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 π are π" from "Only π are π." 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 π are π:
Only π are π = All π are π
I understand the note to ask for a similar translation principle.
I'm not sure that I entirely understand how only π are π are to be distinguished from the only π are π. When considering the above statement about senators, I might symbolise it like this:
βx(Tx β 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 π are π 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.
Best Answer
Ah, rubber-ducking apparently works with mathematics as well! At least, just after hitting the post button, another symbolisation occurred to me:
Translating back into words, no
xexists such thatxwas expelled, andxwasn't found guilty of treason.Does that seem reasonable?