A note on fuzzy predicate logic. Petr H jek 1. Academy of Sciences of the Czech Republic
|
|
- Sharleen Armstrong
- 5 years ago
- Views:
Transcription
1 A note on fuzzy predicate logic Petr H jek 1 Institute of Computer Science, Academy of Sciences of the Czech Republic Pod vod renskou v 2, Prague. Abstract. Recent development of mathematical fuzzy logic is briey surveyed. The set of all formulas of predicate logic that are tautologies with respect to all continuous t-norms is shown to be heavily non-recursive ( 2 -hard). 1 Introduction Recently considerable progres has been made in strictly mathematical (formal, symbolic) aspects of fuzzy logic as a logic with a comparative notion of truth, logic of vague propositions that may be more true or less true. My book [4] Metamathematics of fuzzy logic contains a unied theory of logics based on continuous t-norms (as truth functions of conjunction); a basic fuzzy logic is introduced and studied in depth, together with three stronger logics ( Lukasiewicz, Godel and product logic) corresponding to three most important continuous t-norms. Needless to say, several results were obtained by other authors; let we mention at least the books [1, 3]. My [5] is a critical self-review of my book and a survey of further results in mathematical fuzzy logic. The development of mathematical fuzzy logic is expected to help in building a bridge between fuzzy logic in the broad sense (term used by L. Zadeh, the founder of fuzzy set theory) and pure logicians. In the present paper I oer the reader a short survey of some selected fundamental results serving as preliminaries to just one (important) new undecidability result concerning t-norm predicate logics. I hope that the paper helps the reader to get some insight into mathematical fuzzy logic in its present state of development. 2 Preliminaries Our standard set of truth values is [0; 1] { the real unit interval. Recall that a t-norm in a binary operation in [0; 1] which is commutative, associative, nondecreasing in both arguments and satisfying 0x = 0 and 1x = x for each x: We restrict ourselves to continuous t-norms. Each continuous t-norm determines uniquely its residuum ) satisfying, for each x; y; z; the condition x x ) y i x z y: () 1 Partial support by COST Action 15 is acknowledged. 1
2 Note that x ) y = 1 i x y: The following are three important continuous t-norms and their residua: name x + y x ) y for a > y negation x ) 0 Lukasiewicz max(0; x + y? 1) 1? x + y ( 1? x Godel min(x; y) y 1 for x = 0 product x y y=x 0 for x > 0 For each continuous t-norm ; the structure ([0; 1]; ; ; ); 0; 1); i.e. [0; 1] with its natural ordering, operations ; ) and designated elements 0, 1 is the t-algebra determined by : Note that minimum is denable from ; ); indeed, min(x; y) = x (x ) y): More generally, a BL-algebra is a structure L = (L; ; ; ); 0 L ; 1 L ) where (L; ; 0 L ; 1 L ) is a lattice with the least element 0 L and largest element 1 L ; is commutative and associative, 1 x = x for each x; ) is a residuum of (i.e. the equivalence () above holds for each x:y:z) and x \ y = x (x ) y); (x ) y) [ (y ) x) = 1 L where \; [ are the lattice operation of inmum and supremum. Each t-algebra is a particular BL-algebra; and each BL-algebra L can serve as the algebra of truth functions of a propositional calculus, i.e. we may evaluate propositional atoms p by elements e(p) 2 L and compute the value el(') of each formula ' built up from propositional variables using conjunction & and implication!: we take to be the truth function if & and ) the truth function of! : (Note that :' is dened as '! 0 where 0 is the truth constant with the value 0 L :) A propositional formula ' is an L-tautology if el(') = 1 L for each evaluation e of atoms in L; ' is a t-tautology of it is an L-tautology for each t-algebra L, and ' is a BL-tautology if it is an L-tautology for each BL-algebra L. There is a simple set of 7 (schemas of) axioms of BL { particular BLtautologies, complete (together with the rule of modus ponens) with respect to BL-tautologies. Note a recent result [2] saying that t-tautologies coincide with BL-tautologies; thus those axioms are also complete for t-tautologies. (See Appendix for the axioms.) Extending axioms of BL by simple additions axioms we get axioms of Lukasiewicz logic L (additional axiom: ::' '); Godel logic G (('&') ') and product logic (2 additional axioms) complete for L-tautologies, L being given by Lukasiewicz, Godel and product t-norm respectively. (Notation: L = [0; 1] L ; [0; 1] G; [0; 1] respectively.) See [4] for details.) Additional axioms lead to particular classes of BL-algebras: BL-algebras L such that the additional axiom of Lukasiewicz logic is and L-tautology are called MValgebras; similarly for Godel logic (G-algebras) and product logic (-algebras). One can prove that a formula ' is a [0; 1] L-tautology i it is an L-tautology for each MV-algebra L; similarly for [0; 1] G (G-algebras) and [0; 1] (-algebras). 2
3 Consider the language of predicate logic with some predicates P 1 ; : : : P n (each with its arity), object variables, connectives &!; truth constant 0 and quantiers 8; 9: Given a BL-algebra L, an L-interpretation of our language is a structure M = (M; (r P ) P predicate ) where M 6= ; and for each predicate P of arity n; r P is an n-ary L-fuzzy relation on M; i.e. r P : M n! L: The truth value k'k L M;v of a formula ' in M for an evaluation v (of object variables by elements of M) given by the truth functions is given by the usual Tarski style conditions, e.g. kp (x; y)k L M;v = r P (v(x); v(y)); k'& k L M;v = k'kl M;v k kl M;v ; k(8x)'kl M;v = inffk'k L M;v jv0 x vg (where v 0 x v means that v 0 (y) = v(y) for all y dierent from x:) This is always dened if L is a t-algebra (all inma and suprema exist). For a general BL-algebra L we call M L-safe if all truth values k'k L are well M;v dened. A formula ' of predicate logic is an L-tautology if k'k L M;v = 1 L for all L-safe M and all v: The notion of a t-tautology (BL-tautology) is obvious. (Let us recall that the condition of safeness is superuous in the case of t-algebras.) Axioms of the basic predicate fuzzy logic BL8 are those of BL plus 5 axiom schemas for quantiers (see Appendix). Deduction rules are modus ponens and generalization. The completeness theorem says that a predicate formula ' is provable in BL8 i it is a BL-tautology. Axioms of Lukasiewicz predicate logic L8 are those of BL8 plus the additional axiom schema of Lukasiewicz propositional logic ::' '): Similarly for G8; 8: We have analogous completeness theorems: L8 proves ' i ' is an L-tautology for each MV-algebra L. Similarly for G8 (G-algebras), 8 (-algebras). For G8 it can be proved that ' is an L-tautology for each G-algebra i it is a [0; 1] G -tautology (where, once more, [0; 1] G is the t-algebra given by Godel t-norm min). But the analogous statement is false both for [0; 1] L (MV-algebras) and [0; 1] (-algebras): whereas MV-tautologies equal L8-provable formulas and hence form a 1 (rec. enumerable) set, [0; 1] L -tautologies from a 2-complete set (rst proved by Ragaz). The set of [0; 1] -tautologies is 2 -hard. Thus neither the set of (predicate) [0; 1] L -tautologies nor the set of (predicate) [0; 1] -tautologies is recursively axiomatizable. A natural question remains if the set of t-tautologies is 1 (recursively axiomatizable). In the next section we show that it is not. It follows that, in contradistinction to the propositional calculus (recall the result of [2] mentioned above), t-tautologies of the predicate calculus form a proper subclass of predicate BL-tautologies. 3 Complexity of predicate t-tautologies We are going to present a recursive reduction of [0; 1] L-tautologies to t-tautologies, i.e. we associate in an eective way with each (closed) formula ' another formula ' ## such that ' is a [0; 1] L -tautology i '## is an L-tautology for each t-algebra 3
4 L. For this purpose we shall need some known facts on the structure of continuous t-norms. Let us start with an example. Take n + 1 numbers 0 = a 0 < : : : < a n = 1 and decide if on [a i ; a i+1 ] your t-norm should be isomorphic to Lukasiewicz, Godel or product t-norm; choose isomorphisms f i : [0; 1] $ [a i ; a i+1 ]: For x; y from the same interval [a i ; a i+1 ] let x y = f i (f?1 (x) i f?1 (y)) where i is the t-norm you have chosen for the i-th interval; for x; y not from the same interval let x y = min(x; y): This is a continuous t-norm; the corresponding residuum ) looks as follows: for x y; x ) y = 1; for x > y from the same interval x ) y = f i (f?1 (x) ) i f?1 (y)) (where ) i is the residuum of i ); for x > y not from the same interval x ) y = y: Each continuous t-norm is like this, but not necesary with a nite set of \division points" a i ; in general the division points closed nowhere dense subset of [0; 1]: (For example, think of the set f0g[f 1 jn positive naturalg; being isomorphic to Lukasiewicz on [ ; 1 ] for n even and to product for n odd.) We call n 1 n+1 n this representation of the t-norm Mostert-Shields representation. (See [4] for details and references.) One more thing we need is the fact that in Lukasiewicz predicate logic the existential quantier is denable from 8 as in classical logic: (9x)' :(8x):' is a tautology. Thus in L8 each formula is equivalent to a formula not containing the existential quantier. Last preliminary thing: let us say that the rst summand of a continuous t-norm is Lukasiewicz if there is a least positive division point a and on [0; a]; is isomorphic to Lukasiewicz t-norm on [0; 1] as above. Similarly for Godel and product; note that there may be no rst summand, as in the example with 1 n above. The reader easily veries the following Lemma. Let be a continuous t-norm and [0; 1] the corresponding t-algebra; let x 2 [0; 1]; x > 0: (1) (?)(?)x = x i x = 0 or x = 1 or the rst summand [0; a] is Lukasiewicz and 0 < x < a: (2) In all remaining cases (?)x = 0 and hence (?)(?)x = 1: Now we are ready to present our reduction. Denition. For each '; let ' # be the result of replacing, in '; each atom P (t 1 ; : : : ; t n ) by its double negation ::P (t 1 ; : : : ; t n ): Furthermore, let Q be a unary predicate and c a constant not occuring in '; let ' ## be the formula :Q(c) _ ::Q(c) _ ' # : Theorem. Let ' be a formula not containing the existential quantier. Then (i), (ii), (iii) are mutually, equivalent, where (i) ' is a [0; 1] L -tautology; 4
5 (ii) ' # is an L-tautology for each L given by a continuous t-norm whose rst summand (in the Mostert-Shields representation) is Lukasiewicz; (iii) ' ## is a t-tautology. Proof. First let be a continuous t-norm whose rst summand on [0; a] 2 is isomorphic to Lukasiewicz t-norm (a 1); let L be the BL-algebra on [0; 1] given by : Then [0; a)[f1g is the domain of a BL-subalgebra L 1 of L and the mapping f dened by f(x) = (?)(?)x is a homomorphism of L onto L 1 (identical on [0; a) and mapping the rest to 1). Moreover, for each L-structure M, formula ' and evaluation v of variables, k' # k L M;v = f(k'k L M;v) = k'k L 1 M ; 1 ;v where M 1 results from M by replacing the interpretation r P by the L-interpretation r 0 of ::P; i.e. of each predicate P r 0 (a 1 ; : : : ; a n ) = (?)(?)r(a 1 ; : : : ; a n ): This is shown by induction on the complexity of '; observing that f preserves even innite infs. (Caution: this would fail if we did not assume that the rst factor is Lukasiewicz. If it is not then f(0) = 0 and f(x) = 1 for x > 0 and innite infs are not preserved!) Recall that L 1 is isomorphic to [0; 1] L ; thus if ' is a [0; 1] L-tautology, then ' is an L 1 -tautology and ' # is an L-tautology. Thus (i) implies (ii). Now consider ' ##, i.e. :Q(c)_::Q(c)_' # and let L be given by any continuous t-norm ; let ' be a [0; 1] L-tautology. If the rst factor of is Lukasiewicz then ' # is an L-tautology and so is ' ## : In all other cases for (the rst factor is product or 0 is the inmum if positive idempotents), (8x)(:Q(x) _ ::Q(x)) is an L-tautology and so is ' ## : Thus (ii) implies (iii). Finally let us show that (iii) implies (i). Let ' be such that ' ## is a t- tautology and hence a [0; 1] L -tautology. Let L stand for [0; 1] L ; let M be any L-interpretation of the language of ' and let m 2 M: We may interpret c as m and set the truth value of Q(c) to be 1 ; interpreting Q arbitrarily for other elements of 2 M: Let M 0 be the expanded structure; then kq(c) _ :Q(c)k L = 1 and therefore M0 2 k' # k L = k' # k L M0 M = 1: Moreover, since L = [0; 1] L we get k'# k L M = k'kl M (due to the L-validity of the axiom of double negation). Thus k'k L M = 1 for arbitrarily M, and ' is a [0; 1] L-tautology. This completes the proof. Corollary. The set of all t-tautologies of the predicate calculus is 2 -hard and hence not recursively axiomatizable. Remark. Thus formulas provable in the basic predicate logic BL8 form a proper subset of the set of all t-tautologies of predicate calculus. It would be very interesting to nd a natural formula ' which is a predicate t-tautology but is not an L-tautology for an appropriate BL-algebra L. 5
6 4 Appendix. Axioms of the basic fuzzy predicate calculus. Axioms for connectives: (A1) ('! )! ((! )! ('! )) (A2) ('& )! ' (A3) ('& )! ( &') (A4) ('&('! ))! ( &(! ')) (A5a) ('! (! ))! (('& )! ) (A5b) (('& )! )! ('! (! )) (A6) (('! )! )! (((! ')! )! ) (A7) 0! ' Axioms for quantiers: (81) (8x)'(x)! '(y) (91) '(y)! (8x)'(x) (82) (8x)(! )! (! (8x) ) (92) (8x)('! )! ((9x)'! ) (83) (8x)(' _ )! ((8x)' _ where y is a constant or a variable substitutable for x in ' and the formula does not contain free occurences of x: References [1] Cignoli R., d'ottaviano I. M. L., Mundici D.: Algebraic Foundations of Many-valued Reasoning, Kluwer (to appear). [2] Cignoli R., Esteva F., Godo L., Torrens A.: Basic fuzzy logic is the logic of continuous t-norms and their residua, submitted. [3] Gottwald S.: Fuzzy sets and fuzzy logic, Viehweg Wiesbaden [4] H jek P.: Metamathematics of fuzzy logic, Kluwer 1998 [5] H jek P.: Mathematical fuzzy logic { state of art. Proc. Logic Colloquium'98 (Buss at al., ed.) Lect. Notices in Logic, Springer-Verlag, to appear. [6] H jek P.: Basic fuzzy logic and BL-algebras, Soft Computing 2 (1998) 124{128 6
Fleas and fuzzy logic a survey
Fleas and fuzzy logic a survey Petr Hájek Institute of Computer Science AS CR Prague hajek@cs.cas.cz Dedicated to Professor Gert H. Müller on the occasion of his 80 th birthday Keywords: mathematical fuzzy
More informationEmbedding logics into product logic. Abstract. We construct a faithful interpretation of Lukasiewicz's logic in the product logic (both
1 Embedding logics into product logic Matthias Baaz Petr Hajek Jan Krajcek y David Svejda Abstract We construct a faithful interpretation of Lukasiewicz's logic in the product logic (both propositional
More informationSome consequences of compactness in Lukasiewicz Predicate Logic
Some consequences of compactness in Lukasiewicz Predicate Logic Luca Spada Department of Mathematics and Computer Science University of Salerno www.logica.dmi.unisa.it/lucaspada 7 th Panhellenic Logic
More informationResiduated fuzzy logics with an involutive negation
Arch. Math. Logic (2000) 39: 103 124 c Springer-Verlag 2000 Residuated fuzzy logics with an involutive negation Francesc Esteva 1, Lluís Godo 1, Petr Hájek 2, Mirko Navara 3 1 Artificial Intelligence Research
More informationOmitting Types in Fuzzy Predicate Logics
University of Ostrava Institute for Research and Applications of Fuzzy Modeling Omitting Types in Fuzzy Predicate Logics Vilém Novák and Petra Murinová Research report No. 126 2008 Submitted/to appear:
More informationWhat is mathematical fuzzy logic
Fuzzy Sets and Systems 157 (2006) 597 603 www.elsevier.com/locate/fss What is mathematical fuzzy logic Petr Hájek Institute of Computer Science, Academy of Sciences of the Czech Republic, 182 07 Prague,
More informationExtending the Monoidal T-norm Based Logic with an Independent Involutive Negation
Extending the Monoidal T-norm Based Logic with an Independent Involutive Negation Tommaso Flaminio Dipartimento di Matematica Università di Siena Pian dei Mantellini 44 53100 Siena (Italy) flaminio@unisi.it
More informationOn Very True Operators and v-filters
On Very True Operators and v-filters XUEJUN LIU Zhejiang Wanli University School of Computer and Information Technology Ningbo 315100 People s Republic of China ZHUDENG WANG Zhejiang Wanli University Institute
More informationThe Erwin Schrödinger International Boltzmanngasse 9 Institute for Mathematical Physics A-1090 Wien, Austria
ESI The Erwin Schrödinger International Boltzmanngasse 9 Institute for Mathematical Physics A-1090 Wien, Austria Algebras of Lukasiewicz s Logic and their Semiring Reducts A. Di Nola B. Gerla Vienna, Preprint
More informationThe Blok-Ferreirim theorem for normal GBL-algebras and its application
The Blok-Ferreirim theorem for normal GBL-algebras and its application Chapman University Department of Mathematics and Computer Science Orange, California University of Siena Department of Mathematics
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 informationIntroduction to Fuzzy Sets and Fuzzy Logic
Introduction to Fuzzy Sets and Fuzzy Logic Luca Spada Department of Mathematics and Computer Science University of Salerno www.logica.dmi.unisa.it/lucaspada REASONPARK. Foligno, 17-19 September 2009 1/
More informationSome properties of residuated lattices
Some properties of residuated lattices Radim Bělohlávek, Ostrava Abstract We investigate some (universal algebraic) properties of residuated lattices algebras which play the role of structures of truth
More informationConstructions of Models in Fuzzy Logic with Evaluated Syntax
Constructions of Models in Fuzzy Logic with Evaluated Syntax Petra Murinová University of Ostrava IRAFM 30. dubna 22 701 03 Ostrava Czech Republic petra.murinova@osu.cz Abstract This paper is a contribution
More informationFirst-order t-norm based fuzzy logics with truth-constants: distinguished semantics and completeness properties
First-order t-norm based fuzzy logics with truth-constants: distinguished semantics and completeness properties Francesc Esteva, Lluís Godo Institut d Investigació en Intel ligència Artificial - CSIC Catalonia,
More information23.1 Gödel Numberings and Diagonalization
Applied Logic Lecture 23: Unsolvable Problems in Logic CS 4860 Spring 2009 Tuesday, April 14, 2009 The fact that Peano Arithmetic is expressive enough to represent all computable functions means that some
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 informationMaking fuzzy description logic more general
Making fuzzy description logic more general Petr Hájek Institute of Computer Science Academy of Sciences of the Czech Republic 182 07 Prague, Czech Republic Abstract A version of fuzzy description logic
More informationBoolean Algebra and Propositional Logic
Boolean Algebra and Propositional Logic Takahiro Kato June 23, 2015 This article provides yet another characterization of Boolean algebras and, using this characterization, establishes a more direct connection
More informationBoolean Algebra and Propositional Logic
Boolean Algebra and Propositional Logic Takahiro Kato September 10, 2015 ABSTRACT. This article provides yet another characterization of Boolean algebras and, using this characterization, establishes a
More informationThe logic of perfect MV-algebras
The logic of perfect MV-algebras L. P. Belluce Department of Mathematics, British Columbia University, Vancouver, B.C. Canada. belluce@math.ubc.ca A. Di Nola DMI University of Salerno, Salerno, Italy adinola@unisa.it
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 informationStates of free product algebras
States of free product algebras Sara Ugolini University of Pisa, Department of Computer Science sara.ugolini@di.unipi.it (joint work with Tommaso Flaminio and Lluis Godo) Congreso Monteiro 2017 Background
More informationOn varieties generated by Weak Nilpotent Minimum t-norms
On varieties generated by Weak Nilpotent Minimum t-norms Carles Noguera IIIA-CSIC cnoguera@iiia.csic.es Francesc Esteva IIIA-CSIC esteva@iiia.csic.es Joan Gispert Universitat de Barcelona jgispertb@ub.edu
More informationMultiplicative Conjunction and an Algebraic. Meaning of Contraction and Weakening. A. Avron. School of Mathematical Sciences
Multiplicative Conjunction and an Algebraic Meaning of Contraction and Weakening A. Avron School of Mathematical Sciences Sackler Faculty of Exact Sciences Tel Aviv University, Tel Aviv 69978, Israel Abstract
More informationTowards Formal Theory of Measure on Clans of Fuzzy Sets
Towards Formal Theory of Measure on Clans of Fuzzy Sets Tomáš Kroupa Institute of Information Theory and Automation Academy of Sciences of the Czech Republic Pod vodárenskou věží 4 182 08 Prague 8 Czech
More informationOn some Metatheorems about FOL
On some Metatheorems about FOL February 25, 2014 Here I sketch a number of results and their proofs as a kind of abstract of the same items that are scattered in chapters 5 and 6 in the textbook. You notice
More informationcse371/mat371 LOGIC Professor Anita Wasilewska Fall 2018
cse371/mat371 LOGIC Professor Anita Wasilewska Fall 2018 Chapter 7 Introduction to Intuitionistic and Modal Logics CHAPTER 7 SLIDES Slides Set 1 Chapter 7 Introduction to Intuitionistic and Modal Logics
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 informationEQ-algebras: primary concepts and properties
UNIVERSITY OF OSTRAVA Institute for Research and Applications of Fuzzy Modeling EQ-algebras: primary concepts and properties Vilém Novák Research report No. 101 Submitted/to appear: Int. Joint, Czech Republic-Japan
More informationFuzzy Logic in Narrow Sense with Hedges
Fuzzy Logic in Narrow Sense with Hedges ABSTRACT Van-Hung Le Faculty of Information Technology Hanoi University of Mining and Geology, Vietnam levanhung@humg.edu.vn arxiv:1608.08033v1 [cs.ai] 29 Aug 2016
More informationPropositional Logics and their Algebraic Equivalents
Propositional Logics and their Algebraic Equivalents Kyle Brooks April 18, 2012 Contents 1 Introduction 1 2 Formal Logic Systems 1 2.1 Consequence Relations......................... 2 3 Propositional Logic
More informationPREDICATE LOGIC. Schaum's outline chapter 4 Rosen chapter 1. September 11, ioc.pdf
PREDICATE LOGIC Schaum's outline chapter 4 Rosen chapter 1 September 11, 2018 margarita.spitsakova@ttu.ee ICY0001: Lecture 2 September 11, 2018 1 / 25 Contents 1 Predicates and quantiers 2 Logical equivalences
More informationAdding truth-constants to logics of continuous t-norms: axiomatization and completeness results
Adding truth-constants to logics of continuous t-norms: axiomatization and completeness results Francesc Esteva, Lluís Godo, Carles Noguera Institut d Investigació en Intel ligència Artificial - CSIC Catalonia,
More informationPREDICATE LOGIC: UNDECIDABILITY AND INCOMPLETENESS HUTH AND RYAN 2.5, SUPPLEMENTARY NOTES 2
PREDICATE LOGIC: UNDECIDABILITY AND INCOMPLETENESS HUTH AND RYAN 2.5, SUPPLEMENTARY NOTES 2 Neil D. Jones DIKU 2005 14 September, 2005 Some slides today new, some based on logic 2004 (Nils Andersen) OUTLINE,
More informationChapter I: Introduction to Mathematical Fuzzy Logic
Chapter I: Introduction to Mathematical Fuzzy Logic LIBOR BĚHOUNEK, PETR CINTULA, AND PETR HÁJEK This chapter provides an introduction to the field of mathematical fuzzy logic, giving an overview of its
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 informationEvery formula evaluates to either \true" or \false." To say that the value of (x = y) is true is to say that the value of the term x is the same as th
A Quick and Dirty Sketch of a Toy Logic J Strother Moore January 9, 2001 Abstract For the purposes of this paper, a \logic" consists of a syntax, a set of axioms and some rules of inference. We dene a
More informationLogic via Algebra. Sam Chong Tay. A Senior Exercise in Mathematics Kenyon College November 29, 2012
Logic via Algebra Sam Chong Tay A Senior Exercise in Mathematics Kenyon College November 29, 2012 Abstract The purpose of this paper is to gain insight to mathematical logic through an algebraic perspective.
More informationForcing in Lukasiewicz logic
Forcing in Lukasiewicz logic a joint work with Antonio Di Nola and George Georgescu Luca Spada lspada@unisa.it Department of Mathematics University of Salerno 3 rd MATHLOGAPS Workshop Aussois, 24 th 30
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 informationThis is logically equivalent to the conjunction of the positive assertion Minimal Arithmetic and Representability
16.2. MINIMAL ARITHMETIC AND REPRESENTABILITY 207 If T is a consistent theory in the language of arithmetic, we say a set S is defined in T by D(x) if for all n, if n is in S, then D(n) is a theorem of
More information2.2 Lowenheim-Skolem-Tarski theorems
Logic SEP: Day 1 July 15, 2013 1 Some references Syllabus: http://www.math.wisc.edu/graduate/guide-qe Previous years qualifying exams: http://www.math.wisc.edu/ miller/old/qual/index.html Miller s Moore
More information1. Propositional Calculus
1. Propositional Calculus Some notes for Math 601, Fall 2010 based on Elliott Mendelson, Introduction to Mathematical Logic, Fifth edition, 2010, Chapman & Hall. 2. Syntax ( grammar ). 1.1, p. 1. Given:
More informationOn the set of intermediate logics between the truth and degree preserving Lukasiewicz logics
On the set of intermediate logics between the truth and degree preserving Lukasiewicz logics Marcelo Coniglio 1 Francesc Esteva 2 Lluís Godo 2 1 CLE and Department of Philosophy State University of Campinas
More informationCHAPTER 11. Introduction to Intuitionistic Logic
CHAPTER 11 Introduction to Intuitionistic Logic Intuitionistic logic has developed as a result of certain philosophical views on the foundation of mathematics, known as intuitionism. Intuitionism was originated
More informationGödel Negation Makes Unwitnessed Consistency Crisp
Gödel Negation Makes Unwitnessed Consistency Crisp Stefan Borgwardt, Felix Distel, and Rafael Peñaloza Faculty of Computer Science TU Dresden, Dresden, Germany [stefborg,felix,penaloza]@tcs.inf.tu-dresden.de
More informationWell-behaved Principles Alternative to Bounded Induction
Well-behaved Principles Alternative to Bounded Induction Zofia Adamowicz 1 Institute of Mathematics, Polish Academy of Sciences Śniadeckich 8, 00-956 Warszawa Leszek Aleksander Ko lodziejczyk Institute
More informationLogic Part II: Intuitionistic Logic and Natural Deduction
Yesterday Remember yesterday? classical logic: reasoning about truth of formulas propositional logic: atomic sentences, composed by connectives validity and satisability can be decided by truth tables
More informationA Fuzzy Formal Logic for Interval-valued Residuated Lattices
A Fuzzy Formal Logic for Interval-valued Residuated Lattices B. Van Gasse Bart.VanGasse@UGent.be C. Cornelis Chris.Cornelis@UGent.be G. Deschrijver Glad.Deschrijver@UGent.be E.E. Kerre Etienne.Kerre@UGent.be
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 informationHandbook of Logic and Proof Techniques for Computer Science
Steven G. Krantz Handbook of Logic and Proof Techniques for Computer Science With 16 Figures BIRKHAUSER SPRINGER BOSTON * NEW YORK Preface xvii 1 Notation and First-Order Logic 1 1.1 The Use of Connectives
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 informationLogic, Sets, and Proofs
Logic, Sets, and Proofs David A. Cox and Catherine C. McGeoch Amherst College 1 Logic Logical Operators. A logical statement is a mathematical statement that can be assigned a value either true or false.
More informationSoft set theoretical approach to residuated lattices. 1. Introduction. Young Bae Jun and Xiaohong Zhang
Quasigroups and Related Systems 24 2016, 231 246 Soft set theoretical approach to residuated lattices Young Bae Jun and Xiaohong Zhang Abstract. Molodtsov's soft set theory is applied to residuated lattices.
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 informationWhat are the recursion theoretic properties of a set of axioms? Understanding a paper by William Craig Armando B. Matos
What are the recursion theoretic properties of a set of axioms? Understanding a paper by William Craig Armando B. Matos armandobcm@yahoo.com February 5, 2014 Abstract This note is for personal use. It
More informationNonclassical logics (Nichtklassische Logiken)
Nonclassical logics (Nichtklassische Logiken) VU 185.249 (lecture + exercises) http://www.logic.at/lvas/ncl/ Chris Fermüller Technische Universität Wien www.logic.at/people/chrisf/ chrisf@logic.at Winter
More informationAN ALGEBRAIC STRUCTURE FOR INTUITIONISTIC FUZZY LOGIC
Iranian Journal of Fuzzy Systems Vol. 9, No. 6, (2012) pp. 31-41 31 AN ALGEBRAIC STRUCTURE FOR INTUITIONISTIC FUZZY LOGIC E. ESLAMI Abstract. In this paper we extend the notion of degrees of membership
More informationOn Urquhart s C Logic
On Urquhart s C Logic Agata Ciabattoni Dipartimento di Informatica Via Comelico, 39 20135 Milano, Italy ciabatto@dsiunimiit Abstract In this paper we investigate the basic many-valued logics introduced
More informationPRESERVATION THEOREMS IN LUKASIEWICZ MODEL THEORY
Iranian Journal of Fuzzy Systems Vol. 10, No. 3, (2013) pp. 103-113 103 PRESERVATION THEOREMS IN LUKASIEWICZ MODEL THEORY S. M. BAGHERI AND M. MONIRI Abstract. We present some model theoretic results for
More informationLogic Part I: Classical Logic and Its Semantics
Logic Part I: Classical Logic and Its Semantics Max Schäfer Formosan Summer School on Logic, Language, and Computation 2007 July 2, 2007 1 / 51 Principles of Classical Logic classical logic seeks to model
More informationApproximating models based on fuzzy transforms
Approximating models based on fuzzy transforms Irina Perfilieva University of Ostrava Institute for Research and Applications of Fuzzy Modeling 30. dubna 22, 701 03 Ostrava 1, Czech Republic e-mail:irina.perfilieva@osu.cz
More informationKamila BENDOVÁ INTERPOLATION AND THREE-VALUED LOGICS
REPORTS ON MATHEMATICAL LOGIC 39 (2005), 127 131 Kamila BENDOVÁ INTERPOLATION AND THREE-VALUED LOGICS 1. Three-valued logics We consider propositional logic. Three-valued logics are old: the first one
More informationCHAPTER 2 INTRODUCTION TO CLASSICAL PROPOSITIONAL LOGIC
CHAPTER 2 INTRODUCTION TO CLASSICAL PROPOSITIONAL LOGIC 1 Motivation and History The origins of the classical propositional logic, classical propositional calculus, as it was, and still often is called,
More informationOn the filter theory of residuated lattices
On the filter theory of residuated lattices Jiří Rachůnek and Dana Šalounová Palacký University in Olomouc VŠB Technical University of Ostrava Czech Republic Orange, August 5, 2013 J. Rachůnek, D. Šalounová
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 informationBL-Functions and Free BL-Algebra
BL-Functions and Free BL-Algebra Simone Bova bova@unisi.it www.mat.unisi.it/ bova Department of Mathematics and Computer Science University of Siena (Italy) December 9, 008 Ph.D. Thesis Defense Outline
More informationOn Hájek s Fuzzy Quantifiers Probably and Many
On Hájek s Fuzzy Quantifiers Probably and Many Petr Cintula Institute of Computer Science Academy of Sciences of the Czech Republic Lukasiewicz logic L Connectives: implication and falsum (we set ϕ = ϕ
More informationFoundations of Mathematics MATH 220 FALL 2017 Lecture Notes
Foundations of Mathematics MATH 220 FALL 2017 Lecture Notes These notes form a brief summary of what has been covered during the lectures. All the definitions must be memorized and understood. Statements
More informationVague and Uncertain Entailment: Some Conceptual Clarifications
Conditionals, Counterfactuals, and Causes in Uncertain Environments Düsseldorf, May 2011 Vague and Uncertain Entailment: Some Conceptual Clarifications Chris Fermüller TU Wien, Austria Overview How does
More informationLecture 14 Rosser s Theorem, the length of proofs, Robinson s Arithmetic, and Church s theorem. Michael Beeson
Lecture 14 Rosser s Theorem, the length of proofs, Robinson s Arithmetic, and Church s theorem Michael Beeson The hypotheses needed to prove incompleteness The question immediate arises whether the incompleteness
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 informationExamples: P: it is not the case that P. P Q: P or Q P Q: P implies Q (if P then Q) Typical formula:
Logic: The Big Picture 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
More informationLecture 2: Syntax. January 24, 2018
Lecture 2: Syntax January 24, 2018 We now review the basic definitions of first-order logic in more detail. Recall that a language consists of a collection of symbols {P i }, each of which has some specified
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 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 informationFuzzy logic Fuzzyapproximate reasoning
Fuzzy logic Fuzzyapproximate reasoning 3.class 3/19/2009 1 Introduction uncertain processes dynamic engineering system models fundamental of the decision making in fuzzy based real systems is the approximate
More informationPartial Collapses of the Σ 1 Complexity Hierarchy in Models for Fragments of Bounded Arithmetic
Partial Collapses of the Σ 1 Complexity Hierarchy in Models for Fragments of Bounded Arithmetic Zofia Adamowicz Institute of Mathematics, Polish Academy of Sciences Śniadeckich 8, 00-950 Warszawa, Poland
More informationArithmetical classification of the set of all provably recursive functions
Arithmetical classification of the set of all provably recursive functions Vítězslav Švejdar April 12, 1999 The original publication is available at CMUC. Abstract The set of all indices of all functions
More informationThe nite submodel property and ω-categorical expansions of pregeometries
The nite submodel property and ω-categorical expansions of pregeometries Marko Djordjevi bstract We prove, by a probabilistic argument, that a class of ω-categorical structures, on which algebraic closure
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 informationMathematica Slovaca. Ján Jakubík On the α-completeness of pseudo MV-algebras. Terms of use: Persistent URL:
Mathematica Slovaca Ján Jakubík On the α-completeness of pseudo MV-algebras Mathematica Slovaca, Vol. 52 (2002), No. 5, 511--516 Persistent URL: http://dml.cz/dmlcz/130365 Terms of use: Mathematical Institute
More informationFrom Constructibility and Absoluteness to Computability and Domain Independence
From Constructibility and Absoluteness to Computability and Domain Independence Arnon Avron School of Computer Science Tel Aviv University, Tel Aviv 69978, Israel aa@math.tau.ac.il Abstract. Gödel s main
More informationGeneralized continuous and left-continuous t-norms arising from algebraic semantics for fuzzy logics
Generalized continuous and left-continuous t-norms arising from algebraic semantics for fuzzy logics Carles Noguera Dept. of Mathematics and Computer Science, University of Siena Pian dei Mantellini 44,
More informationApplied Logics - A Review and Some New Results
Applied Logics - A Review and Some New Results ICLA 2009 Esko Turunen Tampere University of Technology Finland January 10, 2009 Google Maps Introduction http://maps.google.fi/maps?f=d&utm_campaign=fi&utm_source=fi-ha-...
More informationPropositional and Predicate Logic - V
Propositional and Predicate Logic - V Petr Gregor KTIML MFF UK WS 2016/2017 Petr Gregor (KTIML MFF UK) Propositional and Predicate Logic - V WS 2016/2017 1 / 21 Formal proof systems Hilbert s calculus
More information1) Totality of agents is (partially) ordered, with the intended meaning that t 1 v t 2 intuitively means that \Perception of the agent A t2 is sharper
On reaching consensus by groups of intelligent agents Helena Rasiowa and Wiktor Marek y Abstract We study the problem of reaching the consensus by a group of fully communicating, intelligent agents. Firstly,
More informationFuzzy filters and fuzzy prime filters of bounded Rl-monoids and pseudo BL-algebras
Fuzzy filters and fuzzy prime filters of bounded Rl-monoids and pseudo BL-algebras Jiří Rachůnek 1 Dana Šalounová2 1 Department of Algebra and Geometry, Faculty of Sciences, Palacký University, Tomkova
More informationIntroduction to Metalogic
Philosophy 135 Spring 2008 Tony Martin Introduction to Metalogic 1 The semantics of sentential logic. The language L of sentential logic. Symbols of L: Remarks: (i) sentence letters p 0, p 1, p 2,... (ii)
More informationEquational Logic. Chapter Syntax Terms and Term Algebras
Chapter 2 Equational Logic 2.1 Syntax 2.1.1 Terms and Term Algebras The natural logic of algebra is equational logic, whose propositions are universally quantified identities between terms built up from
More informationPacket #2: Set Theory & Predicate Calculus. Applied Discrete Mathematics
CSC 224/226 Notes Packet #2: Set Theory & Predicate Calculus Barnes Packet #2: Set Theory & Predicate Calculus Applied Discrete Mathematics Table of Contents Full Adder Information Page 1 Predicate Calculus
More informationCOMP 182 Algorithmic Thinking. Proofs. Luay Nakhleh Computer Science Rice University
COMP 182 Algorithmic Thinking Proofs Luay Nakhleh Computer Science Rice University 1 Reading Material Chapter 1, Section 3, 6, 7, 8 Propositional Equivalences The compound propositions p and q are called
More information03 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 informationFeatures of Mathematical Theories in Formal Fuzzy Logic
Features of Mathematical Theories in Formal Fuzzy Logic Libor Běhounek and Petr Cintula Institute of Computer Science, Academy of Sciences of the Czech Republic Pod Vodárenskou věží 2, 187 02 Prague 8,
More informationModal systems based on many-valued logics
Modal systems based on many-valued logics F. Bou IIIA - CSIC Campus UAB s/n 08193, Bellaterra, Spain fbou@iiia.csic.es F. Esteva IIIA - CSIC Campus UAB s/n 08193, Bellaterra, Spain esteva@iiia.csic.es
More informationPropositional Logic Language
Propositional Logic Language A logic consists of: an alphabet A, a language L, i.e., a set of formulas, and a binary relation = between a set of formulas and a formula. An alphabet A consists of a finite
More informationWhen does a semiring become a residuated lattice?
When does a semiring become a residuated lattice? Ivan Chajda and Helmut Länger arxiv:1809.07646v1 [math.ra] 20 Sep 2018 Abstract It is an easy observation that every residuated lattice is in fact a semiring
More informationINDEPENDENCE OF THE CONTINUUM HYPOTHESIS
INDEPENDENCE OF THE CONTINUUM HYPOTHESIS CAPSTONE MATT LUTHER 1 INDEPENDENCE OF THE CONTINUUM HYPOTHESIS 2 1. Introduction This paper will summarize many of the ideas from logic and set theory that are
More informationChapter 1 Elementary Logic
2017-2018 Chapter 1 Elementary Logic The study of logic is the study of the principles and methods used in distinguishing valid arguments from those that are not valid. The aim of this chapter is to help
More information