Why is A=F and B=T in the Truth Table for Implication?

[Gregg]

The truth table for

A\Rightarrow B

Means If A then B else Not B?

But the truth table is supposedly looking like this:

<br /> <br /> \begin{array}{ccc}<br /> A &amp; B &amp; A\Rightarrow B \\<br /> T &amp; T &amp; T \\<br /> T &amp; F &amp; F \\<br /> F &amp; T &amp; T \\<br /> F &amp; F &amp; T<br /> \end{array}<br />

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

 

[HallsofIvy]

I like to think of it as "innocent until proven guilty"! 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]

The truth table for

A\Rightarrow B

Means If A then B else Not B?

But the truth table is supposedly looking like this:

<br /> <br /> \begin{array}{ccc}<br /> A &amp; B &amp; A \Rightarrow B \\<br /> T &amp; T &amp; T \\<br /> T &amp; F &amp; F \\<br /> F &amp; T &amp; T \\<br /> F &amp; F &amp; T<br /> \end{array}<br />

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

The other one,

<br /> \begin{array}{ccc}<br /> A &amp; B &amp; A ? B \\<br /> T &amp; T &amp; T \\<br /> T &amp; F &amp; F \\<br /> F &amp; T &amp; F \\<br /> F &amp; F &amp; T<br /> \end{array}<br />

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

 

[Gregg]

The other one,

<br /> \begin{array}{ccc}<br /> A &amp; B &amp; A ? B \\<br /> T &amp; T &amp; T \\<br /> T &amp; F &amp; F \\<br /> F &amp; T &amp; F \\<br /> F &amp; F &amp; T<br /> \end{array}<br />

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

thanks