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1. We have started doing the semantics of FOL. We can now say what it is for any sentence of FOL to be true or false in an interpretation, unless that sentence of FOL has a quantier in it. On the left below are eight FOL sentences without quantiers, and on the right is an interpretation that I provided. Please say whether these FOL sentences are true or false in the interpretation provided. (a) (F(b) ∧ G(b)) (b) a = c (c) b = c (d) (S(a, b) ∨ S(d, c)) (e) (G(d) → G(c)) (f) ((R(b, a) ∧ R(b, b)) → F(d)) (g) (a = b ↔ ¬c = d) (h) (a = a ∧ (F(a) ∧ ¬G(a))) Domain: All the numbers. a: 2 b: 3 c: 2 d: 5 F(x): x is even. G(x): 3, 5, 7, 10 R(x, y): x is greater than y. S(x, y): < 2, 3 >, < 3, 2 > 2. Pretty soon, we will talk about validity for argumentsin FOL. It is going to be exactly what you would expect: since we have said that interpretations will be our “cases” in FOL, an FOL argument will be valid if and only if there is no interpretation in which the premises are true and the conclusion is false. And so: below are four arguments in FOL. For each one, there is an interpretation in which the premise(s) are true and the conclusion is false: as we will say, the argument is not valid in FOL. For each argument, please provide such an interpretation: one in which the premise(s) are true and the conclusion is false. (a) F (a) ∴ G(a) (b) a = a ∴ ¬b = a (c) (F (a) → G(a)), G(a) ∴ (F (a) ↔ G(a)) (d) (R(a, b) ∧ R(b, c)) ∴ R(a, c) 3. More proof practice! This is the same as on Assignment 7: please just prove these things in our deduction system for FOL, using the rules that were also in our system for TFL: both the basic and the derived rules are allowed. (a) (S(a, b) ∨ P ), (P ∨ S(b, a)), ¬P ` (S(a, b) ∧ S(b, a)) (b) (a = b ∨ (b = c → a = b)) ` (¬a = b → ¬b = c) (c) (G(b) ∧ (G(a) ∨ G(c))) ` ((G(b) ∧ G(a)) ∨ (G(b) ∧ G(c))) (d) (R(a, b) → R(b, c)), (R(a, b) ∧ R(b, a)) ` (R(a, b) → (R(b, c) ∧ R(b, a)))

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1.
a. False
b. True
c. False
d. True
e. False
f. True
g. False
h. True

2.
a.
Domain: All the numbers.
a: 2
F(x): _____x is even.
G(x): 3, 5, 7, 10...
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