Review of Predicate Logic

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1 Review of Predicate Logic Martin Held FB Computerwissenschaften Universität Salzburg A-5020 Salzburg, Austria 19. Jänner 2016 COMPUTERWISSENSCHAFTEN

3 Table of Content 1 Review of Propositional and Predicate Logic c M. Held (Univ. Salzburg) Predicate Logic (WS 2015/16) 3

4 1 Review of Propositional and Predicate Logic Propositional Logic Predicate Logic Special Quantifiers c M. Held (Univ. Salzburg) Predicate Logic (WS 2015/16) 4

5 1 Review of Propositional and Predicate Logic Propositional Logic Predicate Logic Special Quantifiers c M. Held (Univ. Salzburg) Predicate Logic (WS 2015/16) 5

6 Propositional Logic Goal: specification of a language for formally expressing theorems and proofs. Aka: propositional calculus, logic of statements, statement logic; Dt.: Aussagenlogik. Definition 1 (Proposition, Dt.: Aussage) A proposition is a statement that is either true or false. Propositions can be atomic, like The sun is shining, or compound, like The sun is shining and the temperature is high. In the latter case, the proposition is a composition of atomic or compound propositions by means of logical connectives. c M. Held (Univ. Salzburg) Predicate Logic (WS 2015/16) 6

7 Language of Propositional Logic Definition 2 (Propositional formula, Dt.: aussagenlogische Formel) A propositional formula is constructed inductively from a set of propositional variables (typically p, q, r or p 1, p 2,...); connectives (operators):,,,, ; parentheses: (, ); constants (truth values):, (or F, T ); based on the following rules: A propositional variable is a propositional formula. The constants and are propositional formulas. If φ 1 and φ 2 are propositional formulas then so are the following: ( φ 1 ), (φ 1 φ 2 ), (φ 1 φ 2 ), (φ 1 φ 2 ), (φ 1 φ 2 ). c M. Held (Univ. Salzburg) Predicate Logic (WS 2015/16) 7

8 Precedence Rules Precedence rules (Dt.: Vorrangregeln) are used frequently to avoid the burden of too many parentheses. From highest to lowest precedence, the following order is common.,,, Unfortunately, different precedence rules tend to be used by different authors. Thus, make it clear which order you use, or in case of doubt, insert parentheses! It is common to represent the truth values of a proposition under all possible assignments to its variables by means of a truth table. In addition to the standard connectives we also define another operator, NAND, denoted by. c M. Held (Univ. Salzburg) Predicate Logic (WS 2015/16) 8

9 Truth Tables p q p p q p q p q p q p q T T F T T T T F T F F F T F F T F T T F T T F T F F T F F T T T Common names for the operators in natural language: p: NOT, negation; p q: AND, conjunction; p q: OR, disjunction; p q: IMPLIES, conditional, if p then q, q if p, p sufficient for q, q necessary for p; p q: IFF, equivalence, biconditional, p if and only if q, p necessary and sufficient for q. Note: The truth table (Dt.: Wahrheitstabelle) of a formula with n variables has 2 n rows. c M. Held (Univ. Salzburg) Predicate Logic (WS 2015/16) 9

10 Tautologies, Contradictions Definition 3 (Tautology, Dt.: Tautologie) A propositional formula is a tautology if it is true under all truth assignments to its variables. Definition 4 (Contradiction, Dt.: Widerspruch) A propositional formula is a contradiction if it is false under all truth assignments to its variables. Standard examples: (p p) and (p p). Easy to prove: The negation of a tautology yields a contradiction, and vice versa. c M. Held (Univ. Salzburg) Predicate Logic (WS 2015/16) 10

11 Logical Equivalence Definition 5 (Logical equivalence, Dt.: logische Äquivalenz) Two propositional formulas are logically equivalent if they have the same truth table. Logical equivalence of formulas φ 1, φ 2 is commonly denoted by φ 1 φ 2. Theorem 6 Two propositional formulas φ 1, φ 2 are logically equivalent iff φ 1 φ 2 is a tautology. Definition 7 (Complete set of connectives, Dt.: vollständige Junktorenmenge) A set S of connectives is said to be complete (or truth-functionally adequate/complete) if, for any given propositional formula, a logically equivalent one can be written using only connectives of S. Note: The set { } is a complete set of connectives. c M. Held (Univ. Salzburg) Predicate Logic (WS 2015/16) 11

12 Laws for Logical Equivalence Theorem 8 Let φ 1, φ 2 be propositional formulas. Then the following equivalences hold: Identity: φ 1 T φ 1 φ 1 F φ 1 Domination: φ 1 T T φ 1 F F Idempotence: φ 1 φ 1 φ 1 φ 1 φ 1 φ 1 Double negation: φ 1 φ 1 Commutativity: φ 1 φ 2 φ 2 φ 1 φ 1 φ 2 φ 2 φ 1 φ 1 φ 2 φ 2 φ 1 Distributivity: (φ 1 φ 2 ) φ 3 (φ 1 φ 3 ) (φ 2 φ 3 ) (φ 1 φ 2 ) φ 3 (φ 1 φ 3 ) (φ 2 φ 3 ) Associativity: (φ 1 φ 2 ) φ 3 φ 1 (φ 2 φ 3 ) (φ 1 φ 2 ) φ 3 φ 1 (φ 2 φ 3 ) De Morgan s laws: (φ 1 φ 2 ) φ 1 φ 2 (φ 1 φ 2 ) φ 1 φ 2 Trivial tautology: φ 1 φ 1 T Trivial contradiction: φ 1 φ 1 F Contraposition: φ 1 φ 2 φ 1 φ 2 φ 2 φ 1 φ 1 φ 2 Implication as Disj.: φ 1 φ 2 φ 1 φ 2 c M. Held (Univ. Salzburg) Predicate Logic (WS 2015/16) 12

13 Logical Implication and Proofs Definition 9 (Logical implication, Dt.: logische Implikation) A formula φ 1 logically implies φ 2, denoted by φ 1 = φ 2, if φ 1 φ 2 is a tautology. Definition 10 (Proof, Dt.: Beweis) A proof of ψ based on premises φ 1,..., φ n is a finite sequence of propositions that ends in ψ such that each proposition is either a premise or a logical implication of the previous proposition. Note: Logical implication rather than logical equivalence! Thus, note that it need not be possible to revert a proof! pay close attention to which steps are actual equivalences if you intend to argue both ways! c M. Held (Univ. Salzburg) Predicate Logic (WS 2015/16) 13

14 Rules of Inference Aka: proof rules (Dt.: Schlußregeln). In addition to the following inference rules for propositional formulas φ 1, φ 2, all the equivalence rules apply: Each equivalence can be written as two inference rules since they are valid in both directions. φ 1 φ 2 φ 1 φ 1 φ 1 φ 2 φ 1 φ 2 φ 2 φ 1 (CONTRAPOSITION) φ 1 φ 1 φ 2 φ 2 (MODUS PONENS) φ 1 φ 1 φ 2 φ 2 (MODUS TOLLENDO PONENS) φ 1 φ 2 φ 1 φ 2 φ 2 (RULE OF CASES) φ 1 φ 2 φ 2 φ 3 φ 1 φ 3 (CHAIN RULE) c M. Held (Univ. Salzburg) Predicate Logic (WS 2015/16) 14

15 Satisfiability Definition 11 (Satisfiability, Dt.: Erfüllbarkeit) A formula φ is satisfiable if there exists at least one truth assignment to the variables of φ that makes φ true. Definition 12 (Satisfiability equivalent) Two formulas are satisfiability equivalent if both formulas are either satisfiable or not satisfiable. c M. Held (Univ. Salzburg) Predicate Logic (WS 2015/16) 15

16 Conjunctive Normal Form In mathematics, normal forms are canonical representations of objects such that all equivalent objects have the same representation. Definition 13 (Literal, Dt.: Literal) A literal is a propositional variable or the negation of a propositional variable. A clause is a disjunction of literals. E.g., if p, q are variables then p and q are literals, and (p q) is a clause. Definition 14 (Conjunctive normal form, Dt.: konjunktive Normalform) A propositional formula is in (general) conjunctive normal form (CNF) if it is a conjunction of clauses. E.g., p 1 (p 2 p 5 p 6 ) ( p 3 p 4 p 6 ) is a CNF formula. Definition 15 (k-cnf) A CNF formula is a k-cnf formula if every clause contains at most k literals. c M. Held (Univ. Salzburg) Predicate Logic (WS 2015/16) 16

17 Conjunctive Normal Form Note: Some textbooks demand exactly k literals rather than at most k literals. Note: It is common to demand that no variable may appear more than once in a clause. Note: For k 3, a general CNF formula can easily be converted in polynomial time (in the number of literals) into a k-cnf formula with exactly k literals per clause such that no variable appears more than once in a clause and such that the two formulas are satisfiability equivalent. c M. Held (Univ. Salzburg) Predicate Logic (WS 2015/16) 17

18 1 Review of Propositional and Predicate Logic Propositional Logic Predicate Logic Special Quantifiers c M. Held (Univ. Salzburg) Predicate Logic (WS 2015/16) 18

19 Predicate Logic Definition 16 (n-place relation, Dt.: n-stellige Relation) Let A 1, A 2,..., A n be sets, for some n N. An n-place relation R on A 1, A 2,..., A n is a subset of their Cartesian product, i.e., R A 1 A 2 A n. c M. Held (Univ. Salzburg) Predicate Logic (WS 2015/16) 19

20 Predicate Logic Definition 16 (n-place relation, Dt.: n-stellige Relation) Let A 1, A 2,..., A n be sets, for some n N. An n-place relation R on A 1, A 2,..., A n is a subset of their Cartesian product, i.e., R A 1 A 2 A n. Definition 17 (n-place function, Dt.: n-stellige Funktion) Let A 1, A 2,..., A n, B be sets, for some n N. An n-place function F from A 1 A 2 A n to B is an (n + 1)-place relation on A 1, A 2,..., A n, B such that for any (a 1, a 2,..., a n) A 1 A 2 A n there exists a unique b B such that (a 1, a 2,..., a n, b) F. c M. Held (Univ. Salzburg) Predicate Logic (WS 2015/16) 19

21 Predicate Logic Definition 16 (n-place relation, Dt.: n-stellige Relation) Let A 1, A 2,..., A n be sets, for some n N. An n-place relation R on A 1, A 2,..., A n is a subset of their Cartesian product, i.e., R A 1 A 2 A n. Definition 17 (n-place function, Dt.: n-stellige Funktion) Let A 1, A 2,..., A n, B be sets, for some n N. An n-place function F from A 1 A 2 A n to B is an (n + 1)-place relation on A 1, A 2,..., A n, B such that for any (a 1, a 2,..., a n) A 1 A 2 A n there exists a unique b B such that (a 1, a 2,..., a n, b) F. It is common to write y = F(a 1,..., a n) for pick y such that (a 1,..., a n, y) F. The set A 1 A 2 A n is called the domain and the set B is called the codomain of F. c M. Held (Univ. Salzburg) Predicate Logic (WS 2015/16) 19

22 Predicate Logic Definition 16 (n-place relation, Dt.: n-stellige Relation) Let A 1, A 2,..., A n be sets, for some n N. An n-place relation R on A 1, A 2,..., A n is a subset of their Cartesian product, i.e., R A 1 A 2 A n. Definition 17 (n-place function, Dt.: n-stellige Funktion) Let A 1, A 2,..., A n, B be sets, for some n N. An n-place function F from A 1 A 2 A n to B is an (n + 1)-place relation on A 1, A 2,..., A n, B such that for any (a 1, a 2,..., a n) A 1 A 2 A n there exists a unique b B such that (a 1, a 2,..., a n, b) F. It is common to write y = F(a 1,..., a n) for pick y such that (a 1,..., a n, y) F. The set A 1 A 2 A n is called the domain and the set B is called the codomain of F. An n-place relation/function over a set A is a relation/function where A 1 = A 2 =... = A n = A, i.e., A 1 A 2 A n = A n. It is also called an n-ary relation/function. An 1-ary relation/function is called unary, and a 2-ary relation/function is called binary. c M. Held (Univ. Salzburg) Predicate Logic (WS 2015/16) 19

23 Predicate Logic: Predicates Definition 18 (Predicate, Dt.: Prädikat) For an n-ary relation R over A, an n-ary predicate over A is the n-ary function f R : A n {T, F}, where { T if (a1,..., a n) R, f R(a 1,..., a n) := F otherwise. Thus, a predicate is a Boolean function. Note: This is a slight abuse of notation since the symbols : and in f : M N actually form already a 3-ary predicate! An 1-ary predicate is called unary, and a 2-ary predicate is called binary. A sample unary predicate { on R is T if x 0, x is non-negative := F otherwise. Dt.: Prädikatenlogik. c M. Held (Univ. Salzburg) Predicate Logic (WS 2015/16) 20

24 Language of Predicate Logic Definition 19 (Predicate vocabulary, Dt.: Symbolmenge) A predicate vocabulary consists of a set C of constant symbols, a set F of function symbols, a set V of variables, typically {x 1, x 2,...} or {a, b,...}, a set P of predicate symbols, including the 0-ary predicate symbols (truth values), or F, T, together with logical connectives,,,,, quantifiers,, parentheses. c M. Held (Univ. Salzburg) Predicate Logic (WS 2015/16) 21

25 Language of Predicate Logic Definition 20 (Term) A term over (C, V, F) is defined inductively as follows: Every constant c C is a term. Every variable x V is a term. If t 1,..., t n are terms and f is an n-ary function symbol then f (t 1,..., t n) is a term. Note: Constants can be thought of as 0-ary function symbols. Thus, a set C of constants need not be considered when defining the language of predicate logic. c M. Held (Univ. Salzburg) Predicate Logic (WS 2015/16) 22

26 Language of Predicate Logic Definition 21 (Formulas) The set of formulas over (C, V, F, P) is defined inductively as follows: and are formulas. If t 1,..., t n are terms and P P is an n-ary predicate, then P(t 1,..., t n) is a (so-called atomic) formula. If φ and ψ are formulas then ( φ), (φ ψ), (φ ψ), (φ ψ) and (φ ψ) are formulas. If φ is a formula then ( x φ) and ( x φ) are formulas. In both cases, the scope of the quantifier is given by the formula φ to which the quantifier is applied. Definition 22 (Quantifier-free formula, Dt.: quantorenfreie Formel) A quantifier-free formula is a formula which does not contain a quantifier. c M. Held (Univ. Salzburg) Predicate Logic (WS 2015/16) 23

27 Quantifiers Definition 23 (Universe of discourse, Dt.: Wertebereich, Universum) The universe of discourse specifies the set of values that the variable x may assume in ( x φ) and ( x φ). Definition 24 (Universal quantifier, Dt.: Allquantor) ( x P(x)) is the statement P(x) is true for all x (in the universe of discourse). Definition 25 (Existential quantifier, Dt.: Existenzquantor) ( x P(x)) is the statement there exists x (in the universe of discourse) such that P(x) is true. The notation (!x P(x)) is a convenience short-hand for there exists exactly one x such that P(x) is true, i.e., for denoting existence and uniqueness of a suitable x. c M. Held (Univ. Salzburg) Predicate Logic (WS 2015/16) 24

28 Precedence Rules for Quantified Formulas No universally accepted precedence rule exists. Thus, you have to make your specific order very clear. Even better, use parentheses or (significant) spaces between coherent parts of the expression. First-order logic versus higher-order logic: In first-order predicate logic, predicate quantifiers or function quantifiers are not permitted, and variables are the only objects that may be quantified. Also, predicates are not allowed to have predicates as arguments. c M. Held (Univ. Salzburg) Predicate Logic (WS 2015/16) 25

29 Free Variables Definition 26 (Free variables, Dt.: freie Variable) The free variables of a formula φ or a term t, denoted by FV (φ) and FV (t), are defined inductively as follows: For a constant c C: FV (c) := {}; For a variable x V: For a term f (t 1,..., t n): For a formula P(t 1,..., t n): FV (x) := {x}; FV (f (t 1,..., t n)) := FV (t 1 )... FV (t n); FV (P(t 1,..., t n)) := FV (t 1 )... FV (t n); Also, FV ( ) := {}, For formulas φ and ψ: FV ( ) := {}; FV (( φ)) := FV (φ), FV ((φ ψ)) := FV (φ) FV (ψ), FV ((φ ψ)) := FV (φ) FV (ψ), FV ((φ ψ)) := FV (φ) FV (ψ), FV ((φ ψ)) := FV (φ) FV (ψ); For a formula φ: FV (( x φ)) := FV (φ) \ {x}, FV (( x φ)) := FV (φ) \ {x}. c M. Held (Univ. Salzburg) Predicate Logic (WS 2015/16) 26

30 Bound Variables Definition 27 (Bound variables, Dt.: gebundene Variable) The bound variables of a formula φ or a term t, denoted by BV (φ) and BV (t), are defined inductively as follows: For a constant c C: BV (c) := {}; For a variable x V: BV (x) := {}; For a term f (t 1,..., t n): BV (f (t 1,..., t n)) := {}; For a formula P(t 1,..., t n): BV (P(t 1,..., t n)) := {}; Also, BV ( ) := {}, For formulas φ and ψ: BV ( ) := {}; BV (( φ)) := BV (φ), BV ((φ ψ)) := BV (φ) BV (ψ), BV ((φ ψ)) := BV (φ) BV (ψ), BV ((φ ψ)) := BV (φ) BV (ψ), BV ((φ ψ)) := BV (φ) BV (ψ); For a formula φ: BV (( x φ)) := BV (φ) {x}, BV (( x φ)) := BV (φ) {x}. c M. Held (Univ. Salzburg) Predicate Logic (WS 2015/16) 27

31 Free and Bound Variables Note: Technically speaking, one variable symbol may denote both a free and a bound variable of a formula! However, common sense dictates to use a different symbol if a different variable is meant, even if not required by the syntax of predicate logic: Do not use the same symbol for bound and free variables! E.g., (P(x) ( x Q(x))) is syntactically correct but extremely difficult to parse for a human. Also, do not re-use symbols of bound variables inside nested quantifiers! E.g., ( x (P(x) ( x Q(x)))) is syntactically correct but horrible to parse. Definition 28 (Sentence, Dt.: geschlossener Ausdruck) A formula φ is a sentence if FV (φ) = {}. c M. Held (Univ. Salzburg) Predicate Logic (WS 2015/16) 28

32 Substitutions Definition 29 (Substitution, Dt.: Ersetzung) For a formula φ, variable x and term t, we obtain the substitution of x by t, denoted as φ[t/x], by replacing each free occurrence of x in φ by t. Definition 30 (Valid substitution, Dt.: gültige Ersetzung) A substitution of t for x in a formula φ is valid if and only if no free variable of t ends up being bound in φ[t/x]. Not a valid substitution of x: φ ( y N y > 10 x < y) and t := 2y + 5. Again, it is very poor practice to substitute x by t if t contains any variable that also is a bound variable of φ! φ ( z N z 2 > 0) ( y N y > 10 x < y) and t := 2z + 5. c M. Held (Univ. Salzburg) Predicate Logic (WS 2015/16) 29

33 Equivalence Rules Theorem 31 Let x be a variable, and φ and ψ be formulas which normally contain x as a free variable. Then the following equivalences hold: De Morgan s laws: ( ( x φ)) ( x ( φ)) ( ( x φ)) ( x ( φ)) Trivial conjunction: ( x (φ ψ)) (( x φ) ( x ψ)) Only if x FV (ψ): ( x (φ ψ)) (( x φ) ψ) ( x (φ ψ)) (( x φ) ψ) c M. Held (Univ. Salzburg) Predicate Logic (WS 2015/16) 30

34 Rules of Inference Let x, y be variables and φ, ψ be propositional formulas. The following inference rules allow to deduce new formulas. (( x φ) ( x ψ)) ( x (φ ψ)) ( x (φ ψ)) ( x φ) ( x ψ) ( x ( y φ)) ( y ( x φ)) Note that the other direction does not hold for any of these inference rules! In addition to these three inference rules all the equivalence rules apply: Each equivalence can be written as two inference rules since they are valid in both directions. c M. Held (Univ. Salzburg) Predicate Logic (WS 2015/16) 31

35 1 Review of Propositional and Predicate Logic Propositional Logic Predicate Logic Special Quantifiers c M. Held (Univ. Salzburg) Predicate Logic (WS 2015/16) 32

36 Special Quantifiers What is the syntactical meaning of n f (i)? i=m Apparently, this is the common short-hand notation for n f (i) = i=m m i n f (i) = P(i,m,n) f (i) = f (m) + f (m + 1) + + f (n 1) + f (n), where f (i) is a term with the free variable i and (m i n) is a formula with free variables i, m, n, and P(i, m, n) : [(i m) (i n)]. c M. Held (Univ. Salzburg) Predicate Logic (WS 2015/16) 33

37 Special Quantifiers Thus, the -quantifier takes a predicate, P(i, m, n), and and a term, f (i), and converts it to the new term (f (m) + f (m + 1) + f (m + 2) + + f (n 1) + f (n)), By convention, the variable i is bound inside of n i=m f (i), while m and n remain free. Similarly, n f (i) := f (m) f (m + 1) f (m + 2)... f (n 1) f (n). i=m Again, by convention, if n < m then n f (i) := 0 i=m and n f (i) := 1. i=m Union ( ) and intersection ( ) of several sets are further examples of special quantifiers: n i=1a i. c M. Held (Univ. Salzburg) Predicate Logic (WS 2015/16) 34

38 Special Quantifiers: Sets Standard notation for a set with a finite number of elements: {,,..., }; e.g., {1, 2, 3, 4}. Obvious disadvantage: explicit enumeration of all elements of a set allows to specify only finite sets! Infinite sets require us to give a statement A to specify a characteristic property of the set: S := {x : A} or S := {f (x) : A}, where S shall contain those elements x, or those terms f (x), for some universe of discourse, for which the statement A holds. Typically, x will be a free variable of A. Thus, the three symbols { and : and } together act as a quantifier that binds x. c M. Held (Univ. Salzburg) Predicate Logic (WS 2015/16) 35

39 Convenient Short-Hand Notations The following short-hand notations are convenient for using the predicate x X in conjunction with sets or quantifiers: {x X : A(x)} is a short-hand notation for {x : x X A(x)} ( x X A(x)) is a short-hand notation for ( x (x X A(x))) ( x X A(x)) is a short-hand notation for ( x (x X A(x))) If x is a typed variable e.g., a real number and P is a simple unary predicate e.g., P(x) : (x > 3) then the following notations are also used commonly: ( P(x) A(x)) is a short-hand notation for ( x (P(x) A(x))) ( P(x) A(x)) is a short-hand notation for ( x (P(x) A(x))) Other wide-spread notations include the following variations: dropping the parentheses: x P(x) instead of ( x P(x)) colon instead of space as separator: x: P(x) instead of x P(x) c M. Held (Univ. Salzburg) Predicate Logic (WS 2015/16) 36

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