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Classifying Material Implications Over Minimal Logic
Classifying Material Implications over Minimal Logic Hannes Diener and Maarten McKubre-Jordens March 28, 2018 Abstract The so-called paradoxes of material implication have motivated the development of many non- classical logics over the years [2–5, 11]. In this note, we investigate some of these paradoxes and classify them, over minimal logic. We provide proofs of equivalence and semantic models separating the paradoxes where appropriate. A number of equivalent groups arise, all of which collapse with unrestricted use of double negation elimination. Interestingly, the principle ex falso quodlibet, and several weaker principles, turn out to be distinguishable, giving perhaps supporting motivation for adopting minimal logic as the ambient logic for reasoning in the possible presence of inconsistency. Keywords: reverse mathematics; minimal logic; ex falso quodlibet; implication; paraconsistent logic; Peirce’s principle. 1 Introduction The project of constructive reverse mathematics [6] has given rise to a wide literature where various the- orems of mathematics and principles of logic have been classified over intuitionistic logic. What is less well-known is that the subtle difference that arises when the principle of explosion, ex falso quodlibet, is dropped from intuitionistic logic (thus giving (Johansson’s) minimal logic) enables the distinction of many more principles. The focus of the present paper are a range of principles known collectively (but not exhaustively) as the paradoxes of material implication; paradoxes because they illustrate that the usual interpretation of formal statements of the form “. → . .” as informal statements of the form “if. then. ” produces counter-intuitive results. Some of these principles were hinted at in [9]. Here we present a carefully worked-out chart, classifying a number of such principles over minimal logic. -
Two Sources of Explosion
Two sources of explosion Eric Kao Computer Science Department Stanford University Stanford, CA 94305 United States of America Abstract. In pursuit of enhancing the deductive power of Direct Logic while avoiding explosiveness, Hewitt has proposed including the law of excluded middle and proof by self-refutation. In this paper, I show that the inclusion of either one of these inference patterns causes paracon- sistent logics such as Hewitt's Direct Logic and Besnard and Hunter's quasi-classical logic to become explosive. 1 Introduction A central goal of a paraconsistent logic is to avoid explosiveness { the inference of any arbitrary sentence β from an inconsistent premise set fp; :pg (ex falso quodlibet). Hewitt [2] Direct Logic and Besnard and Hunter's quasi-classical logic (QC) [1, 5, 4] both seek to preserve the deductive power of classical logic \as much as pos- sible" while still avoiding explosiveness. Their work fits into the ongoing research program of identifying some \reasonable" and \maximal" subsets of classically valid rules and axioms that do not lead to explosiveness. To this end, it is natural to consider which classically sound deductive rules and axioms one can introduce into a paraconsistent logic without causing explo- siveness. Hewitt [3] proposed including the law of excluded middle and the proof by self-refutation rule (a very special case of proof by contradiction) but did not show whether the resulting logic would be explosive. In this paper, I show that for quasi-classical logic and its variant, the addition of either the law of excluded middle or the proof by self-refutation rule in fact leads to explosiveness. -
[The PROOF of FERMAT's LAST THEOREM] and [OTHER MATHEMATICAL MYSTERIES] the World's Most Famous Math Problem the World's Most Famous Math Problem
0Eft- [The PROOF of FERMAT'S LAST THEOREM] and [OTHER MATHEMATICAL MYSTERIES] The World's Most Famous Math Problem The World's Most Famous Math Problem [ THE PROOF OF FERMAT'S LAST THEOREM AND OTHER MATHEMATICAL MYSTERIES I Marilyn vos Savant ST. MARTIN'S PRESS NEW YORK For permission to reprint copyrighted material, grateful acknowledgement is made to the following sources: The American Association for the Advancement of Science: Excerpts from Science, Volume 261, July 2, 1993, C 1993 by the AAAS. Reprinted by permission. Birkhauser Boston: Excerpts from The Mathematical Experience by Philip J. Davis and Reuben Hersh © 1981 Birkhauser Boston. Reprinted by permission of Birkhau- ser Boston and the authors. The Chronicleof Higher Education: Excerpts from The Chronicle of Higher Education, July 7, 1993, C) 1993 Chronicle of HigherEducation. Reprinted by permission. The New York Times: Excerpts from The New York Times, June 24, 1993, X) 1993 The New York Times. Reprinted by permission. Excerpts from The New York Times, June 29, 1993, © 1993 The New York Times. Reprinted by permission. Cody Pfanstieh/ The poem on the subject of Fermat's last theorem is reprinted cour- tesy of Cody Pfanstiehl. Karl Rubin, Ph.D.: The sketch of Dr. Wiles's proof of Fermat's Last Theorem in- cluded in the Appendix is reprinted courtesy of Karl Rubin, Ph.D. Wesley Salmon, Ph.D.: Excerpts from Zeno's Paradoxes by Wesley Salmon, editor © 1970. Reprinted by permission of the editor. Scientific American: Excerpts from "Turing Machines," by John E. Hopcroft, Scientific American, May 1984, (D 1984 Scientific American, Inc. -
'The Denial of Bivalence Is Absurd'1
On ‘The Denial of Bivalence is Absurd’1 Francis Jeffry Pelletier Robert J. Stainton University of Alberta Carleton University Edmonton, Alberta, Canada Ottawa, Ontario, Canada [email protected] [email protected] Abstract: Timothy Williamson, in various places, has put forward an argument that is supposed to show that denying bivalence is absurd. This paper is an examination of the logical force of this argument, which is found wanting. I. Introduction Let us being with a word about what our topic is not. There is a familiar kind of argument for an epistemic view of vagueness in which one claims that denying bivalence introduces logical puzzles and complications that are not easily overcome. One then points out that, by ‘going epistemic’, one can preserve bivalence – and thus evade the complications. James Cargile presented an early version of this kind of argument [Cargile 1969], and Tim Williamson seemingly makes a similar point in his paper ‘Vagueness and Ignorance’ [Williamson 1992] when he says that ‘classical logic and semantics are vastly superior to…alternatives in simplicity, power, past success, and integration with theories in other domains’, and contends that this provides some grounds for not treating vagueness in this way.2 Obviously an argument of this kind invites a rejoinder about the puzzles and complications that the epistemic view introduces. Here are two quick examples. First, postulating, as the epistemicist does, linguistic facts no speaker of the language could possibly know, and which have no causal link to actual or possible speech behavior, is accompanied by a litany of disadvantages – as the reader can imagine. -
How the Law of Excluded Middle Pertains to the Second Incompleteness Theorem and Its Boundary-Case Exceptions
How the Law of Excluded Middle Pertains to the Second Incompleteness Theorem and its Boundary-Case Exceptions Dan E. Willard State University of New York .... Summary of article in Springer’s LNCS 11972, pp. 268-286 in Proc. of Logical Foundations of Computer Science - 2020 Talk was also scheduled to appear in ASL 2020 Conference, before Covid-19 forced conference cancellation S-0 : Comments about formats for these SLIDES 1 Slides 1-15 were presented at LFCS 2020 conference, meeting during Janaury 4-7, 2020, in Deerfield Florida (under a technically different title) 2 Published as refereed 19-page paper in Springer’s LNCS 11972, pp. 268-286 in Springer’s “Proceedings of Logical Foundations of Computer Science 2020” 3 Second presentaion of this talk was intended for ASL-2020 conference, before Covid-19 cancelled conference. Willard recommends you read Item 2’s 19-page paper after skimming these slides. They nicely summarize my six JSL and APAL papers. Manuscript is also more informative than the sketch outlined here. Manuscript helps one understand all of Willard’s papers 1. Main Question left UNANSWERED by G¨odel: 2nd Incomplet. Theorem indicates strong formalisms cannot verify their own consistency. ..... But Humans Presume Their Own Consistency, at least PARTIALLY, as a prerequisite for Thinking ? A Central Question: What systems are ADEQUATELY WEAK to hold some type (?) knowledge their own consistency? ONLY PARTIAL ANSWERS to this Question are Feasible because 2nd Inc Theorem is QUITE STRONG ! Let LXM = Law of Excluded Middle OurTheme: Different USES of LXM do change Breadth and LIMITATIONS of 2nd Inc Theorem 2a. -
Cantor's Paradise Regained: Constructive Mathematics From
Cantor's Paradise Regained: Constructive Mathematics from Brouwer to Kolmogorov to Gelfond Vladik Kreinovich Department of Computer Science University of Texas at El Paso El Paso, TX 79968 USA [email protected] Abstract. Constructive mathematics, mathematics in which the exis- tence of an object means that that we can actually construct this object, started as a heavily restricted version of mathematics, a version in which many commonly used mathematical techniques (like the Law of Excluded Middle) were forbidden to maintain constructivity. Eventually, it turned out that not only constructive mathematics is not a weakened version of the classical one { as it was originally perceived { but that, vice versa, classical mathematics can be viewed as a particular (thus, weaker) case of the constructive one. Crucial results in this direction were obtained by M. Gelfond in the 1970s. In this paper, we mention the history of these results, and show how these results affected constructive mathematics, how they led to new algorithms, and how they affected the current ac- tivity in logic programming-related research. Keywords: constructive mathematics; logic programming; algorithms Science and engineering: a brief reminder. One of the main objectives of science is to find out how the world operates, to be able to predict what will happen in the future. Science predicts the future positions of celestial bodies, the future location of a spaceship, etc. From the practical viewpoint, it is important not only to passively predict what will happen, but also to decide what to do in order to achieve certain goals. Roughly speaking, decisions of this type correspond not to science but to engineering. -
Constructive Algorithm Alex Tung 27/2/2016
Constructive Algorithm Alex Tung 27/2/2016 Constructive Algorithm • Gives an answer by explicit construction of one • Designing a constructive algorithm usually requires a lot of rough work and/or insight and/or intuition Construction Problems in OI • Construction problems are problems solvable using a constructive algorithm • Usually oe iteestig ad less stadad Hee popula i OI • Many difficult constructive problems are brilliant :D • Personal experience (in OI/ACM): • Theorem 1. It feels awesome to solve a nontrivial construction problem in contest. • Theorem 2. If I am able to solve a construction problem in contest, it is trivial. Examples from HKOI 2015/16 • Junior Q1 (Model Answer) • Find a model answer to make Alice pass and others fail • Senior Q3 (Arithmetic Sequence) • Find a sequence without length-3 arithmetic subsequence Example from Math • Q1. Prove that for all positive integers N, there exists N consecutive composite numbers. • Hint: consider (N+1)! • A. N+! + , N+! + , …, N+! + N, N+! + N+ ae N consecutive composite numbers. Non-example from Math • Q. Betad’s postulate Pove that fo all positive iteges N, thee exists a prime number in the interval [N, 2N]. • Do you think a constructive proof exists for Q2? • A2. For those interested: https://en.wikipedia.org/wiki/Proof_of_Bertrand's_postulate How to solve a construction problem • Step 1: Identify it as a construction problem • Step 2a: Look for concepts related to this problem • Step 2b: Try small examples, observe a pattern • Step 3: Make a guess • Step 4: Convince yourself that it is correct • Step 5: Code! • If AC, congratulations • Otherwise, debug/go back to step 3 Step 1: Identify a construction problem • ad-ho, i.e. -
Constructivity in Homotopy Type Theory
Ludwig Maximilian University of Munich Munich Center for Mathematical Philosophy Constructivity in Homotopy Type Theory Author: Supervisors: Maximilian Doré Prof. Dr. Dr. Hannes Leitgeb Prof. Steve Awodey, PhD Munich, August 2019 Thesis submitted in partial fulfillment of the requirements for the degree of Master of Arts in Logic and Philosophy of Science contents Contents 1 Introduction1 1.1 Outline................................ 3 1.2 Open Problems ........................... 4 2 Judgements and Propositions6 2.1 Judgements ............................. 7 2.2 Propositions............................. 9 2.2.1 Dependent types...................... 10 2.2.2 The logical constants in HoTT .............. 11 2.3 Natural Numbers.......................... 13 2.4 Propositional Equality....................... 14 2.5 Equality, Revisited ......................... 17 2.6 Mere Propositions and Propositional Truncation . 18 2.7 Universes and Univalence..................... 19 3 Constructive Logic 22 3.1 Brouwer and the Advent of Intuitionism ............ 22 3.2 Heyting and Kolmogorov, and the Formalization of Intuitionism 23 3.3 The Lambda Calculus and Propositions-as-types . 26 3.4 Bishop’s Constructive Mathematics................ 27 4 Computational Content 29 4.1 BHK in Homotopy Type Theory ................. 30 4.2 Martin-Löf’s Meaning Explanations ............... 31 4.2.1 The meaning of the judgments.............. 32 4.2.2 The theory of expressions................. 34 4.2.3 Canonical forms ...................... 35 4.2.4 The validity of the types.................. 37 4.3 Breaking Canonicity and Propositional Canonicity . 38 4.3.1 Breaking canonicity .................... 39 4.3.2 Propositional canonicity.................. 40 4.4 Proof-theoretic Semantics and the Meaning Explanations . 40 5 Constructive Identity 44 5.1 Identity in Martin-Löf’s Meaning Explanations......... 45 ii contents 5.1.1 Intensional type theory and the meaning explanations 46 5.1.2 Extensional type theory and the meaning explanations 47 5.2 Homotopical Interpretation of Identity ............ -
Lecture Notes on Constructive Logic: Overview
Lecture Notes on Constructive Logic: Overview 15-317: Constructive Logic Frank Pfenning Lecture 1 August 29, 2017 1 Introduction According to Wikipedia, logic is the study of the principles of valid infer- ences and demonstration. From the breadth of this definition it is immedi- ately clear that logic constitutes an important area in the disciplines of phi- losophy and mathematics. Logical tools and methods also play an essential role in the design, specification, and verification of computer hardware and software. It is these applications of logic in computer science which will be the focus of this course. In order to gain a proper understanding of logic and its relevance to computer science, we will need to draw heavily on the much older logical traditions in philosophy and mathematics. We will dis- cuss some of the relevant history of logic and pointers to further reading throughout these notes. In this introduction, we give only a brief overview of the goal, contents, and approach of this class. 2 Topics The course is divided into four parts: I. Proofs as Evidence for Truth II. Proofs as Programs III. Proof Search as Computation IV. Substructural and Modal Logics LECTURE NOTES AUGUST 29, 2017 L1.2 Constructive Logic: Overview Proofs are central in all parts of the course, and give it its constructive na- ture. In each part, we will exhibit connections between proofs and forms of computations studied in computer science. These connections will take quite different forms, which shows the richness of logic as a foundational discipline at the nexus between philosophy, mathematics, and computer science. -
On the Constructive Content of Proofs
On the Constructive Content of Proofs Monika Seisenberger M¨unchen 2003 On the Constructive Content of Proofs Monika Seisenberger Dissertation an der Fakult¨atf¨urMathematik und Informatik der Ludwig–Maximilians–Universit¨atM¨unchen vorgelegt von Monika Seisenberger M¨arz 2003 Erstgutachter: Prof. Dr. H. Schwichtenberg Zweitgutachter: Prof. Dr. W. Buchholz Tag der m¨undlichen Pr¨ufung:10. Juli 2003 Abstract This thesis aims at exploring the scopes and limits of techniques for extract- ing programs from proofs. We focus on constructive theories of inductive definitions and classical systems allowing choice principles. Special emphasis is put on optimizations that allow for the extraction of realistic programs. Our main field of application is infinitary combinatorics. Higman’s Lemma, having an elegant non-constructive proof due to Nash-Williams, constitutes an interesting case for the problem of discovering the constructive content behind a classical proof. We give two distinct solutions to this problem. First, we present a proof of Higman’s Lemma for an arbitrary alphabet in a theory of inductive definitions. This proof may be considered as a constructive counterpart to Nash-Williams’ minimal-bad-sequence proof. Secondly, using a refined A-translation method, we directly transform the classical proof into a constructive one and extract a program. The crucial point in the latter is that we do not need to avoid the axiom of classical dependent choice but directly assign a realizer to its translation. A generalization of Higman’s Lemma is Kruskal’s Theorem. We present a constructive proof of Kruskal’s Theorem that is completely formalized in a theory of inductive definitions. -
Proof Theory of Constructive Systems: Inductive Types and Univalence
Proof Theory of Constructive Systems: Inductive Types and Univalence Michael Rathjen Department of Pure Mathematics University of Leeds Leeds LS2 9JT, England [email protected] Abstract In Feferman’s work, explicit mathematics and theories of generalized inductive definitions play a cen- tral role. One objective of this article is to describe the connections with Martin-L¨of type theory and constructive Zermelo-Fraenkel set theory. Proof theory has contributed to a deeper grasp of the rela- tionship between different frameworks for constructive mathematics. Some of the reductions are known only through ordinal-theoretic characterizations. The paper also addresses the strength of Voevodsky’s univalence axiom. A further goal is to investigate the strength of intuitionistic theories of generalized inductive definitions in the framework of intuitionistic explicit mathematics that lie beyond the reach of Martin-L¨of type theory. Key words: Explicit mathematics, constructive Zermelo-Fraenkel set theory, Martin-L¨of type theory, univalence axiom, proof-theoretic strength MSC 03F30 03F50 03C62 1 Introduction Intuitionistic systems of inductive definitions have figured prominently in Solomon Feferman’s program of reducing classical subsystems of analysis and theories of iterated inductive definitions to constructive theories of various kinds. In the special case of classical theories of finitely as well as transfinitely iterated inductive definitions, where the iteration occurs along a computable well-ordering, the program was mainly completed by Buchholz, Pohlers, and Sieg more than 30 years ago (see [13, 19]). For stronger theories of inductive 1 i definitions such as those based on Feferman’s intutitionic Explicit Mathematics (T0) some answers have been provided in the last 10 years while some questions are still open. -
A Constructive Proof of Dependent Choice, Compatible with Classical Logic Hugo Herbelin
A Constructive Proof of Dependent Choice, Compatible with Classical Logic Hugo Herbelin To cite this version: Hugo Herbelin. A Constructive Proof of Dependent Choice, Compatible with Classical Logic. LICS 2012 - 27th Annual ACM/IEEE Symposium on Logic in Computer Science, Jun 2012, Dubrovnik, Croatia. pp.365-374. hal-00697240 HAL Id: hal-00697240 https://hal.inria.fr/hal-00697240 Submitted on 15 May 2012 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. A Constructive Proof of Dependent Choice, Compatible with Classical Logic Hugo Herbelin INRIA - PPS - Univ. Paris Diderot Paris, France e-mail: [email protected] Abstract—Martin-Löf’s type theory has strong existential elim- and P(t2) in the right-hand side, leading to an unexpected de- ination (dependent sum type) that allows to prove the full axiom generacy of the domain of discourse3 [19]. This first problem of choice. However the theory is intuitionistic. We give a condition is solved by using higher-level reduction rules such as on strong existential elimination that makes it computationally compatible with classical logic. With this restriction, we lose the E[prf (callccα(t1; φ(throwα(t2; p))))] full axiom of choice but, thanks to a lazily-evaluated coinductive .