Systems of Combinatory Logic Related to Quine's 'New Foundations'
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The Lambda Calculus Is Algebraic
Under consideration for publication in J. Functional Programming 1 2 P. Selinger point of view. We suggest another way of looking at the problem, which yields a sense in which the lambda calculus is equivalent to an algebraic theory. The Lambda Calculus is Algebraic The basic observation is that the failure of the ξ-rule is not a deficiency of the lambda calculus itself, nor of combinatory algebras, but rather it is an artifact of the way free variables are interpreted in a model. Under the usual interpretation, free variables are in- PETER SELINGER terpreted as elements of a lambda algebra A. Thus, an equation M = N between terms Department of Mathematics and Statistics University of Ottawa, Ottawa, Ontario K1N 6N5, Canada is said to be satisfied if it holds whenever the free variables of M and N are replaced by (e-mail: [email protected]) elements of A. We call this the local interpretation. We suggest a different interpretation, called the absolute interpretation, under which free variables are interpreted as indetermi- nates. Let A[x1 ...xn] be the lambda algebra obtained from A by freely adjoining elements Abstract x1 ...xn. Under the absolute interpretation, an equation M = N is said to be satisfied if it holds in A[x1 ...xn]. This paper serves as a self-contained, tutorial introduction to combinatory models of the untyped The fundamental observationof this paper is that the two interpretations do not coincide, lambda calculus. We focus particularly on the interpretation of free variables. We argue that free and that the absolute interpretation satisfies all rules of the lambda calculus, including the variables should not be interpreted as elements in a model, as is usually done, but as indeterminates. -
Nonstandard Set Theories and Information Management
Nonstandard Set Theories and Information Management VAROL AKMAN akmantroycsbilkentedutr Department of Computer Engineering and Information Science Bilkent University Bilkent Ankara Turkey MUJDATPAKKAN pakkantrb ounbitnet Department of Computer Engineering Bosphorus University Bebek Istanbul Turkey Abstract The merits of set theory as a foundational to ol in mathematics stimulate its use in various areas of articial intelligence in particular intelligent information systems In this pap er a study of various nonstandard treatments of set theory from this p ersp ective is oered Applications of these alternative set theories to information or knowledge management are surveyed Keywords set theory knowledge representation information management commonsense rea soning nonwellfounded sets hyp ersets Intro duction Set theory is a branch of mo dern mathematics with a unique place b ecause various other branches can b e formally dened within it For example Book of the inuential works of N Bourbaki is devoted to the theory of sets and provides the framework for the remaining 1 volumes Bourbaki said in Goldblatt All mathematical theories may b e regarded as extensions of the general theory of sets On these foundations I can state that I can build up the whole of the mathematics of the present day This brings up the p ossibility of using set theory in foundational studies in AI Mc Carthy has emphasized the need for fundamental research in AI and claimed that AI needs mathematical and logical theory involving conceptual innovations In an -
Deriving Logical Relations from Interpretations of Predicate Logic
Replace this file with prentcsmacro.sty for your meeting, or with entcsmacro.sty for your meeting. Both can be found through links on the ENTCS Web Page. Deriving Logical Relations from Interpretations of Predicate Logic Claudio Hermida1 Uday S. Reddy2 University of Birmingham Edmund P. Robinson3;4 Queen Mary, University of London Abstract This paper extends the results of Hermida's thesis about logical predicates to more general logical relations and a wider collection of types. The extension of type constructors from types to logical relations is derived from an interpretation of those constructors on a model of predicate logic. This is then further extended to n-ary relations by pullback. Hermida's theory shows how right adjoints in the category of fibrations are composed from a combination of Cartesian lifting and a local adjunction. This result is generalised to make it more applicable to left adjoints, and then shown to be stable under pullback, deriving an account of n-ary relations from standard predicate logic. A brief discussion of lifting monads to predicates includes the existence of an initial such lifting, generalising existing results. Keywords: logical relations, fibrations, categorical type theory, monadic types 1 Introduction The purpose of this paper is to illustrate a link between logical relations and predicate logics. We begin by revisiting the conditions under which the predicates in a predicate logic for reasoning about a type theory can be shown to carry the structure of that type theory. We then show that similar structure on binary and more generally n-ary relations can be derived from that on unary predicates. -
The Proper Treatment of Variables in Predicate Logic
The Proper Treatment of Variables in Predicate Logic Kai F. Wehmeier University of California, Irvine Abstract Bertrand Russell (1903, x93) observed that “the variable is a very complicated logi- cal entity, by no means easy to analyze correctly”. This assessment is borne out by the fact that even now we have no fully satisfactory understanding of the role of variables in a compositional semantics for first-order logic. In standard Tarskian semantics, variables are treated as meaning-bearing entities; moreover, they serve as the basic building blocks of all meanings, which are constructed out of variable assignments. But this has disquieting consequences, including Fine’s antinomy of the variable and an undue dependence of meanings on language (representation- alism). Here I develop an alternative, Fregean version of predicate logic that uses the traditional quantifier–variable apparatus for the expression of generality, pos- sesses a fully compositional, non-representational semantics, and is not subject to the antinomy of the variable. The advantages of Fregean over Tarskian predicate logic are due to the former’s treating variables not as meaningful lexical items, but as mere marks of punctuation, similar to parentheses. I submit that this is indeed how the variables of predicate logic should be construed. 1 1 Introduction In standard, Tarski-style syntax for first-order logic, atomic formulas are constructed from predicate symbols and an appropriate number of variables and names. Truth- functional connectives can be used to form new formulas out of ones already con- structed. From any formula already constructed, and any individual variable, a new formula can be obtained by first writing a quantifier symbol, appending the chosen variable, and then appending the original formula. -
New Foundations of Cost–Benefit Analysis
Regulation & Governance (2009) 3, 72–83 doi:10.1111/j.1748-5991.2009.01045.x RESEARCH FORUM New foundations of cost–benefit analysis A reply to Professors Sinden, Kysar, and Driesen Matthew Adler University of Pennsylvania Law School, Philadelphia, PA, USA Eric A. Posner University of Chicago Law School, Chicago, IL, USA Abstract This article responds to the criticisms of New Foundations of Cost–Benefit Analysis that appeared in a review by Amy Sinden, Douglas A. Kysar, and David M. Driesen. We argue that their criticisms are either based on misunderstandings of our approach or are too demanding, in the sense that no reasonable decision procedure would satisfy them. We illustrate this second argument by demon- strating that their preferred approach – feasibility analysis – has little to recommend it. Keywords: cost–benefit analysis, regulation, welfarism. Introduction The basic thrust of New Foundations of Cost–Benefit Analysis is to detach cost–benefit analysis (CBA) from Kaldor–Hicks efficiency,which we argue lacks moral relevance (Adler & Posner 2006, pp. 19–24), and instead to see CBA as a rough, administrable proxy for overall wellbeing. We defend the moral relevance of overall wellbeing by appealing, not to utilitarianism,but to“weak welfarism”– a much more catholic moral view that sees overall wellbeing as one among a possible plurality of foundational moral criteria, along with distributive considerations, deontological rights, and non-welfare values (Adler & Posner 2006, pp. 52–61). We also reject the simple equation of wellbeing and preference- satisfaction. An outcome benefits a person only if she prefers it and her preferences are self-interested and survive idealization (Adler & Posner 2006, pp. -
On the Brink of New Promise the FUTURE of U.S
On the Brink of New Promise THE FUTURE OF U.S. COMMUNITY FOUNDATIONS By Lucy Bernholz, Katherine Fulton, and Gabriel Kasper Blueprint Research & Design, Inc. and the Monitor Institute, a member of Monitor Group Created with funding support from the Charles Stewart Mott Foundation and the Ford Foundation © Copyright 2005 Blueprint Research & Design, Inc. and Monitor Company Group, LLP. We encourage readers to use and share the contents of this report, with the understanding that it is the intellectual property of Blueprint Research & Design and the Monitor Group, and that full attribution is required. An invitation from the Charles Stewart Mott and Ford foundations Our two foundations have been privileged to have worked with community foundations across the United States over the past quarter century. For both of us, this is an area of philanthropy we care about deeply. Th e Charles Stewart Mott and Ford foundations work at national and indeed international levels. Yet we both know how valuable it is for large private foundations to have strong, dynamic community founda- tions as partners working at the local level. Th is is why we embraced the opportunity to look deep into the future of the fi eld through the expert eyes of Lucy Bernholz, Katherine Fulton, and Gabriel Kasper. Over those years, Mott and Ford have made grants to community foundations to build general and administrative endowments, to provide peer-to-peer learning opportunities and technical assistance, and to strengthen programming expertise in areas such as low-income neighborhoods, the environment, and minority communities. We have also assisted the fi eld to develop abroad, and we have helped build the infrastructure for community foundations We recommend this report to community foundation staff, boards of directors, across the nation. -
The Problem of the Consistency of “New Foundations”
The Problem of the Consistency of \New Foundations" Randall Holmes September 10, 2013 1 Abstract: I will explain the nature of the long- standing problem of the consistency of the set theory New Foundations proposed by the philoso- pher W. v. O. Quine in 1937 both in the prior context of the development of set theory, its usefulness in mathematics, and the problem of the "paradoxes" of set theory, and in the pos- terior context of partial solutions to the consis- tency problem and related results. I do claim to have solved this problem (this is not generally agreed yet) but I am not going to talk about that on this occasion. The talk should be ac- cessible to a general audience of mathemati- cians; I hope that a graduate student or mature undergraduate would get something out of it too. Plan of the talk I'm going to talk about the problem of the consistency of Quine's set theory \New Foun- dations", which is the central issue of my tiny area of set theory. I currently believe that I have solved this prob- lem, but this has nothing to do with the present talk, or very little. What I propose to do is explain what the prob- lem is and put it in some kind of context. Why does one need a set theory? Why is there a problem of consistency of set theories? What is New Foundations anyway and why is there a problem with it in particular? What are the relevant related results and partial solutions to the problem? 2 What is a set? When I tell a layman with no maths that I work in set theory, they ask \What is a set?". -
Exceptionally Monadic Error Handling
Exceptionally Monadic Error Handling Looking at bind and squinting really hard Jan Malakhovski∗ IRIT, University of Toulouse-3 and ITMO University February 2014 - October 2018 Abstract We notice that the type of catch :: c a -> (e -> c a) -> c a operator is a special case of monadic bind operator (>>=) :: m a -> (a -> m b) -> m b, the semantics (surprisingly) matches, and this observation has many interesting consequences. For instance, the reader is probably aware that the monadic essence of the (>>=) operator of the error monad λA.E ∨ A is to behave like identity monad for "normal" values and to stop on "errors". The unappreciated fact is that handling of said "errors" with a catch operator of the "flipped" "conjoined" error monad λE.E ∨ A is, too, a monadic computation that treats still unhandled "errors" as "normal" values and stops when an "error" is finally handled. We show that for an appropriately indexed type of computations such a "conjoined" struc- ture naturally follows from the conventional operational semantics of throw and catch operators. Consequently, we show that this structure uniformly generalizes all conventional monadic error handling mechanisms we are aware of. We also demonstrate several more interesting instances of this structure of which at least bi-indexed monadic parser combinators and conventional exceptions implemented via continuations have immediate practical applications. Finally, we notice that these observations provide surprising perspectives on error handling in general and point to a largely unexplored trail in programming language design space. Contents 1 Extended Abstract 3 2 Preliminaries 4 3 Introduction 5 4 Not a Tutorial: Side A 7 arXiv:1810.13430v3 [cs.PL] 30 Jan 2019 4.1 Before-Monadic................................. -
By WV QUINE Only the Truth Functions, Quantification, and Membership
538 MATHEMATICS: W. V. QUINE PROC. N. A. S. ON THE CONSISTENC Y OF "NEW FOUNDA TIONS" By W. V. QUINE HARVARD UNIVERSITY Communicated by Hassler Whitney, June 12, 1951 The question of the consistency of the system of set theory in my "New Foundations,"' briefly NF, has long been mooted.2 3 The question takes on added interest with Wang's new revision4 of the system of my Mathe- matical Logic;5 for Wang has proved that this revision, which is to be used in the forthcoming third printing of Mathematical Logic, is consistent if NF is consistent. The purpose of the present note is to point out that NF is consistent if and only if the system of Whitehead and Russell's Principia Mathematica,6 briefly PM, is consistent in a certain one of its natural ver- sions. PM may be conceived in a form in which the variables carry indices indi- cating logical type. Alternatively PM may be conceived under a conven- tion which Whitehead and Russell called typical ambiguity; under this con- vention the indices are suppressed but still only those formulae are ad- mitted as meaningful which are stratified, i.e., so constituted that type in- dices could be inserted everywhere without violating the theory of types. Another point on which we have a choice, in framing a version of PM, is the status of relations and relational predication: we may assume relational predication as primitive on a par with set-membership, and decree a sprawling annex to the theory of types to accommodate relations, as was done in effect by Whitehead and Russell; or alternatively we may, follow- ing Wiener,7 reduce relations to certain sets of sets of sets (at the cost of giving up relations between things of different types). -
New Foundations of Transnational Private Regulation Author(S): Fabrizio Cafaggi Source: Journal of Law and Society, Vol
Cardiff University New Foundations of Transnational Private Regulation Author(s): Fabrizio Cafaggi Source: Journal of Law and Society, Vol. 38, No. 1, The Challenge of Transnational Private Regulation: Conceptual and Constitutional Debates (MARCH 2011), pp. 20-49 Published by: Wiley on behalf of Cardiff University Stable URL: https://www.jstor.org/stable/23030395 Accessed: 21-01-2019 16:18 UTC JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at https://about.jstor.org/terms Cardiff University, Wiley are collaborating with JSTOR to digitize, preserve and extend access to Journal of Law and Society This content downloaded from 154.59.124.94 on Mon, 21 Jan 2019 16:18:47 UTC All use subject to https://about.jstor.org/terms JOURNAL OF LAW AND SOCIETY VOLUME 38, NUMBER 1, MARCH 2011 ISSN: 0263-323X, pp. 20-49 New Foundations of Transnational Private Regulation Fabrizio Cafaggi* In section I oj this article, the factors driving towards the emergence of new transnational private regulation (TPR) are identified in com parison with, on the one hand, merchant law and, on the other, inter national public regimes. In section II, the focus is on the private sphere, looking at both the different conflicts of interests arising in the regulatory relationships and the need for governance responses. -
New Foundations for Physical Geometry
New Foundations for Physical Geometry Tim Maudlin (First Draft: Please do not cite) Socrates: I do not insist that my argument is right in all other respects, but I would contend at all costs in both word and deed as far as I could that we will be better men, braver and less idle, if we believe that one must search for the things one does not know, rather than if we believe that it is not possible to find out what we do not know and that we must not look for it. Plato, Meno 86c 2 NEW FOUNDATIONS: INTRODUCTION 3 Introduction The thesis of this book is both simple and audacious. It is so simple that the basic claims can be boiled down into two sentences. First: the most fundamental geometrical structure that organizes physical points into a space is the line. Second: what endows space‐time with its geometry is time. The remainder of this volume does nothing but elucidate these two sentences. Everything flows from them in such a straightforward way that I am almost convinced that the reader could close the book forthwith and, with sufficient patience and diligence, reconstruct most of the content from these two propositions. As for the audacity, acceptance of either of these propositions demands the rejection of widely held and deeply entrenched alternatives. Consider a collection of objects that we wish to regard as forming not merely a set (which it does automatically) but as forming a space. Organizing the set into a space requires something more than the set‐theoretic structure. -
New Foundations for Information Theory: Probability Theory Information Theory Subset Logic = Partition Logic
New Foundations for Information Theory: Probability Theory Information Theory Subset Logic = Partition Logic David Ellerman University of California-Riverside University of Ljubljana, Slovenia Probability Theory = Information Theory David EllermanUniversityNew Foundations of California-RiversideUniversity for Information Theory: Subset of Ljubljana, Logic SloveniaPartition () Logic 1 / 34 Duality of Subsets and Partitions:I “The dual notion (obtained by reversing the arrows) of ‘part’ is the notion of partition.” [Lawvere]; mono S X dualizes to epi X Y. ! ! Duality of Elements and Distinctions ("Its" & "Dits") Partition p = B1, ..., B6 on set U = u1, ..., un . f –g f g Probability Theory = Information Theory David EllermanUniversityNew Foundations of California-RiversideUniversity for Information Theory: Subset of Ljubljana, Logic SloveniaPartition () Logic 2 / 34 Duality of Subsets and Partitions:II Dual Logics: Boolean subset logic of subsets and partition logic Probability Theory = Information Theory David EllermanUniversityNew Foundations of California-RiversideUniversity for Information Theory: Subset of Ljubljana, Logic SloveniaPartition () Logic 3 / 34 Duality of Subsets and Partitions: III Published papers on partition logic: The Logic of Partitions: Introduction to the Dual of the Logic of Subsets. Review of Symbolic Logic, 3(2 June), 287–350, 2010. An introduction of partition logic. Logic Journal of the IGPL, 22(1), 94–125, 2014. Birkhoff & von Neumann created quantum logic by linearizing logic of subsets to logic of subspaces of a vector space. Hence the dual form of quantum logic created by linearing logic of partitions to logic of direct-sum decompositions of a vector space: The Quantum Logic of Direct-Sum-Decompositions: The Dual to the Quantum Logic of Subspaces. Logic Journal of the IGPL.