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everyone.

Thank you very much in advance for any help you can give me with this issue.

I'm enrolled in an Introduction to Logic course, and we're currently working on Categorical Propositions. I've run into a problem that's baffling me, and I was hoping I might get some help with it.

The instructions for the problem are the following:

Translate the premise and conclusion of the following immediate inferences into standard form categorical propositions. Then use conversion, obversion, contraposition, or the traditional square of opposition to determine whether each is valid or invalid.

The problem with which I'm having trouble is this one:

Flu vaccines are never completely effective. Therefore, not ever flu vaccine is completely effective.

This is one of the exercises for which an answer is provided at the back of the book. The answer provided is this:

No flu vaccines are completely effective medicines. Therefore, some flu vaccines are not completely effective medicines. Valid.

While I think I can understand the rationales for converting the two statements into categorical propositions, I'm having a lot of trouble figuring out how the immediate inference formed from the propositions can be valid.

We have learned by this point about conversion, obversion, and contraposition, and while I think I understand these means of converting the propositions into equivalent statements, none of them (as I understand it, anyway) will work to convert a universal proposition into a particular one.

So, I'm stuck here. There's obviously something I'm missing, but I can't figure out what it is. Any guidance you might be able to offer would be much appreciated.

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  • We have learned by this point about conversion, obversion, and contraposition, and while I think I understand these means of converting the propositions into equivalent statements, none of them (as I understand it, anyway) will work to convert a universal proposition into a particular one. The question you quoted also tells you that you might need to use "the traditional square of opposition", have you looked into that?
    – Hypnosifl
    Jul 10 at 11:32
  • No flu vaccines are completely effective medicines. Therefore, some flu vaccines are not completely effective medicines. Valid. Do you understand this in words? No vaccine is completely effective. What is meant by this? Jul 10 at 15:20
  • No are completely is equivalent to some (or all) are not completely. The meaning of completely has changed though. Jul 10 at 15:27
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You have likely wrote the question incorrectly. The terminology is extremely important and you can’t use it Willy nilly. All categorical arguments MUST have three propositions whether you SEE THEM PHYSICALLY there or not. You were given “Flu vaccines are never completely effective. Therefore not every flu vaccine is completely effective”. None of these are in standard categorical form. They are just propositions and not really syllogisms. Your job is to put them in standard form & then translate one into the other. You should be able to justify one proposition into the other. The first premise can be put in the form “No flu vaccines are completely effective medications.” There is an inference rule from the Square of Opposition called subalternation. With subalterns you can begin with a universal and go to a particular proposition in the Aristotelian Square. [There is also a modern Square of Opposition that math uses where subalternation is not allowed.]. You were too focused on the other inferences conversion, obversion and contraposition and forgot about the Square of Opposition.

When dealing with categorical syllogisms you are not allowed to use modern language phrases. All you are allowed to do is use exactly four forms: All s are P, No s are P, Some s are p, and Some s are not P. There is no such thing as a form All s are not p. I mention this only because people commit this error often. If you see something like that you should know it has to be translated into one of the forms above. All s are not p is VAGUE. There are two possible translations: NO s are p and Some s are not p. Notice the Some s are not p is the subaltern of No s are p. In this way the safest way to translate All s are not p is to treat it as a n O type proposition over an E type proposition.

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  • All syllogisms involve three propositions, but there are other types of inferences involving categorical propositions in classical Aristotelian logic, that's what "conversion, obversion, contraposition, or the traditional square of opposition" are all about (see here for the first three, and here for the square of opposition). For example, obversion allows you to start with the single premise "All S are P" and reach the conclusion "No S are non-P" without any third proposition.
    – Hypnosifl
    Jul 10 at 18:42
  • @Hypnosifl , I actually submitted accidentally before I added text. I would like to mention the phrase “Reach the conclusion [or conclude] without a third proposition” seems to be used in a different context. I was taught there is NO CONCLUSION involved here because there is no argument in the first place. We only have ONE proposition and NOT TWO. By obversion we prove the two declarative sentences are IDENTICAL to each other. If the first declarative sentence is true then the other declarative sentence written in different words must also have the same truth value as the original.
    – Logikal
    Jul 10 at 18:59
  • Hello, everyone. Thank you for your answers. We have not yet progressed to the point of dealing with categorical syllogisms. We're still dealing with categorical propositions only, and working with immediate inferences as part of that. I understand the suggestion that I have transcribed the question incorrectly, and I have checked again to make sure. The wording of the question is transcribed correctly, as is the wording of the answer. I think that the answer likely lies with subalternation. Can subalternation be used to demonstrate the two categorical propositions are equivalent?
    – sixo33
    Jul 10 at 20:51
  • @sixo33 - Yes, you've got it--if you're using a textbook which discusses how to use subalternation to show two propositions are equivalent you can refer to that, otherwise note the parts of the SEP article that say ‘Some S is not P’ is a subaltern of ‘No S is P’ along with A proposition is a subaltern of another iff it must be true if its superaltern is true, and the superaltern must be false if the subaltern is false.
    – Hypnosifl
    Jul 10 at 21:03
  • @sixo33, to be pedantic here is needed. The inferences we are discussing are not EQUIVALENT. Once you translate the propositions are IDENTICAL. That is the same message is being relayed in communication but with different words used.
    – Logikal
    Jul 10 at 21:19

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