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A set of formulas is consistent if there exists an interpretation satisfying all of them simultaneously; it is inconsistent if no such interpretation exists. Consistency is essential for logical systems—an inconsistent set of axioms allows deriving any formula, making the system useless for sound reasoning.
You already know that a formula is satisfiable if some interpretation makes it true, and unsatisfiable (a contradiction) if none does. Consistency extends this idea from single formulas to sets: a set Σ of formulas is consistent if there is at least one interpretation that satisfies every formula in Σ simultaneously. It is inconsistent if no such interpretation exists — the formulas collectively impose contradictory demands on the world.
The simplest inconsistency arises from a direct contradiction: the set {P, ¬P} is inconsistent because any interpretation that makes P true makes ¬P false, and vice versa. But inconsistency can be more subtle. Consider {P → Q, P, ¬Q}: any interpretation making P true and Q false satisfies ¬Q but falsifies P → Q; any interpretation making Q true satisfies P → Q but falsifies ¬Q; you cannot satisfy all three at once. The set is inconsistent even though no single formula in it is a contradiction.
Why does inconsistency matter so much? Because of ex falso quodlibet — the principle of explosion: from a contradiction, you can derive any formula whatsoever. If an axiomatic system contains an inconsistency, every formula is provable in it, including both a statement and its negation. The system says everything and therefore says nothing useful. This is why consistency is the *minimum* requirement for any formal theory to be meaningful. When mathematicians worried in the early 20th century about the foundations of mathematics — leading to Gödel's incompleteness theorems, Hilbert's program, and Russell's type theory — the central concern was whether their axiom systems were consistent.
Consistency and satisfiability are deeply linked: by the completeness theorem (which you will encounter later), a set of first-order sentences is consistent — meaning no contradiction is derivable — if and only if it is satisfiable — meaning some interpretation makes all of them true. This equivalence is non-trivial and connects the syntactic notion of derivability with the semantic notion of truth. For propositional logic, you can check consistency by truth tables: build the combined truth table for all formulas in Σ and see whether any row makes every formula true. For infinite sets, you need deeper tools — but the concept is the same: does a coherent picture of the world exist in which all the claims hold?