Absorption (logic)

Absorption is a valid argument form and rule of inference of propositional logic. The rule states that if $P$ implies $Q$, then $P$ implies $P$ and $Q$. The rule makes it possible to introduce conjunctions to proofs. It is called the law of absorption because the term $Q$ is "absorbed" by the term $P$ in the consequent.Russell and Whitehead, ''Principia Mathematica'' The rule can be stated:

:$\frac$

where the rule is that wherever an instance of "$P \to Q$" appears on a line of a proof, "$P \to (P \land Q)$" can be placed on a subsequent line.

Formal notation

The ''absorption'' rule may be expressed as a sequent:

: $P \to Q \vdash P \to (P \land Q)$

where $\vdash$ is a metalogical symbol meaning that $P \to (P \land Q)$ is a syntactic consequence of $(P \rightarrow Q)$ in some logical system;

and expressed as a truth-functional tautology or theorem of propositional logic. The principle was stated as a theorem of propositional logic by Russell and Whitehead in ''Principia Mathematica'' as:

:$(P \to Q) \leftrightarrow (P \to (P \land Q))$

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

Examples

If it will rain, then I will wear my coat.

Therefore, if it will rain then it will rain and I will wear my coat.

Proof by truth table

Formal proof

See also

*Absorption law

References

{{Reflist

Rules of inference
Theorems in propositional logic