In predicate calculus there is no distinction between properties and relations---they are both just predicates. A property is a 1-ary predicate, and an n-ary relation is just an n-ary predicate. However, many analytic philosophers continue to speak of properties and relations rather than just using the combined terminology of predicates. What is the reason for this? Is it merely convenient terminology or are there principled reasons to distinguish properties from relations?

  • Please provide some evidence of research. I don't think the premise of your question is true. Aug 18 at 19:17
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    Reduction of relations to unary predicates is impossible, you need at least binary predicates for that. Some binary relations are irreducible, this is why full predicate calculus is strictly stronger than monadic predicate calculus. Under more strict rules, even some ternary predicates can be irreducible, see Peirce’s Reduction Thesis.
    – Conifold
    Aug 18 at 21:09
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    Is your question just one of terminology, i.e. why properties are not called unary relations? That's just because that's the meaning the words have in ordinary English, 'relation' suggests that there are at least two entities involved; technically a property is nothing else than a unary relation.
    – lemontree
    Aug 18 at 22:02
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    See Royse, Leibniz and the Reducibility of Relations to Properties for a historical account:""We owe to Russell, more than to any other single philosopher, a clear understanding of the nature and importance of relations... he made it abundantly clear that... dyadic relations are not reducible to monadic attributes of their terms". Specifically, Russell showed that asymmetric dyadic relations cannot be so reduced on the natural understanding of "reduction". Even on Royse's sympathetic reading of Leibniz, divine intellect is required to reduce them.
    – Conifold
    Aug 18 at 22:05
  • @Conifold, I believe you have misunderstood the direction of the proposed reduction because of the confusing way that the question was worded. He is not proposing reducing relations to properties, but properties to (unary) relations. Or rather, he is proposing a combined entity, the "predicate". of which both properties and relations are subsets. Aug 19 at 15:48

In first-order logic, it is possible to reduce n-ary predicates to (n-1)-ary predicates for n greater than two. But it is not possible to reduce binary predicates to unary predicates.

However, if you are willing to step beyond first-order logic, there are resources for accomplishing this. The problem is that there is no obvious way to specify a general semantics for higher order logics. One could at a pinch say that the relation 'loves' in "John loves Mary" is a unary attribute of the ordered pair <John, Mary>. But it is problematic to generalise this. We could accomplish it using plural quantification and say that binary relations are equivalent to unary attributes of pluralities. But again, it is difficult to find a general way to specify the semantics of plural quantification over binary relations.

David Lewis had a go at addressing this problem using third-order logic and mereology (the logic of wholes and parts) in his book Parts of Classes (Blackwell, 1991).

  • I believe you are misunderstanding his question because of his confusing wording. I believe his proposal is to view both properties and relations as instances of a single sort of entity, the predicate, of which properties are 1-ary predicates and n-ary relations are n-ary predicates. My comment to the question was to ask him why he thinks analytic philosophers have "refused" to do this. Aug 19 at 15:52
  • Well, indeed I take it as read that properties are unary predicates and relations are n-ary predicates. For the question to mean anything substantial it has to be concerned with whether n-ary predicates are reducible to unary ones.
    – Bumble
    Aug 19 at 17:27
  • I took it as an ontological question. One could ask whether, for example, "coats and hats" is a natural category or whether it is just an artificial conglomerate of two natural classes. Similarly, a predicate could be viewed as a formal or linguistic entity artificially joining two very distinct things--properties and relations--into an artificial conglomerate. My comment to the question was asking for evidence that analytical philosophers view predicates this way. Aug 19 at 21:57

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