Exercise $2.28$ Goldrei

Question

My question is about part $(b)-(i)$. Is there an easier way to solve it other than proving that for any truth assignment $v:Form(P,S) \rightarrow \{T,F\}$, $v^{**} = v$? thanks.

Answer 1

In some sense you can say that there is no simpler way, although technically you don't need the actual equality $v^{**} = v$, just that for every truth assignment $v$, there is some truth assignment $w$ such that $w^* = v$.

Expanding the definitions gives the proof:

$φ$ is a tautology

  iff ( $v(φ) = T$ for every truth assignment $v$ )

  iff ( $v^*(φ^*) = T$ for every truth assignment $v$ )

  iff ( $v(φ^*) = T$ for every truth assignment $v$ )   [(1)]

  iff ( $φ^*$ is a tautology ).

(1) is trivial for the backward implication but uses the earlier remark for the forward implication.