# What is this rule of inference called?

Let P, Q, R be three statements. Is there a name for the following rule of inference?

If P implies Q, and if P implies R, then P implies both Q and R.

• Just for clarity's sake, do you mean for your conclusion to be `P -> Q & R` (i.e. how the "and" operator works), or do you mean something more like `P -> Q,R` (i.e. multiple consequents)? Jul 2 '18 at 7:32

R&W in their landmark work in formal logic : Principia Mathematica, page 110, called it "Principle of Composition" :

if a proposition implies each of two propositions, then it implies their logical product. This is called by Peano the "principle of composition."

The reference is to Giuseppe Peano; see e.g. Logique mathématique (1897).

• Did the phrase “principle of composition” enter into general use? I found an apparently related idea on SEP at “Propositional Dynamic Logic” > Hoare calculus. Otherwise I have not been able to find the phrase. The most typical search result is “fallacy of composition”. I was able to find several references to Distribution as a rule of equivalence. E.g., math SE math.stackexchange.com/questions/1318235/… May 28 '18 at 2:30
• @Mark Andrews - I do not think so... but there is no reason to expect that every tautology has a name. May 28 '18 at 5:50

If P implies Q, and if P implies R, then P implies both Q and R.

The name of the rule is Distribution. See Barker, Stephen (1965), The elements of logic, p. 124-25 (Truth-functional principles for use in deduction).

• The question asks for the name of this rule of inference. May 27 '18 at 3:12
• In short, the name is Distribution. But it takes the two steps to show why distribution works. May 27 '18 at 3:30