Abstract

This paper discusses Leibniz’s treatment of the term ‘nihil’ that appears in some logical papers about the notion of Real Addition. First, the paper argues that the term should be understood as an empty (singular) term and that sentences with empty terms can be true (§2). Second, it sketches a positive free logic to describe the logical behaviour of empty terms (§3). After explaining how this approach avoids a contradiction that threatens the introduction of the term ‘nihil’ in the Real Addition calculus (§4), and how this approach should be understood within Leibniz’s philosophy (§5), the paper assesses the prospects of such an approach with regard to two fundamental issues in Leibniz’s thought: the fictional nature of infinitesimals (§6), and the occurrence of the term ‘nothing’ in the proof of the existence of God that we find in the New Essays (§7).

Highlights

  • In the literature there are different logical systems that allow for empty terms; in our case the system known as Positive Free Logic (PFL) will do

  • Based on the passage of the Generales Inquisitiones quoted at the beginning of this paper, Mates (1972) argues that sentences with non-denoting terms are always considered false by Leibniz

  • We have here developed a different approach, according to which Leibniz holds that some sentences with empty terms can be true

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Summary

Introduction

A proposition which contains a non-denoting subject-term cannot be true, because there is no object to which we can attribute the property expressed by the predicate. It is less clear what Leibniz had in mind with non-denoting terms. Is he speaking of terms that refer to something which is not actual, but still possible, or he is speaking of terms which are empty by logical necessity, i.e. terms which imply a contradiction such as ‘the greatest velocity’ or ‘the infinite number’? Propositions that contain contradictory terms, such as ‘the greatest velocity’ or ‘the infinite number’, are always false

The Empty Term ‘Nihil’
Another Characterization of ‘nihil’
A Logic for Nothing!
Language of PFL
Syntax of PFL
Semantics of PFL
Discriminating Actual from Merely Possible Objects
The Formal Machinery at Work 1
The Formal Machinery at Work 2
The Formal Machinery at Work 3
Conclusion
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