Maximal consistency proof for set of propositional logic with specific restriction?

Question

I ran into struggle when I comes to one sentence on logic.

Why the set of all propositional that under any valuation has value 1 is not maximal consistent ?

I read it on my books, without adding detail as a result.

Answer 1

If I interpret your question correctly, isn't this just that your set of sentences, $\Gamma$, contains non-controversial sentences (valuated always true). Now take a sentence, say $A \ne B$, or $\exists x \exists y (x \ne y)$, depending if we're talking about propositional or first-order logic. Then there's a way to make valuation or interpretation true or false, proving that the sentence is not in $\Gamma$ but also allowing it to be extended consistently.

Answer 2

Let $\Gamma$ be the set of all logical tautologies(i.e. formulas which are evaluated to 1 under every interpretation), and A a propositional variable. The formula A is no logical tautology, so $A \notin \Gamma$ but $\Gamma \cup \{A\}$ is consistent.