Perhaps the word doesn't exist, or it's just in the back of my head where I can't reach it currently.
It's a word supporting the proposition that two statements tell the same story, but about different subjects, if you will. A less exact word that is often used is "as", or "is to what ... is to ...", but I'm after a word that implies identity between the two that is in nature close to mathematical exactness.
So they (usually) come in pairs. Good examples:
a) The Earth revolves around the Sun
b) The Moon revolves around the Earth.
(concept: orbit.) => "The Earth revolves around the Sun as The Moon revolves around the Earth."
a) The main working tool of a smith is a hammer.
b) The main working tool of a tailor is a needle.
(concept: tools of the trade) => "A hammer is to a smith what a needle is to a tailor."
Edit: this is an example of a strict analogy. However, expressions such as "Hand is to palm what foot is to sole" is called an analogy, while not presenting a relation. (As I see an intelligent Wikipedia editor points out, "While most competent English speakers will immediately give the right answer to the analogy question (sole), it is more difficult to identify and describe the exact relation that holds both between hand and palm, and between foot and sole").
I'm looking for a term that conveys the exactness of swapping terms in an equation in mathematics, in such a strict way as to preserve the form of the equation and the relation it states.
In other words, a synonym for "both statements are non-identical expressions of an identical concept".
Let us suppose that you wanted to show such strict formal equality for two statements,
a) "You are kind to your brother"
b) "Your brother is kind to you."
In this context, the relations would be expressed in pseudo-language as:
a) [self] [perform-act-assumed-beneficial] [in-direction-of] [human]
b) [human] [perform-act-assumed-beneficial] [in-direction-of] [self]
I'm looking for the word or expression that makes clear this strict formal equality with the exception for the swapping of the terms.
Note that here, there are just two terms and two statements involved, and they are swapped symmetrically. In more proper examples, there may be relations expressed between three terms, where all three are thrown away and replaced by new terms in the second statement, and more than two statements.
Further alternatives to the above,
c) [human] [perform-act-assumed-beneficial] [in-direction-of] [other-human]
d) [local-community-decision-maker] [perform-act-assumed-beneficial] [in-direction-of] [local-charity]
Feel free to inquire further, as I seem unable to concretize this at the moment. I hope you understand I mean something stronger than "similarity" and weaker than "equality" and "identity" - but that exact matching of concepts is involved.
Edit: As per some of the suggestions,
Analogy does not relate two statements.
A Syllogism is what it may formally become to argue the truth of the common concept, but the premises of a syllogism do not have to be of the same form; here they must.
The statements certainly share a Template (showing that such a template can be ascribed to all statements requires at least one successfully argued conversion to pseudo-language), but it is to be expressed that they are homologous via a common concept (the strength of which is to be determined by experience, evidence).
Would calling the multiple statements homomorphisms for the common concept be transparent to well-read academics in Philosophy?