Computation in Aristotle's Syllogistic

The Syllogistic of Aristotle’s Analytics is widely interpreted either as a theory of deduction or as a specialized logic of the a-e-i-o predication relations. I argue that it should instead be understood as a computationalist epistemology: an algorithmic treatment of evident validity, or what Aristotle calls ‘perfection’. Against the prevailing opinion that Aristotle meant for the class of syllogisms to be coextensive with the class of deductions, I suggest that he understood syllogisms to constitute a particular kind of deduction. Roughly, he took them to be either computational deductive procedures, or data structures susceptible to such procedures, where both are dependably linked to the epistemic feature of evident validity.

Using contemporary techniques in the theory of computation, I show how the underlying data structure of syllogisms can be represented as full binary trees. Further, I elucidate several mechanical procedures that are found in the Analytics, most notably, the procedure of perfecting imperfect syllogisms.

The exposure of these computational features is found to raise an interesting combinatorial problem for syllogisms with more than two premises. Informally, the problem may be posed as a question: for such a syllogism, what are all the distinct ways in which its conclusion may be obtained from its premises? This is algorithmically interpreted as the task of finding all the possible syllogistic full binary trees of a multi-premise syllogism.

This turns out to be a computationally challenging task. For such syllogisms whose premises are ordered, I show both that the problem is computationally solvable, and executable in polynomial time. For multi-premise syllogisms with unordered premises, I suggest that the problem of finding all the relevant trees may be NP-complete.

In closing, this thesis suggests that Aristotle should be credited with the innovation of computationalist analysis as a philosophical method. It further observes that the algorithmic features of the Syllogistic deserve further study, and should not be occluded by the more purely logical aspects of the Analytics.