If no homotypic circles of explanation, no heterotypic ones either

Most people agree that one cannot have circularity in the order of explanation when one keeps the type of explanation fixed, i.e., there are no homotypic circles of explanation. Some like me think one cannot have circularity in the order of explanation at all. Why? One intuition might be that explanations of all types are still explanations, and so the circularity is still an explanatory circularity. :-) (Yes, that begs the question.) More seriously, heterotypic explanations (namely, explanations of different types) can be combined, sometimes chainwise (A explains B and B explains C, and thereby A explains C) and sometimes in parallel (A explains B and C explains D so A-and-C explains B-and-D). This means that the types of explanation are not quite as separate as they might seem.

Here is an argument building on the second intuition. We need two concepts. First, we can talk of two sets of explanatory relations as independent, namely without any interaction between the explanatory relations in the two sets. Second, given two type of explanation₁ and explanation₂, I will say that explanation_1|2 is a type of explanation where explanation₁ and explanation₂ are combined in parallel.

1. If circularity in explanation is possible, it is possible to have a two-item heterotypic explanatory loop.

2. If it is possible to have a two-item heterotypic explanatory loop, it is possible to have two independent two-item heterotypic explanatory loops where each loop involves the same pair of explanation types as the other loop.

3. Necessarily, if A explains₁ B and C explains₂ D, and the two explanatory relations here are independent, then A-and-C explains_1|2 B-and-D.

4. Necessarily, the relations explains_1|2 and explains_2|1 are the same.

5. It is not possible to have a circle of explanations of the same type.

6. Suppose circularity in explanation is possible. (Assume for reductio)

7. There is a possible world w, such that at w: there are A, B, C and D such that A explains₁ B, B explains₂ A, D explains₁ C and C explains₂ D, and the above explanatory relations between A and B are independent of the above explanatory relations between C and D. (6,1,2)

8. At w: A-and-C explains_1|2 B-and-D. (3,7)

9. At w: B-and-D explains_2|1 A-and-C. (3,7)

10. At w: B-and-D explains_1|2 A-and-C. (4,9)

11. At w: there is a circle of explanations of type 1|2. (8,10)

12. Contradiction! (5,11)

13. So, circularity in explanation is impossible.

I think the most problematic premise in this argument is (4). However, if (4) is not true, we have a vast multiplication in types of explanation.