# How do you represent the following sentences in first-order logic?

Represent the following sentences in first-order logic, using a consistent vocabulary(which you must define):

a) Some students took French in spring 2001.
b) Every student who takes French passes it.
c) Only one student took Greek in spring 2001.
d) The best score in Greek is always higher than the best score in French.
e) Every person who buys a policy is smart.
f) No person buys an expensive policy.
g) There is an agent who sells policies only to people who are not insured.

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For a) you can write something like $\exists x\; \operatorname{Student}(x) \land \operatorname{FrenchInSpring2001}(x)$ – martini Oct 1 '12 at 9:30
Have you just pasted in your homework assignment??? This site isn't here to get someone else to do your thinking for you. If you are in genuine difficulties understanding how to render simple English sentences into QL then you need to look at some more presentations in standard logic books. Paul Teller's very nice Primer is now freely available at tellerprimer.ucdavis.edu (or there's my Intro to Formal Logic which spends a lot of time on this). – Peter Smith Oct 1 '12 at 11:20
@ Peter Smith well okay, i'll keep that in mind. Thanks for the references- and by the way, it's not my homework assignment, just something i do out of enthusiasm. – Ghost Oct 1 '12 at 12:35
@Ghost Obviously I have a nasty suspicious mind :-) Teller's Primer has lots of worked examples, and also then exercises with the answers to the exercises available on line. (My book too has lots of worked examples, answers to the exercises available on line, and should be in most libraries.) – Peter Smith Oct 1 '12 at 12:56

Let me address the first three sentences (since they form a maximal coherent collection of sentences).

We have a use for the following five predicates:

• $\mathsf{Student}(x)$: $x$ is a student;
• $\mathsf{French}(y), \mathsf{Greek}(y)$: $y$ is the course in French, resp. Greek;
• $\mathsf{Take}(x,y)$: $x$ (student) takes $y$ (course);
• $\mathsf{Pass}(x,y)$: $x$ (student) passes $y$ (course);
• $\mathsf{TakeInSpring2001}(x,y)$: $x$ (student) takes $y$ (course) in Spring 2001.

Now we can formulate the sentences as follows:

a) $\exists x\exists y: \mathsf{Student}(x) \land \mathsf{French}(y) \land \mathsf{TakeInSpring2001}(x,y)$

b) $\forall x\forall y: (\mathsf{Student}(x) \land \mathsf{French}(y) \land \mathsf{Take}(x,y)) \implies \mathsf{Pass}(x,y)$

c) $\exists x\exists y\forall z: \mathsf{Student}(x) \land \mathsf{Greek}(y) \land \mathsf{TakeInSpring2001}(x,y) \land ((\mathsf{Student}(z) \land \mathsf{TakeInSpring2001}(z,y))\implies x = z)$

The other five shouldn't be hard once one understands the above three. A different approach would be to use more specific predicates (e.g. $\mathsf{FrenchInSpring2001}(x)$ as suggested in the comments), but that would defeat the "consistent vocabulary" requirement of the question IMO.

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Your (b) is wrong, it implies students who do not take French also pass French. – user282882 Oct 22 '15 at 16:56
@YuxiangZhang Thanks, fixed. – Lord_Farin Oct 22 '15 at 21:41