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first order logic (Definition)

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.



"first order logic" is owned by Henry.
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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
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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 17474 times total.

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