Tuesday, February 05, 2008

Analogy as an Equivalence Relation

Due to lack of sleep and tiring work, my brain has been officially MIA since about 1 o'clock this afternoon, so it's very risky to do a logic post, even a simple one. But since my brain has been lost somewhere since about 1 o'clock, my ability to assess risks is impaired. So here goes.

An equivalence relation is a relation that has the following characteristics:

It is reflexive (a is related to a).
It is symmetrical (if a is related to b, b is related to a).
It is transitive (if a is related to b, and b is related to c, a is related to c).

One might regard analogy, understood as a relation, as meeting each of these criteria.

A is analogous to A. This can be taken as a tautology, so we can add the analogy a : a to any inference at any time, for any a. So if you are reasoning analogically you can always take something as its own analogue.

If A is analogous to B, B is analogous to A. If there is something in or about A that is similar enough to something in or about B that we can say that A is analogous to B, for that very reason we can say that B is analogous to A.

So analogy in general is certainly reflexive and symmetrical.

If A is analogous to B, and B is analogous to C, A is analogous to C. Slightly trickier. Consider:

(1) A is analogous to B. (Premise)
(2) B is analogous to C. (Premise)
(3) C is analogous to B. (from 2 by Symmetry)

It is clear from (1) and (3) that A and C are analogous in at least one respect: namely, they both are analogous to B. Thus if we take 'is analogous to' to mean 'is analogous to in any way', analogy is always transitive. Sometimes when we are talking about analogy we restrict to the transitivity of analogy to what might be called relevant analogy, i.e., cases where the analogy of A to B and of B to C meet some special condition that allows us to say the two are relevant to each other. This is an entirely legitimate way to go, of course; but we set aside such approaches for now.

So analogy is an equivalence relation. On the basis of this you can build an axiom system using analogy as your only equivalence relation. (I'll simply write 'x is analogous to y' as 'xy'.)

We'll take the above properties and analogize them:

Reflexivity: (aa)
Symmetry: (ab)(ba) [i.e., 'a is analogous to b' is analogous to 'b is analogous to a']
Transitivity: (((ab)c)(a(bc)))

Some other axioms you might have:

Verisimilitude: T(aa)
Falselikeness: F(~T)
Double negation (~(~a))(a)
Analogical Distributivity of Disanalogy: (~(ab))((~a)b))
Analogical Distributivity of Disjunction: ((a v (b v c))((a v b) v (a v c))

And so forth. Because analogy's being taken as an equivalence relation we can substitute analogue for analogue. There's nothing distinctively interesting about this, since it's just an ordinary, humdrum sort of logic, with the only qualification being that it's all done with analogies. Actually, unless I'm mistaken, it's a straightforward equational logic in which all of the equations are analogies. Proofs are straightforward, e.g.:

(1) (~(F(~(aa)))) [premise to be refuted]
(2) (~(~(aa)F)))) [1, symmetry]
(3) (aa)F [2, double negation]
(4) T(aa) [axiom]
(5) F(~T) [axiom]
(6) (aa)(~T) [3,5 substitution]
(7) (T)(~T) [4,7 substitution]

(7) is the analogical version of a contradiction.