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theory (Definition)

If $ L$ is a logical language for some logic $ \mathcal{L}$, a set $ T$ of formulas with no free variables is called a theory (of $ \mathcal{L}$). If $ \mathcal{L}$ is a first-order logic, then $ T$ is called a first-order theory.

We write $ T\vDash \phi$ for any formula $ \phi$ if every model $ \mathcal{M}$ of $ \mathcal{L}$ such that $ M\vDash T$, $ M\vDash\phi$.

We write $ T\vdash\phi$ is for there is a proof of $ \phi$ from $ T$.

Remark. Let $ S$ be an $ L$-structure for some signature $ L$. The theory of $ S$ is the set of formulas satisfied by $ S$:

$\displaystyle \lbrace \varphi \mid S\models \varphi \rbrace,$
and is denoted by $ \operatorname{Th}(S)$.



"theory" is owned by CWoo. [ full author list (2) | owner history (2) ]
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Cross-references: signature, first-order theory, free variables, formulas, logic, logical language
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This is version 5 of theory, born on 2002-08-28, modified 2008-01-16.
Object id is 3383, canonical name is Theory.
Accessed 5714 times total.

Classification:
AMS MSC03B05 (Mathematical logic and foundations :: General logic :: Classical propositional logic)
 03B10 (Mathematical logic and foundations :: General logic :: Classical first-order logic)

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