What's the name of the rule of inference where we conclude $p$ implies $q$, given the premise $p$ and $q$?

Question

What's the name of the rule of inference where we conclude $p$ implies $q$, given the premise $p$ and $q$?

Premise: $p$ and $q$.

Conclusion: $p$ implies $q$ (or the converse).

What's the name of this rule of inference?

Answer 1

It has no name. It can be thought of as either part of the definition of logical implication or as a theorem derived from "first principles."

In many introductory textbooks, the following truth table effectively defines logical implication:

Source: https://www.erpelstolz.at/gateway/TruthTable.html

The first line tells us that if $P$ and $Q$ are true, then the implication $P\to Q$ is also true.

If you are familiar with the basic methods of proof, you can prove the $P \land Q \to (P\to Q)$ using a form of natural deduction as follows (screenshot from my proof checker):

The other lines of the truth table, among other things, are similarly proven here.

Answer 2

Going from $p$ and $q$ to $p \to q$ is not a very interesting inference, and it has no name.

A more interesting inference is going from $q$ alone to $p \to q$. And that one is sometimes called 'Conditionalization'

Indeed, the fact that you can get to $p \to q$ from $q$ alone is exactly why going to $p \to q$ on the basis of $q$ and $p$ is uninteresting, as the $p$ turns out to be completely unncessary in this inference. You might as well ask the name for:

$q \to r$

$q$

$p$

$\therefore r$