Standard | Formal | NonStandard | Summary | Review
Propositions are either standard form or nonstandard. If a proposition is not standard form, it is classified as nonstandard. We first consider the four standard form propositions, then discuss nonstandard propositions in the last section of this Study.
There are only four standard form propositions. Each consists of a subject and a predicate. In each form, the subject and the predicate are joined together by is or are, the copula. The relation between the subject and predicate is identified by the use of: All, No, Some, or Some ... not.... If a and b stand for the subject and predicate terms, respectively, the four forms are: (1) All a is b, (2) No a is b, (3) Some a is b, and (4) Some a is not b.
The proposition "All men are mortal" asserts a relation of inclusion between the class of men and the class of mortals. More plainly, it states that all members of the class men fall within the class mortal. The form of all such propositions is All a is b, or A(ab) where a stands for the subject term and b stands for the predicate term. Note that in A propositions, the subject is included in the predicate, but not the predicate in the subject. For example, from "All men are mortals" (true) it does not follow that all mortals are men (false).
The proposition "No Christian is an atheist" asserts a relation of exclusion between two classes, Christians and atheists. No member of the class Christians is a member of the class atheists, and conversely, no atheist is a Christian. The classes of E propositions are mutually exclusive. The form is No a is b, or E(ab), where a stands for any subject, and b stands for any predicate. Thus, with E propositions all members of one class are excluded from the other, and vice versa.
The proposition "Some Americans are Calvinists" asserts a relation of partial inclusion between the class Americans and the class Calvinists. Something less than all members of the subject-class is included in the predicate-class, and conversely, some members of the class Calvinists are included in the class Americans. The form of the I proposition is Some a is b, or I(ab), where, as before, a stands for any subject, b for any predicate. Ordinarily, some can mean a few in number. In logic, the word can also mean as few as one or any number less than all.
The proposition "Some men are not Christian" asserts a relation of partial exclusion between the two classes, men and Christians. Some men are entirely excluded from all of the class of Christians. Does it follow then that some Christians are not men? Perhaps some angels are Christian? No, the converse of an O proposition does not follow from the original. Its form is Some a is not b, or O(ab). Remember, there is no converse for an O proposition.
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The following chart serves as a summary of the foregoing descriptions of the four forms. Do not be confused in that the letters a and b are used throughout, even when the propositions contain different subject matter. Recall that the letters, a and b, stand for any subject and any predicate, respectively. Indeed, we could have used x and y or any other pair of letters to stand for subjects and predicates.
|All men are mortal||All a is b.||A(ab)|
|No Christian is an atheist.||No a is b.||E(ab)|
|Some Americans are Calvinists.||Some a is b.||I(ab)|
|Some men are not Christian.||Some a is not b.||O(ab)|
The source of the letters for the four forms is of historical interest. From affirmo (I affirm), meaning affirmative in quality, we get A and I; E and O come from nego (I deny), meaning negative in quality.
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The four forms share three important properties: distribution, quantity, and quality defined just below.
The formal properties, quality and quantity, of A, E, I, and O forms depend on the definition of distribution. We distinguish a distributed term (subject or predicate) from an undistributed term in this manner: A distributed term is one modified by All or No. When a term is modified by "some," it is undistributed. Using the subscripts "d" for distributed and "u" for undistributed, the four forms distribute their terms as indicated below in Chart 1.2.
|Forms||Subject Term||Predicated Term|
|A||All sd is pu||Distributed||Undistributed|
|E||No sd is pd.||Distributed||Distributed|
|I||Some su is pu.||Undistributed||Undistributed|
|O||Some su is not pd.||Undistributed||Distributed|
Where, s = subject term; p = predicate term.
To recapitulate: With the A form, only the subject term is distributed; the predicate is undistributed, since, as noted previously, all of the predicate is not included in the subject. The E form distributes both subject and predicate terms, since No s is p; and No p is s. With the I form, some part of the subject term class is included in some part of the predicate term class; therefore, both terms are undistributed. Last, in the O form, some part of the subject term class is excluded from all of the predicate term class (Some s is not p); therefore, only the predicate term is distributed, the subject term, undistributed.
Previously we indicated that the A and I letters came from the Latin affirmo, and E and O from the Latin nego. Remembering the sources of the letters may help to recall that the A and I forms are affirmative in quality; E and O, negative in quality. An affirmative form is one that does not distribute its predicate. The A and I forms do not distribute the predicates; therefore, they are affirmative in quality. A negative form is one that does distribute its predicate. The E and O forms distribute the predicates; therefore, they are negative in quality.
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Each of the four forms is either universal or particular in quantity. If a form distributes its subject term, it is universal in quantity. The A and E forms are universal, since each distributes its subject term. On the other hand, a form is particular in quantity if its subject term is undistributed. The I and the O forms have undistributed subject terms; therefore, these are particular.
|All sd is pu||universal||affirmative|
|No sd is pd||universal||negative|
|Some su is pu.||particular||affirmative|
|Some su is not pd||particular||negative|
Chart 1.3 may serve the student as a memory device for reinforcing how quantity and quality is determined by distribution of terms in standard form propositions. The chart is no substitute for memorizing the definition of distribution and understanding what it means. The importance of distribution of terms cannot be overemphasized, for it not only serves as the basis for defining the quality and quantity of the four forms, but is the basis for some of the rules that test the validity of deductive inference.
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Only standard form propositions are candidates for the premises and conclusion for the type of argument (syllogism) discussed in Study Three. Most nonstandard propositions are easily translated to standard form. Others will require practice and careful attention to the meaning of the proposition in question. This may result in some awkward formulations of English. The goal is clarity of meaning, not elegant prose.
In the case of an English verb other than the present tense of the verb to be, change the verbs into predicate adjectives. For example, "All competent students know logic" becomes "All competent students are knowers-of-logic.
When the language of the sentence contains clauses or prepositional phrases as well as a verb other than the English copula, the use of parameters will help make the sense of the proposition clear. For example, "All persons-who-are-competent-students are persons-who-are-knowers-of-logic." Here the word, persons, appears in both the subject and predicate, and together with hyphens assists in reading the proposition as an A proposition. The purpose is to make the sense of the proposition crystal clear.
More effort is required with two other classes of propositions: exclusive and exceptive propositions.
How can we make clear the sense of this exclusive proposition? "Only atheists will be ejected." What does it mean? It means "All persons-who-are-ejected are persons-who-are-atheists." Thus the sense of exclusive propositions (only x is y) is the A form, the result obtained when subject and predicate are interchanged.
Exceptive propositions (all except x is y) are really two in one form. For example, "All except the soldiers gave up the fight" means (1) All persons who are non-soldiers (civilians) are persons who gave up the fight; and (2) No person who is a soldier is a person who gave up the fight. Not to anticipate the material of the next Study, let it be merely noted for now that neither one of these can be deduced from the other.
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Some logic books label propositions with proper names, singular propositions. We make no distinction between singular and other universal propositions. All propositions using proper names are either Form A or Form E, depending on the quality. The name Socrates, in "Socrates is mortal" is the entire subject class, which happens to have only one member. An example of an E form is "Socrates is not immortal," or, "No Socrates is immortal." These are not the only propositions wherein "all" or "no" is implied. Some propositions appear to name only some members of a class, when all members of a class are either included or excluded. Example: "Dinosaurs are extinct" does not mean that some are or some may not be extinct. The sense of the statement is that all dinosaurs are extinct. In other words, the "all" is implied, and when the context calls for "all," or "no" the result is either an A Form or an E Form, depending on the quality of the original. Also, an implied "some" proposition is translated as the I Form or O Form, depending on quality.
The grammatical and logical subjects of some propositions sometimes need to be distinguished, if one is to achieve the correct sense of a proposition. An example cited in a logic book is: "You always squirm out of an argument." The grammatical subject, "you," is not the logical subject. Rather, always meaning "every time you get into an argument" is the logical subject. The sense of the original is "All times-you-get-into-an-argument are times-you-squirm-out-of-it." (The statement may appear to be awkward, but the meaning is accurately worded and that's what matters!)
Application of tests to determine the validity of inference depends on the clear sense of standard form propositions. However, the job of re-wording nonstandard propositions into standard form A, E, I, and O has benefits beyond the requirements of deductive inference. Where testing for validity is not an issue, rewording nonstandard into standard forms will avoid misunderstandings, mistakes, and confusion. If you can't reword a nonstandard proposition into standard form, you probably do not know what it means. Therefore, it is essential that you develop translation skills to achieve clarity of thought and to avoid misunderstanding or mistakes in reasoning.
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Standard form propositions consists of subject and predicate terms joined by the copula "is" or "are" and qualified by "All," "No," "Some," or "Some ... not ...." These requirements yield four forms: (1) All a is b, (2) No a is b, (3) Some a is b, and (4) Some a is not b known as A, E, I, and O forms, respectively. (The forms are also expressed as A(ab), E(ab), I(ab), and O(ab).) The formal properties of distribution, quality, and quantity of the four standard forms were explained and illustrated. A distributed term is one modified by "all" or "no." If a term is modified by "some," it is undistributed. If a proposition's predicate term is distributed, the proposition is said to be negative in quality; if the predicate of a proposition is not distributed, then it is affirmative in quality. This definition of quality distinguishes E(ab) and O(ab), both negative, from A(ab) and I(ab), both affirmative. If a proposition distributes its subject term, it is universal in quantity. On the other hand, if a proposition's subject term is undistributed, it is particular in quantity. By this definition, we distinguish A(ab) and E(ab), both universal, from I(ab) and O(ab), both particular. Finally, some guidelines for translating nonstandard propositions into standard form were described.
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|1||Of the four standard forms, which distribute their subject terms? Which do not distribute their subject terms? What formal property is defined in each case?|
|2||Of the four standard forms, which distribute their predicate terms? Which do not distribute their predicate terms? What formal property is defined in each case?|
|3||Which of the other three forms differ in both quantity and quality from A(ab)? From I(ab)?|
|4||What is the general formulation of exclusive propositions? What is the procedure for transforming an exclusive proposition into standard form. Of the four standard forms, which distribute their subject terms? Which do not distribute their subject terms?|
|5||Compose some examples of exceptive propositions. Identify the two component sentences embedded in each.|