Well, the way I did my truth tables for the second question neitehr "if/then" nor "If and only if" seem to be working...
I had already determined that neither "and" nor "or" would work because p#P had to be a tautology (# being my mysterious connective...)...
P#P has to be a tautology, while Q#P is not a tautological consequence of the sentences P#Q and P...
In a truth table for
([P->Q] ^ P] -> [Q -> P]
I got all T's, as I did when I replaced the "if/then" with "if and only if" which makes them both tautological consequences...
Horace, I underestimated the question and may have misled you. For that you have my deep, sincere, heartfelt apology. Sorry.
I incorrectly assumed that the mystery connective ("#") stood in place for one of the commonly used connectives of propositional logic. Propositional logic commonly uses four binary connectives (and, or, if/then, iff) and one unary connective (not).
But there are technically a total of four unary connectives and sixteen binary connectives, as in boolean algebra.
So it appears that your assignment is to find the meaning of the mystery connective ("#") from all the possible binary connectives
, not just the four commonly used in propositional logic.
It seems that your mystery connective is that of converse implication, which has the form:
", which is equivalent to P v ~Q
or in the case of
", it is equivalent to P v ~P
I will leave it to you to work out the truth table for the other part of the problem, but it does check out.
And since R is a logical consequence
Your first post says that 'A v ~A' is a logical consequence of R, not that R is a logical consequence of 'A v ~A'.
But I don't see what the point of putting this on a tree would be...
The law of the excluded middle is foundational in bivalent classical propositional logic. The point of the question is probably to show that the "P v ~P" form follows from any sentence in propositional logic, and that the sentence tableau method has a specific rule just to allow this in a truth tree, to show that any sentence of the form "P v ~P" follows from any other sentence in propositional logic