Commutativity of conjunction

Commutativity of conjunction can be expressed in sequent notation as:

: $(P\; \backslash land\; Q)\; \backslash vdash\; (Q\; \backslash land\; P)$

and

: $(Q\; \backslash land\; P)\; \backslash vdash\; (P\; \backslash land\; Q)$

where $\backslash vdash$ is a metalogical symbol meaning that $(Q\; \backslash land\; P)$ is a syntactic consequence of $(P\; \backslash land\; Q)$, in the one case, and $(P\; \backslash land\; Q)$ is a syntactic consequence of $(Q\; \backslash land\; P)$ in the other, in some logical system;

or in rule form:

:$\backslash frac\{P\; \backslash land\; Q\}\{\backslash therefore\; Q\; \backslash land\; P\}$

and

:$\backslash frac\{Q\; \backslash land\; P\}\{\backslash therefore\; P\; \backslash land\; Q\}$

where the rule is that wherever an instance of "$(P\; \backslash land\; Q)$" appears on a line of a proof, it can be replaced with "$(Q\; \backslash land\; P)$" and wherever an instance of "$(Q\; \backslash land\; P)$" appears on a line of a proof, it can be replaced with "$(P\; \backslash land\; Q)$";

or as the statement of a truth-functional tautology or theorem of propositional logic:

:$(P\; \backslash land\; Q)\; \backslash to\; (Q\; \backslash land\; P)$

and

:$(Q\; \backslash land\; P)\; \backslash to\; (P\; \backslash land\; Q)$

where $P$ and $Q$ are propositions expressed in some formal system.

For any propositions H₁, H₂, ... H_n, and permutation σ(n) of the numbers 1 through n, it is the case that:

:H₁ $\backslash land$ H₂ $\backslash land$ ... $\backslash land$ H_n

is equivalent to

:H_σ(1) $\backslash land$ H_σ(2) $\backslash land$ H_σ(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.

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{{Classical logic}}

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Category:Classical logic

Category:Rules of inference

Category:Theorems in propositional logic