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?