Predicate Logic 16. Quantifiers. The Lecture

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1 Predicate Logic 16. Quantifiers The Lecture

2 First order (predicate logic) formulas Quantifiers are the final elements that first order (i.e. predicate logic) formulas are built up from. First order formulas are built up from atomic formulas by means of logical operations: negation, conjunction, disjunction, implication, equivalence, existential quantifier, and universal quantifier. Parentheses (,) are used for clarity.

3 First order formulas are of the form where A and B are first order formulas. Parentheses (,) are used for clarity.

4 First order formulas are of the form atomic where A and B are first order formulas. Parentheses (,) are used for clarity.

5 First order formulas are of the form atomic A where A and B are first order formulas. Parentheses (,) are used for clarity.

6 First order formulas are of the form atomic A where A and B are first order formulas. Parentheses (,) are used for clarity.

7 First order formulas are of the form atomic A where A and B are first order formulas. Parentheses (,) are used for clarity.

8 First order formulas are of the form atomic A where A and B are first order formulas. Parentheses (,) are used for clarity.

9 First order formulas are of the form atomic A where A and B are first order formulas. Parentheses (,) are used for clarity.

10 First order formulas are of the form atomic A xa where A and B are first order formulas. Parentheses (,) are used for clarity.

11 First order formulas are of the form atomic A xa xa where A and B are first order formulas. Parentheses (,) are used for clarity.

12 Examples P 0 (x) P 1 (x) (x<y y<x) x(xey z(xez zey)) x(b(x) z(y(z) z<x))

13 Universal quantifier explained

14 Universal quantifier explained xa: Every value of x satisfies A.

15 Universal quantifier explained xa: Every value of x satisfies A. Every tile is red.

16 Universal quantifier explained xa: Every value of x satisfies A. Every tile is red. Every x satisfies x 2 0.

17 Universal quantifier explained xa: Every value of x satisfies A. Every tile is red. Every x satisfies x 2 0. All vertices x and y are neighbors.

18 Universal quantifier explained xa: Every value of x satisfies A. Every tile is red. Every x satisfies x 2 0. All vertices x and y are neighbors. All men are mortal.

19 Universal quantifier explained xa: Every value of x satisfies A. Every tile is red. Every x satisfies x 2 0. All vertices x and y are neighbors. All men are mortal. Everybody loves her.

20 Existential quantifier explained

21 Existential quantifier explained xa: Some value of x satisfies A.

22 Existential quantifier explained xa: Some value of x satisfies A. Some tiles are red.

23 Existential quantifier explained xa: Some value of x satisfies A. Some tiles are red. Some reals x satisfy x 2 =2.

24 Existential quantifier explained xa: Some value of x satisfies A. Some tiles are red. Some reals x satisfy x 2 =2. Some vertices x and y are neighbors.

25 Existential quantifier explained xa: Some value of x satisfies A. Some tiles are red. Some reals x satisfy x 2 =2. Some vertices x and y are neighbors. There is a yellow tile.

26 Existential quantifier explained xa: Some value of x satisfies A. Some tiles are red. Some reals x satisfy x 2 =2. Some vertices x and y are neighbors. There is a yellow tile. There is a vertex with two neighbors.

27 Assignments and quantifiers In order to define when an assignment satisfies a quantified formula, we need the concept of a modified assignment. This row is a modified assignment x y z S S(2/x) S(8/z) This row is another modified assignment

28 Modified assignments Assignment s(a/x) is like assignment s except that the value of x is changed to a. x y z S S(2/x) S(8/z) 1 5 8

29 Assignment satisfying a quantified formula

30 Assignment satisfying a quantified formula Assignment s satisfies xa in M if the modified assignment s(a/x) satisfies A in M for every a in M.

31 Assignment satisfying a quantified formula Assignment s satisfies xa in M if the modified assignment s(a/x) satisfies A in M for every a in M. Assignment s satisfies xa in M if the modified assignment s(a/x) satisfies A in M for some a in M.

32 Satisfaction We have defined when an assignment s satisfies a formula A in a structure M. When this is the case, we write M s A. This is called the Tarski Truth Definition.

Tarski Truth Definition Atomic Atomic Atomic

Pn(x) if and only if s(x) Pn M M s R(x,y) if

if M s A Conjunction M s if and only if M s A

M s A or M s B Implication M s if and only if

33 Tarski Truth Definition Atomic Atomic Atomic Negation M s x=y if and only if s(x)=s(y) M s Pn(x) if and only if s(x) Pn M M s R(x,y) if and only if (s(x),s(y)) R M M s A if and only if M s A Conjunction M s if and only if M s A and M s B Disjunction M s AvB if and only if M s A or M s B Implication M s if and only if M s A or M s B Equivalence M s if and only if [M s A and M s B] or [M s A and M s B] Universal quantifier M s xa if and only if M s(a/x) A for all a in M Existential quantifier M s xa if and only if M s(a/x) A for some a in M 11

Introduction to Sets and Logic (MATH 1190)

Introduction to Sets and Logic (MATH 1190) Introduction to Sets Logic () Instructor: Email: shenlili@yorku.ca Department of Mathematics Statistics York University Sept 18, 2014 Outline 1 2 Tautologies Definition A tautology is a compound proposition