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Source: A Concise Introduction to Logic (12 Ed, 2014), by Patrick J. Hurley.
Please observe my amelioration (and so change) of Hurley's notation;
I use MIN, MAJ, and MID to abbreviate the Minor, Major, and Middle Terms.

[p 287:] Rule 4: [...] a Negative Conclusion Requires a Negative Premise.
Fallacy: [...] Drawing a negative conclusion from affirmative premises

[p 288:] Conversely, a negative conclusion asserts that the MIN class is separate either wholly or partially from the MAJ class. But if both premises are affirmative, they assert class inclusion rather than separation. Thus, a negative conclusion cannot be drawn from affirmative premises.

I seek only intuition; so please do not answer with formal proofs or arguments.

The bolded sentence does not convince me. The only affirmative Categorical Propositions are A (in which only the Subject is distributed) and I (in which neither the Subject nor Predicate is distributed). So for any Affirmative Proposition, at least one term is not distributed and we know nothing about this one term. Then how is the bolded true?
How can you assert something about a term about which you know nothing?

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Intuitively, you might understand it better with an example from Wikipedia.

P1: We don't read that trash.
P2: People who read that trash don't appreciate real literature.
C:  Therefore, we appreciate real literature.

The conclusion is not logically derivative of the premises. All we can say is that if someone reads 'that trash,' then they do not appreciate real literature. This tells us nothing of someone who does not read it. There could be a multitude of different factors at play to determine ones state of appreciation of real literature other than simply reading one thing. This example is ultimately a form of Denying the Antecedent.

P1: !P
P2: P -> Q
C:  !Q
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In Aristotle's Logic :

Syllogisms are structures of sentences each of which can meaningfully be called true or false: assertions (apophanseis), in Aristotle’s terminology. According to Aristotle, every such sentence must have the same structure: it must contain a subject (hupokeimenon) and a predicate and must either affirm or deny the predicate of the subject. Thus, every assertion is either the affirmation (kataphasis) or the denial (apophasis) of a single predicate of a single subject.

The affermative sentences are of two kind :

Universal : P is affirmed of all of S, i.e. "Every S is P ["All S is (are) P"];

Particular : P is affirmed of some of S, i.e. "Some S is (are) P".

Thus, in both cases we have a clear relation of inclusion : the "set [or class] of Ss" is totally (partly) included into the "set of Ps".

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