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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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{{MSC|03}}
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{{TEX|done}}
  
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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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A ''propositional formula'' is
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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 $(A\mathbin\&B)$, $(A\lor B)$, $(A\supset B)$, and $(\neg A)$.
  
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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.
  
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====References====
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|valign="top"|{{Ref|Wó}}||valign="top"|  R. Wójcicki,  "Theory of logical calculi", Kluwer  (1988)  pp. 13; 61    {{MR|1009788}}  {{ZBL|0682.03001}}
  
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====Comments====
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|valign="top"|{{Ref|Zi}}||valign="top"| Z. Ziembinski,  "Practical logic", Reidel  (1976)  pp. Chapt. V, §5  {{ZBL|0372.02001}}
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====References====
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<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> Z. Ziembinski,  "Practical logic" , Reidel  (1976)  pp. Chapt. V, §5</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  R. Wójcicki,   "Theory of logical calculi" , Kluwer  (1988)  pp. 13; 61</TD></TR></table>
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Latest revision as of 13:57, 30 December 2018

2020 Mathematics Subject Classification: Primary: 03-XX [MSN][ZBL]

A propositional formula is 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 $(A\mathbin\&B)$, $(A\lor B)$, $(A\supset B)$, 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.

References

[Wó] R. Wójcicki, "Theory of logical calculi", Kluwer (1988) pp. 13; 61 MR1009788 Zbl 0682.03001
[Zi] Z. Ziembinski, "Practical logic", Reidel (1976) pp. Chapt. V, §5 Zbl 0372.02001
How to Cite This Entry:
Propositional formula. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Propositional_formula&oldid=15015
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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