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Click on the phrases to see them in context. The original texts by Immanuel Kant and David Hume are available from the Gutenberg Projet.

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Now we have demonstrated this necessity of a free first beginning of a series of phenomena, only in so far as it is required for the comprehension of an origin of the world, all following states being regarded as a succession according to laws of nature alone.

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The apodeictical proposition cogitates the assertorical as determined by these very laws of the understanding, consequently as affirming a priori, and in this manner it expresses logical necessity.
 The assertorical speaks of logical reality or truth; as, for example, in a hypothetical syllogism, the antecedens presents itself in a problematical form in the major, in an assertorical form in the minor, and it shows that the proposition is in harmony with the laws of the understanding.