Syllogism
Etymology
From Greek "syllogismos," "a reckoning together" or "inference," from "syn," "together," and "logizesthai," "to reckon" or "to reason," itself from "logos." Aristotle made it the centerpiece of his logic, giving the classic pattern: all men are mortal; Socrates is a man; therefore Socrates is mortal. The word still marks the moment reasoning becomes checkable machinery rather than persuasion.
noun · /SIL-uh-jiz-um/
A form of deductive argument in which a conclusion is drawn from two premises that share a common term.
What it names
The skeleton of valid deduction — the structure that lets you check whether a conclusion actually follows, separate from whether the premises happen to be true.
Example
His case looked airtight until you noticed the syllogism smuggled its conclusion into the first premise.
Related idea
A full syllogism states both premises; an enthymeme is the same reasoning with one premise left silent, which is how arguments actually appear in the wild.