Aristotle's Syllogistic Logic as a Theory of an Arithmetic Kind

A certain paradox is associated with Aristotle's logic. The theory of syllogisms is, on the one hand, generally considered to be the first system of formal logic in history, but, on the other hand, this logic was not used by such ancient scholars as Euclid, Archimedes or Ptolemy, and Aristotle himself did not use it in his writings about natural science. The objective of this article is to attempt to clarify this paradox by means of an analysis of the epistemological structure of the language in which Aristotle's logic is formulated. In the first two sections, we will introduce the concepts of relational, compositional and deductive synthesis and the phenomenal, ontological and causal reduction of language. On the basis of these concepts, we distinguish theories of three kinds - physical theories, mathematical theories and arithmetical theories. We will try to show that syllogistic logic is a theory of the arithmetical kind. If our interpretation is correct, it shows why the creators of modern science had to reject Aristotelian logic and the methodology based on it.