Abstract
Even if Leibniz didn’t have the opportunity to actually conceive an explicit modal logic system, remains the fact that he had worked out a modal metaphysics, of which the inaugural act, in his Elementa juris naturalis (c. 1671) was an obvious reference to the Apulean square of opposition. Later, scholars acknowledged in this passage probably one of the first sketch of deontic logic of norms. His modal metaphysics rather deals with the so-called alethic modalities, sometimes expounded through a language such as R.M. Adams wondered whether Leibniz could be “a sort of grandfather of possible worlds semantics for modal logic”. In the following study, the Apulean square is used as a hermeneutic tool: however, is the square really well-fitted to express some basic implications of the modal metaphysics? The most remarkable point is that Leibniz was well aware of the K-distributive axiom □(p⊃q)⊃(□p⊃□q) common to the main modal systems today. This awareness dissuaded him to trust the too easy solution of the necessitarianism issue grounded on the well-known distinction coming from Boethius between a “necessitas consequentiae” and a “necessitas consequentis”.
Thus Leibniz must consider a proof-theory solution style (for a contingent proposition, the reducibility of the predicate to its subject cannot be achieved). A new square could be making up, however implying a difficulty: if the modal metaphysics cannot admit the logical convertibility of the contingent propositions of which the dictum of one is exactly the negation of the dictum of the other, known since Aristotle. This is the reason for what the Apulean square could represent a Procrustean bed for Leibniz’ modal metaphysics. At the end, a new modality square may be drawn, according to which each modality in the corner is expressed by quantifications on possible worlds. Whether possible worlds semantics could supersedes Leibniz’ own explanation of contingency thanks to this new square of modalities, i.e. a possible worlds square of modalities, will be the topic of a second part of this study.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
“Omnes ergo Modalium complicationes et transpositiones et oppositiones, ab Aristotle alisque in Logicis demonstratae ad haec nostra Juris Modalia non inutiliter transferri possint” [14, VI, 1: 465–485]. This text came to the fore thanks to Blanché, as it is remembered by Kalinowski, post-scriptum of ([11]: 207) (in the same book, see also chapters ii and iii, about the analogy between the alethic and deontic modalities). See further [10, 26].
- 2.
Aphorismi de felicitate, sapientia, caritate, justitia [14, VI, 4-D: 2800].
- 3.
De rerum originatione radicali (1697) [16, 3: 302–318].
- 4.
Origo veritatum contingentium (1689) ([15]: 325–326).
- 5.
“Ajo igitur Existens esse quod cum plurimis compatibile est; seu Ens maxime possibile. Itaque omnia coexistentia aeque possibilia sunt” (Generales inquisitiones de analysi notionum et veritate, 1686, §73, [19]: 248).
- 6.
- 7.
([24]: 40) Boethius supported a similar version in his own commentary of Aristotle’s De interpretatione.
- 8.
De mente, de universo, de Deo, December 1675 [14, VI, 3: 464].
- 9.
“Impossibilis duplex notio, id quod essentiam non habet, et id quod Existentiam non habet seu quod nec fuit nec erit, quod incompatibile est deo, sive existentiae sive rationi quae facit ut res sint potius quam non sint. Videndum an demonstrari possit esse Essentias quae Existentia careant. Ne quis dicat nihil concipi posse quod non aliquando futurum sit in tota aeternitate. Omnia quae sunt erunt et fuerunt totum constituunt. Quicquid necessario incompatibile est, impossibile est” [14, VI, 3: 463–464].
- 10.
“Quare errant quicunque <absolutè id est per se> impossibile pronuntiant, quicquid nec fuit, nec est, nec erit” ([17]: 56).
- 11.
“Quia sua natura possibile definivimus, quod in se non implicat contradictionem, etsi ejus coexistentia cum Deo aliquo modo dici possit implicare contradictionem. Sed opus erit constantes adhibere vocabulorum significationes ut species omnis absurdae locutionis evitetur” [15, VI, 4: 1447]. This text appeared in ([14]: 289), but cut of the part of the sentence here underlined, which is of the highest importance for our purpose, happily restored in the academic edition (equally in its English translation in ([18]: 21)).
- 12.
“[…] indeed, here, we call necessary only but necessary by itself, namely that which possesses in itself the reason of its existence and of its truth, such are truths of geometry” ([17]: 57).
- 13.
“ […] pays des réalités possibles”: to Arnauld, 1st July 1686 [16, 2: 55].
- 14.
If A is “man” and B is “mortal”, A=AB expresses well the equality of the extensions “man” and “mortal man”, hence the inclusion of “mortal man” in “man”, thus, of “mortal” in “man” in virtue of the idempotent law. If then A is “man” and B is “wise”, AY=B: every man associated to an undetermined quality is wise. If Y=B, this quality may be but wisdom, so all men are not wise.
- 15.
General Notations [14, VI 4-A: 752].
- 16.
Note on freedom (1689?) [14, VI 4-B: 1655–1656].
- 17.
By opposition to the modum, of which negation refers to what is necessary (Aristotle, First Analytics, I, chapter 13, 32a-30, [2]: 61).
- 18.
Gen. Inq. §§74, 134, 143, 144. “An existential proposition tertii adjecti is: every man is (namely exists) exposed to the sin; and this proposition certainly is existential namely contingent” (§144, [19]: 285).
- 19.
“However, in this case, this is a contingent proposition, namely it is possible that it would be true or that it would be false” (note of §66, [19]: 245).
- 20.
[19]: 237, 239. The French translation is: “Est possible ce dont on peut montrer…” [ibid.]. However the corresponding Latin verb exactly is: “demonstrare”, that appears to be more demanding (see also §§66, 134).
- 21.
This point is widely commentated by J. Bouveresse ([6]: 91).
- 22.
Sleigh ([27]: 88) believes that Leibniz did not confess to Arnauld his mathematical analogy on the topic of the contingency considering the strong reluctances of his interlocutor for agreeing the containment theory; however nothing, in the Discours de métaphysique as in the correspondence with Arnauld, would be incoherent with this analogy.
References
Adams, R.M.: Leibniz. Determinist, Theist, Idealist. Oxford University Press, Oxford (1994)
Aristotle: First Analytics. Tricot, J. (French trans.). Organon III. Librairie philosophique Jules Vrin, Paris (1971)
Aristotle: De Interpretatione. Tricot, J. (French trans.). Organon II. Librairie philosophique Jules Vrin, Paris (1977)
Blanché, R.: Structures intellectuelles. Essai sur l’organisation systématique des concepts. Librairie philosophique Jules Vrin-reprises, Paris (1986)
Boethius: La consolation philosophique. Moreschini, C. (French edition), Vanpeteghem, É. (translated and annotated), Tilliette, J.-Y. (introduction). Le livre de Poche, Lettres gothiques, Paris (2005)
Bouveresse, J., Essais, V.: Descartes, Leibniz, Kant. Agone, Marseille (2006)
Calvin, J.: L’institution de la religion chrétienne. Kerygma/Farel, Marne-la-Vallée/Aix-en-Provence (1978)
Calvin, J.: De aeterna Dei praedestinatione. Neuser, W.H. (ed.), Fatio O. (French traduction). Scripta Ecclesiastica, vol. 1. Droz, Geneva (1998)
Gardies, J.-L.: Essai sur les fondements a priori de la rationalité morale et juridique. Bibliothèque de Philosophie du Droit, vol. 14. Librairie Générale de Droit et de Jurisprudence R. Pichon and R. Durand-Auzias, Paris (1972)
Gardies, J.-L., Kalinowski, G.: Un logicien déontique avant la lettre: Gottfried Wilhelm Leibniz. Arch. Rechts- Soz.philos. 60, 79–112 (1974)
Kalinowski, G.: La logique des normes. Presses Universitaires de France, Paris (1972)
Kalinowski, G.: Un aperçu élémentaire des modalités déontiques. Langages 10(43), 10–18 (1976)
Kalinowski, G.: La logique juridique de Leibniz. Stud. Leibnit. 9, 168–189 (1977)
Leibniz, G.-W.: Sämtliche Schriften und Briefe, Hrsg. von der Berlin-Brandenburgischen und der Göttinger Akademie der Wissenschaften, vormals Preussische Akademie, danach Akademie der Wiss. der DDR, 8 Serien. Darmstadt, spätter Leipzig, Berlin (1923). Quoted: [14] followed by the number a series, volume and pagination
Leibniz, G.-W.: Textes inédits d’après les manuscrits de la bibliothèque provinciale de Hanovre. Grua, G. (published and annotated), 2 vols. Presses Universitaires de France, Paris (1948)
Leibniz, G.-W.: Philosophischen Schriften, 7 vol. G. Olms, Hildesheim (1968). Gerhardt, v.C.I. (herausg.). Quoted: [16] followed by the number of the book and pagination
Leibniz, G.-W.: La profession de foi du philosophe-Confessio philosophi. Belaval, Y. (established, presented, translated and annotated). Librairie philosophique Jules Vrin, Paris (1970)
Leibniz, G.-W.: Philosophical Essays. Ariew, R., Garber, D. (trans.). Hackett, Indianapolis (1989)
Leibniz, G.-W.: Recherches générales sur l’analyse des notions et des vérités. 24 thèses métaphysiques et autres textes logiques et métaphysiques. Rauzy, J.-B. (introductions and notes). Presses Universitaires de France, Paris (1998)
Lenzen, W.: Leibniz’s calculus of strict implication. In: Srzednicki, J. (ed.) Initiatives in Logic (Reason and Argument I), pp. 1–35. Nijhoff, Dordrecht (1987)
Lenzen, W.: Das System der Leibniz’schen Logik. de Gruyter, Berlin (1990)
Lenzen, W.: Leibniz’s logic. In: Gabbay, D.M., Woods, J. (eds.) Handbook of the History of Logic. The Rise of Modern Logic: From Leibniz to Frege, pp. 1–83 (2004)
Lenzen, W.: Leibniz on alethic and deontic modal logic. In: Berlioz, D., Nef, F. (eds.) Leibniz et les puissances du langage, pp. 341–362. Librairie philosophique Jules Vrin, Paris (2005)
Marenbon, J.: Le temps, l’éternité et la prescience de Boèce à Thomas d’Aquin. Librairie philosophique Jules Vrin, Paris (2005)
Russell, B.: A Critical Exposition of the Philosophy of Leibniz. Routledge, London (1996)
Schepers, H.: Zum Problem der Kontingenz bei Leibniz: Die beste der möglichen Welten. In: Collegium Philosophicum. Studien, Joachim Ritter zum 60, pp. 326–350. Geburstag, Basel (1965)
Sleigh, R.: Leibniz & Arnauld: A Commentary on Their Correspondence. Yale University Press, New Haven (1990)
Tarski, A.: Some observations on the concepts of ω-consistency and ω-completeness. In: Logic, Semantics, Metamathematics: Papers from 1923–1938, pp. 279–295 Hackett, Indianapolis (1983)
Von Wright, G.H.: On the logic of norms and action. In: Hilpinen, R. (ed.) New Studies in Deontic Logic, pp. 3–35. Reidel, Dordrecht (1981)
Vuillemin, J.: Nécessité ou contingence. L’aporie de Diodore et les systèmes philosophiques. Les éditions de Minuit, Paris (1984)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer Basel
About this chapter
Cite this chapter
Alcantara, JP. (2012). Leibniz, Modal Logic and Possible World Semantics: The Apulean Square as a Procrustean Bed for His Modal Metaphysics. In: Béziau, JY., Jacquette, D. (eds) Around and Beyond the Square of Opposition. Studies in Universal Logic. Springer, Basel. https://doi.org/10.1007/978-3-0348-0379-3_4
Download citation
DOI: https://doi.org/10.1007/978-3-0348-0379-3_4
Publisher Name: Springer, Basel
Print ISBN: 978-3-0348-0378-6
Online ISBN: 978-3-0348-0379-3
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)