## Tuesday, September 30, 2008

### Subalternation

It is common to divide categorical propositions into four groups:

A: Universal Affirmative
E: Universal Negative
I: Particular Affirmative
O: Particular Negative

But this is arguably not precise enough. There are actually two different ways each description -- Universal Affirmative, etc. -- could be understood, a weak way and a strong way. Using what I've called Welton diagrams:

Universal Affirmative, Weak Interpretation
|  |  |X|  |

Universal Affirmative, Strong Interpretation
|O|  |X|  |

Universal Negative, Weak Interpretation
|X|  |  |  |

Universal Negative, Strong Interpretation
|X|  |O|  |

Particular Affirmative, Weak Interpretation
|O|  |  |  |

Particular Affirmative, Strong Interpretation
|O|X|X|X|

Particular Negative, Weak Interpretation
|  |  |O|  |

Particular Negative, Strong Interpretation
|X|X|O|X|

You can see these differences operating in different 'squares' of opposition.* For instance, if you look at a standard textbook discussion of the 'modern' and the 'traditional' square of opposition, you'll find that they both take I and O under the weak interpretation, but whereas the 'traditional' takes A and E under the strong interpretation, the 'modern' takes A and E under the weak interpretation. These are far from exhausting the options. In Lewis Carroll's system, which arguably fits natural language better than either, A is taken under the strong interpretation while E, I, and O are all take under the weak interpretation. If you handle propositional logic syllogistically
you end up taking A and E as weak but I and O as strong.

This is very visible in the case of subalternation. Subalternation can occur only when flowing from a stronger to a weaker claim. Thus you can potentially have it in the following directions:

A-Strong to I-Weak
E-Strong to O-Weak
I-Strong to A-Strong
I-Strong to A-Weak
O-Strong to E-Strong
O-Strong to E-Weak

In the 'modern' square of opposition, universals are treated as hypotheticals; they are weak, so no subalternation is possible. In the 'traditional' square, universals are treated as categoricals; they are strong, so subalternation is possible. In Carroll's system, A is treated as a double proposition (an E + an I) so it is strong, and allows the withdrawal, so to speak, of the I proposition; E propositions, on the other hand, simply exclude a single term from another, and therefore are much weaker than the corresponding O proposition. In syllogistic propositional logic. A is treated as a hypothetical again, and E as a disjunction; both of these are low-information, and therefore weak. But I (and with it O) is conjunction, which is high-information, and therefore strong. So subalternation flows in the opposite direction we expect.

For my part, I'm currently inclined to think that we should regard the weak universals and the strong particulars as not really categorical propositions at all; rather, they are non-categorical propositions that bear certain very important analogies to the standard categoricals. But this is an issue I'm still thinking out.

_______
* I put 'square' in quotes here because a great many discussions of the square of opposition would be improved if it were recognized that the 'square' is really just an abbreviation for a cube, one with the following vertices:

A (False)
E (False)
I (False)
O (False)
A (True)
E (True)
I (True)
O (True)

All operations in the square are determined entirely by combinations of relations of consistency and inconsistency among these vertices. For instance, subalternation, in which I is subalternate to A, follows when the following inconsistencies hold

A (True) and A (False)
I (True) and I (False)
A (True) and I (False)

given that the following is consistent:

A (False) and I (True).

Meddle with these relations of consistency and inconsistency in the cube and you change the operations of the 'square'.