without having to rely at all on the testimony of the first mathematician. In this paper, I argue that collective epistemic goals are critical to understanding the methodological choices of mathematicians. But I argue that the collective epistemic goals promoted by transferability do not explain the complete rejection of probabilistic proofs. (shrink)

As a counterpoint to demonstrative proofs in metaphysics, Robert Nozick presented the case for God’s existence based on the value of personal experiences. Personal experiences shape one’s life, but this is even more evident with extraordinary experiences, such can be religious ones. In the next step, says the argument, if those experiences can be explained only by invoking the concept of the Supreme Being, then God exists. The second step mirrors scientific explanation constituting what Nozick calls the “argument to the (...) best explanation.” The argument is set against the background of Nozick’s methodology which rejects demonstrative proofs in metaphysics. Its purpose is threefold. It aims to establish a certain sort of experience (religious/spiritual) as a legitimate basis for the argumentation; it aims to show that it is not philosophically blasphemous to explain such experiences by introducing a concept of divinity. Finally, it seeks to showcase the non-dogmatic, investigative nature of the argument. By exploring the merits of Nozick’s proposal, I will try to elucidate all three components, which should pave the way for a broader discussion on the role of non-demonstrative arguments in metaphysics. (shrink)

Ontologically, brains are more basic than mental representations. Epistemologically, mental representations are more basic than brains and, indeed, all other non-mental entities: it is, and must be, on the basis of mental representations that we know anything about non-mental entities. Since, consequently, mental representations are epistemically more fundamental than brains, the former cannot possibly be explained in terms of the latter, notwithstanding that the latter are ontologically more fundamental than the former. There is thus an explanatory gap, notwithstanding (...) the presumptive truth of materialism. (shrink)

The goals of reduction andreductionism in the natural sciences are mainly explanatoryin character, while those inmathematics are primarily foundational.In contrast to global reductionistprograms which aim to reduce all ofmathematics to one supposedly ``universal'' system or foundational scheme, reductive proof theory pursues local reductions of one formal system to another which is more justified in some sense. In this direction, two specific rationales have been proposed as aims for reductive proof theory, the constructive consistency-proof rationale and the foundational (...) reduction rationale. However, recent advances in proof theory force one to consider the viability of these rationales. Despite the genuine problems of foundational significance raised by that work, the paper concludes with a defense of reductive proof theory at a minimum as one of the principal means to lay out what rests on what in mathematics. In an extensive appendix to the paper,various reduction relations betweensystems are explained and compared, and arguments against proof-theoretic reduction as a ``good'' reducibilityrelation are taken up and rebutted. (shrink)

Here I describe a game that I use in my logic classes once we begin derivations. The game can help improve class dynamics, help struggling students recognizes they are not alone, open lines of communication between students, and help students of all levels prepare for exams. The game can provide struggling students with more practice with the fundamental rules of a logical system while also challenging students who excel at derivations. If students are struggling with particular rules or strategies (...) in the system, the game can be tailored to address them. I explain how the game has evolved since I started using it, highlighting the pedagogical benefits of the changes I have made, and I provide examples of the handouts I distribute and a “checklist” to use before, during, and after the game. (shrink)

Moore’s proof of an external world is a piece of reasoning whose premises, in context, are true and warranted and whose conclusion is perfectly acceptable, and yet immediately seems flawed. I argue that neither Wright’s nor Pryor’s readings of the proof can explain this paradox. Rather, one must take the proof as responding to a sceptical challenge to our right to claim to have warrant for our ordinary empirical beliefs, either for any particular empirical belief (...) we might have, or for belief in the existence of an external world itself. I show how Wright’s and Pryor’s positions are of interest when taken in connection with Humean scepticism, but that it is only linking it with Cartesian scepticism which can explain why the proof strikes us as an obvious failure. (shrink)

Constructive mathematics is based on the thesis that the meaning of a mathematical formula is given, not by its truth-conditions, but in terms of what constructions count as a proof of it. However, the meaning of the terms 'construction' and 'proof' has never been adequately explained (although Kreisel, Goodman and Martin-Löf have attempted axiomatisations). This monograph develops precise (though not wholly formal) definitions of construction and proof, and describes the algorithmic substructure underlying intuitionistic logic. Interpretations of (...) Heyting arithmetic and constructive analysis are given. The philosophical basis of constructivism is explored thoroughly in Part I. The author seeks to answer objections from platonists and to reconcile his position with the central insights of Hilbert's formalism and logic. Audience: Philosophers of mathematics and logicians, both academic and graduate students, particularly those interested in Brouwer and Hilbert; theoretical computer scientists interested in the foundations of functional programming languages and program correctness calculi. (shrink)

Why can testimony alone be enough for findings of liability? Why statistical evidence alone can’t? These questions underpin the “Proof Paradox” (Redmayne 2008, Enoch et al. 2012). Many epistemologists have attempted to explain this paradox from a purely epistemic perspective. I call it the “Epistemic Project”. In this paper, I take a step back from this recent trend. Stemming from considerations about the nature and role of standards of proof, I define three requirements that any successful (...) account in line with the Epistemic Project should meet. I then consider three recent epistemic accounts on which the standard is met when the evidence rules out modal risk (Pritchard 2018), normic risk (Ebert et al. 2020), or relevant alternatives (Gardiner 2019 2020). I argue that none of these accounts meets all the requirements. Finally, I offer reasons to be pessimistic about the prospects of having a successful epistemic explanation of the paradox. I suggest the discussion on the proof paradox would benefit from undergoing a ‘value-turn’. (shrink)

The essay collection "Kant on Proofs for God's Existence" provides a highly needed, comprehensive analysis of the radical turns of Kant's views on proofs for God's existence.— In the "Theory of Heavens" (1755), Kant intends to harmonize the Newtonian laws of motion with a physico-theological argument for the existence of God. But only a few years later, in the "Ground of Proof" essay (1763), Kant defends an ontological ('possibility' or 'modal') argument on the basis of its logical exactitude while (...) he praises the physico-theological argument for its beauty and appeal to the common sense. In the first "Critique" (1781/7), Kant replaces traditional constitutive ontological, cosmological, and physico-theological proofs with his own regulative theoretical and moral-practical religious arguments. He continues to defend a moral argument in the second "Critique" (1788). But in the third "Critique" (1790), Kant reintroduces a physico-theological besides an ethicotheological argument in order to unify the critical system of philosophy. Kant develops further moral arguments and arguments from evil in the "Theodicy" essay (1791) and the "Religion" (1793/4), and still searches for the right kind of proof for God's existence in the "Opus postumum" (1796–1804).—Part one of this volume is dedicated to an analysis of Kant's proofs for God's existence in their historical order that explains which proofs Kant favors or rejects in various periods of his thought. Part two contains a systematic classification of main kinds of proof for God's existence in Kant that outlines the argumentative structure of particular kinds of proof and discusses Kant's potential reasons for their variations and modifications. The essay collection speaks to Kant specialists, philosophers, and theologians, but introduces the topic to non-academic readers also. -/- Contributers: Uygar Abaci (Pennsylvania State University), Craig Bacon (University of South Carolina), Andrew Chignell (Princeton University), Eckart Förster (Humboldt University, Johns Hopkins University), Ina Goy (Beijing Normal University) Paul Guyer (Brown University), Graham Oppy (Monash University), Lara Ostaric (Temple University), Stephen Palmquist (Independent Scholar), Lawrence Pasternack (Oklahoma State University), Konstantin Pollok (University of Mainz), Oliver Sensen (University of Tulane), Allen Wood (Bloomington University, Stanford University). (shrink)

What explains change? Edward Feser argues in his ‘Aristotelian proof’ that the only adequate answer to these questions is ultimately in terms of an unchangeable, purely actual being. In this paper, I target the cogency of Feser’s reasoning to such an answer. In particular, I present novel paths of criticism—both undercutting and rebutting—against one of Feser’s central premises. I then argue that Feser’s inference that the unactualized actualizer lacks any potentialities contains a number of non-sequiturs.

This text makes a great supplement and provides a systematic approach for teaching undergraduate and graduate students how to read, understand, think about, and do proofs. The approach is to categorize, identify, and explain (at the student's level) the various techniques that are used repeatedly in all proofs, regardless of the subject in which the proofs arise. How to Read and Do Proofs also explains when each technique is likely to be used, based on certain key words (...) class='Hi'>that appear in the problem under consideration. Doing so enables students to choose a technique consciously, based on the form of the problem. (shrink)

The European Union is making increased efforts to find simpler and more effective ways to function adequately in the eyes of its citizens by using ‘soft law’ instruments such as recommendations. Although they have no legally binding force, recommendations have practical and legal effects occurring partly due to their normative content in which a course of action is prescribed and further supported by arguments intended to persuade the addressees of a political position. Although recommendations function as persuasive instruments due to (...) their argumentation, the characteristics of argumentation and how it is employed to convince the addressee to comply with certain measures have not been investigated at all. The main goal of the paper is to explain how arguments are used by the European Commission when recommending Member States to follow a new course of action. First, we will unravel the justificatory reasons employed by the Commission in order to make Member States comply with new measures and we will show how these reasons are combined into an argumentative pattern : 1–23, 2016; J Argum Context 6: 3–26, 2017). This pattern basically prescribes a course of action to Member States, which is further supported by arguments in which the necessity and advantages of following the proposed course of action are justified. Second, we will explain how and why the way in which the arguments are combined in this complex pattern could be potentially persuasive for the Member States despite the legally non-binding character of recommendations. We will show that the European Commission tries to persuade the Member States to take new measures by evading the burden of proof imposed by the legislative framework. At a more specific level of analysis, we will delve into the implicit premises in the argumentation, which enable us to identify cases of evasion of the burden of proof due to the Commission’s use of implicit starting points which might not be accepted by the Member States. (shrink)

Leibniz propôs que demonstrações fossem reformuladas como deduções a partir de identidades, e que proposições do tipo A = A fossem a fonte única de verdade. Neste artigo, procuro explicar essa teoria da prova (e do conhecimento), assim como seus conceitos elementares, ou seja, os conceitos de identidade, verdade (ou possibilidade) e proposição (inclusive a teoria leibniziana da redutibilidade a proposições sujeito-predicado). Leibniz proposed that demonstrations be reformulated as deductions from identities, and that propositions of the type A (...) = A be the only source of truth. In this article, I aim to explain this theory of proof (and knowledge), as well as its elementary concepts, such as identity, truth (or possibility) and proposition (including Leibniz's theory of reducibility of propositions to subject-predicate form). (shrink)

In a 2022 paper, Hamami claimed that the orthodox view in mathematics is that a proof is rigorous if it can be translated into a derivation. Hamami then developed a descriptive account that explains how mathematicians check proofs for rigor in this sense and how they develop the capacity to do so. By exploring introductory texts in computability theory, we demonstrate that Hamami’s descriptive account does not accord with actual mathematical practice with respect to computability (...) theory. We argue instead for an alternative account in which imagination, anticipation, and interpretations of natural language play roles in establishing mathematical rigor. (shrink)

§1. Introduction. This paper deals with aspects of my doctoral dissertation which contributed to the early development of model theory. What was of use to later workers was less the results of my thesis, than the method by which I proved the completeness of first-order logic—a result established by Kurt Gödel in his doctoral thesis 18 years before.The ideas that fed my discovery of this proof were mostly those I found in the teachings and writings of Alonzo Church. (...) This may seem curious, as his work in logic, and his teaching, gave great emphasis to the constructive character of mathematical logic, while the model theory to which I contributed is filled with theorems about very large classes of mathematical structures, whose proofs often by-pass constructive methods.Another curious thing about my discovery of a new proof of Gödel's completeness theorem, is that it arrived in the midst of my efforts to prove an entirely different result. Such “accidental” discoveries arise in many parts of scientific work. Perhaps there are regularities in the conditions under which such “accidents” occur which would interest some historians, so I shall try to describe in some detail the accident which befell me.A mathematical discovery is an idea, or a complex of ideas, which have been found and set forth under certain circumstances. The process of discovery consists in selecting certain input ideas and somehow combining and transforming them to produce the new output ideas. The process that produces a particular discovery may thus be represented by a diagram such as one sees in many parts of science; a “black box” with lines coming in from the left to represent the input ideas, and lines going out to the right representing the output. To describe that discovery one must explain what occurs inside the box, i.e., how the outputs were obtained from the inputs. (shrink)