Currently I'm learning some logic and while surfing the net I found this post that states:
"A samurai without a sword is like a samurai with the sword but without the sword"
So I was wondering if I could prove that the statement is a Tautology. But first I wasn't sure how to interpret the "is like". What I ended up doing is treating it as biconditional. So let P = Samurai with a sword then, the expression would be $\neg{P} \iff (P \wedge \neg{P})$. Using a truth table I was able to confirm this to be a tautology:
$$\begin{array}{c|c|c|c|c|} P & \neg{P} & P \wedge \neg{P} & \neg{P} \implies (P \wedge \neg{P}) & (P \wedge \neg{P}) \implies \neg{P} \\ \hline \text{T} & \text{F} & \text{F} & \text{T} & \text{T} \\ \hline \text{F} & \text{T} & \text{F} & \text{T} & \text{T} \\ \hline \end{array}$$
Is my reasoning correct? Should I treate the "is like" as biconditional
Update: I just realized that I was wrong and the statement is not a tautology. The table should be:
$$\begin{array}{c|c|c|c|c|} P & \neg{P} & P \wedge \neg{P} & \neg{P} \implies (P \wedge \neg{P}) & (P \wedge \neg{P}) \implies \neg{P} \\ \hline \text{T} & \text{F} & \text{F} & \text{T} & \text{T} \\ \hline \text{F} & \text{T} & \text{F} & \text{F} & \text{T} \\ \hline \end{array}$$
But I'm still not sure if my interpretation of the statement is correct.
Let $S(x): \text{x is a samurai}$, P(x): $\text{x has a sword}$.
My suggestion is:
$$\forall x, S(x) \implies (\neg P(x) \iff P(x) \land \neg P(x))$$
This is a tautology iff there are no samurais or if every samurai has got a sword (which even makes some sense).