We show that, when the number of participating agents n tends to infinity, all classical social choice rules are asymptotically strategy -proof with the proportion of manipulable profiles being of order O.

Which rules for aggregating judgments on logically connected propositions are manipulable and which not? In this paper, we introduce a preference-free concept of non-manipulability and contrast it with a preference-theoretic concept of strategy-proofness. We characterize all non-manipulable and all strategy-proof judgment aggregation rules and prove an impossibility theorem similar to the Gibbard--Satterthwaite theorem. We also discuss weaker forms of non-manipulability and strategy-proofness. Comparing two frequently discussed aggregation rules, we show that “conclusion-based voting” is less vulnerable (...) to manipulation than “premise-based voting”, which is strategy-proof only for “reason-oriented” individuals. Surprisingly, for “outcome-oriented” individuals, the two rules are strategically equivalent, generating identical judgments in equilibrium. Our results introduce game-theoretic considerations into judgment aggregation and have implications for debates on deliberative democracy. (shrink)

This paper deals with designing a harvesting control strategy for a predator–prey dynamical system, with parametric uncertainties and exogenous disturbances. A feedback control law for the harvesting rate of the predator is formulated such that the population dynamics is asymptotically stabilized at a positive operating point, while maintaining a positive, steady state harvesting rate. The hierarchical block strict feedback structure of the dynamics is exploited in designing a backstepping control law, based on Lyapunov theory. In order to account for (...) unknown parameters, an adaptive control strategy has been proposed in which the control law depends on an adaptive variable which tracks the unknown parameter. Further, a switching component has been incorporated to robustify the control performance against bounded disturbances. Proofs have been provided to show that the proposed adaptive control strategy ensures asymptotic stability of the dynamics at a desired operating point, as well as exact parameter learning in the disturbance-free case and learning with bounded error in the disturbance prone case. The dynamics, with uncertainty in the death rate of the predator, subjected to a bounded disturbance has been simulated with the proposed control strategy. (shrink)

In the division problem with single-peaked preferences, an allocation rule is strategy-proof for same tops if no one can gain by reporting a false preference relation having the true peak. This new condition is so weak that it is implied by strategy-proofness and tops-only. We show that the uniform rule is the only rule satisfying this mild property under efficiency and envy-freeness. We then analyze how largely the preference domain can be extended with admitting a rule satisfying (...) the three axioms, and show that the single-plateaued domain is the unique such maximal domain. (shrink)

herent and rational way. Several proposals have been made for information merging in which it is possible to encode the preferences of sources (Benferhat, Dubois, Prade, & Williams, 1999; Benferhat, Dubois, Kaci, & Prade, 2000; Lafage & Lang, 2000; Meyer, 2000, 2001; Andreka, Ryan, & Schobbens, 2001). Information merging has much in common with social choice theory, which aims to define operations reflecting the preferences of a society from the individual preferences of the members of the society. Given this connection, (...) frameworks for information merging should provide satisfactory resolutions of problems raised in social choice theory. We investigate the link between the merging of epistemic states and two important results in social choice theory. We show that Arrow’s well-known impossibility theorem (Arrow, 1963) does not hold in merging frameworks when the preferences of sources are represented in terms of epistemic states. This is achieved by providing a consistent set of properties for merging from which Arrow-like properties can be derived. Similarly, by extending these to a consistent framework which includes properties corresponding to the notion of being strategy-proof, we show that results due to Gibbard and Satterthwaite (Gibbard, 1973; Satterthwaite, 1973, 1975) and other (Benoit, 2000; Barber´a, Dutta, & Sen, 2000) do not hold in merging frameworks. (shrink)

An important appeal of strategy-proofness is the robustness that it implies. Under a strategy-proof voting rule, every individual has an optimal strategy independently of the behavior of all other voters, namely truth-telling. In particular, optimal play is robust with respect to the beliefs voters may have about the type and the behavior of the other voters. Following Blin and Satterthwaite (Economet J Economet Soc 45(4):881–888, 1977), we call this logically weaker property “belief-independence.” In this paper, we (...) give a number of examples of voting rules that are belief-independent but not strategy-proof. However, we also show that belief-independence implies strategy-proofness under a few natural additional conditions. The notion of belief-independence naturally leads to a the strengthening of strategy-proofness to “robust” strategy-proofness which requires that no voter whose true preference may come from a restricted domain can benefit by submitting any unrestricted preference ordering given any unrestricted preference profile for all other voters. There are examples of strategy-proof voting rules (on a restricted domain) that are not robustly strategy-proof, but under natural additional conditions the two properties are shown to be equivalent. (shrink)

The paper works with a formal model of referenda, where a finite number of voters can choose between two options and abstention. A referendum will be invalid if too many voters abstain, otherwise the referendum will return one of the two options. We consider quorum rules where an option is chosen if it is preferred by the majority of voters and if at least a certain number of voters (the quorum) votes for the alternative. The paper characterizes these rules as (...) the only referenda which are strategy-proof over certain preferences. (shrink)

We study the existence of a group of individuals which has some decisive power for social choice correspondences that satisfy a monotonicity property which we call modified monotonicity. And we examine the relation between modified monotonicity and strategy-proofness of social choice correspondences according to the definition by Duggan and Schwartz (2000). We will show mainly the following two results. (1) Modified monotonicity implies the existence of an oligarchy. An oligarchy is a group of individuals such that it has (...) some decisive power (semi-decisiveness), and at least one of the most preferred alternatives of every its member is always chosen by any social choice correspondence. (2) Strategy-proofness of social choice correspondences is equivalent to modified monotonicity. (shrink)

The notion of proof is central to mathematics yet it is one of the most difficult aspects of the subject to teach and master. In particular, undergraduate mathematics students often experience difficulties in understanding and constructing proofs. Understanding Mathematical Proof describes the nature of mathematical proof, explores the various techniques that mathematicians adopt to prove their results, and offers advice and strategies for constructing proofs. It will improve students’ ability to understand proofs and construct correct proofs of their own. The (...) first chapter of the text introduces the kind of reasoning that mathematicians use when writing their proofs and gives some example proofs to set the scene. The book then describes basic logic to enable an understanding of the structure of both individual mathematical statements and whole mathematical proofs. It also explains the notions of sets and functions and dissects several proofs with a view to exposing some of the underlying features common to most mathematical proofs. The remainder of the book delves further into different types of proof, including direct proof, proof using contrapositive, proof by contradiction, and mathematical induction. The authors also discuss existence and uniqueness proofs and the role of counter examples. (shrink)

Emphasizing the creative nature of mathematics, this conversational textbook guides students through the process of discovering a proof as they transition to advanced mathematics. Using several strategies, students will develop the thinking skills needed to tackle mathematics when there is no clear algorithm or recipe to follow.

In this paper, we consider three different search strategies for a cut-free sequent system formalizing orthologic, and estimate the respective search spaces. Applying backward search, there are classes of formulae for which both the minimal proof length and the search space are exponential. In a combined forward and backward approach, all proofs are polynomial, but the potential search space remains exponential. Using a forward strategy, the potential search space becomes polynomial yielding a polynomial decision procedure for orthologic and the (...) word problem for free ortholattices. (shrink)

In this paper, we consider three different search strategies for a cut-free sequent system formalizing orthologic, and estimate the respective search spaces. Applying backward search, there are classes of formulae for which both the minimal proof length and the search space are exponential. In a combined forward and backward approach, all proofs are polynomial, but the potential search space remains exponential. Using a forward strategy, the potential search space becomes polynomial yielding a polynomial decision procedure for orthologic and the (...) word problem for free ortholattices. (shrink)

A heuristic proof is given that the divergent QCD perturbation series is, asymptotic. By treating it as an asymptotic expansion we show that it makes sense to keep only the first few terms. The example of e+e− annihilation is considered. It is shown that by keeping only the first few terms one can get within a per cent (or smaller) of the complete sum of the series even at very low momenta where the coupling is large. More generally, (...) this affords an explanation of the phenomena of precocious scaling and why keeping only leading order corrections generally works so well. (shrink)

§1. Introduction. Asymptotic cones of metric spaces were first invented by Gromov. They are metric spaces which capture the ‘large-scale structure’ of the underlying metric space. Later, van den Dries and Wilkie gave a more general construction of asymptotic cones using ultrapowers. Certain facts about asymptotic cones, like the completeness of the metric space, now follow rather easily from saturation properties of ultrapowers, and in this survey, we want to present two applications of the van den Dries-Wilkie (...) approach. Using ultrapowers we obtain an explicit description of the asymptotic cone of a semisimple Lie group. From this description, using semi-algebraic groups and non-standard methods, we can give a short proof of the Margulis Conjecture. In a second application, we use set theory to answer a question of Gromov.§2. Definitions. The intuitive idea behind Gromov's concept of an asymptotic cone was to look at a given metric space from an ‘infinite distance’, so that large-scale patterns should become visible. In his original definition this was done by gradually scaling down the metric by factors 1/nfornϵ ℕ. In the approach by van den Dries and Wilkie, this idea was captured by ultrapowers. Their construction is more general in the sense that the asymptotic cone exists for any metric space, whereas in Gromov's original definition, the asymptotic cone existed only for a rather restricted class of spaces. (shrink)

In this paper, the author examines the burden of proof in the argu- mentative confrontations taking part in practices of political accountability. She does so by explaining how politicians maneuver strategically with the burden of proof in an attempt at winning the discussion in which they are involved. After making clear the role of the burden of proof in defining the difference of opinion in argumentative confrontations, the author outlines the constraints imposed by practices of political accountability on the burden (...) of proof. Finally, she analyzes in detail a concrete case in which a politician maneuvers in such a way that his burden of proof is significantly diminished. (shrink)

We show that the theory Tlog of the asymptotic couple of the field of logarithmic transseries is distal. As distal theories are NIP, this provides a new proof that Tlog is NIP.

There exist valuable methods for theorem proving in non classical logics based on translation from these logics into first-order classical logic (abbreviated henceforth FOL). The key notion in these approaches istranslation from aSource Logic (henceforth abbreviated SL) to aTarget Logic (henceforth abbreviated TL). These methods are concerned with the problem offinding a proof in TL by translating a formula in SL, but they do not address the very important problem ofpresenting proofs in SL via a backward translation. We propose a (...) framework for presenting proofs in SL based on a partial backward translation of proofs obtained in a familiar TL: Order-Sorted Predicate Logic. The proposed backward translation transfers some formulasF TL belonging to the proof in TL into formulasF SL , such that the formulasF SL either (a) belong to a corresponding deduction in SL (in the best case) or, (b) are semantically related in some precise way, to formulas in the corresponding deduction in SL (in the worst case). The formulasF TL andF SL can obviously be considered aslemmas of their respective proofs. Therefore the transfer of lemmas of TL gives at least a skeleton of the corresponding proof in SL. Since the formulas of a proof keep trace of the strategy used to obtain the proof, clearly the framework can also help in solving another fundamental and difficult problem:the transfer of strategies from classical to non classical logics. We show how to apply the proposed framework, at least to S5, S4(p), K, T, K4. Two conjectures are stated and we propose sufficient (and in general satisfactory) conditions in order to obtain formulas in the proof in SL. Two particular cases of the conjectures are proved to be theorems. Three examples are treated in full detail. The main lines of future research are given. (shrink)

We consider two versions of truth as grounded in verification procedures: Dummett's notion of proof as an effective way to establish the truth of a statement and Hintikka's GTS notion of truth as given by the existence of a winning strategy for the game associated with a statement. Hintikka has argued that the two notions should be effective and that one should thus restrict one's attention to recursive winning strategies. In the context of arithmetic, we show that the two (...) notions do not coincide: on the one hand, proofs in PA do not yield recursive winning strategies for the associated game; on the other hand, there is no sound and effective proof procedure that captures recursive GTS truths. We then consider a generalized version of Game Theoretical Semantics by introducing games with backward moves. In this setting, a connection is made between proofs and recursive winning strategies. We then apply this distinction between two kinds of verificationist procedures to a recent debate about how we recognize the truth of Gödelian sentences. (shrink)

The derivation on the differential-valued field Tlogof logarithmic transseries induces on its value group${{\rm{\Gamma }}_{{\rm{log}}}}$a certain mapψ. The structure${\rm{\Gamma }} = \left$is a divisible asymptotic couple. In [7] we began a study of the first-order theory of$\left$where, among other things, we proved that the theory$T_{{\rm{log}}} = Th\left$has a universal axiomatization, is model complete and admits elimination of quantifiers in a natural first-order language. In that paper we posed the question whetherTloghas NIP. In this paper, we answer that question in (...) the affirmative:Tlogdoes have NIP. Our method of proof relies on a complete survey of the 1-types ofTlog, which, in the presence of QE, is equivalent to a characterization of all simple extensions${\rm{\Gamma }}\left\langle \alpha \right\rangle$of${\rm{\Gamma }}$. We also show thatTlogdoes not have the Steinitz exchange property and we weigh in on the relationship between models ofTlogand the so-calledprecontraction groupsof [9]. (shrink)

We present a setting in which the search for a proof of B or a refutation of B can be carried out simultaneously: in contrast, the usual approach in automated deduction views proving B or proving ¬B as two, possibly unrelated, activities. Our approach to proof and refutation is described as a two-player game in which each player follows the same rules. A winning strategy translates to a proof of the formula and a counter-winning strategy translates to a (...) refutation of the formula. The game is described for multiplicative and additive linear logic . A game theoretic treatment of the multiplicative connectives is intricate and our approach to it involves two important ingredients. First, labeled graph structures are used to represent positions in a game and, second, the game playing must deal with the failure of a given player and with an appropriate resumption of play. This latter ingredient accounts for the fact that neither player might win. (shrink)

This book introduces students to the art and craft of writing proofs, beginning with the basics of writing proofs and logic, and continuing on with more in-depth issues and examples of creating proofs in different parts of mathematics, as well as introducing proofs-of-correctness for algorithms. The creation of proofs is covered for theorems in both discrete and continuous mathematics, and in difficulty ranging from elementary to beginning graduate level. Just beyond the standard introductory courses on calculus, theorems and proofs become (...) central to mathematics. Students often find this emphasis difficult and new. This book is a guide to understanding and creating proofs. It explains the standard "moves" in mathematical proofs: direct computation, expanding definitions, proof by contradiction, proof by induction, as well as choosing notation and strategies. (shrink)

The completeness of the modal logic S4 for all topological spaces as well as for the real line , the n-dimensional Euclidean space and the segment etc. was proved by McKinsey and Tarski in 1944. Several simplified proofs contain gaps. A new proof presented here combines the ideas published later by G. Mints and M. Aiello, J. van Benthem, G. Bezhanishvili with a further simplification. The proof strategy is to embed a finite rooted Kripke structure for S4 into a (...) subspace of the Cantor space which in turn encodes . This provides an open and continuous map from onto the topological space corresponding to . The completeness follows as S4 is complete with respect to the class of all finite rooted Kripke structures. (shrink)

The objective of this chapter is to introduce some of the views that have been put forward in almost 20 years of research on diagrammatic proofs in the philosophy of mathematical practice. In Sect. 1, some contextual elements will be presented on the reasons why diagrammatic proofs have attracted so much philosophical attention in the past years. In Sect. 2, the “first wave” in the research on diagrammatic proofs based on the analysis of case studies will be described: this is (...) how it started. In Sect. 3, the “second wave,” which is calling for “big pictures” and divides into two different strategies, will be discussed: this is where it is going. In Sect. 4, some conclusions will be drawn. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Proof and Persuasion in the Philosophical Debate about AbortionChris KaposyPhilosophers involved in debating the abortion issue often assume that the arguments they provide can offer decisive resolution.1 Arguments on the prolife side of the debate, for example, usually imply that it is rationally mandatory to view the fetus as having a right to life, or full moral standing.2 Such an account assumes that philosophical argument can compel the reader (...) to see the fetus in a certain way and that dissent risks irrationality.I wish to question this image of the use of philosophical argument in the abortion debate. I focus on the question of fetal moral standing, which is one of the subproblems in the debate. I am especially concerned with prolife arguments for fetal moral standing, since the prolife side must argue for a position that is compulsory. The prochoice side, in contrast, typically holds that people are entitled to believe what they want about the fetus’s moral standing.My position is that philosophical arguments should be understood as tools for persuasion in the abortion debate rather than as a method for discovering decisive proofs about the morality of abortion. I argue that because any one position on the moral standing of fetuses cannot be established decisively, law and policy should favour the prochoice position. In support of these conclusions, I present two common argumentative strategies used to suggest that the prolife view of the fetus is rationally [End Page 139] mandatory, and I illustrate why arguments involving these strategies do not compel us to view the fetus as having full moral standing.Framing the IssueIn order to show that abortion is wrong (in cases in which it is considered wrong), one must first establish that fetuses have full moral standing.3 It is a serious moral wrong to take the life of someone with full moral standing—this is part of what it means to have this status. The possessor of full moral standing is entitled to have his or her life protected. If abortion is wrong, then the fetus must have full moral standing, because without such standing, killing a fetus could be justified by the pregnant woman’s strong interest in ending an unwanted pregnancy. Those arguing against the moral permissibility of abortion must show that it is mandatory (in some way) to view the fetus as having full moral standing. There are a range of positions one could take with regard to the fetus’s status, from viewing the fetus as an incipient organism with no moral standing to seeing the fetus as gaining moral standing as the pregnancy progresses. But if it is mandatory to view the fetus as having full moral standing, all other positions must then be mistaken, incorrect, or somehow unjustified.This paper analyzes why philosophers opposed to abortion think it is mandatory to attribute full moral standing to fetuses. There are two popular argumentative strategies offering different accounts of the source of this obligation. The first strategy suggests that fetal full moral standing derives from morally relevant facts about the fetus. According to this line of argument, we are rationally obligated to view the fetus as having full moral standing because we cannot deny (without courting irrationality) that there are properties, characteristics, or attributes of the fetus that entitle it to this status.The second strategy is to claim that the fetus has full moral standing because this position is most consistent with our intuitions in cases unrelated to abortion. Philosophers taking this approach are fond of analogies comparing fetuses to comatose adults or to people at the end of life, for example. This strategy of appealing to your interlocutor’s sense of consistency I call the “shared-value” strategy.The two strategies map very well onto the distinction some have made between argumentum ad rem and argumentum ad hominem. The philosopher of rhetoric Henry Johnstone Jr. draws this distinction in very clear terms, and others in the history of philosophy make note of the difference [End Page 140] between these two types of arguments as well.4 The appeal to the facts (or argumentum ad rem) seeks to establish the truth of its... (shrink)

In this article, we propose a series of control strategies to synchronize two chaotic financial systems. Due to the characteristics of chaotic systems, the system is very sensitive to its initial values. Thus, the behaviour of two systems with different initial values will be completely different. In order to realize the synchronization of two financial chaotic systems, we designed a series of controls including controllers to realize global asymptotic synchronization and controllers to realize global exponential synchronization to make the (...) two systems fully synchronized. We provide mathematical proofs to show that the designed controls are effective. Numerical methods are used to verify the effectiveness of the controls. (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 the paper we examine the use of non-classical truth values for dealing with computation errors in program specification and validation. In that context, 3-valued McCarthy logic is suitable for handling lazy sequential computation, while 3-valued Kleene logic can be used for reasoning about parallel computation. If we want to be able to deal with both strategies without distinguishing between them, we combine Kleene and McCarthy logics into a logic based on a non-deterministic, 3-valued matrix, incorporating both options as a (...) non-deterministic choice. If the two strategies are to be distinguished, Kleene and McCarthy logics are combined into a logic based on a 4-valued deterministic matrix featuring two kinds of computation errors which correspond to the two computation strategies described above. For the resulting logics, we provide sound and complete calculi of ordinary, two-valued sequents. (shrink)

This book represents a new departure in science studies: an analysis of a scientific style of writing, situating it within the context of the contemporary style of literature. Its philosophical significance is that it provides a novel way of making sense of the notion of a scientific style. For the first time, the Hellenistic mathematical corpus - one of the most substantial extant for the period - is placed centre-stage in the discussion of Hellenistic culture as a whole. Professor Netz (...) argues that Hellenistic mathematical writings adopt a narrative strategy based on surprise, a compositional form based on a mosaic of apparently unrelated elements, and a carnivalesque profusion of detail. He further investigates how such stylistic preferences derive from, and throw light on, the style of Hellenistic poetry. This important book will be welcomed by all scholars of Hellenistic civilization as well as historians of ancient science and Western mathematics. (shrink)

In this paper we extend to non-classical set theories the standard strategy of proving independence using Boolean-valued models. This extension is provided by means of a new technique that, combining algebras (by taking their product), is able to provide product-algebra-valued models of set theories. In this paper we also provide applications of this new technique by showing that: (1) we can import the classical independence results to non-classical set theory (as an example we prove the independence of $\mathsf {CH}$ (...) ); and (2) we can provide new independence results. We end by discussing the role of non-classical algebra-valued models for the debate between universists and multiversists and by arguing that non-classical models should be included as legitimate members of the multiverse. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Proof and Persuasion in the Philosophical Debate about AbortionChris KaposyPhilosophers involved in debating the abortion issue often assume that the arguments they provide can offer decisive resolution.1 Arguments on the prolife side of the debate, for example, usually imply that it is rationally mandatory to view the fetus as having a right to life, or full moral standing.2 Such an account assumes that philosophical argument can compel the reader (...) to see the fetus in a certain way and that dissent risks irrationality.I wish to question this image of the use of philosophical argument in the abortion debate. I focus on the question of fetal moral standing, which is one of the subproblems in the debate. I am especially concerned with prolife arguments for fetal moral standing, since the prolife side must argue for a position that is compulsory. The prochoice side, in contrast, typically holds that people are entitled to believe what they want about the fetus’s moral standing.My position is that philosophical arguments should be understood as tools for persuasion in the abortion debate rather than as a method for discovering decisive proofs about the morality of abortion. I argue that because any one position on the moral standing of fetuses cannot be established decisively, law and policy should favour the prochoice position. In support of these conclusions, I present two common argumentative strategies used to suggest that the prolife view of the fetus is rationally [End Page 139] mandatory, and I illustrate why arguments involving these strategies do not compel us to view the fetus as having full moral standing.Framing the IssueIn order to show that abortion is wrong (in cases in which it is considered wrong), one must first establish that fetuses have full moral standing.3 It is a serious moral wrong to take the life of someone with full moral standing—this is part of what it means to have this status. The possessor of full moral standing is entitled to have his or her life protected. If abortion is wrong, then the fetus must have full moral standing, because without such standing, killing a fetus could be justified by the pregnant woman’s strong interest in ending an unwanted pregnancy. Those arguing against the moral permissibility of abortion must show that it is mandatory (in some way) to view the fetus as having full moral standing. There are a range of positions one could take with regard to the fetus’s status, from viewing the fetus as an incipient organism with no moral standing to seeing the fetus as gaining moral standing as the pregnancy progresses. But if it is mandatory to view the fetus as having full moral standing, all other positions must then be mistaken, incorrect, or somehow unjustified.This paper analyzes why philosophers opposed to abortion think it is mandatory to attribute full moral standing to fetuses. There are two popular argumentative strategies offering different accounts of the source of this obligation. The first strategy suggests that fetal full moral standing derives from morally relevant facts about the fetus. According to this line of argument, we are rationally obligated to view the fetus as having full moral standing because we cannot deny (without courting irrationality) that there are properties, characteristics, or attributes of the fetus that entitle it to this status.The second strategy is to claim that the fetus has full moral standing because this position is most consistent with our intuitions in cases unrelated to abortion. Philosophers taking this approach are fond of analogies comparing fetuses to comatose adults or to people at the end of life, for example. This strategy of appealing to your interlocutor’s sense of consistency I call the “shared-value” strategy.The two strategies map very well onto the distinction some have made between argumentum ad rem and argumentum ad hominem. The philosopher of rhetoric Henry Johnstone Jr. draws this distinction in very clear terms, and others in the history of philosophy make note of the difference [End Page 140] between these two types of arguments as well.4 The appeal to the facts (or argumentum ad rem) seeks to establish the truth of its... (shrink)

This paper is the final part of the syntactic demonstration of the Arithmetical Completeness of the modal system G; in the preceding parts [9] and [10] the tools for the proof were defined, in particular the notion of syntactic countermodel. Our strategy is: PA-completeness of G as a search for interpretations which force the distance between G and a GL-LIN-theorem to zero. If the GL-LIN-theorem S is not a G-theorem, we construct a formula H expressing the non G-provability of (...) S, so that ⊢GL-LIN ∼ H and so that a canonical proof T of ∼ H in GL-LIN is a syntactic countermodel for S with respect to G, which has the height θ(T) equal to the distance d(S, G) of S from G. Then we define the interpretation ξ of S which represents the proof-tree T in PA. By induction on θ(T), we prove that ⊢PA Sξ and d(S, G) > 0 imply the inconsistency of PA. (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)

The prime number theorem, established by Hadamard and de la Vallée Poussin independently in 1896, asserts that the density of primes in the positive integers is asymptotic to 1/ln x. Whereas their proofs made serious use of the methods of complex analysis, elementary proofs were provided by Selberg and Erdos in 1948. We describe a formally verified version of Selberg's proof, obtained using the Isabelle proof assistant.

We introduce a cyclic proof system for the two-way alternation-free modal μ-calculus. The system manipulates one-sided Gentzen sequents and locally deals with the backwards modalities by allowing analytic applications of the cut rule. The global effect of backwards modalities on traces is handled by making the semantics relative to a specific strategy of the opponent in the evaluation game. This allows us to augment sequents by so-called trace atoms, describing traces that the proponent can construct against the opponent’s (...) class='Hi'>strategy. The idea for trace atoms comes from Vardi’s reduction of alternating two-way automata to deterministic one-way automata. Using the multi-focus annotations introduced earlier by Marti and Venema, we turn this trace-based system into a path-based system. We prove that our system is sound for all sequents and complete for sequents not containing trace atoms. (shrink)

This paper is a continuation of dag-like proof compression research initiated in [9]. We investigate proof compression phenomenon in a particular, most transparent case of propositional DNF Logic. We define and analyze a very efficient semi-analytic sequent calculus SEQ*0 for propositional DNF. The efficiency is achieved by adding two special rules CQ and CS; the latter rule is a variant of the weakened substitution rule WS from [9], while the former one being specially designed for DNF sequents. We show that (...) dag-like SEQ*0 has good deterministic proof search strategy. Furthermore, we expose six examples showing that dag-like deducibility in SEQ*0 admits exponential-size proof compression, as compared to familiar proof systems like cutfree sequent calculi, resolution and cutting plane proof systems. (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)

An extensive literature overlapping economics, statistical decision theory and finance, contrasts expected utility [EU] with the more recent framework of mean–variance (MV). A basic proposition is that MV follows from EU under the assumption of quadratic utility. A less recognized proposition, first raised by Markowitz, is that MV is fully justified under EU, if and only if utility is quadratic. The existing proof of this proposition relies on an assumption from EU, described here as “Buridan’s axiom” after the French philosopher’s (...) fable of the ass that starved out of indifference between two bales of hay. To satisfy this axiom, MV must represent not only “pure” strategies, but also their probability mixtures, as points in the (σ, μ) plane. Markowitz and others have argued that probability mixtures are represented sufficiently by (σ, μ) only under quadratic utility, and hence that MV, interpreted as a mathematical re-expression of EU, implies quadratic utility. We prove a stronger form of this theorem, not involving or contradicting Buridan’s axiom, nor any more fundamental axiom of utility theory. (shrink)

In the following paragraphs, I will describe ten strategies through which we can show the weaknesses of every form of theism based on the "One God", while postulating that the Trinity is a good solution. This approach follows up on Swinburne’s claims about the existence of a priori and a posteriori proofs for the existence of the Trinity (his proofs are part of the sixth strategy). Clearly, these strategies are not “new”: they have been advocated by many thinkers in (...) the past and in the present. I merely revived them, and brought them together in a kind of cumulative reasoning: the strength of them arises when these strategies are considered together, showing that the Trinity is a reasonable hypothesis even though it is contradictory. The proposed strategies lead to the conclusion that there must exist in God something similar to what we call ‘real relations’ and ‘multiplicity’; and in order for God to be relational, there must exist in Him some “distincts” that relate to one another. This is postulated by philosophical reasoning, not just by Revelation, and regardless of the choice to support a process metaphysics. As Trinitarian theology contains the mystery of eternal generation, the strategies do account for the fact that in philosophy we contemplate the mystery of eternal self-distinction and of all the other ‘self-actions’ of the One. The eternal generation remains mysterious, but it is the idea that best helps us describe how God is (One and Triune) and how he creates the world. God must be Triune in every theistic system. The One God is, as One, also Triune. The Unity-Trinity of the Principle is the only apophatic point that we can reach from many quarters. Once autonomous paths of reason have established that God is a Person and Persons, One-Multiple, Creator (communicated), Free, Relational and Infinite possibility, it therefore emerges that our most reliable hypothesis is that of the Trinitarian God. (shrink)

This paper uses some modest claims about knowledge to identify a significant problem for contemporary American trial procedure. First, suppose that legal proof requires knowledge. In particular, suppose that the defendant in a jury trial is proven guilty only if the jury knows that the defendant is guilty. Second, suppose that knowledge is subject to pragmatic encroachment. In particular, whether the jury knows the defendant is guilty depends on what’s at stake in their decision to convict, including the consequences that (...) the defendant may face if convicted. Then in order to know whether a defendant has been proven guilty, jurors may need to know something about the potential consequences of conviction. But in nearly every American criminal trial, this information is withheld from jurors. -/- In §1, I lay out the philosophical premises of my argument. In §2, I say more about why these premises present a problem for American trial procedure, and I identify social and political structures that exacerbate the problem. I describe the reasoning that has led courts to withhold sentencing information from jurors, and I diagnose the flaw in this reasoning. In §3, I expand my initial argument, strengthening its conclusions and offering alternative sets of premises that still entail them. I argue that the legal ramifications of pragmatic encroachment depend on highly controversial questions in epistemology, questions about the precise nature of practical stakes. In §4, I propose strategies for legal reform. (shrink)

We introduce a new class of population games called stable games. These games are characterized by self-defeating externalities: when agents revise their strategies, the improvements in the payoffs of strategies to which revising players are switching are always exceeded by the improvements in the payoffs of strategies which revising players are abandoning. We show that stable games subsume many well-known classes of examples, including zero-sum games, games with an interior ESS, wars of attrition, and concave potential games. We prove that (...) the set of Nash equilibria of any stable game is convex, and offer an elementary proof of existence of equilibrium. Finally, we show that the set of Nash equilibria of a stable game is globally asymptotically stable under a variety of evolutionary dynamics. These convergence results are proved by constructing Lyapunov functions defined in terms of revision potentials—that is, potential functions for the protocols agents follow when they consider switching strategies. (shrink)

The halting problem for Turing machines is decidable on a set of asymptotic probability one. The proof is sensitive to the particular computational models.

This paper explores applications of concepts from argumentation theory to mathematical proofs. Note is taken of the various contexts in which proofs occur and of the various objectives they may serve. Examples of strategic maneuvering are discussed when surveying, in proofs, the four stages of argumentation distinguished by pragma-dialectics. Derailments of strategies (fallacies) are seen to encompass more than logical fallacies and to occur both in alleged proofs that are completely out of bounds and in alleged proofs that are at (...) least mathematical arguments. These considerations lead to a dialectical and rhetorical view of proofs. (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)

This article applies the analytic rigor of philosophy to the vexed topic of business strategy, and uses the objective, public evidence of business strategy as an existence proof for the possibility of free will and purpose in the private realm of subjective intentionality. The first part distinguishes three types of intentionality in philosophy—purposive intentionality, referential intentionality, and the problematic intentionality of a godlike, miraculous “inner intender.” After rejecting this third type of intentionality, and noting that its rejection saves (...) the first two types of intentionality from guilt by association, the second part draws parallels with three types of strategy in business: purposive, referential, and godlike. The first defines the goals and objectives of a company; the second picks out and targets consumers in market driven strategy; and the third, with the help of philosophical reflections, demands a rethinking of the function of leadership without reliance on a single, godlike leader. In the third part of this article, the existence proof from the public world of business is used to shed light on the possibility of intentionality in the private world of subjective intentionality. Finally, the article draws conclusions for its three audiences: for the philosophers, with credit to Nietzsche who saw it all, a greater clarity about intentionality and free will; for business people, greater clarity about the importance of purposiveness and strategic intent; and for business philosophers, a demonstration showing how—through strategy and intentionality—we can both create value and give meaning to the lives of our employees, ourselves, and our customers. (shrink)

Propositional logic -- Predicate logic -- Proof strategies and diagrams -- Mathematical induction -- Set theory -- Functions -- Relations -- Core concepts in abstract algebra -- Core concepts in real analysis.