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  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?

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  • 2
    1., as phrased, does not fit your translation, and this argument is invalid. 1. only says that virtue makes Peter patient, it does not say that he needs virtue to be patient. He may well manage the household with patience without being virtuous (and colloquially, virtue is a much stronger condition than mere patience). In other words, it should be translated as "if virtue then patience", not the other way around. And nothing non-trivial then follows from premises that both have patience in the consequent.
    – Conifold
    Jul 23 at 4:29
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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.”)

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    This is very clear and helpful, thank you. Let's assume that 1 is generalized. So, at this point of the argument, we don't have enough information to infer any relation between the virtuous and the good. But can we come up with a premise that can assume this relation? Perhaps a premise that can allow us to make the virtuous and the good equivalent? This would be in the form of a necessary and sufficient condition, I think?
    – part-two
    Jul 24 at 3:10
  • If I could trouble you with one more question: I want to set aside 1 and generalize 2 to "all humans." 2 thus becomes "if all humans are good, they are patient." Can I infer that "all humans are good in the same way"? On the one hand, it seems that I can, because 2 says that all humans can't be good without being patient. But on the other hand, it seems that I can't, because 2 also says that all humans can be patient without being good.
    – part-two
    Jul 24 at 7:07
  • Your first question, I didnt mention one more statement above that is the same as the others: “A only if B”. You could add that to the list. So “Necessary and sufficient” means the two always go together. A is sufficient for B (If A, then B) and A is necessary for B (B only if A). This can be said as A if and only if B (or B if and only if A). Or A is equivalent to B. If we assume that virtuous always goes with goodness, then A and B are technically equivalent in logic (even though technically that’s not really the same as saying they mean the exact same thing, just they always exist together
    – Al Brown
    Jul 24 at 18:06
  • To clarify this part: “A is sufficient for B (If A, then B) and A is necessary for B (B only if A).” Here: A is sufficient for B (If A, then B. or B if A. or A only if B). and A is necessary for B (If B, then A. or A if B. or B only if A).” You might be able to think how A if B and A only if B are different. First is B implies A and second is A implies B. So we have suff “A only if B”, and ness “A if B”, so can be said A if and only if B, written A iff B sometimes. So if you see “iff” that means equivalent in logic (if and only if)
    – Al Brown
    Jul 24 at 18:23
  • I might check your second question tonight. Have a great Saturday 🙏🏻👍🏻
    – Al Brown
    Jul 24 at 18:31

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