The Kripkean explanation of aposteriori necessity: in the case of identity statements about chemical substances

ABSTRACT

In the addenda to his Naming and Necessity, Kripke provides an account of how necessary aposteriori statements are possible. In such a case, there is an apriori general principle telling us that it is necessary if true at all. Though straightforward in its broad compass, this account faces two obvious questions in its application: in each case of necessary aposteriori statements, what is the underlying principle and how is it established apriori? I treat these questions with respect to theoretical identity statements concerning chemical substances, such as ‘water is ${\mathrm{H}}_2\mathrm{O}$’. I argue that the general principle underlying the necessity of the statements is that if a chemical substance has a certain chemical composition, then it could not have had any other chemical composition. Then I defend the view that the principle is a conceptual truth by providing a novel derivation of it from the theoretical concept of chemical substance with a sufficient level of formal rigor. The logical principles required for the derivation will also be stated and defended.

KEYWORDS:

• Aposteriori necessity
• theoretical identity statements
• chemical substances
• property identity
• de re and de dicto modalities

Acknowledgments

This paper derives largely from chapter four of my dissertation Necessity, Essence, and Analyticity: Toward an Analytic Essentialist Account of Necessity delivered to the Graduate Faculty in Philosophy at the Graduate Center, City University of New York, 2022. I owe special thanks to Kit Fine for pressing me to show how (NC) might be derived from what may plausibly be taken as a definition of chemical substance and for helping me to see some of the logical problems involved therein. I would also like to express my sincerest gratitude to Graham Priest, my advisor, Michael Devitt, Melvin Fitting, and David Papineau for reading multiple versions of this paper and providing many insightful comments and discussions. This paper also benefitted from questions and comments from participants in the Fall Conference of the Korean Society of Analytic Philosophy at Sungkyunkwan University. Finally, I would like to thank two anonymous referees from this journal; their constructive comments led to a significant improvement.

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Correction Statement

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Notes

1 See, for example, Miller (2007, 323), LaPorte (2013, 9–10 and §3.2 of 2022), Nimtz (2017, 953).

2 Notice here that I am not claiming that (NC) holds true in virtue of the theoretical concept of chemical substance. In this respect, the notion of a conceptual truth is different from the traditional conception of analyticity. Principle (NC) may well hold true in virtue of what it is to be a chemical substance. But that does not preclude it from being a conceptual truth in my sense of the term. In this connection, see the discussion towards the end of Section 4 below.

3 For example, see Hale (2013, 280):

To be a pure substance, in the chemical sense of the term, just is to be matter having a certain chemical composition – i.e. to be wholly composed of atoms of a certain element, or of molecules of a certain compound. There is, accordingly, no mystery about how we may know, a priori, the general principle needed for the application of Kripke's inferential model to substances. Our knowledge that gold is a certain element – the element with atomic number 79, and that water is a certain compound – the compound composed of molecules in which two hydrogen atoms covalently bond with an oxygen atom – is of course a posteriori, but we know, courtesy of the very meaning of the term, that if a substance has a certain chemical composition, it necessarily does so.

Here Hale writes as if it is obvious that (NC) follows from the concept of chemical substance. But how exactly is it supposed to follow? With this said, the influence of Hale's discussion on my own account will become clearer later on.

4 The proof should be familiar to the present readership. See Hale (2013, 260–263) for a detailed discussion. It should also be noted that the proof can be, and has in fact been, challenged. See Priest (2021, 1882–1884).

5 For this line of analysis, see also Barnett (2000, 109), Burgess (2013, 68–69; 2021, 401) and Soames (2007, 36).

6 See Needham (2000, 2002) for discussion.

7 It should be noted that Salmon himself does not discuss the case of sets in his 1979 paper. His main concern is with Kripke's argument for the so-called necessity of material origin, which states that if a material object has its material origin in a certain hunk of matter then it could not have had another material origin (a cross-world identification criterion for material objects). His analysis is basically that this cannot be derived from the principle that in any possible world w, two material objects are identical only if they have the same material origin (the intra-world criterion of identification for material objects) without additional substantive modal assumptions. It should be clear that this intra-world identification criterion for material objects is completely analogous in its logical form to the one for sets given above, namely (In). So, I discuss the case of sets for the sake of simplicity. See pp.705–707 p.711 of his (1979) where he claims that the same consideration applies to the case of chemical substances as well. For Salmon's discussion of chemical substance, see pp.166–169 and pp.183–189 of his (1981).

8 See Hughes and Creswell (1996, 250–254) for a detailed discussion.

9 See Burgess (2013, 67–68), Salmon (1981, 177–178) and Soames (2006, 714) for a similar analysis.

10 The idea here is that a kind may be ascribed a property in virtue of its members having a corresponding property. See Liebesman (2011, 419) for a similar view. I thank an anonymous referee for calling my attention to this paper.

11 Strictly, the equivalence between (2) and (3) requires an additional assumption that there is an actual instance of ${\mathrm{H}}_2\mathrm{O}$. See the derivation of (NC) below.

12 Here I am using the term ‘de re modality’ in the broad sense that a statement involves de re modality just in case it quantifies into a modal context, regardless of whether the quantification is first-order or second-order. However, it is perhaps helpful to make a distinction between the two cases. Note that (Cr) contains a first-order quantification into a modal context; so, it states what is necessary of individual sets. In contrast, (2*) contains a second-order quantification into a modal context. So, it states what is necessary of water as a kind of matter, but it has no implication on what is necessary of particular instances of water. So, there is an important difference between first-order and second-order de re modalities. For want of a better term, I should perhaps call the latter de qualitate modality.

13 I thank an anonymous referee from this journal for pressing me on this objection.

14 Here, of course, the descriptions should be given the wide scope reading.

15 I also think that that the distinction between specificative and descriptive identifications of properties can help clarify some of the issues concerning what it is for a predicate to be rigid as discussed in Soames (2002, 241–263) and LaPorte (2013, 89–117) For they concern in part how we may understand statements involving descriptive identifications of properties, such as ‘her eyes are the color of the cloudless sky at noon’, ‘white is the color of the Antarctica’, etc. But lack of space prevents me from discussing their views in detail here; a proper investigation into this must wait for another day.

16 This so-called essentialist idea has become increasingly popular in the recent literature on metaphysical necessity under the influence of Fine (1994), Hale (2002, 2013) and Lowe (2008, 2012). It should be noted that Lowe thinks that that the identity statements concerning natural kinds are in general not necessary, though he is certainly in the essentialist camp. See his (2007) for detailed discussion.

17 The argument assumes that an interpretation of an open formula is given by its satisfaction condition at each possible world.

18 A bit more formally, we need to show the validity of the following principle: $\mathit{if}\lambda \mathit{x}\varphi \mathit{\left(}\mathit{x}\mathit{\right)}\approx \lambda \mathit{x}\psi \mathit{\left(}\mathit{x}\mathit{\right)}\mathrm{then}\lambda \mathit{x}\xi \mathit{\left(}\mathit{x}\mathit{\right)}\approx \lambda \mathit{x}\xi \mathit{\left[}\varphi \mathit{/}\psi \mathit{\right]}\mathit{\left(}\mathit{x}\mathit{\right)}\mathit{,}$where $\lambda x\xi \left[\varphi /\psi \right]\left(x\right)$ is the result of substituting ϕ in zero or more (but not necessarily all) places for ψ in $\lambda x\xi \left(x\right)$. The principle is valid if we give an open formula $\varphi \left(x\right)$ and the corresponding lambda term $\lambda x\varphi \left(x\right)$ the same interpretation, namely the satisfaction condition at each possible world. To see this, suppose that $\lambda x\varphi \left(x\right)\approx \lambda x\psi \left(x\right)$. Then $\varphi \left(x\right)$ and $\psi \left(x\right)$ have exactly the same satisfaction condition at each possible world, which in turn implies that $\xi \left(x\right)$ and $\xi \left[\varphi /\psi \right]$ have the same satisfaction condition at each possible world. Hence the corresponding lambda terms $\lambda x\xi \left(x\right)$ and $\lambda x\xi \left[\varphi /\psi \right]\left(x\right)$ will receive the same interpretation. Therefore, $\lambda x\xi \left(x\right)\approx \lambda x\xi \left[\varphi /\psi \right]\left(x\right)$. It might be objected that this proof relies on the implicit understanding of properties as intensions, which is suspicious. Though I am somewhat sympathetic to this objection, I also believe that any reasonable calculus of properties should validate the above principle. For our purpose, it is sufficient to note that the principle is valid in a natural extension of the standard semantics for second-order modal logic.

Additional information

Funding

This work was supported by the Ministry of Education of the Republic of Korea and the National Research Foundation of Korea (NRF-2022S1A5B5A17044347).