A Logic Solver Without Rules of Inference: Aristotle's Syllogistic Classified by Letting a System Rest

Aristotle's categorical syllogistic is the logic on which the tradition of reasoning by rules of inference was built. This paper reports a system that contains no rule of inference and that matched the independently established validity of all 512 cases of that syllogistic: the statements are declared as laws on a space of cells, the premises are held, the conclusion is denied, and a network of elements is let come to rest. A resting state that satisfies every premise and denies the conclusion is read at its ends and re-checked, and is then an exact countermodel; a search that ends without one is reported as a search that ended, with the facts it could not reconcile named, and is never reported as a proof. The network runs the one operation of the operational representation of systems introduced in an earlier paper. Its three ways of composing elements, series, parallel and transposition, are the product t-norm, its dual and standard negation, and they let a logic's statements be written as a system without writing the logic's rules. The same system produced exactly checked countermodels on three larger finite fragments, beyond the reach of its own enumeration. Compiling the connectives instead as weighted sums through a saturating function, the usual way, missed some countermodels, every miss behind a wide disjunction; the composition laws found all but one, and a control shows why. When a search ends without a countermodel, the system names the fact it could not get past, and that statement is then used to search again. Version 1.1: the abstract states that the representation was built to describe systems in general and that the logic is found in it; §8.1 gains a paragraph on what the result means for the representation; no change of results.