Is this translation correct?

Question

If I say some real numbers are rational it can be denoted in first order logic,

$(\exists x)$ $(real(x) \land rational(x))$

Where $real(x)$ - x is a real number.

$rational(x)$ - x is a rational number.

But if I say all the computers are fast, it denotes by,

$(\forall x)$ $(computer(x) \to fast(x))$

Where $computer(x) $ - x is a computer.

$fast(x)$ - x is fast.

So I want to know we can use implication in second example because that says "for all"? If there was existential quantifier should I use "and" for "implies"?

Please someone tell me the reason behind using implication there.

Answer 1

My answer here should clarify the meaning of quantifiers for you, and also help you answer your question here. It is not a matter of anyhow choosing some symbols. The symbols are supposed to make perfect sense!

When you want to say "Some real number is rational", note that it means the same thing as "Something is both a real number and a rational number" (think about it and you will agree), and hence also "Some rational number is real". That is precisely why it corresponds to:

$\exists x\ ( real(x) \land rational(x) )$.

Similarly, when you want to say "Every computer is fast", it means the same as "Given anything, if it is a computer then it is fast". Why? Again you can simply think through it to convince yourself that they are the same, but for this case I want to add a note. You see, the former concerns only computers, right? That means that you don't care about non-computers. Now the latter is on the surface about everything, but the inner conditional statement only concerns computers too! So it is:

$\forall x\ ( computer(x) \rightarrow fast(x) )$.

Got it?

For your question hiding in your comment about:

Every person has at least one secret which is not shared by that person.

Your attempt is incorrect. You clearly intend "$secret$" to be a property, so "$secret(y)$" is a true-false statement about $y$. So "$shared(secret(y),x)$" makes no sense. Instead you would want "$sharedby(y,x)$". (I've also changed the name to better reflect the intended meaning.) Furthermore, it is technically not right to put the quantifiers all at the front. They are supposed to reflect the structure of what you want to assert!

For any person $x$:

  There is some secret $y$:

    $y$ is not shared by $x$.

Translate accordingly:

$\forall x\ ( person(x) \rightarrow$

  $\exists y\ ( secret(y) \land$

    $\cdots$

  $)$

$)$