Friday, February 18, 2011

One-Stack, Two-Stack

I've been thinking a bit about James's post on a visual representation of universality relations for syllogisms. Aristotle's basic syllogistic is based on the universality of terms. So suppose you have the following premises: "All mammals are vertebrates, all felines are mammals". Then we can arrange the terms in a stack, going from the term that is capable of being the most universal at the top to the term that is capable of being the least universal at the bottom. In a first figure syllogism, which this would be, we would get the following stack:


And every Barabara (AAA-1) syllogism would look like this, whatever its terms. In order to handle all first figure syllogisms, we'd need to add two conventions. First, we need to be able to indicate that we are allowing exceptions in the application of the term. This is just what particular quantity is, so another way to put it is that we need to be able to indicate particular quantity. James's suggestion of an asterisk works well. So the premises "All pets are lovable, some cats are pets" would then get us:


This, of course, would be Darii (AII-1).

Second, we need to be able to indicate negations, and this is, I think, easy enough. Since the stacks make it reasonable to think of terms falling under terms, we can ask ourselves, "If we say, 'No dogs are allowed,' what term is 'dogs' being put under?" And the answer to that is easy: Not Allowed, or Unallowed, or Nonallowed, however one wishes to say it. That is, it's being placed under the negative complement. So we can have Celarent:


Which would represent the premises, "No dogs are allowed, all chihuahuas are dogs." Ferio follows quite easily: it uses both of our conventions:


Thus we have all of the first figure. In essence, what we're showing is the dictum de omni et nullo in action. Our two conventions will handle most of what's needed for the rest of the figures, but the first figure, of course, is the only figure that is easily represented in a single stack without additional conventions. Rather than try to find conventions that would allow single-stack representation of the other figures, I suggest we start with two-stack representation. In the Barbara syllogism above, we could just as easily have represented our premises in two stacks:

Vertebrates Mammals
Mammals Felines

The first stack is the major premise and the second stack is the minor premise; the resulting representation, therefore, will work in a very similar way to the standard ways of handling syllogistic that we've inherited from the scholastics, although our conventions will allow some minor simplifications.

Since we're interested in relations of universality, we can add one more convention: major stack is always on the left, minor stack is always on the right. As James notes, this rules out any fourth figure diagram, because the fourth figure breaks this rule. Representing all the rest of the figures and moods becomes quite easy. Here is Cesare (EAE-2):

-Middle Middle
Major Minor

Notice that in this syllogism, the Middle term tops both stacks, and one of the middle terms is negated. This will characterize all second figure syllogisms. Here is Darapti:

Major Minor
Middle Middle

Here the middle terms bottom both stacks; and this, of course, will be true of all third figure syllogisms. Every non-weakened mood of each of the three figures can be handled by adding the following rules:

(1) Only a middle term or minor term at the bottom of a stack can receive *.
(2) There can be one but only one *.
(3) Only a major term or middle term at the top of a stack can receive -.
(4) There can be one but only one -.
(5) If there are two middles at the top of the stack, one must receive -.

(2) and (4), of course, are just standard rules about particular and negative premises: you can't get anything from two particular premises, and you can't get anything from two negative premises. (1), (3), and (5) each help guarantee that distribution works, or, to put it another way, they prevent violations of the dictum de omni. All the valid moods, then, are permutations of the figures (neither * nor - ; * but not - ; - but not * ; both * and -) that follow these five rules. And that's actually a nice feature. When you just teach Barbara, Celarent, Darii, Ferioque, &c. it's often not really obvious why these moods are the ones we select out, nor why there are four in the first figure, four in the second figure, but six in the third figure.

As one would expect, you can do standard transformations with these stacks, and, say, turn Cesare into Celarent. It becomes very much like a puzzle: given moves like conversion, transmutation, and partial conversion, how do you get from one double-stack diagram to a first figure double-stack diagram (which allows, of course, the simple one-stack diagram where the conclusion can just be read off)? The representation doesn't add any logical power, but it is capable of handling all of the syllogisms Aristotle would have considered non-defective, and also makes clear why these were the syllogisms he thought non-defective, and one reason why Aristotelians have usually considered the first figure to be the complete figure: it allows the most economical representation of all the universality relations between terms.

One could certainly do more with this; not being Tom, I lack the ability to do anything fancy with logical systems without much slow thought, so this is all I have at the moment.

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