We come now to the core reasoning technology for this course called Standard Form Categorical Reasoning. It is the 'core' of this course in that the success of your work from this point forward will depend on your mastery of it. It is a 'reasoning technology' in that it is a codification of principles, rules and methods that guarantee valid reasoning whenever and where ever they are applied. Here are the fundamental concepts of this reasoning technology. These fundamentals are expanded in REASON ARGUE REFUTE on pages 43-131 inclusive.
1. Classical deductive (or Aristotelian) logic is based on the idea of membership in categories, classes and groups. It is known in the western culture then as Aristotelian Categorical Logic. In this topic, we take up the task of mastering this form of reasoning by formal argument. Categorical Logic constitutes the very core of this course in Critical Thinking and Writing. From this point forward, we will it use in our essays, exams, term paper, and final examination.
Our study begins with the notion of a 'standard-form-categorical-proposition'.
A categorical proposition (truth claim) affirms or denies, in whole or in part, that members of one class or category are included or excluded in another class or category. Since categorical propositions deal with two states (affirmative and negative) of two classes (universal and particular), there are, then, just four possible categorical propositions.
These four categorical propositions are known by 'Types', A, E, I and O that have nothing to do with vowels.
The universal affirmative, Type (A) proposition states that every member of one class is also a member of the second class.
The universal negative, Type (E) proposition states that no member of one class is a member of the second.
In a particular affirmative, Type (I) proposition, some members (at least one) of one class are members of the second.
In the particular negative, Type (O) proposition, some members of one class are not members of the second.
THE CONCEPT THAT MEMBERS OF ONE CATEGORY INCLUDE OR EXCLUDE ALL OR SOME MEMBERS OF ANOTHER CATEGORY IS CALLED: DISTRIBUTION
This notion of DISTRIBUTION is the most essential concept in Aristotelian logic. In fact, it is the very analytic mechanism of categorical deductive VALIDITY itself. Therefore, the following precise definition must be fully understood in order to master classical deductive logic.
"A categorical term is distributed, if and only if, the categorical proposition that contains it tells us something about ALL members of that categorical term."
"Conversely, a categorical term is undistributed, if and only if, the categorical proposition that contains it does NOT tell us something about ALL members of that categorical term."
2. Propositions are said to have quality-either affirmative or negative-and quantity-either universal or particular. They may also be distributed or undistributed: a proposition is said to distribute a term if it refers to all members of the class designated by the term. In an A proposition, for example, the subject term is distributed, but the predicate term is not.
Examples of categorical propositions by Type ( A, E, I, and O):
||Quantity and Quality
||Distribution of Terms
||Every S is P
||S is distributed, P is not distributed
||No S is P
||Both S and P are distributed
||Some S is P
||Neither S nor P are distributed
||Some S is not P
||S is not distributed, P is distributed
3. The traditional square of opposition graphically displays the relationships that exist between the four different standard form categorical propositions. Propositions can be contradictories, contraries, sub contraries, subalterns, or superalterns. Each of these relationships leads to certain possible immediate inferences, which the square of opposition outlines. Three of these inferences are conversion, obversion, and contraposition. The traditional Aristotelian Square of Opposition looks like this:
4. The problem of existential import presents some problems for the relationships suggested by the traditional square of opposition. As a result, most modern logicians adopt a different interpretation of the square, called Boolean. Under this interpretation, particular propositions (I and O) have existential import; but universal propositions (A and E) do not. This is so because both Type I and Type O categorical propositions actually claim that "at least one" member of a class or category exists. In Logic, the word 'some' means "at least one". More on this later.
5. Diagrams and symbolizing techniques are useful in helping to visualize the relationships of categorical propositions. Venn diagrams are especially effective at exhibiting the relationships between classes by marking and shading overlapping circles. When 3 standard form Categorical Propositions are formulated into an argument, then that argument is called a Standard Form Categorical Syllogism.
Example of a Venn Diagram for a valid categorical argument (syllogism):
All foundations of American society are based upon our reliance on self-preservation. Some freedoms are the foundations of American society. Some freedoms are our reliance on self-preservation.
Class / Category, Universal affirmative (A), Particular negative (O), Opposition, Sub contraries, Conversion, Existential import, Categorical proposition, Universal negative (E), Quality/Quantity, Contraries, Sub/Superalterns, Obversion, Boolean interpretation, Particular affirmative (I), Distributed/Undistributed terms, Subject/Predicate, Contradictories, The Square of Opposition, Contraposition, Venn diagrams.
If we expand the scope of our investigation to include shared terms and their complements, we can identify logical relationships of three additional varieties. Since each of these new cases involves a pair of categorical propositions that are logically equivalent to each other, that is, either both of them are true or both are false, they enable us to draw an immediate inference from the truth (or falsity) of either member of the pair to the truth (or falsity) the other.
The converse of any categorical proposition is the new categorical proposition that results from putting the predicate term of the original proposition in the subject place of the new proposition and the subject term of the original in the predicate place of the new proposition. Thus, for example, the converse of "No dogs are felines" is "No felines are dogs," and the converse of "Some snakes are poisonous animals" is "Some poisonous animals are snakes."
Conversion grounds an immediate inference for both E and I propositions That is, the converse of any /i>E or /i>I proposition is true if and only if the original proposition was true. Thus, in each of the pairs noted as examples in the previous paragraph, either both propositions are true or both are false.
In addition, if we first perform a sub alternation and then convert our result, then the truth of an A proposition may be said, in "conversion by limitation," to entail the truth of an I proposition with subject and predicate terms reversed: If "All singers are performers" then "Some performers are singers." But this will work only if there really is at least one singer.
Generally speaking, however, conversion doesn't hold for A and O propositions: it is entirely possible for "All dogs are mammals" to be true while "All mammals are dogs" is false, for example, and for "Some females are not mothers" to be true while "Some mothers are not females" is false. Thus, conversion does not warrant a reliable immediate inference with respect to A and O propositions.
In order to form the obverse of a categorical proposition, we replace the predicate term of the proposition with its complement and reverse the quality of the proposition, either from affirmative to negative or from negative to affirmative. Thus, for example, the obverse of "All ants are insects" is "No ants are non-insects"; the obverse of "No fish are mammals" is "All fish are non-mammals"; the obverse of "Some musicians are males" is "Some musicians are not non-males"; and the obverse of "Some cars are not sedans" is "Some cars are non-sedans."
Obversion is the only immediate inference that is valid for categorical propositions of every form. In each of the instances cited above, the original proposition and its obverse must have exactly the same truth-value, whether it turns out to be true or false.
The contrapositive of any categorical proposition is the new categorical proposition that results from putting the complement of the predicate term of the original proposition in the subject place of the new proposition and the complement of the subject term of the original in the predicate place of the new. Thus, for example, the contrapositive of "All crows are birds" is "All non-birds are non-crows," and the contrapositive of "Some carnivores are not mammals" is "Some non-mammals are not non-carnivores."
Contraposition is a reliable immediate inference for both A and O propositions; that is, the contrapositive of any A or O proposition is true if and only if the original proposition was true. Thus, in each of the pairs in the paragraph above, both propositions have exactly the same truth-value.
In addition, if we form the contrapositive of our result after performing sub alternation, then an E proposition, in "contraposition by limitation," entails the truth of a related O proposition: If "No bandits are biologists" then "Some non-biologists are not non-bandits," provided that there is at least one member of the class designated by "bandits."
In general, however, contraposition is not valid for E and I propositions: "No birds are plants" and "No non-plants are non-birds" need not have the same truth-value, nor do "Some spiders are insects" and "Some non-insects are non-spiders." Thus, contraposition does not hold as an immediate inference for E and I propositions.
Omitting the troublesome cases of conversion and contraposition "by limitation," then, there are exactly two reliable operations that can be performed on a categorical proposition of any form:
A proposition: All S are P. Obverse: No S are non-P. Contrapositive: All non-P are non-S.
E proposition: No S are P. Converse: No P are S. Obverse: All S are non-P.
I proposition: Some S are P. Converse: Some P are S. Obverse: Some S are not non-P.
O proposition: Some S are not P. Obverse: Some S are non-P. Contrapositive: Some non-P are not non-S.
It is time to express more explicitly an important qualification regarding the logical relationships among categorical propositions. You may have noticed that at several points in these two lessons we declared that there must be some things a certain kind. This special assumption, that the class designated by the subject term of a universal proposition has at least one member, is called existential import. Classical logicians typically presupposed that universal propositions do have existential import.
But modern logicians have pointed that the system of categorical logic is more useful if we deny the existential import of universal propositions while granting, of course, that particular propositions do presuppose the existence of at least one member of their subject classes. It is sometimes very handy, even for non-philosophers, to make a general statement about things that don't exist. A sign that reads, "All shoplifters are prosecuted to the full extent of the law," for example, is presumably intended to make sure that the class designated by its subject term remains entirely empty. In the remainder of our discussion of categorical logic, we will exclusively employ this modern interpretation of universal propositions.
Although it has many advantages, the denial of existential import does undermine the reliability of some of the truth-relations we've considered so far. In the traditional square of opposition, only the contradictories survive intact; the relationships of the contraries, the sub contraries, and sub alternation no longer hold when we do not suppose that the classes designated by the subject terms of A and E propositions have members. (And since conversion and contraposition "by limitation" derive from sub alternation, they too must be forsworn.) From now on, therefore, we will rely only upon the immediate inferences in the table at the end of the previous section of this lesson and suppose that A and O propositions and E and I propositions are genuinely contradictory.
The modern interpretation of categorical logic also permits a more convenient way of assessing the truth-conditions of categorical propositions, by drawing Venn diagrams, topological representations of the logical relationships among the classes designated by categorical terms. The basic idea is fairly straightforward :
Each categorical term is represented by a labeled circle. The area inside the circle represents the extension of the categorical term, and the area outside the circle its complement. Thus, members of the class designated by the categorical term would be located within the circle, and everything else in the world would be located outside it.
We indicate that there is at least one member of a specific class by placing an × inside the circle; an × outside the circle would indicate that there is at least one member of the complementary class.
To show that there are no members of a specific class, we shade the entire area inside the circle; shading everything outside the circle would indicate that there are no members of the complementary class.
Notice that diagrams of these two sorts are incompatible: no area of a Venn diagram can both be shaded and contain an ×; either there is at least one member of the represented class, or there are none.
In order to represent a categorical proposition, we must draw two overlapping circles, creating four distinct areas corresponding to four kinds of things: those that are members of the class designated by the subject term but not of that designated by the predicate term; those that are members of both classes; those that are members of the class designated by the predicate term but not of that designated by the subject term; and those that are not members of either class.
Categorical propositions of each of the four varieties may then be diagrammed by shading or placing an × in the appropriate area.
The universal negative (E) proposition asserts that nothing is a member of both classes designated by its terms, so its diagram shades the area in which the two circles overlap.
The particular affirmative (I) proposition asserts that there is at least one thing that is a member of both classes, so its diagram places an X in the area where the two circles overlap.
Notice that the incompatibility of these two diagrams models the contradictory relationship between E and I propositions; one of them must be true and the other false, since either there is at least one member that the two classes have in common or there are none.
The particular negative (O) proposition asserts that there is at least one thing that is a member of the class designated by its subject term but not of the class designated by its predicate term, so its diagram places an × in the area inside the circle that represents the subject term but outside the circle that represents the predicate term.
Finally, the universal affirmative (A) proposition asserts that every member of the subject class is also a member of the predicate class. Since this entails that there is nothing that is a member of the subject class that is not a member of the predicate class, an A proposition can be diagrammed by shading the area inside the subject circle but outside the predicate circle. Again, the incompatibility of the diagrams for A and O propositions represents the fact that they are logically contradictory; one of them must be true and the other false.
Questions for Study:
1. What are the properties of A, E, I, and O propositions? Come up with examples of propositions for each of these types of standard form categorical propositions.
2. What do affirmative propositions have in common? What do particular propositions have in common? What about universal and negative propositions? How do the terms quality and quantity come into play in these considerations?
3. What is the difference between contraries and contradictories? Between contraries and sub contraries?
4. When does conversion result in valid inferences? Why does it work then, but not in other cases? Consider the same question with contraposition and obversion as well.
5. Why is existential import so problematic for Aristotelian logic? What changes does it require to the square of opposition?