# how come false AND false is true?

• Sep 23rd 2010, 09:09 PM
Truthbetold
how come false <--> false is true?
For truth tables and the biconditional proposition operator, how comes false <--> false is true?

It seems counter intuitive, if I may make a stab at sounding mathematical.

Thanks!
• Sep 23rd 2010, 09:49 PM
undefined
Quote:

Originally Posted by Truthbetold
For truth tables and the AND operator, how comes false AND false is true?

It seems counter intuitive, if I may make a stab at sounding mathematical.

Thanks!

false AND false is false
• Sep 23rd 2010, 10:12 PM
Truthbetold
Yeah....
I'm getting loopy or something. That's the second time today I haven't made sense.

I confused AND with the biconditional proposition.

If I may rephrase my question, how come the truth value of p <--> q where p = q = false is true?
• Sep 23rd 2010, 10:16 PM
undefined
Quote:

Originally Posted by Truthbetold
Yeah....
I'm getting loopy or something. That's the second time today I haven't made sense.

I confused AND with the biconditional proposition.

If I may rephrase my question, how come the truth value of p <--> q where p = q = false is true?

Biconditional indicates logical equivalence. Note that p <--> q is the same as (p -> q) AND (q -> p). Other ways you might hear it described are "if and only if", "is a necessary and sufficient condition for", and "is equivalent to".
• Sep 23rd 2010, 11:25 PM
Soroban
Hello, Truthbetold!

Quote:

For truth tables and the biconditional proposition operator,
how come $F \leftrightarrow F$ is true?

The biconditional $p \leftrightarrow q$ means that $\,p$ and $\,q$ have the same truth value.

Hence: . $\begin{Bmatrix} T\leftrightarrow T\, \text{ is true.} \\ F \leftrightarrow F\,\text{ is true.} \end{Bmatrix}$
• Sep 24th 2010, 08:41 AM
tonio
Quote:

Originally Posted by Truthbetold
Yeah....
I'm getting loopy or something. That's the second time today I haven't made sense.

I confused AND with the biconditional proposition.

If I may rephrase my question, how come the truth value of p <--> q where p = q = false is true?

I think the real problem here, and the one on which yours lays, might be: why the value of $A\rightarrow B$ is true

whenever the value of $A$ is false, no matter what the value of $B$ is?

The short answer is that it is so because it is defined so. Period. The long answer may pass through all kinds

of explantions including what truth really is and stuff. You may want to grab a book with some history of logic in it and read it.

Tonio