Show that if S is a proposition, where S is the conditional statement “If S is true, then unicorns live," then“Unicorns live” is true. Show that it follows that S can not be a proposition.
Here is my guess
<---> means equivalent
---> means implies
1. “If S is true, then unicorns live," then“Unicorns live” is true <---> S
2. Let: unicorns live <--> P
3. “If S is true, then unicorns live," <---> S ---> P
4. From 1. ( S ---> P) ---> P <---> S
5. But from 3. S--->P is true
6. from 5 & 4 P is true
7. But since S <---> P therefore S is true
8. So S is an assertion and can't be a proposition, which is required to proof
Hope this helps
I, on the other hand, don't understand what amrmuhammed said!
He seems to be saying S is an "assertion" (alway true? I would call that a "tautology") and so not a proposition. My understanding is that a "proposition" is any statement that is either true of false (regardless of whether we know which). A statement that is always true is certainly a proposition. I think, rather, that we need to look at it like this:
If S is true, then the statement "if S is true, then unicorns live" which IS S is a true statement. But then since whenever "if P then Q " is true and "P" is true, it follows that Q is true, we must have "unicorns live".
What about if S is false? The only way a conditional statement can be false is if the hypothesis is true and the conclusion false. But since the hypothesis is "S is true", that does not hold here. S cannot be false!
So far that is what amrmuhammed said. But the fact that S must be true does not mean it cannot be proposition!
But it does follow that, since S is true, the statement "unicorns live" is also true. And now we need one more fact: there are NO living unicorns! For that reason, the statement S cannot be true and since it also cannot be false, it cannot be a proposition at all.