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I have some doubts regarding categorial propositions used in syllogism.

  1. Does "No A is B" implies "Some A is not B" and "Some B is not A"? Is there any counterexamples to this fact? In math, "No A is B" is equivalent to saying "the intersection of sets A and B is empty." This is equivalent to saying A is a subset of complement of B and B is a subset of complement of A. So there is no doubt that this trivial fact is true?

  2. The necessary and sufficient condition for "Some A is B" is "Some A is not B"?

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    Voting to close because there are hundreds of easily findable articles that cover this in detail. Dec 3, 2023 at 8:34
  • Aristotles Organon goes through categorical propositions in great detail. one thing I noticed is that in Aristotles logic "not B" is not allowed to be a predicate at all Dec 10, 2023 at 19:19

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In modern quantifier logic "no A is B" does not imply "some A is not B". That is because "some A is not B" is taken to entail the existence of at least one A, while "no A is B" does not. For example, "no unicorns live on Mars" is true, while "some unicorns do not live on Mars" is false, since it would imply that at least one unicorn exists that does not live on Mars.

However, it is different in Aristotelian logic. There, "no A is B" does imply "some A is not B". Depending on how you interpret the logic, you have to suppose either that both propositions entail the existence of As, or that both do not. In Aristotelian logic it may help to think of "some A is not B" as meaning "either there are no As, or if there are, at least some and possibly all of them are not B".

"Some A is B" and "some A is not B" are not equivalent. In both modern and aristotelian logic, "some A is B" does not exclude "all A is B", and "some A is not B" does not exclude "all A is not B". The reason it is tempting to think that "some A is B" implies "some A is not B" is that an utterance of "some A is B" often carries the conversational implicature that not all are. If you know "all A is B" then it may be a violation of the cooperative principle to state the weaker "some A is B" and hence the utterance may be misleading, though not actually false.

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  • Thank you. Can you tell me what you mean by "some A is B" does not exclude "all A is B"? There is a possibility that "some A is B" actually means "all A is B" and in this case "some A is not B" is false?
    – Ray Siplao
    Dec 3, 2023 at 4:35
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    I mean that "some A is B" is consistent with "all A is B". So if in some case "all A is B" is true, then "some A is not B" is false. There is no contradiction in saying, "Some of the people in this room speak Spanish. In fact all of them do."
    – Bumble
    Dec 3, 2023 at 5:27
  • Regarding the difference between modern quantified logic and Aristotelian logic, I have doubts about whether "all A is B" is true or false when A is empty. In modern quantified logic, "all A is B" is true when A is empty because its negation "some A is not B" entails the existence of some element in A that is not B, which is false. In Aristotelian logic, I am not sure if "all A is B" is necessarily true when A is empty. I think the answer is yes. Following what you said in the second paragraph, "all A is B" is true when either A is empty or A is not empty and all members in A belong to B?
    – Ray Siplao
    Dec 3, 2023 at 22:46
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    In Aristotelian logic, "all A is B" implies "some A is B". So, either both entail the existence of As, or both do not. Most commentators go with the former interpretation and hold that both presuppose the existence of As. The latter interpretation is also possible. I have an answer to a related question here where I go into a lot more detail.
    – Bumble
    Dec 4, 2023 at 0:21
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Questions about categorical propositions.

1.Does "No A is B" imply "Some A is not B" and "Some B is not A"?

Answer: Yes. On the Square of Opposition, "No A is B" is an E statement; this means that the categories A and B are both distributed. Generally, when a category is distributed, some quality applies to everything in that category.

Here, whatever an A might be, it cannot be a B. The same is true in reverse: a B cannot be an A.

1a. Is there any counterexample to this fact? In math, "No A is B" is equivalent to saying "the intersection of sets A and B is empty." This is equivalent to saying A is a subset of complement of B and B is a subset of complement of A. So there is no doubt that this trivial fact is true?

Answer: Ido not understand the comment about complementarity, and I will simply have to say that I do not know.

2.The necessary and sufficient condition for "Some A is B" is "Some A is not B"?

Answer: No. The two statements are subcontraries: both can be true at the same time, but both cannot be false at the same time. In other words, "Some" and "Some-not" can be true together, but their respective negations, “All A is B" and “No A is B", cannot be so.

"Some A is B" and "Some A is not B" are not otherwise related.

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  • Conclusion 1 is false:

    “No A is B” is equivalent to “A intersection B = empty set”.

    If both A and B are empty, then “No A is B” is satisfied, but neither “Some A is not B” nor “Some B is not A”.

  • Statement 2 is false:

    2a. “Some A is B” is equivalent to “A intersection B = not empty”.

    2b. “Some A is not B” is equivalent to “(A intersection complement of B) not empty”.

    If (A not empty) and (A contained in B), then 2a is satisfied, but 2b not.

  • I agree with @Bumble that “some“ presupposes the existence of at least one corresponding element.

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