Saturday, August 26, 2006

Sommers-Englebretsen Term Logic, Part I

The purpose of this post and its sequels is to provide a basic introduction to elements in the Sommers-Englebretsen Term-Functor Logic, also called TFL, or, as I will tend to call it, SETL. This is a way of handling propositions that has a great many advantages, being more closely allied to natural language than modern predicate logic, and being surprisingly flexible and powerful, given how simple it is.

Three Kinds of Opposition

The most natural place to start if you want to understand how SETL works is to look at what sorts of opposition can be logically relevant. SETL starts from the basic idea that every categorical assertion is the affirmation or denial of a simple or complex predicate of all or some of a subject. When you look at assertions in this way, we can see that they admit of three basic kinds of oppositions

1. Opposition of Quality. 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. 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 SETL is that there is no significant distinction between term quality and predicate quality; 'S is non-P' is not significantly different from 'S isn't P'. Changing the quality does make an important difference to a logical argument, so this opposition, which I will (following Englebretsen) call C-opposition, because it is the foundation of logically contrary propositions.

2. Opposition of Quantity. Every predicate is predicated of some or all of a subject. This sort of opposition, which will be called Q-opposition, is an opposition between a universal subject and a particular subject. So 'Some S is P' is Q-opposed to 'All S is P'.

3. Predicative Opposition. Every predicate is affirmed or denied of its subject. This is the third opposition, which we will call P-opposition. 'It is not the case that S is P' is P-opposed to '(It is the case that) S is P'.

The upshot is as follows.

(a) Every categorical assertion has a subject (S) and a predicate (P).
(b) Every term, independently of its role in the assertion, has a mark of C-opposition.
(c) Every P as a complete term has a mark of C-opposition and as a predicate has a mark of P-opposition.
(d) Every S is a term with a mark of Q-opposition.

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

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 mark of C-opposition. Thus:

(±S)...(±P)

We know from (d) that every subject has its mark of Q-opposition. Thus:

±(±S)...(±P)

And since each predicate may be itself a complex term, it has as predicate another C-opposition mark (c). Thus:

±(±S)±(±P)

And we know from (c) as well that every predicate, as predicate, has a P-opposition mark, which we can symbolize, putting the P-opposition mark over the whole predication (and thus 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 (thus their positive C-opposition signs); P is affirmed of S (thus its positive P-opposition sign); and S is of universal quantity (thus its negative Q-opposition sign). 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. More technically, 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., All S is P is equivalent by contraposition to All nonP is not 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. But that's the only tricky thing about this basic format: + and - simply indicate an opposition, and '-' in particular shouldn't be confused with negation.

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. And that's where it gets neat. 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 Baraba (AAA) 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) would be:

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

This is all quite cool. But I'm partly getting ahead of myself here. SETL 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 SETL is capable of handling. So in the next post on this subject we'll look at how SETL handles various key issues (singular terms, relations, identities, meta-propositions, existence). And then we'll handle arguments.

1 comment:

  1. Orangrutan10:31 PM

    Thanks for this. I started on the book recently and there aren't a lot of resources on this topic. Except for the Englebretson book and this site there is nothing else I have found yet.

    Orangrutan

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