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According to the Stanford Encyclopedia of Philosophy, the following holds: Relations and relational properties can be distinguished. A relation is borne from one thing to another thing. A relational property is the property of bearing a relation to something. To illustrate what is meant, consider the following example: Marriage is a relation between two people. Yet being married to Cleopatra is a relational property held by Anthony. While being married to Anthony is a relational property had by Cleopatra (https://plato.stanford.edu/entries/relations/).

With this said, consider the following from Euclid’s Elements: For all X and Y, if X is prime to Y, then the greatest common factor between X and Y is 1. If I were to translate this statement into a formula of first order logic, there seems to be an ambiguity as to how to translate it, particularly the part where it says X is prime to Y. I can either translate X is prime to Y as a predicate of arity 2 or I could translate it into two unary predicates, i.e. X has the property of being prime to Y and Y has the property of being prime to X. Which would be better, translating a relation into a predicate of arity 2 or translating a relation into two unary predicates?

Or consider the statement X equals Y: Should it be translated as a predicate of arity 2 or two unary predicates?

Or consider the statement, X is the greatest prime number, should it be translated as a predicate of multiple arities or should it be translated as a unary predicate?

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    Equality: it ia binary predicate Eq(x,y), but we can obviously consider the property if being "equal to y": Eq_y(x). Commented May 29 at 6:00
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    You can only translate it into an unary predicate when one of your variables, X or Y, is fixed. And you can never translate it into a pair of unary predicates because then both of them have to be fixed. If you want to keep both X and Y as variables then you have no choice but to translate it into a binary predicate P(X,Y). You will then get relational properties (as unary predicates) by replacing one of the variables with a constant, say P(X,C) is a property domain elements can have (in relation to C). Predicate logic does not allow you to index predicates by domain variables.
    – Conifold
    Commented May 29 at 6:20

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In predicate logic you can take an (n+1)-arity predicate and rewrite it using n-arity predicates without loss, but only down as far as n=2. So, for example, you can rewrite a ternary predicate using binary predicates, but you cannot translate from a binary predicate to unary predicates without loss of expressive ability. The procedure for doing this translation is related to the technique of currying. It is also somewhat more distantly related to the process of normalising a relational database to sixth normal form.

To take your example, the binary predicate, Married(cleopatra, anthony) is not equivalent to the conjunction of the unary predicates Married-to-anthony(cleopatra) and Married-to-cleopatra(anthony) since the unary predicates do not allow us to express propositions such as, "there are two persons such that each is married to the other", or, "for any two persons x and y, x is married to y if and only if y is married to x". A similar consideration applies to the binary predicate 'equals'. It cannot be replaced with unary predicates. On the other hand, 'greatest prime number' is a unary predicate: it is a property of numbers, albeit one that is not satisfied.

That said, in some cases it is possible to express relations using unary predicates by extending the logic. This is the approach taken in plural quantification, which is an extension of standard logic. But even this does not provide a fully general way to translate statements about relations into statements about properties.

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A relation between two sets X and Y is a subset R of the Cartesian product

R subset X x Y

Here R is the set of all ordered pairs (x,y) from XxY such that x relates to y.

Not all relations are symmetric like the relations “is married to” or “greatest common divisor of x and y” or “equality”. In general the inverse relation R^(-1) is different from the relation R. As a consequence, in general one cannot split R into a set of elements satisfying a given unary predicate.

The predicate “is the greatest prime” is not a relation but a unary predicate. Moreover, it is an empty predicate: There is no greatest prime.

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  • Your phrasing of “greatest prime” is accurate but I feel we lose a lot of information with that word choice. Something more informative is the following: X is prime and X is greater than all prime numbers excluding itself. Thus, when you negate the predicate it should translate to if X is prime then X is less than or equal to some prime number, including itself.
    – AUTIST INC
    Commented May 29 at 9:17
  • In other words, you can distribute the negation in the response I just wrote.
    – AUTIST INC
    Commented May 29 at 9:18
  • @AUTISTINC I did not agree with your negation in your previous question philosophy.stackexchange.com/questions/113401/…
    – Jo Wehler
    Commented May 29 at 9:24
  • You didn't provide a good reason why you didn't agree with it though. I mean if we consider the predicate X is the greatest prime number, the negation of it would be X is not the greatest prime number. Yet that phrasing of the negation doesn't allow us to distribute the negation though.
    – AUTIST INC
    Commented May 29 at 9:36
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    @AUTISTINC I now added to my answer to your previous post the negation.
    – Jo Wehler
    Commented May 29 at 9:44

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