The lessons present the rich history of the realization of Leibniz’s slogan “Calculemus!”. He made numerous attempts to build an adequate arithmetical semantics of the syllogistic, translating the terms into integers and the syllogistic propositions into arithmetical relations. All those attempts were unsuccessful. Finally, Leibniz constructed a working model using pairs of relatively prime integers. However, this model did not satisfy him because it was more complicated and did not include the negation of terms. The initial and most intuitive idea of using single integers and their divisibility revived in two adequate (and isomorphic) semantics. They include the full Boolean syllogistic with term negation and all term operations. In such a way, Leibniz’s dream “to replace disputations by calculations” was realized according to his primary plan.

1. From Aristotle to Leibniz

The Aristotelian syllogistic included 256 combinations of three propositions, each of which was in one of four forms: “All S are P” (SaP), “Some S are P” (SiP), and their negations SoP and SeP. Leibniz extended the classical syllogistic in two directions: he introduced the Boolean operations with terms (negation, conjunction, etc.), and generalized the syllogisms to arbitrary Boolean constructions of basic propositions. Two important innovations were SaS and SiS. In the reconstruction of the primary Leibniz idea, the entire new system obtained two adequate translations into arithmetic corresponding to the extensional and intensional interpretations of term relations. Moreover, the arithmetical models were extended to cover the full monadic predicate calculus (the logic of properties) with equality. It became clear that the cause of Leibniz’s unsuccessful attempts was his preference for the intensional semantics, which was definitely anti-intuitive in representing the particular affirmative propositions SiP. The successful “binary” model, using pairs of mutually prime integers, will be discussed shortly.

2. Leibniz and the algebraic semantics 

In parallel with the arithmetical models, Leibniz developed a dozen completely new syllogistic constructions based on the algebra of terms. All of them will be presented in the lesson. A large list of well-known equalities is observed in his manuscript, as AA = A, AB = BA, ~~A = A, etc. Generally speaking, Leibniz discovered that syllogistic term relations SaP and SiP might be represented by algebraic term relations using term operations. E.g., SaP is equivalent to S = SP, where SP is an idempotent, commutative, and associative term composition (conjunction). Adding disjunction and a minimal element (0), Leibniz obtained consecutive algebraic systems, which G. Birkhoff named partially ordered structures, semi-lattices, lattices, and Boolean algebras. Observing the variations in the language of syllogistic, one can notice a transfer from syllogistic relations to term operations. This transfer means, in fact, a consecutive elimination of the traditional syllogistic: while only specific term relations (A and I) have appeared in the beginning, only term operations (composition and negation plus a minimal term 0) together with term equality appear in the end. It is amazing to find in Leibniz’s esquisses syllogistic axioms 160 years before G. Boole and 250 years before J. Shephersdson!

3. After Leibniz 

Negation of predicates is an ordinary procedure, while the negation of objects meets some vagueness. “He is not a painter” is clear, but what does a “non-painter” mean? Probably, this is the cause of the lack of syllogistic propositions of the types ~SaP (“All non-S are P”) and ~So~P (“Some non-S are not P”). Although their doubtful sense, they obtain an acceptable meaning in another, logically equivalent reading: “Everything is either S or P” and “Something is neither S nor P”. Those gaps in the full disposition of negations over the subjects and predicates were observed by G. Boole, but without further investigation. In the last lesson, the two new propositions will be introduced and denoted by u and y, respectively. The arithmetical semantics of Lesson 1 will be extended to the new propositions. 52 new non-trivial true syllogisms appear in addition to the traditional 24, and their full list will be shown. Moreover, the 6 syllogistic relations are enough to express all relations. Having well-known Łukasiewicz’s 4 axioms of the general syllogistic based on a- and i-propositions, 3 additional axioms of u provide for the complete axiomatization of the full syllogistic. When two new “vertices”, corresponding to the u- and y-propositions, have been added to the classical “logical square”, the adequate spatial “logical octahedron” is obtained. Its 3 diagonal planes of intersection represent the 3 “logical squares” composed by a-i-o-e, a-u-o-y, and e-u-o-y, respectively. Finally, the syllogisms Barbara and SaS suggest to recognize in a a transitive and reflexive relation, producing an S4-necessity modality []1 in the relating Kripke semantics. Analogously, the axioms of i Datisi and SiS produce a second modality []2, and so do the axioms of u, which generate a third modality []3. All three modalities are connected. In such a way, the Kripke semantics (anticipated by Leibniz's ideas of “possible worlds”) reveals a “hidden” modal logic lying under the full syllogistic. This fact waits for its philosophical interpretation.

References

 [1] Beziau, J.-Y.  The Hexagon of Intelligence. In Z. Shi, M. Chakraborty and S. Kar (Eds), Intelligence Science III, Springer, Cham, (25-34), (2021).

[2] Birkhoff G., Lattice Theory, Amer. Math. Soc., Providence, 3d ed. 1967.

[3] Boole G., The Mathematical Analysis of Logic, Being an Essay towards a Calculus of Deductive Reasoning, Macmillan, Cambridge, 1847.

[4] Boole, G. An Investigation of the Laws of Thought. London: Walton&Maberly, 1854.

[5] Boolos G., R. Jeffrey, Computability and Logic, Cambridge Univ. Press, 3rd. ed., 1989.

[6] Couturat L., Opuscules et fragments inédits de Leibniz, éxtraits des manuscrits de la Bibliothèque royale de Hanovre, F. Alcan, Paris, 1903 (reprint: Olms, Hildesheim, 1961).

[7] Leibniz G. W., Sämtliche Schriften und Briefe, 6. Reiche, Philosophische Schriften, 4. Band (1677 – Juni 1690), Teil A, Berlin, Academie Verlag, 1999.

[8] Łukasiewicz J., Aristotle’s Syllogistic from the Standpoint of Modern Formal Logic, Clarendon Press, Oxford, 2nd ed. 1957.

[9] Shepherdson J., On the interpretation of Aristotelian syllogistic, J. Symb. Logic, 21 (1956), pp. 137−147.

[10] Sotirov V., Arithmetizations of syllogistic à la Leibniz, J. Appl. Non-Class. Logics, 9 (1999), n. 2−3, pp. 387−405. (On my webpage.)

[11] Sotirov V., Monadic predicate calculus with equality arithmetized à la Leibniz, C. r. Acad. bulgare Sci., 54 (2001), n. 1, 9–10. (On my webpage.)

[12] Vakarelov, D.  Information Systems, Similarity Relations and Modal Logics. In E. Orłowska (Ed.), Rough Set Analysis (pp. 493-550) (1998), Springer.