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The Jonathan Dolhenty Archive |
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INTRODUCTION: Part 6 - Page 2 by Jonathan Dolhenty, Ph.D.
Logical Opposition of Modals The Square of Opposition refers solely to categorical propositions which do not contain a mode affecting the copula. The way we treat modal propositions is similar to ordinary categorical propositions, but the logical opposition affects the mode itself. (Remember there are four modes we learned about.)
 Obviously, the subject of the modal proposition may be either a universal term or a particular term (all, no, or some). The opposition becomes more complicated thereby, but the general scheme must be carried out according to the logical opposition intended. If the logical opposition intended affects both the quantity and the mode of the propositions, the Square of Opposition would be as it appears here. Thus,  On the other hand, if the logical opposition affects only the mode, but not the quantity, of the propositions, the quantity (all or some) will remain the same and only the mode will change.  Or, From these relations of opposites it will be clear that we are often entitled to conclude from the truth or falsity of one proposition to the truth or falsity of another. This method of concluding from the truth or falisty of one statement to the truth or falsity of another is called immediate inference; it is called immediate because we can pass directly from the one to the other, without the necessity of adducing any other idea or judgment as proof. The Square of Opposition and the Three Laws of Thought are sufficient to make their truth or falsity evident, provided we know beforehand that one of these opposites is true or false. The Square of Opposition (with its relations of subalternation, contradiction, contrariety, subcontrariety) will act as a powerful aid toward correct thinking.
Equivalent Sentences Besides the immediate inference of logical opposition, we have the immediate inference of eduction. Eduction is a mental process whereby, from any proposition taken as true, we derive another proposition implied in it, though differing from the first proposition in subject or predicate or both. There are three main forms of eduction: obversion, conversion, and contraposition. The purpose of this technique is to transform certain sentences into other sentences which are equivalent in meaning, but may have a different logical form. The advantage of this is that an argument which may not be in strict syllogistic form can be transformed into a syllogism. Consider the following argument:
This argument appears to be valid but we can't test it by the rules already discussed because it contains more than three terms. It appears, in fact, to have five terms: unwise people, trustworthy people, wise people, unaggressive people, and aggressive people. But look at the second premise. It means the same thing as "All aggressive people are unwise." So, if we substitute this latter sentence for the original sentence, we get the following argument:
The argument now contains only three terms (unwise, trustworthy, aggressive) and is a syllogism. It can now be tested using the standard rules for testing the validity of syllogisms. We can then see that the argument is valid. Main Forms of Eduction Obversion Obversion is an eduction in which the inferred judgment, while retaining the original subject, has for its predicate the contradictory of the original predicate. The original proposition is called the obvertend and the inferred proposition is called the obverse. In obverting a given sentence we do two things:
Example: Consider the sentence "All men are mortal." First, we change the quality, and the sentence becomes "No men are mortal." Then we negate the predicate and the sentence becomes "No men are non-mortal." The sentence "No men are non-mortal" is equivalent to the sentence "All men are mortal." Every A, E, I, and O sentence can be obverted. Study the following diagram:
Care must be exercised in obverting sentences in ordinary language since some negative English terms can be confusing and some terms, which may appear to negate, do not negate at all. For instance, "large" is not the negation of "small." Certain prefixes ("im," "un," "in") do not always express negation. Logicians prefer, therefore, to use the prefix "non" in order to negate the predicate. Therefore, the negation of "rich" is not "poor," but "non-rich." When obverting, there must be a change in the quality of the sentence only. Do not change the quantity. Then a universal sentence remains a universal sentence and a particular sentence remains a particular sentence. Conversion Conversion is an eduction in which the inferred judgment takes the subject of the original proposition for its predicate, and the predicate of the original proposition for its subject. In other words, when we convert we merely interchange subject and predicate. The original proposition we call the convertend and the inferred proposition we call the converse. Example: Consider the sentence "No dogs are horses." This sentence is equivalent to the sentence "No horses are dogs." Conversion is unlike obversion because not every standard sentence has an equivalent converse. Only the E and the I sentences can be converted. Example: The O sentence cannot be converted. From "Some woman are not nuns," we cannot infer "Some nuns are not woman." Example: The A sentence cannot be converted simply. From "All dogs are animals," we cannot infer that "All animals are dogs." It is possible, however, to partially convert the A sentence, using a technique logicians call Conversion by Limitation. When we convert a true A sentence, we can transform it into a true I sentence. The sentence "All dogs are animals" can be partially converted into "Some animals are dogs." Partial conversion, however, does not give us a sentence which is exactly equivalent in meaning to the original. This is because the quantity of the original sentence is changed. Study this diagram of permissible conversions:
Contraposition Contraposition is an eduction in which the subject of the inferred proposition is the contradictory of the predicate of the original proposition. It is (read carefully!) the obverse of a converted obverse. In order to obtain the contraposition of a sentence, three operations must be performed:
Example: Consider the sentence "All dogs are animals."
Contraposition cannot be applied to all four standard sentences. The A sentence and the O sentence have contrapositives. The I sentence has no contrapositive and the E sentence has only a partial contrapositive. Usually, contraposition is only applied to A sentences. You should now be able to transform large parts of ordinary language into arguments of a syllogistic form, making it possible to test the validity of the argument. The chart below will help you see how to form equivalent sentences.
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