Why is the condition of a conditionally valid categorical syllogism determined by the term in the syllogism that is "superfluously" distributed?

Can anyone give me, or point me to, a simple explanation of why this is so?

Why is the condition of a conditionally valid categorical syllogism determined by the term in the syllogism that is "superfluously" distributed?

Can anyone give me, or point me to, a simple explanation of why this is so?

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Hi Colin,

I’ll take a stab at the explanation. I think Venn Diagrams of the conditionally valid argument forms (eg. AAI-1, EAO-3, AEO-4) can go a long way in explaining the pattern. The basic pattern is that (explanation to come) every superfluously distributed term has three of its four Venn areas shaded, and hence, if it has members, those members will be in the unshaded area. Consequently, particular propositions will (conditionally) follow from premises composed entirely of universals.

A term is superfluously distributed if it’s distributed in the premises, but not necessarily so. So M-terms that are distributed twice are superfluously distributed, S- and P-terms distributed in the premises but not in the conclusion are superfluously distributed. In each case, the term superfluously distributed has Venn Diagrams with only one place for objects in the class to reside, and so if it has members, those members will be in that area.

Two examples (AAI-1 and EAO-4) of how this works.

AAI-1 looks like:

All M are P

All S are M

Therefore: Some S are P

S is distributed in the premises, but unnecessarily so. So it’s a superfluously distributed term. A Venn Diagram of the syllogism with standard numbering of the areas (1-7) has premise 1 shade areas 4&7, and premise 2 shade 1&2. This leaves only area 5 for the S-term’s members. If Aristotle’s assumptions are right that terms denote classes with members, then there are some S’s… and so Some S are P follows. Again, on the condition that S has members.

AEO-4 looks like:

No P are M

All M are S

Therefore: Some S are not P

M is distributed twice in the premises, and consequently is unnecessarily so once. So M is the superfluously distributed term. The shading for premise 1 is areas 5&6, and that for 2 is areas 6&7. Area 5 is the only left open for members of M, and consequently, Some S are not P follows if M has members (which could only be in area 5 … which is an area of S outside of P).

If you do Venn diagrams for any of the conditionally valid syllogisms, you’ll see the same pattern – the superfluously distributed terms each have three of their four areas shaded, and as a consequence, if they have members, have them in the remaining fourth areas. The explanation, again with Venn-language, is that because the term is distributed in the premises guarantees that two of the areas will be shaded, and since a distributed M (either if M is distributed twice or if S or P are the unnecessarily distributed terms), three areas of the term in question will be shaded… consequently, if there are members of the class denoted by the term, they will be in the shaded area… and so particularly quantified propositions follow regarding them. (Sorry for the long post, and also sorry for not having the visual media to help with the diagrams)

Thanks, that helps a lot. The idea, if I follow correctly, is that the superfluously distributed term provides “more information” of the sort that allows us to narrow the “location” of the instances of that term, so that if they exist, then we can infer the particular conclusion from the premises.

I think I get that.

But, I still struggle to express this in terms of the meaning of distribution. For example, a fallacy of the illicit minor occurs when the conclusion asserts something about every member of the subject term, while the minor premises does not.

Here the idea seems to be something like–the premises contain more information about the classes, such that we can determine the existential condition for the valid inference of a particular premise from two universals. But I still can’t get that last clause crystal clear.

Right, clarifying the last clause, “we can determine the existential condition for the valid inference of a particular premise from two universals” can be done simply, because in some cases the syllogism form would be unconditionally valid if the conclusion were, instead of a particular (I or O), a universal. AAA-1 is unconditionally valid, and so AAI-1 is conditionally valid. (They have the same Venn diagrams, it’s just that AAI-1 requires an object in area 5 and AAA-a requires only area 5 be unshaded. I follows from A by subalternation)

Though this pattern of subalternation between unconditionally and conditionally valid arguments (A to I in our case above) doesn’t hold up in all cases. For example, EAO-4 doesn’t have the same relation to EAE-4, because EAE-4 isn’t valid.

I think the way you put it is right — the premises contain enough information about the casses for inferences about existential assumption to be valid. This is because with distributed terms, we get information about the class as a whole… and as a consequence, in conjunction with the information about the other intersecting terms, have enough information to make conditional inferences about where at least one member of the class belongs.

Existential fallacy?

Jem, precisely. Conditionally valid syllogisms are valid on the condition that at least one of the classes named in the syllogism has actual members (specifically, the class named by the superfluously distributed term). If you’re a Boolean (better read as non-Aristotelian) about classes, this is never to be assumed, and so these inferences are called ‘existential fallacies.’ But Aristotle’s logic was based on the thought that classes are distinguishable by their contents, and so different classes must have different members. Which means that for a class to be a distinct class at all, it has to have members. For Aristotle, then, existential assumption isn’t illicit.

Think of the difference between conditionally and unconditionally valid syllogisms on analogy with the difference between the two instantiations of the square of opposition. Without the assumption that classes have members, the only valid relation is between contradictories. But with the assumption that classes have members, A and E validly bear the relation of contrariety, I and O of subcontrariety, and A-I and E-O bear the relation of subalternation. Without the assumption that we are talking about things that exist, these would be illicit inferences, but so long as we assume that we are talking about what is instead of what isn’t, they are perfectly fine.

Does something like this play into the reason that mathematicians tend to be realists about sets?

As far as I can tell, Aristotle didn’t talk in terms of “distribution” in his account of the categorical syllogism. This notion seems to have been thunk up later–in particular, in the Middle Ages (cf. Pseudo Scotus,

Super Librum I Analyticorum Priorum, quaestiovii [5]). According to Kneale and Kneale (source of the above reference, see p. 273), it is about the only thing that survived from the long, intense, and nuanced stuy of the logical properties of terms (including supposition, etc.). Aside from this, the existential rules (as some logic books put it) seem somewhat arbitrary, since, for instance, the rules of conversion suggest that any A-statent can be subalternated, then converted without loss of meaning.Then of course, there’s this article on distribution.