Monday, February 08, 2010

Sommers Notation, Part I

For my students I'm putting together a new guide for using Sommers-Englebretsen Term-Functor Logic, also known as Sommers-Englebretsen Term Logic, and which I will call here, for simplicity, Sommers notation. I thought I would put up a draft version.

Parts of a Categorical Proposition

If we want to talk about categorical propositions, it’s helpful to break them down into six parts which may be briefly described as follows:

(1) Subject Term: This is what the proposition is directly about.
(2) Predicate Term: This is what is being said about the subject term.
(3) Universe (or domain) of discourse: This is something the subject and predicate term share that allows them to be compared in the proposition.
(4) Quantity of subject: how much of the subject the predicate covers.
(5) Quality of predication: how the predicate term applies to the subject term.
(6) Judgment: how the whole proposition is being put forward.

There is much more that could be said about each of these things, both separately and in their relations, but this will do for our purposes. Because the universe of discourse is already implicit in the subject term and the predicate term, we don’t usually need to include it specifically in a notation, although (as we shall see) it can be useful to have a way to do it if we need it. But, in general, we want a notation that allows us to keep track of all these parts.

Sommers notation starts from the basic idea that every categorical proposition is the affirmation or denial of a simple or complex predicate of all or some of a subject. When you look at categorical propositions in this way, we can see that they admit of three basic kinds of oppositions.

1. Term Valence. Every term has either a positive or a negative term quality. 'Red' would be an example of a term with positive term quality; 'Nonred' would be an example of a term with negative term quality.

2. Opposition of Quality. Likewise, every predicate has either a positive or a negative predicate quality. 'Is red' has a positive predicate quality; 'Isn't red' has a negative predicate quality. (One of the features of Sommers notation is that there is no practically significant distinction between term quality and predicate quality; 'S is non-P' is not significantly different from 'S isn't P'. But when starting out is useful to treat them as distinct.)

3. Opposition of Quantity. Every predicate is predicated of at least some or definitely all of a subject. This sort of opposition is an opposition between a universal subject and a particular subject. So 'Some S is P' is opposite in quantity to 'All S is P'.

3. Opposition of Judgment. Every proposition is put forward as true or false, that is, it is asserted or denied. 'It is not the case that S is P' is opposed in judgment to '(It is the case that) S is P'.

The upshot is as follows.

(a) Every categorical proposition has a subject (S) and a predicate (P).
(b) Every term, independently of its role in the proposition, has a mark indicating term valence.
(c) Every P as a complete term has a mark indicating opposition of quality.
(d) Every S is a term with a mark indicating opposition of quantity.
(e) The entire proposition linking S and P is itself a term with a mark indicating opposition of judgment.

Given these four basics, which I will not argue for here, we can develop the basic format of Sommers notation.

Plus and Minus

We have three oppositions. Sommers's great idea was to take these oppositions and note them down as plus and minus in a subject-predicate proposition. So we have (on the basis of (a) above)

S...P

as our assertion. However, we know from (b) that every term has its own valence. Thus:

(±S)...(±P)

We know from (d) that we want every subject to have its mark of quantity. Thus:

±(±S)...(±P)

And since each predicate may be itself a complex term, it has as predicate a mark of quality, which allows us to take (c) into account. Thus:

±(±S)±(±P)

And we know from (e) that every proposition has a judgment that (so to speak)links the predicate and subject together so that the proposition either asserts something or denies something. This we can symbolize, putting the mark of judgment at the beginning, as:

±(±(±S)±(±P))

Of course, this is just a general format. Let's take a basic assertion: All S is P. This can be symbolized by:

+(-(+S)+(+P))

S and P are both of positive quality, so the sign linked to each term is positive. P is affirmed of S and so there is a plus sign linking (+S) and (+P); S is of universal quantity, which we indicate with the minus in front of (+S); and the whole proposition is being asserted as true, so we have a plus sign in front of the whole proposition. This is pretty intuitive, except, perhaps, for the reason why the universal quantity is given a minus and the particular quantity is given a plus. The reason for this is (if you want the crude and read version) is that if we do it this way the whole thing works. If you want a more technical answer, however, we make the universal minus and the particular plus in order to preserve the contraposition of the A categorical (All S is P) and the conversion of the I categorical (Some S is P). That is, we want these two equivalences:

+(-(+S)+(+P)) = + (-(-P)+(-S)) [i.e., Every S is P is equivalent by contraposition to Every nonP is nonS]

+(+(+S)+(+P)) = +(+(+P)+(+S)) [i.e., Some S is P is equivalent by conversion to Some P is S]

It's easy to recognize these equivalences if we give the universal a minus and the particular a plus. Another way to put the same point in technical terms would be to say that the minus for quantity tracks what the traditional theory of the syllogism calls distribution. But quantity's the only tricky thing about this basic format: + and - simply indicate an opposition, and '-' in particular shouldn't be confused with negation, which it only sometimes has.

In the above format, all we've marked are the terms and their oppositions. Which opposition is relevant is entirely a matter of where it is positioned in the assertion, so we don't have to worry about distinguishing them in any other way; + and - will do for everything. The really neat stuff we'll get to later. For now, we'll note just one neat feature that this way of symbolizing yields us. If we treat +'s as we usually treat +'s (e.g., in math), we can contract a string of plus signs. Thus

+(+(+S)+(+P))

can be written as

+S+P

without any loss of logical function. Likewise, we can treat -'s in a complementary way, such that two minuses together become a plus, and a minus and plus contract to a minus. Thus

-(+S)-(-P)

can be written as

-S+P without any loss of logical function. They are always logically equivalent, although in their expanded forms they may look different. We can then give a simplifed form to all the basic Aristotelian categoricals:

A (All S is P) -S+P
E (No S is P) -S-P
I (Some S is P) +S+P
O (Some S is not P) +S-P

But, given that we can do all Aristotelian syllogisms. A syllogism works when it can be formulated as a true equation and both sides are similar. Take the famous Barbara (AAA-1) syllogism:

All S is M
All M is P
Therefore, All S is P.

This has the equation:

(-S+M) + (-M+P) = -S+P

Just treat it as you would treat it if it were an algebra equation. You'll see that the left side is indeed equal to the right. All we have to do in order to be certain that it is a valid syllogism is to make sure that the two sides are similar. The two sides are said to be similar if (a) they have the same extremes (i.e., terms that are not arithmetically eliminable); and (b) they have the same quantity (the conjunction including a particular always being particular). In the Barbara case, the sides are clearly similar. Therefore it is valid. We can even handle 'weakened syllogisms' (syllogisms with universal premises that have particular conclusions) if we assume that they have the hidden premise +S+S (which, as we'll see, is a tautology and can be introduced at will). Thus Camestrop (AEO-2) would be:

(-S-M) + (-P+M) + (+S+S) = +S-P

It is useful to summarize this in a procedure that will help us to determine validity in any categorical syllogism using Sommers notation:

(1) Check to see if the argument is algebraically acceptable – that the premises add up to the conclusion.
(2) Check to see if the argument is regular. There are two, and only two, ways an argument can be regular. Either it has only universal propositions, or, if it has a particular proposition, it has one (and only one) particular proposition in the premises and a particular conclusion.
(3) To handle weakened syllogisms, if an argument fails at (1) and (2), check to see if it can pass if we add (+S+S) to the premises, where S is the subject term of the conclusion,

That is, a syllogism is valid if its premises add up to the conclusion and it is regular, either in itself or with the premise ‘Some S is S’ is added.

This is all quite cool. But I'm partly getting ahead of myself here. Sommers notation is more powerful than I've suggested so far, and we need to introduce a few additional tools if we are to see this and handle all the kinds of argument Sommers notation is capable of handling. So now we'll go on to look at how Sommers notation handles various key issues (singular terms, relations, identities, meta-propositions, existence). Then we'll handle arguments.

(Part II)

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