Tuesday, February 22, 2011

Stacking Categorical Propositions

An advantage of the stack method of representing logical propositions is that it allows for the easy tracking of distribution, and shows quite clearly its relation to universality.

Suppose we take a standard A (universal affirmative) proposition: Every beagle is Snoopy-like. We can represent this, stackwise, in the following way:

 Snoopy-like Beagle

That is, Snoopy-like is here the term under which Beagle falls, with no allowance for exceptions. We can then represent the I (particular affirmative) proposition Some beagle is Snoopy-like as:

 Snoopy-like *Beagle

Snoopy-like is still the term under which Beagle falls, but the asterisk lets us know that we are allowing exceptions to this relationship (not guaranteeing any, of course, just allowing them). We continue on with the E (universal negative) proposition, No beagle is Snoopy-like:

 ~Snoopy-like Beagle

(I use the tilde instead of the minus here because I think it stands out better.) This lets us know that the term under which Beagle falls is non-Snoopy-like, and no allowance is made for exceptions. And to conclude the cycle, the O (particular negative) proposition, Some beagle is not Snoopy-like:

 ~Snoopy-like *Beagle

Just like E, of course, but this time allowing exceptions.

Now the old way of handling syllogisms, which once was common in logic textbooks is by way of distribution. It fell out of favor for quite a while as an unclear concept, but has slowly been making a comeback as people have begun to think that there was something to the idea after all. In Sommers-style term functor logic, for instance, distributed terms are associated with minus signs and undistributed terms with plus signs, and it is precisely this that makes the whole thing work. The standard way of assigning distribution to the four families of categorical proposition is as follows:

 Subject Term Predicate Term A Distributed Undistributed E Distributed Distributed I Undistributed Undistributed O Undistributed Distributed

That is, subject terms are distributed when quantity is universal and predicate terms are distributed when quality is negative. (Asebinop was the mnemonic suggested sometimes in the nineteenth century: A subject, E both, I neither, O predicate; which, I confess, has never seemed to me a particularly memorably mnemonic.)

Keeping this in mind we can see that our rules for stacking guarantee that distribution is respected. Terms at the top of a stack are undistributed unless they have ~ (because they are predicate terms) and terms at the bottom of a stack are distributed unless they have * (because they are subject terms). So far, so simple. But it's worth keeping in mind that the basic idea of stacking is to show relations of universality: those terms are more universal or encompassing, at least in principle, that are on top or to the left (although this latter is not perfectly straightforward), while ~ so to speak makes a top term more encompassing and * so to speak makes a bottom term less. Distribution is what it is because of the three properties that determine how expansive or restrictive a term is in its relation to another term: position in the stack (i.e., status as subject or predicate), ~ (i.e., status as negative or positive), * (i.e., status as exception-allowing or non-exception-allowing).

Properly speaking, of course, a term is distributed, or used distributively, when it applies to everything for which a term has 'personal supposition'; so this all makes quite a bit of sense, although full discussion would require getting into details of supposition theory. But supposition theory itself is a detailed theory of what we might call the capacity of a term to stand for something, and terms are included in or fall under other terms to the extent that what they stand for is part of what the other term stands for.

Thinking of our stacks in terms of the universality relations among terms, we can also see why something like subalternation makes sense. If I have a proposition like this:

 Snoopy-like Beagle

I am saying that Beagle falls under or is included within Snoopy-like, no exceptions; if I add a * to the bottom of the stack:

 Snoopy-like *Beagle

all I am doing is saying that Beagle falls under or is included within Snoopy-like, and I am not committed to there being no exceptions: the * just indicates that the term might, for all I'm claiming, cover less than it does when unrestricted. This is a weaker claim, and by the dictum de omni (what applies without qualification to a term still applies to the term under qualification) the one follows from another.

The other oppositions, of course, work much the same way: contrariety and subcontrariety add or subtract a ~, and contradiction adds or subtracts ~ and also *. One potential disadvantage is that conversion is less perspicuous.