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Is a biconditional necessarily a tautology? For instance, in the proposition (K--->N) iff (N--->K), "iff" is a type of equivalence, correct? So, if a tautology is also an equivalence, then this statement would be tautologous, would it not?

Sage Hopkins
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A biconditional, say, $K \longleftrightarrow N\equiv (K \rightarrow N) \land (N\rightarrow K)$ is true only when both $K$ and $N$ are true, or when both $K$ and $N$ are false.

However if one is true and the other false, the biconditional is not true. Hence, it is not true for all truth-value assignments for $K, N$. Hence, it is not a tautology.

We do have that $K \iff N \equiv (K\; \text{ XNor } N)$ and shares the following truth-table:

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amWhy
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  • Ok, that is a helpful explanation. Would it be accurate, then, to call the above proposition a contingency, given that the veracity of the proposition as a whole is contingent upon both sides being true? – Sage Hopkins Dec 17 '16 at 21:58
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    Exactly. It is indeed a contingency. However, $K \longleftrightarrow N$ is also true when both K and N are false. $K \longleftrightarrow N$ is true exactly when K and N share the same truth-value, and false otherwise. – amWhy Dec 17 '16 at 22:08