how come false AND false is true?

**Truthbetold** · September 23rd 2010, 08:09 PM

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!

**undefined** · September 23rd 2010, 08:49 PM

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

**Truthbetold** · September 23rd 2010, 09:12 PM

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?

**undefined** · September 23rd 2010, 09:16 PM

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".

**Soroban** · September 23rd 2010, 10:25 PM

Hello, Truthbetold!

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: . $\begin{Bmatrix} T\leftrightarrow T\, \text{ is true.} \\ F \leftrightarrow F\,\text{ is true.} \end{Bmatrix}$

**tonio** · September 24th 2010, 07:41 AM

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

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