Tuesday, February 09, 2010

Sommers Notation, Part II

(Part I)

Singular Terms

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

-D+C
-D-F

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

+D+T
+D-H

But what if we have a singular term, like, 'Fido is a dog'? Every singular subject can be treated as having a 'wild quantity', because as far as the term itself goes it doesn’t matter whether you treat it as universal or as particular. Thus 'Fido is a dog' would be symbolized as:

±F+D

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. Sometimes, to help keep track, it’s useful to mark a term with a star to indicate that it is singular, e.g.,

±F*+D*

It would be possible to handle singular terms without this convention, however; one could treat singular propositions as always universal, but as implicitly paired with corresponding particular propositions. This would, in effect, be treating every singular proposition as a noncategorical proposition, but for practical purposes it would amount to the same thing.

Relations

Take the proposition, "All sophists take money from some fools". This is called a relational proposition because it has a term that relates other terms to each other. The basic format of this proposition is:

-S+P

But the P term is a complex term consisting of other terms, and sometimes these terms play a role in inference. What can we do? Wecan expand the predicate in this way:

-S+(T+M+F)

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:

-S1+(T123+G2+F3)

The 1, 2, 3, indicates that the action of the term, T, is going from S to F through G.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, since we can use them as pronouns, for which see below. But for the most part we don’t need them to do so.)

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

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 logically equivalent, which is precisely the result you should get. There are more complicated terms 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 later. But even without this we can do a lot.

Pronouns

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

The first conjunct is easy: +B1+(K12+G2). The second conjunct has a pronoun, however. How will we handle this? Given this, we can represent the whole sentence as:

(+B1+(K12+G2))+(±2+(C21±1)

Singular pronouns are just singular terms, and are treated as such. Nonsingular pronouns act like normal terms. But the use of the subscripts in this way is just a matter of convenience, to keep track of the fact that we are dealing with pronouns.

(Singular) Identity and Existence

Because singular terms are indifferent to quantity and can be qualified, we can handle identities between singular terms very easily. 'Socrates is Socrates' becomes:

±S+S or
±S*+S*

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 will not discuss it here.)

Just as identity is handled by normal predication in Sommers notation, so, too, are existential statements: existence is a predicate in Sommers notation. If I say, “Socrates exists,” I can represent it as:

±S+E

There are other ways to handle this, as well, but we won’t get into them here.

Having looked at some of these we will get into actual arguments, starting with some simple ones. Then we will look at how Sommers can handle whole propositions as terms.

(Part III)

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