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2. Sentence (mathematical logic) - Wikipedia

   en.wikipedia.org/wiki/Sentence_(mathematical_logic)

   In mathematical logic, a sentence (or closed formula) [1] of a predicate logic is a Boolean-valued well-formed formula with no free variables. A sentence can be viewed as expressing a proposition , something that must be true or false.

3. Closed-form expression - Wikipedia

   en.wikipedia.org/wiki/Closed-form_expression

   The quadratic formula =. is a closed form of the solutions to the general quadratic equation + + =. More generally, in the context of polynomial equations, a closed form of a solution is a solution in radicals; that is, a closed-form expression for which the allowed functions are only n th-roots and field operations (+,,, /).

4. Well-formed formula - Wikipedia

   en.wikipedia.org/wiki/Well-formed_formula

   A closed formula, also ground formula or sentence, is a formula in which there are no free occurrences of any variable. If A is a formula of a first-order language in which the variables v 1, …, v n have free occurrences, then A preceded by ∀v 1 ⋯ ∀v n is a universal closure of A.

5. Extension by new constant and function names - Wikipedia

   en.wikipedia.org/wiki/Extension_by_new_constant...

   In mathematical logic, a theory can be extended with new constants or function names under certain conditions with assurance that the extension will introduce no contradiction. Extension by definitions is perhaps the best-known approach, but it requires unique existence of an object with the desired property. Addition of new names can also be ...

6. Deductive closure - Wikipedia

   en.wikipedia.org/wiki/Deductive_closure

   In mathematical logic, a set ⁠ ⁠ of logical formulae is deductively closed if it contains every formula ⁠ ⁠ that can be logically ...

7. Quantifier (logic) - Wikipedia

   en.wikipedia.org/wiki/Quantifier_(logic)

   In logic, a quantifier is an operator that specifies how many individuals in the domain of discourse satisfy an open formula. For instance, the universal quantifier ∀ {\displaystyle \forall } in the first order formula ∀ x P ( x ) {\displaystyle \forall xP(x)} expresses that everything in the domain satisfies the property denoted by P ...

8. Closure (mathematics) - Wikipedia

   en.wikipedia.org/wiki/Closure_(mathematics)

   Conversely, if closed sets are given and every intersection of closed sets is closed, then one can define a closure operator C such that () is the intersection of the closed sets containing X. This equivalence remains true for partially ordered sets with the greatest-lower-bound property , if one replace "closed sets" by "closed elements" and ...

9. Complete theory - Wikipedia

   en.wikipedia.org/wiki/Complete_theory

   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 ...