Example of a complete and consistent set of formulas in propositional logic

Question

So, I am aware that every set which is inconsistent is complete (every formula can be derived from it) and that a set is consistent if and only if it is satisfiable. But what is an example of consistent and complete set?

Answer 1

Let $\textit{Var}$ be the (finite or infinite) set of propositional variables in your propositional language.

This set of formulas is:

1. consistent: indeed, it is satisfiable, by the valuation $v$ that associates true with each element of $\textit{Var}$;

2. complete: for each formula $A$, either $A$ is consequence of $\textit{Var}$, or $\lnot A$ is a consequence of $\textit{Var}$; indeed, there is only one valuation that satisfies $\textit {Var}$ (the valuation $v$ that associates true with each element of $\mathit{Var}$) and for such a valuation either $A$ is true or $\lnot A$ is true).

See also here for a similar question.

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Is $\mathit{Var}$ the only consistent and complete set of formulas in propositional logic? Not at all! Consider the sets:

• $\mathit{Var}'$ which is like $\mathit{Var}$ except that the first propositional variable $X$ has been replaced by $\lnot X$;
• $\mathit{Var}''$ which is like $\mathit{Var}$ except that the second propositional variable $Y$ has been replaced by $\lnot Y$;
• $\mathit{Var}'''$ which is like $\mathit{Var}$ except that the first and the second propositional variables $X$ and $Y$ have been replaced by $\lnot X$ and $\lnot Y$, respectively;
• $\ldots$

An argument analogous to the one for $\mathit{Var}$ proves that $\mathit{Var}'$, $\mathit{Var}''$, $\mathit{Var}'''$, $\dots$ are consistent and complete set of formulas. Note that $\mathit{Var}$, $\mathit{Var}'$, $\mathit{Var}''$, $\mathit{Var}'''$, $\dots$ are pairwise non-equivalent (no one is a consequence no one else). Actually, it can be proved that, if $|\mathit{Var}|$ is the (finite or infinite) cardinality of the set $\mathit{Var}$, you have $2^{|\mathit{Var}|}$ different, non-equivalent consistent and complete sets of formulas.