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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. 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 deduced from ⁠ ⁠, formally: if ⁠ ⁠ always implies ⁠ ⁠.

  5. 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 ...

  6. Quantifier (logic) - Wikipedia

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

    For example, the existential closure of the open formula n>2 ∧ x n +y n =z n is the closed formula ∃n ∃x ∃y ∃z (n>2 ∧ x n +y n =z n); the latter formula, when interpreted over the positive integers, is known to be false by Fermat's Last Theorem.

  7. Deduction theorem - Wikipedia

    en.wikipedia.org/wiki/Deduction_theorem

    The deduction theorem for predicate logic is similar, but comes with some extra constraints (that would for example be satisfied if is a closed formula). In general a deduction theorem needs to take into account all logical details of the theory under consideration, so each logical system technically needs its own deduction theorem, although ...

  8. 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 ...

  9. Theory (mathematical logic) - Wikipedia

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

    In mathematical logic, a theory (also called a formal theory) is a set of sentences in a formal language.In most scenarios a deductive system is first understood from context, after which an element of a deductively closed theory is then called a theorem of the theory.

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