In logic, a categorical proposition, or categorical statement, is a proposition that asserts or denies that all or some of the members of one category (the subject term) are included in another (the predicate term).
[1] The study of arguments using categorical statements (i.e., syllogisms) forms an important branch of deductive reasoning that began with the Ancient Greeks.
The Ancient Greeks such as Aristotle identified four primary distinct types of categorical proposition and gave them standard forms (now often called A, E, I, and O).
If, abstractly, the subject category is named S and the predicate category is named P, the four standard forms are: A large number of sentences may be translated into one of these canonical forms while retaining all or most of the original meaning of the sentence.
Greek investigations resulted in the so-called square of opposition, which codifies the logical relations among the different forms; for example, that an A-statement is contradictory to an O-statement; that is to say, for example, if one believes "All apples are red fruits," one cannot simultaneously believe that "Some apples are not red fruits."
Modern understanding of categorical propositions (originating with the mid-19th century work of George Boole) requires one to consider if the subject category may be empty.
The existential viewpoint is a stronger stance than the hypothetical and, when it is appropriate to take, it allows one to deduce more results than otherwise could be made.
The hypothetical viewpoint, being the weaker view, has the effect of removing some of the relations present in the traditional square of opposition.
Arguments consisting of three categorical propositions — two as premises and one as conclusion — are known as categorical syllogisms and were of paramount importance from the times of ancient Greek logicians through the Middle Ages.
Although formal arguments using categorical syllogisms have largely given way to the increased expressive power of modern logic systems like the first-order predicate calculus, they still retain practical value in addition to their historic and pedagogical significance.
[2] Quantity refers to the number of members of the subject class (A class is a collection or group of things designated by a term that is either subject or predicate in a categorical proposition.
Quality It is described as whether the proposition affirms or denies the inclusion of a subject within the class of the predicate.
[4] For instance, an A-proposition ("All S is P") is affirmative since it states that the subject is contained within the predicate.
The two terms (subject and predicate) in a categorical proposition may each be classified as distributed or undistributed.
Although not developed here, Venn diagrams are sometimes helpful when trying to understand the distribution of terms for the four forms.
The empty set is a particular case of subject and predicate class distribution.
Note the ambiguity in the statement: It could either mean that "Some Americans (or other) are conservatives" (de dicto), or it could mean that "Some Americans (in particular, Albert and Bob) are conservatives" (de re).
But if, as an example, this group of "some politicians" were defined to contain a single person, Albert, the relationship becomes clearer; This is the de re interpretation of the intensional statement (
This is a definition that applies to every member of the "corrupt people" group, and is, therefore, distributed.
[5] Peter Geach and others have criticized the use of distribution to determine the validity of an argument.
There are several operations (e.g., conversion, obversion, and contraposition) that can be performed on a categorical statement to change it into another.
The simplest operation is conversion where the subject and predicate terms are interchanged.
Obversion changes the quality (that is the affirmativity or negativity) of the statement and the predicate term.
, is interpreted as a predicate term 'non-P' in each categorical statement in obversion.
Contraposition is the process of simultaneous interchange and negation of the subject and predicate of a categorical statement.
It is also equivalent to converting (applying conversion) the obvert (the outcome of obversion) of a categorical statement.