can’t be c

1. can’t be b this is b can’t be c Could be c could be c

4. Names • Constants are used to name existing objects • a, b, c, d, e, f • max, claire, carl • No constant can name more than one object • An object can have more than one name or no name at all

5. Leonard Euler Tiberius Sempronius Gracchus Gaius Sempronius Gracchus Lincoln Honest Abe Examples

7. Predicates • A (determinate) property possessed by an object • Shape • Size • A (determinate) relationship among objects • Shape relationship • Size relationship • Positional relationship • Equality =

8. Atomic Sentences • A sentence formed by a single predicate followed by one or more names • Max is tall Tall(max) • e is larger than b Larger(e,b) • e is identical to a e = a • A sentence expresses a claim that is either true or false

9. Atomic Sentences in FOL • Predicate(arg1, arg2,…, argn) • Predicates have names beginning with an uppercase letter or are represented by an operator symbol • The number of arguments is called the predicate’s arity • The order of the arguments is importantLarger(e,c) – e is larger than cLarger(c,e) – c is larger than eBetween(a,b,e) – a is between b and eBetween(b,a,e) – b is between a and e • =(a,b) • a and b are identical • Usually, written in infix form a = b

10. Function Symbols • A function is used to express complex names (a reference to an individual without using a name) • father(b) – b’s father • password(c) – c’s password • A function may be nested • Max’s father’s father • father(father(max)) • A function is never a predicate • Can’t nest predicates • Tall(Tall(max)) • A predicate forms a sentence, while a function names an individual

11. Functions in FOL • function(arg1, arg2,…, argn) • Function names begin with a lowercase letter or are expressed with a symbol • father(max) Max’s father • father(mother(max)) Max’s mother’s father • youngestChild(max,ann) Max and Ann’s youngest child • *(5,+(2,4)) 30 • starship(son(dr_crusher)) Dr_Crusher’s son’s starship

12. Connectives • Not  • And, Or ,  • Material Conditional  • Biconditional 

13. Examples • Larger(e,c) • Cube(b)  Large(b) • SameRow(e,c)  BackOf(e,b) e is not larger than c b is a cube or b is large e and c are in the same row and e is in back of b

14. First Order Logic • Names • Predicates • Functions • Connectives Atomic Sentences Are there more?

15. Example FOL

16. Brando is Nancy’s favorite actor. brando = favoriteActor(nancy) Translation • Nancy’s favorite actor is better than Max’s favorite actor. • BetterActor(favoriteActor(nancy), favoriteActor(max)) • Sean is his own favorite actor. • sean = favoriteActor(sean) • Brando is someone’s favorite actor. • x(brando = favoriteActor(x))

17. Quantifiers and Variables • For every x x • There exists y y

18. First Order Logic • Names • Predicates • Functions • Connectives • Quantifiers and variables Revised List

19. Translation using functions • c is the front-most block in b’s column. • c is in the same row as the front-most block in b’s column. • The right-most block in the same row as the front-most block in b’s column is small. c = fm(b) SameRow(c,fm(b)) Small(rm(fm(b)))

20. First-order Arithmetic • Names • Zero 0 • One 1 • Predicates • Equality = • Less than < • Functions • Addition + • Multiplication  • 0 and 1 are terms. • If t1 and t2 are terms then so are (t1 + t2) and (t1 t2). • 3) Nothing is a term unless formed from the above rules.

21. What is an argument? • A series of statements in which one (called the conclusion) is meant to follow from or be supported by the others (called the premises).

22. Fitch-style Argument • P1 P2... PnQ premises conclusion

23. Valid Argument • A valid argument is one that guarantees the truth of its conclusion on the assumption that the premises are true. • A valid argument ensures the conclusion is true provided the premises are true. • A valid argument does not depend on any world for its validity

24. premises conclusion Valid Argument • Large(b) v Cube(b)Cube(b) Large(b) 

25. premises conclusion Invalid Argument • Large(b) v Cube(b) Cube(b) Large(b)

26. Sound Argument • If an argument is valid and its premises are true, then the argument is said to be sound. • The soundness or unsoundness of an argument is determined with respect to some world

27. Sound Argument

28. Argument is not sound

29. Methods of Proof • Formal • We will use a Fitch-style proof employed in the text and software of the same name. “Formal” proof evokes images of being rigorous. In fact, it has to do creating a proof with strict syntax rules. • Informal • This style of proof , used by mathematicians, is just as rigorous. It consists of sentences describing the situation at hand, the inferences being made, and the justification of each inference.

30. What constitutes a proof? • A proof that sentence Q follows from the premises P1, P2, …, Pn is a step-by-step demonstration that shows Q must be true in any circumstances in which the premises are all true.

31. Types of Proof • Direct • Indirect • Proof by cases • Proof by contradiction • Proof by induction • Proof by counterexample

32. Fitch Bar Fitch-style Proof • P1 P2 … Pn S1 S2 … Sn Q Premises Deductions & Justificationsmay contain sub-proofs Conclusion

33. These follow from above Rules/Axioms • = EliminationIf b = c and P(b) then P(c). • = Introductiona = a • Symmetry of IdentityIf a = b then b = a. • Transitivity of IdentityIf a = b and b = c then a = c

34. = Elimination • P(n) n = m P(m)

35. = Introduction • n = n

36. Symmetry of Identity • a = b 1)2)3) a = a = Introduction b = a = Elimination 1, 2

38. Example Formal Proof • Smaller(a,b) c = b 1)2)3)4) 5)6) Larger(b,a) Ana Con 1 = Introduction c = c b = c = Elim 2, 4 Larger(c,a) = Elim 5, 3

39. Example Informal Proof Prove: If a is smaller than b and c is identical to b then c is larger than a. Since a is smaller than b, it follows that b must be larger than a. Moreover, since c is identical to b, it follows that c must be larger than a. QED

40. Consequence Rules • There are three consequence “rules” in Fitch • Tautological Consequence (Taut Con) • First-order Consequence (FO Con) • Analytic Consequence (Ana Con) Cons rules are proof seekers that work behind the scenes. Success is indicated by the beloved blue check mark . Failure is indicated by the dreaded red x.

41. Taut Con • Weakest of the three Cons “rules” • Attempts to prove if the current step follows from the cited statements by virtue of the truth-functional connectives. E.g., p  q can be replaced by p  q.

42. FO Con • More powerful than Taut Con but weaker than Ana Con • Attempts to prove if the current step follows from the cited steps by virtue of the truth-functional connectives, the quantifiers, and the identity predicate. E.g., a=b  b=c can be replaced by a=c.

43. Ana Con • Most powerful Cons “rule” • Attempts to prove if the current step follows from the cited steps by virtue of the truth-functional connectives, the quantifiers, the the identity predicate and the meanings of each predicate of Tarski’s World. E.g., Larger(a,b) can be replaced by Smaller(b,a).

44. Showing Non-consequence • To show Q is not a consequence of premises P1, P2, …, Pn, create a world where the premises are simultaneously true and the conclusion Q is false. • This shows the argument below is invalid P1 P2 … Pn Q

45. b c a d Invalid Argument • SameRow(b,c) SameRow(a,d) SameRow(d,f) LeftOf(a,b) LeftOf(f,c) f f

47. Boolean Connectives • Negation  • Conjunction  • Disjunction  It is not the case that And, but, moreover Or

48. Negation

49. Negation Facts • P is translated as It is not the case that P. • (a = b) is equivalent to a  b • P is equivalent to P

50. The Game • Used to understand the truth value of a complex sentence • Strategy: Given a sentence of the form P that you believe to be True (False) implies  P is False (True)implies  P is True (False)implies P is False (True)