Introduction to Theoretical Computer Science, Lesson 2: Formalization in a logical language

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1 Introduction to Theoretical Computer Science, Lesson 2: Formalization in a logical language Marie Duží marie.duzi@vsb.cz Introduction to Logic

2 Two basic logical systems:. Propositional logic (PL) 2. First-order predicate logic (FOL) Ad (): PL makes it possible to combine atomic propositions into molecular propositions by means of logical connectives: Conjunction: (p q); p and q Disjunction: (p q); p or q Implication: (p q); if p then q Equivalence: (p q); p if and only if q Introduction to Logic 2

3 Two fundamental logical systems: Tom is singing (p) and he went to cinema (q) (p q) Tom is singing (p) or he went to cinema (q) (p q) If Tom is singing (p) then he went to cinema (q) (p q) An atomic proposition is a statement that can be evaluated as true or false. The two-values principle (tercium non datur) (This is not the only possibility; there are many-valued logics, logics of partial functions with truth-value gaps, fuzzy logics, etc.) Is the definition of atomic proposition trivial? Try to evaluate: The King of France is bald. Did you stop beating your wife? (answer yes/no in case you have never been married or have never beaten your wife) Introduction to Logic 3

4 Two fundamental logical systems Ad (2): In addition, First-order predicate logic (FOL) makes it possible to analyze a structure of atomic propositions up to the level of properties of individuals and relations between them All monkeys like bananas. Judy is a monkey. Judy likes bananas. From the viewpoint of Propositional Logic (PL) the above are simple (atomic) sentences: p, q, r, and {p, q} does not entail r. r PL does not make it possible to prove the validity of this simple e argument. All students are clever Charles is not clever Charles is not a student What are the valid schemata of these arguments? What are the valid schemata of these arguments? We need to analyze the structure of atomic propositions; thus FOL is needed to prove the validity of such arguments. Introduction to Logic 4

5 Propositional logic: There are two kinds of Sentences: Atomic (Elementary) no proper part of the sentence is a sentence as well Molecular (Composed) the sentence has its own part(s) that is (are) a sentence(s) as well The Compositionality Principle: meaning of a composed sentence is a function (depends only on) the meanings of its components. The meaning of sentences in propositional logic is reduces to: True (), False (). A Booleon algebra of truth values. Introduction to Logic 5

6 Examples of composed sentences It is raining in Prague and it is a sunshine in Brno. S connective S2 It is not true that it is raining in Prague. connective S Introduction to Logic 6

7 Definition: language of PL A formal language is defined by an alphabet (a set of symbols) and a grammar (a set of rules that define the way of forming Well Formed Formulas - WFF) Language of Propositional Logic (PL). alphabet: a) Symbols for propositions: p, q, r,... b) Symbols for logical connectives:,,,, c) Auxiliary symbols: (, ), [, ], {, } Symbols ad a) stand for elementary sentences Symbols ad b), i.e.,,,,, are called, respectively: negation (), disjunction (), conjunction (), implication (), equivalence (). Introduction to Logic 7

8 Definition: language of PL 2. Grammar (Inductive definition of an infinite set of WFF). Symbols p, q, r,... are (well-formed) formulas (the definition base). 2. If A, B are formulas, then expressions A, A B, A B, A B, A B are (well-formed) formulas (inductive definition step). 3. Only expressions due to. and 2. are WFFs. (the definition closure). The language of PL is the set of well-formed formulas. Note: Formulas ad () are atomic formulas Formulas ad (2) are composed formulas Introduction to Logic 8

9 Well-formed formulas of PL (notes) Symbols A, B are metasymbols. We can substitute for them any WFF created according to the definition. The outermost parentheses can be omitted. Some other symbols are often used in the notation of logical connectives: Symbol variant notation ,, & ~ Example: (p q) p is a WFF (the outermost parentheses omitted) (p ) q is not a WFF Introduction to Logic 9

10 Definition: semantics (meaning) of formulas The truth-value valuation of propositional symbols is a mapping v that to each propositional symbol p assigns a truth value, i.e., a value from the set {,}, which encodes the set {True, False}; v: {p i } {,} The truth-value function of a PL formula is a function w, which for each valuation v of propositional symbols p i associates the formula with its truth value in the following way: The truth value of an elementary formula p: wp v = vp for any propositional variable p. If the truth values of formulas A, B are given, then the truth value of the formulas A, A B, A B, A B, A B are defined by the following Introduction Table: to Logic

11 Introduction to Logic The truth value functions A B A B A B A B A B A

12 Transforming natural language to the PL language Atomic sentences: p, q, r,... Logical connectives: Negation: it is not true that (unary connective) Conjunction: and, but, : (binary, commutative connection) Prague is a capital and 2+2=4: p q Note: not every and denotes a logical connective! Example: Peter came home and opened the window. Disjunction: or (binary, commutative connection) Prague or Brno is a great city. p q non-exclusive or In a natural language we often use or as an exclusive either, or : I ll go to the cinema or I ll stay at home Exclusive either, or is a non-equivalence Introduction to Logic 2

13 Implication if A then B : A B; A is the antecedent, B is the consequent. (binary, non-commutative connective) Material implication (middle ages: suppositio materialis ) does not take into account any semantic relation between antecedent and consequent Hence implication does not render a causal or chronological relation: If +=2, then iron is a metal (a true proposition): p q If UFO (flying saucers) exist, then I am a pope : p r (What does a speaker want to say? Since obviously I am not a pope, UFO do not exist) Connectives because, therefore, since do not correspond to implication! Since the team lost the match, players left the world championship earlier. Because I am sick, I stay at home. sick home? But then it would have to be true even if I am not sick (see the definition of implication) We might analyze it by means of the modus ponens: [p (p q)] q 3

14 The equivalence connective if and only if (iff) The Greek army used to win if and only if the result of the battle depended on their physical strength : p q It is often used in mathematical definitions; in a natural language its use is not so frequent Example: a) I ll slap you if you cheat on me cheat slap b) I ll slap you if and only if you cheat on me cheat slap Situation: You did not cheat. When can you be slapped? Ad (a) You may be slapped; Ad (b) You might not be slapped. Introduction to Logic 4

15 Satisfiable formulas, tautology, contradiction, model A model of a formula A: valuation v such that A is true in v: w(a) v =. A formula is satisfiable iff it has at least one model A formula is a contradiction iff it has no model A formula is a tautology iff any valuation v is its model. A set of formulas {A,,A n } is satisfiable iff there is a valuation v such that v is a model of every formula A i, i =,...,n. The valuation v is then a model of the set {A,,A n }. Introduction to Logic 5

16 Satisfiable formulas, tautology, contradiction, model (example) A: (p q) (p q) Formula A is a tautology, A is a contradiction, (p q), (p q) are satisfiable. p q p q p q (p q) (p q) (p q) A Introduction to Logic 6

17 Formal language of FOL Alphabet Logical symbols individual variables: x, y, z,... Symbols for truth-connectives:,,,, Symbols for quantifiers:, Special symbols Predicates: P n, Q n,... n arity = number of arguments Functional: f n, g n, h n, Auxiliary symbols: (, ), [, ], {, },... Introduction to Logic 7

18 Formal language of FOL Grammar terms: i. each variable symbol x, y,... is a term ii. iii. if t,,t n (n ) are terms and if f is an n-ary functional symbol, then the expression f(t,,t n ) is a term; If n =, then we talk about individual constant (denoted a, b, c, ) only expressions due to i. and ii. are terms Introduction to Logic 8

19 Formal language of FOL Grammar atomic formulas: If P is an n-ary predicate symbol and if t,,t n are terms, then P(t,,t n ) is an atomic formula (molecular) formulas: each atomic formula is a formula if A is a formula, then A is a formula if A and B are formulas, then (A B), (A B), (A B), (A B) are formulas if x is a variable and A a formula, then x A and x A are formulas Introduction to Logic 9

20 Formal language of FOL st order We can quantify only over individual variables We cannot quantify over properties or functions Example: Leibniz s definition of identity: If two individuals have all the properties identical, then it is one and the same individual P [P(x) = P(y)] (x = y) here we need a 2 nd -order language, because we quantify over properties Introduction to Logic 2

21 Example: language of arithmetic We need special functional symbols: (the constant zero) constant is a -ary functional symbol unary symbol: s (the successor function) binary symbols: + and (adding and multiplying) Examples of terms (using infix notation for + and ):, s(x), s(s(x)), (x + y) s(s()), etc. Formulas (= is here a special predicate symbol): s() = ( x) + s(), x (y = x z), x [(x = y) y (x = s(y))] Introduction to Logic 2

22 Transforming natural language into the language of FOL all, every, none, nobody, any,... somebody, something, some, there is,... A sentence often needs to be equivalently reformulated No student is retired It holds for every individual x that if x is a student then x is not retired : x [S(x) R(x)] But: Not all students are retired It is not true that any student is retired : x [S(x) R(x)] x [S(x) R(x)] Introduction to Logic 22

23 Transforming natural language into the language of FOL An auxiliary rule: +, + (almost always) x [P(x) Q(x)] x [P(x) Q(x)] It is not true that all P s are Q s Some P s are not Q s x [P(x) Q(x)] x [P(x) Q(x)] It is not true that some P s are Q s No P is a Q de Morgan laws in FOL Introduction to Logic 23

24 Transforming natural language into the language of FOL The lift is used only by employees: x [L(x) E(x)] All employees use the lift: Mary likes only winners: x [E(x) L(x)] Hence, for all individuals x it holds that if Mary likes x then x must be a winner: x [L(m, x) W(x)] to like is a binary relation, not a property!!! Introduction to Logic 24

25 Transforming natural language into the language of FOL Everybody loves somebody sometimes x y t L(x, y, t) Everybody loves somebody sometimes but Hitler doesn t like anybody x y t L(x, y, t) zl (h, z) Everybody loves nobody ambiguous Nobody loves anybody ambiguous; Everybody dislikes anybody: x y L (x, y) x yl (x, y) Introduction to Logic 25

26 Free, bound variables x yp(x, y, t) xq(y, x) bound, free free, bound Formula in a clear form: each variable has only free occurrences, or only bound occurrences; each quantifier quantifies its own variables. For instance, the above formula is not in the clear form: x in the second conjunct is another variable than the x in the first conjunct, similarly for y. Clear formula: x yp(x, y, t) zq(u, z) Introduction to Logic 26

27 Substitution of terms for variables Ax/t arises from A by a correct (i.e., collisionless) substitution of a term t for the variable x. There are two rules for a correct substitution: We can substitute a term t only for free occurrences of a variable x in a formula A, and we have to substitute for all the free occurrences. No individual variable that occurrs in the term t can become bound in A (in such a case the term t is not substitutable for x in the formula A). Introduction to Logic 27

28 Substitution, example A(x): P(x) yq(x, y), term t = f(y) After executing the substitution A(x/f(y)), we obtain: P(f(y)) yq(f(y), y). The term f(y) is not substitutable for x in A We d change the sense of the formula Introduction to Logic 28

29 Semantics of FOL!!! P(x) yq(x, y) is this formula true? A non-reasonable question; For, we do not know what the symbols P, Q mean, what they stand for. They are only symbols which can stand for any predicate (property). P(x) P(x) is this formula true? YES, it is; and it is always so, in all the circumstances. It is a tautology, i.e. a logically valid formula. Introduction to Logic 29

30 Semantics of FOL!!! x P(x, f(x)) First, we have to specify x P(x, f(x)) how to understand these formulas: ) What do they talk about; we have to choose the universe of discourse: any non-empty set U 2) What does the predicate symbol P denote; it is a binary symbol, with two arguments; thus it has to denote a binary relation R U U 3) What does the symbol functional f denote; it is an unary, one-argument symbol; thus it has to denote a function F U U, denoted F: U U Introduction to Logic 3

31 Semantics of FOL (example) A: x P(x, f(x)) we have to specify B: x P(x, f(x)) how to understand these formulas: ) Let U = N (the set of natural numbers) 2) Let P denote the relation < (i.e., the set of pairs, where the first element is strictly less than the second one: {,,,2,,,2, }) 3) Let f denote the function second power x 2, i.e., the set of pairs where the second element is the power of the first one: {,,,, 2,4,,5,25, } Now we can evaluate the truth values of the formulas A, B Introduction to Logic 3

32 Semantics of FOL (example) A: x P(x, f(x)) B: x P(x, f(x)) We evaluate from the inside : First evaluate the term f(x). Each term denotes an element of the universe. Which one? It depends on the valuation e of the variable x. Let e(x) =, then f(x) = x 2 =. Let e(x) =, then f(x) = x 2 =, Let e(x) = 2, then f(x) = x 2 = 4, etc. Now by evaluating P(x, f(x)) we have to obtain a truth value: e(x) =, is not < False e(x) =, is not < False, e(x) = 2, 2 is < 4 True. Introduction to Logic 32

33 Semantics of FOL (example) A: x P(x, f(x)) B: x P(x, f(x)) The formula P(x, f(x)) is in the given interpretation True for some valuations of the variable x, and False for other valuations. The meaning of x (x): the formula is true for all (some) valuations of x Formula A: False in our interpretation I: I A Formula B: True in the interpretation I: = I B Introduction to Logic 33

34 Model of a formula, interpretation A: x P(x, f(x)) B: x P(x, f(x)) We have found an interpretation I in which the formula B is true. The Interpretation structure N, <, x 2 satisfies the formula B; it is a model of the formula B. How to adjust the interpretation in order it were a model of the formula A? There are infinitely many possibilities, infinitely many models. For instance: N, <, x+, {N/{,}, <, x 2, N,, x 2, All the models of the formula A are also models of the formula B ( what holds for all, holds also for some ) Introduction to Logic 34

35 Model of a formula, interpretation C: x P(x, f(y)) what are the models of this formula (with a free variable y)? Let us interpret again. Choose a Universe U = N 2. Assign a relation to the symbol P: 3. Assign a function to the symbol f: x 2 Is the structure IS = N,, power a model of the formula C? In order to be so, the formula C must be true in IS for all the valuations of the variable y. Hence the formula P(x, f(y)) must be true for all valuations of x and y. But it is not so, for instance, if e(x) = 5, e(y) = 2, then 5 is not 2 2 Introduction to Logic 35

36 Model of a formula, interpretation C: x P(x, f(y)) what are the models of this formula (with a free variable y)? The structure N,, x 2 is not a model of formula C. A (trivial) model is, e.g., N, N N, x 2. The whole Cartesian product N N, i.e. the set of all the pairs of natural numbers, is also a relation over N. Or, the structure N,, F, where F is the function, mapping N N, such that F associates all the natural numbers with the number. Introduction to Logic 36

37 Equivalent formulation, negation, de Morgan laws It is raining It is not true that it is not raining p p It is raining or snowing It is not true that it is neither raining nor snowing (p q) (p q) It is raining and snowing It is not true that it is not raining or not snowing (p q) (p q) It is not true that It is raining and snowing It is not raining or not snowing (p q) (p q) It is not true that It is raining or snowing It is not raining and not snowing (p q) (p q) It is not true that if it is raining then it is snowing It is raining and not snowing (p q) (p q) If it is raining then it is snowing It is not raining or it is snowing (p q) (p q) Be careful of implication! 37

38 De Morgan laws It is not true that all A are B Some A are not B x [A(x) B(x)] x [A(x) B(x)] x [A(x) B(x)] It is not true that some A is B No A is B x [A(x) B(x)] x [A(x) B(x)] x [A(x) B(x)] x [A(x) B(x)] It is not true that no A is B Some A are B x [A(x) B(x)] x [A(x) B(x)] x [A(x) B(x)] It is not true that some A are not B All A are B x [A(x) B(x)] x [A(x) B(x)] x [A(x) B(x)] In general: x A(x) x A(x); x A(x) x A(x) (A B) (A B); (A B) (A B); (A B) (A B); Introduction to Logic 38