Solving a puzzle of definition

ABSTRACT

This paper concerns the question of which logical principles hold for real definitions. Recently, Samuel Elgin has presented five principles concerning real definitions that seem initially plausible. He has shown them to be jointly inconsistent. This gives rise to a puzzle that can only be solved by denying one of the principles. In this paper, I argue against Elgin's principle of expansion, which concerns substituting a definiens for its definiendum within the definiens of a further definition. I show that this principle fails for every irreflexive notion of definition and proposes a replacement that allows to restore consistency and solve the puzzle.

KEYWORDS:

• Essence
• real definition
• logic
• transparency

Acknowledgments

The author wishes to thank an anonymous referee for excellent comments that significantly improved this paper.

Disclosure statement

No potential conflict of interest was reported by the author.

Notes

1 Note that from a dialectical point of view, denying Coextensionality is not a viable route for someone who wishes to save Elgin's puzzle, for Coextensionality (apart from being very hard to deny) is an integral ingredient of the puzzle.

2 The analogous claim that factive grounding should be defined in terms of non-factive grounding is endorsed in e.g. Litland (2017).

3 Even if you are inclined to hold that there are some true reflexive explanations, you presumable will not be prepared to accept every reflexive explanation that a variant of this example might commit you to.

4 Thanks to an anonymous referee for pressing this point.

5 It is for a closely related reason that in his account of real definition, Fabrice Correia calls for a restriction of Leibniz's Law (see Correia 2017, 57).

6 Note that y is supposed to be of the type of a and that, according to Elgin, Application Congruence and the other principles used in the derivation of Expansion ‘are to be read either as schemata with applications in every type, or else as terms whose type is contextually evident’ (Elgin 2022, 11). This legitimises choosing an instance for F that yields a further predicate when applied to a (rather than a sentence, as the occurrence of ‘Fa’ in the consequent of Application Congruence might suggest).

7 Thanks to an anonymous referee for pressing this point.