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A formal system S is syntactically complete or deductively complete or maximally complete or negation complete if for each sentence (closed formula) φ of the language of the system either φ or ¬φ is a theorem of S. Syntactical completeness is a stronger property than semantic completeness.
The completeness theorem applies to any first-order theory: If T is such a theory, and φ is a sentence (in the same language) and every model of T is a model of φ, then there is a (first-order) proof of φ using the statements of T as axioms. One sometimes says this as "anything true in all models is provable".
In mathematical logic, a theory is complete if it is consistent and for every closed formula in the theory's language, either that formula or its negation is provable. That is, for every sentence φ , {\displaystyle \varphi ,} the theory T {\displaystyle T} contains the sentence or its negation but not both (that is, either T ⊢ φ ...
The universal quantifier "for every" in this sentence expresses the idea that the claim "if x is a philosopher, then x is a scholar" holds for all choices of x. The negation of the sentence "For every x, if x is a philosopher, then x is a scholar" is logically equivalent to the sentence "There exists x such that x is a philosopher and x is
The converse of the soundness property is the semantic completeness property. A deductive system with a semantic theory is strongly complete if every sentence P that is a semantic consequence of a set of sentences Γ can be derived in the deduction system from that set. In symbols: whenever Γ ⊨ P, then also Γ ⊢ P.
President Donald Trump issued a sweeping series of pardons for defendants charged in the Jan. 6, 2021, attack on the U.S. Capitol, commuting the sentences of 14 individuals and offering a “a ...
In 1949 "The completeness of the first order functional calculus" [12] was published, as well as "Completeness in the theory of types" [27] in 1950. Both presented part of the results exposed in the dissertation "The completeness of formal systems" with which Henkin received his Ph.D. degree at Princeton in 1947. One of Henkin's best known ...
More than 1,580 defendants have been charged and about 1,270 have been convicted in a sprawling investigation that has resulted in more than 660 prison sentences, according to statistics released ...