The Four Types of Categorical Propositions - My Notes on 5.2 in "The Reasonable Person" by Mark Grannis

Formal Logic With Propositions

Grannis begins by talking about the difference between matter and form in logic. The matter refers to the nature of the subject and predicate as to what they are. The form refers to how they are logically being related to one another in the structure of this formulation. Here we are going to be looking at the formal aspect of logic, i.e. the structure, when talking about propositions. There are four fundamental categorical propositions (four structures): two dealing with quality, and two with quantity. 

Quality of Propositions - Affirmative or Negative

"The quality of the proposition is affirmative or negative, depending on whether the proposition affirms or denies the predicate of the subject." The first two types of categorial propositions are determined by the affirming or negating quality of the copula (the verb to be). "Copula: the verb of being that relates the predicate to the subject, either affirmatively or negatively." For example, S is P or S is not P. 

Now this simple form is not always present in the way we see expressions in the real world, but we can reorganize any of these sentences to fit this simple form. This has to be done by turning complex parts of sentences into unified subjects or predicates. In English we can do this by adding "that which" or "the type of thing".

Quantity of Propositions - Universal, (Singular), and Particular

"The quantity of the proposition in universal, particular, or singular, depending on whether the proposition applies to all inferiors of the subject class, some inferiors of the subject class, or to a singular subject." The quantity of a proposition deals with the amount of something being predicated. There can be propositions which claim to include all cases (Universal), some which include a certain set of cases (Particular), and some which only make a claim about one subject (Singular). 

The Quantifier

The quantity of a proposition is determined by the "quantifier" word, like "all" or "some" or "no". 

Universal and Singular Are Treated the Same

In regards to form, universal and singular propositions can be treated the same because just as a universal proposition includes all of the group mentioned, so too the individual exhausts its own group as there is only one of them. 

Four Categorical Propositions

And so we have ended up with four types of categorical propositions. (1) A universal affirmative, (2) a universal negation, (3) a particular affirmative, and (4) a particular negation. 

Standard Logical Form

Q{S}c{P}

Q = Quantifier (All, Some, No)

{S} = Subject 

c = copula (Verb To Be)

{P} = Predicate 

Affirmative Propositions 

"Why do we call them 'A' propositions and 'I' propositions? Because A and I are the first two vowels in the Latin word affirmo, which in English means 'I affirm'."

The A proposition = "All S is P"

The I proposition = "Some S is P"

The Negative Propositions

"And it will probably not surprise you to learn that the letters O and E come from Latin, just like the letters A and I. O and E are vowels in Latin nego, which in English means 'I deny.'" 

The E proposition = "No S is P"

The O proposition = "Some S is not P"