Why is the darapti syllogism invalid?

Question

I found the following exposition of darapti on wikipedia:

  All squares are rectangles. (MaP)
   All squares are rhombuses. (MaS)
∴ Some rhombuses are rectangles. (SiP)

As far as I can see, this is of the form:

all A's are B's

all A's are C's

∴some C's are B's

I was watching a talk by the logician Graham Priest, where he stated that darapti is now invalid but he didn't offer an explanation.

I was wondering if anyone could show why it is invalid, by way of an example?

Thanks.

Answer 1

In Aristotle's Logic Darapti is a valid figure.

I suspect that the issue is with the so-called "existential import" :

From a modern standpoint, [we infer] "Some monsters are chimeras" from [...] "All chimeras are monsters"; but the former is often construed as implying in turn "There is something which is a monster and a chimera", and thus that there are monsters and there are chimeras. In fact, this simply points up something about Aristotle's system: Aristotle in effect supposes that all terms in syllogisms are non-empty.

In "modern term" ∀x(Fx → Gx) and ∀x(Fx → ¬Gx) are both true when ¬∃xFx is true, that is, when there are no Fs. (These are the so-called “vacuously true” universal generalizations.) So ‘All Fs are Gs’, on the modern reading, does not imply that there are Fs, and so does not imply that some Fs are Gs.

We can see :

• Harry Gensler, Introduction to Logic (2nd ed - 2010), page 32 :

Historically, “Aristotelian” and “modern” logicians disagree about the validity of some syllogism forms. They disagree because of differing policies about allowing empty terms (general terms that don’t refer to any existing beings).

Compare these two arguments:

All cats are animals. Therefore : Some animals are cats.

All unicorns are animals. Therefore : Some animals are unicorns.

The first seems valid while the second seems invalid. Yet both have the same form [...]. What’s going on here?

When we read the first argument, we tend to presuppose that there’s at least one cat. Given this as an assumed additional premise, it follows validly that some animals are cats. When we read the second argument, we don’t assume that there’s at least one unicorn. Without this additional assumption, it doesn’t follow that some animals are unicorns.

The Aristotelian view, which assumes that each general term in a syllogism refers to at least one existing being, calls the argument “valid.” The modern view, which allows empty terms like “unicorn” that don’t refer to existing beings, calls the argument “invalid.”

Consider this pair of arguments with the same form (a form that’s valid on the Aristotelian view but invalid on the modern view):

All cats are mammals. All cats are furry. Therefore : Some mammals are furry.

All square circles are squares. All square circles are circles. Therefore : Some squares are circles.

The first inference is sensible, because there are cats. The second inference isn’t sensible, because there are no square circles. Some logic books use the Aristotelian view, but most use the modern view. It makes a difference in very few cases.

Answer 2

Here is an excerpt from Russell's fifth lecture on Logical Atomism:

"This notion, of course, of general propositions not involving existence is one which is not in the traditional doctrine of the syllogism. In the traditional doctrine of the syllogism, it was assumed that when you have such a statement as “All Greeks are men,” that implies that there are Greeks, and this produced fallacies. For instance, “All chimeras are animals, and all chimeras breathe flame, therefore some animals breathe flame.” This is a syllogism in Darapti, but that mood of the syllogism is fallacious, as this instance shows. That was a point, by the way, which had a certain historical interest, because it impeded Leibniz in his attempts to construct a mathematical logic. He was always engaged in trying to construct such a mathematical logic as we have now, or rather such a one as Boole constructed, and he was always failing because of his respect for Aristotle. Whenever he invented a really good system, as he did several times, it always brought out that such moods as Darapti are fallacious. If you say “All A is B and all A is C, therefore some B is C” – if you say this you incur a fallacy, but he could not bring himself to believe that it was fallacious, so he began again. That shows you that you should not have too much respect for distinguished men."

Answer 3

The problem with Darapti is the existential fallacy. The reasoning uses two universal premises to reach a particular conclusion. From the site Logically Fallacious:

A formal logical fallacy, which is committed when a categorical syllogism employs two universal premises (“all”) to arrive at a particular (“some”) conclusion.

In a valid categorical syllogism, if the two premises are universal, then the conclusion must be universal, as well.

The fallacy claims to validate syllogisms when some terms are known to be empty. Again from Logically Fallacious:

All forest creatures live in the woods.

All leprechauns are forest creatures.

Therefore, some leprechauns live in the woods.

The site cautions, "Just because the conclusion might be true, does not mean the logic used to produce it, was valid."