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There are two ways, because two prove a hypothesis depends on the theoretical metaphysical frame used and on the logical structure that allows to proof something.
Example: proof through derivation of essence in aristotelian logical frame (syllogism)
Hypothesis: Peter is mortal
Process of demonstration:
All human are mortal.
Peter is a human.
QED: Peter is mortal
Another way is to use Plato’s dichotomie: usually demonstration through contradiction in essence in a logical frame allowing the proof through ‘tiers exclu’ (third excluded)
Hypothesis: Peter is wet, thus he has come in touch with water.
Demonstration: It is not possible to be wet without getting in touch with water, thus Peter has come in touch with water.
Both are a priori (without empirical verification, through the theory of essences)
These are little in use nowaday, as proof supposed to derive from experience (empiria = empirical) only. In order to frame a hypothesis nowadays, you do not only need the logical frame (symbolic logic) inside of the given metaphysical frame (only through experience) and the instrumental frame, which may include possible diformation resulting from the instruments used to measure or verify (chemistry, physics, partly biology). The problem is, as well objected already Hume, that the conclusion is never decisive but relative to percentage (probable). You can never conclude from the fact that 93% of people smoking are affected by cancer, that ‘smoking causes cancer’ as the latter implies a consequence in essence which can only be a priori. In fact nowaday there are neither real proofs nor real demonstrations: there is cumulation of data, whose value is difficult to evaluate.
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