Abstract

There being no limit to the number of categories, there is no limit to the number of Aristotelian syllogisms. Aristotle showed that this potential infinity of syllogisms could be obtained as substitution instances of finitely many syllogistic forms, further reduced by exploiting symmetry, in particular the premises’ independence of their order. A consensus subsequently emerged that there were 24 valid assertoric syllogistic forms. A more modern concern, completeness, showed that there could be no more. Using no additional principles beyond substitution and symmetry, we further reduce these 24 to four forms. A third principle, contraposition, allows a further reduction to two forms, namely the unconditional form and the conditional form, conditioned on one of the terms being inhabited. We achieve these reductions via a regularly organized proof system in the form of a graph with 24 vertices and three kinds of edges corresponding to the three principles.