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Difference between revisions of "Propositional formula"

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An expression constructed from propositional variables (cf. [[Propositional variable|Propositional variable]]) by means of the propositional connectives (cf. [[Propositional connective|Propositional connective]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075510/p0755101.png" /> (and possibly others) in accordance with the following rules: 1) each propositional variable is a propositional formula; and 2) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075510/p0755102.png" /> are propositional formulas, then so are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075510/p0755103.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075510/p0755104.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075510/p0755105.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075510/p0755106.png" />.
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An expression constructed from propositional variables (cf. [[Propositional variable|Propositional variable]]) by means of the propositional connectives (cf. [[Propositional connective|Propositional connective]]) $\&,\lor,\supset,\neg,\equiv$ (and possibly others) in accordance with the following rules: 1) each propositional variable is a propositional formula; and 2) if $A,B$ are propositional formulas, then so are $(A\&B)$, $(A\lor B)$, $(A\supset B)$, and $(\neg A)$.
  
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If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075510/p0755107.png" /> is a set of propositional connectives (a fragment), then a propositional formula in the fragment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075510/p0755108.png" /> is a propositional formula in whose construction rule 2) only connectives from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075510/p0755109.png" /> are used.
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If $\sigma$ is a set of propositional connectives (a fragment), then a propositional formula in the fragment $\sigma$ is a propositional formula in whose construction rule 2) only connectives from $\sigma$ are used.
  
  

Revision as of 15:23, 9 April 2014

An expression constructed from propositional variables (cf. Propositional variable) by means of the propositional connectives (cf. Propositional connective) $\&,\lor,\supset,\neg,\equiv$ (and possibly others) in accordance with the following rules: 1) each propositional variable is a propositional formula; and 2) if $A,B$ are propositional formulas, then so are $(A\&B)$, $(A\lor B)$, $(A\supset B)$, and $(\neg A)$.

If $\sigma$ is a set of propositional connectives (a fragment), then a propositional formula in the fragment $\sigma$ is a propositional formula in whose construction rule 2) only connectives from $\sigma$ are used.


Comments

References

[a1] Z. Ziembinski, "Practical logic" , Reidel (1976) pp. Chapt. V, §5
[a2] R. Wójcicki, "Theory of logical calculi" , Kluwer (1988) pp. 13; 61
How to Cite This Entry:
Propositional formula. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Propositional_formula&oldid=31453
This article was adapted from an original article by S.K. Sobolev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article
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