Distinguish 'the only f are g' from 'only f are g' in predicate logic

Question

I'm trying to teach myself some predicate logic by reading Howard Pospesel's Predicate Logic and doing the exercises.

In chapter 2, exercise 7 contains some optional, challenging exercises, and one of them has me a bit stumped. In its entirety, the exercise is this:

(e) (newspaper) "The only senators ever (e)xpelled were those found guilty of (t)reason." (Ex = x is an expelled senator, Tx = x is found guilty of treason) (Note: Distinguish "The only are " from "Only are ." Can you formulate a translation principle for statements such as (e)?)

The newspaper in brackets in the beginning is just the original source of the text. That's how the book writes the problems. Likewise, the bracketed letters are suggestions for the predicate symbol, which in this exercise is further expanded to an explicit dictionary.

It's the note that confuses me. The book has previously formulated a translation principle for statements of the type only are :

Only are = All are

I understand the note to ask for a similar translation principle.

I'm not sure that I entirely understand how only are are to be distinguished from the only are . When considering the above statement about senators, I might symbolise it like this:

∀x(Tx → Ex)

Meaning that for all x, if x was found guilty of treason, then x was expelled.

I have doubts about this answer, though, because that's the exact same way that one symbolises only are statements, and the note seems to indicate that it's not the same.

English is my second language, so it may be that there's some linguistic subtlety that escapes me.

If anyone can make this clearer for me, I'd appreciate it.

Answer 1

Long comment

Why

Only are = All are ?

Because "Only are " means that there is no that is not an (only women are mothers, because there is no mother which is not a woman).

And "there is no that is not an " is $\lnot \exists x (Gx \land \lnot Fx)$.

But the last one is equivalent to : $\forall x \lnot (Gx \land \lnot Fx)$ and in turn $\lnot (P \land \lnot Q)$ is equivalent to $(P \to Q)$.

Conclusion :

"Only are " can be symbolized ad : $\forall x (Gx \to Fx)$, that reads : "All are ".

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What about "The only are " ?

It is dubtful how to read it... but we can try to follow the author's suggestion to distinguish it form the previous one.

"The only senators ever (e)xpelled were those found guilty of (t)reason" means that the senators expelled (if any) were those guilty of treason (which does not implies that every senator guilty of treason has been expelled...)

This is :

$\forall x (Ex \to Tx)$

that, by the same equivalences above, is (as you said) is equivalent to : $\lnot \exists x (Ex \land \lnot Tx)$.

Answer 2

Ah, rubber-ducking apparently works with mathematics as well! At least, just after hitting the post button, another symbolisation occurred to me:

-∃x(Ex & -Tx)

Translating back into words, no x exists such that x was expelled, and x wasn't found guilty of treason.

Does that seem reasonable?