p q p-->q
T T T
T F F
F T T
F F T
Why do the last two columns in the truth table equal True? This really does not make any sense to me.
Hello, IDontunderstand!
Plato is absolutely correct . . .
I too was puzzled when I first learned about Implication.
This is how my professor explained it.
Consider this scenario.
Your professor makes you this promise:
. . If you get 100% on the final exam, then you will get an "A" for the course.
(Imagine further than you don't trust him, so you get this in writing,
. . signed and notarized before witnesses!)
Under what circumstances has he broken his promise,
. . (and you can start a class-action suit)?
Consider the four possible outcomes:
(1) You got 100% on the Final and you got an "A" for the course.
. . .He kept his promise . . . no problem!
(2) You got 100% on the Final and you did not get an "A".
. . .He broke his promise . . . call your lawyer!
(3) You did not get 100% and you got an "A".
. . .Maybe you got "only" 99% and your average is still "A".
. . .No basis for a lawsuit.
(4) You did not get 100% and you did not get an "A".
. . .Maybe you got only 60% and your average dropped to a "B".
. . .Again, no basis for a lawsuit.
In brief: his promise involves only what would happen if you got 100% on the final.
No promises were made about what would happen if you did not get 100%.
And that is why .$\displaystyle \begin{Bmatrix} T \to T \\ F \to T \\ F \to F\end{Bmatrix}$ are all True.