Commutativity of conjunction

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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[edit]

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[edit]

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[edit]

  1. ^ Elliott Mendelson (1997). Introduction to Mathematical Logic. CRC Press. ISBN 0-412-80830-7.