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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 ...
In theories of arithmetic, such as Peano arithmetic, there is an intricate relationship between the consistency of the theory and its completeness.A theory is complete if, for every formula φ in its language, at least one of φ or ¬φ is a logical consequence of the theory.
Oxymorons in the narrow sense are a rhetorical device used deliberately by the speaker and intended to be understood as such by the listener. In a more extended sense, the term "oxymoron" has also been applied to inadvertent or incidental contradictions, as in the case of "dead metaphors" ("barely clothed" or "terribly good").
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.
It is inconsistent if it is not consistent. [ 66 ] [ 68 ] An inconsistent formula is also called self-contradictory , [ 1 ] and said to be a self-contradiction , [ 1 ] or simply a contradiction , [ 81 ] [ 82 ] [ 83 ] although this latter name is sometimes reserved specifically for statements of the form ( p ∧ ¬ p ) {\displaystyle (p\land ...
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
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 ...
A paradox is a logically self-contradictory statement or a statement that runs contrary to one's expectation. [1] [2] It is a statement that, despite apparently valid reasoning from true or apparently true premises, leads to a seemingly self-contradictory or a logically unacceptable conclusion.