How Involved do You Want to be in a Non-symmetric Relationship?

Abstract

There are three different degrees to which we may allow a systematic theory of the world to embrace the idea of relatedness—supposing realism about non-symmetric relations as a background requirement. (First Degree) There are multiple ways in which a non-symmetric relation may apply to the things it relates—for the binary case, aRb ≠ bRa. (Second Degree) Every such relation has a distinct converse—for every R such that aRb there is another relation R* such that bR*a. (Third Degree) Each one of them applies in an order to the things it relates—with regard to the state that results from R's applying to a and b, either R applies to a first and b second, or it applies to b first and a second. Whereas the first degree is near-indubitable, embracing the second or third generates unwholesome consequences. The second degree embodies a commitment to the existence of a superfluity of distinct converses and states to which such relations give rise. The third degree embodies commitment to recherché facts of the matter about how the states that arise from the application of one non-symmetric relation compare to any other. It is argued that accounts that purport to offer an analysis of the first degree generate unwelcome second or third degree consequences. This speaks in favour of our adopting an account of the application of relations that's not an analysis at all, an account that takes the first degree as primitive.

Keywords:

• relations
• converse
• analysis
• Fine
• Russell
• Williamson

Notes

¹ For a defence of the view that contemporary physics relies upon relations, see Butterfield [2006, 2011]. Butterfield argues that a physical theory's fundamental quantities cannot be construed as intrinsic properties of space-time points but must be conceived as essentially relational in character. Butterfield extends his case not only to velocity but to other mechanical notions such as stress, strain, and elasticity, that are represented using vectors and tensors. For the corresponding claim that contemporary mathematics relies upon relations see Shapiro [1991: 96–118, 221].

² The idea that aRb is different from bRa relies upon an appreciation of the fact that there are two distinct states that potentially arise from the application of non-symmetric R to a and b. Appreciation of this fact doesn't require us to speak a language that relies upon word order to convey information about how relations configure their terms. Latin speakers also have this appreciation when they understand that ‘Marius amat Mariam’ describes a quite different state from ‘Maria amat Marium’ where the difference between aRb and bRa is conveyed by means of declining the nouns.

³ Contra, e.g., the treatment of passive sentences presented in Parsons [1990: 91–2].

⁴ Deploying Church's lambda notation to represent converse functions [1941: 7].

⁵ Here, a language like English which exploits word-order to encode information about how a and b are related may give rise to an appearance of (third-degree) order where, e.g., Latin or Sanskrit avoids doing so because this information is carried using inflected nouns that can appear in any order.

⁶ See Williamson's remark that a binary relation ‘R has two argument positions’ and that we may ‘speak of putting the entities which R is said to relate themselves into these argument positions’ [1985: 257].

⁷ A similar difficulty afflicts Bergmann [1981: 146–7] and Chisholm [1996: 51–3] who seek to mimic the Wiener-Kuratowski account of sequences as classes of classes.

⁸ See MacBride [2007: 47–53] for further criticism of Fine's analysis.

⁹ Referring to an earlier draft of the present paper, Gaskin and Hill [2012: 184] also recommend taking order as primitive but their reasons for doing so differ from those presented here. They reject Russell's ‘directionalism’ because of the problem of converses but they do not distinguish the second from the third degree, fail to recognize that Russell may avoid converses anyway, nor acknowledge directionalism's independent third-degree consequences. Following Fine, they reject positionalism because of its commitment to argument positions and the problems it encounters with the identity of symmetric states; but they fail to note the vicious regress that also threatens positionalism. They reject Fine's ‘anti-positionalism’ because the substitution relation ‘cannot create’ but merely preserves structure. This last objection may beg the question against Fine because it's not obvious that anti-positionalism aims to reduce facts about structure to facts about substitution. But for present purposes nothing hangs upon this. My objection to Fine, that anti-positionalism violates the possibility of lonely applications of non-symmetric relations, applies even if their objection to Fine fails.

¹⁰ I am especially grateful to Kit Fine and Herbert Hochberg. I would also like to thank audiences at the Universities of Barcelona, East Anglia, Geneva, Liverpool, London and Stirling, and Jeremy Butterfield, Ghislain Ghuighon, Jane Heal, Frédérique Janssen-Lauret, Nick Jones, E. J. Lowe, Laurie Paul, Bryan Pickel, Hugh Mellor, Stewart Shapiro, Alan Weir, Tim Williamson and two anonymous AJP referees for discussion. This paper was written during a period of leave supported by the AHRC.