Conventions, Symbolism, and Relations among Categorical Propositions in Experimental Research

The four forms of categorical propositions (1) through (4) men­tioned above are named, and often referred to, respectively, as A, E, I, and O propositions.

The (verb) terms of affirmation (e.g., “is,” “are,” and “were”) and negation (e.g., “is not,” “are not,” and “was not”) are referred to as indicating the quality of affirmation or negation of a given proposition.

The (adjective) terms (“all,” “some,” and “no”) are referred to as indicating the quantity in a given proposition; “all” and “no” are universal (meaning complete) in quantity, and “some” is par­ticular in quantity.

The cupola refers to the term between and connecting the subject term and predicate term in a given proposition, such as “is” in “All S is P,” or any other word or words (together), usually a verb variation.

Following the conventions relative to terms, the format com­mon to each of the four propositions, relative to the sequence of terms, can be generalized as follows:

Quantifier, subject term, cupola, predicate term

The term distribution, in relation to propositions, is used in a very special sense, far removed from its literal one. A given prop­osition is said to distribute the subject term, the predicate term, or both if the reference made by the proposition applies to all members of either class. For instance, in the E proposition “No S are P” reference is made to all members of the class S, with no exception; hence, this proposition is said to distribute its subject term. And, reference is made to the class P, again as a whole, meaning none of the members of P is included in the class S; hence, the proposition is said to distribute its predicate term as well. In contrast, if any exception is inferred in the class of either S or P, then that term is said to be undistributed.

Based on such analysis, Table 13.1 summarizes the four forms of categorical propositions.

1. Opposition

Categorical propositions having the same subject and predicate terms, but differing in quality, quantity (defined earlier), or both, present different truth relations; the differences are collectively given the traditional name oppositions, some variations of which follow.

Two propositions are said to be contradictories if one is the denial or negation of the other, meaning that they both cannot be true and both cannot be false.

Example 1:

All wrestlers are athletes.

Some wrestlers are athletes.

Example 2:

No wrestlers are runners. Some wrestlers are runners.

Example 1 has A and O propositions, and Example 2 has E and I propositions. Whereas the propositions of Example 1 differ both in quality and quantity, those of Example 2 differ only in quantity. Both exemplify contradictories.

Two propositions are said to be contraries if the truth of one entails the falsity of the other, although both can be false (together).

Example:

All wrestlers are giants. No wrestlers are giants.

Two propositions are said to be subcontraries if they cannot both be false, although both can be true.

Example:

Some tennis champions are tall (people). Some tennis champions are not tall (people).

The opposition between a universal proposition and the cor­responding particular proposition is named as subalternation, between which the universal proposition is known as superaltern and the particular one as the subaltern.

Example 1:

All living things are animals.

Some living things are animals.

Example 2:

All living things are not animals.

Some living things are not animals.

These various kinds of opposition, traditionally diagrammed in a so-called Square of Opposition (named after George Boole), are presented in Figure 13.1.

  • A being given as true: E is false, I is true, O is false.
  • E being given as true: A is false, I is false, O is true.
  • I being given as true: E is false, while A and O are undetermined.
  • O being given as true: A is false, while E and I are undetermined.
  • A being given as false: O is true, while E and I are undeter­mined.
  • E being given as false: I is true, while A and O are undeter­mined.
  • I being given as false: A is true, E is false, O is true.
  • O being given as false: A is true, E is false, I is true.

2. More Sources of Immediate Inferences

An application limited to E and I propositions is shown in Table 13.2. By simply interchanging the places of subject and predicate terms, the truth-value of the converted proposition is retained. This application is called conversion, and each proposition is then referred to as the converse of the other.

Obversion

Applicable to all the four standard forms of propositions, the pro­cess of obversion yields a proposition of equivalent truth-value. The process is more involved than conversion. Firstly, we need to note the meaning of complement as used here. Every category, that is, every class, has a complementary class, or simply a comple­ment, which is the collection (or class) of all things that do not belong to the original class.

Example: The complement of (the class of) “animal” is “nonanimal,” and the complement of (the class of) “nonanimal” is “non-non-animal,” which is (the class of) “animal.” Obverting a given proposition is done by

  1. Keeping the subject term unchanged
  2. Keeping the quantity of the proposition unchanged
  3. Changing the quality of the proposition
  4. Replacing the predicate term by its complement

Examples: A few typical propositions to be obverted, known as the obvertend, and the corresponding propositions, after doing the obversion, known as obverse are shown in Table 13.3.

Source: Srinagesh K (2005), The Principles of Experimental Research, Butterworth-Heinemann; 1st edition.

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