# Number-free Mathematics Based on T-norm Fuzzy Logic

Save this PDF as:

Size: px
Start display at page:

## Transcription

3 4 Distribution functions Classical distribution functions present a special way how to define a probabilistic measure on Borel sets, i.e., on the σ- algebra B of subsets of the real line generated by all intervals (,x]. A function f : R [, 1] defines a measure on B with µ(,x]=f(x) and µ(r) =1iff it satisfies the following conditions, which can thus be taken as the axioms for distribution functions: 1. Monotony: if x y then f(x) f(y), for all x, y R 2. Margin conditions: lim f(x) =, lim f(x) =1 x x + 3. Right-continuity: lim f(x) =f(x ) for all x R x x + Let us translate these conditions into the number-free language. 5 The function f : R [, 1] represents a standard fuzzy set of reals, i.e., in FCT over any expansion of MTL, the standard model of a predicate A on reals. 6 By the standard semantics of MTL, the above conditions translate into the following axioms on the predicate A: 7 ( xy)(x y (Ax Ay)) (1) ( x)ax, ( x)ax (11) ( x )[( x >x )Ax Ax ] (12) In FCT, these conditions express, respectively, the upperness of A in R, the null plinth and full height of A, and the left-closedness of the fuzzy upper class A. Consequently, number-free distribution functions are left-closed upper sets in R with full height and null plinth, i.e., (weakly bounded) fuzzy Dedekind cuts on R. The number-free rendering of distribution functions as fuzzy Dedekind cuts corresponds to the known fact that distribution functions represent Hutton fuzzy reals (cf., e.g., [15]). A use of fuzzy Dedekind cuts for the development of a logic-based theory of fuzzy intervals (or fuzzy numbers) is hinted at in [16]. 5 Continuous functions on R In Section 4 we abused the presence of monotony for the number-free rendering of right-continuity. Yet, if we aim at a graded theory where monotony can be satisfied to partial degrees, we need a different number-free characterization of continuity that does not rely on monotony. We shall work with functions R [, 1] only, even though various generalizations are easy to obtain. For a number-free rendering of right-continuity, we shall use the following classical characterization. A function f : R [, 1] is right-continuous in x iff lim sup x x + f(x) = lim inf x x + f(x) =f(x ), (13) 5 Of course, the translation is by no means unique: we always select one which is sufficiently straightforward and which results in meaningful concepts of fuzzy mathematics. 6 Recall that R as well as other crisp mathematical structures are available in FCT by means of the -interpretation, see [2, 7] and [14, 4]. In this and the next section, we shall understand all firstorder quantifications relativized to R, unless specified otherwise. 7 We use the fact that due to the monotony assumed, the margin conditions reduce to inf x f(x) = and sup x f(x) = 1, and the right-continuity to f(x ) inf x>x f(x). The representation theorem is then immediate by the standard semantics of MTL. Observe that as the axioms are required to degree 1, they are (due to the crispness of ) independent of the particular left-continuous t-norm used. where lim sup x x + f(x) = inf x1>x sup x1>x>x f(x) (14) lim inf x x + f(x) =sup x1>x inf x1>x>x f(x). (15) This translates into the following number-free definitions in FCT over MTL : 8 LimSup + (A, x ) df ( x 1 >x )( x (x,x 1 ))Ax (16) LimInf + (A, x ) df ( x 1 >x )( x (x,x 1 ))Ax (17) Cont + (A, x ) df (Ax = LimSup + (A, x )) & (Ax = LimInf + (A, x )) (18) The left-sided predicates LimSup and LimInf are defined dually (with < for >), and the both-sided ones as LimSup(A, x ) df LimSup (A, x ) LimSup + (A, x ) (19) LimInf(A, x ) df LimInf (A, x ) LimInf + (A, x ). (2) These definitions reconstruct the classical notions in a number-free way in the framework of FCT; the representation theorems follow directly from the above considerations and the standard semantics of MTL. The following observation shows that many properties of the classical notions can be reconstructed in FCT as well. Observation 5.1. By shifts of crisp relativized quantifiers valid in first-order MTL, the following theorems are easily provable in FCT: 9 1. LimInf(A, x ) LimSup(A, x ) 2. A B (LimSup(A, x ) LimSup(B,x )) and analogously for LimInf. 3. LimInf( A, x ) LimSup(A, x ) LimSup( A, x ) LimInf(A, x ) (Equality holds in logics with involutive negation, but not generally in MTL.) 4. LimInf(A B,x ) = LimInf(A, x ) LimInf(B,x ) LimInf(A B,x ) = LimInf(A, x ) LimInf(B,x ) and analogously for LimSup. Since the definitions reconstruct classical notions, we have retained the classical terminology and notation referring to limits and continuity, even though these regard membership functions (i.e., semantic models of fuzzy classes), rather than fuzzy classes themselves. In FCT, the fuzzy predicate LimSup + (A, x ) actually expresses the condition that x is a right-limit point of the fuzzy class A, and LimInf + (A, x ) that x is an interior point of A {x } in the left halfopen interval topology, 1 as these are the properties expressed by the defining formulae if all sets involved are crisp. 8 Again it can be observed that the definitions are independent of a particular t-norm and are the same in all expansions of MTL. 9 The theorems are stated for both-sided limits only, but hold equally well for one-sided limits. 1 I.e., the topology with the open base of all half-open intervals [a, b), also known as the lower-limit topology or the Sorgenfrey line. 451

4 Consequently, the formulae ( x A) LimInf(A, x ) and ( x A) LimSup(A, x ) express the notions of openness resp. closedness of A in a fuzzy interval topology on R. The study of this fuzzy topology and its relationship to the fuzzy interval topologies of [17, 18] is left for future work. Like the definitions of LimInf and LimSup, the theorems of Observation 5.1 have double meanings, too. On the one hand they can be understood as number-free reconstructions and graded generalizations of the classical theorems on (membership) functions. On the other hand, they can be interpreted as fuzzy-mathematical theorems on (fuzzy) sets, under the above fuzzy interval topology on reals. In particular, 1. says that an interior point of a fuzzy set of reals is also its limit point; 2. that a limit point of a fuzzy set is also a limit point of a larger fuzzy set (and dually for interior points); 3. that an interior point of the complement of a fuzzy set A is not a limit point of A (and vice versa); and 4. that x is a limit point of A B exactly to the degree it is a limit point of A and ( ) a limit point of B (and dually for ). (The theorems are graded, thus is represents fuzzy implication.) 6 Operations on R In the previous examples, only the codomain of real-valued functions of reals was rendered numberless. Obviously, the domain R (or more conveniently, R {± }) can be re-scaled into [, 1] and regarded as the standard set of truth degrees as well. The functions R n R then become n-ary functions from truth values to truth values, i.e., truth functions of fuzzy propositional connectives. An apparatus for internalizing truth values and logical connectives in FCT was developed in [4, 3]. As shown there, the truth values can be internalized as the elements of the crisp class L = Ker Pow{a}, i.e., subclasses of a fixed crisp singleton. The class L of internalized truth values is ordered by crisp inclusion, and the correspondence between internal and semantical truth values is given as follows: 11 α L represents the semantic truth value of α, and the semantic truth value of ϕ is represented by the class ϕ = df {a ϕ}; the correspondences ϕ (a ϕ) and ϕ ψ (ϕ ψ) then hold. Logical connectives are then internalized by crisp functions c: L n L (which can be called internal, inner, or formal connectives). In particular, definable connectives c of the logic are represented by the corresponding class operations c = {x L c(x X 1,...,x X n )} on L (e.g., & by ; by ; etc.). Since n-ary internal connectives are crisp functions valued in L, they can as well be regarded as fuzzy subsets of L n, i.e., n-ary fuzzy relations on L. Usual fuzzy class operations and graded predicates, e.g., the graded inclusion c d df ( x 1...x n )(cx 1...x n dx 1...x n ), (21) then apply to them and make their theory graded. The number-free theory of functions R n R is thus in fact the fuzzy-logical theory of internal connectives, i.e., of fuzzy relations on internal truth values. An elaboration of the theory of unary and binary internal connectives has been sketched in [19, 2]. These preliminary papers focus on the defining properties of t-norms (i.e., monotony, commutativity, associativity, 11 However, see [4, Rem. 3.3] for certain metamathematical qualifications regarding this correspondence. and the unit) and the relation of domination between internal connectives, making them graded by reinterpretation of their defining formulae in fuzzy logic (cf. [2, 7], [5, 2.3], or [6, 4]) and studying their graded properties. A full paper on the topic (by the authors of [2]) is currently under construction. 7 Metrics Recall that a pseudometric 12 on a set X is a function d: X 2 [, + ] such that d(x, x) = (22) d(x, y) =d(y, x) (23) d(x, z) d(x, y)+d(y, z). (24) The numberless reduction will first need to normalize the range of pseudometrics from [, + ] to [, 1], e.g., by setting c(x, y) =2 d(x,y). (25) The defining conditions on pseudometric then become the following equivalent conditions on c: 13 c(x, x) =1 (26) c(x, y) =c(y,x) (27) c(x, z) c(x, y) c(y, z) (28) These conditions are nothing else but the defining conditions of fuzzy equivalences, also known as similarity relations [21], in the standard semantics of product fuzzy logic [1]. We can thus equate number-free pseudometrics with product similarities, i.e., standard models of the following axioms in product fuzzy logic: Cxx (29) Cxy Cyx (3) Cxy & Cyz Cxz (31) The definition of a metric strengthens the first condition to d(x, y) =iff x = y, which is equivalent to c being a fuzzy equality (also called separated similarity), i.e., c(x, y) =1iff x = y, thus replacing the first axiom by Cxy x = y. Using a different left-continuous t-norm represents a different way of combining the distances d(x, y) and d(y, z) in the triangle inequality: e.g., with the minimum t-norm, c represents a (pseudo)ultrametrics under the same transformation (25), while with the Łukasiewicz t-norm, c represents a bounded pseudometric d under a different transformation c(x, y) =(1 d(x, y))/d max, where d max < + is an upper bound on the distances. (The obvious both-way representation theorems are left to the reader.) Since furthermore many theorems on number-free metrics hold generally over MTL,itis quite natural to generalize the notion of number-free metrics to any similarity, not only the product one. Various notions based on such (generalized) number-free metrics can be defined and their properties investigated in the framework of FCT (over MTL or stronger). Only a few 12 For simplicity, we shall work with extended pseudometrics, allowing the value Notice that since the function 2 x reverses the order, the fuzzy relation c: X 2 [, 1] expresses closeness rather than distance. 452

5 observations on number-free limits are given here as a concluding illustration of the numberless approach. Fix a metric d rendered in the numberless way by a closeness predicate C under the transformation (25). The limit lim n x n of a sequence {x n } n N (abbreviated x) under C can be defined as follows: 14 Lim C ( x, x) df ( n )( n >n )Cxx n (32) Theorem 7.1. Standard models over product logic validate Lim C ( x, x) iff x = lim n x n under d. Proof: lim x n = x under d iff lim sup d(x, x n ) =,iff lim inf 2 d(x,xn) =1,iffsup n inf n>n c(x, x n )=1, which is the semantics of Lim C ( x, x). As noted above, the meaning of Lim C is natural not only in product logic, but also in other t-norm logics, esp. if the relation C is interpreted as indistinguishability rather than mere closeness: 15 then (32) expresses the condition that from some index on, x n is indistinguishable from x. Similarly, Theorem 7.2 below expresses the fact that all limits of x are indistinguishable (to the degree the indistinguishability relation is symmetric and transitive). Discarding the in (32) yields a graded number-free notion of limit: 16 Lim C ( x, x) df ( n )( n >n )Cxx n (33) Interestingly, Lim C ( x, x) coincides with G. Soylu s notion of similarity-based fuzzy limit [23]. Even without employing explicitly the formalism of t-norm fuzzy logic, Soylu was able to prove graded theorems such as [23, Prop. 3.5], Lim C ( x, x) & Lim C ( y, y) Lim C ( x + y, x + y). (34) With the apparatus of FCT, the gradedness of Soylu s results can be extended even further by not requiring the full satisfaction of the defining properties of the similarity C (this conforms to the standard methodology of constructing graded theories [6, 7]). An example of such graded results is the following theorem on the fuzzy uniqueness of the limit: Theorem 7.2. FCT over MTL proves: Sym C & Trans C & Lim C ( x, x) & Lim C ( x, x ) Cxx (35) Proof. By Trans C we obtain Cxx n & Cx n x Cxx ; thus Cx n x & Cx n x Cxx by Sym C, whence ((n >n ) Cx n x)& ((n >n ) Cx n x ) Cxx (36) follows propositionally. By generalization on n and distribution of the quantifier, ( n >n )Cx n x &( n >n )Cx n x Cxx (37) is obtained (as in the consequent the quantification is void). Generalization on n and the shift of the quantifier to the antecedent (as ) then yields the required formula. 14 The in Lim C refers to the in the defining formula, which will later be dropped. 15 Cf. Menger s probabilistic indistinguishability relations [22]. 16 Observe that Lim C is a Σ 2-formula: compare it with the classical Π 3-definition and the Π 1-definition in non-standard analysis. A more detailed investigation of convergence based on fuzzy indistinguishability in the formal framework of FCT exceeds the scope of the present paper, and is therefore left for future research. Appendix: Fuzzy Class Theory Fuzzy Class Theory (FCT) is a formal theory over a given first-order fuzzy logic aiming at axiomatic approximation of Zadeh s fuzzy sets of all orders over a fixed crisp domain. It can be characterized as Henkin-style higher-order fuzzy logic, or fuzzified Russell-style simple type theory. FCT can be regarded as a foundational theory for fuzzy mathematics [24], since other axiomatic mathematical theories over fuzzy logic can be formalized within its framework. For more details on FCT see [2, 5]; the relevant definitions of [5] are briefly repeated here for reference. The reader s familiarity with the logic MTL and its main extensions is assumed; for details on these logics see [1, 25]. Here we only recapitulate its standard real-valued semantics, which is crucial for number-free mathematics: &... a left-continuous t-norm... the residuum of, defined as x y = df sup{z z x y},... min, max... x... the bi-residuum: min(x y, y x)... x =1 sgn(1 x),... inf, sup Łukasiewicz logic further specifies x y =(x + y 1), product logic sets x y = x y, and Gödel logic sets x y = x y. Fuzzy Class Theory FCT is a formal theory over a given multi-sorted first-order fuzzy logic L (at least as strong as MTL ), with sorts of variables for: atomic objects (lowercase letters x,y,...), fuzzy classes of atomic objects (uppercase letters A,B,...), fuzzy classes of fuzzy classes of atomic objects (calligraphic letters A, B,...), etc., in general for fuzzy classes of the n-th order (X (n),y (n),...). Besides the crisp identity predicate =, the language of FCT contains: The membership predicate between objects of successive sorts The class terms {x ϕ} of order n +1, for any variable x of any order n and any formula ϕ The symbols x 1,...,x k for k-tuples of individuals x 1,...,x k of any order In formulae of FCT we employ usual abbreviations and defined notions known from classical mathematics or traditional fuzzy mathematics, including those listed in Table 1, for all orders of fuzzy classes. FCT has the following axioms, for all formulae ϕ and variables of all orders: The logical axioms of multi-sorted first-order logic L The axioms of crisp identity: x = x; x = y & ϕ(x) ϕ(y); and x 1,...,x k = y 1,...,y k x i = y i 453

### Features 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,

### A Semantics for Counterfactuals Based on Fuzzy Logic

A Semantics for Counterfactuals Based on Fuzzy Logic In M. Peliš, V. Punčochář (eds.): The Logica Yearbook 2010, pp. 25 41, College Publications, 2011. Libor Běhounek Ondrej Majer Abstract Lewis Stalnaker

### On the difference between traditional and deductive fuzzy logic

On the difference between traditional and deductive fuzzy logic Libor Běhounek Institute of Computer Science, Academy of Sciences of the Czech Republic Pod Vodárenskou věží 2, 182 07 Prague 8, Czech Republic

### The quest for the basic fuzzy logic

Petr Cintula 1 Rostislav Horčík 1 Carles Noguera 1,2 1 Institute of Computer Science, Academy of Sciences of the Czech Republic Pod vodárenskou věží 2, 182 07 Prague, Czech Republic 2 Institute of Information

### Many-Valued Semantics for Vague Counterfactuals

Many-Valued Semantics for Vague Counterfactuals MARCO CERAMI AND PERE PARDO 1 1 Introduction This paper is an attempt to provide a formal semantics to counterfactual propositions that involve vague sentences

### CHAPTER 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

### On 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

### Chapter 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

### Making 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

### Introduction 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)

### Some 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

### On the Minimum Many-Valued Modal Logic over a Finite Residuated Lattice

On the Minimum Many-Valued Modal Logic over a Finite Residuated Lattice arxiv:0811.2107v2 [math.lo] 2 Oct 2009 FÉLIX BOU, FRANCESC ESTEVA and LLUÍS GODO, Institut d Investigació en Intel.ligència Artificial,

### Non-classical Logics: Theory, Applications and Tools

Non-classical Logics: Theory, Applications and Tools Agata Ciabattoni Vienna University of Technology (TUV) Joint work with (TUV): M. Baaz, P. Baldi, B. Lellmann, R. Ramanayake,... N. Galatos (US), G.

### CHAPTER 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

### Fuzzy 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

### In Which Sense Is Fuzzy Logic a Logic for Vagueness?

In Which Sense Is Fuzzy Logic a Logic for Vagueness? Libor Běhounek National Supercomputing Center IT4Innovations, Division University of Ostrava, Institute for Research and Applications of Fuzzy Modeling

### Advances in the theory of fixed points in many-valued logics

Advances in the theory of fixed points in many-valued logics Department of Mathematics and Computer Science. Università degli Studi di Salerno www.logica.dmi.unisa.it/lucaspada 8 th International Tbilisi

### Boolean Algebras. Chapter 2

Chapter 2 Boolean Algebras Let X be an arbitrary set and let P(X) be the class of all subsets of X (the power set of X). Three natural set-theoretic operations on P(X) are the binary operations of union

### Left-continuous t-norms in Fuzzy Logic: an Overview

Left-continuous t-norms in Fuzzy Logic: an Overview János Fodor Dept. of Biomathematics and Informatics, Faculty of Veterinary Sci. Szent István University, István u. 2, H-1078 Budapest, Hungary E-mail:

### DE MORGAN TRIPLES REVISITED

DE MORGAN TRIPLES REVISITED Francesc Esteva, Lluís Godo IIIA - CSIC, 08913 Bellaterra, Spain, {esteva,godo}@iiia.csic.es Abstract In this paper we overview basic nown results about the varieties generated

### Well-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

### Partial 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

### Chapter 1. Logic and Proof

Chapter 1. Logic and Proof 1.1 Remark: A little over 100 years ago, it was found that some mathematical proofs contained paradoxes, and these paradoxes could be used to prove statements that were known

### Equational 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

### Logical Foundations of Fuzzy Mathematics

Logical Foundations of Fuzzy Mathematics Logické základy fuzzy matematiky Libor Běhounek PhD Thesis / Disertační práce Charles University in Prague Faculty of Arts Department of Logic Univerzita Karlova

### Propositional Logic Arguments (5A) Young W. Lim 11/8/16

Propositional Logic (5A) Young W. Lim Copyright (c) 2016 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version

### 02 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

### Modal Logic. UIT2206: The Importance of Being Formal. Martin Henz. March 19, 2014

Modal Logic UIT2206: The Importance of Being Formal Martin Henz March 19, 2014 1 Motivation The source of meaning of formulas in the previous chapters were models. Once a particular model is chosen, say

### Embedding 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

### Peano 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

### Logic 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.

### Fuzzy 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

### Modal Logic XX. Yanjing Wang

Modal Logic XX Yanjing Wang Department of Philosophy, Peking University May 6th, 2016 Advanced Modal Logic (2016 Spring) 1 Completeness A traditional view of Logic A logic Λ is a collection of formulas

### A logic for reasoning about the probability of fuzzy events

A logic for reasoning about the probability of fuzzy events Tommaso Flaminio Dip. di Matematica e Scienze Informatiche, University of Siena 53100 Siena, Italy flaminio@unisi.it Lluís Godo IIIA - CSIC 08193

### Introduction 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

### First order Logic ( Predicate Logic) and Methods of Proof

First order Logic ( Predicate Logic) and Methods of Proof 1 Outline Introduction Terminology: Propositional functions; arguments; arity; universe of discourse Quantifiers Definition; using, mixing, negating

### Logic. Propositional Logic: Syntax

Logic Propositional Logic: Syntax Logic is a tool for formalizing reasoning. There are lots of different logics: probabilistic logic: for reasoning about probability temporal logic: for reasoning about

### COMP219: Artificial Intelligence. Lecture 19: Logic for KR

COMP219: Artificial Intelligence Lecture 19: Logic for KR 1 Overview Last time Expert Systems and Ontologies Today Logic as a knowledge representation scheme Propositional Logic Syntax Semantics Proof

### Logic: 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

### Adding 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,

### Semantics of intuitionistic propositional logic

Semantics of intuitionistic propositional logic Erik Palmgren Department of Mathematics, Uppsala University Lecture Notes for Applied Logic, Fall 2009 1 Introduction Intuitionistic logic is a weakening

### Applied 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

### Nonclassical 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

### THE INVERSE FUNCTION THEOREM FOR LIPSCHITZ MAPS

THE INVERSE FUNCTION THEOREM FOR LIPSCHITZ MAPS RALPH HOWARD DEPARTMENT OF MATHEMATICS UNIVERSITY OF SOUTH CAROLINA COLUMBIA, S.C. 29208, USA HOWARD@MATH.SC.EDU Abstract. This is an edited version of a

### Propositional 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

### Classical Propositional Logic

The Language of A Henkin-style Proof for Natural Deduction January 16, 2013 The Language of A Henkin-style Proof for Natural Deduction Logic Logic is the science of inference. Given a body of information,

### Lecture 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

### Formally real local rings, and infinitesimal stability.

Formally real local rings, and infinitesimal stability. Anders Kock We propose here a topos-theoretic substitute for the theory of formally-real field, and real-closed field. By substitute we mean that

### Explicit Logics of Knowledge and Conservativity

Explicit Logics of Knowledge and Conservativity Melvin Fitting Lehman College, CUNY, 250 Bedford Park Boulevard West, Bronx, NY 10468-1589 CUNY Graduate Center, 365 Fifth Avenue, New York, NY 10016 Dedicated

### Russell s logicism. Jeff Speaks. September 26, 2007

Russell s logicism Jeff Speaks September 26, 2007 1 Russell s definition of number............................ 2 2 The idea of reducing one theory to another.................... 4 2.1 Axioms and theories.............................

### 5-valued Non-deterministic Semantics for The Basic Paraconsistent Logic mci

5-valued Non-deterministic Semantics for The Basic Paraconsistent Logic mci Arnon Avron School of Computer Science, Tel-Aviv University http://www.math.tau.ac.il/ aa/ March 7, 2008 Abstract One of the

### Cauchy Sequences. x n = 1 ( ) 2 1 1, . As you well know, k! n 1. 1 k! = e, = k! k=0. k = k=1

Cauchy Sequences The Definition. I will introduce the main idea by contrasting three sequences of rational numbers. In each case, the universal set of numbers will be the set Q of rational numbers; all

### Propositional 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

### Chapter 2 Background. 2.1 A Basic Description Logic

Chapter 2 Background Abstract Description Logics is a family of knowledge representation formalisms used to represent knowledge of a domain, usually called world. For that, it first defines the relevant

### Some Background Material

Chapter 1 Some Background Material In the first chapter, we present a quick review of elementary - but important - material as a way of dipping our toes in the water. This chapter also introduces important

### Algebraizing 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..........................

### Computational Intelligence Winter Term 2017/18

Computational Intelligence Winter Term 2017/18 Prof. Dr. Günter Rudolph Lehrstuhl für Algorithm Engineering (LS 11) Fakultät für Informatik TU Dortmund Plan for Today Fuzzy relations Fuzzy logic Linguistic

### 1 Completeness Theorem for Classical Predicate

1 Completeness Theorem for Classical Predicate Logic The relationship between the first order models defined in terms of structures M = [M, I] and valuations s : V AR M and propositional models defined

### A 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

### Advanced Calculus: MATH 410 Real Numbers Professor David Levermore 5 December 2010

Advanced Calculus: MATH 410 Real Numbers Professor David Levermore 5 December 2010 1. Real Number System 1.1. Introduction. Numbers are at the heart of mathematics. By now you must be fairly familiar with

### Preservation of t-norm and t-conorm based properties of fuzzy relations during aggregation process

8th Conference of the European Society for Fuzzy Logic and Technology (EUSFLAT 2013) Preservation of t-norm and t-conorm based properties of fuzzy relations during aggregation process Urszula Dudziak Institute

### Interpreting Lattice-Valued Set Theory in Fuzzy Set Theory

Interpreting Lattice-Valued Set Theory in Fuzzy Set Theory Petr Hájek and Zuzana Haniková Institute of Computer Science, Academy of Sciences of the Czech Republic, 182 07 Prague, Czech Republic email:

### Propositional and Predicate Logic

Propositional and Predicate Logic CS 536-05: Science of Programming This is for Section 5 Only: See Prof. Ren for Sections 1 4 A. Why Reviewing/overviewing logic is necessary because we ll be using it

### BL-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

### CHAPTER 1 INTRODUCTION TO BRT

CHAPTER 1 INTRODUCTION TO BRT 1.1. General Formulation. 1.2. Some BRT Settings. 1.3. Complementation Theorems. 1.4. Thin Set Theorems. 1.1. General Formulation. Before presenting the precise formulation

### First Order Logic: Syntax and Semantics

CS1081 First Order Logic: Syntax and Semantics COMP30412 Sean Bechhofer sean.bechhofer@manchester.ac.uk Problems Propositional logic isn t very expressive As an example, consider p = Scotland won on Saturday

### Topos Theory. Lectures 17-20: The interpretation of logic in categories. Olivia Caramello. Topos Theory. Olivia Caramello.

logic s Lectures 17-20: logic in 2 / 40 logic s Interpreting first-order logic in In Logic, first-order s are a wide class of formal s used for talking about structures of any kind (where the restriction

### Filters in Analysis and Topology

Filters in Analysis and Topology David MacIver July 1, 2004 Abstract The study of filters is a very natural way to talk about convergence in an arbitrary topological space, and carries over nicely into

### Lecture 9. Model theory. Consistency, independence, completeness, categoricity of axiom systems. Expanded with algebraic view.

V. Borschev and B. Partee, October 17-19, 2006 p. 1 Lecture 9. Model theory. Consistency, independence, completeness, categoricity of axiom systems. Expanded with algebraic view. CONTENTS 0. Syntax and

### Něco málo o logice. Vydatná motivace jako předkrm, pořádná porce Gödelových vět a trocha fuzzy logiky jako zákusek. Petr Cintula

Něco málo o logice Vydatná motivace jako předkrm, pořádná porce Gödelových vět a trocha fuzzy logiky jako zákusek Petr Cintula Ústav informatiky Akademie věd České Republiky Petr Cintula (ÚI AV ČR) Logika

### KRIPKE 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

### are Banach algebras. f(x)g(x) max Example 7.4. Similarly, A = L and A = l with the pointwise multiplication

7. Banach algebras Definition 7.1. A is called a Banach algebra (with unit) if: (1) A is a Banach space; (2) There is a multiplication A A A that has the following properties: (xy)z = x(yz), (x + y)z =

### Subtractive Logic. To appear in Theoretical Computer Science. Tristan Crolard May 3, 1999

Subtractive Logic To appear in Theoretical Computer Science Tristan Crolard crolard@ufr-info-p7.jussieu.fr May 3, 1999 Abstract This paper is the first part of a work whose purpose is to investigate duality

### Equivalents of Mingle and Positive Paradox

Eric Schechter Equivalents of Mingle and Positive Paradox Abstract. Relevant logic is a proper subset of classical logic. It does not include among itstheoremsanyof positive paradox A (B A) mingle A (A

### Hypersequent calculi for non classical logics

Tableaux 03 p.1/63 Hypersequent calculi for non classical logics Agata Ciabattoni Technische Universität Wien, Austria agata@logic.at Tableaux 03 p.2/63 Non classical logics Unfortunately there is not

### A LITTLE REAL ANALYSIS AND TOPOLOGY

A LITTLE REAL ANALYSIS AND TOPOLOGY 1. NOTATION Before we begin some notational definitions are useful. (1) Z = {, 3, 2, 1, 0, 1, 2, 3, }is the set of integers. (2) Q = { a b : aεz, bεz {0}} is the set

### Introduction to fuzzy logic

Introduction to fuzzy logic Andrea Bonarini Artificial Intelligence and Robotics Lab Department of Electronics and Information Politecnico di Milano E-mail: bonarini@elet.polimi.it URL:http://www.dei.polimi.it/people/bonarini

### AI Principles, Semester 2, Week 2, Lecture 5 Propositional Logic and Predicate Logic

AI Principles, Semester 2, Week 2, Lecture 5 Propositional Logic and Predicate Logic Propositional logic Logical connectives Rules for wffs Truth tables for the connectives Using Truth Tables to evaluate

### Propositional Logic. Fall () Propositional Logic Fall / 30

Propositional Logic Fall 2013 () Propositional Logic Fall 2013 1 / 30 1 Introduction Learning Outcomes for this Presentation 2 Definitions Statements Logical connectives Interpretations, contexts,... Logically

### Chapter 4. Basic Set Theory. 4.1 The Language of Set Theory

Chapter 4 Basic Set Theory There are two good reasons for studying set theory. First, it s a indispensable tool for both logic and mathematics, and even for other fields including computer science, linguistics,

### A MODAL EXTENSION OF FIRST ORDER CLASSICAL LOGIC Part I

Bulletin of the Section of Logic Volume 32/4 (2003), pp. 165 177 George Tourlakis 1 Francisco Kibedi A MODAL EXTENSION OF FIRST ORDER CLASSICAL LOGIC Part I Abstract We formalize a fragment of the metatheory

### A 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

### Axioms of Kleene Algebra

Introduction to Kleene Algebra Lecture 2 CS786 Spring 2004 January 28, 2004 Axioms of Kleene Algebra In this lecture we give the formal definition of a Kleene algebra and derive some basic consequences.

### First Order Logic: Syntax and Semantics

irst Order Logic: Syntax and Semantics COMP30412 Sean Bechhofer sean.bechhofer@manchester.ac.uk Logic Recap You should already know the basics of irst Order Logic (OL) It s a prerequisite of this course!

### Intuitionistic Fuzzy Sets - An Alternative Look

Intuitionistic Fuzzy Sets - An Alternative Look Anna Pankowska and Maciej Wygralak Faculty of Mathematics and Computer Science Adam Mickiewicz University Umultowska 87, 61-614 Poznań, Poland e-mail: wygralak@math.amu.edu.pl

### This section will take the very naive point of view that a set is a collection of objects, the collection being regarded as a single object.

1.10. BASICS CONCEPTS OF SET THEORY 193 1.10 Basics Concepts of Set Theory Having learned some fundamental notions of logic, it is now a good place before proceeding to more interesting things, such as

### CSE 1400 Applied Discrete Mathematics Proofs

CSE 1400 Applied Discrete Mathematics Proofs Department of Computer Sciences College of Engineering Florida Tech Fall 2011 Axioms 1 Logical Axioms 2 Models 2 Number Theory 3 Graph Theory 4 Set Theory 4

### On the Craig interpolation and the fixed point

On the Craig interpolation and the fixed point property for GLP Lev D. Beklemishev December 11, 2007 Abstract We prove the Craig interpolation and the fixed point property for GLP by finitary methods.

### 2 (Bonus). Let A X consist of points (x, y) such that either x or y is a rational number. Is A measurable? What is its Lebesgue measure?

MA 645-4A (Real Analysis), Dr. Chernov Homework assignment 1 (Due 9/5). Prove that every countable set A is measurable and µ(a) = 0. 2 (Bonus). Let A consist of points (x, y) such that either x or y is

### A new Approach to Drawing Conclusions from Data A Rough Set Perspective

Motto: Let the data speak for themselves R.A. Fisher A new Approach to Drawing Conclusions from Data A Rough et Perspective Zdzisław Pawlak Institute for Theoretical and Applied Informatics Polish Academy

### FROM AXIOMS TO STRUCTURAL RULES, THEN ADD QUANTIFIERS.

FROM AXIOMS TO STRUCTURAL RULES, THEN ADD QUANTIFIERS. REVANTHA RAMANAYAKE We survey recent developments in the program of generating proof calculi for large classes of axiomatic extensions of a non-classical

### 3 The Semantics of the Propositional Calculus

3 The Semantics of the Propositional Calculus 1. Interpretations Formulas of the propositional calculus express statement forms. In chapter two, we gave informal descriptions of the meanings of the logical

### An 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

### CS2742 midterm test 2 study sheet. Boolean circuits: Predicate logic:

x NOT ~x x y AND x /\ y x y OR x \/ y Figure 1: Types of gates in a digital circuit. CS2742 midterm test 2 study sheet Boolean circuits: Boolean circuits is a generalization of Boolean formulas in which

### There are infinitely many set variables, X 0, X 1,..., each of which is

4. Second Order Arithmetic and Reverse Mathematics 4.1. The Language of Second Order Arithmetic. We ve mentioned that Peano arithmetic is sufficient to carry out large portions of ordinary mathematics,

### Lecture 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

### Review of Predicate Logic

Review of Predicate Logic Martin Held FB Computerwissenschaften Universität Salzburg A-5020 Salzburg, Austria held@cosy.sbg.ac.at 19. Jänner 2016 COMPUTERWISSENSCHAFTEN Legal Fine Print and Disclaimer

### HANDOUT AND SET THEORY. Ariyadi Wijaya

HANDOUT LOGIC AND SET THEORY Ariyadi Wijaya Mathematics Education Department Faculty of Mathematics and Natural Science Yogyakarta State University 2009 1 Mathematics Education Department Faculty of Mathematics