(related to Lemma: Every Proposition Implies Itself)

We want to show that for every proposition \(x\) the implication \(x\Rightarrow x\) is a tautology. This can be easily verified: By definition of implication, any possible semantics of \(A\) makes the compound proposition \(x \Rightarrow x\) a tautology:

\([[x]]_I\) | \([[x]]_I\) | \([[x \Rightarrow x]]_I\) |
---|---|---|

\(1\) | \(1\) | \(1\) |

\(0\) | \(0\) | \(1\) ∎ |