# The Biconditional Truth Table

Hi Mr. Nance!

I had a student ask me a question about the biconditional in Lesson 5 of Intermediate Logic the other day that had not occurred to me. She asked for written explanation of why if both sides of the biconditional were false then the truth table was true. Conceptually I understood it because of the definition of equivalency and from working through the truth tables, but verbally, I could not give her an example. Could you give a verbal example of each possibility of true and false like you did for the conditional, disjunction, and conjunction? I told her just to memorize the truth values, but honestly it would make more sense if I could give her an example that would explain why. Thanks!

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Some of the difficulty in producing examples for the biconditional truth table stem from similar difficulties with the conditional, such as why a conditional with a false antecedent and a true consequent is considered true; e.g. why is it true that “If a triangle has four sides then the Seahawks lost Super Bowl XLIX”?

But let me plunge ahead boldly and try to give some examples that might help. Consider the true statements “I am a parent” and “I have children” and the false statements “I am a Martian” and “I am from Mars.”

T ≡ T:  “I am a parent if and only if I have children.” This is clearly true.
T ≡ F:  “I am a parent if and only if I am from Mars.” Clearly false.
F ≡ T:  “I am a Martian if and only if I have children.” Clearly false.
F ≡ F:  “I am a Martian if and only if I am from Mars.” Also clearly true.

Now, to make that work neatly, I had to use pairs of statements that I knew were equivalent. Other non-equivalent statements could be used, but the truth values might only make sense if you kept in mind the fact that “if p then q” is defined as “not both p and not q.”

Blessings!