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
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