Commutativity of conjunction
In propositional logic, the commutativity of conjunction is a valid argument form and truth-functional tautology. It is considered to be a law of classical logic. It is the principle that the conjuncts of a logical conjunction may switch places with each other, while preserving the truth-value of the resulting proposition.[1]
Formal notation
Commutativity of conjunction can be expressed in sequent notation as:
and
where is a metalogical symbol meaning that is a syntactic consequence of , in the one case, and is a syntactic consequence of in the other, in some logical system;
or in rule form:
and
where the rule is that wherever an instance of "" appears on a line of a proof, it can be replaced with "" and wherever an instance of "" appears on a line of a proof, it can be replaced with "";
or as the statement of a truth-functional tautology or theorem of propositional logic:
and
where and are propositions expressed in some formal system.
Generalized principle
For any propositions H1, H2, ... Hn, and permutation σ(n) of the numbers 1 through n, it is the case that:
- H1 H2 ... Hn
is equivalent to
- Hσ(1) Hσ(2) Hσ(n).
For example, if H1 is
- It is raining
H2 is
- Socrates is mortal
and H3 is
- 2+2=4
then
It is raining and Socrates is mortal and 2+2=4
is equivalent to
Socrates is mortal and 2+2=4 and it is raining
and the other orderings of the predicates.
References
- ^ Elliott Mendelson (1997). Introduction to Mathematical Logic. CRC Press. ISBN 0-412-80830-7.
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