The categorical proposition is a basic concept in Aristotelian or traditional logic (also sometimes called syllogistic or categorical logic). Aristotelian logic, albeit with substantial revisions over the course of almost 2,000 years, was accepted as the definitive logical system until developments in the late nineteenth century with Gottlob Frege and Bertrand Russell supplanted it and ushered in modern mathematical logic.
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Logic is the study of correct, or valid inferences. Aristotle’s logical system is based upon a form of argument called a syllogism. The syllogism is an argument with two premises and a conclusion that follows from these premises. Each of the three propositions (i.e. two premises and a conclusion) in a syllogism is a categorical proposition. A categorical proposition is a type of proposition that uses the logical expressions "all," "some," "is," and "is not," to link "terms," which refer to some set, class or kind. This reference to categories, sets, or classes, is why they are called categorical propositions. An example of a categorical proposition is “All whales are mammals.” Aristotelian logic regards four basic types of categorical propositions as lying at the heart of all correct reasoning. These are the universal affirmative proposition—“All S is p,” the universal negative proposition—“No S is p,” the particular affirmative proposition—“Some S is p,” and the particular negative proposition—“Some S is not p.”
Understanding the concept of a categorical proposition requires some discussion of the notion of a proposition. A proposition is usually defined as a thought or content expressed by a sentence, when it is used to say something true or false. Propositions are, roughly, thoughts about how things are, and are appraisable as true or false depending on whether the thought corresponds to how the world it. Consider the following example:
This sentence expresses a proposition for it makes a specific claim about the world, which may or may not be true.
One reason for distinguishing sentences from propositions is that not all sentences are appraisable as true or false. Commands (e.g. ‘Shut the door’) and questions (“Is the door open?”) are perfectly legitimate ‘’sentences’’ and do not say anything about how things stand in the world. For this reason commands and questions are sentences but not propositions. Another reason for distinguishing sentences from propositions is that the very same thought, i.e., proposition, may be expressed in many different ways and in a variety of different languages.
Propositions are the sorts of things that can be (e.g.) thought, believed, asserted, doubted, mentioned, and imagined. For example, one could think to oneself, “The black dog bit the white rabbit”; or one could ‘’assert’’ it; or one could doubt it by saying “I doubt that the black dog bit the white rabbit.” Or taking another example, the sentence “It seems quite unlikely that a man will land on Mars by 2009” expresses the proposition "a man will land on Mars by 2009" although the speaker does not commit to the truth of that proposition. Rather, the speaker doubts whether the state of affairs represented by the propositions will ever obtain.
Propositions are sometimes identified with statements or judgments, but it seems best to keep these separate. Assuming that the expressions "statement" and "judgment" are interchangeable, we can say that people make statements when they assert propositions. Making a statement is essentially adopting a certain attitude toward a proposition. A statement consists of (1) a thought or meaning called a proposition and (2) the speaker or writer’s endorsement of the proposition (the assertion). So all judgments assert propositions but not all propositions are asserted (e.g. A proposition which is doubted is not asserted).
We have now considered the notion of a proposition in general. A categorical proposition is a proposition of a special sort. It is a proposition with two [1] "terms," one of two [2] "copulas," and one of two [3] "quantifiers." Explanation of each of these is as follows.
Categorical propositions contain two "terms." Terms are the constituents of propositions, and not whole propositions themselves. A term picks out a set or class of objects, either real or imagined. Examples of terms include chickens, people, Martians, dogs, and carnivores.
The term of a categorical proposition picks out a group of things. This group of things is called a set, or a class, or a category. The (groups of) objects that the term picks out do not have to really exist in our world. So the term "Martians" is perfectly legitimate even though Martians don’t actually exist.
A categorical proposition is made up of two terms. The first term, which occurs in the subject position, is called the minor term. The second term, which is occurs in the predicate position is called the major term.
Categorical propositions admit only one verb, and this the verb "to be." The verb "to be" is called a copula. For example, the sentence "The dog is black" employs the copula. In a categorical propositions, the copula links the subject term with the predicate term. In other words, it links up two terms, which each pick out categories of objects, with one another. The term, "whales," may (e.g.) be linked with another term, "mammals," in the proposition, “all whales are mammals.”
In Aristotelian logic, the negation of the verb "to be" came to known as the "negative copula." So when one says, “the dog is not black,” one employs the negative copula. Ultimately, it makes no difference whether we say that there are two copulas, one positive and one negative, or only one copula, which is negated or not negated. One should adhere to the convention, which says that there are two copulas, one positive and one negative.
Categorical propositions are said to have a "quality" and a "quantity" (the notion of quantity will be discussed in a moment). The quality of the categorical proposition is determined by the copula. If the copula is negative then the proposition is said to be a negative proposition; if the copula is positive the proposition is said to be an affirmative proposition.
All Categorical propositions contain one (and only one) of two quantifiers. A quantifier, as the name suggests, specifies the number of a given class. There are only two quantifiers. The first quantifier is called the "universal quantifier," usually represented by "all" or "every." The universal quantifier picks out every member of a particular class, such as "all men," or ‘all whales’. The second quantifier is the existential quantifier, usually represented by ‘some’, or ‘at least one’. The existential quantifier picks at least one member of the class, such as ‘some men’ or some ‘whales’. Every categorical proposition is said to have a quantity. The quantity of the categorical proposition is either universal (all, every) or particular (some).
Now that the meaning of the components of a categorical proposition has been considered, it is time to see how they operate together. Here are some examples of categorical propositions:
These examples illustrate the basic form of the categorical proposition. Each involves a quantifier, two terms (i.e. the "subject" and the "predicate"), which are linked by a copula.
Two quantifiers (“all,” “some””) and two copulas ("is," "is not") can be combined in only four ways. In other words, there are only four basic forms of categorical proposition. The quantity of the categorical proposition is either universal ("all," "every") or particular ("some"). The quality of the categorical proposition is either affirmative ("is," "are") or negative ("is not," "are not").
Two of the basic forms are universal propositions (i.e. they say something about a whole domain). Universal propositions use the universal quantifier. One of the universal forms is an affirmative statement (i.e. it affirms something about the subject; and one of the universal forms is a negative statement; i.e. it denies something about the subject). The other two categorical forms are particular propositions, (i.e. they say something about particular individuals in a domain). Particular propositions use the existential quantifier. Particulars also come in affirmative and negative form or quality.
In the Middle Ages each of these four basic forms of categorical proposition came to be called by first four vowels "A," "E," "I," and "O." This practice is continued in many logic books even today, so that you will often see (i.e.) universal affirmative propositions called "A" propositions, and so on. The following table will clarify this further.
A. All A are B [Universal affirmative proposition]
A. All {term} are {term}.
E. No A are B [Universal negative proposition]
E. No {term} are {term}.
I. Some A are B [Particular affirmative proposition]
I. Some {term} are {term}.
O. Some A are not B [Particular negative proposition]
O. Some {term} are not {term}.
There are logical relations between the categorical propositions such that when these propositions are combined, categorical syllogisms are generated. The categorical syllogism is an argument with two premises and a conclusion that follows from these premises. Each of the three propositions (i.e. two premises and a conclusion) in a syllogism is a categorical proposition. The following is an example of a categorical syllogism: (1) all chickens are birds; (2) all birds are feathered creatures, therefore, (3) all chickens are feathered creatures, is a categorical syllogism. (See categorical logic for more on the concept of a syllogism.)
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