Vtyjkyuk

I tried to disprove rule "A valid syllogism must have at the least one positive premise" and shockingly it looks like I succeeded. Even more, the syllogism which I found violates other rules too, namely it has five terms ("non-living thing" and "non-mortal" make it five terms) and has positive conclusion despite having negative premise.

Let's take this example:

All humans are living things.
All livings things are mortal.
Thus all humans are mortal.

It seems valid for me. Now let's transform its premises. "All humans are living things." is equivalent to "No human is a non-living thing." and "All livings things are mortal." means the same as "No living thing is non-mortal.". If we applied transformations what doesn't change the meaning of our premises, then it's natural to expect that validity of the whole
syllogism will stay the same (so if it was invalid it will stay invalid and if it was valid it'll stay valid).

Now we have VALID syllogism:

No human is a non-living thing.
No living thing is non-mortal.
Thus all humans are mortal.

Let's double check it with a diagram. Just to be sure.

Here we have three circles, H means "humans", LT stands for "living things" and M contains everything that is mortal. I used gray color to indicate empty intersections.

Now back to premises. "No human is a non-living thing." means that intersection between set of humans and set of non-living things is empty. In other words, areas #1 and #5 are empty because together they form intersection between humans and non-living things. "No living thing is non-mortal." means that intersection between living things and non-mortals is empty, thus making areas #2 and #4 empty.

And now, the conclusion. "Thus all humans are mortal." seems legit considering that the only area where the set of all humans is still white is #7.

The classification you link to tacitly assumes that the claims that make up a "syllogism" have a very particular form, connecting two statements of belonging to particular categories.

When you reword "All humans are living things" to "No human is not a living thing", you're moving beyond that assumption (since now one of the statements is of not belonging to the category that appears in the next premise). It should therefore not surprise you that rules designed to work for the standard form of the claims stop working.

(Indeed if one considers claims of the form "no human is not a living thing", then it appears that distinguishing between "positive" and "negative" claim stops being useful at all).

It should also be noted that mathematical logic over the last 100-150 years has definitively abandoned the idea that categorical syllogisms are a useful way to analyze and structure mathematical arguments in general. They're now seen as supporting only a narrow range of weak reasoning which doesn't come close to what one needs for actual mathematical reasoning or proofs.

Your link's high-strung claim that "for your life of ideas mastery of the 15 valid forms of standard form categorical syllogisms is indispensable" finds no support in modern logic -- I daresay only a tiny minority of working mathematicians would be able to name any of the classical syllogisms, much less reproduce the elaborate systems the ancients used to categorize them.

"No human is a non-living thing" is not an A, E, I, or O form. Henning Makholm points out in an answer that each part of the syllogism must be one of those four forms, in order for the overall argument to even have the ability to be called a "syllogism". (Another way of reaching the same conclusion, if we treat "no human is a non-living thing" as an E, is that in that case we cannot link "non-living thing" in the first premise with "living thing" in the second premise - there is no common "middle term" in the argument presented, just two E premises.).

Separately, the "two negative premises" argument in the question would not be considered a valid syllogism because the first two sentences are consistent with there being no humans and no living things, but in categorical logic "All humans are mortal" has existential import and implies "some humans are mortal".

Using the syntax of first-order logic, the basic form of the syllogism is:

$(forall x (A(x) implies B(x))) land (forall x (B(x) implies C(x))) implies (forall x (A(x) implies C(x)))$

But $forall x(A(x) implies B(x))$ can be rewritten as $nexists x(A(x) land lnot B(x))$, so the syllogism can be rewritten as

$(nexists x(A(x) land lnot B(x))) land (nexists x(B(x) land lnot C(x))) implies (forall x (A(x) implies C(x)))$

which is a syllogism with two negative premises.

In your example you have "$x$ is human" as $A(x)$, "$x$ is a living thing" as $B(x)$ and "$x$ is mortal" as $C(x)$.

So "Rule 4" is incorrect. And "Rule 1" is incorrect if $B(x)$ and $lnot B(x)$ are counted as different terms.