Dialectical foundationalists, including Adler, Brandom, Leite, and Williams, claim that some asserted propositions do not require defense just because an interlocutor challenges them. By asserting such a proposition, the speaker shifts the burden of proof to her interlocutor. Dialectical egalitarians claim that all asserted propositions require defense when challenged. I elucidate the dispute between dialectical foundationalists and egalitarians, and I defend a broadly egalitarian stance against several prominent objections.

Recent years have seen fresh impetus brought to debates about the proper role of statistical evidence in the law. Recent work largely centres on a set of puzzles known as the ‘proof paradox’. While these puzzles may initially seem academic, they have important ramifications for the law: raising key conceptual questions about legal proof, and practical questions about DNA evidence. This article introduces the proof paradox, why we should care about it, and new work attempting to resolve (...) it. (shrink)

The standard approach to what I call “proof-theoretic semantics”, which is mainly due to Dummett and Prawitz, attempts to give a semantics of proofs by defining what counts as a valid proof. After a discussion of the general aims of proof-theoretic semantics, this paper investigates in detail various notions of proof-theoretic validity and offers certain improvements of the definitions given by Prawitz. Particular emphasis is placed on the relationship between semantic validity concepts and validity concepts used (...) in normalization theory. It is argued that these two sorts of concepts must be kept strictly apart. (shrink)

Explores how to identify future-proof science. Peter Vickers takes a transdisciplinary approach in his analysis of 'scientific fact' in order to defend science against potentially dangerous scepticism.

The axiomatic presentation of modal systems and the standard formulations of natural deduction and sequent calculus for modal logic are reviewed, together with the difficulties that emerge with these approaches. Generalizations of standard proof systems are then presented. These include, among others, display calculi, hypersequents, and labelled systems, with the latter surveyed from a closer perspective.

Covers first-order language in method appropriate for first and second courses in logic. CD-ROM consists of a new book, 3 programs,and an Internet-based grading service.

This chapter provides broad coverage of the notion of logical consequence, exploring its modal, semantic, and epistemic aspects. It develops the contrast between proof-theoretic notion of consequence, in terms of deduction, and a model-theoretic approach, in terms of truth-conditions. The main purpose is to relate the formal, technical work in logic to the philosophical concepts that underlie reasoning.

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)

Questions concerning the proof-theoretic strength of classical versus nonclassical theories of truth have received some attention recently. A particularly convenient case study concerns classical and nonclassical axiomatizations of fixed-point semantics. It is known that nonclassical axiomatizations in four- or three-valued logics are substantially weaker than their classical counterparts. In this paper we consider the addition of a suitable conditional to First-Degree Entailment—a logic recently studied by Hannes Leitgeb under the label HYPE. We show in particular that, by formulating the (...) theory PKF over HYPE, one obtains a theory that is sound with respect to fixed-point models, while being proof-theoretically on a par with its classical counterpart KF. Moreover, we establish that also its schematic extension—in the sense of Feferman—is as strong as the schematic extension of KF, thus matching the strength of predicative analysis. (shrink)

How many serious mistakes can a brilliant philosopher make in a single paragraph? Many think that Mill answers this question by example—in the third paragraph of Chapter IV of Utilitarianism. Here is the notorious paragraph: The only proof capable of being given that an object is visible, is that people actually see it. The only proof that a sound is audible, is that people hear it: and so of the other sources of our experience. In like manner, I (...) apprehend, the sole evidence it is possible to produce that anything is desirable, is that people do actually desire it. If the end which the utilitarian doctrine proposes to itself were not, in theory and in practice, acknowledged to be an end, nothing could ever convince any person that it was so. No reason can be given why the general happiness is desirable, except that each person, so far as he believes it to be attainable, desires his own happiness. This, however, being a fact, we have not only all the proof which the case admits of, but all which it is possible to require, that happiness is a good: that each person's happiness is a good to that person, and the general happiness, therefore, a good to the aggregate of all persons. Happiness has made out its title as one of the ends of conduct, and consequently one of the criteria of morality. (shrink)

This volume is the first ever collection devoted to the field of proof-theoretic semantics. Contributions address topics including the systematics of introduction and elimination rules and proofs of normalization, the categorial characterization of deductions, the relation between Heyting's and Gentzen's approaches to meaning, knowability paradoxes, proof-theoretic foundations of set theory, Dummett's justification of logical laws, Kreisel's theory of constructions, paradoxical reasoning, and the defence of model theory. The field of proof-theoretic semantics has existed for almost 50 years, (...) but the term itself was proposed by Schroeder-Heister in the 1980s. Proof-theoretic semantics explains the meaning of linguistic expressions in general and of logical constants in particular in terms of the notion of proof. This volume emerges from presentations at the Second International Conference on Proof-Theoretic Semantics in Tübingen in 2013, where contributing authors were asked to provide a self-contained description and analysis of a significant research question in this area. The contributions are representative of the field and should be of interest to logicians, philosophers, and mathematicians alike. (shrink)

Public and professional debates have come to rely heavily on a special type of reasoning: the argument-from-ignorance, in which conclusions depend on the _lack_ of compelling information. "I win my argument," says the skillful advocate, "unless you can prove that I am wrong." This extraordinary gambit has been largely ignored in modern rhetorical and philosophical studies. Yet its broad force can be demonstrated by analogy with the modern legal system, where courts have long manipulated burdens of proof with skill (...) and subtlety. This legal, philosophical, and rhetorical study by Richard H. Gaskins provides the first systematic treatment of arguments-from-ignorance across a wide range of modern discourse—from constitutional law, scientific inquiry, and moral philosophy to organizational behavior, computer operation, and personal interaction. Gaskins reviews the historic shifts in constitutional proof burdens that have shaped public debate on fundamental rights and, by analogy, on the fundamental status of intellectual and cultural authority. He shows how similar shifts have dominated polemical battles between scientific and ethical modes of authority, affecting both academic and popular discussion. Finally, he discovers the philosophical roots of default reasoning strategies in the arguments of Kant and nineteenth-century Kantian schools. Concluding that shifting proof burdens are inescapable in a world of scientific and moral uncertainty, Gaskins emphasizes the common strategic ground shared by dogmatic and skeptical reasoning. Using Hegelian strategies, he describes a more pluralistic temper that can move critical thinking beyond polemics and strengthen our capacities for common discourse. (shrink)

Is science getting at the truth? The sceptics – those who spread doubt about science – often employ a simple argument: scientists were sure in the past, and then they ended up being wrong. Such sceptics draw on dramatic quotes from eminent scientists such as Lord Kelvin, who reportedly stated at the turn of the 20th century “There is nothing new to be discovered in physics now,” shortly before physics was dramatically transformed. They ask: given the history of science, wouldn’t (...) it be naïve to think that current scientific theories reveal ‘the truth’, and will never be discarded in favour of other theories? Through a combination of historical investigation and philosophical-sociological analysis, Identifying Future-Proof Science defends science against such potentially dangerous scepticism. It is argued that we can confidently identify many scientific claims that are future-proof: they will last forever, so long as science continues. How do we identify future-proof claims? This appears to be a new question for science scholars, and not an unimportant one. It is argued that the best way to identify future-proof science is to avoid any attempt to analyse the relevant first-order scientific evidence, instead focusing purely on second-order evidence. Specifically, a scientific claim is future-proof when the relevant scientific community is large, international, and diverse, and at least 95% of that community would describe the claim as a ‘scientific fact’. In the entire history of science, no claim meeting these criteria has ever been overturned, despite enormous opportunity. (shrink)

I present an account of truth values for classical logic, intuitionistic logic, and the modal logic S5, in which truth values are not a fundamental category from which the logic is defined, but rather, an idealisation of more fundamental logical features in the proof theory for each system. The result is not a new set of semantic structures, but a new understanding of how the existing semantic structures may be understood in terms of a more fundamental notion of logical (...) consequence. (shrink)

We consider the following problem: Given a proof of the Skolemization of a formula F, what is the length of the shortest proof of F? For the restriction of this question to cut-free proofs we prove corresponding exponential upper and lower bounds.

Paley’s ‘proof’ of the existence of God, or some supposed version of it, is well known. In this paper I offer the real thing and two objections to it. One objection is my own, and the other is provided by Darwin.

Paul Cohen’s method of forcing, together with Saul Kripke’s related semantics for modal and intuitionistic logic, has had profound effects on a number of branches of mathematical logic, from set theory and model theory to constructive and categorical logic. Here, I argue that forcing also has a place in traditional Hilbert-style proof theory, where the goal is to formalize portions of ordinary mathematics in restricted axiomatic theories, and study those theories in constructive or syntactic terms. I will discuss the (...) aspects of forcing that are useful in this respect, and some sample applications. The latter include ways of obtaining conservation results for classical and intuitionistic theories, interpreting classical theories in constructive ones, and constructivizing model-theoretic arguments. (shrink)

In an attempt to improve upon Alexander Pruss’s work (The principle of sufficient reason: A reassessment, pp. 240–248, 2006), I (Weaver, Synthese 184(3):299–317, 2012) have argued that if all purely contingent events could be caused and something like a Lewisian analysis of causation is true (per, Lewis’s, Causation as influence, reprinted in: Collins, Hall and paul. Causation and counterfactuals, 2004), then all purely contingent events have causes. I dubbed the derivation of the universality of causation the “Lewisian argument”. The Lewisian (...) argument assumed not a few controversial metaphysical theses, particularly essentialism, an incommunicable-property view of essences (per Plantinga’s, Actualism and possible worlds, reprinted in: Davidson (ed.) Essays in the metaphysics of modality, 2003), and the idea that counterfactual dependence is necessary for causation. There are, of course, substantial objections to such theses. While I think a fight against objections to the Lewisian argument can be won, I develop, in what follows, a much more intuitive argument for the universality of causation which takes as its inspiration a result from Frederic B. Fitch’s work (J Symb Logic 28(2):135–142, 1963) [with credit to who we now know was Alonzo church’s, Referee Reports on Fitch’s Definition of value, in: (Salerno (ed.), New essays on the knowability paradox, 2009)] that if all truths are such that they are knowable, then (counter-intuitively) all truths are known. The resulting Church–Fitch proof for the universality of causation is preferable to the Lewisian argument since it rests upon far weaker formal and metaphysical assumptions than those of the Lewisian argument. (shrink)

In the last few years there has been a resurgence of interest in Moore’s Proof of the existence of an external world, which is now often rendered as follows:1 (I) Here’s a hand (II) If there is a hand here, there is an external world Therefore (III) There is an external world The contemporary debate has been mostly triggered by Crispin Wright’s influential—conservative —“Facts and certainty” and further fostered by Jim Pryor’s recent—liberal—“What’s wrong with Moore’s argument?”.2 This debate is (...) worth surveying with care because—so I shall contend—it will help us see that, in fact, it allows for an important view that, so far, hasn’t been explicitly considered. The critical survey will be the task of the next two sections, while, in the remaining.. (shrink)

A natural deduction formulation is given for the intermediate logic called MH by Gabbay in [4]. Proof-theoretic methods are used to show that every deduction can be normalized, that MH is the weakest intermediate logic for which the Glivenko theorem holds, and that the Craig-Lyndon interpolation theorem holds for it.

Prouver des théorèmes est une pratique mathématique qui semble clairement améliorer notre compréhension mathématique. Ainsi, prouver et reprouver des théorèmes en mathématiques, vise à apporter une meilleure compréhension. Cependant, comme il est bien connu, les preuves mathématiques totalement formalisées sont habituellement inintelligibles et, à ce titre, ne contribuent pas à notre compréhension mathématique. Comment, alors, comprendre la relation entre prouver des théorèmes et améliorer notre compréhension mathématique. J'avance ici que nous avons d'abord besoin d'une notion différente de preuve (formelle), qui (...) ne tienne pas la forme pour opposée au contenu. La pratique de la preuve algébrique au xviiie siècle fournit un exemple de preuve à la fois pleinement rigoureuse et porteuse de contenu, et dans ce cas, il est possible de voir comment une preuve mathématique apporte une compréhension mathématique. Il s'agira alors de mobiliser les enseignements de cet exemple pour étudier le type de raisonnement déductif à partir de concepts, qui a constitué la norme dans la pratique mathématique depuis le xixe siècle. (shrink)

This is the first book-length treatment of hybrid logic and its proof-theory. Hybrid logic is an extension of ordinary modal logic which allows explicit reference to individual points in a model. This is useful for many applications, for example when reasoning about time one often wants to formulate a series of statements about what happens at specific times. There is little consensus about proof-theory for ordinary modal logic. Many modal-logical proof systems lack important properties and the relationships (...) between proof systems for different modal logics are often unclear. In the present book we demonstrate that hybrid-logical proof-theory remedies these deficiencies by giving a spectrum of well-behaved proof systems for a spectrum of different hybrid logics. (shrink)

We define a class of functions, the descent recursive functions, relative to an arbitrary elementary recursive system of ordinal notations. By means of these functions, we provide a general technique for measuring the proof-theoretic strength of a variety of systems of first-order arithmetic. We characterize the provable well-orderings and provably recursive functions of these systems, and derive various conservation and equiconsistency results.

We give a simplified proof of elimination of imaginaries in ACVF, based on ideas of Hrushovski. This proof manages to avoid many of the technical issues which arose in the original proof by Haskell, Hrushovski, and Macpherson.

Dialogical egalitarianism is the thesis that any proposition asserted in dialogue, if questioned, must be supported or else retracted. Dialogical foundationalism is the thesis that some propositions are privileged over this burden of proof, standing in no need of support unless and until support for their negation is provided. I first discuss existing arguments for either thesis, dismissing each one of them. Absent a successful principled argument, I then examine which thesis it is pragmatically more advantageous to adopt in (...) analytic philosophical dialogue. This requires identifying the goal of such dialogue, to the attainment of which the thesis would be so advantageous. To identify this goal, I draw on Douglas Walton’s typology of dialogues for an analysis of the types of dialogue of 110 representatively selected journal articles in current analytic philosophy. 95% of articles are found to instantiate persuasion dialogue. In light of the thus prevalent goal of persuading one’s opponent, I argue that the adoption of dialogical egalitarianism in analytic philosophical dialogue is pragmatically inescapable. (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)

In 1969, De Jongh proved the “maximality” of a fragment of intuitionistic predicate calculus forHA. Leivant strengthened the theorem in 1975, using proof-theoretical tools (normalisation of infinitary sequent calculi). By a refinement of De Jongh's original method (using Beth models instead of Kripke models and sheafs of partial combinatory algebras), a semantical proof is given of a result that is almost as good as Leivant's. Furthermore, it is shown thatHA can be extended to Higher Order Heyting Arithmetic+all trueΠ (...) 2 0 -sentences + transfinite induction over primitive recursive well-orderings. As a corollary of the proof, maximality of intuitionistic predicate calculus is established wrt. an abstract realisability notion defined over a suitable expansion ofHA. (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)

A general definition theory should serve as a foundation for the mathematical study of definitional structures. The central notion of such a theory is a precise explication of the intuitively given notion of a definitional structure. The purpose of this paper is to discuss the proof theory of partial inductive definitions as a foundation for this kind of a more general definition theory. Among the examples discussed is a suggestion for a more abstract definition of lambda-terms (derivations in natural (...) deduction) that could provide a basis for a more systematic definitional approach to general proof theory. (shrink)

"I began to ask myself whether there might be found a single argument which would require no other for its proof than itself alone, and would suffice to demonstrate that God truly exists." - St. Anselm.

Solovay's 1976 completeness result for modal provability logic employs the recursion theorem in its proof. It is shown that the uses of the recursion theorem can in this proof replaced by the diagonalization lemma for arithmetic and that, in effect, the proof neatly fits the framework of another, enriched, system of modal logic so that any arithmetical system for which this logic is sound is strong enough to carry out the proof, in particular $\text{I}\Delta _{0}+\text{EXP}$ . (...) The method is adapted to obtain a similar completeness result for the Rosser logic. (shrink)

The paper ends with an argument that says: necessarily, if there are finitely spatially extended particulars, then there are monadic universals. Before that, in order to characterize the distinction between particulars and universals, Roman Ingarden’s notions of existential moments and modes (ways) of being are presented, and a new pair of such existential moments is introduced: multiplicity–monadicity. Also, it is argued that there are not only real universals, but instances of universals (tropes) and fictional universals too.

The interpretation of propositions in Lukasiewicz's infinite-valued calculus as answers in Ulam's game with lies--the Boolean case corresponding to the traditional Twenty Questions game--gives added interest to the completeness theorem. The literature contains several different proofs, but they invariably require technical prerequisites from such areas as model-theory, algebraic geometry, or the theory of ordered groups. The aim of this paper is to provide a self-contained proof, only requiring the rudiments of algebra and convexity in finite-dimensional vector spaces.

Masini, A., 2-Sequent calculus: a proof theory of modalities, Annals of Pure and Applied Logic 58 229–246. In this work we propose an extension of the Getzen sequent calculus in order to deal with modalities. We extend the notion of a sequent obtaining what we call a 2-sequent. For the obtained calculus we prove a cut elimination theorem.

To appear in the Proceedings of Logic Colloquium 2006. (28 pages).

Deductive proof checking programs are the most popular form of logic CAI. Whatever the reason for their widespread use, the proliferation and continuous development of these programs is evident. Contemporary proof checkers cover a wider variety of texts and rule sets, and offer more helpful editing, diagnostic, and remedial features than were once provided. These programs appear to be prime candidates for developing in the direction of "intelligent" CAI (ICAI). The primary thrust of ICAI is to build programs (...) that make use of information about learner strengths and weaknesses, the content of the subject matter being taught, and techniques for teaching various kinds of subject matter. This is a tall order by any standard, but there are signs that some initial progress is being made in the area of logic CAI. In particular, the expert system approach for offering strategic advice during proof construction is being explored by some projects. (shrink)

In this paper we apply proof theoretic methods used for classical systems in order to obtain upper bounds for systems in partial logic. We focus on a truth predicate interpreted in a Kripke style way via strong Kleene; whereas the aim is to connect harmoniously the partial version of Kripke–Feferman with its intended semantics. The method we apply is based on infinitary proof systems containing an ω-rule.

Decisions concerning proof of facts in criminal law must be rational because of what is at stake, but the decision‐making process must also be cognitively feasible because of cognitive limitations, and it must obey the relevant legal–procedural constraints. In this topic three approaches to rational reasoning about evidence in criminal law are compared in light of these demands: arguments, probabilities, and scenarios. This is done in six case studies in which different authors analyze a manslaughter case from different theoretical (...) perspectives, plus four commentaries on these case studies. The aim of this topic is to obtain more insight into how the different approaches can be applied in a legal context. This will advance the discussion on rational reasoning about evidence in law and will contribute more widely to cognitive science on a number of topics, including the value of probabilistic accounts of cognition and the problem of dealing with cognitive biases in reasoning under uncertainty in practical contexts. (shrink)

Argues for a minimal level of quantification for the "proof beyond reasonable doubt" standard of criminal law: if a jury asks "Is 60% enough?", the answer should be "No.".