# Predicate Logic Problem

Given the following predicates:

Truck(X): X is a truck

Person(X): X is a person

IsBlue(X): X is of blue color

Like(X,Y): X likes Y

Expensive(X): X is expensive

Write following statement in predicate logic (In each case Ross and Rachel are persons)

a) Ross likes all blue trucks

b) Rachel does not like any blue trucks at all

c) Ross and Rachel both like the same blue truck (not all blue trucks)

d) No person likes expensive blue trucks.

Can anyone help?

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I can't really give any hints for your concrete examples without solving them. To demonstrate, I will formalize three different statements:

Alice likes Bob: $\mathrm{Like}(\mathrm{Alice}, \mathrm{Bob})$

Alice likes some expensive trucks: $\exists t: \mathrm{Truck}(t) \wedge \mathrm{Expensive}(t) \wedge \mathrm{Like}(\mathrm{Alice}, t)$, read: there is a $t$ which is a truck and expensive, and Alice likes it. This means that there is at least one expensive truck that Alice likes.

Alice does not like blue persons: $\forall p: (\mathrm{Person}(p) \wedge \mathrm{IsBlue}(p)) \to \neg \mathrm{Like}(\mathrm{Alice}, p)$, read: if $p$ is a person and blue, this implies that is not liked by Alice. In other words, if $p$ is any blue person, Alice does not like it.

I hope this helps you.

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For the first statement should it not be Alice likes some expensive truck not "trucks" as trucks would be more than one truck? –  hmmmm Mar 21 '12 at 14:41
If you want to be precise, yes. But then, we need to discuss how many objects constitute "some"... –  Johannes Kloos Mar 21 '12 at 14:50
Well I thought it might be a small source of confusion for the op that's all –  hmmmm Mar 21 '12 at 14:51

So here's what I came up with

a. ∀ t : Truck(t) ∧ IsBlue(t) ∧ Like(Ross,t)

b. ∀t : (Truck(t) ∧ IsBlue(t)) → ¬Like(Rachel,t))

c. ∃t : Truck(t) ∧ IsBlue(t) ∧ Like(Ross,t) ∧ Like(Ross,t)

d. ∀t and ∀p : Person(p) ∧ ((Truck(t) ∧ Expensive(t) ∧ isBlue(t)) → ¬Like(p,c))

I'm pretty sure I got a and c, but the last two I was not sure. For C, should it be all ANDs connecting the conditions or an if then statement in there somewhere.

Also, I wasn't sure for d if I need two ∀ or how to set it up for all trucks and all persons.

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Your (b) is right. Why would (a) and (d) be different from (b), aside from the negation? Why do you think they are not also implications? Right now, (a) says that everything is a blue truck that Ross likes, and (d) says that everything is a person, neither of which is what you want to say. You need to put some of these predications in an antecedent (or some equivalent construction). Also, there is a typo in (c) -- you're missing Rachel. –  Rachel Mar 22 '12 at 0:57