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first order logic
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(Definition)
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A logic is first order if it has exactly one type. Usually the term refers specifically to the logic with connectives $\neg$ , $\vee$ , $\wedge$ , $\rightarrow$ , and $\leftrightarrow$ and the quantifiers $\forall$ and
$\exists$ , all given the usual semantics:
- $\neg\phi$ is true iff $\phi$ is not true
- $\phi\vee\psi$ is true if either $\phi$ is true or $\psi$ is true
- $\forall x\phi(x)$ is true iff $\phi^t_x$ is true for every object $t$ (where $\phi^t_x$ is the result of replacing every unbound occurrence of $x$ in $\phi$ with $t$ )
- $\phi\wedge\psi$ is the same as $\neg(\neg\phi\vee\neg\psi)$
- $\phi\rightarrow\psi$ is the same as $(\neg\phi)\vee\psi$
- $\phi\leftrightarrow\psi$ is the same as $(\phi\rightarrow\psi)\wedge(\psi\rightarrow\phi)$
- $\exists x\phi(x)$ is the same as $\neg\forall x\neg\phi(x)$
However languages with slightly different quantifiers and connectives are sometimes still called first order as long as there is only one type.
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"first order logic" is owned by Henry.
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(view preamble | get metadata)
| Other names: |
classical first order logic, FO |
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Cross-references: languages, occurrence, object, iff, semantics, quantifiers, connectives, term, type, first order, logic
There are 22 references to this entry.
This is version 4 of first order logic, born on 2002-08-28, modified 2003-12-02.
Object id is 3379, canonical name is FirstOrderLogic.
Accessed 19541 times total.
Classification:
| AMS MSC: | 03B10 (Mathematical logic and foundations :: General logic :: Classical first-order logic) |
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Pending Errata and Addenda
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