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For instance, for any two sentences A, B, the sentence A & B is satisfied if and only if A and B are satisfied (where '&' stands for conjunction), for any sentence A, ~A is satisfied if and only if A fails to be satisfied, and for any open sentence A where x is free in A, (x)A is satisfied if and only if for every substitution of an item of the ...
If φ has no free variables, that is, if φ is an atomic sentence, and it is satisfied by A, then one writes A ⊧ φ. In this case, one may also say that A is a model for φ, or that φ is true in A. If T is a collection of atomic sentences (a theory) satisfied by A, one writes A ⊧ T
A sentence is consistent if it is true under at least one interpretation; otherwise it is inconsistent. A sentence φ is said to be logically valid if it is satisfied by every interpretation (if φ is satisfied by every interpretation that satisfies ψ then φ is said to be a logical consequence of ψ).
The complete theory of a structure A is the set of all first-order sentences over the signature of A that are satisfied by A. It is denoted by Th(A). More generally, the theory of K, a class of σ-structures, is the set of all first-order σ-sentences that are satisfied by all structures in K, and is denoted by Th(K). Clearly Th(A) = Th({A ...
Löwenheim proved the existence of this cardinal for any logic in which the collection of sentences forms a set, using the following argument. Given such a logic, for each sentence φ, let κ φ be the smallest cardinality of a model of φ, if φ has any model, and let κ φ be 0 otherwise. Then the set of cardinals { κ φ : φ is a sentence in L}
Justice James Mellor hit Wright with a one-year prison sentence that was suspended for two years and ordered him to pay 145,000 British pounds in costs. ... I am entirely satisfied that Dr. Wright ...
The definition of tautology can be extended to sentences in predicate logic, which may contain quantifiers—a feature absent from sentences of propositional logic. Indeed, in propositional logic, there is no distinction between a tautology and a logically valid formula. In the context of predicate logic, many authors define a tautology to be a ...
In an encore “20/20” airing Dec. 27 at 9 p.m. ET, the show, which originally aired in 2023, tells the story of Julie Jensen, the mother of two who was found dead in her bed in 1998.
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