QuestionQuestion

Since we have now done the full semantics for FOL, we can now say what it takes for any sentence of FOL to be true or false in an interpretation, and also, we can symbolize lots of English sentences in FOL, using the semantics of FOL and a symbolization key to re-produce the kind of dependence described on handout 2.5: the kind of dependence that leads to First-order validity.On the left below are eight FOL sentences. On the right is an interpretation that I provided.Please say whether these FOL sentences are true or false in the interpretation provided.

(a)(Q∨G(g))
(b)∃x(R(x, e))
(c)T(e, a)
(d)∀x(G(x)→F(x))
(e)∀yF(y)
(f )∃xR(x, x)
(g)∀x(¬x=a→¬G(x))
(h)(P→∀x∀yR(x, y))

Domain: All the animals in the world.
a: Justin Trudeau
e: The largest mouse in Vancouver.
g: Moosh, a cat owned by Jim’s friend
Sophie.
F(x):xis a mammal.
G(x):a, e
R(x, y):xis larger thany.
T(x, y):< a, e >
P: True
Q: False

And now, it is time to symbolize some things. Please symbolize the following English sentences in FOL, providing an appropriate symbolization key.
(a) Somebody in Burnaby is a liar and a cheater!
(b) Either everyone is foolish, or Justin Trudeau is an okay Prime Minister.
(c) Nobody knows Wilhelm Windelband.
(d) All of the movies that were made by Howard Hawks are good.

Here are some proofs. To do them, however, you will need the new Basic Rules that we will add to our system: the rules we did not have in when we were working with TFL. On Monday the 25th, we will talk about these new rules for FOL. If you’d like to try them before Monday, take a look at the six new rules on page 381 of Magnus and described in chapter 32.
(a)∀xG(x)∴∀yG(y)
(b)∀x(F(x)→x=a), F(b)∴a=b
(c)∀x∀y(R(x, y)∨R(y, x)),¬R(a, b)∴((F(a)→F(a))∧R(b, a))
(d)∀x(F(x)→G(x))∴∃x(¬F(x)∨G(x))

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1.
a. False
[because both sides of the disjunction are false, i.e. Q is false and G(g) is false because g is not something to which G applies]
b. True
[there is something larger than the largest mouse in Vancouver…presumably]
c. False
[the order of e and a is reversed]
d. True
[because JT and a mouse are mammals]
e. False
[because all the animals in the world are not mammals]
f. False...
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