Robinson's Consistency Theorem in Soft Model Theory by Daniele Mundici
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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. -
Truth Definitions and Consistency Proofs
TRUTH DEFINITIONS AND CONSISTENCY PROOFS BY HAO WANG 1. Introduction. From investigations by Carnap, Tarski, and others, we know that given a system S, we can construct in some stronger system S' a criterion of soundness (or validity) for 5 according to which all the theorems of 5 are sound. In this way we obtain in S' a consistency proof for 5. The consistency proof so obtained, which in no case with fairly strong systems could by any stretch of imagination be called constructive, is not of much interest for the purpose of understanding more clearly whether the system S is reliable or whether and why it leads to no contradictions. However, it can be of use in studying the interconnection and relative strength of different systems. For example, if a consistency proof for 5 can be formalized in S', then, according to Gödel's theorem that such a proof cannot be formalized in 5 itself, parts of the argument must be such that they can be formalized in S' but not in S. Since S can be a very strong system, there arises the ques- tion as to what these arguments could be like. For illustration, the exact form of such arguments will be examined with respect to certain special systems, by applying Tarski's "theory of truth" which provides us with a general method for proving the consistency of a given system 5 in some stronger system S'. It should be clear that the considerations to be presented in this paper apply to other systems which are stronger than or as strong as the special systems we use below. -
The Development of Mathematical Logic from Russell to Tarski: 1900–1935
The Development of Mathematical Logic from Russell to Tarski: 1900–1935 Paolo Mancosu Richard Zach Calixto Badesa The Development of Mathematical Logic from Russell to Tarski: 1900–1935 Paolo Mancosu (University of California, Berkeley) Richard Zach (University of Calgary) Calixto Badesa (Universitat de Barcelona) Final Draft—May 2004 To appear in: Leila Haaparanta, ed., The Development of Modern Logic. New York and Oxford: Oxford University Press, 2004 Contents Contents i Introduction 1 1 Itinerary I: Metatheoretical Properties of Axiomatic Systems 3 1.1 Introduction . 3 1.2 Peano’s school on the logical structure of theories . 4 1.3 Hilbert on axiomatization . 8 1.4 Completeness and categoricity in the work of Veblen and Huntington . 10 1.5 Truth in a structure . 12 2 Itinerary II: Bertrand Russell’s Mathematical Logic 15 2.1 From the Paris congress to the Principles of Mathematics 1900–1903 . 15 2.2 Russell and Poincar´e on predicativity . 19 2.3 On Denoting . 21 2.4 Russell’s ramified type theory . 22 2.5 The logic of Principia ......................... 25 2.6 Further developments . 26 3 Itinerary III: Zermelo’s Axiomatization of Set Theory and Re- lated Foundational Issues 29 3.1 The debate on the axiom of choice . 29 3.2 Zermelo’s axiomatization of set theory . 32 3.3 The discussion on the notion of “definit” . 35 3.4 Metatheoretical studies of Zermelo’s axiomatization . 38 4 Itinerary IV: The Theory of Relatives and Lowenheim’s¨ Theorem 41 4.1 Theory of relatives and model theory . 41 4.2 The logic of relatives . -
First-Order Logic
Chapter 5 First-Order Logic 5.1 INTRODUCTION In propositional logic, it is not possible to express assertions about elements of a structure. The weak expressive power of propositional logic accounts for its relative mathematical simplicity, but it is a very severe limitation, and it is desirable to have more expressive logics. First-order logic is a considerably richer logic than propositional logic, but yet enjoys many nice mathemati- cal properties. In particular, there are finitary proof systems complete with respect to the semantics. In first-order logic, assertions about elements of structures can be ex- pressed. Technically, this is achieved by allowing the propositional symbols to have arguments ranging over elements of structures. For convenience, we also allow symbols denoting functions and constants. Our study of first-order logic will parallel the study of propositional logic conducted in Chapter 3. First, the syntax of first-order logic will be defined. The syntax is given by an inductive definition. Next, the semantics of first- order logic will be given. For this, it will be necessary to define the notion of a structure, which is essentially the concept of an algebra defined in Section 2.4, and the notion of satisfaction. Given a structure M and a formula A, for any assignment s of values in M to the variables (in A), we shall define the satisfaction relation |=, so that M |= A[s] 146 5.2 FIRST-ORDER LANGUAGES 147 expresses the fact that the assignment s satisfies the formula A in M. The satisfaction relation |= is defined recursively on the set of formulae. -
The History of Logic
c Peter King & Stewart Shapiro, The Oxford Companion to Philosophy (OUP 1995), 496–500. THE HISTORY OF LOGIC Aristotle was the first thinker to devise a logical system. He drew upon the emphasis on universal definition found in Socrates, the use of reductio ad absurdum in Zeno of Elea, claims about propositional structure and nega- tion in Parmenides and Plato, and the body of argumentative techniques found in legal reasoning and geometrical proof. Yet the theory presented in Aristotle’s five treatises known as the Organon—the Categories, the De interpretatione, the Prior Analytics, the Posterior Analytics, and the Sophistical Refutations—goes far beyond any of these. Aristotle holds that a proposition is a complex involving two terms, a subject and a predicate, each of which is represented grammatically with a noun. The logical form of a proposition is determined by its quantity (uni- versal or particular) and by its quality (affirmative or negative). Aristotle investigates the relation between two propositions containing the same terms in his theories of opposition and conversion. The former describes relations of contradictoriness and contrariety, the latter equipollences and entailments. The analysis of logical form, opposition, and conversion are combined in syllogistic, Aristotle’s greatest invention in logic. A syllogism consists of three propositions. The first two, the premisses, share exactly one term, and they logically entail the third proposition, the conclusion, which contains the two non-shared terms of the premisses. The term common to the two premisses may occur as subject in one and predicate in the other (called the ‘first figure’), predicate in both (‘second figure’), or subject in both (‘third figure’). -
An Institutional View on Categorical Logic and the Curry-Howard-Tait-Isomorphism
An Institutional View on Categorical Logic and the Curry-Howard-Tait-Isomorphism Till Mossakowski1, Florian Rabe2, Valeria De Paiva3, Lutz Schr¨oder1, and Joseph Goguen4 1 Department of Computer Science, University of Bremen 2 School of Engineering and Science, International University of Bremen 3 Intelligent Systems Laboratory, Palo Alto Research Center, California 4 Dept. of Computer Science & Engineering, University of California, San Diego Abstract. We introduce a generic notion of propositional categorical logic and provide a construction of an institution with proofs out of such a logic, following the Curry-Howard-Tait paradigm. We then prove logic-independent soundness and completeness theorems. The framework is instantiated with a number of examples: classical, intuitionistic, linear and modal propositional logics. Finally, we speculate how this framework may be extended beyond the propositional case. 1 Introduction The well-known Curry-Howard-Tait isomorphism establishes a correspondence between { propositions and types { proofs and terms { proof reductions and term reductions. Moreover, in this context, a number of deep correspondences between cate- gories and logical theories have been established (identifying `types' with `objects in a category'): conjunctive logic cartesian categories [19] positive logic cartesian closed categories [19] intuitionistic propositional logic bicartesian closed categories [19] classical propositional logic bicartesian closed categories with [19] ::-elimination linear logic ∗-autonomous categories [33] first-order logic hyperdoctrines [31] Martin-L¨of type theory locally cartesian closed categories [32] Here, we present work aimed at casting these correspondences in a common framework based on the theory of institutions. The notion of institution arose within computer science as a response to the population explosion among logics in use, with the ambition of doing as much as possible at a level of abstraction independent of commitment to any particular logic [16, 18]. -
Critique of the Church-Turing Theorem
CRITIQUE OF THE CHURCH-TURING THEOREM TIMM LAMPERT Humboldt University Berlin, Unter den Linden 6, D-10099 Berlin e-mail address: lampertt@staff.hu-berlin.de Abstract. This paper first criticizes Church's and Turing's proofs of the undecidability of FOL. It identifies assumptions of Church's and Turing's proofs to be rejected, justifies their rejection and argues for a new discussion of the decision problem. Contents 1. Introduction 1 2. Critique of Church's Proof 2 2.1. Church's Proof 2 2.2. Consistency of Q 6 2.3. Church's Thesis 7 2.4. The Problem of Translation 9 3. Critique of Turing's Proof 19 References 22 1. Introduction First-order logic without names, functions or identity (FOL) is the simplest first-order language to which the Church-Turing theorem applies. I have implemented a decision procedure, the FOL-Decider, that is designed to refute this theorem. The implemented decision procedure requires the application of nothing but well-known rules of FOL. No assumption is made that extends beyond equivalence transformation within FOL. In this respect, the decision procedure is independent of any controversy regarding foundational matters. However, this paper explains why it is reasonable to work out such a decision procedure despite the fact that the Church-Turing theorem is widely acknowledged. It provides a cri- tique of modern versions of Church's and Turing's proofs. I identify the assumption of each proof that I question, and I explain why I question it. I also identify many assumptions that I do not question to clarify where exactly my reasoning deviates and to show that my critique is not born from philosophical scepticism. -
Interpolation: Theory and Applications
Interpolation: Theory and Applications Vijay D’Silva Google Inc., San Francisco UC Berkeley 2016 Wednesday, March 30, 16 1 A Brief History of Interpolation 2 Verification with Interpolants 3 Interpolant Construction 4 Further Reading and Research Wednesday, March 30, 16 Craig Interpolants For two formulae A and B such that A implies B B,aCraig interpolant is a formula I such that 1. A implies I, and I 2. I implies B, and 3. the non-logical symbols in I occur in A and in B. A P (P = Q)=Q = R = Q ^ ) ) ) ) Wednesday, March 30, 16 Reverse Interpolants or “Interpolants” For a contradiction A B,areverse interpolant is ^ a formula I such that B 1. A implies I, and 2. I B is a contradiction, and ^ I 3. the non-logical symbols in I occur in A and in B. Note: In classical logic, A = B is valid exactly ) A if A B is a contradiction. ^¬ In the verification literature, “interpolant” usually means “reverse interpolant.” Wednesday, March 30, 16 International Business Machines Corporation 2050 Rt 52 Hopewell Junction, NY 12533 845-892-5262 October 7, 2008 Dear Andreas, I would like to congratulate Cadence Research Labs on their 15th Anniversary. In these 15 years, Cadence Research Labs has worked at several frontiers of Electronic Design Automation. They focus on hard problems that when solved significantly push the state of the art forward. They found novel solutions to system, synthesis and formal verification problems. Formal verification is the process of exhaustively validating that a logic entity behaves correctly. In contrast to testing-based approaches, which may expose flaws though generally cannot yield a proof of correctness, the exhaustiveness of formal verification ensures that no flaw will be left unexposed. -
Axiomatic Foundations of Mathematics Ryan Melton Dr
Axiomatic Foundations of Mathematics Ryan Melton Dr. Clint Richardson, Faculty Advisor Stephen F. Austin State University As Bertrand Russell once said, Gödel's Method Pure mathematics is the subject in which we Consider the expression First, Gödel assigned a unique natural number to do not know what we are talking about, or each of the logical symbols and numbers. 2 + 3 = 5 whether what we are saying is true. Russell’s statement begs from us one major This expression is mathematical; it belongs to the field For example: if the symbol '0' corresponds to we call arithmetic and is composed of basic arithmetic question: the natural number 1, '+' to 2, and '=' to 3, then symbols. '0 = 0' '0 + 0 = 0' What is Mathematics founded on? On the other hand, the sentence and '2 + 3 = 5' is an arithmetical formula. 1 3 1 1 2 1 3 1 so each expression corresponds to a sequence. Axioms and Axiom Systems is metamathematical; it is constructed outside of mathematics and labels the expression above as a Then, for this new sequence x1x2x3…xn of formula in arithmetic. An axiom is a belief taken without proof, and positive integers, we associate a Gödel number thus an axiom system is a set of beliefs as follows: x1 x2 x3 xn taken without proof. enc( x1x2x3...xn ) = 2 3 5 ... pn Since Principia Mathematica was such a bold where the encoding is the product of n factors, Consistent? Complete? leap in the right direction--although proving each of which is found by raising the j-th prime nothing about consistency--several attempts at to the xj power. -
Consistency, Contradiction and Negation. Logic, Epistemology, and the Unity of Science Series
BOOK REVIEW: CARNIELLI, W., CONIGLIO, M. Paraconsistent Logic: Consistency, Contradiction and Negation. Logic, Epistemology, and the Unity of Science Series. (New York: Springer, 2016. ISSN: 2214-9775.) Henrique Antunes Vincenzo Ciccarelli State University of Campinas State University of Campinas Department of Philosophy Department of Philosophy Campinas, SP Campinas, SP Brazil Brazil [email protected] [email protected] Article info CDD: 160 Received: 01.12.2017; Accepted: 30.12.2017 DOI: http://dx.doi.org/10.1590/0100-6045.2018.V41N2.HV Keywords: Paraconsistent Logic LFIs ABSTRACT Review of the book 'Paraconsistent Logic: Consistency, Contradiction and Negation' (2016), by Walter Carnielli and Marcelo Coniglio The principle of explosion (also known as ex contradictione sequitur quodlibet) states that a pair of contradictory formulas entails any formula whatsoever of the relevant language and, accordingly, any theory regimented on the basis of a logic for which this principle holds (such as classical and intuitionistic logic) will turn out to be trivial if it contains a pair of theorems of the form A and ¬A (where ¬ is a negation operator). A logic is paraconsistent if it rejects the principle of explosion, allowing thus for the possibility of contradictory and yet non-trivial theories. Among the several paraconsistent logics that have been proposed in the literature, there is a particular family of (propositional and quantified) systems known as Logics of Formal Inconsistency (LFIs), developed and thoroughly studied Manuscrito – Rev. Int. Fil. Campinas, v. 41, n. 2, pp. 111-122, abr.-jun. 2018. Henrique Antunes & Vincenzo Ciccarelli 112 within the Brazilian tradition on paraconsistency. A distinguishing feature of the LFIs is that although they reject the general validity of the principle of explosion, as all other paraconsistent logics do, they admit a a restrcited version of it known as principle of gentle explosion. -
Warren Goldfarb, Notes on Metamathematics
Notes on Metamathematics Warren Goldfarb W.B. Pearson Professor of Modern Mathematics and Mathematical Logic Department of Philosophy Harvard University DRAFT: January 1, 2018 In Memory of Burton Dreben (1927{1999), whose spirited teaching on G¨odeliantopics provided the original inspiration for these Notes. Contents 1 Axiomatics 1 1.1 Formal languages . 1 1.2 Axioms and rules of inference . 5 1.3 Natural numbers: the successor function . 9 1.4 General notions . 13 1.5 Peano Arithmetic. 15 1.6 Basic laws of arithmetic . 18 2 G¨odel'sProof 23 2.1 G¨odelnumbering . 23 2.2 Primitive recursive functions and relations . 25 2.3 Arithmetization of syntax . 30 2.4 Numeralwise representability . 35 2.5 Proof of incompleteness . 37 2.6 `I am not derivable' . 40 3 Formalized Metamathematics 43 3.1 The Fixed Point Lemma . 43 3.2 G¨odel'sSecond Incompleteness Theorem . 47 3.3 The First Incompleteness Theorem Sharpened . 52 3.4 L¨ob'sTheorem . 55 4 Formalizing Primitive Recursion 59 4.1 ∆0,Σ1, and Π1 formulas . 59 4.2 Σ1-completeness and Σ1-soundness . 61 4.3 Proof of Representability . 63 3 5 Formalized Semantics 69 5.1 Tarski's Theorem . 69 5.2 Defining truth for LPA .......................... 72 5.3 Uses of the truth-definition . 74 5.4 Second-order Arithmetic . 76 5.5 Partial truth predicates . 79 5.6 Truth for other languages . 81 6 Computability 85 6.1 Computability . 85 6.2 Recursive and partial recursive functions . 87 6.3 The Normal Form Theorem and the Halting Problem . 91 6.4 Turing Machines . -
Abstract Algebraic Logic an Introductory Chapter
Abstract Algebraic Logic An Introductory Chapter Josep Maria Font 1 Introduction . 1 2 Some preliminary issues . 5 3 Bare algebraizability . 10 3.1 The equational consequence relative to a class of algebras . 11 3.2 Translating formulas into equations . 13 3.3 Translating equations into formulas . 15 3.4 Putting it all together . 15 4 The origins of algebraizability: the Lindenbaum-Tarski process . 18 4.1 The process for classical logic . 18 4.2 The process for algebraizable logics . 20 4.3 The universal Lindenbaum-Tarski process: matrix semantics . 23 4.4 Algebraizability and matrix semantics: definability . 27 4.5 Implicative logics . 29 5 Modes of algebraizability, and non-algebraizability . 32 6 Beyond algebraizability . 37 6.1 The Leibniz hierarchy . 37 6.2 The general definition of the algebraic counterpart of a logic . 45 6.3 The Frege hierarchy . 49 7 Exploiting algebraizability: bridge theorems and transfer theorems . 55 8 Algebraizability at a more abstract level . 65 References . 69 1 Introduction This chapter has been conceived as a brief introduction to (my personal view of) abstract algebraic logic. It is organized around the central notion of algebraizabil- ity, with particular emphasis on its connections with the techniques of traditional algebraic logic and especially with the so-called Lindenbaum-Tarski process. It Josep Maria Font Departament of Mathematics and Computer Engineering, Universitat de Barcelona (UB), Gran Via de les Corts Catalanes 585, E-08007 Barcelona, Spain. e-mail: [email protected] 1 2 Josep Maria Font goes beyond algebraizability to offer a general overview of several classifications of (sentential) logics—basically, the Leibniz hierarchy and the Frege hierarchy— which have emerged in recent decades, and to show how the classification of a logic into any of them provides some knowledge regarding its algebraic or its metalogical properties.