Consistency — Definition, Formula & Examples

Consistency is the property of a set of statements (or a formal system) in which no contradiction can be derived. If you can never prove both a statement and its negation from the same set of assumptions, that set is consistent.

A set of formulas $\Gamma$ is consistent if and only if there exists no formula $\varphi$ such that $\Gamma \vdash \varphi$ and $\Gamma \vdash \neg\varphi$ . Equivalently, $\Gamma$ is consistent if $\Gamma \nvdash \bot$ , where $\bot$ denotes a contradiction (falsehood).

How It Works

To check whether a set of statements is consistent, you look for whether any contradiction can be logically derived from them. If you assume a collection of axioms and, through valid inference rules, arrive at both

P

and

\neg P

for some proposition

P

, the system is inconsistent. An inconsistent system is considered trivial in classical logic because the principle of explosion (

\textit{ex falso quodlibet}

) allows you to derive any statement whatsoever from a contradiction. This is why consistency is a minimal requirement for any useful formal system.

Example

Problem: Determine whether the following set of statements is consistent: (1) All mammals are warm-blooded. (2) Whales are mammals. (3) Whales are not warm-blooded.

Step 1: From statements (1) and (2), derive a conclusion using modus ponens: since all mammals are warm-blooded and whales are mammals, whales are warm-blooded.

\text{Mammal}(\text{whale}) \land (\forall x)(\text{Mammal}(x) \to \text{WarmBlooded}(x)) \vdash \text{WarmBlooded}(\text{whale})

Step 2: Statement (3) asserts the negation of this conclusion.

\neg\text{WarmBlooded}(\text{whale})

Step 3: We have derived both WarmBlooded(whale) and ¬WarmBlooded(whale) from the set, which is a contradiction.

\text{WarmBlooded}(\text{whale}) \land \neg\text{WarmBlooded}(\text{whale}) \equiv \bot

Answer: The set of statements is inconsistent because it entails a contradiction.

Why It Matters

Consistency is a foundational requirement in any axiomatic system — from Euclidean geometry to set theory (ZFC). Gödel's second incompleteness theorem famously shows that sufficiently powerful consistent systems cannot prove their own consistency, a result central to mathematical logic and the philosophy of mathematics. In computer science, consistency checks underpin database theory and formal verification of software.

Common Mistakes

Mistake: Confusing consistency with truth. Students sometimes assume that a consistent set of axioms must describe something "real" or that all its statements are true.

Correction: Consistency only means no contradiction can be derived. A consistent system can describe entirely fictional or abstract structures. Truth depends on interpretation; consistency is a purely syntactic (or structural) property.