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 1 / 50

Outline 1 What is logic? (And what is it good for?) 2 Classical propositional logic (the simplest logic there is) 3 Gödel s theorems (a rough and dirty account) 4 Mathematical fuzzy logic (the stuff I do) 5 The future of fuzzy logic (and your role in it) Petr Cintula (ÚI AV ČR) Logika 2 / 50

Logic is not about solving logical puzzles Petr Cintula (ÚI AV ČR) Logika 3 / 50

Logic is not about solving logical puzzles http://www.xkcd.com/246/ Petr Cintula (ÚI AV ČR) Logika 3 / 50

What is logic? (paraphrasing Wikipedia entry) Logic is generally held to be the systematic study of valid arguments There is no universal agreement as to the exact scope and subject matter of logic, but it has traditionally included the classification of arguments, the systematic exposition of the logical form common to all valid arguments, and the study of fallacies and paradoxes. Petr Cintula (ÚI AV ČR) Logika 4 / 50

Logic throughout the history Petr Cintula (ÚI AV ČR) Logika 5 / 50

Logic throughout the history Logic started as a part of philosophy in IV century BCE Aim: to recognize valid arguments in philosophical discussion All men are mortal and Socrates in a man, therefore Socrates is mortal Petr Cintula (ÚI AV ČR) Logika 5 / 50

Logic throughout the history Since 19th century a part of logic has evolved into mathematical logic Aim: to solve the crises of the foundation of mathematics PA Pr(0 = S(0)) Petr Cintula (ÚI AV ČR) Logika 5 / 50

Logic throughout the history Nowadays logic is applied mainly in computer science Aim: to describe and perform reasoning in various formal settings [α](x = 4) [α; (x := 2x)](x = 8) Petr Cintula (ÚI AV ČR) Logika 5 / 50

The three layers of logic I Natural language and natural reasoning scenarios cognitive science, psychology, linguistics, and philosophy Aim: to understand valid reasoning in natural reasoning scenarios and their transformations into formal one II Formal interpreted languages and artificial reasoning scenarios mathematics and (theoretical) computer science Aim: to develop various logical systems to describe and perform reasoning in various formal scenarios Petr Cintula (ÚI AV ČR) Logika 6 / 50

What is a valid reasoning? Petr Cintula (ÚI AV ČR) Logika 8 / 50

What is a valid reasoning? Example 1 If God exists, He must be good and omnipotent. If God was good and omnipotent, He would not allow human suffering. There is human suffering. Therefore, God does not exist. Is this a valid reasoning? Petr Cintula (ÚI AV ČR) Logika 8 / 50

What is a valid reasoning? Formalization Atomic parts: The form of the reasoning: p: God exists q: God is good r: God is omnipotent s: There is human suffering p q r q r s s p Is this a valid reasoning? Petr Cintula (ÚI AV ČR) Logika 9 / 50

Classical logic: Syntax Primitive connectives: binary and unary Defined connectives: binary,, : ϕ ψ = ϕ ψ ϕ ψ = (ϕ ψ) (ψ ϕ) ϕ ψ = (ϕ ψ) Formulas are built from fixed countable set of atoms using the connectives Let us by For denote the set of all formulas. Petr Cintula (ÚI AV ČR) Logika 10 / 50

Classical logic: Semantics Bivalence Principle: Every formula is either true or false. Definition 2 An evaluation is a mapping e from For to {0, 1} such that: e( ϕ) = 1 e(ϕ) { 1 if e(ϕ) e(ψ) e(ϕ ψ) = 0 otherwise. Note that: e(ϕ ψ) = min{e(ϕ), e(ψ)} e(ϕ ψ) = max{e(ϕ), e(ψ)} { 1 if e(ϕ) = e(ψ) e(ϕ ψ) = 0 otherwise Petr Cintula (ÚI AV ČR) Logika 11 / 50

Valid reasoning in classical logic Definition 3 A formula ϕ is a logical consequence of set of formulas Γ, Γ = ϕ, if for every evaluation e: if e(γ) = 1 for every γ Γ, then e(ϕ) = 1. Valid reasoning = logical consequence ψ 1. ψ n ϕ is a valid reasoning iff there is no interpretation making the premises true and the conclusion false. Petr Cintula (ÚI AV ČR) Logika 12 / 50

Valid reasoning in classical logic Example 4 Modus ponens: p q p q It is a valid reasoning (if e(p q) = e(p) = 1, then e(q) = 1). Petr Cintula (ÚI AV ČR) Logika 13 / 50

Valid reasoning in classical logic Example 4 Modus ponens: p q p q It is a valid reasoning (if e(p q) = e(p) = 1, then e(q) = 1). Example 5 Abduction: p q q p It is NOT a valid reasoning (take: e(p) = 0 and e(q) = 1). Petr Cintula (ÚI AV ČR) Logika 13 / 50

Back to our God example Example 6 p q r q r s s p Assume e(p q r) = e(q r s) = e(s) = 1. Then e( s) = 0 and so e(q r) = 0. Thus, we must have e(p) = 0, and therefore: e( p) = 1. It is a valid reasoning! Petr Cintula (ÚI AV ČR) Logika 14 / 50

Back to our God example Example 6 p q r q r s s p Assume e(p q r) = e(q r s) = e(s) = 1. Then e( s) = 0 and so e(q r) = 0. Thus, we must have e(p) = 0, and therefore: e( p) = 1. It is a valid reasoning! BUT, is this really a proof that God does not exist? OF COURSE NOT! We only know that if the premisses were true, then the conclusion would be true as well. Petr Cintula (ÚI AV ČR) Logika 14 / 50

Structurality of logical reasoning Compare our God example with other ones of the same structure: If God exists, If God was good and omnipotent, But, there is human suffering. He must be good and omnipotent He would not allow human suffering. Therefore, God does not exist. Petr Cintula (ÚI AV ČR) Logika 15 / 50

Structurality of logical reasoning Compare our God example with other ones of the same structure: If our politicians were ideal, they would be intelligent and honest. If politicians were intelligent and honest, there would be no corruption. But, there is corruption. Therefore, our politicians are not ideal. Petr Cintula (ÚI AV ČR) Logika 15 / 50

Structurality of logical reasoning Compare our God example with other ones of the same structure: If X is the set of rationals, If a set X is denumerable and dense, But we cannot embed integers in X. X is denumerable and dense. we can embed integers in X Therefore, X is not the set of rationals. Petr Cintula (ÚI AV ČR) Logika 15 / 50

Structurality of logical reasoning Compare our God example with other ones of the same structure: p q r q r s s p Petr Cintula (ÚI AV ČR) Logika 15 / 50

Classical proposition logic: Semantics Definition 7 An evaluation is a mapping e from For to {0, 1} such that: e( ϕ) = 1 e(ϕ) { 1 if e(ϕ) e(ψ) e(ϕ ψ) = 0 otherwise. Definition 8 A formula ϕ is a logical consequence of set of formulas Γ, Γ = ϕ, if for every evaluation e: if e(γ) = 1 for every γ Γ, then e(ϕ) = 1. Petr Cintula (ÚI AV ČR) Logika 16 / 50

Classical proposition logic: Axiomatic system Axioms (for any ϕ, ψ, χ For): A1 ϕ (ψ ϕ) A2 ( ϕ ψ) (ψ ϕ) A3 (ϕ ψ) ((ψ χ) (ϕ χ)) A4 ((ϕ ψ) ψ) ((ψ ϕ) ϕ) A5 ϕ ϕ Deduction rule Modus Ponens (for any ϕ, ψ For): ϕ ϕ ψ ψ Proof: a proof of ϕ from a set of formulas Γ is a finite sequence of formulas ψ 1,..., ψ n such that ψ n = ϕ and for every i n, either ψ i is is an instance of an axiom or an element of Γ, or there are j, k < i such that ψ k = ψ j ψ i. We write Γ ϕ if there is a proof of ϕ from Γ. Petr Cintula (ÚI AV ČR) Logika 17 / 50

Classical proposition logic: The completeness theorem Theorem 9 (Completeness theorem) For each set of formulas Γ {ϕ} we have: Γ ϕ iff Γ = ϕ Petr Cintula (ÚI AV ČR) Logika 18 / 50

They seem to have far reaching consequences... Amazon book description: Kurt Gödel was an intellectual giant. His Incompleteness Theorem turned not only mathematics but also the whole world of science and philosophy on its head. Shattering hopes that logic would, in the end, allow us a complete understanding of the universe, Gödel s theorem also raised many provocative questions: What are the limits of rational thought? Can we ever fully understand the machines we build? Or the inner workings of our own minds? How should mathematicians proceed in the absence of complete certainty about their results? Petr Cintula (ÚI AV ČR) Logika 20 / 50

A word of warning... Petr Cintula (ÚI AV ČR) Logika 21 / 50

A word of warning... Amazon book description: Probing the life and work of Kurt Gödel, Incompleteness indelibly portrays the tortured genius whose vision rocked the stability of mathematical reasoning and brought him to the edge of madness. Petr Cintula (ÚI AV ČR) Logika 21 / 50

Language of arithmetics Object variables OV: x, y, z,... Petr Cintula (ÚI AV ČR) Logika 22 / 50

Language of arithmetics Object variables OV: x, y, z,... Terms: Term = OV 0 Term + Term Term Term S(Term) Petr Cintula (ÚI AV ČR) Logika 22 / 50

Language of arithmetics Object variables OV: x, y, z,... Terms: Term = OV 0 Term + Term Term Term S(Term) Formulas: For = Term Term Term = Term For For For OV(For) Petr Cintula (ÚI AV ČR) Logika 22 / 50

Language of arithmetics Object variables OV: x, y, z,... Terms: Term = OV 0 Term + Term Term Term S(Term) Formulas: For = Term Term Term = Term For For For OV(For) Examples of formulas: x y(x + y y) x( y) (x = y) x y(x y v(v + x = y))) Examples of non-formulas: x y S( x(x = x)) S(x)(x y) Petr Cintula (ÚI AV ČR) Logika 22 / 50

A standard model of arithmetics Consider natural numbers {0, 1,... } equipped with usual addition + multiplication successor function S(i) = i + 1 equality = inequality We denote this model as N We write N = ϕ if a sentence ϕ is valid in N Petr Cintula (ÚI AV ČR) Logika 23 / 50

Axiomatizing N: Peano arithmetics PA x y(s(x) = S(y) x = y) x(x 0 = 0) x(s(x) 0) x y(x S(y) = x y + x) x(x 0 y(x = S(y))) x y(x y v(v + xy)) x(x + 0 = x) x y(x + S(y) = S(x + y)) ϕ(0) x(ϕ(x) ϕ(s(x))) xϕ(x) for arbitrary formula ϕ We write PA ϕ if a sentence ϕ is provable in PA Clearly all axioms of PA are valid in N Actually: PA ϕ implies N = ϕ Petr Cintula (ÚI AV ČR) Logika 24 / 50

Gödel completeness theorem Note: there could be non-standard models of PA, i.e., M = ϕ whenever PA ϕ Theorem 10 (Completeness theorem) PA ϕ IFF M = ϕ for each model M of PA He actually proved that for arbitrary language and arbitrary set of sentences T Petr Cintula (ÚI AV ČR) Logika 25 / 50

1st incompleteness theorem Theorem 11 (1st incompleteness theorem, simple, mode 1) There is a sentence ϕ such that N = ϕ but PA ϕ Petr Cintula (ÚI AV ČR) Logika 26 / 50

1st incompleteness theorem Theorem 11 (1st incompleteness theorem, simple, mode 1) There is a sentence ϕ such that N = ϕ but PA ϕ Theorem 13 (1st incompleteness theorem, simple, mode 2) There is a sentence ϕ such that PA ϕ and PA ϕ Petr Cintula (ÚI AV ČR) Logika 26 / 50

1st incompleteness theorem Theorem 12 (1st incompleteness theorem, general, mode 1) Let T be a set of sentences such that: N is a model of T, i.e., N = ϕ for each ϕ T there is an algorithm to recognize for each ϕ whether ϕ T There is a sentence ϕ such that N = ϕ but T ϕ Theorem 13 (1st incompleteness theorem, simple, mode 2) There is a sentence ϕ such that PA ϕ and PA ϕ Petr Cintula (ÚI AV ČR) Logika 26 / 50

1st incompleteness theorem Theorem 12 (1st incompleteness theorem, general, mode 1) Let T be a set of sentences such that: N is a model of T, i.e., N = ϕ for each ϕ T there is an algorithm to recognize for each ϕ whether ϕ T There is a sentence ϕ such that N = ϕ but T ϕ Theorem 14 (1st incompleteness theorem, general, mode 2) Let T be a set of sentences such that: T contains PA, i.e., T ϕ whenever PA ϕ T is consistent, i.e., T 0 = S(0) there is an algorithm to recognize for each ϕ whether ϕ T There is a sentence ϕ such that T ϕ and T ϕ Petr Cintula (ÚI AV ČR) Logika 26 / 50

The main proof idea: coding things by natural numbers Numerals - 1 = S(0) and (n + 1) = S( n) Coding of sequences - sequence 2, 3, 0 is uniquely coded by 2 3 3 4 5 1 a more complex but better coding is actually used Coding of formulas - any formula is a sequence of symbols, we write ϕ for the numeral corresponding to the code of ϕ Coding of proofs - any proof is a sequence of formulas, so... Theorem 15 (Arithmetization of syntax) For each sets of formulas describable by an algorithm there are formulas τ(x) and Pr τ (x) st. N = τ( m) iff m is a code of a sentence from T N = Pr τ ( ϕ) iff T ϕ Petr Cintula (ÚI AV ČR) Logika 27 / 50

The main proof idea (cont.) Theorem 16 (Diagonalization) For each formula ψ(x) there is a sentence ϕ such that PA ϕ ψ( ϕ) Petr Cintula (ÚI AV ČR) Logika 28 / 50

The main proof idea (cont.) Theorem 16 (Diagonalization) For each formula ψ(x) there is a sentence ϕ such that PA ϕ ψ( ϕ) Theorem 17 (1st incompleteness theorem) Let T be a set of sentences such that: N is a model of T, i.e., N = ϕ for each ϕ T there is an algorithm to recognize for each ϕ whether ϕ T There is a formula ϕ such that N = ϕ but T ϕ Proof: take a sentence ϕ such that T ϕ Pr τ ( ϕ) Assume that N = ϕ then N = Pr τ ( ϕ) thus T ϕ Petr Cintula (ÚI AV ČR) Logika 28 / 50

2nd incompleteness theorem Theorem 18 (2nd incompleteness theorem) Let T be a set of formulas such that: T contains PA, i.e., T ϕ whenever PA ϕ there is an algorithm to recognize for each ϕ whether ϕ T T is consistent, i.e., T 0 = S(0) Then T Pr τ (0 = S(0)) Petr Cintula (ÚI AV ČR) Logika 29 / 50

Are the consequences of Gödel s theorems really so earth-shattering? Petr Cintula (ÚI AV ČR) Logika 30 / 50

Are the consequences of Gödel s theorems really so earth-shattering? Amazon book description: Among the many expositions of Gödel s incompleteness theorems written for non-specialists, this book stands apart. With exceptional clarity, Franzén gives careful, non-technical explanations both of what those theorems say and, more importantly, what they do not. No other book aims, as his does, to address in detail the misunderstandings and abuses of the incompleteness theorems that are so rife in popular discussions of their significance. As an antidote to the many spurious appeals to incompleteness in theological, anti-mechanist and post-modernist debates. Petr Cintula (ÚI AV ČR) Logika 30 / 50

Two more pieces of recommended literature Petr Cintula (ÚI AV ČR) Logika 31 / 50

Two valued logic is boring... Petr Cintula (ÚI AV ČR) Logika 33 / 50

Two valued logic is boring... http://www.xkcd.com/74/ Petr Cintula (ÚI AV ČR) Logika 33 / 50

Lets start from some serious observations... Petr Cintula (ÚI AV ČR) Logika 34 / 50

Lets start from some serious observations... There are heaps of sand Petr Cintula (ÚI AV ČR) Logika 34 / 50

Lets start from some serious observations... There are heaps of sand If I remove one grain from a heap of sand, I still have a heap of sand. Petr Cintula (ÚI AV ČR) Logika 34 / 50

Lets start from some serious observations... There are heaps of sand If I remove one grain from a heap of sand, I still have a heap of sand. Therefore, one grain of sand is a heap of sand. Petr Cintula (ÚI AV ČR) Logika 34 / 50

Sorites paradox [Eubulides of Miletus, IV century BCE] If I remove one grain from any heap, I still have a heap. One million grains of sand is a heap. Therefore, one grain of sand is a heap. Formalization p n : n grains of sand is a heap of sand p 1000000 and p 1000000 p 999999 thus p 999999 p 999999 and p 999999 p 999998 thus p 999998. p 2 and p 2 p 1 thus p 1 There is no doubt that the premise p 1000000 is true. There is no doubt that the conclusion p 1 is false. For each i, the premise p i p i 1 seems to be true. The reasoning is logically valid. We have a paradox! Petr Cintula (ÚI AV ČR) Logika 35 / 50

Vagueness and degrees of truth The predicates that generate this kind of paradoxes are called vague. Possible solution: assume that truth comes in degrees Petr Cintula (ÚI AV ČR) Logika 36 / 50

Vagueness and degrees of truth The predicates that generate this kind of paradoxes are called vague. Possible solution: assume that truth comes in degrees Define an evaluation e as e(p n ) = nε, where ε = 10 6. Note that e(p 0 ) = 0 and e(p 10 6) = 1, i.e. first premise is completely true, the conclusion is completely false. e(p n p n 1 ) = min{1, 1 e(p n ) + e(p n 1 )} = 1 ε. All premises have the same, almost completely true, truth value! Petr Cintula (ÚI AV ČR) Logika 36 / 50

Fuzzy logic in broad sense: fuzzy set theory Fuzzy set a mapping µ: U [0, 1] Zadeh 1965 Fuzzy set theory a.k.a. fuzzy logic in a broad sense : works with degrees of truth combines them using operations corresponding to connectives is a bunch of engineering methods which are usually tailored to particular purposes sometime are a major success at certain applications but is NOT an area of mathematical logic e.g., has no special notions of truth or valid reasoning Petr Cintula (ÚI AV ČR) Logika 37 / 50

Mathematical Fuzzy Logic stems from those origins It was established as a respectable branch of mathematical logic by Petr Hájek in Metamathematics of fuzzy logic Kluwer,1998. Citations: 1500+ (WOS) and 3800+ (Google Scholar) Petr Cintula (ÚI AV ČR) Logika 38 / 50

Recall the classical propositional logic (CL) Formulas are built from atoms using connectives and Semantics of CL: an evaluation is a mapping e: Form {0, 1} st e( ϕ) = 1 e(ϕ) e(ϕ ψ) = min{1, 1 e(ϕ) + e(ψ)} Axiomatic system of CL: deduction rule modus ponens (from ϕ and ϕ ψ infer ψ) and axioms: ϕ ϕ ϕ (ψ ϕ) (ϕ ψ) ((ψ χ) (ϕ χ)) ( ϕ ψ) (ψ ϕ) ((ϕ ψ) ψ) ((ψ ϕ) ϕ) Theorem 19 (Completeness theorem) Γ CL ϕ iff Γ = CL ϕ. Petr Cintula (ÚI AV ČR) Logika 39 / 50

Łukasiewicz logic: An example of fuzzy logic Łukasiewicz logic Š has the same formulas as classical logic Semantics of Ł: an evaluation is a mapping e: Form [0, 1] st e( ϕ) = 1 e(ϕ) e(ϕ ψ) = min{1, 1 e(ϕ) + e(ψ)} Axiomatic system of Ł: deduction rule modus ponens (from ϕ and ϕ ψ infer ψ) and axioms: ϕ (ψ ϕ) (ϕ ψ) ((ψ χ) (ϕ χ)) ( ϕ ψ) (ψ ϕ) ((ϕ ψ) ψ) ((ψ ϕ) ϕ) Theorem 20 (Completeness theorem) Γ Š ϕ iff Γ = Š ϕ. Petr Cintula (ÚI AV ČR) Logika 40 / 50

Since then the theory of MFL got deeper... The following disciplines of mathematical logic were, and still are, developed in MFL proof theory model theory set theory recursion theory complexity theory The connections of MFL with the following areas of mathematics were, and still are, explored: lattice theory group/field theory geometry game theory topology category theory measure theory Petr Cintula (ÚI AV ČR) Logika 41 / 50

... and broader Petr Cintula (ÚI AV ČR) Logika 42 / 50

As witnessed by 1300 pages of... Petr Cintula (ÚI AV ČR) Logika 43 / 50

The future of fuzzy logic Problem: logic got divorced from the study of (human) reasoning Petr Cintula (ÚI AV ČR) Logika 45 / 50

The future of fuzzy logic Problem: logic got divorced from the study of (human) reasoning There is a movement trying to amend the rift Petr Cintula (ÚI AV ČR) Logika 45 / 50

The future of fuzzy logic Problem: logic got divorced from the study of (human) reasoning There is a movement trying to amend the rift The key component: the transformation of natural reasoning scenarios into formalized ones where various logics are directly applicable My program: join the movement and concentrate on scenarios of both kinds involving vague notions: tall, red, young,... Petr Cintula (ÚI AV ČR) Logika 45 / 50

The three layers of my program I To study natural reasoning scenarios involving graded notions and their natural transformations into formalized scenarios preserving the graduality II To utilize logical methods to analyze and perform reasoning in formalized scenarios III To advance the logic of graded notions to meet the demands of the previous goals Petr Cintula (ÚI AV ČR) Logika 46 / 50

The three layers of my program and their three aims (one more ambitious than the other :)) I To study natural reasoning scenarios involving graded notions and their natural transformations into formalized scenarios preserving the graduality II To utilize logical methods to analyze and perform reasoning in formalized scenarios III To advance the logic of graded notions to meet the demands of the previous goals Petr Cintula (ÚI AV ČR) Logika 46 / 50

The three layers of my program and their three aims (one more ambitious than the other :)) I To study natural reasoning scenarios involving graded notions and their natural transformations into formalized scenarios preserving the graduality II To utilize logical methods to analyze and perform reasoning in formalized scenarios More powerful formal methods (mainly for CS) III To advance the logic of graded notions to meet the demands of the previous goals Advancement of mathematical logic Petr Cintula (ÚI AV ČR) Logika 46 / 50

The three layers of my program and their three aims (one more ambitious than the other :)) I To study natural reasoning scenarios involving graded notions and their natural transformations into formalized scenarios preserving the graduality Better understanding of (human) reasoning II To utilize logical methods to analyze and perform reasoning in formalized scenarios More powerful formal methods (mainly for CS) III To advance the logic of graded notions to meet the demands of the previous goals Advancement of mathematical logic Petr Cintula (ÚI AV ČR) Logika 46 / 50

I am not alone... Akademie věd České republiky Ústav Informatiky v.v.i. Oddělení teoretické informatiky Petr Cintula (ÚI AV ČR) Logika 47 / 50

Our team Marta Bílková: modal logics, proof theory, co-algebraic logic Petr Cintula: abstract algebraic logic Matěj Dostál (PhD student): modal logics, co-algebraic logic Zuzana Haniková: computational complexity Rostislav Horčík: logic in computer science, algebraic logic Tomáš Lávička (PhD student): abstract algebraic logic Ondrej Majer: game theory Tommaso Moraschini: abstract algebraic logic Adam Přenosil (PhD student): paraconsistent logic Igor Sedlár: substructural logics, philosophical aspects Amanda Vidal: modal logic Petr Cintula (ÚI AV ČR) Logika 48 / 50

Our close collaborators Libor Běhounek, Ostrava: formal fuzzy mathematics Rudolfo Ertola, Buenos Aires: paraconsistent logic Chris Fermuller, Vienna: game theory, philosophical aspects Nick Galatos, Denver: substructural logics Lluis Godo, Barcelona: applied fuzzy logic Tomáš Kroupa, Prague, game theory, probability Carles Noguera: Prague, abstract algebraic logic George Metcalfe, Bern: proof theory, automated deduction James Raftery, Johannesburg: abstract algebraic logic Nick Smith, Sydney: philosophical aspects Kazushige Terui, Kyoto: substructural logics. Petr Cintula (ÚI AV ČR) Logika 49 / 50

We offer topics for bachelor, master and PhD theses Petr Cintula (ÚI AV ČR) Logika 50 / 50

We offer topics for bachelor, master and PhD theses On different levels of the hierarchy: III Mostly in pure of mathematical fuzzy logic II Recently also in applications of fuzzy logic (mainly in computer science) I In the future also in the quest for understanding human reasoning Petr Cintula (ÚI AV ČR) Logika 50 / 50