The Logic of Attributional and Relational Similarity

In a previous post, I discussed the distinction between attributes and relations:

An attribute is a characteristic of an entity, whereas a relation is a connection between two or more entities. In logic, we can define an attribute as a predicate with one argument and a relation as a predicate with two or more arguments. The distinction between attributes and relations can be unclear. For example, the colour red may be seen as an attribute, RED(X), or a relation, REDDER_THAN(X, Y).

This leads to a distinction between attributional similarity and relational similarity. Two things, X and Y, are attributionally similar when the attributes of X are similar to the attributes of Y. Two pairs, A:B and C:D, are relationally similar when the relations between A and B are similar to the relations between C and D. I’ve been thinking about the rules that govern attributional and relational similarity.

Let A:B::C:D be the assertion that A:B and C:D are relationally similar. The expression A:B::C:D is usually read as “A is to B as C is to D”. This is called a proportional analogy. Aristotle knew that proportional analogies obey certain logical rules:

(1) A:B::C:D → B:A::D:C
(2) A:B::C:D → A:C::B:D

From (1) and (2), we can derive:

A:B::C:D → C:D::A:B

More recently, these rules have been studied by Yves Lepage and others. Rule (1) seems clear, but rule (2) is sometimes odd. It seems that rule (2) is reasonable when the items (A, B, C, and D) are all of the same general type, but it is less reasonable when they are different types:

uncle:aunt::brother:sister → aunt:uncle::sister:brother [same types: kin, rule (1)]
uncle:aunt::brother:sister → uncle:brother::aunt:sister [same types: kin, rule (2)]
dog:bark::cow:moo → bark:dog::moo:cow [different types, rule (1)]
dog:bark::cow:moo → dog:cow::bark:moo [different types, rule (2)]

Let X~Y be the assertion that X and Y are attributionally similar. When X and Y have a very high degree of attributional similarity, we call them synonyms. I have argued that attributional similarity can be reduced to relational similarity, but not vice versa. Here is one way to do the reduction:

(3) X~Y = (by definition) for all Z, X:Z::Y:Z

In other words, X and Y are attributionally similar when, for all Z, the relations between X and Z are similar to the relations between Y and Z. This definition of attributional similarity results in a score of 83.75% on the TOEFL synonyms. (The paper is about tensors, not similarity measures, so it does not explain that the tensor is computing similarity by using definition (3).) Rules (1) and (2) give us:

X:Z::Y:Z → Z:X::Z:Y → Z:Z::X:Y [assuming X, Y, and Z are the same type?]

Hence:

X~Y = for all Z, Z:Z::X:Y

Transitivity seems plausible for proportional analogies:

(4) A:B::C:D & A:B::E:F → C:D::E:F

Now we have a little theorem:

Theorem: A~C & A:B::C:D → B~D

Proof:

1. A~C [by assumption]
2. A:B::C:D [by assumption]
3. A:C::B:D [from 2 by rule (2)]
4. for all X, A:X::C:X [from 1 by definition (3)]
5. for all X, X:X::A:C [from 4 by rules (1) and (2)]
6. for all X, X:X::B:D [from 3 and 5 by rule (4)]
7. for all X, B:X::D:X [from 6 by rules (1) and (2)]
8. B~D [from 7 by definition (3); QED]

I’m not sure about the significance of this, but I thought it was worth writing down.

This post was partly inspired by a blog post by Gustavo Lacerda. Suppose we have:

A = me
B = my lover
C = my lover’s past lovers
D = my lover’s past lovers’ lovers

Then A~C & A:B::C:D → B~D. If I am similar to my lover’s past lovers (A~C) and I am attracted by my lover like my lover’s past lovers are attracted by my lover’s past lovers’ lovers (A:B::C:D), then my lover should be similar to my lover’s past lovers’ lovers (B~D). Thus I am likely to be attracted by my lover’s past lovers’ lovers.

I believe that similarity is a matter of degree, but the above discussion assumes that it is binary (true/false).