The July budget lowered the UK’s controversial benefit cap. The coalition government justified the cap with the following claim: ‘no family should receive welfare income larger than the average family’s in-work income’.
The cap claim makes a serious logical mistake: it slips from an absolute term, ‘family’, to a relative term, ‘average family’. Whether my family is average depends on our relation to other families; but whether we are a family doesn’t. That’s why a family receiving benefits might be larger than the average family and be made destitute by the cap.
Clearly, it’s crucial to distinguish absolute terms, like being a family, being a human, being an animal, from relative ones like being a parent, being larger or being average. So: what is a relative term?
I’m looking at how ancient philosophers answered this question for my British Academy Post-Doctoral Fellowship. In this post, I explain my new research (Duncombe 2015) into Aristotle’s answer(s).
Aristotle starts with a simple idea. A parent (e.g.) is parent of something: an offspring. To be a parent is to be parent of an offspring. So, in general, relative terms are of something (Categories 7 6a36). Parents and offspring need not be matched one-to-one, like knives and forks in a table setting. Each of my parents has three children; each of us offspring has two parents. But you’ll never find a parent without offspring and never find an offspring without parents!
Taken with a fundamental view Aristotle has about what there is in the world, this simple idea about relative terms ends up causing him serious trouble.
Aristotle thinks that there are basic things in the world, ‘substances’. Substances are basic because, while other things depend on substances, substances depend on nothing. Being red (e.g.) depends on things; there is no redness without a red table, red chair, red animal (etc.). But there could be a table, chair, animal (etc.), without any redness. Tables, chairs, animals (etc.) are substances because they’re independent.
Can a relative term be a substance? No. Being a parent is a relative term. But being a parent depends on something being an offspring. Since being a parent depends on something else, it is not independent. Since it’s not independent, a parent cannot be a substance. And the same goes for any other relative: no substance is a relative.
Now we can see how relatives get Aristotle into trouble. His simple idea makes some relatives substances. In particular, parts of substances. A human substance has parts: hands, arms, head (etc.). These parts are substances too, so hand is a substance. But parts are of something. A human hand is a hand of a human. According to Aristotle’s simple idea, if a hand (etc.) is of something then a hand is a relative. So a hand is a substance and a relative. Aristotle hits a contradiction.
What does Aristotle do next? Until now, scholars have thought that Aristotle replaces his simple idea with a narrower idea of relatives (at Categories 7 8a31-2).  But this doesn’t wash: Aristotle returns to the simple idea again and again, as if he sees it as unproblematic (Cat. 8, 11a24-5; Topics 146b3-4;Metaphysics v 15). My research offers a new approach.
Aristotle develops two ways to think about relatives: top-down and bottom-up. I formulate this precisely in Duncombe 2015, but here is rough idea. Looking at a parent top-down, we don’t know who the parent is, only that s/he is a parent. But, since s/he is a parent, we know that s/he has an offspring, although we, of course, don’t know who that offspring is, or even if the child is a son or daughter. On the other hand, looked at from the bottom-up, we do know who the parent is. If we think about ‘parents’ from the bottom-up we are thinking about some specific parent, say my mum. And if you know my mum, you know that she has sons.
How does this help Aristotle? Take his troublesome case, the hand. Looked at ‘top-down’ what do we know about a hand? Not much. We don’t even know who the hand belongs to. We do know that whoever or whatever the hand belongs to has a hand, but that’s about it. We don’t know whether the thing that has a hand is a human – it could be a robot or a monkey. The handed thing need not be a substance. In this case, the contradiction does not arise: the hand is a relative, but not (part of) a substance, so it’s not both a relative and a substance.
Viewed ‘bottom-up’, of course, we do pick a specific hand, (e.g.) my hand. My specific hand is just a hand; in so far as it is specific, it is not of something (Categories 7, 8a18-19). So my hand does not meet the standard for being a relative. Viewed ‘bottom up’ my hand is not a relative, so, again, Aristotle avoids trouble: my hand is not both a relative and a substance.
It seems that the perspective from which we view a relative term makes the difference to whether Aristotle’s simple idea works. Is this a problem? Aristotle talks about ‘relative terms’ rather than ‘relations’. What’s the difference? Can we do away with these troublesome relative terms in favour of better-behaved relations? But whatever the answer to these questions, we can see why the benefits cap claim goes wrong.
Matthew Duncombe is a British Academy Post-Doctoral Fellow at Durham University. He has published on a range of topics in ancient philosophy and is working on a book on relatives and relativity. You can follow him on Twitter: @mbduncombe.
Ackrill, J. L. (1963) (tr.), Aristotle’s Categories and De Interpretatione. Oxford.
Duncombe, M. (2015) ‘Aristotle’s Two Accounts of Relatives in Categories 7’, Phronesis 50:
Harari, O. (2011), ‘The Unity of Aristotle’s Category of Relatives’, Classical Quarterly 61: 521-37.
Hood, P. (2004), Aristotle on the Category of Relation. Lanham, Maryland.
Sedley, D. (2002), ‘Aristotelian Relativities’ in J. Brunschwig, M. Canto-Sperber and P. Pellegrin (eds.), Le style de la pensée: Recueil des texts en homage à Jacques Brunschwig (Paris), 324-52.
 (Ackrill 1963, 102; Sedley 2002, 334; Hood 2004, 38; Harari 2011, 535).