Originally posted by sonhouse
Can you explain the symbols you used in your logic exercise, what exactly they mean? I haven't studied logic and don't know what they mean. I think the ~ sign means 'not'? That's about as far as I got.
P -> Q
Not-Q.
Therefore, not-P.
I thought in math the > symbol meant the symbol on the left is greater than the symbol directly to the right of it but it seems to mean something different here.
Sure thing. P, Q and R, are propositional variables. They stand for propositions ('God exists' 'I will punished in Hell' 'I am evil'😉 but it is not necessarily important to know the propositional content of these variables is. ~ is an operator which you rightly note means 'not'. --> is a conjunction meaning 'if'. The syntax works like this, P-->Q means 'If P, then Q'; ~P-->Q means 'If not-P, then Q'; ~(P-->Q) means 'It is not the case that if P, then Q'.
It is important to understand for this exercise that a conditional P-->Q is true in either two ways: if P is false or Q is true. This may sound strange because we think of a conditional as being true only if Q follows as a logical consequence of P. But think of it this way, a conditional is only false if the antecedent, P, is true and the consequent is false. So a truth table is like this where 0=false, 1=true.
P Q P-->Q
1 1 1
1 0 0
0 1 1
0 0 1
Only in one of four cases is the conditional P-->Q false.
Now in LemonJello's case the logic works like this:
(1) ~P --> ~(Q-->R)
(2) ~Q
(3) P
~Q, therefore Q-->R (if Q is false, then the conditional is trivially true, as in the table above).
So ~(Q-->P) is false.
But from (1) if ~(Q-->P) is false, then the conditional could only still be true if ~P is also false. So ~~P. hence P, God exists.
Clearly this is a failure of classical logic and we would basically want to reject triviality like ~Q, therefore Q-->R.