Deduction in Propositional Logics
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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. -
Revisiting Gödel and the Mechanistic Thesis Alessandro Aldini, Vincenzo Fano, Pierluigi Graziani
Theory of Knowing Machines: Revisiting Gödel and the Mechanistic Thesis Alessandro Aldini, Vincenzo Fano, Pierluigi Graziani To cite this version: Alessandro Aldini, Vincenzo Fano, Pierluigi Graziani. Theory of Knowing Machines: Revisiting Gödel and the Mechanistic Thesis. 3rd International Conference on History and Philosophy of Computing (HaPoC), Oct 2015, Pisa, Italy. pp.57-70, 10.1007/978-3-319-47286-7_4. hal-01615307 HAL Id: hal-01615307 https://hal.inria.fr/hal-01615307 Submitted on 12 Oct 2017 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. Distributed under a Creative Commons Attribution| 4.0 International License Theory of Knowing Machines: Revisiting G¨odeland the Mechanistic Thesis Alessandro Aldini1, Vincenzo Fano1, and Pierluigi Graziani2 1 University of Urbino \Carlo Bo", Urbino, Italy falessandro.aldini, [email protected] 2 University of Chieti-Pescara \G. D'Annunzio", Chieti, Italy [email protected] Abstract. Church-Turing Thesis, mechanistic project, and G¨odelian Arguments offer different perspectives of informal intuitions behind the relationship existing between the notion of intuitively provable and the definition of decidability by some Turing machine. One of the most for- mal lines of research in this setting is represented by the theory of know- ing machines, based on an extension of Peano Arithmetic, encompassing an epistemic notion of knowledge formalized through a modal operator denoting intuitive provability. -
An Un-Rigorous Introduction to the Incompleteness Theorems∗
An un-rigorous introduction to the incompleteness theorems∗ phil 43904 Jeff Speaks October 8, 2007 1 Soundness and completeness When discussing Russell's logical system and its relation to Peano's axioms of arithmetic, we distinguished between the axioms of Russell's system, and the theorems of that system. Roughly, the axioms of a theory are that theory's basic assumptions, and the theorems are the formulae provable from the axioms using the rules provided by the theory. Suppose we are talking about the theory A of arithmetic. Then, if we express the idea that a certain sentence p is a theorem of A | i.e., provable from A's axioms | as follows: `A p Now we can introduce the notion of the valid sentences of a theory. For our purposes, think of a valid sentence as a sentence in the language of the theory that can't be false. For example, consider a simple logical language which contains some predicates (written as upper-case letters), `not', `&', and some names, each of which is assigned some object or other in each interpretation. Now consider a sentence like F n In most cases, a sentence like this is going to be true on some interpretations, and false in others | it depends on which object is assigned to `n', and whether it is in the set of things assigned to `F '. However, if we consider a sentence like ∗The source for much of what follows is Michael Detlefsen's (much less un-rigorous) “G¨odel'stheorems", available at http://www.rep.routledge.com/article/Y005. -
Q1: Is Every Tautology a Theorem? Q2: Is Every Theorem a Tautology?
Section 3.2: Soundness Theorem; Consistency. Math 350: Logic Class08 Fri 8-Feb-2001 This section is mainly about two questions: Q1: Is every tautology a theorem? Q2: Is every theorem a tautology? What do these questions mean? Review de¯nitions, and discuss. We'll see that the answer to both questions is: Yes! Soundness Theorem 1. (The Soundness Theorem: Special case) Every theorem of Propositional Logic is a tautol- ogy; i.e., for every formula A, if A, then = A. ` j Sketch of proof: Q: Is every axiom a tautology? Yes. Why? Q: For each of the four rules of inference, check: if the antecedent is a tautology, does it follow that the conclusion is a tautology? Q: How does this prove the theorem? A: Suppose we have a proof, i.e., a list of formulas: A1; ; An. A1 is guaranteed to be a tautology. Why? So A2 is guaranteed to be a tautology. Why? S¢o¢ ¢A3 is guaranteed to be a tautology. Why? And so on. Q: What is a more rigorous way to prove this? Ans: Use induction. Review: What does S A mean? What does S = A mean? Theorem 2. (The Soun`dness Theorem: General cjase) Let S be any set of formulas. Then every theorem of S is a tautological consequence of S; i.e., for every formula A, if S A, then S = A. ` j Proof: This uses similar ideas as the proof of the special case. We skip the proof. Adequacy Theorem 3. (The Adequacy Theorem for the formal system of Propositional Logic: Special Case) Every tautology is a theorem of Propositional Logic; i.e., for every formula A, if = A, then A. -
Proof of the Soundness Theorem for Sentential Logic
The Soundness Theorem for Sentential Logic These notes contain a proof of the Soundness Theorem for Sentential Logic. The Soundness Theorem says that our natural deduction proofs represent a sound (or correct) system of reasoning. Formally, the Soundness Theorem states that: If Γ├ φ then Γ╞ φ. This says that for any set of premises, if you can prove φ from that set, then φ is truth-functionally entailed by that set. In other words, if our rules let you prove that something follows from some premises, then it ‘really does’ follow from those premises. By definition Γ╞ φ means that there is no TVA that makes all of the members of Γ true and also makes φ false. Note that an equivalent definition would be “Γ╞ φ if any TVA that makes all the members of Γ true also makes φ true”. By contraposition, the soundness theorem is equivalent to “If it is not true that Γ╞ φ then it is not true that Γ├ φ.” Unpacking the definitions, this means “If there is a TVA that makes all the members of Γ true but makes φ false, then there is no way to prove φ from Γ.” We might call this “The Negative Criterion.” This is important since the standard way of showing that something is not provable is to give an invalidating TVA. This only works assuming that the soundness theorem is true. Let’s get started on the proof. We are trying to prove a conditional claim, so we assume its antecedent and try to prove its consequent. -
Logic and Proof
Logic and Proof Computer Science Tripos Part IB Lent Term Lawrence C Paulson Computer Laboratory University of Cambridge [email protected] Copyright c 2018 by Lawrence C. Paulson I Logic and Proof 101 Introduction to Logic Logic concerns statements in some language. The language can be natural (English, Latin, . ) or formal. Some statements are true, others false or meaningless. Logic concerns relationships between statements: consistency, entailment, . Logical proofs model human reasoning (supposedly). Lawrence C. Paulson University of Cambridge I Logic and Proof 102 Statements Statements are declarative assertions: Black is the colour of my true love’s hair. They are not greetings, questions or commands: What is the colour of my true love’s hair? I wish my true love had hair. Get a haircut! Lawrence C. Paulson University of Cambridge I Logic and Proof 103 Schematic Statements Now let the variables X, Y, Z, . range over ‘real’ objects Black is the colour of X’s hair. Black is the colour of Y. Z is the colour of Y. Schematic statements can even express questions: What things are black? Lawrence C. Paulson University of Cambridge I Logic and Proof 104 Interpretations and Validity An interpretation maps variables to real objects: The interpretation Y 7 coal satisfies the statement Black is the colour→ of Y. but the interpretation Y 7 strawberries does not! A statement A is valid if→ all interpretations satisfy A. Lawrence C. Paulson University of Cambridge I Logic and Proof 105 Consistency, or Satisfiability A set S of statements is consistent if some interpretation satisfies all elements of S at the same time. -
Satisfiability 6 the Decision Problem 7
Satisfiability Difficult Problems Dealing with SAT Implementation Klaus Sutner Carnegie Mellon University 2020/02/04 Finding Hard Problems 2 Entscheidungsproblem to the Rescue 3 The Entscheidungsproblem is solved when one knows a pro- cedure by which one can decide in a finite number of operations So where would be look for hard problems, something that is eminently whether a given logical expression is generally valid or is satis- decidable but appears to be outside of P? And, we’d like the problem to fiable. The solution of the Entscheidungsproblem is of funda- be practical, not some monster from CRT. mental importance for the theory of all fields, the theorems of which are at all capable of logical development from finitely many The Circuit Value Problem is a good indicator for the right direction: axioms. evaluating Boolean expressions is polynomial time, but relatively difficult D. Hilbert within P. So can we push CVP a little bit to force it outside of P, just a little bit? In a sense, [the Entscheidungsproblem] is the most general Say, up into EXP1? problem of mathematics. J. Herbrand Exploiting Difficulty 4 Scaling Back 5 Herbrand is right, of course. As a research project this sounds like a Taking a clue from CVP, how about asking questions about Boolean fiasco. formulae, rather than first-order? But: we can turn adversity into an asset, and use (some version of) the Probably the most natural question that comes to mind here is Entscheidungsproblem as the epitome of a hard problem. Is ϕ(x1, . , xn) a tautology? The original Entscheidungsproblem would presumable have included arbitrary first-order questions about number theory. -
Solving the Boolean Satisfiability Problem Using the Parallel Paradigm Jury Composition
Philosophæ doctor thesis Hoessen Benoît Solving the Boolean Satisfiability problem using the parallel paradigm Jury composition: PhD director Audemard Gilles Professor at Universit´ed'Artois PhD co-director Jabbour Sa¨ıd Assistant Professor at Universit´ed'Artois PhD co-director Piette C´edric Assistant Professor at Universit´ed'Artois Examiner Simon Laurent Professor at University of Bordeaux Examiner Dequen Gilles Professor at University of Picardie Jules Vernes Katsirelos George Charg´ede recherche at Institut national de la recherche agronomique, Toulouse Abstract This thesis presents different technique to solve the Boolean satisfiability problem using parallel and distributed architec- tures. In order to provide a complete explanation, a careful presentation of the CDCL algorithm is made, followed by the state of the art in this domain. Once presented, two proposi- tions are made. The first one is an improvement on a portfo- lio algorithm, allowing to exchange more data without loosing efficiency. The second is a complete library with its API al- lowing to easily create distributed SAT solver. Keywords: SAT, parallelism, distributed, solver, logic R´esum´e Cette th`ese pr´esente diff´erentes techniques permettant de r´esoudre le probl`eme de satisfaction de formule bool´eenes utilisant le parall´elismeet du calcul distribu´e. Dans le but de fournir une explication la plus compl`ete possible, une pr´esentation d´etaill´ee de l'algorithme CDCL est effectu´ee, suivi d'un ´etatde l'art. De ce point de d´epart,deux pistes sont explor´ees. La premi`ereest une am´eliorationd'un algorithme de type portfolio, permettant d'´echanger plus d'informations sans perte d’efficacit´e. -
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 . -
Constructing a Categorical Framework of Metamathematical Comparison Between Deductive Systems of Logic
Bard College Bard Digital Commons Senior Projects Spring 2016 Bard Undergraduate Senior Projects Spring 2016 Constructing a Categorical Framework of Metamathematical Comparison Between Deductive Systems of Logic Alex Gabriel Goodlad Bard College, [email protected] Follow this and additional works at: https://digitalcommons.bard.edu/senproj_s2016 Part of the Logic and Foundations Commons This work is licensed under a Creative Commons Attribution-Noncommercial-No Derivative Works 4.0 License. Recommended Citation Goodlad, Alex Gabriel, "Constructing a Categorical Framework of Metamathematical Comparison Between Deductive Systems of Logic" (2016). Senior Projects Spring 2016. 137. https://digitalcommons.bard.edu/senproj_s2016/137 This Open Access work is protected by copyright and/or related rights. It has been provided to you by Bard College's Stevenson Library with permission from the rights-holder(s). You are free to use this work in any way that is permitted by the copyright and related rights. For other uses you need to obtain permission from the rights- holder(s) directly, unless additional rights are indicated by a Creative Commons license in the record and/or on the work itself. For more information, please contact [email protected]. Constructing a Categorical Framework of Metamathematical Comparison Between Deductive Systems of Logic A Senior Project submitted to The Division of Science, Mathematics, and Computing of Bard College by Alex Goodlad Annandale-on-Hudson, New York May, 2016 Abstract The topic of this paper in a broad phrase is \proof theory". It tries to theorize the general notion of \proving" something using rigorous definitions, inspired by previous less general theories. The purpose for being this general is to eventually establish a rigorous framework that can bridge the gap when interrelating different logical systems, particularly ones that have not been as well defined rigorously, such as sequent calculus. -
6.3 Mathematical Induction 6.4 Soundness
6.3 Mathematical Induction This principle is often used in logical metatheory. It is a corollary of mathematical induction. Actually, it To complete our proofs of the soundness and com- is a version of it. Let φ′ be the property a number has pleteness of propositional modal logic, it is worth if and only if all wffs of the logical language having briefly discussing the principles of mathematical in- that number of logical operators and atomic state- duction, which we assume in our metalanguage. ments are φ. If φ is true of the simplest well-formed formulas, i.e., those that contain zero operators, then A natural number is a nonnegative whole number, ′ 0 has φ . Similarly, if φ holds of any wffs that are or a number in the series 0, 1, 2, 3, 4, ... constructed out of simpler wffs provided that those The principle of mathematical induction: If simpler wffs are φ, then whenever a given natural number n has φ′ then n +1 also has φ′. Hence, by (φ is true of 0) and (for all natural numbers n, ′ if φ is true of n, then φ is true of n + 1), then φ mathematical induction, all natural numbers have φ , is true of all natural numbers. i.e., no matter how many operators a wff contains, it has φ. In this way wff induction simply reduces to To use the principle mathematical induction to arrive mathematical induction. at the conclusion that something is true of all natural Similarly, this principle is usually utilized by prov- numbers, one needs to prove the two conjuncts of the ing: antecedent: 1. -
Foundations of Mathematics
Foundations of Mathematics November 27, 2017 ii Contents 1 Introduction 1 2 Foundations of Geometry 3 2.1 Introduction . .3 2.2 Axioms of plane geometry . .3 2.3 Non-Euclidean models . .4 2.4 Finite geometries . .5 2.5 Exercises . .6 3 Propositional Logic 9 3.1 The basic definitions . .9 3.2 Disjunctive Normal Form Theorem . 11 3.3 Proofs . 12 3.4 The Soundness Theorem . 19 3.5 The Completeness Theorem . 20 3.6 Completeness, Consistency and Independence . 22 4 Predicate Logic 25 4.1 The Language of Predicate Logic . 25 4.2 Models and Interpretations . 27 4.3 The Deductive Calculus . 29 4.4 Soundness Theorem for Predicate Logic . 33 5 Models for Predicate Logic 37 5.1 Models . 37 5.2 The Completeness Theorem for Predicate Logic . 37 5.3 Consequences of the completeness theorem . 41 5.4 Isomorphism and elementary equivalence . 43 5.5 Axioms and Theories . 47 5.6 Exercises . 48 6 Computability Theory 51 6.1 Introduction and Examples . 51 6.2 Finite State Automata . 52 6.3 Exercises . 55 6.4 Turing Machines . 56 6.5 Recursive Functions . 61 6.6 Exercises . 67 iii iv CONTENTS 7 Decidable and Undecidable Theories 69 7.1 Introduction . 69 7.1.1 Gödel numbering . 69 7.2 Decidable vs. Undecidable Logical Systems . 70 7.3 Decidable Theories . 71 7.4 Gödel’s Incompleteness Theorems . 74 7.5 Exercises . 81 8 Computable Mathematics 83 8.1 Computable Combinatorics . 83 8.2 Computable Analysis . 85 8.2.1 Computable Real Numbers . 85 8.2.2 Computable Real Functions .