Biconditionals and Conjunctions in Truth Tables

Question

Given that a biconditional $p\iff q$ is True what can be concluded from the statement $\lnot p\land \lnot q$?

In a worded example:

I wear my running shoes if and only if I exercise. (True)

I am not exercising AND I am not wearing my running shoes. (?)

If we set up a truth table, the biconditional is True in two of the four occurrences, but we see that $\lnot p\land \lnot q$ is both True and False, which would mean there is no conclusion, correct?

Answer 1

1. $p\iff q$ means that $p$ and $q$ have identical truth values, which means that $\lnot p$ and $\lnot q$ have identical truth values, which means that $\lnot p \land \lnot q$ is a conjunction of propositions with identical truth vaulues (i.e., both true or both false).
2. $\lnot p \land \lnot q$ is, by definition, true precisely when both conjuncts are true.

So, $(2)$ entails that both $p$ and $q$ are false.

And, $(1)$ neither entails that $(2)$ is true nor that $(2)$ is false.