10. SUMMARY.
(1) A logical proposition is a judgment expressed in words.
(2) The three kinds of logical propositions are categorical, hypothetical, disjunctive.
A categorical proposition is one in which the assertion is made unconditionally.
A hypothetical proposition is one in which the assertion depends upon a condition.
A disjunctive proposition is one which asserts an alternative.
The most common word-signs of the categorical proposition are “all,” “no,” “some” and “some-not,” of the hypothetical, “if” and of the disjunctive, “either-or.”
(3) Every logical categorical proposition has the four elements: quantity sign, subject, copula and predicate.
The quantity sign indicates the extension of the proposition; the logical subject is that of which something is affirmed or denied; the logical predicate is the term which is affirmed or denied of the subject; the copula is always some form of “to be” and is used to connect subject and predicate. “Not” is sometimes used with the copula.
The statements of ordinary conversation are usually not expressed in terms of the four elements, but must be, before they can be used in testing arguments.
(4) One word usually constitutes the grammatical subject while a word with all its modifiers goes to make up the logical subject. The verb with any predicate word is the grammatical predicate. The logical predicate is all which follows the copula—it may include the predicate-word and all its modifiers as well as the modified object.
(5) Categorical propositions are divided into four kinds; universal affirmative (A), universal negative (E), particular affirmative (I), particular negative (O). For the sake of brevity these four are respectively denoted by the vowels A, E, I, O.
An A proposition is one in which the predicate affirms something of all of the logical subject.
An E proposition is one in which the predicate denies something of all of the logical subject.
An I proposition is one in which the predicate affirms something of a part of the logical subject.
An O proposition is one in which the predicate denies something of a part of the logical subject.
Every proposition must be reduced to one of the four types before it can be used as a basis of argumentation.
It is incumbent on the student to recognize these four types with precision and accuracy.
(6) There are a few proposition types which are recognized as being illogical in form. These may be defined as follows:
(1) An indefinite proposition is one without the quantity sign. It usually may be classed as universal.
(2) An elliptical proposition is one in which the copula is suppressed.
(3) An individual proposition is one which has a singular subject. It is universal in content.
(4) Plurative propositions are those introduced by “most,” “a few” or some equivalent quantity sign. These are particular in meaning.
(5) Partitive propositions are particulars which imply a complementary opposite. These arise through the ambiguous use of “all-not,” “some” and “few.”
“All-not” sometimes means “no,” while at other times it may mean “not-all.” If the quantity sign means the latter, then it introduces a partitive proposition.
“Some” may mean “some only,” or “some at least.” The latter is the logical meaning. The former interpretation makes the proposition partitive. When “few” means “few only,” it is partitive in nature.
(6) Exceptive propositions are those introduced by such signs as “all except,” “all but,” “all save,” etc. They are universal only when the exceptions are mentioned.
(7) Exclusive propositions are those introduced by such words as “only,” “alone,” “none but” and “except.” In an exclusive the predicate and not the subject is distributed. Consequently the easiest way to make an exclusive logical is to interchange subject and predicate and call it an A.
(8) An inverted proposition is one where the predicate precedes the subject. Interchanging them gives the logical form.
Of the grammatical sentences only the declarative is logical.
The relative clause, though out of place, must be used with the word it modifies.
(7) There are other propositions, though not illogical, to which the logician usually gives some attention. These may be defined as follows:
(1) An analytical proposition is one in which the predicate gives information already implied in the subject.
(2) A synthetic proposition is one in which the predicate gives information not implied in the subject.
(3) A modal proposition is one which states the manner in which the predicate belongs to the subject. The adverbs of time, place, degree and manner are the signs of the modal proposition.
(4) A pure proposition simply states that the predicate belongs or does not belong to the subject.
(5) A truistic or tautologous proposition is one in which the predicate repeats the words and meaning of the subject.
(8) In considering the relation which may exist between subject and predicate, the two terms are employed in extension only, as this use best serves the interests of inference.
The extensional relation between subject and predicate of the four logical propositions may be stated as follows:
Ordinary A—All of the subject belongs to a part of the predicate.
Co-extensive A—All of the subject belongs to all of the predicate.
E—None of the subject belongs to any part of the predicate.
I—Some of the subject belongs to some of the predicate.
O—Some of the subject is excluded from the whole of the predicate.
In general it may be said that the affirmative propositions are inclusive while the negatives are exclusive.
A term is said to be distributed when it is referred to as a definite whole.
“A” distributes the logical subject only, “E” both logical subject and logical predicate, “I” neither logical subject nor logical predicate, “O” the logical predicate only. The co-extensive “A” distributes both subject and predicate.
It is essential that the student know by heart the distribution of the terms of the logical propositions. Some keyword like uaesneop may be used as an aid to the memory. This means the universals A and E distribute their subjects, while the negatives E and O distribute their predicates.