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  "body": "\n## Summary\n\nIn deductive logic, a **consistent** theory is one that does not lead to a logical contradiction. A theory $T$ is consistent if there is no formula $\\varphi$ such that both $\\varphi$ and its negation $\\lnot\\varphi$ are elements of the set of consequences of $T$. Let $A$ be a set of closed sentences (informally \"axioms\") and $\\langle A\\rangle$ the set of closed sentences provable from $A$ under some (specified, possibly implicitly) formal deductive system. The set of axioms $A$ is **consistent** when there is no formula $\\varphi$ such that $\\varphi \\in \\langle A \\rangle$ and $\\lnot \\varphi \\in \\langle A \\rangle$. A *trivial* theory (i.e., one which proves every sentence in the language of the theory) is clearly inconsistent. Conversely, in an explosive formal system (e.g., classical or intuitionistic propositional or first-order logics) every inconsistent theory is trivial. Consistency of a theory is a syntactic notion, whose semantic counterpart is satisfiability. A theory is satisfiable if it has a model, i.e., there exists an interpretation under which all axioms in the theory are true. This is what *consistent* meant in traditional Aristotelian logic, although in contemporary mathematical logic the term *satisfiable* is used instead.\n",
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