The Mother of All Paradoxes
|
|
- Benedict Booth
- 5 years ago
- Views:
Transcription
1 The Mother of All Paradoxes Volker Halbach Truth and Intensionality Amsterdam 3rd December 2016
2 A theory of expressions The symbols of L are: 1. infinitely many variable symbols v 0, v 1, v 2, v 3, predicate symbols = and, 3. function symbols q, and sub, 4. the connectives, and the quantifier symbol, 5. auxiliary symbols ( and ), 6. possibly finitely many further function and predicate symbols of arbitrary arity but at least the sentence letter p, and 7. for each string e of symbols there is exactly one constant in the language L. The constant for e is called the quotation constant of e. We write e for the quotation constant of e. I write x for v 0, y for v 1, and z for v 2 to make the notation less cluttered.
3 A theory of expressions I distinguish three kinds of nonlogical vocabulary (beyond the logical vocabulary): the syntactic (quotations constants, sub etc) and the contingent vocabulary (p etc), and.
4 A theory of expressions All instances of the following schemata and rules are axioms of the theory A: a.1. all axioms and rules of first-order predicate logic including the identity axioms. a.2. a b = ab, where a and b are arbitrary strings of symbols. a.3. q(a) = a a.4. sub(a, b, c) = d, where a and c are arbitrary strings of symbols, b is a symbol (or, equivalently, a string of symbols of length 1), and d is the string of symbols obtained from a by replacing all occurrences of the symbol b with c. a.5. x y z((x y) z) = (x (y z)) a.6. x y(x y = 0 x = 0 y = 0) a.7. x y(x y = x y = 0) x y(y x = x y = 0) a.8. v 3 v 4 y z(sub(v 3, y, z) sub(v 4, y, z)) = sub(v 3 v 4, y, z) a.9. a = b, if a and b are distinct expressions.
5 A theory of expressions A standard model for the language L without interprets the syntactic vocabulary in the intended way and assigns some interpretations to the other vocabulary, in particular to p. The domain of a standard model is always the set of all L-expressions (finite strings of symbols). Let S be a set of expressions. A sentence φ of L holds in E, S formally E, S φ if and only if φ holds when all nonlogical symbols of L except are interpreted according to the model E and is interpreted by S. S is called the extension of in the model E, S. We have: E, S e iff e S
6 A theory of expressions A standard model for the language L without interprets the syntactic vocabulary in the intended way and assigns some interpretations to the other vocabulary, in particular to p. The domain of a standard model is always the set of all L-expressions (finite strings of symbols). Let S be a set of expressions. A sentence φ of L holds in E, S formally E, S φ if and only if φ holds when all nonlogical symbols of L except are interpreted according to the model E and is interpreted by S. S is called the extension of in the model E, S. We have: E, S e iff e S
7 A theory of expressions A standard model for the language L without interprets the syntactic vocabulary in the intended way and assigns some interpretations to the other vocabulary, in particular to p. The domain of a standard model is always the set of all L-expressions (finite strings of symbols). Let S be a set of expressions. A sentence φ of L holds in E, S formally E, S φ if and only if φ holds when all nonlogical symbols of L except are interpreted according to the model E and is interpreted by S. S is called the extension of in the model E, S. We have: E, S e iff e S
8 A theory of expressions Definition dia(x) = sub(x, x, q(x))
9 A theory of expressions Lemma Assume φ(x) is a formula not containing bound occurrences of x. Then the following holds: A dia(φ(dia(x))) = φ(dia(φ(dia(x)))) dia(φ(dia(x))) = sub(φ(dia(x)), x, q(φ(dia(x)))) = sub(φ(dia(x)), x, φ(dia(x))) = φ(dia(φ(dia(x))))
10 A theory of expressions Lemma (strong diagonalization) If φ(x) is a formula of L with no bound occurrences of x, then one can find a term t such that the following holds: A t = φ(t) and therefore A φ(t) φ(φ(t))
11 Possible worlds semantics Definition A frame is an ordered pair W, R where W is nonempty and R is a binary relation on W.
12 Possible worlds semantics Definition A valuation V for a frame W, R is a function that assigns to every w W a standard model E. Definition A pw-model is a quadruple W, R, V, B such that W, R is a frame, V is a valuation for W, R and B is a -interpretation for W, R satisfying the following condition: B(w) = {φ L for all u W (if wru, then V(u), B(u) φ)} V(u), B(u) is always a standard model and V(u), B(u) φ means that φ is true in the standard model V(u), B(u) in the usual sense of first-order predicate logic.
13 Possible worlds semantics Definition A valuation V for a frame W, R is a function that assigns to every w W a standard model E. Definition A pw-model is a quadruple W, R, V, B such that W, R is a frame, V is a valuation for W, R and B is a -interpretation for W, R satisfying the following condition: B(w) = {φ L for all u W (if wru, then V(u), B(u) φ)} V(u), B(u) is always a standard model and V(u), B(u) φ means that φ is true in the standard model V(u), B(u) in the usual sense of first-order predicate logic.
14 Possible worlds semantics Definition A frame W, R admits a pw-model on every valuation iff for every valuation V on W, R there is a such that B such that W, R, V, B is a pw-model. Definition A frame admits a pw-model iff the frame admits a pw-model on some valuation, that is, iff there is a valuation V such that W, R, V, B is a pw-model model. Strong Characterization problem Which frames admit a pw-model on every valuation?
15 Possible worlds semantics Definition A frame W, R admits a pw-model on every valuation iff for every valuation V on W, R there is a such that B such that W, R, V, B is a pw-model. Definition A frame admits a pw-model iff the frame admits a pw-model on some valuation, that is, iff there is a valuation V such that W, R, V, B is a pw-model model. Strong Characterization problem Which frames admit a pw-model on every valuation?
16 Possible worlds semantics Definition A frame W, R admits a pw-model on every valuation iff for every valuation V on W, R there is a such that B such that W, R, V, B is a pw-model. Definition A frame admits a pw-model iff the frame admits a pw-model on some valuation, that is, iff there is a valuation V such that W, R, V, B is a pw-model model. Strong Characterization problem Which frames admit a pw-model on every valuation?
17 Possible worlds semantics Lemma (normality) Suppose W, R, V, B is a pw-model, w W and φ, ψ sentences of L. Then the following holds: 1. If V(u), B(u) φ for all u W, then V(w), B(w) φ. 2. V(w), B(w) φ ψ ( φ ψ)
18 Possible worlds semantics Lemma 1. If a frame W, R is transitive and W, R, V, B a pw-model on that frame, we have for all sentences φ in L and worlds w W: V(w), B(w) φ φ 2. If a frame W, R is reflexive and W, R, V, B a pw-model on that frame, we have for all sentences φ in L and worlds w W: V(w), B(w) φ φ
19 The paradoxes w Theorem (liar paradox) The frame W 1, R 1 does not admit a pw-model.
20 The paradoxes Example (Montague) If W, R admits a pw-model, then W, R is not reflexive.
21 The paradoxes Example The frame two worlds see each another displayed above does not admit a pw-model.
22 The paradoxes The following frame does not admit a pw-model.
23 The paradoxes Example The frame one world sees itself and one other world does not admit a pw-model. We call the frame W 3, R 3. w 1 w 2
24 The paradoxes Example The frame ω, Pre does not admit a pw-model. Here ω is the set of all natural numbers and Pre is the successor relation. Hence every world n sees n + 1 but no other world. The frame ω, Pre can be displayed by the following diagram: 0 1 2
25 The paradoxes Theorem The frame ω, < does not admit a pw-model. Here < is the usual smaller than relation on the natural numbers: The frame ω, < can be displayed by the following diagram: 0 1 2
26 Frames with possible worlds models Example The frame {w}, Ø admits a pw-model on every valuation.
27 Frames with possible worlds models By Suc we denote the successor relation { k, n k = n + 1} on the set ω of natural numbers. Example The frame ω, Suc admits a pw-model on every valuation
28 Frames with possible worlds models Definition A frame W, R is converse wellfounded (or Noetherian) iff for every non-empty M W there is a w M that is R-maximal in M. Lemma Every converse wellfounded frame W, R admits a pw-model on every valuation.
29 Frames with possible worlds models Definition A frame W, R is converse wellfounded (or Noetherian) iff for every non-empty M W there is a w M that is R-maximal in M. Lemma Every converse wellfounded frame W, R admits a pw-model on every valuation.
30 The Strong Characterization Problem Theorem (Strong Characterization theorem) A frame admits a pw-model on every valuation iff it is converse wellfounded.
31 The Strong Characterization Problem We define x as n..... x. n can be defined properly using the diagonal lemma. corresponds ot the transitive closure of R. Lemma For all sentences φ and ψ und pw-models W, R, V, B the following holds: 1. If V(w), B(w) φ for all w W, then V(w), B(w) φ. 2. V(w), B(w) φ ψ ( φ ψ) 3. V(w), B(w) φ φ 4. V(w), B(w) φ φ φ
32 The Strong Characterization Problem Lemma The transitive closure R of the accessibility relation R of any pw-model that admits a pw-model on every valuation is converse wellfounded. Lemma A frame W, R is converse wellfounded iff its transitive closure W, R is converse wellfounded. This concludes the proof of the Strong Characterization theorem.
33 All worlds share the same domain, the set of all expressions. The definition of a standard model can be generalized, so that a standard model also specifies a set of contingent objects. Since some of them may lack names, the unary predicate is replaced with a binary predicate applying to formulae and sequences of objects (variable assignments). This requires a theory sequences of arbitrary objects (symbols or contingent objects). Extensions
34 Löb s theorem is the mother of all paradoxes as long as the modality is normal. Various predicate modalities aren t normal in this sense, for instance, KF. Conclusion
35 Thank you. Conclusion
36 Now for something completely different... Conclusion
37 Classical Determinate Truth The theory CD is formulated in the language of arithmetic augmented with the new predicate symbols D and T. The axioms below are added to the axioms of PA with the schema of induction is expanded to the language with D and T.
38 Classical Determinate Truth Determinateness axioms (d1) s t D(s=. t) (d2) x ( Sent(x) (D(. x) Dx)) (d3) x y ( Sent(x. y) (D(x. y) Dx Dy)) (d4) v x ( Sent(. vx) (D(. vx) t Dx[t/v])) (d5) s(dṭs Ds Sent(s ))) Truth axioms (t1) s t(ts=. t s = t ) (t2) s( Sent(s ) Ds (TṬs Ts )) (t3) x ( Sent(x) (T(. x) Tx)) (t4) x y ( Sent(x. y) (T(x. y) Tx Ty)) (t5) v x ( Sent(. vx) (T(. vx) t Tx[t/v]))
39 Classical Determinate Truth Optional axioms (o1) s(dḍs Ds Sent(s ))) (o1) Can we add s( Sent(s ) Ds TḌs Ds ))
03 Review of First-Order Logic
CAS 734 Winter 2014 03 Review of First-Order Logic William M. Farmer Department of Computing and Software McMaster University 18 January 2014 What is First-Order Logic? First-order logic is the study of
More informationTRUTH TELLERS. Volker Halbach. Scandinavian Logic Symposium. Tampere
TRUTH TELLERS Volker Halbach Scandinavian Logic Symposium Tampere 25th August 2014 I m wrote two papers with Albert Visser on this and related topics: Self-Reference in Arithmetic, http://www.phil.uu.nl/preprints/lgps/number/316
More informationKRIPKE S THEORY OF TRUTH 1. INTRODUCTION
KRIPKE S THEORY OF TRUTH RICHARD G HECK, JR 1. INTRODUCTION The purpose of this note is to give a simple, easily accessible proof of the existence of the minimal fixed point, and of various maximal fixed
More informationSyntax. Notation Throughout, and when not otherwise said, we assume a vocabulary V = C F P.
First-Order Logic Syntax The alphabet of a first-order language is organised into the following categories. Logical connectives:,,,,, and. Auxiliary symbols:.,,, ( and ). Variables: we assume a countable
More informationAn Introduction to Modal Logic III
An Introduction to Modal Logic III Soundness of Normal Modal Logics Marco Cerami Palacký University in Olomouc Department of Computer Science Olomouc, Czech Republic Olomouc, October 24 th 2013 Marco Cerami
More informationFinal Exam (100 points)
Final Exam (100 points) Honor Code: Each question is worth 10 points. There is one bonus question worth 5 points. In contrast to the homework assignments, you may not collaborate on this final exam. You
More informationInformal Statement Calculus
FOUNDATIONS OF MATHEMATICS Branches of Logic 1. Theory of Computations (i.e. Recursion Theory). 2. Proof Theory. 3. Model Theory. 4. Set Theory. Informal Statement Calculus STATEMENTS AND CONNECTIVES Example
More informationClassical First-Order Logic
Classical First-Order Logic Software Formal Verification Maria João Frade Departmento de Informática Universidade do Minho 2008/2009 Maria João Frade (DI-UM) First-Order Logic (Classical) MFES 2008/09
More informationTRUTH-THEORIES FOR FRAGMENTS OF PA
TRUTH-THEORIES FOR FRAGMENTS OF PA RICHARD G. HECK, JR. The discussion here follows Petr Hájek and Pavel Pudlák, Metamathematics of First-order Arithmetic (Berlin: Springer-Verlag, 1993). See especially
More informationA simplified proof of arithmetical completeness theorem for provability logic GLP
A simplified proof of arithmetical completeness theorem for provability logic GLP L. Beklemishev Steklov Mathematical Institute Gubkina str. 8, 119991 Moscow, Russia e-mail: bekl@mi.ras.ru March 11, 2011
More informationCHAPTER 2. FIRST ORDER LOGIC
CHAPTER 2. FIRST ORDER LOGIC 1. Introduction First order logic is a much richer system than sentential logic. Its interpretations include the usual structures of mathematics, and its sentences enable us
More information3. Only sequences that were formed by using finitely many applications of rules 1 and 2, are propositional formulas.
1 Chapter 1 Propositional Logic Mathematical logic studies correct thinking, correct deductions of statements from other statements. Let us make it more precise. A fundamental property of a statement is
More informationA Quantified Logic of Evidence
A Quantified Logic of Evidence Melvin Fitting Dept. Mathematics and Computer Science Lehman College (CUNY), 250 Bedford Park Boulevard West Bronx, NY 10468-1589 e-mail: melvin.fitting@lehman.cuny.edu web
More informationIntroduction to first-order logic:
Introduction to first-order logic: First-order structures and languages. Terms and formulae in first-order logic. Interpretations, truth, validity, and satisfaction. Valentin Goranko DTU Informatics September
More informationFormal Methods for Java
Formal Methods for Java Lecture 12: Soundness of Sequent Calculus Jochen Hoenicke Software Engineering Albert-Ludwigs-University Freiburg June 12, 2017 Jochen Hoenicke (Software Engineering) Formal Methods
More informationINTRODUCTION TO LOGIC. Propositional Logic. Examples of syntactic claims
Introduction INTRODUCTION TO LOGIC 2 Syntax and Semantics of Propositional Logic Volker Halbach In what follows I look at some formal languages that are much simpler than English and define validity of
More informationPropositional and Predicate Logic - XIII
Propositional and Predicate Logic - XIII Petr Gregor KTIML MFF UK WS 2016/2017 Petr Gregor (KTIML MFF UK) Propositional and Predicate Logic - XIII WS 2016/2017 1 / 22 Undecidability Introduction Recursive
More informationFirst-Order Logic. 1 Syntax. Domain of Discourse. FO Vocabulary. Terms
First-Order Logic 1 Syntax Domain of Discourse The domain of discourse for first order logic is FO structures or models. A FO structure contains Relations Functions Constants (functions of arity 0) FO
More informationPOSSIBLE-WORLDS SEMANTICS FOR MODAL NOTIONS CONCEIVED AS PREDICATES
POSSIBLE-WORLDS SEMANTICS FOR MODAL NOTIONS CONCEIVED AS PREDICATES VOLKER HALBACH, HANNES LEITGEB, AND PHILIP WELCH Abstract. If is conceived as an operator, i.e., an expression that gives applied to
More informationFuzzy Does Not Lie! Can BAŞKENT. 20 January 2006 Akçay, Göttingen, Amsterdam Student No:
Fuzzy Does Not Lie! Can BAŞKENT 20 January 2006 Akçay, Göttingen, Amsterdam canbaskent@yahoo.com, www.geocities.com/canbaskent Student No: 0534390 Three-valued logic, end of the critical rationality. Imre
More informationAn Introduction to Modal Logic V
An Introduction to Modal Logic V Axiomatic Extensions and Classes of Frames Marco Cerami Palacký University in Olomouc Department of Computer Science Olomouc, Czech Republic Olomouc, November 7 th 2013
More informationThis paper is also taken by Combined Studies Students. Optional Subject (i): Set Theory and Further Logic
UNIVERSITY OF LONDON BA EXAMINATION for Internal Students This paper is also taken by Combined Studies Students PHILOSOPHY Optional Subject (i): Set Theory and Further Logic Answer THREE questions, at
More informationNeighborhood Semantics for Modal Logic Lecture 3
Neighborhood Semantics for Modal Logic Lecture 3 Eric Pacuit ILLC, Universiteit van Amsterdam staff.science.uva.nl/ epacuit August 15, 2007 Eric Pacuit: Neighborhood Semantics, Lecture 3 1 Plan for the
More informationMathematics 114L Spring 2018 D.A. Martin. Mathematical Logic
Mathematics 114L Spring 2018 D.A. Martin Mathematical Logic 1 First-Order Languages. Symbols. All first-order languages we consider will have the following symbols: (i) variables v 1, v 2, v 3,... ; (ii)
More informationTR : Binding Modalities
City University of New York (CUNY) CUNY Academic Works Computer Science Technical Reports Graduate Center 2012 TR-2012011: Binding Modalities Sergei N. Artemov Tatiana Yavorskaya (Sidon) Follow this and
More informationNotes on Satisfiability-Based Problem Solving. First Order Logic. David Mitchell October 23, 2013
Notes on Satisfiability-Based Problem Solving First Order Logic David Mitchell mitchell@cs.sfu.ca October 23, 2013 Preliminary draft. Please do not distribute. Corrections and suggestions welcome. In this
More informationAxiomatic Theories of Partial Ground II Partial Ground and Typed Truth
Axiomatic heories of Partial Ground II Partial Ground and yped ruth Johannes Korbmacher jkorbmacher@gmail.com June 2016 Abstract his is part two of a two-part paper, in which we develop an axiomatic theory
More informationPropositional and Predicate Logic - VII
Propositional and Predicate Logic - VII Petr Gregor KTIML MFF UK WS 2015/2016 Petr Gregor (KTIML MFF UK) Propositional and Predicate Logic - VII WS 2015/2016 1 / 11 Theory Validity in a theory A theory
More informationHerbrand Theorem, Equality, and Compactness
CSC 438F/2404F Notes (S. Cook and T. Pitassi) Fall, 2014 Herbrand Theorem, Equality, and Compactness The Herbrand Theorem We now consider a complete method for proving the unsatisfiability of sets of first-order
More informationGödel s Completeness Theorem
A.Miller M571 Spring 2002 Gödel s Completeness Theorem We only consider countable languages L for first order logic with equality which have only predicate symbols and constant symbols. We regard the symbols
More informationAlgebraizing Hybrid Logic. Evangelos Tzanis University of Amsterdam Institute of Logic, Language and Computation
Algebraizing Hybrid Logic Evangelos Tzanis University of Amsterdam Institute of Logic, Language and Computation etzanis@science.uva.nl May 1, 2005 2 Contents 1 Introduction 5 1.1 A guide to this thesis..........................
More informationCSE 1400 Applied Discrete Mathematics Definitions
CSE 1400 Applied Discrete Mathematics Definitions Department of Computer Sciences College of Engineering Florida Tech Fall 2011 Arithmetic 1 Alphabets, Strings, Languages, & Words 2 Number Systems 3 Machine
More informationHENNESSY MILNER THEOREM FOR INTERPRETABILITY LOGIC. Abstract
Bulletin of the Section of Logic Volume 34/4 (2005), pp. 195 201 Mladen Vuković HENNESSY MILNER THEOREM FOR INTERPRETABILITY LOGIC Abstract Interpretability logic is a modal description of the interpretability
More informationA generalization of modal definability
A generalization of modal definability Tin Perkov Polytechnic of Zagreb Abstract. Known results on global definability in basic modal logic are generalized in the following sense. A class of Kripke models
More information185.A09 Advanced Mathematical Logic
185.A09 Advanced Mathematical Logic www.volny.cz/behounek/logic/teaching/mathlog13 Libor Běhounek, behounek@cs.cas.cz Lecture #1, October 15, 2013 Organizational matters Study materials will be posted
More informationPredicate Logic: Sematics Part 1
Predicate Logic: Sematics Part 1 CS402, Spring 2018 Shin Yoo Predicate Calculus Propositional logic is also called sentential logic, i.e. a logical system that deals with whole sentences connected with
More informationFirst-Order Predicate Logic. Basics
First-Order Predicate Logic Basics 1 Syntax of predicate logic: terms A variable is a symbol of the form x i where i = 1, 2, 3.... A function symbol is of the form fi k where i = 1, 2, 3... und k = 0,
More informationHandout: Proof of the completeness theorem
MATH 457 Introduction to Mathematical Logic Spring 2016 Dr. Jason Rute Handout: Proof of the completeness theorem Gödel s Compactness Theorem 1930. For a set Γ of wffs and a wff ϕ, we have the following.
More information5. Peano arithmetic and Gödel s incompleteness theorem
5. Peano arithmetic and Gödel s incompleteness theorem In this chapter we give the proof of Gödel s incompleteness theorem, modulo technical details treated in subsequent chapters. The incompleteness theorem
More informationLogic: The Big Picture
Logic: The Big Picture A typical logic is described in terms of syntax: what are the legitimate formulas semantics: under what circumstances is a formula true proof theory/ axiomatization: rules for proving
More informationRestricted truth predicates in first-order logic
Restricted truth predicates in first-order logic Thomas Bolander 1 Introduction It is well-known that there exist consistent first-order theories that become inconsistent when we add Tarski s schema T.
More informationPeano Arithmetic. CSC 438F/2404F Notes (S. Cook) Fall, Goals Now
CSC 438F/2404F Notes (S. Cook) Fall, 2008 Peano Arithmetic Goals Now 1) We will introduce a standard set of axioms for the language L A. The theory generated by these axioms is denoted PA and called Peano
More informationChapter 3. Formal Number Theory
Chapter 3. Formal Number Theory 1. An Axiom System for Peano Arithmetic (S) The language L A of Peano arithmetic has a constant 0, a unary function symbol, a binary function symbol +, binary function symbol,
More informationBetween proof theory and model theory Three traditions in logic: Syntactic (formal deduction)
Overview Between proof theory and model theory Three traditions in logic: Syntactic (formal deduction) Jeremy Avigad Department of Philosophy Carnegie Mellon University avigad@cmu.edu http://andrew.cmu.edu/
More informationINF3170 Logikk Spring Homework #8 For Friday, March 18
INF3170 Logikk Spring 2011 Homework #8 For Friday, March 18 Problems 2 6 have to do with a more explicit proof of the restricted version of the completeness theorem: if = ϕ, then ϕ. Note that, other than
More informationClassical First-Order Logic
Classical First-Order Logic Software Formal Verification Maria João Frade Departmento de Informática Universidade do Minho 2009/2010 Maria João Frade (DI-UM) First-Order Logic (Classical) MFES 2009/10
More informationMotivation. CS389L: Automated Logical Reasoning. Lecture 10: Overview of First-Order Theories. Signature and Axioms of First-Order Theory
Motivation CS389L: Automated Logical Reasoning Lecture 10: Overview of First-Order Theories Işıl Dillig Last few lectures: Full first-order logic In FOL, functions/predicates are uninterpreted (i.e., structure
More informationLogic and Modelling. Introduction to Predicate Logic. Jörg Endrullis. VU University Amsterdam
Logic and Modelling Introduction to Predicate Logic Jörg Endrullis VU University Amsterdam Predicate Logic In propositional logic there are: propositional variables p, q, r,... that can be T or F In predicate
More informationGödel s Incompleteness Theorems
Seminar Report Gödel s Incompleteness Theorems Ahmet Aspir Mark Nardi 28.02.2018 Supervisor: Dr. Georg Moser Abstract Gödel s incompleteness theorems are very fundamental for mathematics and computational
More informationRelations to first order logic
An Introduction to Description Logic IV Relations to first order logic Marco Cerami Palacký University in Olomouc Department of Computer Science Olomouc, Czech Republic Olomouc, November 6 th 2014 Marco
More informationThe logic of Σ formulas
The logic of Σ formulas Andre Kornell UC Davis BLAST August 10, 2018 Andre Kornell (UC Davis) The logic of Σ formulas BLAST August 10, 2018 1 / 22 the Vienna Circle The meaning of a proposition is the
More informationFirst Order Logic (FOL) 1 znj/dm2017
First Order Logic (FOL) 1 http://lcs.ios.ac.cn/ znj/dm2017 Naijun Zhan March 19, 2017 1 Special thanks to Profs Hanpin Wang (PKU) and Lijun Zhang (ISCAS) for their courtesy of the slides on this course.
More informationAxiomatic Theories of Truth
Axiomatic Theories of Truth Graham Leigh University of Leeds LC 8, 8th July 28 Graham Leigh (University of Leeds) Axiomatic Theories of Truth LC 8, 8th July 28 1 / 15 Introduction Formalising Truth Formalising
More informationProseminar on Semantic Theory Fall 2013 Ling 720 First Order (Predicate) Logic: Syntax and Natural Deduction 1
First Order (Predicate) Logic: Syntax and Natural Deduction 1 A Reminder of Our Plot I wish to provide some historical and intellectual context to the formal tools that logicians developed to study the
More informationPart II. Logic and Set Theory. Year
Part II Year 2018 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2018 60 Paper 4, Section II 16G State and prove the ǫ-recursion Theorem. [You may assume the Principle of ǫ- Induction.]
More informationAutomata, Logic and Games: Theory and Application
Automata, Logic and Games: Theory and Application 1. Büchi Automata and S1S Luke Ong University of Oxford TACL Summer School University of Salerno, 14-19 June 2015 Luke Ong Büchi Automata & S1S 14-19 June
More informationLecture 7. Logic. Section1: Statement Logic.
Ling 726: Mathematical Linguistics, Logic, Section : Statement Logic V. Borschev and B. Partee, October 5, 26 p. Lecture 7. Logic. Section: Statement Logic.. Statement Logic..... Goals..... Syntax of Statement
More informationFormal Epistemology: Lecture Notes. Horacio Arló-Costa Carnegie Mellon University
Formal Epistemology: Lecture Notes Horacio Arló-Costa Carnegie Mellon University hcosta@andrew.cmu.edu Logical preliminaries Let L 0 be a language containing a complete set of Boolean connectives, including
More informationPredicate Logic. CSE 595 Semantic Web Instructor: Dr. Paul Fodor Stony Brook University
Predicate Logic CSE 595 Semantic Web Instructor: Dr. Paul Fodor Stony Brook University http://www3.cs.stonybrook.edu/~pfodor/courses/cse595.html 1 The alphabet of predicate logic Variables Constants (identifiers,
More informationSeptember 13, Cemela Summer School. Mathematics as language. Fact or Metaphor? John T. Baldwin. Framing the issues. structures and languages
September 13, 2008 A Language of / for mathematics..., I interpret that mathematics is a language in a particular way, namely as a metaphor. David Pimm, Speaking Mathematically Alternatively Scientists,
More informationVAUGHT S THEOREM: THE FINITE SPECTRUM OF COMPLETE THEORIES IN ℵ 0. Contents
VAUGHT S THEOREM: THE FINITE SPECTRUM OF COMPLETE THEORIES IN ℵ 0 BENJAMIN LEDEAUX Abstract. This expository paper introduces model theory with a focus on countable models of complete theories. Vaught
More informationFormal Methods for Java
Formal Methods for Java Lecture 20: Sequent Calculus Jochen Hoenicke Software Engineering Albert-Ludwigs-University Freiburg January 15, 2013 Jochen Hoenicke (Software Engineering) Formal Methods for Java
More informationDual-Intuitionistic Logic and Some Other Logics
Dual-Intuitionistic Logic and Some Other Logics Hiroshi Aoyama 1 Introduction This paper is a sequel to Aoyama(2003) and Aoyama(2004). In this paper, we will study various proof-theoretic and model-theoretic
More informationGödel s Incompleteness Theorem. Overview. Computability and Logic
Gödel s Incompleteness Theorem Overview Computability and Logic Recap Remember what we set out to do in this course: Trying to find a systematic method (algorithm, procedure) which we can use to decide,
More informationFirst-order logic Syntax and semantics
1 / 43 First-order logic Syntax and semantics Mario Alviano University of Calabria, Italy A.Y. 2017/2018 Outline 2 / 43 1 Motivation Why more than propositional logic? Intuition 2 Syntax Terms Formulas
More informationAxiomatisation of Hybrid Logic
Imperial College London Department of Computing Axiomatisation of Hybrid Logic by Louis Paternault Submitted in partial fulfilment of the requirements for the MSc Degree in Advanced Computing of Imperial
More informationOverview of Topics. Finite Model Theory. Finite Model Theory. Connections to Database Theory. Qing Wang
Overview of Topics Finite Model Theory Part 1: Introduction 1 What is finite model theory? 2 Connections to some areas in CS Qing Wang qing.wang@anu.edu.au Database theory Complexity theory 3 Basic definitions
More informationNotes for Math 601, Fall based on Introduction to Mathematical Logic by Elliott Mendelson Fifth edition, 2010, Chapman & Hall
Notes for Math 601, Fall 2010 based on Introduction to Mathematical Logic by Elliott Mendelson Fifth edition, 2010, Chapman & Hall All first-order languages contain the variables: v 0, v 1, v 2,... the
More informationApplied Logic. Lecture 1 - Propositional logic. Marcin Szczuka. Institute of Informatics, The University of Warsaw
Applied Logic Lecture 1 - Propositional logic Marcin Szczuka Institute of Informatics, The University of Warsaw Monographic lecture, Spring semester 2017/2018 Marcin Szczuka (MIMUW) Applied Logic 2018
More informationAn Introduction to Modal Logic. Ordinary logic studies the partition of sentences1 into two categories, true and false.
An Introduction to Modal Logic Ordinary logic studies the partition of sentences1 into two categories, true and false. Modal logic investigates a finer classification. A sentence can be either necessary
More informationInterpretability Logic
Interpretability Logic Logic and Applications, IUC, Dubrovnik vukovic@math.hr web.math.pmf.unizg.hr/ vukovic/ Department of Mathematics, Faculty of Science, University of Zagreb September, 2013 Interpretability
More informationNIP Summer School 2011 Truth and Paradox Lecture #1
NIP Summer School 2011 Truth and Paradox Lecture #1 Colin Caret & Aaron Cotnoir 1 Tarskianism 2 Basic Paracomplete Theories 3 Basic Paraconsistent Theories Definition 1.1 (Formal Languages) A formal language
More informationPropositional Logic: Syntax
Logic Logic is a tool for formalizing reasoning. There are lots of different logics: probabilistic logic: for reasoning about probability temporal logic: for reasoning about time (and programs) epistemic
More informationC. Modal Propositional Logic (MPL)
C. Modal Propositional Logic (MPL) Let s return to a bivalent setting. In this section, we ll take it for granted that PL gets the semantics and logic of and Ñ correct, and consider an extension of PL.
More informationA MODEL-THEORETIC PROOF OF HILBERT S NULLSTELLENSATZ
A MODEL-THEORETIC PROOF OF HILBERT S NULLSTELLENSATZ NICOLAS FORD Abstract. The goal of this paper is to present a proof of the Nullstellensatz using tools from a branch of logic called model theory. In
More informationMaximal Introspection of Agents
Electronic Notes in Theoretical Computer Science 70 No. 5 (2002) URL: http://www.elsevier.nl/locate/entcs/volume70.html 16 pages Maximal Introspection of Agents Thomas 1 Informatics and Mathematical Modelling
More informationCOMP 409: Logic Homework 5
COMP 409: Logic Homework 5 Note: The pages below refer to the text from the book by Enderton. 1. Exercises 1-6 on p. 78. 1. Translate into this language the English sentences listed below. If the English
More informationMATH 770 : Foundations of Mathematics. Fall Itay Ben-Yaacov
MATH 770 : Foundations of Mathematics Fall 2005 Itay Ben-Yaacov Itay Ben-Yaacov, University of Wisconsin Madison, Department of Mathematics, 480 Lincoln Drive, Madison, WI 53706-1388, USA URL: http://www.math.wisc.edu/~pezz
More informationOn the Complexity of the Reflected Logic of Proofs
On the Complexity of the Reflected Logic of Proofs Nikolai V. Krupski Department of Math. Logic and the Theory of Algorithms, Faculty of Mechanics and Mathematics, Moscow State University, Moscow 119899,
More informationAxiomatic Theories of Partial Ground I
J Philos Logic (2018) 47:161 191 DOI 10.1007/s10992-016-9423-9 Axiomatic Theories of Partial Ground I The Base Theory Johannes Korbmacher 1 Received: 24 June 2016 / Accepted: 22 December 2016 / Published
More informationLogic Michælmas 2003
Logic Michælmas 2003 ii Contents 1 Introduction 1 2 Propositional logic 3 3 Syntactic implication 5 3.0.1 Two consequences of completeness.............. 7 4 Posets and Zorn s lemma 9 5 Predicate logic
More informationthe logic of provability
A bird s eye view on the logic of provability Rineke Verbrugge, Institute of Artificial Intelligence, University of Groningen Annual Meet on Logic and its Applications, Calcutta Logic Circle, Kolkata,
More informationMODEL THEORY FOR ALGEBRAIC GEOMETRY
MODEL THEORY FOR ALGEBRAIC GEOMETRY VICTOR ZHANG Abstract. We demonstrate how several problems of algebraic geometry, i.e. Ax-Grothendieck, Hilbert s Nullstellensatz, Noether- Ostrowski, and Hilbert s
More informationPart 2: First-Order Logic
Part 2: First-Order Logic First-order logic formalizes fundamental mathematical concepts is expressive (Turing-complete) is not too expressive (e. g. not axiomatizable: natural numbers, uncountable sets)
More informationA Tableau Calculus for Minimal Modal Model Generation
M4M 2011 A Tableau Calculus for Minimal Modal Model Generation Fabio Papacchini 1 and Renate A. Schmidt 2 School of Computer Science, University of Manchester Abstract Model generation and minimal model
More informationLINDSTRÖM S THEOREM SALMAN SIDDIQI
LINDSTRÖM S THEOREM SALMAN SIDDIQI Abstract. This paper attempts to serve as an introduction to abstract model theory. We introduce the notion of abstract logics, explore first-order logic as an instance
More informationPredicate Calculus - Semantics 1/4
Predicate Calculus - Semantics 1/4 Moonzoo Kim CS Dept. KAIST moonzoo@cs.kaist.ac.kr 1 Introduction to predicate calculus (1/2) Propositional logic (sentence logic) dealt quite satisfactorily with sentences
More information3. The Logic of Quantified Statements Summary. Aaron Tan August 2017
3. The Logic of Quantified Statements Summary Aaron Tan 28 31 August 2017 1 3. The Logic of Quantified Statements 3.1 Predicates and Quantified Statements I Predicate; domain; truth set Universal quantifier,
More informationFirst-Order Logic First-Order Theories. Roopsha Samanta. Partly based on slides by Aaron Bradley and Isil Dillig
First-Order Logic First-Order Theories Roopsha Samanta Partly based on slides by Aaron Bradley and Isil Dillig Roadmap Review: propositional logic Syntax and semantics of first-order logic (FOL) Semantic
More informationSOCRATES DID IT BEFORE GÖDEL
Logic and Logical Philosophy Volume 20 (2011), 205 214 DOI: 10.12775/LLP.2011.011 Josef Wolfgang Degen SOCRATES DID IT BEFORE GÖDEL Abstract. We translate Socrates famous saying I know that I know nothing
More informationGödel s Incompleteness Theorem. Overview. Computability and Logic
Gödel s Incompleteness Theorem Overview Computability and Logic Recap Remember what we set out to do in this course: Trying to find a systematic method (algorithm, procedure) which we can use to decide,
More informationGS03/4023: Validation and Verification Predicate Logic Jonathan P. Bowen Anthony Hall
GS03/4023: Validation and Verification Predicate Logic Jonathan P. Bowen www.cs.ucl.ac.uk/staff/j.bowen/gs03 Anthony Hall GS03 W1 L3 Predicate Logic 12 January 2007 1 Overview The need for extra structure
More information02 Propositional Logic
SE 2F03 Fall 2005 02 Propositional Logic Instructor: W. M. Farmer Revised: 25 September 2005 1 What is Propositional Logic? Propositional logic is the study of the truth or falsehood of propositions or
More informationFirst Order Logic (FOL)
First Order Logic (FOL) Testing, Quality Assurance, and Maintenance Winter 2018 Prof. Arie Gurfinkel based on slides by Prof. Ruzica Piskac, Nikolaj Bjorner, and others References Chpater 2 of Logic for
More informationPeano Arithmetic. by replacing the schematic letter R with a formula, then prefixing universal quantifiers to bind
Peano Arithmetic Peano Arithmetic 1 or PA is the system we get from Robinson s Arithmetic by adding the induction axiom schema: ((R(0) v (œx)(r(x) 6 R(sx))) 6 (œx)r(x)). What this means is that any sentence
More informationPřednáška 12. Důkazové kalkuly Kalkul Hilbertova typu. 11/29/2006 Hilbertův kalkul 1
Přednáška 12 Důkazové kalkuly Kalkul Hilbertova typu 11/29/2006 Hilbertův kalkul 1 Formal systems, Proof calculi A proof calculus (of a theory) is given by: A. a language B. a set of axioms C. a set of
More informationFrom Logic Programming Semantics to the Consistency of Syntactical Treatments of Knowledge and Belief
From Logic Programming Semantics to the Consistency of Syntactical Treatments of Knowledge and Belief Thomas Bolander Informatics and Mathematical Modelling Technical University of Denmark tb@imm.dtu.dk
More information12th Meeting on Mathematics of Language. 6, September, 2011
A Co-inductive Collaborate Research Team for Verification, National Institute of Advanced Industrial Science and Technology, Japan 12th Meeting on Mathematics of Language 6, September, 2011 1 / 17 Outline
More informationCS156: The Calculus of Computation
Page 1 of 31 CS156: The Calculus of Computation Zohar Manna Winter 2010 Chapter 3: First-Order Theories Page 2 of 31 First-Order Theories I First-order theory T consists of Signature Σ T - set of constant,
More information