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Hi Avi,

In a discussion on another forum, someone said:

It is universally acknowledged that Aristotle's 2,300-year-old logic is inadequate: for example, it is unable to demonstrate, from the fact that a horse is an animal, that the head of a horse is the head of an animal.

Another poster countered with this :

All horses are animals.
Each horsehead belongs to a horse.
Each horsehead belongs to an animal.

Is this not a valid syllogism?

But it seems to me that it isn't, because there are too many terms : horses, animals, horseheads, things that belong to a horse, and things that belong to an animal. 'belongs to' is a relation, and so strictly speaking this isn't a valid categorical syllogism.

My solution would be this, which uses only the requisite 3 terms :

H = horses
A = animals

1. All H are A
2. All h of H are h of H (tautology)
3. therefore, All h of H are h of A

Since A subsumes H, we can replace H in the predicate of the 2nd premise by A (Dictum de Omni).

Any comments? I note that you mention this kind of argument in FL.

Actually, I believe Aristotle's original formulation of categorical propositions used 'belongs to' as the copula, so the form was :

A does/doesn't belong to All/Some B

But 'belongs to' has a different meaning here than in the first argument above, where it really means 'is a part of' rather than subsumption. If you used 'belongs to' as the copula throughout the argument then there would only be 3 terms, but then the fallacy would be equivocation.

Hi Joe.

“a horse is an animal, therefore the head of a horse is the head of an animal” does have an underlying syllogism, as follows (3rd figure AAI):

All horses are animals,
whence some animals (namely the horses, at least) have heads.

I mention this case in my Future Logic, chapter 19.1. http://thelogician.net/FUTURE-LOGIC/Quantity-More-Details-19.htm - where I wrote:

“What logicians call immediate inference by added determinants (e.g. ‘horses are animals: therefore, the heads of horses are heads of animals’) or complex conception (e.g. ‘Physics is a science: therefore, physical treatises are scientific treatises’), involve substitutive syllogism, with a tacit minor premise (e.g. ‘horses have heads’ or ‘some treatises are about physics’), which enables the conclusion to be drawn. These processes are illicit when the rules of substitution are not properly obeyed (e.g. ‘horses are animals: the majority of horses are the majority of animals’ or ‘physics is fun, physical treatises are funny’).”

Notice the possibility of error if the wrong type of term is involved.

Your proposed syllogism actually has four terms, not three. A, H, h of A, and h of H. But you are right to say a replacement of A for H is used.

Anti-Aristotelians assume he did not know substitution. Though he did not deal with it in formal terms, as I recall he does discuss it or at least use it somewhere(s). If you look at the modern logic treatment, it comes down to about the same. There is no new magic; they are just making an explicit principle out of it by means of symbols. Then they make a big deal of it, to give themselves importance. Check it out.

Avi

Joe, I should clarify the role played by this underlying syllogism in the given 'added determinants' argument.

All horses are animals,
whence some animals (namely the horses, at least) have heads.

What is the precise utility of this syllogism? It serves to equate the respective units of the two terms involved - namely (all) horses and (some) animals. The common reference of these two terms is what makes logically possible the substitution of one term for the other.

Thus, the head of a (any) horse is at the same time the head of (some) animal. Thus, we can licitly argue that the head of a horse is the head of an animal.

And this is made possible entirely by means of Aristotelian syllogism. The modern critics of the latter are thus shown to be boasting of innovation without justification.

On the other hand, for instance, the argument "horses are animals: the majority of horses are the majority of animals" is invalid, because we cannot form an equivalent syllogism. We cannot predicate the term majority of horses; i.e. we cannot produce the minor premise "all horses are majority" - because, of course, majority here is a quantity of horses, not a predicate of theirs. Whence the desired conclusion "some animals are majority" is also nonsensical.

Regards, Avi

Avi, as usual, your input has shed light on the subject - thanks. Chapter 19 of your book is interesting and seems to extend the rather narrow province of the traditional categorical syllogism into many more forms of argument. Obviously, those who criticize Aristotle's logic are not aware of these 'extensions', which is why they think it is so limited. It's somewhat surprising to me that when the new and "improved" logic came along early in the 20th century, there was not more resistance to it, but perhaps that's because so few people knew (or really cared) that the traditional logic has essentially the same inferential power. Too many people are impressed by the apparent "rigor" of mathematical notation, it seems.

Returning to the horses head argument, the guy who posted the syllogism has responded to my objection that it had too many terms (by the way, you are right that my solution has 4 terms, not 3 - my bad) :

Joe,

'A thing which belongs to a horse' and 'a thing which belongs to an animal' aren't actual terms. To show why, let's reformulate the syllogism.

Horses are a kind of animal.

This says precisely the same thing as the other syllogism, except now, everything uses 'is.'

But this still has 5 terms, so isn't a valid categorical syllogism, although it is a valid argument which contains an implicit substitution.

Anyway, these are comments following an interesting blog post by philosopher Ed Feser, who I've mentioned before in relation to arguments about the existence of God. You might like to read the post, and maybe even add a comment.

The metaphysical presuppositions of formal logic

Hi Joe,

As you say, "too many people are impressed by the apparent "rigor" of mathematical notation." They do not realize that symbols are merely abbreviations of longer verbal statements - they add nothing to the the accuracy of the information they transmit. Indeed, they are poisonous to knowledge, in that they befuddle people, who tend to forget just what the symbol contains and what it does not contain. The mind, if we are careful, is forced to artificially keep returning to the underlying verbal discourse, to truly understand the symbol. Why cause people such fatiguing and prone-to-error detours? Many people give up on logic nowadays because of this unnecessary complication.

Symbolization should only occur, if at all, at the very end of the theoretical research and discovery process. It is not a reliable tool to start with.

Regarding the link you gave me - I am sorry but cannot follow it, because I do not want to get entangled in endless conversations with hostile elements. I am retired, and enjoy my rest. You are welcome to pursue the conversation with them, using the replies proposed above. I always appreciate your questions, because they get me thinking.

Best regards, Avi