Demonstration is literally a showing. For Aristotle, its main purpose is associated with learning and teaching, rather than proof. Its real objective is not Stoic or Cartesian certainty “that” something is true, but the clearest possible understanding of the substantive basis for definite conclusions, based on a grasping of reasons.
Aristotle’s main text dealing with demonstration, the Posterior Analytics, is not about epistemology or foundations of knowledge, although it touches on these topics. Rather, it is about the pragmatics of improving our informal semantic understanding by formal means.
For Aristotle, demonstration uses the same logical forms as dialectic, but unlike dialectic — which does not make assumptions ahead of time whether the hypotheses or opinions it examines are true, but focuses on explicating their inferential meaning — demonstration is about showing reasons and reasoning behind definite conclusions. Dialectic is a kind of conditional forward-looking interpretation based on consequences, while demonstration is a kind of backward-looking interpretation based on premises. Because demonstration’s practical purpose has to do with exhibiting the basis for definite conclusions, it necessarily seeks sound premises, or treats its premises as sound, whereas dialectic is indifferent to the soundness of the premises it analyzes in terms of their consequences.
We are said to know something in Aristotle’s stronger sense when we can clearly explain why it is the case, so demonstration is connected with knowledge. This connection has historically led to much misunderstanding. In the Arabic and Latin commentary traditions, demonstration was interpreted as proof. The Posterior Analytics was redeployed as an epistemological model for “science” based on formal deduction, understood as the paradigm for knowledge, while the role of dialectic and practical judgment in Aristotle was greatly downplayed. (See also Demonstrative “Science”?; Searching for a Middle Term; Plato and Aristotle Were Inferentialists; The Epistemic Modesty of Plato and Aristotle; Belief; Foundations?; Brandom on Truth.)