Consistency

Absurdum

It is sometimes convenient to have a sentence, usually denoted by "$\perp$" and called falsum, absurdity or contradiction, and which is false in all structures for the language $\mathcal{L}$. We define $\perp$ by $\perp :=\phi \wedge \neg \phi$ for any sentence $\phi$. For example, $\phi$ can be taken to be $\forall x(x=x)$.

Consistency

A set of sentences $\Sigma$ is consistent if a contradiction cannot be derived from $\Sigma$. That is, if $\Sigma /⊢\perp$. $\Sigma$ is inconsistent if $\Sigma ⊢\perp$.

Consistency, Equivalent Versions

Suppose $\Sigma$ is a set of sentences in the language $\mathcal{L}$. Then $\begin{array}{rl}\Sigma \text{ is consistent}& ⟺\text{for every sentence }\phi ,\Sigma ⊬\phi \text{ or }\Sigma ⊬\neg \phi \\ & ⟺\text{for some sentence }\psi ,\Sigma ⊬\psi \\ & ⟺\Sigma ⊬\perp .\end{array}$ The set $\Sigma$ is inconsistent if it is not consistent. That is, $\begin{array}{rl}\Sigma \text{ is inconsistent}& ⟺\text{for some sentence }\phi ,\Sigma ⊢\phi \text{ and }\Sigma ⊢\neg \phi \\ & ⟺\text{for every sentence }\psi ,\Sigma ⊢\psi \\ & ⟺\Sigma ⊢\perp .\end{array}$

Proof For some sentence $\phi ,\Sigma ⊢\phi \text{ and }\Sigma ⊢\neg \phi$ $⟹$for every sentence $\psi ,\Sigma ⊢\psi$. $\Sigma ⊢\phi$ and $\Sigma ⊢\neg \phi$, then for any sentence $\psi$ with propositional tautology $⊢\phi \to (\neg \phi \to \psi ),$ use modus ponens twice, we get $\Sigma ⊢\psi$ for every $\psi$. For every every sentence $\psi ,\Sigma ⊢\psi$, $⟹$ $\Sigma ⊢\perp$. If $\Sigma ⊢\psi$ for every $\psi$, then in particular $\Sigma ⊢\psi \wedge \neg \psi$ for every sentence $\psi$. So $\Sigma ⊢\perp$. $\Sigma ⊢\perp ⟺$ $\Sigma$ is inconsistent.

Properties of Consistency

Suppose $\Sigma$ is a set of sentences and $\sigma$ is a sentence.

1. $\Sigma$ is consistent iff every finite subset of $\Sigma$ is consistent;
2. If $\Sigma ⊢\sigma$ then $\Sigma \cup \{\sigma \}$ is consistent, but not conversely.
3. $\Sigma \cup \{\sigma \}$ is inconsistent iff $\Sigma ⊢\neg \sigma ,\Sigma \cup \{\sigma \}$ is consistent iff $\Sigma ⊬\neg \sigma$;
4. $\Sigma \cup \{\neg \sigma \}$ is inconsistent iff $\Sigma ⊢\sigma ,\Sigma \cup \{\neg \sigma \}$ is consistent iff $\Sigma ⊬\sigma$.

Maximal Consistency

A set of sentences $\Sigma$ is maximal consistent in the language $\mathcal{L}$ iff $\Sigma$ is consistent and has no proper consistent extension in $\mathcal{L}$. That is $\Sigma$ is consistent and $\Sigma \cup \{\sigma \}$ is consistent iff $\sigma \in \Sigma$.

Maximal Consistency

A set of sentences $\Sigma$ is maximal consistent iff $\Sigma$ is consistent and for each sentence $\sigma$ either $\sigma \in \Sigma$ or $\neg \sigma \in \Sigma$.