Formal Proofs & F PHIL /31/2001

Outline Homework Problems The FOL of F and Fitch notation Formal Proofs Assignment

Problem 18 Give a proof of a = c from the premises a = b and b = c using only the indiscernability of identicals. Indiscernability of Identicals: If a = b, then whatever we can say about a must be true of b as well

Problem 18 Continued Suppose that a = b and b = c. If a = b, then, anything we can say about b must be true of a as well. Since we know that b = c, a = c. Therefore, a = c.

Problem Premise: LeftOf(a, b); Conclusion: RightOf (b, a) Because LeftOf and RightOf are complimentary, the conclusion follows. We know this because there is no way of constructing a world in which LeftOf(a,b) is true and RightOf(b, a) is false.

Problem 19 Continued 2. Premises: LeftOf(a, b), b = c; Conclusion: RightOf(c, a). If b = c, anything we can say about b must be true of c as well, so we know LeftOf(a, c) is true. Because LeftOf(a, c) is true, RightOf(c, a) must be true as well.

Problem 19 Continued 3. Premises: LeftOf(a, b), RightOf(c, a); Conclusion: LeftOf(b, c) If we draw this on a sheet of paper or use Tarski’s world, we see that both premises could be true and the conclusion false because c could be between a and b. Two possibilities: a b c (conclusion is true) a c b (conclusion is false) A conclusion is not a logical consequences of the premises if there is at least one case where the premises could be true and the conclusion false.

Problem 19 Continued 4. Premises: BackOf(a, b), FrontOf(a, c); Conclusion: FrontOf(b, c) Because BackOf(a, b) means the same thing as FrontOf(b, a) and because if FrontOf(b, a) and FrontOf(a, c) we know that FrontOf(b, c)

Problem 19 Continued 5. Premises: Between(b, a, c), LeftOf(a, c); Conclusion: LeftOf(a, b) Recall the definition of Between… If a and c are in adjacent rows and if we draw a line from the midpoint of c to the midpoint of a, b is between if the line passes through it.

Problem 19 Continued Between(b, a, c) is true LeftOf(a, c) is true LeftOf(a, b) is false So LeftOf(a, b) does not follow from our premises. Remember: Our conclusion does not follow if there is at least one case in which the premises are true but the conclusion can be false. a b c

Problem Folly was Claire’s disk at 2 pm. 2. Silly was Max’s disk at 2 pm. 3. Silly was not Claire’s disk at 2 pm. Does 3 from 1 and 2? No. Consider: Owns(Claire, Silly, 2:00) and Owns(Claire,Folly, 2:00). (Nothing says she can’t own two disks at the same time).

Problem Folly was Claire’s disk at 2 pm. 2. Silly was Max’s disk at 2 pm. 3. Silly was not Claire’s disk at 2 pm. Does 2 follow from 1 and 3? Owns(Claire, Folly, 2:00) ~ Owns(Claire,Silly, 2:00) Do either of the propositions tell us anything about Max? No, because there is no rules that says that every student must own a disk.

Problem Folly was Claire’s disk at 2 pm. 2. Silly was Max’s disk at 2 pm. 3. Silly was not Claire’s disk at 2 pm. Does 1 follow from 2 and 3? Owns(Max, Silly, 2:00) ~ Owns(Claire,Silly, 2:00) Do either of the propositions tell us anything about Claire owning Folly? 2 does not follow because there’s no rule that says every disk must be owned by someone.

Fitch-style Notation For now, the important things to remember are Each statement gets its own line. A horizontal bar must separate premises from the rest of the proof. Every statement is numbered to aid in providing its rationale. You must provide a rationale (a rule and one or more numbers referring to a previously ascertained statement) for every new statement in the proof that is not a premise.

Fitch-style notation We need not worry about the vertical bar for now, but it will become important later. For now, just notice that the vertical bar means that all of the statements are part of the same proof.

Fitch notation 1. a = b 2. a = aRefl = 3. b = aInd Id: 2, 1 Premise(s) Conclusion Intermediate step(s) Rationales

Rules in F Reflexivity of Identity (Refl =) Informal: If a name “n” appears somewhere in your proof, you are entitled to say “n = n” Formal: n = n The arrow indicates the “conclusion” of the rule. That is, what you’re entitled to say.

Indiscernibility of Identicals (Ind Id): Informal: If you know that a = b, whatever you can say about a is true of b. Formal: P(n)    n = m    P(m) Rules in F

Rules using Word When you state rules in F … Don’t worry about the vertical bar (for now) Use “>” (greater than) for the arrow: P(n)    n = m    > P(m)

Notes on Proofs The “premises” of a rule must appear in the proof before its conclusion. There can be steps in between the “premises” of a rule. The “premises” of a rule do not have to be in any particular order.

Example Proof Given the rule, Trans LeftOf LeftOf(a, b)  LeftOf(b, c)  LeftOf(a, c) We can construct the following proof

Example proof Given LeftOf(a, b), LeftOf(b, c), and c = d, prove LeftOf(a, d). 1. LeftOf(a, b) 2. LeftOf(b, c) 3. c = d 4. LeftOf(a, c)Trans LeftOf: 1, 2 5. LeftOf(a, d)Ind Id: 3, 4

Assignment Problems due by 5pm Friday I will go over these problems on Friday. I’ll hand out a set of study questions on Friday that we will go over Monday and Wednesday. Exam 1 will be Friday, Feb. 9.