Difference between revisions of "Propositional formula"
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A ''propositional formula'' is | A ''propositional formula'' is | ||
| − | 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 $(\neg A)$. | + | 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 $(\neg A)$. |
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. | 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. | ||
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=43578
Propositional formula. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Propositional_formula&oldid=43578
This article was adapted from an original article by S.K. Sobolev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article