Towards leibnizian possibility. Formal frame of modal theory of individual concepts

Studia Philosophiae Christianae UKSW 49(2013)3 KORDULA ŚWIĘTORZECKA Instytut Filozofii UKSW, Warszawa Towards leibnizian possibility. Formal frame of modal theory of individual concepts1* Abstract. In the presented analyses we propose a formal complement to a certain version of the semantics of possible worlds inspired by Leibniz’s ideas and provide an adequate logic of it. As the starting point we take the approach of Benson Mates (Leibniz on possible worlds). Mates refers to Leibniz’s philosophy, but also uses tools of contemporary semantics of possible worlds and elaborates on an original conception of predication due to which possible worlds can be identified with collections of certain concepts, and not individuals. We complete a fragmentary description given by Mates in order to analyze if his conception allows for the establishment of this specific idea of a possible world. Our first step is to define a notion of the individual concept and describe possible world semantics in which possible worlds consist of individual concepts of compossible individuals (s-worlds). Our second step is to choose some version of modal free logic with the identity (S5MFLID), which is complete in our reformulation of Mates’ semantics. The connections between standard interpretation of S5MFLID and semantics inspired by Mates show that our logic does not distinguish s-worlds from i-worlds – counterparts of s-worlds that are collections of individuals. Keywords: formal ontology, Leibniz, theory of concepts, possible world semantics, modal free logic The paper is a slightly modified translation of my Polish text O możliwych światach pojęć jednostkowych. Formalna rekonstrukcja koncepcji B. Matesa [On possible worlds of individual concepts. A formal reconstruction of the idea of B. Mates] in: Nauka i język. (seria druga). Księga pamiątkowa Marianowi Przełęckiemu w darze na dziewiędziesięciolecie urodzin [Science and Language (Series IInd). Commemorative book for Marian Przełęcki as a gift of the ninetieth anniversary of the birth], ed. by A. Brożek, J. Jadacki, Norbertinum, Lublin (in print), 125-137. 1 72 Kordula Świętorzecka [2] 1. Introduction. 2. Possible s-worlds. 3. Languages and interpretation. 4. Logic. 4.1 System MS5C. 4.2 System S5MFLID 5. From worlds of individual concepts to worlds of individuals. 1. Introduction The following analyses raise the issue of modality, which is a compound of the subject and metatheoretical matters: we propose a formal complement to a certain version of the semantics of possible worlds inspired by Leibniz’s ideas and provide a logic adequate to it. Contemporary modal philosophical logics and their set-theoretical interpretations meet with skepticism among philosophers, who claim that such approaches do not capture the intended philosophical content of what is traditionally understood as modality. It is worth highlighting that sometimes such arguments are justified, however, in general, the matter is complicated enough because this philosophical content keeps escaping attempts to be satisfactorily precise. One such attempt was undertaken by B. Mates2, whose conception is of interest here3. Mates’ idea is interesting, because it refers to Leibniz’s philosophy and also uses tools of contemporary semantics of possible worlds and elaborates on an original conception of predication, due to which possible worlds can be identified with collections of certain concepts and not individuals. Mates, however, provides only a fragmentary account of his idea, and the present work is aimed at completing it in order to analyze if Mates’ conception allows for the establishment of this specific idea of a possible world. 2. Possible σ-worlds The conception considered here is based on the assumption that possible worlds are determined by so called individual concepts. AccorB. Mates, Leibniz on possible worlds, in: Logic, Methodology and Philosophy of Science, III, Studies in Logic and the Foundations of Mathematics, Logic, ed. B. van Rootselaar, J. F. Staal, Amsterdam 1968, 507–529. 2 My inspiration comes from the paper of Prof. M. Przełęcki, On Possibility and Possible Worlds, Poznań Studies in the Philosophy of the Sciences and the Humanities 4(2010)1–4, 27–36. 3 The conception considered here is based on the assumption that pos The conception considered here is based on the assumption that pos determined by so called individual concepts. According to Mates, such an ap determined by so called individual concepts. According to Mates, 73 such an ap Towards Leibnizian possibility [3] the justification of certain key theses of Leibnizian theory of possible worlds the justification of certain key theses of Leibnizian theory of possible worlds 4 Leibnizian theory concepts. ding to Mates, suchof anindividual approach allows for 4the justification of certain Leibnizian theory of individual concepts. key theses of Leibnizian theory of possible worlds together with the Let us reconstruct the initial steps sketched by Mates. 4 Let us reconstruct initial steps sketched by Mates. Leibnizian theory ofthe individual concepts. us reconstruct thei}initial by Mates. WeLet assume that D={d a setsketched of individual entities, which constitute t i∈I is steps WeWe assume is aaset setofofindividual individual entities, which assumethat thatD={d D={di}i}i∈I is entities, which con-constitute t iÎI real world, and are the realizations of individual concepts. Mates assumes th stitute the domain of the the realizations real world, and are the realizations of Mates indivi- assumes th real world, and are of individual concepts. dual concepts. Mates d assumes that Ditsisindividual infinite. concept: For each individual i, we define ForFor each individual concept: each individualdid, iwe , wedefine defineits its individual individual concept: Def (σ). σ(di) ={X ⊆ D: di ∈ X}. Def (σ). σ(di) ={X ⊆ D: di ∈ X}. CON is a set of all individual CON a set individualconcepts: concepts: CON is aisset ofof allallindividual concepts: Def (CON). CON = {X:∃i∈I (X=σ(di))}. Def (CON). CON = {X:∃i∈I (X=σ(di))}. as as well as course, infinitethen then also also every Of Of course, if if DDisisinfinite every concept conceptofofdidisi isinfinite infinite Of course, if D is infinite then also every concept of d is infinite well as CON. Let us note that there are no concepts in CONi which are as well as that there are no concepts in CON which are unrealized. unrealized. that there are no concepts in CON which are unrealized. Mates adoptsLeibniz’s Leibniz’s idea idea of of compossibility of of thethe objects Mates adopts compossibilityofofexistence existence Mates adopts Leibniz’spossible idea ofworlds, compossibility of existence objects which constitute and introduces, in the setofofthe objects possible worlds, and relation introduces, in the setof of CON, conthe twoargum CON, the two-argument of compossibility individual possible worlds, and introduces, in the set of CON, the twoargum cepts Γ Ì CON´CON. (...truncated)