The assertion 2 is false.
Proof: ¬(p & ¬q) ⊢ p → q
Proof2: "My brother John is not a bachelor" ⊢ "My brother John is married"
Both are valid arguments. Both have a negative premise, and an affirmative conclusion.
The latter is an illustration that you can rephrase probably every negative premise into an affirmative one, and vice versa.
Therefore it does not even seem to make sense to distinguish negative premises from affirmative ones.
That said, it is probably possible to explain why the authors think that 2 is true:
In ancient Aristotelian logic, there are only 2 types of "affirmative" propositions ("All F are G", and "Some F are G"), and 2 types of "negative" propositions ("No F are G", "Some F are not G").
Inside this system, the authors are indeed correct: an "affirmative" conclusion cannot be derived from the "negative" premises that this system allows.
As to your question regarding 2 and 3, consider the following argument:
No fish is a mammal
Some aquatic animals are mammals
Ergo: Some aquatic animals are not fishes
This argument satisfies both condition 2 and condition 3. I don't see how these contradict each other.
I find all 3 statement true and non-contradictory. The premises and conclusion form and AND function, with:
an invalid conclusion when both premises are negative,
a negative conclusion when either premise is negative, and
a positive conclusion when both premises are positive.
When one premise is negative, the other must be positive for the conclusion to be negative, otherwise the conclusion becomes invalid (two negatives).
Here is the question:
Why's 2 true please? I can't intuit why, because 3 outwardly contradicts it.
I prefer an intuitive explanation, not one with Truth Tables or Formal Deduction.
Consider the three statements:
1: As soon as there are two negative premises, the argument is automatically invalid.
2: If the conclusion is affirmative, there can be no negative premises. If there is a negative premise, there must be a negative conclusion.
3: However, negative conclusions can be arrived at by a combination of both negative and positive information. Even a negative conclusion requires some positive backing.
A potentially helpful intuitive explanation might be this analogous argument replacing the premise or conclusion of a syllogism with an affirmative identity, x = y, or its negation, x ≠ y.
There are three cases to consider for three identities or their negations:
Both premises are affirmative: If a = b and b = c then we can make an affirmative conclusion: a = c. So two affirmative premises of this sort give us an affirmative conclusion.
One and only one premise is affirmative: We could have a = b and b ≠ c. Or, we could have a ≠ b and b = c. In both cases there is a definite, but negative conclusion: a ≠ c.
Both premises are negative: In this case we have a ≠ b and b ≠ c. Can we claim that a = c? No. If we let a = 1, b = 2 and c = 3, then we have a counterexample. So that affirmative conclusion would be invalid. Can we claim that a ≠ c? No. If we let a = 1, b = 2 and c = 1, then we have a counterexample. So that negative conclusion would be invalid also.
Although these affirmative identities and their negatives do not represent all possible statements in syllogisms they hopefully show intuitively why the claims in Capaldi and Smit's text are likely correct.
Capaldi, N., Smit, M. The art of deception: an introduction to critical thinking. Prometheus Books.
Why does one negative premise suffice to imply a negative conclusion?
One negative premise is sufficient to require a negative conclusion because of the distribution of terms in the premises. From the Capaldi and Smit book:
If the conclusion is affirmative, there can be no negative premises. If there is a negative premise, there must be a negative conclusion.
In the statement All P are Q, the term P is distributed. This means that the statement says something about the group of all P: every P is a Q. The statement says nothing about the group of Q; All P are Q is not equivalent to All Q are P.
In the statement Some P are Q, neither term is distributed. The statement says that at least one P is a Q, but says nothing more.
Both statements, All and Some, are positive. They add information about Group P in relation to Q.
The negative statements are different. They exclude. A conclusion based on such a premise must account for the exclusion, and so becomes a negative statement itself.
In the statement No P are Q, both terms are distributed. Whatever might be known about P, to a certainty it is not a Q. In the statement Some P are not Q, the term Q is distributed. Whatever else is true about Group P, to a certainty at least one of its members is not a Q.
Distribution, also called Distribution Of Terms, in syllogistics, the application of a term of a proposition to the entire class that the term denotes.
Encyclopaedia Britannica, Distribution. https://www.britannica.com/topic/distribution-logic The Britannica article includes the technical definition of distribution.