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If a system of equations is inconsistent, then the equations cannot be true together leading to contradictory information, such as the false statements 2 = 1, or + = and + = (which implies 5 = 6). Both types of equation system, inconsistent and consistent, can be any of overdetermined (having more equations than unknowns), underdetermined ...
A system will be said to be inconsistent if it yields the assertion of the unmodified variable p [S in the Newman and Nagel examples]. In other words, the notion of "contradiction" can be dispensed when constructing a proof of consistency; what replaces it is the notion of "mutually exclusive and exhaustive" classes.
Conversely, in an explosive formal system (e.g., classical or intuitionistic propositional or first-order logics) every inconsistent theory is trivial. [ 2 ] : 7 Consistency of a theory is a syntactic notion, whose semantic counterpart is satisfiability .
In logic, the law of non-contradiction (LNC) (also known as the law of contradiction, principle of non-contradiction (PNC), or the principle of contradiction) states that contradictory propositions cannot both be true in the same sense at the same time, e. g. the two propositions "the house is white" and "the house is not white" are mutually exclusive.
The contradiction of a belief, ideal, or system of values causes cognitive dissonance that can be resolved by changing the challenged belief, yet, instead of affecting change, the resultant mental stress restores psychological consonance to the person by misperception, rejection, or refutation of the contradiction, seeking moral support from ...
The state of containing contradictory elements, which cannot all be true at the same time within a logical framework. inconsistent arithmetic An arithmetic system in which a contradiction can be derived, violating the principle of consistency. indefinite description
This motivated a great deal of research around the turn of the 20th century to develop a consistent (contradiction-free) set theory. In 1908, Ernst Zermelo proposed an axiomatization of set theory that avoided the paradoxes of naive set theory by replacing arbitrary set comprehension with weaker existence axioms, such as his axiom of separation ...
A set of axioms is (simply) consistent if there is no statement such that both the statement and its negation are provable from the axioms, and inconsistent otherwise. That is to say, a consistent axiomatic system is one that is free from contradiction. Peano arithmetic is provably consistent from ZFC, but not from within itself.