  Prior to about 1865, thinkers on logic commonly had divided arguments into two subclasses: the class of deductive arguments (a.k.a. necessary inferences) and the class of inductive arguments (a.k.a. probable inferences). About this time, Peirce began to hold that there were two utterly distinct classes of probable inferences, which he referred to as inductive inferences and abductive inferences (which he also called hypotheses and retroductive inferences). Peirce reached this conclusion by entertaining what would happen if one were to interchange propositions in the syllogism AAA-1 (Barbara): All Ms are Ps; all Ss are Ms; therefore, all Ss are Ps. This valid syllogism Peirce accepted as representative of deduction. But he also seemed typically to regard it in connection with a problem of drawing conclusions on the basis of taking samples. For let us regard being an M as being a member of a population of some sort, say being a ball of the population of balls in some particular urn. Let us regard P as being some property a member of this population can have, say being red. And, finally, let us regard being an S as being a member of a random sample taken from this population. Then our syllogism in Barbara becomes: All balls in this urn are red; all balls in this particular random sample are taken from this urn; therefore, all balls in this particular random sample are red. Peirce regarded the major premise here as being the Rule, the minor premise as being the particular Case, and the conclusion as being the Result of the argument. The argument is a piece of deduction.  In this example the argument is also an argument from population to random sample that is also a necessary inference.