How to make this argument valid?

Question

1. If Peter is virtuous, he manages a household with patience.
2. If Peter is to be good, he needs patience.

Here's my attempt to understand 1 and 2. I get confused by the wording and necessity-sufficiency in general, so I try to think through them slowly.

1 says that without virtue (being virtuous), Peter cannot have (manage with) patience. That is, virtue is necessary for patience. That is, if Peter has patience, he must have virtue.

2 says that without patience, Peter cannot be good. That is, patience is necessary for goodness. That is, if Peter has goodness, he must have patience.

In simplified terms:

1. If patience, then virtue.
2. If goodness, then patience.
3. If goodness, then virtue.

I can conclude 3 on the basis of 1 and 2. 3 says that virtue is necessary for goodness.

Here's the thing. For 3 to follow, I need to switch the order of 1 and 2. Normally, I believe this is fine. But the 1 and 2 here are part of a bigger argument, and in that argument, 2 must follow from 1, which is itself an intermediate conclusion. (It was simply inferred as a hypothetical syllogism, so there shouldn't be a mistake.)

Alternatively, I thought about adding an implicit premise, *, between 1 and 2, to establish some kind of relation between virtue and goodness:

1. If patience, then virtue. *. If virtue, then goodness.
2. If patience, then goodness.

But then * ends up switching the order of patience and goodness in 2.

What went wrong?

Answer 1

First, on ness and sufficient:

If A, then B. This means that anytime you have A, you will have B. It does not say that anytime you have B, you will have A. There could be times with B but not A. If anytime you have A you have B, then B is necessary for having A. That’s because you can’t have A without B. A is sufficient for B (knowing we have A, that is sufficient to know we have B).

These are all the same:

If A, then B,

B is necessary for A,

A is sufficient for B,

There is no case with A but not B; there may be cases with B but not A,

A implies B,

In a venn diagram, the circle for the cases of A lies entirely within the circle for the cases of B.

Whenever A, then B ,

We can’t have A without B.

————

Item 1 does not say, “If Peter virtuous, then Peter patient.” It says if Peter virtuous, then Peter Manages a household with patience. That doesn’t mean he is generally patient.

But let’s proceed pretending that 1 says Peter virtuous implies Peter patient. The second one says, “If good, then patient.” Why? Because according to the statement there could be a case where he is patient but not good. There cannot be a case where he is good but not patient.

If he’s virtuous, then he’s patient. If he’s good, then he’s patient. But there may be cases where he’s patient and not good and not virtuous, or good but not virtuous, or virtuous but not good, or both good and virtuous. We don’t have enough info to tell. We know that both the good and virtuous circles lie within the patient circle, but we don’t even know if the good and virtuous circles overlap. (Remember we proceeded as if 1 said “If virtuous, then patient.” even though technically it only said, “If virtuous, then patient at Managing a house.”)