Logic statement about a formal system proven in a metalanguage
This article is about logical statements. For theories about theories, see Metatheory.
In logic, a metatheorem is a statement about a formal system proven in a metalanguage. Unlike theorems proved within a given formal system, a metatheorem is proved within a metatheory, and may reference concepts that are present in the metatheory but not the object theory.[citation needed]
A formal system is determined by a formal language and a deductive system (axioms and rules of inference). The formal system can be used to prove particular sentences of the formal language with that system. Metatheorems, however, are proved externally to the system in question, in its metatheory. Common metatheories used in logic are set theory (especially in model theory) and primitive recursive arithmetic (especially in proof theory). Rather than demonstrating particular sentences to be provable, metatheorems may show that each of a broad class of sentences can be proved, or show that certain sentences cannot be proved.[citation needed]
a metatheorem is a statement about a formal system proven in a metalanguage. Unlike theorems proved within a given formal system, a metatheorem is proved...
In mathematical logic, a deduction theorem is a metatheorem that justifies doing conditional proofs from a hypothesis in systems that do not explicitly...
φ0. We also use repeatedly the method of the hypothetical syllogism metatheorem as a shorthand for several proof steps. (1) φ 0 {\displaystyle \varphi...
axiomatisation using an extra rule of generalisation (see the section on Metatheorems), in which case the rules Q6 and Q7 are redundant.[dubious – discuss]...
Statements made in the metatheory about the theory are called metatheorems. A metatheorem is a true statement about a formal system expressed in a metalanguage...
problems influenced mathematics for the rest of the 20th century. A metatheorem is defined as: "a statement about theorems. It usually gives a criterion...
(\neg p\to p)\to p} We also use the method of the hypothetical syllogism metatheorem as a shorthand for several proof steps. p → ( ¬ p → q ) {\displaystyle...
In classical logic, a hypothetical syllogism is a valid argument form, a deductive syllogism with a conditional statement for one or both of its premises...
conclusion. In most logics, weakening is either an inference rule or a metatheorem if the logic doesn't have an explicit rule. Notable exceptions are: Relevance...
Hypothetical syllogism. We also use the method of the hypothetical syllogism metatheorem as a shorthand for several proof steps. The proof is as follows: q →...
stratifying out use and mention, language and metalanguage, and theorem and metatheorem predate key discoveries in western philosophy by millennia." "The Sanskrit...
In mathematics and computer science, the Entscheidungsproblem (German for 'decision problem'; pronounced [ɛntˈʃaɪ̯dʊŋspʁoˌbleːm]) is a challenge posed...
about a formal system (as opposed to within a formal system) is called a metatheorem. Some important theorems in mathematical logic are: Compactness of first-order...
theories. Statements made in the metatheory about the theory are called metatheorems. A political theory is an ethical theory about the law and government...
this fact, or more properly speaking, a metaproof. These examples are metatheorems of our theory of mathematical logic since we are dealing with the very...
In mathematical logic, the compactness theorem states that a set of first-order sentences has a model if and only if every finite subset of it has a model...
formal system, which, in order to avoid confusion, are usually called metatheorems. A logical system is a deductive system (most commonly first order logic)...
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