For propositions that are not always true and known to be true, they may be differentiated either byqualityorquantity. The convention in logic is to express propositions abstractly in terms ofS(subject) andP(predicate). Thus, propositions may differ either in

1. Quality – as the mind either combines or separates:

• a. Combinations/affirmations – S is P
• b. Separations/denials– S is not P

These constitute the two qualities of propositions:affirmativeandnegative.

Propositions may also differ according to how many of a subject the predicate is said. Thus, propositions may differ in

2. Quantity– as the subject applies to all, some (not all), or one of a given type.

• a. Universal – what Aristotle calls “Universal taken universally”:
  
  All are or None are (All are not) – E.g., All men are good. No men are good.
• b. Particular – what Aristotle calls “Universal not taken universally”:
  
  Some (not all) are or some are not – E.g., Some men are good. Some men are not good.
• c. Individual – Individual; Singular:
  
  One is or One is not – E.g., Socrates is good. Socrates is not good

Thus, there are four types of universal or general propositions which have the following vowels traditionally assigned to them:

A – Universal Affirmative: All S is P – Every rose is red, All roses are red.

E – Universal Negative: No S is P – No rose is red, All roses are not red.

I – Particular Affirmative: Some S is P – Some roses are red.
(Avoid “Not all roses are red.” as it is easily confused with O.)

O – Particular Negative: Some S is not P – Some roses are not red. Not every rose is red.

                   3. Square of Opposition

These, then, are the four kinds of proposition and the relations between them:

Aristotle discovered that, based on the relations between propositions, the truth or falsity of certain propositions entailed the truth or falsity of others.

• Contradictorystatements are completely opposed in truth. If one is true, the other must be false. If one is false, the other must be true. For example, if “All swans are white” is true, “Some swans are not white” must be false; if “No men are good” is false, then “Some men are good” must be true. 
• Contrariescannot both be true, but both could be false: If “All men are animals” is true, then “No men are animals” must be false; but if “All men are white” is false, it need not be true (in fact it is not true) that “No men are white.” In fact, some men are white, and some are not.
• Subcontrariescannot both be false. If “Some philosophers are rich” is false, then “Some philosophers are not rich” must be true. In fact, the contradictory must be true, “No philosopher is rich.” Likewise, the contrary of this E proposition, “All philosophers are rich” must be false.

There are then the following relationships concerningsubalternpropositions according to the principle “Truth descends; falsity ascends.” That is, true superaltern propositions (A and E) imply true subalterns (I and O, respectively), but false subaltern propositions imply false superalterns. If “All men are rational” is true, then “Some men are rational” must be true,” and if “No man is an island” is true, then “Some man is not an island” must also be true. But if “Some lawyers are truthful” is false, then “All lawyers are truthful” must be false, and if “Some dog is not loyal” is false, then “No dog is loyal” is false. (In fact, the contradictory claim, “All dogs are loyal” and the subcontrary claim “Some dog is loyal” must both be true.)

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Updated January 18, 2025