Why is the Absorption Law considered a Rule of Inference instead of Replacement?

Solution 1:

In Wikipedia's terminology, a rule of replacement is a type of inference rule that lets one replace arbitrary subexpressions with equivalent subexpressions.

The absorption law is not a rule of replacement, but it could be made into one. If it were, it would be a strictly more powerful inference rule than the Absorption Law.

Absorption, as you noted, is an inference rule. It can be written as $P \to Q \vdash P \to (P \land Q)$ or as follows.

$$ \frac{P \to Q}{P \to (P \land Q)} $$

The Wikipedia article notes that the proposition $(P \to Q) \leftrightarrow (P \to (P \land Q))$ is proven as a theorem in Principia Mathematica. The symbol $\leftrightarrow$ in this context does not refer to some kind of meta-level equivalence; it is a connective like $\to$ and $\land$.

A rule of replacement, according to Wikipedia, a rule of inference (transformation rule is a redirect to inference rule), that targets individual subexpressions.

For example, $A$ is equivalent to $\lnot\lnot A$ in classical logic. So, replacing all or some occurrences of $A$ with $\lnot\lnot A$ in a proposition would be a valid inference.

With absorption, we could express the transformation rule as follows.

$$ P \to Q \;\; \text{is equivalent to} \;\; P \to (P \land Q) $$

This gives us a much more powerful inference rule. For example, the inference below is an application of the "Absorption Replacement Rule" but not of the Absorption Law.

$$ \frac{\lnot (P \to (Q \land R))}{\lnot(P \to (P \land Q \land R))} $$

Solution 2:

There are many formal proof systems, each with their own set of rules. And there is nothing that prevents a creator of such a system to define a rule any which way they want. Of course, the intent is that the rules reflect elementary/basic inferences, and valid inferences at that.

Now, as you point out, some systems have both rules of inference (that go one way only, and can only be applied to whole statements) and rules of replacement (that can go either way, and can be applied to component statements of larger statements). And yes, rules of replacement are in a way more powerful ... or certainly they are more user-friendly. And so, sure, it seems weird that Absorption would be defined as a rule of inference, rather than a rule of replacement, but again, creators of these systems can do whatever they want to do. In a way, only they can answer your question.

Still, I think there actually are some aesthetical, and maybe even some theoretical reasons for treating Absorption as a rule of inference rather than replacement.

In their textbook "Introduction to Logic", Copi and Cohen lay out 9 rules of inference (including Absorption) and 10 rules of replacement. A list of those rules can be found here.

Now, if you look at the rules of replacement that they have, I think you'll find that Absorption does feel a little different from the others: the others seem to be either principles that are nicely 'symmetric', or they reveal some kind of 'definition' or 'connection' between different operators. Absorption is neither of those things.

Also, theoretically it is interesting that these 19 rules together allows one to derive any statement from any other statement of which it is a logical consequence. As such, we can say the system is 'complete' (it is a bit of a weak form of completeness, as it cannot derive tautologies from nothing ... but as long as you have at least one premise, then anything that logically follows from the premises can be derived from those premises). And, with Absorption being defined as a rule of inference, this completeness result is more 'interesting' than if it were a more powerful rule of replacement.