Translation of sentence into predicate logic. "Greeks who fear Romans, fear only romans.".

Question

I am trying to figure out how to translate this sentence into predicate logic: "Greeks who fear Romans, fear only Romans".

R_ _ : _ Fear _

x: Greeks

y: Romans

z: Others

This is my crack at it, and I am almost certain that it is wrong.

$ (Rxy \implies (Rxy \land \lnot Rxz)) $

Help is greatly appreciated.

Answer 1

I am reading this sentence as: "The Greeks who fear Romans, only fear Romans ... but the Greeks who don't fear Romans, well, we don't have anything to say about them (so they may actually fear non-Romans)"

In other words, Greeks who fear at least some Romans fear only Romans (i.e. don;t fear any non-Romans), but Greeks who don't fear any Romans at all may or may not fear non-Romans.

As such, the basic form of the sentence is:

$$\forall x \ (x \text{ fears at least some Romans } \rightarrow x \text{ only fears Romans})$$

Now, to say that $x \text{ fears at least some Romans }$, you can use:

$$\exists y \ Rxy$$

and to say that $x \text{ only fears Romans}$, you can do:

$$\forall z (Rxz \rightarrow \exists y \ z=y)$$

So, plugging those in, you get:

$$\forall x \ (\exists y \ Rxy \rightarrow \forall z (Rxz \rightarrow \exists y \ z=y))$$