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Self-consistent Field
In classical deductive logic, a consistent theory is one that does not lead to a logical contradiction. The lack of contradiction can be defined in either semantic or syntactic terms. The semantic definition states that a theory is consistent if it has a model, i.e., there exists an interpretation under which all formulas in the theory are true. This is the sense used in traditional Aristotelian logic, although in contemporary mathematical logic the term ''satisfiable'' is used instead. The syntactic definition states 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 \varphi, \lnot \varphi \in \langle A \rangle for no formula \varphi. If there e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Classical Logic
Classical logic (or standard logic or Frege-Russell logic) is the intensively studied and most widely used class of deductive logic. Classical logic has had much influence on analytic philosophy. Characteristics Each logical system in this class shares characteristic properties: Gabbay, Dov, (1994). 'Classical vs non-classical logic'. In D.M. Gabbay, C.J. Hogger, and J.A. Robinson, (Eds), ''Handbook of Logic in Artificial Intelligence and Logic Programming'', volume 2, chapter 2.6. Oxford University Press. # Law of excluded middle and double negation elimination # Law of noncontradiction, and the principle of explosion # Monotonicity of entailment and idempotency of entailment # Commutativity of conjunction # De Morgan duality: every logical operator is dual to another While not entailed by the preceding conditions, contemporary discussions of classical logic normally only include propositional and first-order logics. Shapiro, Stewart (2000). Classical Logic. In Stanford ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
