 # Introduction to Logic and Reasoning

## Transcribed Text

1. (2 points) We have started working with FOL. FOL is dierent from TFL in its syntax, even though (as someone pointed out in class) all the sentences of TFL are also sentences of FOL. The dierence is that there are many new sentences in FOL. Handout 2.2 (which summarizes Magnus’ Chapter 26) tells us, ocially, what is and what is not a sentence of FOL. Please say whether or not each of the following is an FOL sentence. (Note: I’m asking about whether it’s a sentence, not a formula!) (a) (F(a) ∧ G(a)) (b) ∀xF(x) (c) ∀x(F(G(H(a)))) (d) (P → Q) (e) (P → (a = b ↔ ∀xF(x))) (f) ∃x(a) (g) P(x, b, a, q, z) (h) a = b = c 2. (2 points) We have started to talk about interpretations, and soon we will use them to provide semantics for FOL, dening entailment, validity, and more in FOL. This will enable us to symbolize English sentences in FOL. We can’t do those things yet. But what counts as a proof in a natural deduction system is just about which symbols the rules allow us to write where. We don’t need to think about truth and validity to do proofs. The natural deduction system we will use for FOL is much like the system for TFL: all the Basic Rules of TFL (as well as the Derived Rules discussed in Assignment 6 and summarized on Magnus’ p380) are rules of the natural deduction system for FOL. This means we can already start practicing proofs in the natural deduction system for FOL. Here are two examples. 1 ¬¬∀xF(x) 2 ¬∀xF(x) 3 ⊥ ¬E, 1, 2 4 ∀xF(x) IP, 2–3 1 P 2 F(a, b) 3 P R, 1 4 P ∧ P ∧I, 1, 3 5 (F(a, b) → (P ∧ P)) →I, 2–4 Below are four FOL arguments. For each, a formal proof of the conclusion of each argument from the premises is possible, using only rules (Basic or Derived) that also appeared in the deduction system for TFL. Please give such a proof. (There is space on the next page.) (a) ∀xF(x),(∀xF(x) → (Q ∨ a = b)), ¬a = b, ` Q (b) ¬∃x(F(x) ∧ G(x)) ` (∃x(F(x) ∧ G(x)) → ∀xH(x)) (c) (¬P(a) → ¬G(b)) ` (G(b) → P(a)) (d) ((R(a, b) ∧ R(b, c)) ∨ (R(b, a) ∧ R(b, c))) ` R(b, c) 3. (1 Point) We have been talking about arguments like this one: (a) Willard is a logician. (b) All logicians wear hats. (c) Therefore: Willard wears a hat. We said: it looks like it doesn’t matter who (or what) Willard is; it doesn’t even matter what most of the words here mean at all. So long as Willard is some thing, and wearing hats is something that a thing might do, and being a logician is also something that something might do, the truth of the premises will guarantee the truth of the conclusion: as long as the premises are true, the conclusion will be true too. But at the end of class on Wednesday, somebody asked: but what if “wear hats” meant: DOES NOT wear hats? In that case, the second sentence would really be saying that all logicians don’t wear hats. And then, the truth of the premises looks like it will not guarantee the truth of the conclusion! This is a very good question! I think, however, that even if “wears hats” meant that, the truth of the premises really would still guarantee the truth of the conclusion! Please explain why that is.

## Solution Preview

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1.
(a) (F(a)∧G(a))   - Yes

(b) ∀xF(x)   - Yes

(c) ∀x(F(G(H(a))))   - No

(d) (P→Q)   - Yes

(e) (P→(a=b↔∀xF(x)))   - Yes

(f) ∃x(a)   - No...
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