1. When I took my elementary logic course many years ago, it was primarily concern with Propositional Logic and Predicate Logic.
The course briefly touched on Traditional Logic (Aristotle Logic or Syllogistic).
I found the Traditional Square of Opposition, being both logical and geometrical, very interesting; but I did not have the time to understand it properly.
I find the idea of subalternation puzzling - the textbook I used was too brief on this idea and this idea was not prominent in propositional logic.
Recently, I have the opportunity to read up on the Traditional Square of Opposition and the following are some reading notes on subalternation.
2. A categorical proposition affirms or denies relationship between two terms: subject (S) and predicate (P).
The two terms are treated as class for objects.
The Square of Opposition embodies the logical relations between four types of categorical propositions:
(a) Symbol: A
Name: Universal Affirmative
Logical form: Every S is P.
(b) Symbol: E
Name: Universal Negative
Logical form: No S is P.
(c) Symbol: I
Name: Particular Affirmative
Logical form: Some S is P.
(d) Symbol: O
Name: Particular Negative
Logical form: Some S is not P.
3. Examples of the four categorical propositions:
Let the subject 'S' be the class of human beings.
(a) A: Universal Affirmative (Every S is P)
Every human being is mortal.
(b) E: Universal Negative (No S is P)
No human being is mortal.
(c) I: Particular Affirmative (Some S is P)
Some human beings are mortal.
(d) O: Particular Negative (Some S is not P)
Some human beings are not mortal.
4. The Traditional Square of Opposition (Terence Parsons 2012):
5. The Traditional Square of Opposition embodies six theses:
* A, E, I and O are used as the name of logical forms and double duty as name of propositions of that logical forms.
(a) Corresponding A and O propositions are contradictories of each other.
(b) Corresponding E and I propositions are also contradictories of each other.
(c) Corresponding A and E propositions are contraries of each other.
(d) Corresponding I and O propositions are subcontraries of each other.
(e) I is a subalternation of the corresponding A proposition.
(f) O is a subalternation of the corresponding E proposition.
6. Definitions:
(a) Two propositions are contradictory iff they cannot both be true and they cannot both be false.
(b) Two propositions are contraries iff they cannot both be true but they can both be false.
(c) Two propositions are subcontraries iff they cannot both be false but they can both be true.
7. Subalternation
Subalternation is a relation between two corresponding categorical propositions: one universal and one particular.
The two corresponding propositions must be of the same quality: either affirmative or negative.
The universal categorical proposition is called the superaltern.
The particular categorical propositional is called the subaltern.
Subalternation is the immediate inference from the superaltern to the subaltern.
The inference is one-directional.
There are only two subalternation inferences:
(a) A --> I
(b) E --> O
A and E are the superalterns.
I and O are the subalterns.
Examples:
(a) A --> I
(Every human being is mortal) --> (Some human beings are mortal)
(b) E --> O
(No human being is mortal) --> (Some human beings are not mortal)
8. Contraposition of subalternation
Contraposition is bi-directional inferences of the form:
(p --> q) <--> (not q --> not p)
Applying contraposition to the two subalternations, we have:
(a) not I --> not A
(b) not O --> not E
So there are four logical relationships between a superaltern and its corresponding subaltern.
The concepts of contradiction, contrary and subcontrary are general and apply to all propositions.
But the concept of subalternation seems to be specific to Traditional Logic.
References:
Parsons, Terence. 2012. The Traditional Square of Opposition. In The Stanford Encyclopedia of Philosophy (Fall 2012 Edition), ed. Edward N. Zalta.
http://plato.stanford.edu/archives/fall2012/entries/square/
(accessed October 9, 2012).
"Square of opposition", Wikipedia - The Free Encyclopedia,
http://en.wikipedia.org/wiki/Square_of_opposition
(accessed October 9, 2012).
"Subalternation", Wikipedia - The Free Encyclopedia,
http://en.wikipedia.org/wiki/Subalternation
(accessed October 9, 2012).
End.
