+Marko Savic It's exactly the same thing. Remember that the proofs are basically conditional proofs (well, actually, you will get to that later) where we assume that the premises are true and see what follows as well. Put differently, IF premises THEN conclusion. So you have P arrow Q and P as premises ((P arrow Q) ^ P) and have Q as a conclusion. Or ((P arrow Q)^P) arrow Q. Clear?
Take a look at the circle diagram we saw at 4:05. For P->Q to be true, P has to be inside Q. For the second row of the truth table, It says a dot is inside P but not inside Q, which cannot be possible because P is supposed to be inside Q. Therefore P->Q is false. For the third row, it says a dot is not inside P but inside Q, which can be possible because the Q circle is bigger than P so a dot can be outside P but inside Q. Therefore P->Q is true. For the fourth row, it says a dot is not inside P nor Q, so the dot is irrelevant to P->Q, therefore P->Q can be true.
I will give you a short intuitive explanation. The expression p->q simply affirms q. So in the table, whenever q is true (T) then the expression p->q will be true. And the only other time it can be true is when p and q are both false (F) since we are simply affirming something that we know is false.
With modus ponens, can i interchange premise 1 and premise 2? Like Premise 1: I am Miley Cyrus. Premise 2: If I am Miley Cyrus, then I am crazy. thanks
The first two columns are the premises which we are using to prove that the third column, the conclusion, is true. -F and -T is a separate premise to F and T, so -F and -T should be a separate premise column. We didn't add that column because the conclusion does not contain -F or -T therefore we have no need to add it in as a column.
@@ChristopherKim I think that is close! I think in this case, the first two columns (P and P => Q) are the premises, and then Q is the conclusion. I am guessing that William wrote the table that way so that the simple sentences P and Q would be in the leftmost columns.
First of all, saying I do not believe x exists is not really an "if then" statement. So logical notation would just be ~b (not believe in x). Saying "you believe/not believe" simply means "it is the case/not the case" So, ~b ~b Is equivalent. For future reference, it's somewhat ambiguous to use the word "believe" especially when we're dealing with logic, because logic doesn't measure opinion just true and false.
