First Order Predicate Logic

Question

I would like some help to check my solution for the translation of a sentence into predicate logic.

The sentence is given as:

Every patient was seen by a cardiologist

I thought of this:

$ ∀x∃y(P(x) ∧ S(x, C(y)) $

where:

P(x): x is a patient
S(x, y): x was seen by y
C(y): y is a cardiologist

Additionally, is it allowed to add a $C(y)$ inside another function as I did above in $S(x, C(y))$?

Thanks in advance

Answer 1

Your statements does not imply "Every patient was seen by a cardiologist".

It implies something like "Everyone is patient and a cardiologist sees them".

The reason is using "$\land$" in this, you should use $\implies$ like this:

$\forall x \exists y \big(P(x) \implies (C(y) \land S(x,y)) \big)$

This implies that" Everyone who is a patient was seen by one who is a cardiologist".