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Is the following argument valid?

  1. If A is to be good, they must be just
  2. If B is to be good, they must be just
  3. Therefore, if C is to be good, they must be just
  4. Therefore, if C is just, they become good

(1) to (3) looks valid to me, but I'm not sure on what grounds I can make this inference.

(3) to (4) also looks valid to me, and I think it has something to do with necessary and sufficient conditions, but I'm not sure how to make the reasoning explicit.

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Edit (7/24/21): Thanks all for the helpful comments, and my intuition above was clearly wrong. If I may, I'd like to turn that into a more specific argument. (Originally, I thought putting the argument in simplified terms would be enough for my purpose and make it easier to get feedback. I apologize for the confusion.)

Folks have rightly pointed out that (1) to (3) are invalid because, basically, there is nothing tying A, B, and C together. Would my following substitution change this judgement?

Let A = adult men and adult women, B = children and old people, and C = all humans.

  1. If adult men and adult women are to be good, they must be just
  2. If children and old people are to be good, they must be just
  3. Therefore, if all humans are to be good, they must be just

For the sake of the argument, let's assume that adult men and women, and children and old people, constitute all humans. So now, there seems to be something tying A, B, and C together. But my question is: Is this enough to make the argument valid? If not, why?

One more uncertainty: Would adding this implicit premise (also pointed by folks) help, something like, "One who is good must be just"? Or would the argument end up being circular, since this is akin to what (3) tries to establish?

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    If you mean that 1. and 2. are assumptions and 3 is to follow from 1+2 and 4 from 3, none of them is valid. Why do you think they are? Which inference rules are you basing your judgement on?
    – lemontree
    Jul 21 at 7:09
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    Your vague argument lacks some implicit premise: "rightness" (to be just) is a property of "goodness", like "every triangle is trilateral". If so, if A,B,C are triangles, they will be necessarily trilateral. Jul 21 at 10:52
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    And 4 is obviously wrong: assuming that "rightness" is a property of "goodness" does not mean that everything right will be good. Apples are red but it is not true that everything red is an apple. Jul 21 at 10:53
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No.

A good -> A just

B good -> B just

C good -> C just ? Non sequitur. C was not mentioned before

C just -> C good ? Another non sequitur. This would be the converse of line 3, but an implication does not entail its converse.

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  • Thanks for the reply. What you said about (1) to (3) makes sense. But can you quickly clarify (3) to (4)? I can see that (4) is the converse of (3) now. But what did you mean by "an implication does not entail its converse"? Suppose that (3) is not an intermediate conclusion and is only a given premise. Can I simply infer (4) from (3) by conversion then? And would this inference be valid?
    – part-two
    Jul 21 at 6:27
  • @part-two no. "conversion" is not a valid step. From A->B you cannot conclude B->A.
    – causative
    Jul 21 at 7:24
  • @part-two even assuming 3 is true doesnt make 4 true. 3 only says just is required for good, but it may not be sufficient. Maybe C must be just AND active to be good: 3 would be true, but 4 would not. Also, we dont even know 3 is true so cant use it for 4 anyway.
    – Al Brown
    Jul 21 at 8:59
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No, it is not.

  1. and 2) are premises. In a vacuum i cannot evaluate them, so i assume them to be true and correct.

  2. is a first conclusion; which does not follow.

  3. would be valid if "all that are good must be just" would be a (true) premise, which was not mentioned. Even if A and B are the only existing cases of whatever they are, the fact that both of them are good and just might be a coincidence

  4. simply does not follow from 1) and 2). With 3) not being valid in this argument, 4) cannot be valid either, since it depends on 3) The step from 3) to 4) is valid by itself, but if 3) isn't valid, 4) cannot be.

A more accurate argument would be: P1) all that are good must be just P2) C becomes "good" Q1) C will automatically become "just"

(Note that neither A nor B are included, since they are examples of P1, and therefore irrelevant to the basic argument)

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Concerning the first argument:

  1. If A is to be good, they must be just
  2. If B is to be good, they must be just
  3. Therefore, if C is to be good, they must be just

This is not valid essentially because there is nothing in the argument making necessary that whatever applies to A or B also applies to C.

Concerning the second argument (without the "all" in the conclusion):

  1. If adult men and adult women are to be good, they must be just
  2. If children and old people are to be good, they must be just
  3. Therefore, if humans are to be good, they must be just

This second argument is not formally valid. The reason is that to try and see it as valid, you need to interpret the words "human", "children" etc. In a formally valid argument, there is no need for interpretation.

The argument is also not clearly informally valid.

Essentially, informal validity depends on what we means by "humans", "children" etc. Informal validity requires that we give the argument a "charitable interpretation". However, this interpretation should be uncontroversial, and here we have a problem with babies. Are babies children? Are they not humans?

Suppose we say babies are human but not children, so the premises don't apply to them but the conclusion does, so you infer a conclusion (about babies) that does not follow from the premises (that are not about babies). This makes the argument invalid, not only formally invalid, but informally invalid.

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  • What about this then? A = adult men and adult women. B = children and old people. C = all humans. For the sake of the argument, it seems that we can reasonably assume that adult men, adult women, children, and old people constitute all human beings. If so, can (1) to (3) be valid?
    – part-two
    Jul 24 at 5:57
  • @part-two "What about this then?" This would be a different argument. The one you asked about is not valid. Update you question and I'll update my answer but it doesn't look good. Jul 24 at 16:01
  • I just updated the question. Thanks in advance for your help.
    – part-two
    Jul 25 at 0:48
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    @part-two, your "reasonable assumption" has to be treated as a premise of the argument. Think about it like a computer - you need to tell it everything that is relevant about the world in order for it to be able to calculate, because it's not a human person furnished with a background of human experience.
    – Paul Ross
    Jul 25 at 13:19

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