Search results
Results from the WOW.Com Content Network
A set of sentences is called a theory; thus, individual sentences may be called theorems. To properly evaluate the truth (or falsehood) of a sentence, one must make reference to an interpretation of the theory. For first-order theories, interpretations are commonly called structures. Given a structure or interpretation, a sentence will have a ...
First-order logic uses quantified variables over non-logical objects, and allows the use of sentences that contain variables. Rather than propositions such as "all men are mortal", in first-order logic one can have expressions in the form "for all x , if x is a man, then x is mortal"; where "for all x" is a quantifier, x is a variable, and "...
In mathematical logic, a first-order language of the real numbers is the set of all well-formed sentences of first-order logic that involve universal and existential quantifiers and logical combinations of equalities and inequalities of expressions over real variables.
Let ψ be a sentence in first-order logic.The spectrum of ψ is the set of natural numbers n such that there is a finite model for ψ with n elements.. If the vocabulary for ψ consists only of relational symbols, then ψ can be regarded as a sentence in existential second-order logic (ESOL) quantified over the relations, over the empty vocabulary.
A relationship between two structures in logic and mathematics where they satisfy the same first-order sentences. elimination of quantifiers A process in logical deduction where quantifiers are removed from logical expressions while preserving equivalence, often used in the theory of real closed fields. elimination rule
Propositional logic, as currently studied in universities, is a specification of a standard of logical consequence in which only the meanings of propositional connectives are considered in evaluating the conditions for the truth of a sentence, or whether a sentence logically follows from some other sentence or group of sentences.
Logical systems extending first-order logic, such as second-order logic and type theory, are also undecidable. The validities of monadic predicate calculus with identity are decidable, however. This system is first-order logic restricted to those signatures that have no function symbols and whose relation symbols other than equality never take ...
Then sentences that were second-order become first-order, with the formerly second-order quantifiers ranging over the second sort instead. This reduction can be attempted in a one-sorted theory by adding unary predicates that tell whether an element is a number or a set, and taking the domain to be the union of the set of real numbers and the ...