Modern Square of Opposition

From Logic


The Modern Square of Opposition and the Existential fallacy

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The Existential Fallacy

This fallacy is committed in any inference based on the relations of the Traditional Square of Opposition that involves propositions that we cannot safely presuppose exist. Thus, inferring from All Unicorns are lucky that some unicorns are lucky commits the existential fallacy. Contradictions are excluded - saying All people from Atlantis have gills, therefore, no people from Atlantis lack gills is still considered valid.

The Problem

Medieval philosophers made a shocking discovery concerning The Traditional Square of Opposition: it breaks down into nonsense when dealing with categorical statements dealing with propositions with empty denotations... i.e., things that do not exist.

According Irving Copi, a proposition is said to have existential import if it typically is uttered to assert the existence of some class of objects. For example, the proposition 'There are books on my desk implies the existence of books. On the other hand, the statement: There are no unicorns does not have any existential implications. This apparently simple observation obliges us to resolve a deep logical problem before we can go on to symbolize and diagram categorical propositions.

The difficulty can be appreciated by reflecting on propositions with a particular quantity: I and O propositions. (See Categorical Propositions). These propositions have existential import. The particular, affirmative I proposition some soldiers are heroes says that there is at least one soldier who is a hero. So Particular Categorical Propositions assert that at least one member of their designated class actually exists, and if you recall from the previous section, we may limit a universal statement, a Universal A or O statement, through subalternation. This would seem to imply that universal statements also have existential import, as particular propositions I and O follow logically from their corresponding universal propositions through subalternation. A and E, propositions must also have existential import, since existential import could not be derived validly from a proposition without existential import.

This creates a serious problem! For example, we know from the Traditional Square of Opposition that Universal A and O propositions are contradictories: All Danes speak English is contradicted by Some Danes do not speak English. Contradictories cannot both be true, since one of the pair must be false, nor can they both be false, since one of the pair must be true. But if corresponding A and O propositions have existential import, then both contradictories could be false! To illustrate: The A proposition All inhabitants of Mars are blond and its corresponding O proposition Some inhabitants of Mars are not blond are contradictories: Now, if they have existential import, then both of these propositions are false if Mars has no inhabitants. But if they can both be false, then they cannot be contradictories. We have the same problem with subcontraries and subalterns.

The Solutions

There are several solutions. The most desparate is turning to the notion of a presupposition. To rescue the Traditional Square of Opposition, we might simply assume that all propostions: A, E, I and O presuppose that the class that they make reference to is not empty. In this way, A and E remain contraries, I and O will remain subcontraries; subalterns will validly follow from their superalterns, and Aand O as well as E and I, will remain contradictories. To hold to this, however, we must insist that all the classes we make reference to are not empty.

Why not just do that?

The Failures of Presuppositionalism

The existential presupposition is both a necessary and sufficient means to rescue the Traditional Square, but it comes at a very high cost. While the presupposition rescues the traditional relations among A, E, I and O propositions, it does so only at the cost of reducing their power to formulate certain assertions. For example, If we presupose that a designated class has members, we will never be able the generate the proposition that denies that it has members! For this reason, we can see that holding to groundless propositions delimits knowledge acquisition. To make any presupposition we must first carefully consider whether we are justified in making the presuppositon, and the existential presupposition as an ad hoc correction, is clearly not a well grounded position.

The other solution is this: we must give up many of the immediate arguments promised to us by the Traditional Square of Opposition. The modern treatment of categorical propositions is called Boolean - after it's inventor, George Boole, one of the founders of modern symbollic logic.

The Modern Boolean Solution

I and O propositions continue to have existential import in the Boolean interpretation. So the I proposition Some S is P is false if the class S is empty, and the O proposition Some S is not P is likewise false if the class S is empty.

The universal propositions A and E remain as contradictories of the particular propositions O and I. For example, the A proposition All S are P contradicts the O proposition Some S are not P.

Universals no longer have existential import: All of this is coherent because in the Boolean interpretation, universal propositions no longer are interpreted as having any existential import. Even when the S class is empty, the statement A statement All S are P can be true.

The Major Shift

Corresponding A and E propositions are no longer interpreted as contraries. Both an A and a corresponding E proposition can be true at the same time. Does this sound paradoxical? Here is the solution: The modern square interprets universal statements (A and E propositions) as conditional statements. According to this interpretation, All S are P becomes If there are any S, then they are P. It follows that in the Boolean interpretation that subalternation, inferring an I from an A proposition, is no longer valid. It also follows that since I and O problems do have existential value, that they can be both be false if the subject class is empty, so I and O propositions are no longer subcontraries either

So poof goes contraries, subcontaries and subalternation. The only relation on the traditional square of opposition that is preserved in the modern interpretation is the relation of contradiction. Thus, the modern square of opposition permits fewer inferences than Aristotle's square. We may use the old square when feel comfortable presupposing that the categorical propositions in question do not have empty denotation, but we now risk running into the existential fallacy.

As for immediate inferences: the Boolean interpretation preserves some of these relationships. Conversion for E and I propositions is still valid. Contraposition for A and E propositions is still valid. And obversion for any proposition remains valid. But conversion by limitation and contraposition by limitation is generally no longer considered to be valid.

The Modern Square of Opposition


The Boolean interpretation transform the Traditional square into the Modern Square, in the following way: relations along the sides of the square are undone, but the diagonal, contradictory relationship, is preserved and remains in force.

Those taking the Course in Logic 101 will now want to proceed to the next section: Classical Logic


  • Copi, I. M, Cohen, C., (2001), "Introduction to Logic", 11th Edition.
  • Hurely, P. J. (2000) A Concise Introduction to Logic - 7th Edition
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