A layman's guide to the mechanics of Gödel's proof together with a lucid discussion of the issues which it raises. Includes an essay discussing the significance of Gödel's work in the light of Wittgenstein's criticisms.

This book discusses how rational choice theory grew out of RAND's work for the US Air Force. It concentrates on the work of William J. Riker, Kenneth J. Arrow, James M. Buchanan, Russel Hardin, and John Rawls. It argues that within the context of the US Cold War with its intensive anti-communist and anti-collectivist sentiment, the foundations of capitalist democracy were grounded in the hyper individualist theory of non-cooperative games.

This paper critically engages Philip Mirowki's essay, "The scientific dimensions of social knowledge and their distant echoes in 20th-century American philosophy of science." It argues that although the cold war context of anti-democratic elitism best suited for making decisions about engaging in nuclear war may seem to be politically and ideologically motivated, in fact we need to carefully consider the arguments underlying the new rational choice based political philosophies of the post-WWII era typified by Arrow's impossibility theorem. A distrust (...) of democratic decision-making principles may be developed by social scientists whose leanings may be toward the left or right side of the spectrum of political practices. (shrink)

Northern Ireland physicist John Stewart Bell's possible understanding of quantum theory.

Charles Chihara's new book develops and defends a structural view of the nature of mathematics, and uses it to explain a number of striking features of mathematics that have puzzled philosophers for centuries. The view is used to show that, in order to understand how mathematical systems are applied in science and everyday life, it is not necessary to assume that its theorems either presuppose mathematical objects or are even true. Chihara builds upon his previous work, in which he presented (...) a new system of mathematics, the constructibility theory, which did not make reference to, or presuppose, mathematical objects. Now he develops the project further by analysing mathematical systems currently used by scientists to show how such systems are compatible with this nominalistic outlook. He advances several new ways of undermining the heavily discussed indispensability argument for the existence of mathematical objects made famous by Willard Quine and Hilary Putnam. And Chihara presents a rationale for the nominalistic outlook that is quite different from those generally put forward, which he maintains have led to serious misunderstandings.A Structural Account of Mathematics will be required reading for anyone working in this field. (shrink)

In a fluent, clear, and lively style this translation by two of Maslov's junior colleagues brings the work of the late Soviet scientist S. Yu. Maslov to a wider audience. Maslov was considered by his peers to be a man of genius who was making fundamental contributions in the fields of automatic theorem proving and computational logic. He published little, and those few papers were regarded as notoriously difficult. This book, however, was written for a broad audience of readers (...) and describes elegant examples of applications in such fields as computer science, artificial intelligence, operations research, economic modeling, and biological modeling, among others. The book also brings to light the work by the American mathematician E. L. Post, which inspired Maslov's own work in the development of a general theory and which has been long neglected by mathematicial logicians and systems theorists in the United States. The book's first chapter introduces the Rules of the Game. Part I, Mathematics of Calculi, covers E. L. Post's canonical systems, deductive systems and algorithms, and probabilistic calculi and deductive information. Part II, Horizonal Modeling, takes up a "toy" economy, the calculi of technological possibilities, and the development of rules. Part III, Vertical Modeling, deals with the topics of "to fight and to search" and the consequences of the asymmetry of cognitive mechanisms. Vladimir Lifschitz is affiliated with the Department of Computer Science at Stanford University, and Michael Gelfond with the Department of Electrical Engineering and Computer Science at the University of Texas, El Paso. Theory of Deductive Systems and Its Applicationsis included in the Foundation of Computing Series, edited by Michael Garey. (shrink)

In 1931 the mathematical logician Kurt Godel published a revolutionary paper that challenged certain basic assumptions underpinning mathematics and logic. A colleague of Albert Einstein, his theorem proved that mathematics was partly based on propositions not provable within the mathematical system and had radical implications that have echoed throughout many fields. A gripping combination of science and accessibility, Godel’s Proof by Nagel and Newman is for both mathematicians and the idly curious, offering those with a taste for logic and (...) philosophy the chance to satisfy their intellectual curiosity. (shrink)

The main theorem of this paper is: If ƒ is a partial function from ℵ 1 to ℵ 1 which is ∑ 1 1 -bounded, then there is a weakly finite primitive recursive dilator D such that for all infinite αϵdom , ƒ ⩽ D . The proof involves only elementary combinatorial constructions of trees. A generalization to ptykes is also given.

Arrow’s axiomatic foundation of social choice theory can be understood as an application of Tarski’s methodology of the deductive sciences—which is closely related to the latter’s foundational contribution to model theory. In this note we show in a model-theoretic framework how Arrow’s use of von Neumann and Morgenstern’s concept of winning coalitions allows to exploit the algebraic structures involved in preference aggregation; this approach entails an alternative indirect ultrafilter proof for Arrow’s dictatorship result. This link also connects Arrow’s seminal result (...) to key developments and concepts in the history of model theory, notably ultraproducts and preservation results. (shrink)

Halin in 1965 proved that if a graph has [Formula: see text] many pairwise disjoint rays for each [Formula: see text] then it has infinitely many pairwise disjoint rays. We analyze the complexity of this and other similar results in terms of computable and proof theoretic complexity. The statement of Halin’s theorem and the construction proving it seem very much like standard versions of compactness arguments such as König’s Lemma. Those results, while not computable, are relatively simple. They only (...) use arithmetic procedures or, equivalently, finitely many iterations of the Turing jump. We show that several Halin-type theorems are much more complicated. They are among the theorems of hyperarithmetic analysis. Such theorems imply the ability to iterate the Turing jump along any computable well ordering. Several important logical principles in this class have been extensively studied beginning with work of Kreisel, H. Friedman, Steel and others in the 1960s and 1970s. Until now, only one purely mathematical example was known. Our work provides many more and so answers Question 30 of Montalbán’s Open Questions in Reverse Mathematics in 2011. Some of these theorems including ones in Halin in 1965 are also shown to have unusual proof theoretic strength as well. (shrink)

It is demonstrated that a non Abelian Stokes Theorem is necessary to describe the B3 field of radiation. A simple form of the theorem is build up from the fundamental definition of B3 in O(3) gauge field theory, which is a gauge field theory applied to electrodynamics with an O(3) internal gauge symmetry bases on a complex basis ((1), (2), (3)). The indices (1) and (2) are complex conjugate pairs based on circular polarization, and the index (3) is (...) aligned with the propagation axis of radiation. (shrink)

A set S of degrees is said to be an initial segment if c ≤ d ∈ S→-c∈S. Shoenfield has shown that if P is the lattice of all subsets of a finite set then there is an initial segment of degrees isomorphic to P. Rosenstein [2] (independently) proved the same to hold of the lattice of all finite subsets of a countable set. We shall show that “countable set” may be replaced by “set of cardinality at most that of (...) the continuum.” This result is also an extension of [3, Corollary 2 to Theorem 15], which states that there is a sublattice of degrees isomorphic to the lattice of all finite subsets of 2N. (A sublattice of degrees is a subset closed under ∪ and ∩; an initial segment closed under ∪ is necessarily a sublattice, but not conversely.)It seems worth noting that the proof of the present result was preceded in time by our proof [4] of the analogous theorem for hyperdegrees, and is in fact an adaptation of that proof. Thus the present work has been influenced much more directly by the Gandy-Sacks forcing construction of a minimal hyperdegree [1] than by previous work on initial segments of degrees. (shrink)

Intuitionistic logic is presented here as part of familiar classical logic which allows mechanical extraction of programs from proofs. to make the material more accessible, basic techniques are presented first for propositional logic; Part II contains extensions to predicate logic. This material provides an introduction and a safe background for reading research literature in logic and computer science as well as advanced monographs. Readers are assumed to be familiar with basic notions of first order logic. One device for making this (...) book short was inventing new proofs of several theorems. The presentation is based on natural deduction. The topics include programming interpretation of intuitionistic logic by simply typed lambda-calculus (Curry-Howard isomorphism), negative translation of classical into intuitionistic logic, normalization of natural deductions, applications to category theory, Kripke models, algebraic and topological semantics, proof-search methods, interpolation theorem. The text developed from materal for several courses taught at Stanford University in 1992-1999. (shrink)

This short book has two main purposes. The first is to explain Kurt Gödel's first and second incompleteness theorems in informal terms accessible to a layperson, or at least a non-logician. The author claims that, to follow this part of the book, a reader need only be familiar with the mathematics taught in secondary school. I am not sure if this is sufficient. A grasp of the incompleteness theorems, even at the level of ‘the big picture’, might require some experience (...) with the rigor of mathematical proof. Moreover, since the incompleteness theorems concern formal deductive systems, it would help for a potential reader of this book to have some familiarity with at least elementary logic.There are, of course, a number of informal expositions of the incompleteness theorems. I do not see the need to engage in a comparative study. The second, and more important, goal of the present book is to discuss some alleged consequences of the incompleteness theorems and, in particular, to debunk thoroughly claims in philosophy, religion, and literary criticism that are supposed to be established, or at least bolstered, by incompleteness. The execution of this part of the book is masterly. The arguments are clear and compelling.The book has eight chapters, each between eight and twenty-eight pages. The opening chapter gives a very brief account of the incompleteness theorems and a sketch of Gödel's life and work. Chapter 2 gives the main overview of the incompleteness theorems, plus …. (shrink)

This is a critical analysis of the first part of Go¨del’s 1951 Gibbs lecture on certain philosophical consequences of the incompleteness theorems. Go¨del’s discussion is framed in terms of a distinction between objective mathematics and subjective mathematics, according to which the former consists of the truths of mathematics in an absolute sense, and the latter consists of all humanly demonstrable truths. The question is whether these coincide; if they do, no formal axiomatic system (or Turing machine) can comprehend the mathematizing (...) potentialities of human thought, and, if not, there are absolutely unsolvable mathematical problems of diophantine form. (shrink)

This book provides an invaluable overview of the reach of logic. It provides reference to some of the most important, well-established results in logic, while at the same time offering insight into the latest research issues in the area. It also has a balance of theory and practice, containing essays in the areas of modal logic, intuitionistic logic, logic and language, nonmonotonic logic and logic programming, temporal logic, logic and learning, combination of logics, practical reasoning, logic and artificial intelligence, abduction, (...) theorem proving and goal-directed reasoning. It will be invaluable reading for researchers and graduate students in logic and computer science, and a fabulous source of inspiration for research students in search of a topic for a PhD in logic and theoretical computer science. (shrink)

Some sufficient conditions on a Symmetric Monoidal Closed category K are obtained such that a diagram in a free SMC category generated by the set A of atoms commutes if and only if all its interpretations in K are commutative. In particular, the category of vector spaces on any field satisfies these conditions . Instead of diagrams, pairs of derivations in Intuitionistic Multiplicative Linear logic can be considered . Two derivations of the same sequent are equivalent if and only if (...) all their interpretations in K are equal. In fact, the assignment of values to atoms is defined constructively for each pair of derivations. Taking into account a mistake in R. Voreadou's proof of the “abstract coherence theorem” found by the author, it was necessary to modify her description of the class of non-commutative diagrams in SMC categories; our proof of S. Mac Lane conjecture also proves the correctness of the modified description. (shrink)

By indirect-deduction theorems introduced in the present paper we mean the theorems that allow to formalize indirect reasonings occurring in deductive practice in general and in mathematics in particular. We discuss the relationship between the introduced theorems and some logical calculi being virtually confined to propositional calculi with implication and negation. It is worth to notice that the above theorems are very handy and effective in proving logical theses.

This fourth edition of one of the classic logic textbooks has been thoroughly revised by John Burgess. The aim is to increase the pedagogical value of the book for the core market of students of philosophy and for students of mathematics and computer science as well. This book has become a classic because of its accessibility to students without a mathematical background, and because it covers not simply the staple topics of an intermediate logic course such as Godel's Incompleteness Theorems, (...) but also a large number of optional topics from Turing's theory of computability to Ramsey's theorem. John Burgess has now enhanced the book by adding a selection of problems at the end of each chapter, and by reorganising and rewriting chapters to make them more independent of each other and thus to increase the range of options available to instructors as to what to cover and what to defer. (shrink)

Quantum cognition is a new theoretical framework for constructing cognitive models based on the mathematical principles of quantum theory. Due to its increasing empirical success, one wonders what it tells us about the underlying process of cognition. Does it imply that we have quantum minds and there is some sort of quantum structure in the brain? In this paper, I address this important issue by using a new result in the research of quantum foundations. Based on the PBR theorem (...) about the reality of the wave function, I show, given certain assumptions, that the wave function assigned to a cognitive system such as our brain, which is used to calculate probabilities of thoughts/judgment outcomes in quantum cognition, is a real representation of the cognitive state of the system, not a mere representation of incomplete knowledge about the state of the system. This result supports a realist interpretation of quantum cognition, according to which the cognitive state of our brain and its dynamics are not classical but quantum in quantum cognition. In short, quantum cognition implies quantum minds. However, this result does not mean that we have quantum minds and our brain is a quantum computer, since quantum cognition by its standard formulation has not been fully confirmed by experiments. The hope is that more crucial experiments can be done in the near future to determine whether or not quantum cognition is real. (shrink)

Using recent results in topos theory, two systems of higher-order logic are shown to be complete with respect to sheaf models over topological spaces-so-called "topological semantics". The first is classical higher-order logic, with relational quantification of finitely high type; the second system is a predicative fragment thereof with quantification over functions between types, but not over arbitrary relations. The second theorem applies to intuitionistic as well as classical logic.

The aim of this paper is to distinguish between, and examine, three issues surrounding Humphreys's paradox and interpretation of conditional propensities. The first issue involves the controversy over the interpretation of inverse conditional propensities — conditional propensities in which the conditioned event occurs before the conditioning event. The second issue is the consistency of the dispositional nature of the propensity interpretation and the inversion theorems of the probability calculus, where an inversion theorem is any theorem of probability that (...) makes explicit (or implicit) appeal to a conditional probability and its corresponding inverse conditional probability. The third issue concerns the relationship between the notion of stochastic independence which is supported by the propensity interpretation, and various notions of causal independence. In examining each of these issues, it is argued that the dispositional character of the propensity interpretation provides a consistent and useful interpretation of the probability calculus. (shrink)

We examine the Dual Ramsey Theorem and two related combinatorial principles VW(k,l) and OVW(k,l) from the perspectives of reverse mathematics and effective mathematics. We give a statement of the Dual Ramsey Theorem for open colorings in second order arithmetic and formalize work of Carlson and Simpson [1] to show that this statement implies ACA 0 over RCA 0 . We show that neither VW(2,2) nor OVW(2,2) is provable in WKL 0 . These results give partial answers to questions (...) posed by Friedman and Simpson [3]. (shrink)

In this article we find some sufficient and some necessary ${\Sigma^1_1}$ -conditions with oracles for a model to be resplendent or chronically resplendent. The main tool of our proofs is internal arguments, that is analogues of classical theorems and model-theoretic constructions conducted inside a model of first-order Peano Arithmetic: arithmetised back-and-forth constructions and versions of the arithmetised completeness theorem, namely constructions of recursively saturated and resplendent models from the point of view of a model of arithmetic. These internal arguments (...) are used in conjunction with Pabion’s theorem that ensures that certain oracles are coded in a sufficiently saturated model of arithmetic. Examples of applications are provided for the theories of dense linear orders and of discrete linear orders. These results are then generalised to other ω-categorical theories and theories with a unique countable recursively saturated model. (shrink)

From a purely formalist viewpoint on the philosophy of mathematics, experiments cannot (and should not) play a role in warranting mathematical statements but must be confined to heuristics. Yet, due to the incorporation of new mathematical methods such as computer-assisted experimentation in mathematical practice, experiments are now conducted and used in a much broader range of epistemic practices such as concept formation, validation, and communication. In this article, we combine corpus studies and qualitative analyses to assess and categorize the epistemic (...) roles experiments are seen—by mathematicians—to have in actual mathematical practice. We do so by text-mining a corpus of reviews from the _Mathematical Reviews_, which include the indicator word “experiment”. Our qualitative, grounded classification of samples from this corpus allows us to explore the various roles played by experiments. We thus identify instances where experiments function as references to established knowledge, as tools for heuristics or exploration, as epistemic warrants, as communication or pedagogy, and instances simply proposing experiments. Focusing on the role of experiments as epistemic warrants, we show through additional sampling that in some fields of mathematics, experiments can warrant theorems as well as methods. We also show that the expressed lack of experiments by reviewers suggests concordant views that experiments could have provided epistemic warrants. Thus, our combination of corpus studies and qualitative analyses has added a typology of roles of experiments in mathematical practice and shown that experiments can and do play roles as epistemic warrants depending on the mathematical field. (shrink)

A survey of works on automatic theorem-proving in the ussr 1964-1970. the philosophical problems are not touched.

We prove the following three theorems on the enumeration degrees of ∑20 sets. Theorem A: There exists a nonzero noncuppable ∑20 enumeration degree. Theorem B: Every nonzero Δ20enumeration degree is cuppable to 0′e by an incomplete total enumeration degree. Theorem C: There exists a nonzero low Δ20 enumeration degree with the anticupping property.

Avec en arrière-plan l’influence d’Edmund Husserl, Alexandre Koyré et Gaston Bachelard, cet article présente l’idée d’une « élucidation ontologique » des sciences, en s’appuyant sur la discussion que donne Dominique Janicaud du principe de contradiction dans son œuvre majeure La puissance du rationnel. De même que le principe de contradiction ne peut être déduit de lui-même, les sciences doivent être fondées sur quelque chose qui se situe au-delà de leur propre sphère de rationalité. De même que le principe de contradiction (...) est fondamental pour tout raisonnement, tout en présupposant simultanément la facticité de ce raisonnement, la validité des sciences est liée à leur genèse. Un résultat scientifique ne peut s’expliquer qu’en fonction du chemin qui y a conduit ; et il ne peut être accepté comme loi, principe ou théorème que pour autant que le même chemin peut à nouveau être parcouru. C’est cette historicité qui est en jeu non seulement dans l’affirmation de Bachelard selon laquelle la raison présuppose l’arithmétique, mais aussi dans le célèbre compte-rendu que donne Husserl de l’origine de la géométrie. En fait, cette question de l’histoire est encore plus ancienne, remontant au-delà du « synthétique a priori » de Kant à l’avant et après que formule Descartes dans ses Méditations et qui s’applique à la fois à la res extensa et à la res cogitans. Ou, comme l’exprime Janicaud dans Chronos : C’est ici que la philosophie rencontre la vérité historique du temps. Cette conception est défendue dans cet article en relation avec les tentatives d’éviter le problème de l’histoire qui peuvent être détectées chez les contemporains de Janicaud. (shrink)

The quantum probability flux of a particle integrated over time and a distant surface gives the probability for the particle crossing that surface at some time. We prove the free fluxacross-surfaces theorem, which was conjectured by Combes, Newton and Shtokhamer [1], and which relates the integrated quantum flux to the usual quantum mechanical formula for the cross section. The integrated quantum flux is equal to the probability of outward crossings of surfaces by Bohmian trajectories in the scattering regime.

Using recent results in topos theory, two systems of higher-order logic are shown to be complete with respect to sheaf models over topological spaces- so -called "topological semantics." The first is classical higher-order logic, with relational quantification of finitely high type; the second system is a predicative fragment thereof with quantification over functions between types, but not over arbitrary relations. The second theorem applies to intuitionistic as well as classical logic.