Square of opposition

In traditional logic it is called square of opposition to certain logical relations that according to the Aristotelian system occured between different categorical propositions of the same subject and predicate, resultig of changing quality, quantity or both.

Remember that so-called categorical propositions affirm or deny relationships between classes, one represented by the term subject, the other by the predicate.

According to quality, a proposition of this kind can include or exclude a class (denoted by the subject term) in another (designated by the predicate). Depending on the quantity, a proposition distributes its subject term or not. The latter means that it refers to all members of the class that the term subject denotes, in which case it would be distributed, or, otherwise not.

Propositions can be contrary, contradictories, subordinate and subcontraries to each other.

Two propositions are contrary if both cannot be true; they are contradictory if, besides being contrary, it is possible for both to be false, they are subcontraries in case they cannot be false at the same time (they can both be true.) One proposion is a subaltern of another when, having both the same subject and predicate, and having also the same quality, the universal one ditributes the first, but the particular does not.

The four typical forms of categorical propositions are:

A: Every A is B
E: No A is B
I: Some A is B
O: Some A is not B

And traditionally were drawn certain 'immediate' inferences from the oppositions between them.

For example (taking, as mentioned, same subject and predicate), from the truth of A was inferred the falsity of O, because they are both contradictory, and E's, as well as I's truth. The falsity of A implies the falsity of O and let the other two indeterminate.

From E it was inferred the falsity of both A and I, and the O's truth. Of its falsity, the truth of I.

It could be inferred from I, that E could not be the case, remaining A and O indeterminated.

As for O, from its truth was inferred the falsity of A (without determining the valuation of the rest): while from its falsity, A's (its contradictory) truth, and I's, as well as the falsity of E.

 

(Image: wikipedia)

But in this post it was not intended to present the traditional system of categorical propositions but to mention a comment on orthodoxy's criticism to that system, comment on which Strawson calls attention in his Introduction to Logical Theory (Ch. 6).

The criticism to which I wanted to refer concludes that there is not a consistent and acceptable interpretation of the system−and in particular the square of oppositions−, in an interpretation of it's constants in a way they approach to the ordinary use of the words that are expressed. To which it is assumed that it must be chosen between only two possible translations of the propositions to the system of predicate logic.

These are:

1:
A: ~ ∃ x (Fx ∧ Gx ~) or ∀ x (Fx ⊃ Gx)
E: ~ ∃ x (Fx ∧ Gx) or ∀ x (Fx ⊃ Gx ~)
I: ∃ x (Fx ∧ Gx) or ~ ∀ x (Fx ⊃ Gx ~)
O: ∃ x (Fx ∧ Gx ~) or ~ ∀ x (Fx ⊃ Gx)

and 2:

A: ~ ∃ x (Fx Gx ∧ ~) ∧ ∃ x Fx
E: ~ ∃ x (Fx ∧ Gx) ∃ x Fx ∧
I: ∃ x (Fx ∧ Gx)
O: ∃ x (Fx ∧ Gx ~)

In both cases some of the rules of the above mentioned, wich are represented in the opposition square, should be resigned .

In Table 1, for example, A and E can be true together, so they are no longer opposed. O and I, in turn, no longer subcontraries.

With respect to the second table, for example, O and I are not subcontraries anymore. Furthermore A and O on the one hand, and I and E on the other, are not contradictory.

Strawson shows how there are at least two different ways to interpret A, E, I, O for which the traditional laws still apply.

One of them, that seems the most important, has to do with criticism of Russell's theory of denotation. Briefly, while the proposition "that books over the table are all in spanish", presupposes for the one who enunciate it that the class of "books over the table" is not empty, in "the king of France is bald " that there is a king of France. If there are no books on the table, the first is not true, if there is no king of France, the second is not false. But I did not wanted to expand on this issue at this time.

The question I wanted to reach was: which formulas of predicate logic can be offered to translate the four forms of categorical propositions A, E, I, O, that retain the traditional laws, if there are any?

 

The dilemma of Anacharsis

Here can be finded a discusion about a Davidson's critic to Strawson philosophy. One of the arguments is that if we, as Kant, supousse there exists a "conceptual scheme", then we cannot reach at it's knowledge as if it was not tracendental but real. In the terms of the mentioned blog: "What right, then, does Strawson have to describe our conceptual scheme, a scheme he repeatedly admits we cannot transcend?".

Well, I think the idea is very similar to another very old one. I.e., certain dilemma attributed to Anacharsis from which it would not be possible any criteria to judge any art.

Anacharsis consider two possibilities: the person who judges about art is a profane or an artist. If he is a profane, he would not be able to do it because he would not know about it. In the other case (which is the similar one to de cited argument), the artist must be the judge at the same time that the judged (as artist himself) and that is not allowed, because then, how can he be a fair judge?

Per sophisma figurae dictionis

Major premise: That which is a subject, is substance

Minor premise: A thinking being can only be thought of as subject

Conclusion: Therefore, every thinking being is substance


Kant describes the fallacy of the syllogism in the context of transcendental philosophy, and therefore making reference to (so to speak) transcendental cognitive structure. This is no accidental, because for him, the transcendental illusion is not merely logical, and therefore he distinguishes it from the fallacy, which is logical. The difference, I think, is established in passages like this:

«For we have here to do with a natural and unavoidable illusion, which rests upon subjective principles and imposes these upon us as objective, while logical dialectic, in the detection of sophisms, has to do merely with an error in the logical consequence of the propositions, or with an artificially constructed illusion, in imitation of the natural error. There is, therefore, a natural and unavoidable dialectic of pure reason--not that in which the bungler, from want of the requisite knowledge, involves himself, nor that which the sophist devises for the purpose of misleading, but that which is an inseparable adjunct of human reason, and which, even after its illusions have been exposed, does not cease to deceive, and continually to lead reason into momentary errors, which it becomes necessary continually to remove.»
(Critique of Pure Reason)

It may be noticed here somehow Cartesian discarded hypothesis on the meditation about the supreme being. The major premise, Kant says, refers to a being that can be given in an intuition as an object, while in the minor it is only considered in relation to the thinking and the unity of consciousness, but not to intuition as it can be given as an object of it. Everything clear up to this point, but in a footnote the obscureness returns. There, what it is said is that in each of both premises, it is the term thinking what is taken in two different senses. That is, again: in the major, as directed at an object of a possible intuition, in the minor in reference to self-consciousness, without reference to any object.

But where does the conviction of Kant on his own hypothesis of inclination for error of reason takes its grounds? It should be noticed that his place is not precisely conditioned by his criticism, but rather the contrary. And on the other hand, considering the proposition in itself: in which way is more profitable, taking the subject in its broadest sense, raising it to the level of pure form of all thought in general, or taking it as a singular term, and related to certain thoughts that are inherent to it in a particular way?

Notice, for example, that the amphibology (which produces the fallacy) does not concern so much to the term thinking but rather to subject, which serves as a medium. The first case, in fact, could not take place according to the syllogism which we have set out here. Kant stated it this way: «That which cannot be cogitated otherwise than as subject, does not exist otherwise than as subject, and is therefore substance». But the ambiguity is in the word subject: first subject as substrate of attributes, then as medium of all thoughts. Maybe because a question of time, i.e. the diachrony inherent to language, today we see the thing more simply, and this leads us to see the seeds of illusory appearance not transcendental, but merely logical. But this does not mean that Kant's intuition on this point has been entirely wrong and that some of its validity is preserved over time, although in other shapes.

Kant says (in the second edition of the Critique in question, which reformulates what concerns the paralogisms of pure reason):

«the proposition 'I think' (in the problematical sense) contains the form of every judgement in general and is the constant accompaniment of all the categories»²

Since it is "taken problematically" this I think might be thought as an idea in itself that could not be known speculatively. This means that it is not, for example, an immediate certainty obtained from an experience (that of consciousness in relation to it self in it's speculation), from which infer it existence through any syllogism. Kant conceived the Ego, on contrary, as:

«nothing but the simple and in itself perfectly contentless representation "I" which cannot even be called a conception, but merely a consciousness which accompanies all conceptions. By this "I," or "He," or "It," who or which thinks, nothing more is represented than a transcendental subject of thought = x, which is cognized only by means of the thoughts that are its predicates, and of which, apart from these, we cannot form the least conception.»³.


This topic of the rational doctrine of the subject or thing that thinks -as a problematic concept- could not then contain any substantiality (immateriality) or simplicity (incorruptibility) or identity (personality).

_______________
(1) Kant, Critique of Pure Reason.
(2) Ibíd.
(3) Ibíd.


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Denoting phrases and definite descriptions

In On denoting, Russell explains what he called denotative phrases and gives examples of them: «a man, some man, any man, every man, all men, the present King of England, the present King of France, the center of mass of the solar system at the first instant of the twentieth century, the revolution of the earth round the sun, the revolution of the sun round the earth».

And groups them into three cases:

Without reference: the present King of France
With a particular reference: the center of mass of the solar system for ...
With ambiguous reference: a man

Such kind of prases can be find within sentences. As he considers 'all', 'nothing' and 'something' the primitive denotative phrases, he proceeds to analyze them schematically. According to Russell, the notion 'C(x) is always true' (corresponding to 'all', ie 'all x is C') is fundamental and indefinable, other notions are defined through it.


C(everything) = "C (x) is always true"
C(nothing) = "«C(x) is false» is always true"
C(something) = "It is false that «C(x) is false» is always true"

He also defines the notion "C(a man)" using the example:

"I met a man", where C(x) is the property "I met x", and "C(a man)" would be:
"«I met x, and x is human» is not always false."

We also define the following expressions (according to that indicated above):

C(all men) = "«if x is human, then C(x) is true» is always true"
C(no men) = "«if x is human, then C(x) is false» is always true"
C(some men) = C(a man) = "It is false that «C(x) and x is human» is always false"

C(every man) = C(all men)

Then there are the cases with a definite article such as "the". An example might be: "The father of Charles II was executed," which can be interpreted as "C(the father of Charles II)" (where: C(x) = "x was executed"). So we obtain:


"It is not always false of x that x begat Charles II and that x was executed and that «if y begat Charles II, y is identical with x» is always true of y".

To simplify all this, that may seem confusing, here is its formal representation.

C(all) = ∀x Cx
C(nothing) = ∀x ¬Cx
C(something) = ¬∀x ¬Cx

C(all men) = ∀x (Hx ⊃ Cx)
C(no men) = ∀x (Hx ⊃ ¬Cx)
C(a man) = ¬∀x ¬(Hx ∧ Cx)

C(the father of Charles II) = ¬∀x ¬[Px ∧ Cx ∧ ∀y (Py ⊃ y = x)]


The foregoing allows a reduction of all propositions in which denotative phrases occur to some where they not. That is, instead of expressions that denote objects in the world as subjects of sentences, we will have sentences that will denote either the truth or falsehood.

According to Strawson, this view was still widely accepted among logicians when he wrote a critical review in On Referring.

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