Implication

**Gregg** · Jun28-09, 07:36 PM

The truth table for

$LaTeX Code: A\\Rightarrow B$

Means If A then B else Not B?

But the truth table is supposedly looking like this:

$LaTeX Code: \\begin{array}{ccc} A & B & A\\Rightarrow B \\\\ T & T & T \\\\ T & F & F \\\\ F & T & T \\\\ F & F & T \\end{array} $

Only problem I have is A=F and B=T? Why is this ?

**HallsofIvy** · Jun28-09, 08:20 PM

I like to think of it as "innocent until proven guilty"! $LaTeX Code: A\\Rightarrow B$ means "if A is true then B is true". It doesn't say what happens if A is false. So we can logically take it either way and we choose to say it is true.

Suppose your professor tells you "If you get an "A" on every test, I will give you an "A" for the course" and then

1) You get an "A" on every test and you get an "A" for the course. Was he telling the truth? Of course he was.

2) You get an "A" on every test and you do NOT get an "A" for the course. Was he telling the truth? No, of course not.

3) You, say, fail every test and do NOT get an "A" for the course. Was he telling the truth? Actually, you can't know since you haven't "tested" what would have happened if you had gotten an "A" on every test. But I could see no reason for accusing your professor of lying.

4) You get an A on every test except one (on which you get a "B") and you get an "A" for the course. Was he telling the truth? Again, he didn't say what would happen if you didn't get an "A" on every test- and it would be very foolish of you to go to your professor and complain! Again, he is "innocent until proven guilty".

**g_edgar** · Jun29-09, 08:48 AM

Originally Posted by Gregg

The truth table for

$LaTeX Code: A\\Rightarrow B$

Means If A then B else Not B?

But the truth table is supposedly looking like this:

$LaTeX Code: \\begin{array}{ccc} A & B & A \\Rightarrow B \\\\ T & T & T \\\\ T & F & F \\\\ F & T & T \\\\ F & F & T \\end{array} $

Only problem I have is A=F and B=T? Why is this ?

The other one,

$LaTeX Code: \\begin{array}{ccc} A & B & A ? B \\\\ T & T & T \\\\ T & F & F \\\\ F & T & F \\\\ F & F & T \\end{array} $

also has a name, it is called $LaTeX Code: A \\Leftrightarrow B$ or "A if and only if B". This one really is "if A then B else not B"

**Gregg** · Jul1-09, 12:08 AM

Originally Posted by g_edgar

The other one,

$LaTeX Code: \\begin{array}{ccc} A & B & A ? B \\\\ T & T & T \\\\ T & F & F \\\\ F & T & F \\\\ F & F & T \\end{array} $

also has a name, it is called $LaTeX Code: A \\Leftrightarrow B$ or "A if and only if B". This one really is "if A then B else not B"

thanks