Saturday, August 26, 2006

Sommers-Englebretsen Term Logic, Part II

In Part I, I discussed some of the basics of SETL. In this post, I want to look at how SETL handles certain tricky issues that are important for the proper handling of propositions in logic. (There will be a later post with additional readings and sources.)

Singular Terms

It's easy enough to see how SETL handles universal quantity, like 'All dogs are canines' or 'No dogs are felines':


It's also easy enough to see how SETL handles particular quantity, like 'Some dogs are tame' or 'Some dogs are not housebroken':


But what if we have a singular term, like, 'Fido is a dog'? In SETL this is handled fairly easily. For singular terms, the distinction between P-opposition and C-opposition turns out not to be significant; and every singular subject can be treated as having a 'wild quantity', because they are indifferent to whether you treat them as universal or as particular. Thus 'Fido is a dog' would be symbolized as:


In an argument you can treat the singular proposition as universal or particular, as you need; the only tricky thing is that you sometimes need to keep track of what you are doing with it. In any case, SETL has no problem with singular terms.


When people give a reason for rejecting traditional term logic in favor of modern predicate logic, one of the reasons at the top of the list is that traditional term logic can't handle relational propositions. As it happens, SETL can handle relational propositions by allowing complex predicate terms. Take the proposition, "All sophists take money from some fools". The basic format of this proposition is:


But the P term is a complex term consisting of other terms. So we can expand the predicate in this way:


Then we can do all sorts of things with this. For instance, suppose we add to it the proposition, "All money is gold." The conclusion is:

-S+(T+G+F) [All sophists take gold from some fools]

Sometimes it is useful to use subscripts, when the direction of the relation is important. So, we could symbolize this proposition as:


This, however, is just a convenience to help us keep track of what the terms mean in complex relational predicates. Subscripts can do a little more than this, for which see below.

On this basis we can translate any relational you could want. Here are some examples and their translations.

Richard loves Richard. ±R +(L±R)
Every boy loves every girl. -B1+(L12-G2)
Every boy loves some girl. -B1+(L12+G2)
Some boy loves every girl. +B1+(L12-G2)
Some boy loves some girl. +B1+(L12+G2)
No boy loves every girl. -B1-(L12-G2)
Every boy sends a rose to some girl. -B1+(S123+R2+G3)
Some girl was sent a rose by every boy. +G3+(S123+R2+B1)

Note that the last two are equivalent, which is precisely the result you should get. There are more complicated predicates that can't be handled so easily, for instance,

Some girls who think that all love is easy are unhappy.

To do this one must introduce propositional nominalization, which we will get to below. But even without this we can do a lot, as we will see in a later post.

Singular Pronouns

Suppose we have a sentence like: "Some boy kissed some girl and she clobbered him."

The first conjunct is easy: +B1+(K12+G2). Given this, we can represent the whole sentence as:


But the use of the subscripts in this way is just a matter of convenience -- to show that we are dealing with pronouns. Singular pronouns are just singular terms, and are treated as such.

(Singular) Identity and Existence

SETL can also handle singular identity very easily. Because singular terms are indifferent to quantity and can be qualified, we can handle such an identity very easily. 'Socrates is Socrates' becomes:


Thus there is no need to bring in any special way of handling identity in order to handle singular identity statements. (Identity between variables is more difficult, and we'll look at that below.)

Just as identity is handled by normal predication in SETL, so, too, are existential statements: existence is a predicate in SETL.

Propositional Nominalization

How would we handle propositions about propositions? The natural way is to treat them as complex terms. Consider the sentence we noted above:

Some girls who think that all love is easy are unhappy.

If we use [p] to indicate the proposition, "All love is easy", we get the following rendering:


But because [p] is a complex term, we can treat it as one, keeping it in square brackets to indicate that it is a nominalized proposition:


In nominalizing, we have embedded one sentence in another by treating it as a term. This turns out to be a means of doing quite a few things that are rather fun. Most important of these is that we can handle propositional logic in our term logic. Consider the following sentences in propositional logic and their categorical SETL forms:

(If p then q) = -[p]+[q]
(p and q) = +[p]+[q]
(If p then if q then r) = -[p]+[-[q]+[r]]

This can be extended. SETL, unlike predicate logic, does not presuppose propositional logic. Likewie, we can handle metapropositions easily:

[p] is false
It is not the case that [p]

And so forth.

Category (Domain) Nominalization

In propositional logic every use of a variable technically requires a domain of discourse (sometimes called a universe of discourse. This is obvious when one consider Lewis Carroll syllogisms. ends up being of this general structure

/domain of discourse for propositions with x/
If it is P, it is Q (where 'it' refers back to the things in the domain of discourse)

In common usage, people don't worry about domain of discourse much; but technically you can't have a variable without it being a variable capable of ranging over a domain of some sort, so it's always there, and necessarily so.

In SETL, as with any term logic, domain of discourse is much less important, but it still can be defined for every sentence, and usefully so, because every statement is true if and only if it denotes its domain of discourse. In SETL every term has a domain (or, if you prefer, category), and the domain of discourse for any sentence is the intersection of the domains (or, if you prefer, categories) of all its terms. Thus

We symbolize the domain (or, if you prefer, category) indicated by the term D with /D/. This is what I am calling 'category nominalization' or 'domain nominalization'. /D/ consists of everything that is D or nonD, where D and nonD are both taken to be part of a category. Thus if D is 'red', then /red/ consists of everything that can be truly characterized by 'red' or 'nonred', where the latter is understood in such a way that it only applies to things falling in the same category as red things. Thus, blue things might fall under this category, but not the number two or the pain in my left hand, because these are a different category (we can meaningfully say of them that they are not the sort of thing that could be either red or nonred). Now, we can use this to handle a particular type of proposition:

Everything is P
Something is P.

In these the subject is the nominalized domain, so they are respectively translated as:


Or in other words, every member of the category or domain associated with P is P; some member of the category or domain associated with P is P. (Of course, things get more complicated if we need to use a larger domain of which P is only part; e.g., if there are lots of sentences, and we need to say that everything in the domain of all the sentences is P. But it works the same way.)

In the next post I will look briefly at some loose ends that are not covered by the above points.

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