Modal Metalogic: Completeness
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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. -
5 Propositional Logic: Consistency and Completeness
5 Propositional Logic: Consistency and completeness Reading: Metalogic Part II, 24, 15, 28-31 Contents 5.1 Soundness . 61 5.2 Consistency . 62 5.3 Completeness . 63 5.3.1 An Axiomatization of Propositional Logic . 63 5.3.2 Kalmar's Proof: Informal Exposition . 66 5.3.3 Kalmar's Proof . 68 5.4 Homework Exercises . 70 5.4.1 Questions . 70 5.4.2 Answers . 70 5.1 Soundness In this section, we establish the soundness of the system, i.e., Theorem 3 (Soundness). Every theorem is a tautology, i.e., If ` A then j= A. Proof The proof is by induction the length of the proof of A. For the Basis step, we show that each of the axioms is a tautology. For the induction step, we show that if A and A ⊃ B are tautologies, then B is a tautology. Case 1 (PS1): AB B ⊃ A (A ⊃ (B ⊃ A)) TT TT TF TT FT FT FF TT Case 2 (PS2) 62 5 Propositional Logic: Consistency and completeness XYZ ABC B ⊃ C A ⊃ (B ⊃ C) A ⊃ B A ⊃ C Y ⊃ ZX ⊃ (Y ⊃ Z) TTT TTTTTT TTF FFTFFT TFT TTFTTT TFF TTFFTT FTT TTTTTT FTF FTTTTT FFT TTTTTT FFF TTTTTT Case 3 (PS3) AB » B » A » B ⊃∼ A A ⊃ B (» B ⊃∼ A) ⊃ (A ⊃ B) TT FFTTT TF TFFFT FT FTTTT FF TTTTT Case 4 (MP). If A is a tautology, i.e., true for every assignment of truth values to the atomic letters, and if A ⊃ B is a tautology, then there is no assignment which makes A T and B F. -
Completeness of the Propositions-As-Types Interpretation of Intuitionistic Logic Into Illative Combinatory Logic
University of Wollongong Research Online Faculty of Engineering and Information Faculty of Engineering and Information Sciences - Papers: Part A Sciences 1-1-1998 Completeness of the propositions-as-types interpretation of intuitionistic logic into illative combinatory logic Wil Dekkers Catholic University, Netherlands Martin Bunder University of Wollongong, [email protected] Henk Barendregt Catholic University, Netherlands Follow this and additional works at: https://ro.uow.edu.au/eispapers Part of the Engineering Commons, and the Science and Technology Studies Commons Recommended Citation Dekkers, Wil; Bunder, Martin; and Barendregt, Henk, "Completeness of the propositions-as-types interpretation of intuitionistic logic into illative combinatory logic" (1998). Faculty of Engineering and Information Sciences - Papers: Part A. 1883. https://ro.uow.edu.au/eispapers/1883 Research Online is the open access institutional repository for the University of Wollongong. For further information contact the UOW Library: [email protected] Completeness of the propositions-as-types interpretation of intuitionistic logic into illative combinatory logic Abstract Illative combinatory logic consists of the theory of combinators or lambda calculus extended by extra constants (and corresponding axioms and rules) intended to capture inference. In a preceding paper, [2], we considered 4 systems of illative combinatory logic that are sound for first order intuitionistic prepositional and predicate logic. The interpretation from ordinary logic into the illative systems can be done in two ways: following the propositions-as-types paradigm, in which derivations become combinators, or in a more direct way, in which derivations are not translated. Both translations are closely related in a canonical way. In the cited paper we proved completeness of the two direct translations. -
An Introduction to First-Order Logic
Outline An Introduction to First-Order Logic K. Subramani1 1Lane Department of Computer Science and Electrical Engineering West Virginia University Completeness, Compactness and Inexpressibility Subramani First-Order Logic Outline Outline 1 Completeness of proof system for First-Order Logic The notion of Completeness The Completeness Proof 2 Consequences of the Completeness theorem Complexity of Validity Compactness Model Cardinality Lowenheim-Skolem¨ Theorem Inexpressibility of Reachability Subramani First-Order Logic Outline Outline 1 Completeness of proof system for First-Order Logic The notion of Completeness The Completeness Proof 2 Consequences of the Completeness theorem Complexity of Validity Compactness Model Cardinality Lowenheim-Skolem¨ Theorem Inexpressibility of Reachability Subramani First-Order Logic Completeness The notion of Completeness Consequences of the Completeness theorem The Completeness Proof Outline 1 Completeness of proof system for First-Order Logic The notion of Completeness The Completeness Proof 2 Consequences of the Completeness theorem Complexity of Validity Compactness Model Cardinality Lowenheim-Skolem¨ Theorem Inexpressibility of Reachability Subramani First-Order Logic Completeness The notion of Completeness Consequences of the Completeness theorem The Completeness Proof Soundness and Completeness Theorem Soundness: If ∆ ⊢ φ, then ∆ |= φ. Theorem Completeness (Godel’s¨ traditional form): If ∆ |= φ, then ∆ ⊢ φ. Theorem Completeness (Godel’s¨ altenate form): If ∆ is consistent, then it has a model. Subramani First-Order Logic Completeness The notion of Completeness Consequences of the Completeness theorem The Completeness Proof Soundness and Completeness (contd.) Theorem The traditional completeness theorem follows from the alternate form of the completeness theorem. Proof. Assume that ∆ |= φ. It follows that any model M that satisfies all the expressions in ∆, also satisfies φ and hence falsifies ¬φ. -
Completeness and Compactness of First-Order Tableaux
CS 486: Applied Logic Lecture 18, March 27, 2003 18 Completeness and Compactness of First-Order Tableaux 18.1 Completeness Proving the completeness of a first-order calculus gives us G¨odel’sfamous completeness result. G¨odelproved it for a slightly different proof calculus, and the proof that we will show here goes back to Beth and Hintikka. Let us briefly resume the propositional case. The key to the completeness proof was the use of Hintikka’s lemma, which states that every downward saturated set, finite or not, is satisfiable. We then showed that every open and complete path is in fact a Hintikka sequence. Putting these two things together we reasoned that the root of an open and complete tableau must be satisfiable. Thus a complete tableau for a valid formula cannot be open which means that every tableau for a valid formula will eventually close. We will prove the first order case along these lines, but have to keep in mind that several things have changed. ² The definition of a valuation now includes quantifiers. ² The definition of Hintikka sets must take γ and ± formulas into account. ² The notion of a complete tableau needs to be adjusted, because there is now the possibility of non-terminating proof attempts. Fortunately, we can easily make the necessary adjustments and then proceed as before. First, let us define first-order Hintikka sets. A Hintikka Set for a universe U is a set S of U-formulas such that for all closed U-formulas A, ®, ¯, γ, and ± the following conditions hold. 2 ¯ 62 H0 : A atomic and A S 7! A S 2 2 2 H1 : ® S 7! ®1 S ^ ®2 S 2 2 2 H2 : ¯ S 7! ¯1 S _ ¯2 S 2 2 2 H3 : γ S 7! 8k U. -
PHIL 677 Lec 01 Metalogic Fall Term 2019 MW 14:00–15:15, Math Sciences 319
PHIL 677 Lec 01 Metalogic Fall Term 2019 MW 14:00–15:15, Math Sciences 319 Course Outline Instructor: Richard Zach (he/him) Email: [email protected] Oce: Social Sciences 1230 Oce Hours: MW 13:00–13:50, or by appointment Phone: (403) 220–3170 Teaching Assistant: Weidong Sun (he/him) Email: [email protected] Oce: Social Sciences 1242 Oce Hours: T 15:40-16:40, R 10:00–11:00 Phone: (403) 220–6464 Email is preferred over phone. However please keep the following in mind: 1. Please ensure that “Phil 677” or some other clearly identifying term occurs in the subject line. Otherwise there is a strong possibility that your message will be deleted unread as spam. 2. If you email to make an appointment please indicate the times when you are available. 3. Please make sure your first and last names are clearly included in the body of any email message. 4. We will not respond to email after 7pm or on weekends. 1 Course Information Formal logic has many applications both within philosophy and outside (espe- cially in mathematics, computer science, and linguistics). This second course will introduce you to the concepts, results, and methods of formal logic neces- sary to understand and appreciate these applications as well as the limitations of formal logic. It will be mathematical in that you will be required to master abstract formal concepts and to prove theorems about logic; but it does not presuppose any (advanced) knowledge of mathematics. We will start from the basics. We will begin by studying some basic formal concepts: sets, relations, and functions, and the sizes of infinite sets. -
Provability, Soundness and Completeness Deductive Rules of Inference Provide a Mechanism for Deriving True Conclusions from True Premises
CSC 244/444 Lecture Notes Sept. 16, 2021 Provability, Soundness and Completeness Deductive rules of inference provide a mechanism for deriving true conclusions from true premises Rules of inference So far we have treated formulas as \given", and have shown how they can be related to a domain of discourse, and how the truth of a set of premises can guarantee (entail) the truth of a conclusion. However, our goal in logic and particularly in AI is to derive new conclusions from given facts. For this we need rules of inference (and later, strategies for applying such rules so as to derive a desired conclusion, if possible). In general, a \forward" inference rule consists of one or more premises and a con- clusion. Both the premises and the conclusion are generally schemas, i.e., they involve metavariables for formulas or terms that can be particularized in many ways (just as we saw in the case of valid formula schemas). We often put a horizontal line under the premises, and write the conclusion underneath the line. For instance, here is the rule of Modus Ponens φ, φ ) MP : This says that given a premise formula φ, and another formula of form φ ) , we may derive the conclusion . It is intuitively clear that this rule leads from true premises to a true conclusion { but this is an intuition we need to verify by proving the rule sound, as illustrated below. An example of using the rule is this: from Dog(Snoopy), Dog(Snoopy) ) Has-tail(Snoopy), we can conclude Has-tail(Snoopy). -
Notes on Incompleteness Theorems
The Incompleteness Theorems ● Here are some fundamental philosophical questions with mathematical answers: ○ (1) Is there a (recursive) algorithm for deciding whether an arbitrary sentence in the language of first-order arithmetic is true? ○ (2) Is there an algorithm for deciding whether an arbitrary sentence in the language of first-order arithmetic is a theorem of Peano or Robinson Arithmetic? ○ (3) Is there an algorithm for deciding whether an arbitrary sentence in the language of first-order arithmetic is a theorem of pure (first-order) logic? ○ (4) Is there a complete (even if not recursive) recursively axiomatizable theory in the language of first-order arithmetic? ○ (5) Is there a recursively axiomatizable sub-theory of Peano Arithmetic that proves the consistency of Peano Arithmetic (even if it leaves other questions undecided)? ○ (6) Is there a formula of arithmetic that defines arithmetic truth in the standard model, N (even if it does not represent it)? ○ (7) Is the (non-recursively enumerable) set of truths in the language of first-order arithmetic categorical? If not, is it ω-categorical (i.e., categorical in models of cardinality ω)? ● Questions (1) -- (7) turn out to be linked. Their philosophical interest depends partly on the following philosophical thesis, of which we will make frequent, but inessential, use. ○ Church-Turing Thesis:A function is (intuitively) computable if/f it is recursive. ■ Church: “[T]he notion of an effectively calculable function of positive integers should be identified with that of recursive function (quoted in Epstein & Carnielli, 223).” ○ Note: Since a function is recursive if/f it is Turing computable, the Church-Turing Thesis also implies that a function is computable if/f it is Turing computable. -
On Synthetic Undecidability in Coq, with an Application to the Entscheidungsproblem
On Synthetic Undecidability in Coq, with an Application to the Entscheidungsproblem Yannick Forster Dominik Kirst Gert Smolka Saarland University Saarland University Saarland University Saarbrücken, Germany Saarbrücken, Germany Saarbrücken, Germany [email protected] [email protected] [email protected] Abstract like decidability, enumerability, and reductions are avail- We formalise the computational undecidability of validity, able without reference to a concrete model of computation satisfiability, and provability of first-order formulas follow- such as Turing machines, general recursive functions, or ing a synthetic approach based on the computation native the λ-calculus. For instance, representing a given decision to Coq’s constructive type theory. Concretely, we consider problem by a predicate p on a type X, a function f : X ! B Tarski and Kripke semantics as well as classical and intu- with 8x: p x $ f x = tt is a decision procedure, a function itionistic natural deduction systems and provide compact д : N ! X with 8x: p x $ ¹9n: д n = xº is an enumer- many-one reductions from the Post correspondence prob- ation, and a function h : X ! Y with 8x: p x $ q ¹h xº lem (PCP). Moreover, developing a basic framework for syn- for a predicate q on a type Y is a many-one reduction from thetic computability theory in Coq, we formalise standard p to q. Working formally with concrete models instead is results concerning decidability, enumerability, and reducibil- cumbersome, given that every defined procedure needs to ity without reference to a concrete model of computation. be shown representable by a concrete entity of the model. -
Classical Metalogic an Introduction to Classical Model Theory & Computability
Classical Metalogic An Introduction to Classical Model Theory & Computability Notes by R.J. Buehler January 5, 2015 2 Preface What follows are my personal notes created in preparation for the UC-Berkeley Group in Logic preliminary exam. I am not a computability theorist, nor a model theorist; I am a graduate student with some knowledge who is–alas–quite fallible. Accordingly, this text is made available as a convenient reference, set of notes, and summary, but without even the slight hint of a guarantee that everything contained within is factual and correct (indeed, some areas are entirely unchanged from the moment I copied them off the blackboard). This said, if you find a serious error, I would greatly appreciate it if you would let me know so that it can be corrected. The material for these notes derives from a wide variety of sources: Lectures by Wes Holliday Lectures by Antonio Montalban Lectures by John Steel Kevin Kelly’s computability theory notes David Marker’s “Model Theory: An Introduction” Wilfrid Hodge’s “A Shorter Model Theory” Robert Soare’s “Recursively Enumerable Sets and Degrees" Richard Kaye’s “Models of Peano Arithmetic” Chang and Keisler’s “Model Theory" While I certainly hope my notes are beneficial, if you’re attempting to learn the contained material for the first time, I would highly suggest picking up (at least) a copy of Marker and Soare’s texts in addition. To those Group in Logic students who may be using these notes to help themselves prepare for their preliminary exam, chapters 1-5, 7, and 9-18 contain relevant material, as well as chapter 8, section 3. -
METALOGIC METALOGIC an Introduction to the Metatheory of Standard First Order Logic
METALOGIC METALOGIC An Introduction to the Metatheory of Standard First Order Logic Geoffrey Hunter Senior Lecturer in the Department of Logic and Metaphysics University of St Andrews PALGRA VE MACMILLAN © Geoffrey Hunter 1971 Softcover reprint of the hardcover 1st edition 1971 All rights reserved. No part of this publication may be reproduced or transmitted, in any form or by any means, without permission. First published 1971 by MACMILLAN AND CO LTD London and Basingstoke Associated companies in New York Toronto Dublin Melbourne Johannesburg and Madras SBN 333 11589 9 (hard cover) 333 11590 2 (paper cover) ISBN 978-0-333-11590-9 ISBN 978-1-349-15428-9 (eBook) DOI 10.1007/978-1-349-15428-9 The Papermac edition of this book is sold subject to the condition that it shall not, by way of trade or otherwise, be lent, resold, hired out, or otherwise circulated without the publisher's prior consent, in any form of binding or cover other than that in which it is published and without a similar condition including this condition being imposed on the subsequent purchaser. To my mother and to the memory of my father, Joseph Walter Hunter Contents Preface xi Part One: Introduction: General Notions 1 Formal languages 4 2 Interpretations of formal languages. Model theory 6 3 Deductive apparatuses. Formal systems. Proof theory 7 4 'Syntactic', 'Semantic' 9 5 Metatheory. The metatheory of logic 10 6 Using and mentioning. Object language and metalang- uage. Proofs in a formal system and proofs about a formal system. Theorem and metatheorem 10 7 The notion of effective method in logic and mathematics 13 8 Decidable sets 16 9 1-1 correspondence. -
The Entscheidungsproblem and Alan Turing
The Entscheidungsproblem and Alan Turing Author: Laurel Brodkorb Advisor: Dr. Rachel Epstein Georgia College and State University December 18, 2019 1 Abstract Computability Theory is a branch of mathematics that was developed by Alonzo Church, Kurt G¨odel,and Alan Turing during the 1930s. This paper explores their work to formally define what it means for something to be computable. Most importantly, this paper gives an in-depth look at Turing's 1936 paper, \On Computable Numbers, with an Application to the Entscheidungsproblem." It further explores Turing's life and impact. 2 Historic Background Before 1930, not much was defined about computability theory because it was an unexplored field (Soare 4). From the late 1930s to the early 1940s, mathematicians worked to develop formal definitions of computable functions and sets, and applying those definitions to solve logic problems, but these problems were not new. It was not until the 1800s that mathematicians began to set an axiomatic system of math. This created problems like how to define computability. In the 1800s, Cantor defined what we call today \naive set theory" which was inconsistent. During the height of his mathematical period, Hilbert defended Cantor but suggested an axiomatic system be established, and characterized the Entscheidungsproblem as the \fundamental problem of mathemat- ical logic" (Soare 227). The Entscheidungsproblem was proposed by David Hilbert and Wilhelm Ackerman in 1928. The Entscheidungsproblem, or Decision Problem, states that given all the axioms of math, there is an algorithm that can tell if a proposition is provable. During the 1930s, Alonzo Church, Stephen Kleene, Kurt G¨odel,and Alan Turing worked to formalize how we compute anything from real numbers to sets of functions to solve the Entscheidungsproblem.