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Leibniz's Theory of Propositional Terms: a Reply to Massimo Leibniz’s Theory of Propositional Terms: A Reply to Massimo Mugnai Marko Malink, New York University Anubav Vasudevan, University of Chicago n his contribution to this volume of the Leibniz Review, Massimo Mugnai provides a very helpful review of our paper “The Logic of Leibniz’s Generales IInquisitiones de Analysi Notionum et Veritatum.” Mugnai’s review, which contains many insightful comments and criticisms, raises a number of issues that are of central importance for a proper understanding of Leibniz’s logical theory. We are grateful for the opportunity to explain our position on these issues in greater depth than we did in the paper. In his review, Mugnai puts forward two main criticisms of our reconstruction of the logical calculus developed by Leibniz in the Generales Inquisitiones. Both of these criticisms relate to Leibniz’s theory of propositional terms. The first concerns certain extensional features that we ascribe to the relation of conceptual containment as it applies to propositional terms. The second concerns the syntactic distinction we draw between propositions and propositional terms in the language of Leibniz’s calculus. In what follows, we address each of these criticisms in turn, beginning with the second. Propositions and Propositional Terms One of the most innovative aspects of the Generales Inquisitiones is Leibniz’s in- troduction into his logical theory of the device of propositional terms. By means of this device, Leibniz proposed to represent propositions of the form A is B by terms such as A’s being B. Leibniz regarded these propositional terms as abstract terms (abstracti termini, A VI 4 987), with the important qualification that they are merely “logical” or “notional” abstracts (A VI 4 740; cf. 992). As Mugnai emphasizes in his review, Leibniz took the theory of propositional terms to be one of his principal contributions to the study of logic, highlighting their philosophical significance in his essay De Abstracto et Concreto and again, several years later, in the Noveaux Essais. The Generales Inquisitiones stands out among Leibniz’s writings in that it contains his first systematic treatment of propositional terms and the most detailed exposition of the principles governing their logical operation. The Leibniz Review, Vol. 27, 2017 139 MARKO MALINK & ANUBAV VASUDEVAN In the Generales Inquisitiones, Leibniz first explores the possibility of assimi- lating propositions to terms in §§61–75. He concludes this preliminary discussion on an optimistic note, proclaiming that, if such an assimilation could be effected, it would be of great value to his logic: If, as I hope, I can conceive of all propositions as terms, and of all hypotheti- cal propositions as categorical, and if I can give a universal treatment of them all, this promises a wonderful ease in my symbolism and analysis of concepts, and will be a discovery of the greatest moment. (Generales Inquisitiones §75) In Leibniz’s view, when a proposition is “conceived of” as a term, the result is a new, propositional term.1 Just like other terms, propositional terms can serve as the syntactic constituents A and B in propositions of the form A is B and A coincides with B. For example, if C’s being D and E’s being F are the propositional terms generated from the propositions C is D and E is F, respectively, then C’s being D is E’s being F is a well-formed proposition of Leibniz’s calculus (§138). By contrast, the propositions C is D and E is F cannot themselves appear as syntactic constituents in further propositions. Thus, for example, C is D is E is F is not a well-formed proposition of Leibniz’s calculus. In his review, Mugnai is skeptical that Leibniz draws such a sharp distinction in the syntax of his calculus between propositions and propositional terms. In particular, he raises doubts that Leibniz consistently enforces the requirement that only propositional terms, and not propositions, can take the place of the schematic letters ‘A’ and ‘B’ in propositional forms such as ‘A is B’. We agree with Mugnai that, in the Generales Inquisitiones, Leibniz does not systematically enforce this requirement prior to §61 of the treatise. In these early portions of the treatise, Leibniz states repeatedly that the schematic letters ‘A’ and ‘B’ can stand for either terms or propositions (A VI 4 748 n. 6, §4, §13, §35, §55). These statements would not be correct if the above syntactic requirement were in force. Instead, what Leibniz ought to have said is that ‘A’ and ‘B’ can stand for either non-propositional or propositional terms (or, equivalently, that they can stand for any terms, propositional or otherwise). In response to this apparent problem with our reconstruction of Leibniz’s calculus, it is worth noting that, in the Generales Inquisitiones, all of Leibniz’s claims to the effect that the schematic letters of the calculus can stand for either terms or propositions occur prior to his discovery (inventum, §75) of propositional terms in §§61–75. Thus, it is perhaps not surprising that, prior to this discovery, the distinction between propositions and propositional terms is virtually absent from the The Leibniz Review, Vol. 27, 2017 140 REPLY TO MASSIMO MUGNAI Generales Inquisitiones. Leibniz does mention propositional terms in the opening paragraph of the treatise, remarking that the only abstract terms to be admitted in his calculus are “those which are logical or notional, such as . A’s being B” (A VI 4 740). However, as can be seen from the apparatus to the text in the Academy Edition, this reference to propositional terms is a later addition by Leibniz absent from the original version of the opening paragraph. In this original version, Leibniz had excluded from consideration all abstract terms without qualification, only later adding the exception for propositional terms. It is not clear when Leibniz added this exception, and it may well be the case that he did so only after the “discovery” of propositional terms in §§61–75. Thus, there is no evidence to suggest that, when Leibniz began writing the Generales Inquisitiones, he already had in mind the distinction between propositions and propositional terms.2 Given this, we should expect the syntactic distinction between propositions and propositional terms to begin to manifest itself in the Generales Inquisitiones only after §§61–75. Now, Mugnai observes that, in §75 itself, immediately after introducing propositional terms, “Leibniz writes that with the letter ‘A’ he under- stands either a term or a proposition” (p. 134). The relevant portion of §75 reads as follows: In general, I call a term false if, in the case of incomplex terms, it is an impossible term … and if, in the case of complex terms, it is an impossible proposition or, at any rate, a proposition which cannot be proved. … Thus, by ‘A’ I understand either an incomplex term or a proposition. (Generales Inquisitiones §75) In the final sentence of this passage, Leibniz states that ‘A’ can stand for either an incomplex term or a proposition. The added qualification ‘incomplex’ refers to the distinction between complex and incomplex terms adduced in the first sentence of the passage. As Leibniz makes clear elsewhere in the Generales Inquisitiones, this distinction corresponds to that between propositional and non-propositional terms.3 Since the natural complement to the class of incomplex terms is that of complex terms, the most plausible reading of the final sentence is that ‘A’ can stand for either an incomplex (non-propositional) term or a complex (propositional) term. This reading is further supported by the fact that Leibniz seems to equate complex terms with propositions in the first sentence of the passage. Indeed, if the label ‘proposition’ in the final sentence was not meant to cover complex terms, Leibniz would be omitting these terms from the scope of the schematic letters of his calculus, which is clearly not his intent in §75. The Leibniz Review, Vol. 27, 2017 141 MARKO MALINK & ANUBAV VASUDEVAN This reading also helps to explain why Leibniz added the qualification ‘incom- plex’ to the heading ‘terms’ in the last sentence of the passage just quoted. Had he omitted this qualification, the classes of ‘terms’ and ‘propositions’ would fail to be disjoint, given that the latter is here understood to include complex terms. If the class of ‘propositions’ was not meant to include complex terms, it would be unclear why Leibniz felt the need to add this qualification instead of simply repeating his earlier claims that ‘A’ can stand for either a term or a proposition.4 Mugnai is, of course, right to point out that the literal reading of the final sentence implies that ‘A’ can stand for a proposition. But given the overall context, it is more likely that Leibniz’s intention here is to make an oblique reference to propositional terms by citing the propositions from which these terms are generated. In the later sections of the Generales Inquisitiones, at any rate, Leibniz emphasizes that the propositions of his calculus express relations between terms rather than proposi- tions. Thus, for example, he writes that “a proposition is that which states what term is or is not contained in another” (§195; similarly, §184). Moreover, in these later sections, Leibniz is careful to ensure that a proposition is first transformed into a propositional term before it is used as a syntactic constituent in the construction of further propositions (§198.7). In our view, such circumspection on Leibniz’s part is indicative of his mature conception of propositional terms, according to which a sharp syntactic distinction is to be drawn between these terms and the propositions from which they are generated. This conception, which evolves gradually over the course of the treatise and represents the considered position finally adopted by Leibniz in the Generales Inquisitiones, leads him eventually to abandon his earlier claims to the effect that ‘A’ can stand for a proposition.
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