Proof Theory: from Arithmetic to Set Theory
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“The Church-Turing “Thesis” As a Special Corollary of Gödel's
“The Church-Turing “Thesis” as a Special Corollary of Gödel’s Completeness Theorem,” in Computability: Turing, Gödel, Church, and Beyond, B. J. Copeland, C. Posy, and O. Shagrir (eds.), MIT Press (Cambridge), 2013, pp. 77-104. Saul A. Kripke This is the published version of the book chapter indicated above, which can be obtained from the publisher at https://mitpress.mit.edu/books/computability. It is reproduced here by permission of the publisher who holds the copyright. © The MIT Press The Church-Turing “ Thesis ” as a Special Corollary of G ö del ’ s 4 Completeness Theorem 1 Saul A. Kripke Traditionally, many writers, following Kleene (1952) , thought of the Church-Turing thesis as unprovable by its nature but having various strong arguments in its favor, including Turing ’ s analysis of human computation. More recently, the beauty, power, and obvious fundamental importance of this analysis — what Turing (1936) calls “ argument I ” — has led some writers to give an almost exclusive emphasis on this argument as the unique justification for the Church-Turing thesis. In this chapter I advocate an alternative justification, essentially presupposed by Turing himself in what he calls “ argument II. ” The idea is that computation is a special form of math- ematical deduction. Assuming the steps of the deduction can be stated in a first- order language, the Church-Turing thesis follows as a special case of G ö del ’ s completeness theorem (first-order algorithm theorem). I propose this idea as an alternative foundation for the Church-Turing thesis, both for human and machine computation. Clearly the relevant assumptions are justified for computations pres- ently known. -
Computability and Incompleteness Fact Sheets
Computability and Incompleteness Fact Sheets Computability Definition. A Turing machine is given by: A finite set of symbols, s1; : : : ; sm (including a \blank" symbol) • A finite set of states, q1; : : : ; qn (including a special \start" state) • A finite set of instructions, each of the form • If in state qi scanning symbol sj, perform act A and go to state qk where A is either \move right," \move left," or \write symbol sl." The notion of a \computation" of a Turing machine can be described in terms of the data above. From now on, when I write \let f be a function from strings to strings," I mean that there is a finite set of symbols Σ such that f is a function from strings of symbols in Σ to strings of symbols in Σ. I will also adopt the analogous convention for sets. Definition. Let f be a function from strings to strings. Then f is computable (or recursive) if there is a Turing machine M that works as follows: when M is started with its input head at the beginning of the string x (on an otherwise blank tape), it eventually halts with its head at the beginning of the string f(x). Definition. Let S be a set of strings. Then S is computable (or decidable, or recursive) if there is a Turing machine M that works as follows: when M is started with its input head at the beginning of the string x, then if x is in S, then M eventually halts, with its head on a special \yes" • symbol; and if x is not in S, then M eventually halts, with its head on a special • \no" symbol. -
Sequent Calculus and the Specification of Computation
Sequent Calculus and the Specification of Computation Lecture Notes Draft: 5 September 2001 These notes are being used with lectures for CSE 597 at Penn State in Fall 2001. These notes had been prepared to accompany lectures given at the Marktoberdorf Summer School, 29 July – 10 August 1997 for a course titled “Proof Theoretic Specification of Computation”. An earlier draft was used in courses given between 27 January and 11 February 1997 at the University of Siena and between 6 and 21 March 1997 at the University of Genoa. A version of these notes was also used at Penn State for CSE 522 in Fall 2000. For the most recent version of these notes, contact the author. °c Dale Miller 321 Pond Lab Computer Science and Engineering Department The Pennsylvania State University University Park, PA 16802-6106 USA Office: +(814)865-9505 [email protected] http://www.cse.psu.edu/»dale Contents 1 Overview and Motivation 5 1.1 Roles for logic in the specification of computations . 5 1.2 Desiderata for declarative programming . 6 1.3 Relating algorithm and logic . 6 2 First-Order Logic 8 2.1 Types and signatures . 8 2.2 First-order terms and formulas . 8 2.3 Sequent calculus . 10 2.4 Classical, intuitionistic, and minimal logics . 10 2.5 Permutations of inference rules . 12 2.6 Cut-elimination . 12 2.7 Consequences of cut-elimination . 13 2.8 Additional readings . 13 2.9 Exercises . 13 3 Logic Programming Considered Abstractly 15 3.1 Problems with proof search . 15 3.2 Interpretation as goal-directed search . -
Chapter 1 Preliminaries
Chapter 1 Preliminaries 1.1 First-order theories In this section we recall some basic definitions and facts about first-order theories. Most proofs are postponed to the end of the chapter, as exercises for the reader. 1.1.1 Definitions Definition 1.1 (First-order theory) —A first-order theory T is defined by: A first-order language L , whose terms (notation: t, u, etc.) and formulas (notation: φ, • ψ, etc.) are constructed from a fixed set of function symbols (notation: f , g, h, etc.) and of predicate symbols (notation: P, Q, R, etc.) using the following grammar: Terms t ::= x f (t1,..., tk) | Formulas φ, ψ ::= t1 = t2 P(t1,..., tk) φ | | > | ⊥ | ¬ φ ψ φ ψ φ ψ x φ x φ | ⇒ | ∧ | ∨ | ∀ | ∃ (assuming that each use of a function symbol f or of a predicate symbol P matches its arity). As usual, we call a constant symbol any function symbol of arity 0. A set of closed formulas of the language L , written Ax(T ), whose elements are called • the axioms of the theory T . Given a closed formula φ of the language of T , we say that φ is derivable in T —or that φ is a theorem of T —and write T φ when there are finitely many axioms φ , . , φ Ax(T ) such ` 1 n ∈ that the sequent φ , . , φ φ (or the formula φ φ φ) is derivable in a deduction 1 n ` 1 ∧ · · · ∧ n ⇒ system for classical logic. The set of all theorems of T is written Th(T ). Conventions 1.2 (1) In this course, we only work with first-order theories with equality. -
CS 173 Lecture 13: Bijections and Data Types
CS 173 Lecture 13: Bijections and Data Types Jos´eMeseguer University of Illinois at Urbana-Champaign 1 More on Bijections Bijecive Functions and Cardinality. If A and B are finite sets we know that f : A Ñ B is bijective iff |A| “ |B|. If A and B are infinite sets we can define that they have the same cardinality, written |A| “ |B| iff there is a bijective function. f : A Ñ B. This agrees with our intuition since, as f is in particular surjective, we can use f : A Ñ B to \list1" all elements of B by elements of A. The reason why writing |A| “ |B| makes sense is that, since f : A Ñ B is also bijective, we can also use f ´1 : B Ñ A to \list" all elements of A by elements of B. Therefore, this captures the notion of A and B having the \same degree of infinity," since their elements can be put into a bijective correspondence (also called a one-to-one and onto correspondence) with each other. We will also use the notation A – B as a shorthand for the existence of a bijective function f : A Ñ B. Of course, then A – B iff |A| “ |B|, but the two notations emphasize sligtly different, though equivalent, intuitions. Namely, A – B emphasizes the idea that A and B can be placed in bijective correspondence, whereas |A| “ |B| emphasizes the idea that A and B have the same cardinality. In summary, the notations |A| “ |B| and A – B mean, by definition: A – B ôdef |A| “ |B| ôdef Df P rAÑBspf bijectiveq Arrow Notation and Arrow Composition. -
The Corcoran-Smiley Interpretation of Aristotle's Syllogistic As
November 11, 2013 15:31 History and Philosophy of Logic Aristotelian_syllogisms_HPL_house_style_color HISTORY AND PHILOSOPHY OF LOGIC, 00 (Month 200x), 1{27 Aristotle's Syllogistic and Core Logic by Neil Tennant Department of Philosophy The Ohio State University Columbus, Ohio 43210 email [email protected] Received 00 Month 200x; final version received 00 Month 200x I use the Corcoran-Smiley interpretation of Aristotle's syllogistic as my starting point for an examination of the syllogistic from the vantage point of modern proof theory. I aim to show that fresh logical insights are afforded by a proof-theoretically more systematic account of all four figures. First I regiment the syllogisms in the Gentzen{Prawitz system of natural deduction, using the universal and existential quantifiers of standard first- order logic, and the usual formalizations of Aristotle's sentence-forms. I explain how the syllogistic is a fragment of my (constructive and relevant) system of Core Logic. Then I introduce my main innovation: the use of binary quantifiers, governed by introduction and elimination rules. The syllogisms in all four figures are re-proved in the binary system, and are thereby revealed as all on a par with each other. I conclude with some comments and results about grammatical generativity, ecthesis, perfect validity, skeletal validity and Aristotle's chain principle. 1. Introduction: the Corcoran-Smiley interpretation of Aristotle's syllogistic as concerned with deductions Two influential articles, Corcoran 1972 and Smiley 1973, convincingly argued that Aristotle's syllogistic logic anticipated the twentieth century's systematizations of logic in terms of natural deductions. They also showed how Aristotle in effect advanced a completeness proof for his deductive system. -
Theorem Proving in Classical Logic
MEng Individual Project Imperial College London Department of Computing Theorem Proving in Classical Logic Supervisor: Dr. Steffen van Bakel Author: David Davies Second Marker: Dr. Nicolas Wu June 16, 2021 Abstract It is well known that functional programming and logic are deeply intertwined. This has led to many systems capable of expressing both propositional and first order logic, that also operate as well-typed programs. What currently ties popular theorem provers together is their basis in intuitionistic logic, where one cannot prove the law of the excluded middle, ‘A A’ – that any proposition is either true or false. In classical logic this notion is provable, and the_: corresponding programs turn out to be those with control operators. In this report, we explore and expand upon the research about calculi that correspond with classical logic; and the problems that occur for those relating to first order logic. To see how these calculi behave in practice, we develop and implement functional languages for propositional and first order logic, expressing classical calculi in the setting of a theorem prover, much like Agda and Coq. In the first order language, users are able to define inductive data and record types; importantly, they are able to write computable programs that have a correspondence with classical propositions. Acknowledgements I would like to thank Steffen van Bakel, my supervisor, for his support throughout this project and helping find a topic of study based on my interests, for which I am incredibly grateful. His insight and advice have been invaluable. I would also like to thank my second marker, Nicolas Wu, for introducing me to the world of dependent types, and suggesting useful resources that have aided me greatly during this report. -
Does Reductive Proof Theory Have a Viable Rationale?
DOES REDUCTIVE PROOF THEORY HAVE A VIABLE RATIONALE? Solomon Feferman Abstract The goals of reduction and reductionism in the natural sciences are mainly explanatory in character, while those in mathematics are primarily foundational. In contrast to global reductionist programs which aim to reduce all of mathematics to one supposedly “univer- sal” system or foundational scheme, reductive proof theory pursues local reductions of one formal system to another which is more jus- tified in some sense. In this direction, two specific rationales have been proposed as aims for reductive proof theory, the constructive consistency-proof rationale and the foundational reduction rationale. However, recent advances in proof theory force one to consider the viability of these rationales. Despite the genuine problems of foun- dational significance raised by that work, the paper concludes with a defense of reductive proof theory at a minimum as one of the principal means to lay out what rests on what in mathematics. In an extensive appendix to the paper, various reduction relations between systems are explained and compared, and arguments against proof-theoretic reduction as a “good” reducibility relation are taken up and rebutted. 1 1 Reduction and reductionism in the natural sciencesandinmathematics. The purposes of reduction in the natural sciences and in mathematics are quite different. In the natural sciences, one main purpose is to explain cer- tain phenomena in terms of more basic phenomena, such as the nature of the chemical bond in terms of quantum mechanics, and of macroscopic ge- netics in terms of molecular biology. In mathematics, the main purpose is foundational. This is not to be understood univocally; as I have argued in (Feferman 1984), there are a number of foundational ways that are pursued in practice. -
Realisability for Infinitary Intuitionistic Set Theory 11
REALISABILITY FOR INFINITARY INTUITIONISTIC SET THEORY MERLIN CARL1, LORENZO GALEOTTI2, AND ROBERT PASSMANN3 Abstract. We introduce a realisability semantics for infinitary intuitionistic set theory that is based on Ordinal Turing Machines (OTMs). We show that our notion of OTM-realisability is sound with respect to certain systems of infinitary intuitionistic logic, and that all axioms of infinitary Kripke- Platek set theory are realised. As an application of our technique, we show that the propositional admissible rules of (finitary) intuitionistic Kripke-Platek set theory are exactly the admissible rules of intuitionistic propositional logic. 1. Introduction Realisability formalises that a statement can be established effectively or explicitly: for example, in order to realise a statement of the form ∀x∃yϕ, one needs to come up with a uniform method of obtaining a suitable y from any given x. Such a method is often taken to be a Turing program. Realisability originated as a formalisation of intuitionistic semantics. Indeed, it turned out to be well-chosen in this respect: the Curry-Howard-isomorphism shows that proofs in intuitionistic arithmetic correspond—in an effective way—to programs realising the statements they prove, see, e.g, van Oosten’s book [18] for a general introduction to realisability. From a certain perspective, however, Turing programs are quite limited as a formalisation of the concept of effective method. Hodges [9] argued that mathematicians rely (at least implicitly) on a concept of effectivity that goes far beyond Turing computability, allowing effective procedures to apply to transfinite and uncountable objects. Koepke’s [15] Ordinal Turing Machines (OTMs) provide a natural approach for modelling transfinite effectivity. -
Notes on Proof Theory
Notes on Proof Theory Master 1 “Informatique”, Univ. Paris 13 Master 2 “Logique Mathématique et Fondements de l’Informatique”, Univ. Paris 7 Damiano Mazza November 2016 1Last edit: March 29, 2021 Contents 1 Propositional Classical Logic 5 1.1 Formulas and truth semantics . 5 1.2 Atomic negation . 8 2 Sequent Calculus 10 2.1 Two-sided formulation . 10 2.2 One-sided formulation . 13 3 First-order Quantification 16 3.1 Formulas and truth semantics . 16 3.2 Sequent calculus . 19 3.3 Ultrafilters . 21 4 Completeness 24 4.1 Exhaustive search . 25 4.2 The completeness proof . 30 5 Undecidability and Incompleteness 33 5.1 Informal computability . 33 5.2 Incompleteness: a road map . 35 5.3 Logical theories . 38 5.4 Arithmetical theories . 40 5.5 The incompleteness theorems . 44 6 Cut Elimination 47 7 Intuitionistic Logic 53 7.1 Sequent calculus . 55 7.2 The relationship between intuitionistic and classical logic . 60 7.3 Minimal logic . 65 8 Natural Deduction 67 8.1 Sequent presentation . 68 8.2 Natural deduction and sequent calculus . 70 8.3 Proof tree presentation . 73 8.3.1 Minimal natural deduction . 73 8.3.2 Intuitionistic natural deduction . 75 1 8.3.3 Classical natural deduction . 75 8.4 Normalization (cut-elimination in natural deduction) . 76 9 The Curry-Howard Correspondence 80 9.1 The simply typed l-calculus . 80 9.2 Product and sum types . 81 10 System F 83 10.1 Intuitionistic second-order propositional logic . 83 10.2 Polymorphic types . 84 10.3 Programming in system F ...................... 85 10.3.1 Free structures . -
Implicit Versus Explicit Knowledge in Dialogical Logic Manuel Rebuschi
Implicit versus Explicit Knowledge in Dialogical Logic Manuel Rebuschi To cite this version: Manuel Rebuschi. Implicit versus Explicit Knowledge in Dialogical Logic. Ondrej Majer, Ahti-Veikko Pietarinen and Tero Tulenheimo. Games: Unifying Logic, Language, and Philosophy, Springer, pp.229-246, 2009, 10.1007/978-1-4020-9374-6_10. halshs-00556250 HAL Id: halshs-00556250 https://halshs.archives-ouvertes.fr/halshs-00556250 Submitted on 16 Jan 2011 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. Implicit versus Explicit Knowledge in Dialogical Logic Manuel Rebuschi L.P.H.S. – Archives H. Poincar´e Universit´ede Nancy 2 [email protected] [The final version of this paper is published in: O. Majer et al. (eds.), Games: Unifying Logic, Language, and Philosophy, Dordrecht, Springer, 2009, 229-246.] Abstract A dialogical version of (modal) epistemic logic is outlined, with an intuitionistic variant. Another version of dialogical epistemic logic is then provided by means of the S4 mapping of intuitionistic logic. Both systems cast new light on the relationship between intuitionism, modal logic and dialogical games. Introduction Two main approaches to knowledge in logic can be distinguished [1]. The first one is an implicit way of encoding knowledge and consists in an epistemic interpretation of usual logic. -
Relevant and Substructural Logics
Relevant and Substructural Logics GREG RESTALL∗ PHILOSOPHY DEPARTMENT, MACQUARIE UNIVERSITY [email protected] June 23, 2001 http://www.phil.mq.edu.au/staff/grestall/ Abstract: This is a history of relevant and substructural logics, written for the Hand- book of the History and Philosophy of Logic, edited by Dov Gabbay and John Woods.1 1 Introduction Logics tend to be viewed of in one of two ways — with an eye to proofs, or with an eye to models.2 Relevant and substructural logics are no different: you can focus on notions of proof, inference rules and structural features of deduction in these logics, or you can focus on interpretations of the language in other structures. This essay is structured around the bifurcation between proofs and mod- els: The first section discusses Proof Theory of relevant and substructural log- ics, and the second covers the Model Theory of these logics. This order is a natural one for a history of relevant and substructural logics, because much of the initial work — especially in the Anderson–Belnap tradition of relevant logics — started by developing proof theory. The model theory of relevant logic came some time later. As we will see, Dunn's algebraic models [76, 77] Urquhart's operational semantics [267, 268] and Routley and Meyer's rela- tional semantics [239, 240, 241] arrived decades after the initial burst of ac- tivity from Alan Anderson and Nuel Belnap. The same goes for work on the Lambek calculus: although inspired by a very particular application in lin- guistic typing, it was developed first proof-theoretically, and only later did model theory come to the fore.