Modern Logic and the Traditional Square of Opposition.

[Louis Bacio]

Anyone. What is your take on the difference between the Traditional Square of Opposition and the Modern Square of Opposition?

[Martin Cothran]

Dear Louis,

The answer to the first part of your question is simple, however, the answer to the second part is a little more involved.

Most basically, the difference between the square of opposition between the two systems of logic--traditional and modern--is that, in traditional logic, it is really a square, whereas in modern logic, it is not. In other words, traditional logic recognized four relations of opposition (the four corners of the square) and the modern system only recognizes one.

Traditional logic--because of the traditional metaphysical assumptions about reality that underlie it--recognizes the following four relations of opposition: contradictory, contrary, subcontrary, and subalternate (although the subalternate relation is not a true relation of opposition). Modern logic, in contrast, only recognizes the contradictory relation.

This difference results from the use in modern logic of "set theory". This is rather complicated to explain, but suffice it to say that what the incorporation of set theory does is to allow modern logic (the progenitors of which were largely materialist in their view of reality) to deal exclusively with existence. In traditional logic an exclusive emphasis on existence is seen as misguided, since meaning must also be taken into account.

Kelly Ross has dealt with this issue in his online article "In Defense of Bramantip" (http://www.friesian.com/syllog.htm). In addition, I am currently writing a pamphlet on the differences between traditional and modern logic.

Here is an excerpt from my pamphlet on the issue dealing with the differences between the traditional and modern systems in regard to the square of opposition:

The Modern View of the Square of Opposition
“The problem begins,” says Ross, “not with the syllogisms, but with other basic relationships among propositions.” This is shown most clearly in the monkey wrench that modern logic throws into the traditional square of opposition. The traditional square of opposition offers students a picture of the relationships of opposition between the four kinds of logical statements.

Traditional logic recognizes four kinds of statements. They are illustrated as follows:

A: All men are mortal
I: Some men are mortal
E: No men are mortal
O: Some men are not mortalThese are called, A, I, E and O statements, as indicated. These statements can be opposed to one another in any one of four different ways: they can be contradictory to one another, contrary to one another, subcontrary to one another and subalternate (although subalternation is not considered to be a relationship of true opposition).

The Contrary Relationship of Opposition
In traditional logic, the A and E statements are contrary. In other words, they can both be false at the same time, but cannot both be true. Take the following two statements:

All men are mortal

No men are mortalIn traditional logic, these two statements may both be false—if, for example, only some men are mortal. But they cannot be true at the same time. In other words, there is no case in which we could say all men are mortal and no men are mortal at the same time. In modern logic, however, two contrary statements can both be true, and are therefore not contrary. The reason for this is that, in modern logic, universal statements (e.g. All S is P and No S is P) do not have existential import—they do not imply the existence of their subjects.

The Subcontrary Relationship of Opposition
In traditional logic the I and O statements are considered subcontrary to one another; in other words, they can both be true at the same time, but cannot both be false. Take the following two statements:

Some men are mortal

Some men are not mortalIn traditional logic, these two statements may both be true—if, as a matter of fact, there are some men who are mortal and some not. But they cannot both be false. In modern logic however, two subcontrary statements can both be false at the same time.

This results from modern logic’s incorporation of existential import. Modern logic interprets all universal statements (e.g. All men are mortal and No men are mortal) as having no existential import. In other words, the subject terms in universal terms are assumed not to be existing things (since, in modern logic, the A statement has been reinterpreted to say, “If there are men, then they are mortal,” and the E statement has been reinterpreted to say, “If there are men, then they are not mortal”).

Likewise particular statements (e.g. Some men are mortal and Some men are not mortal) are taken as having existential import. In other words, the subject terms in particular statements are said to assert the actual existence of their subjects (since, in modern logic the I statements has been reinterpreted to say, “There is an x such that x is a man and x is mortal,” and the O statement has been reinterpreted to say, “There is an x such that x is a man and x is not mortal.”)

Why the Contrary and Subcontrary Relationships Must be Rejected in Modern Logic
This being the case, we can have two statements, such as:

I: Some unicorns have horns

O: Some unicorns do not have hornswhich must both be false, since unicorns do not exist. But two subcontrary statements cannot be false at the same time. Therefore, subcontrariety must be discarded. Furthermore, since, under the above example, both I and O statements are false, their contradictories, the A and E statements:

A: All unicorns have horns

E: No unicorns have hornsmust both be true. But two contrary statements cannot both be true at the same time. Therefore, contrariety must be discarded.

Why the Subaltern Relationship of Opposition Must be Rejected in Modern LogicIt is not only the contrary and subcontrary relationships that are affected; so also is the Subaltern relationship. In traditional logic, if I say:

A: All dodos have feathersThen I can infer the statement:

I: Some dodos have feathersSince, if all dodos have feathers, I know that any one or more dodos will, altogether, also have feathers. But because modern logic insists on taking the actual existence of dodos into account in its formal system, it says that there could be a case in which the subject of the A statement (dodos, in this case) does not actually exist. In this case, the statement All dodos have feathers is still true (because in modern logic the A and E statements do not assert the existence or lack of existence of the subject term), but the I statement is false (since in modern logic the I statement does assert the actual existence of the subject term). Therefore, A statements cannot infer I statements, since A statements can be true while their corresponding I statements are false, and therefore, the relationship of subalternation cannot be accepted.

The reason for this would involve a more complicated discussion of set theory’s acceptance of the possibility of an empty set, but the fundamental reason goes back to the fact that, in traditional logic, predicates refer to concepts (where the nature or essence of a thing is known), whereas in modern logic predicates are sets that are defined by their members. All of which is just another way of saying that traditional logic assumes that words have meaning apart from the actual existence of their objects and modern logic does not.

If words have real meaning, then two contrary statements cannot both be true—and two subcontrary statements cannot both be false—since to say so would amount to nonsense. There is no nonsense in modern logic—because there is no sense.

That's a rather involved explanation, but it is a rather complicated topic.

I said before that modern logic takes only existence into account in its interpretation of words, whereas traditional logic takes meaning into account as well. For example, if I were to say, "All unicorns have a horn," the statement would have meaning, even though unicorns do not exist. Modern logic has difficulty with these kinds of statements, whereas traditional logic does not.

According to traditional logic, terms can have both comprehension (meaning) and exension (existence), whereas modern logic does not take comprehension into account at all. It is like the difference between viewing things with one eye covered, where you can only see things two-dimensionally, rather than with both eyes, where you can view them three-dimensionally.

I hope this helps.

Martin Cothran

[dlkester1@gmail.com]

I dont know if I am doing this right, it is my first time to post on one of these things, so if I am doing something wrong I apologize. I had a question about the partial reduction of Bramantip. Since the P is before the conclusion, if we were to partially convert a Bramantip, we would be doing the operation on an I and it is only said to be valid on A statements. Since the Galenic figure is really just a weak version of the first, do we simply perform the operation on the I conclusion in Bramantip knowing that we are really doing it to a weak verson of the A conclusion like in Barbara? IF this is so, why bother converting Fourths to Firsts at all?

According to the traditional interpretation, the I conclusion to Bramantip is only a weaker form of the A conclusion found in BARBARA -- unless it is given an existential interpretation. In the later way of looking at it, we would be reasoning to a particular existing individual from universals -- perhaps not a move some might want to recognize as legitmate. Can you know anything about me as an individual existing person from some universal generalizations about classes of which I am a part? From the reading of online articles on the importance of Bramantip, it appears that Bramantip is important in the differing interpretations of modern and traditional squares of opposition. I can't help but think that this is important as we consider the question of whether the partial conversion of Bramantip is valid -- or I could just be misreading the directions for partial reduction. I had a student ask me this question in class on Friday, and I am scrambling to get together an answer for her. Any comments?

[Martin Cothran]

The reduction procedure is just a way of backtracking from syllogisms in other figures back to the first figure syllogisms. You're not really deriving a first figure syllogism from (in this case) a 4th figure. Syllogisms in other figures are actually derivable from the 1st figure. You are just tracing it back. A BRAMANTIP is just a BARBARA that has been particularized. BRAMANTIP is what happens when the premises are switched and a universal A conclusion is reduced to an I conclusion. In reduction, you are just rolling back the tape so to speak. You are not going forward, you are going backward.

I hope that helps.

[dlkester1@gmail.com]

Dear Brother Martin, Your comment last time was very helpful, and I was wondering if I could bother you again regarding the possibility of contraposing an O statement, raised by one of my students. If we are not allowed to convert O statements, how can we contrapose O statements since the mediate step in this procedure is to convert? Also, you state on page 84 of chapter that there is a procedure for partial conversion (contraposition?) of E statements, but you will not discuss it in the text. Do you have this discussion elsewhere in one of your texts, or is it online somewhere. I look forward to your reply.