Does the "because" in a premise disrupt the validity of this argument?

Question

1. One is good if and only if one does one's job well
2. One does one's job well if and only if, and because, one is virtuous
3. One is virtuous if and only if one is just
4. From 1 and 2: One is good if and only if, and because, one is virtuous
5. From 3 and 4: One is good if and only if, and because, one is just
6. From 5: One is good because one is just

I have 2 worries here:

• 1 says that being good is logically equivalent to doing a job well. 2 says that doing a job well is logically equivalent to being virtuous. So I can say that being good is logically equivalent to being virtuous. This is captured by the "if and only if" in 4: one is good if and only if one is virtuous. But I'm not sure if the "and because" in 2 will complicate things, or can the 4 above really just follow through, with the "and because" simply tapped on?
• Does 6 follow from 5? As in, can I simply extract that one aspect in 5? But if not, I can allow for an additional premise: one is good if and only if one is just. This should then make the move from 5 to 6 okay?

Answer 1

The difficulty in analyzing this argument is the use of the word, "because." This is not a standard logical connective. There are ways it could be defined, but it's not clear how it is defined here.

Step 4 seems to invoke the rule, "If (X iff Y) and (Y because Z), then (X because Z)." Is that valid? Who can say, without giving axioms for the "because" connective?

Step 5 (and then 6, which just restates a part of 5) seems to invoke the rule, "If (Z iff W) and (Y because Z), then (Y because W)." Is this valid? Who can say?

If I were going to define "because," I would say it involves a directed acyclic graph among the set of all formulas, where we are able to infer a node given its parents, and where the root nodes in the graph are the axioms. But simply because a node can be inferred from a set of other nodes, does not mean those other nodes are its parents; the parents of the node are a "canonical way" to deduce that node. We might say a node B is "because of" a set of nodes S, if every node in S is an ancestor of B, and if deletion of the nodes in S partitions the graph into two parts with all the axiom nodes on one side and B on the other. In this definition, the deductions in step 4 and 5 are not valid; logically equivalent formulas need not occupy similar positions in the graph. If a formula B is much more complicated and hard to deduce than a formula A, but A and B are equivalent and the proof of B resorts to A, then we may say that B is because of A. But it is not so natural to say that A is because of B, and if we want the graph to be acyclic we can't say both.