ABSTRACTIn this paper, we present a generalisation of proof simulation procedures for Frege systems by Bonet and Buss to some logics for which the deduction theorem does not hold. In particular, we study the case of finite-valued Łukasiewicz logics. To this end, we provide proof systems and which augment Avron's Frege system HŁuk with nested and general versions of the disjunction elimination rule, respectively. For these systems, we provide upper bounds on speed-ups w.r.t. both the number of steps in (...) proofs and the length of proofs. We also consider Tamminga's natural deduction and Avron's hypersequent calculus GŁuk for 3-valued Łukasiewicz logic and generalise our results considering the disjunction elimination rule to all finite-valued Łukasiewicz logics. (shrink)

We give proofs of the effective monotone interpolation property for the system of modal logic K, and others, and the system IL of intuitionistic propositional logic. Hence we obtain exponential lower bounds on the number of proof-lines in those systems. The main results have been given in [P. Hrubeš, Lower bounds for modal logics, Journal of Symbolic Logic 72 941–958; P. Hrubeš, A lower bound for intuitionistic logic, Annals of Pure and Applied Logic 146 72–90]; here, we give considerably (...) simplified proofs, as well as some generalisations. (shrink)

We give topological proofs of Görnemann’s adaptation to Heyting algebras of the Rasiowa-Sikorski Lemma for Boolean algebras; and of the Rauszer-Sabalski generalisation of it to distributive lattices. The arguments use the Priestley topology on the set of prime filters, and the Baire category theorem. This is preceded by a discussion of criteria for compactness of various spaces of subsets of a lattice, including spaces of filters, prime filters etc.

Chapter 1 constitutes an introduction to Gentzen calculi from two perspectives, logical and philosophical. It introduces the notion of generalisations of Gentzen sequent calculus and the discussion on properties that characterize good inferential systems. Among the variety of Gentzen-style sequent calculi, I divide them in two groups: syntactic and semantic generalisations. In the context of such a discussion, the inferentialist philosophy of the meaning of logical constants is introduced, and some potential objections – mainly concerning the choice of working with (...) semantic generalizations – are addressed. Finally, I’ll introduce the case studies that I’ll be dealing with in part II. Chapter 2 is concerned with the origins and development of Jaśkowski’s discussive logic. The main idea of this chapter is to systematize the various stages of the development of discussive logic related researches from two different angles, i.e., its connections to modal logics and its proof theory, by highlighting virtues and vices. Chapter 3 focuses on the Gentzen-style proof theory of discussive logic, by providing a characterization of it in terms of labelled sequent calculi. Chapter 4 deals with the Gentzen-style proof theory of relevant logics, again by employing the methodology of labelled sequent calculi. This time, instead of working with a single logic, I’ll deal with a whole family of them. More precisely, I’ll study in terms of proof systems those relevant logics that can be characterised, at the semantic level, by reduced Routley-Meyer models, i.e., relational structures with a ternary relation between states and a unique base element. Chapter 5 investigates the proof theory of a modal expansion of intuitionistic propositional logic obtained by adding an ‘actuality’ operator to the connectives. This logic was introduced also using Gentzen sequents. Unfortunately, the original proof system is not cut-free. This chapter shows how to solve this problem by moving to hypersequents. Chapter 6 concludes the investigations and discusses the future of the research presented throughout the dissertation. (shrink)

Canonical extension has proven to be a powerful tool in algebraic study of propositional logics. In this paper we describe a generalisation of the theory of canonical extension to the setting of first order logic. We define a notion of canonical extension for coherent categories. These are the categorical analogues of distributive lattices and they provide categorical semantics for coherent logic, the fragment of first order logic in the connectives ∧, ∨, 0, 1 and ∃. We describe a universal (...) property of our construction and show that it generalises the existing notion of canonical extension for distributive lattices. Our new construction for coherent categories has led us to an alternative description of the topos of types, introduced by Makkai in [22]. This allows us to give new and transparent proofs of some properties of the action of the topos of types construction on morphisms. Furthermore, we prove a new result relating, for a coherent category C, its topos of types to its category of models. (shrink)

Prawitz conjectured that proof-theoretic validity offers a semantics for intuitionistic logic. This conjecture has recently been proven false by Piecha and Schroeder-Heister. This article resolves one of the questions left open by this recent result by showing the extensional alignment of proof-theoretic validity and general inquisitive logic. General inquisitive logic is a generalisation of inquisitive semantics, a uniform semantics for questions and assertions. The paper further defines a notion of quasi-proof-theoretic validity by restricting proof-theoretic validity to allow double negation (...) elimination for atomic formulas and proves the extensional alignment of quasi-proof-theoretic validity and inquisitive logic. (shrink)

Cerreia-Vioglio et al. :341–375, 2011) have proposed a very general axiomatisation of preferences in the presence of ambiguity, viz. Monotonic Bernoullian Archimedean preference orderings. This paper investigates the problem of Arrovian aggregation of such preferences—and proves dictatorial impossibility results for both finite and infinite populations. Applications for the special case of aggregating expected-utility preferences are given. A novel proof methodology for special aggregation problems, based on model theory, is employed.

In this paper, I provide an argument for rejecting Sarah Moss's recent account of legal proof. Moss's account is attractive in a number of ways. It provides a new version of a knowledge-based theory of legal proof that elegantly resolves a number of puzzles about mere statistical evidence in the law. Moreover, the account promises to have attractive implications for social and moral philosophy, in particular about the impermissibility of racial profiling and other harmful kinds of statistical generalisation. In (...) this paper, I show that Moss's account of legal proof crucially depends on a moral norm called the rule of consideration. I argue that we have a number of reasons to be sceptical of this rule. Once we reject the rule, it is not clear that Moss's account of legal proof is either plausible or attractive. (shrink)

The paper contains an exposition of some non standard approach to gentzenization of modal logics. The first section is devoted to short discussion of desirable properties of Gentzen systems and the short review of various sequential systems for modal logics. Two non standard, cut-free sequent systems are then presented, both based on the idea of using special modal sequents, in addition to usual ones. First of them, GSC I is well suited for nonsymmetric modal logics The second one, GSC II (...) is devised specially for symmetric, i.e. Blogics. GSC I and GSC II are not different formalizations, from the theoretical point of view GSC I may be seen as a simplification of the more general approach present in GSC II. They are considered separately, mainly because Blogics demand different, and more complicated, strategy in completeness proof, whereas non symmetric logics are easily and uniformly characterised by means of Fitting's Consistency Properties. The weakest modal logic captured by this formalization is minimal regular logic C, but many stronger logics are obtainable by addition of suitable structural rules, which conforms to Do en's methodology. Both variants of GSC satisfy also other, besides cut-freedom, desirable properties. (shrink)

The interplay of philosophical ambitions and technical reality have given birth to rich and interesting approaches to explain the oft-claimed special character of mathematical and logical knowledge. Two projects stand out both for their audacity and their innovativeness. These are logicism and proof-theoretic semantics. This dissertation contains three chapters exploring the limits of these two projects. In both cases I find the formal results offer a mixed blessing to the philosophical projects. Chapter 1. Is a logicist bound to the claim (...) that as a matter of analytic truth there is an actual infinity of objects? If Hume’s Principle is analytic then in the standard setting the answer appears to be yes. Hodes’s work pointed to a way out by offering a modal picture in which only a potential infinity was posited. However, this project was abandoned due to apparent failures of cross-world predication. I re-explore this idea and discover that in the setting of the potential infinite one can interpret first-order Peano arithmetic, but not second-order Peano arithmetic. I conclude that in order for the logicist to weaken the metaphysically loaded claim of necessary actual infinities, they must also weaken the mathematics they recover. Chapter 2. There have been several recent results bringing into focus the super-intuitionistic nature of most notions of proof-theoretic validity. But there has been very little work evaluating the consequences of these results. In this chapter, I explore the question of whether these results undermine the claim that proof-theoretic validity shows us which inferences follow from the meaning of the connectives when defined by their introduction rules. It is argued that the super-intuitionistic inferences are valid due to the correspondence between the treatment of the atomic formulas and more complex formulas. And so the goals of proof-theoretic validity are not undermined. Chapter 3. Prawitz (1971) conjectured that proof-theoretic validity offers a semantics for intuitionistic logic. This conjecture has recently been proven false by Piecha and Schroeder-Heister (2019). This chapter resolves one of the questions left open by this recent result by showing the extensional alignment of proof-theoretic validity and general inquisitive logic. General inquisitive logic is a generalisation of inquisitive semantics, a uniform semantics for questions and assertions. The chapter further defines a notion of quasi-proof-theoretic validity by restricting proof-theoretic validity to allow double negation elimination for atomic formulas and proves the extensional alignment of quasi-proof-theoretic validity and inquisitive logic.Abstract prepared by Will Stafford extracted partially from the dissertation. (shrink)

We obtain modal completeness of the interpretability logics IL $\!\!\textsf {P}_{\textsf {0}}$ and ILR w.r.t. generalised Veltman semantics. Our proofs are based on the notion of full labels [2]. We also give shorter proofs of completeness w.r.t. the generalised semantics for many classical interpretability logics. We obtain decidability and finite model property w.r.t. the generalised semantics for IL $\textsf {P}_{\textsf {0}}$ and ILR. Finally, we develop a construction that might be useful for proofs of completeness of extensions (...) of ILW w.r.t. the generalised semantics in the future, and demonstrate its usage with $\textbf {IL}\textsf {W}^\ast = \textbf {IL}\textsf {WM}_{\textsf {0}}$. (shrink)

We consider the task of theorem proving in Lambek calculi and their generalisation to ‘multimodal residuation calculi’. These form an integral part of categorial logic, a logic of signs stemming from categorial grammar, of the basis of which language processing is essentially theorem proving. The demand of this application is not just for efficient processing of some or other specific calculus, but for methods that will be generally applicable to categorial logics.It is proposed that multimodal cases be treated by (...) dealing with the highest common factor of all the connectives as linear validity. The prosodic aspects are encoded in labels, in effect the term-structure of quantified linear logic. The correctness condition on proof nets can be implemented by SLD resolution in linear logic with unification on labels/terms limited to one way matching. A suitable unification strategy is obtained for calculi of discontinuity by normalisation of the ground goal term followed by recursive descent and redex pattern matching on the head term. (shrink)

In recent years, several remarkable results have shown that certain theorems of finite combinatorics are unprovable in certain logical systems. These developments have been instrumental in stimulating research in both areas, with the interface between logic and combinatorics being especially important because of its relation to crucial issues in the foundations of mathematics which were raised by the work of Kurt Godel. Because of the diversity of the lines of research that have begun to shed light on these issues, there (...) was a need for a comprehensive overview which would tie the lines together. This volume fills that need by presenting a balanced mixture of high quality expository and research articles that were presented at the August 1985 AMS-IMS-SIAM Joint Summer Research Conference, held at Humboldt State University in Arcata, California.With an introductory survey to put the works into an appropriate context, the collection consists of papers dealing with various aspects of 'unprovable theorems and fast-growing functions'. Among the topics addressed are: ordinal notations, the dynamical systems approach to Ramsey theory, Hindman's finite sums theorem and related ultrafilters, well quasiordering theory, uncountable combinatorics, nonstandard models of set theory, and a length-of-proof analysis of Godel's incompleteness theorem. Many of the articles bring the reader to the frontiers of research in this area, and most assume familiarity with combinatorics and/or mathematical logic only at the senior undergraduate or first-year graduate level. (shrink)

This article generalises the refined A-translation method for extracting programs from classical proofs [U. Berger,W. Buchholz, H. Schwichtenberg, Refined program extraction from classical proofs, Annals of Pure and Applied Logic 114 3–25] to the scenario where additional assumptions such as choice principles are involved. In the case of choice principles, this is done by adding computational content to the ‘translated’ assumptions, an idea which goes back to [S. Berardi, M. Bezem, T. Coquand, On the computational content of the (...) axiom of choice, JSL 63 600–622] and has been further elaborated in [U. Berger, P. Oliva, Modified bar recursion and classical dependent choice, in: M. Baaz, S.D. Friedman, J. Kraijcek , Logic Colloquium ’01, in: Lecture Notes in Logic, vol. 20, Springer, Berlin, 2005, pp. 89–107]. We further investigate the applicability of this extension by means of a small but non-trivial example, and discuss the formalisation as well as some optimisations necessary in order to extract terminating programs. Both formalisation and term extraction have been implemented in the interactive proof assistant Minlog. (shrink)

This paper gives an overview of the common approach to quantification and generalised quantification in formal linguistics and philosophy of language. We point out how this usual general framework represents a departure from empirical linguistic data. We briefly sketch a different idea for proof theory which is closer to the language itself than standard approaches in many aspects. We stress the importance of Hilbert’s operators—the epsilon-operator for existential and tau-operator for universal quantifications. Indeed, these operators are helpful in the construction (...) of a semantic representation which is close to natural language in particular with quantified noun phrases as individual terms. We also define guidelines for the design of proof rules corresponding to generalized quantifiers. (shrink)

This paper gives an overview of the common approach to quantification and generalised quantification in formal linguistics and philosophy of language. We point out how this usual general framework represents a departure from empirical linguistic data. We briefly sketch a different idea for proof theory which is closer to the language itself than standard approaches in many aspects. We stress the importance of Hilbert’s operators—the epsilon-operator for existential and tau-operator for universal quantifications. Indeed, these operators are helpful in the construction (...) of a semantic representation which is close to natural language in particular with quantified noun phrases as individual terms. We also define guidelines for the design of proof rules corresponding to generalized quantifiers. (shrink)

This work studies the structure of proofs containing non-analytic cuts in the cut-based system, a sequent inference system in which the cut rule is not eliminable and the only branching rule is the cut. Such sequent system is invertible, leading to the KE-tableau decision method. We study the structure of such proofs, proving the existence of a normal form for them in the form of a comb-tree proof. We then concentrate on the problem of efficiently computing non-analytic cuts. (...) For that, we study the generalisation of techniques present in many modern theorem provers, namely the techniques of conflict-driven formula learning. (shrink)

The burden of proof (BOP) fallacy is an informal fallacy involving the failure to recognize or properly assign the BOP in a persuasive reasoned dialogue, that is, an interchange between two or more parties whose aim is to prove or defend a position and, in doing so, persuade the other side of its truth or plausibility. In some such dialogues, the amount or strength of evidence required in order to accomplish this goal reasonably may differ for one of the parties (...) involved. The BOP fallacy can occur in two ways. The first is when one side of the dialogue fails to recognize that its opponent incurs the BOP. The second way the BOP fallacy can occur is when one side of the dialogue assigns the BOP incorrectly. There are still some factors that play a role in determining which side of a dialogue incurs the BOP and this chapter discusses these factors. (shrink)

Who carries the burden of proof in analytic philosophical debates, and how can this burden be satisfied? As it turns out, the answer to this joint question yields a fundamental challenge to the very conduct of metaphysics in analytic philosophy. Empirical research presented in this book indicates that the vastly predominant goal pursued in analytic philosophical dialogues lies not in discovering truths or generating knowledge, but merely in prevailing over one’s opponents. Given this goal, the book examines how most effectively (...) to allocate and discharge the burden of proof. It focuses on premises that must prudently be avoided because a burden of proof on them could never be satisfied, and in particular discusses unsupportable bridge premises across inference barriers, like Hume’s barrier between ‘is’ and ‘ought’, or the barrier between the content of our talk or thought, and the world beyond such content. Employing this content/world barrier for a critical assessment of mainstream analytic philosophical methods, this book argues that we must prudently avoid invoking intuitions or other content of thought or talk in support of claims about the world beyond content, that is, metaphysically significant claims. Yet as content-located evidence is practically indispensable to metaphysical debates throughout analytic philosophy, from ethics to the philosophy of mathematics, this book reaches the startling conclusion that all such metaphysical debates must, prudently, be terminated. (shrink)

We prove that the category of Nachbin’s compact ordered spaces and order-preserving continuous maps between them is dually equivalent to a variety of algebras, with operations of at most countable arity. Furthermore, we observe that the countable bound on the arity is the best possible: the category of compact ordered spaces is not dually equivalent to any variety of finitary algebras. Indeed, the following stronger results hold: the category of compact ordered spaces is not dually equivalent to any finitely accessible (...) category, any first-order definable class of structures, and any class of finitary algebras closed under products and subalgebras. An explicit equational axiomatisation of the dual of the category of compact ordered spaces is obtained; in fact, we provide a finite one, meaning that our description uses only finitely many function symbols and finitely many equational axioms. In preparation for the latter result, we establish a generalisation of a celebrated theorem by Mundici: our result—whose proof is independent of Mundici’s theorem—asserts that the category of unital commutative distributive lattice-ordered monoids is equivalent to the category of what we call MV-monoidal algebras.Abstract taken directly from the thesis.E-mail: [email protected]: https://air.unimi.it/retrieve/handle/2434/812809/1698986/phd_unimi_R11882.pdf. (shrink)

If proofs are nothing more than truth makers, then there is no force in the standard argument against classical logic (there is no guarantee that there is either a proof forA or a proof fornot A). The standard intuitionistic conception of a mathematical proof is stronger: there are epistemic constraints on proofs. But the idea that proofs must be recognizable as such by us, with our actual capacities, is incompatible with the standard intuitionistic explanations of the meanings (...) of the logical constants. Proofs are to be recognizable in principle, not necessarily in practice, as shown in section 1. Section 2 considers unknowable propositions of the kind involved in Fitch''s paradox:p and it will never be known thatp. It is argued that the intuitionist faces a dilemma: give up strongly entrenched common sense intuitions about such unknowable propositions, or give up verificationism. The third section considers one attempt to save intuitionism while partly giving up verificationism: keep the idea that a proposition is true iff there is a proof (verification) of it, and reject the idea that proofs must be recognizable in principle. It is argued that this move will have the effect that some standard reasons against classical semantics will be effective also against intuitionism. This is the case with Dummett''s meaning theoretical argument. At the same time the basic reason for regarding proofs as more than mere truth makers is lost. (shrink)

Generalises the concept of a relational model for modal logic, due to Kripke, so as to obtain a closer correspondence between relational and algebraic models. The generalisation obtained is essentially equivalent to the notion of a "first-order" model that was defined independently by S.K.Thomason.

The papers collected in this volume exemplify some of the trends in current approaches to logic, language and computation. Written by authors with varied academic backgrounds, the contributions are intended for an interdisciplinary audience. The first part of this volume addresses issues relevant for multi-agent systems: reasoning with incomplete information, reasoning about knowledge and beliefs, and reasoning about games. Proofs as formal objects form the subject of Part II. Topics covered include: contributions on logical frameworks, linear logic, and different (...) approaches to formalized reasoning. Part III focuses on representations and formal methods in linguistic theory, addressing the areas of comparative and temporal expressions, modal subordination, and compositionality. (shrink)

How is the burden of proof to be distributed among individuals who are involved in resolving a particular issue? Under what conditions should the burden of proof be distributed unevenly? We distinguish attitudinal from dialectical burdens and argue that these questions should be answered differently, depending on which is in play. One has an attitudinal burden with respect to some proposition when one is required to possess sufficient evidence for it. One has a dialectical burden with respect to some proposition (...) when one is required to provide supporting arguments for it as part of a deliberative process. We show that the attitudinal burden with respect to certain propositions is unevenly distributed in some deliberative contexts, but in all of these contexts, establishing the degree of support for the proposition is merely a means to some other deliberative end, such as action guidance, or persuasion. By contrast, uneven distributions of the dialectical burden regularly further the aims of deliberation, even in contexts where the quest for truth is the sole deliberative aim, rather than merely a means to some different deliberative end. We argue that our distinction between these two burdens resolves puzzles about unevenness that have been raised in the literature. (shrink)

In discussions of standards of proof, a familiar perspective often emerges. According to what we call specificationism, standards of proof are legal rules that specify the quantum of evidence required to determine that a litigant’s claim has been proven. In so doing, they allocate the risk of error among litigants (and potential litigants), minimizing the risk of certain types of error. Specificationism is meant as a description of the way the rules actually function. We argue, however, that its claims are (...) either mistaken or at a minimum deeply misleading, especially when it comes to standard of proof rules (SPRs) that contain indeterminate formulas, as is typical. As against specificationism, we argue that SPRs are best understood as rules that confer competence to decide whether a given standard has been met—according to whatever vague or inchoate interpretation (if any) of the rule in question triers of fact implicitly or explicitly employ. We call this the competence-norm approach. (shrink)

The notion of burden of proof and its companion notion of presumption are central to argumentation studies. This book argues that we can learn a lot from how the courts have developed procedures over the years for allocating and reasoning with presumptions and burdens of proof, and from how artificial intelligence has built precise formal and computational systems to represent this kind of reasoning. The book provides a model of reasoning with burden of proof and presumption, based on analyses of (...) many clearly explained legal and non-legal examples. The model is shown to fit cases of everyday conversational argumentation as well as argumentation in legal cases. Burden of proof determines under what conditions an arguer is obliged to support a claim with an argument that backs it up and how strong that argument needs to be to prove the claim in question. (shrink)

This volume contains articles covering a broad spectrum of proof theory, with an emphasis on its mathematical aspects. The articles should not only be interesting to specialists of proof theory, but should also be accessible to a diverse audience, including logicians, mathematicians, computer scientists and philosophers. Many of the central topics of proof theory have been included in a self-contained expository of articles, covered in great detail and depth. The chapters are arranged so that the two introductory articles come first; (...) these are then followed by articles from core classical areas of proof theory; the handbook concludes with articles that deal with topics closely related to computer science. (shrink)

The Nuts and Bolts of Proofs: An Introduction to Mathematical Proofs, Fifth Edition provides basic logic of mathematical proofs and shows how mathematical proofs work. It offers techniques for both reading and writing proofs. The second chapter of the book discusses the techniques in proving if/then statements by contrapositive and proofing by contradiction. It also includes the negation statement, and/or. It examines various theorems, such as the if and only-if, or equivalence theorems, the existence theorems, (...) and the uniqueness theorems. In addition, use of counter examples, mathematical induction, composite statements including multiple hypothesis and multiple conclusions, and equality of numbers are covered in this chapter. The book also provides mathematical topics for practicing proof techniques. Included here are the Cartesian products, indexed families, functions, and relations. The last chapter of the book provides review exercises on various topics. Undergraduate students in engineering and physical science will find this book accessible as well as invaluable. (shrink)

The present paper proposes a generalisation of the notion of disjunctive sequence, that is, of an infinite sequence of letters having each finite sequence as a subword. Our aim is to give a reasonable notion of disjunctiveness relative to a given set of sequences F. We show that a definition like “every subword which occurs at infinitely many different positions in sequences in F has to occur infinitely often in the sequence” fulfils properties similar to the original unrelativised notion (...) of disjunctiveness. Finally, we investigate our concept of generalised disjunctiveness in spaces of Cantor expansions of reals. (shrink)

This book introduces readers to the art of doing mathematical proofs. Proofs are the glue that holds mathematics together. They make connections between math concepts and show why things work the way they do. This book teaches the art of proofs using familiar high school concepts, such as numbers, polynomials, functions, and trigonometry. It retells math as a story, where the next chapter follows from the previous one. Readers will see how various mathematical concepts are tied, will (...) see mathematics is not a pile of formulas and facts, but has an orderly and beautiful edifice. The author begins with basic rules of logic, and then progresses through the topics already familiar to the students: numbers, inequalities, functions, polynomials, exponents, and trigonometric functions. There are also beautiful proofs for conic sections, sequences, and Fibonacci numbers. Each chapter has exercises for the reader. (shrink)

Fear generalisation is a psychological phenomenon that occurs when fear that is elicited in response to a frightening stimulus spreads to similar or related stimuli. The practical harms of pathological fear generalisation related to trauma are well-documented, but little or no attention has been given so far to its epistemic harms. This paper fills this gap in the literature. It shows how the psychological phenomenon, when it becomes pathological, substantially curbs the epistemic agency of those who experience the (...) fear that generalizes, limiting their ability to respond to evidence, and substantially limiting their epistemic horizons. It is argued that when these epistemic harms are caused by wrongful actions and decisions of individuals or institutions, because the fear is elicited in response to a traumatic experience inflicted by them, the harms should be considered epistemic wrongs. The epistemic wrongs are closely akin to agential epistemic injustice, a variety of distributive epistemic injustice, and sometimes also involve epistemic exclusion. The paper thereby identifies a previously underexplored psychological mechanism that can be a vehicle through which both individuals and institutions can epistemically wrong others. The argument has implications for how both epistemic wrongs and epistemic injustice should be conceived, suggesting that both can occur without being caused by primarily epistemic flaws or errors, or a bad epistemic character. Finally, it highlights the advantage of taking a victim-centred approach to understanding epistemic harm. (shrink)

The article discusses burden of proof rules in social criticism. By social criticism I mean an argumentative situation in which an opponent publicly argues against certain social practices; the examples I consider are discrimination on the basis of species and discrimination on the basis of one's nationality. I argue that burden of proof rules assumed by those who defend discrimination are somewhat dubious. In social criticism, there are no shared values which would uncontroversially determine what is the reasonable presumption and (...) who has the burden of proof, nor are there formal rules which would end the debate and determine the winner at a specific point. (shrink)

How the concept of proof has enabled the creation of mathematical knowledge. The Story of Proof investigates the evolution of the concept of proof--one of the most significant and defining features of mathematical thought--through critical episodes in its history. From the Pythagorean theorem to modern times, and across all major mathematical disciplines, John Stillwell demonstrates that proof is a mathematically vital concept, inspiring innovation and playing a critical role in generating knowledge. Stillwell begins with Euclid and his influence on the (...) development of geometry and its methods of proof, followed by algebra, which began as a self-contained discipline but later came to rival geometry in its mathematical impact. In particular, the infinite processes of calculus were at first viewed as "infinitesimal algebra," and calculus became an arena for algebraic, computational proofs rather than axiomatic proofs in the style of Euclid. Stillwell proceeds to the areas of number theory, non-Euclidean geometry, topology, and logic, and peers into the deep chasm between natural number arithmetic and the real numbers. In its depths, Cantor, Gödel, Turing, and others found that the concept of proof is ultimately part of arithmetic. This startling fact imposes fundamental limits on what theorems can be proved and what problems can be solved. Shedding light on the workings of mathematics at its most fundamental levels, The Story of Proof offers a compelling new perspective on the field's power and progress. (shrink)

A common strategy in bioethics is to posit a prima facie case in favour of one policy, and to then claim that the burden of proof falls on those with opposing views. If the burden of proof is not met, it is claimed, then the policy in question should be accepted. This article illustrates, and critically evaluates, examples of this strategy in debates about the sale of organs by living donors, human enhancement, and the precautionary principle. We highlight general problems (...) with this style of argument, and particular problems with its use in specific cases. We conclude that the burden ultimately falls on decision-makers to choose the policy supported by the best reasons. (shrink)

1. Generalising a given set of examples is essential in many machine learning techniques. In principle, a machine learning algorithm builds an abstract model that represents a set of examples. But...

Some thirty years ago, two proposals were made concerning criteria for identity of proofs. Prawitz proposed to analyze identity of proofs in terms of the equivalence relation based on reduction to normal form in natural deduction. Lambek worked on a normalization proposal analogous to Prawitz's, based on reduction to cut-free form in sequent systems, but he also suggested understanding identity of proofs in terms of an equivalence relation based on generality, two derivations having the same generality if (...) after generalizing maximally the rules involved in them they yield the same premises and conclusions up to a renaming of variables. These two proposals proved to be extensionally equivalent only for limited fragments of logic. The normalization proposal stands behind very successful applications of the typed lambda calculus and of category theory in the proof theory of intuitionistic logic. In classical logic, however, it did not fare well. The generality proposal was rather neglected in logic, though related matters were much studied in pure category theory in connection with coherence problems, and there are also links to low-dimensional topology and linear algebra. This proposal seems more promising than the other one for the general proof theory of classical logic. (shrink)

Excessive fear generalisation is a feature characteristic of clinical anxiety and has been linked to its aetiology. Previous animal studies have shown that the mere passage of time increases fear generalisation and that brief exposure to training cues prior to long-term testing reverses this effect. The current study examined these phenomena in humans. Healthy participants learned the relationship between the presentation of a picture of a neutral male face and the delivery of a mild shock. One group was (...) immediately tested with a novel picture of a somewhat different male face (generalisation test). Another group was tested one week later. A third group was also tested one week later and was additionally exposed to the training picture prior to testing. During picture presentations, shock-expectancy ratings were obtained as a measure of fear. Fear generalisation increased from the immediate test to the 1-week follow-up test. This result could not be attributed to level of neuroticism or a general increase in fear (incubation). Furthermore, the time-dependent increase in fear generalisation vanished following brief exposure to the training picture. Results indicate that human fear generalisation is a temporally dynamic process and that memory for stimulus details can be re-established following a reminder treatment. (shrink)

This paper presents an analysis of the concept of burden of proof in argument. Relationship of burden of proof to three traditional informal fallacies is considered: (i) argumentum ad hominem, (ii) petitio principii, and (iii) argumentum ad ignorantiam. Other topics discussed include persuasive dialoque, pragmatic reasoning, legal burden of proof, plausible reasoning in regulated disputes, rules of dialogue, and the value of reasoned dialogue.