4.5 The Traditional Square of Opposition (old)

immediate inference

An argument having a single premise Testing _______________ ______________ or testing an argument having a single premise. Next, let us see how we can use the traditional square of opposition to test _____________ _______________ for validity. Here is an example: All Swiss watches are true works of art. Therefore, it is false that no Swiss watches are true works of art. To evaluate this argument, we begin, as usual, by assuming the premise is true. Since the premise is an A proposition, by the contrary relation the corresponding E proposition is false. But this is exactly what the conclusion says, so the argument is valid.

subalternation relation

The relation by which a true A or E statement necessarily implies a true I or O statement, respectively, and by which a false I or O statement necessarily implies a false A or E statement, respectively The _______________ ________________ is represented by two arrows: a downward arrow marked with the letter ''T'' (true), and an upward arrow marked with an ''F'' (false). These arrows can be thought of as pipelines through which truth values ''flow.'' The downward arrow ''transmits'' only truth, and the upward arrow only falsity. Thus, if an A proposition is given as true, the corresponding I proposition is true also, and if an I proposition is given as false, the corresponding A proposition is false. But if an A proposition is given as false, this truth value cannot be transmitted downward, so the corresponding I proposition will have logically undetermined truth value. Conversely, if an I proposition is given as true, this truth value cannot be transmitted upward, so the corresponding A proposition will have logically undetermined truth value. Analogous reasoning prevails for the subalternation relation between the E and I propositions.

subalternation valid (0 - E)

Use either the traditional square of opposition or conversion, obversion, or contraposition to determine whether the following argument is valid or invalid. For those that are invalid, name the fallacy committed. ★1. It is false that some jogging events are not aerobic activities. Therefore, it is false that no jogging events are aerobic activities.

Invalid (A - E) Illicit contrary

Use either the traditional square of opposition or conversion, obversion, or contraposition to determine whether the following argument is valid or invalid. For those that are invalid, name the fallacy committed. ★10. It is false that all substances that control cell growth are hormones. Therefore, no substances that control cell growth are hormones.

Invalid (I - O) illicit subcontrary

Use either the traditional square of opposition or conversion, obversion, or contraposition to determine whether the following argument is valid or invalid. For those that are invalid, name the fallacy committed. ★13. Some economists are followers of Ayn Rand. Therefore, some economists are not followers of Ayn Rand.

Invalid (I - I) illicit contraposition

Use either the traditional square of opposition or conversion, obversion, or contraposition to determine whether the following argument is valid or invalid. For those that are invalid, name the fallacy committed. ★4. Some terminally ill patients are patients who do not want to live. Therefore, some patients who want to live are recovering patients.

Valid (I - 0), by subcontrary or by subalternation, through contradiction.

Use either the traditional square of opposition or conversion, obversion, or contraposition to determine whether the following argument is valid or invalid. For those that are invalid, name the fallacy committed. ★7. It is false that some international terrorists are political moderates. Therefore, some international terrorists are not political moderates.

valid (A - I), by subalternation or by subcontrary, through contradictory

Use the traditional square of opposition to determine whether the following argument is valid or invalid. Name any fallacies that are committed. ★1. All advocates of school prayer are individuals who insist on imposing their views on others. Therefore, some advocates of school prayer are individuals who insist on imposing their views on others

Invalid (O - I) Illicit Subcontrary

Use the traditional square of opposition to determine whether the following argument is valid or invalid. Name any fallacies that are committed. ★10. It is false that some orthodox psychoanalysts are not individuals driven by a religious fervor. Therefore, it is false that some orthodox psychoanalysts are individuals driven by a religious fervor.

Invalid (E - A) Existential Fallacy

Use the traditional square of opposition to determine whether the following argument is valid or invalid. Name any fallacies that are committed. ★13. No flying reindeer are animals who get lost in the fog. Therefore, it is false that all flying reindeer are animals who get lost in the fog.

Invalid (O - E) Existential Fallacy

Use the traditional square of opposition to determine whether the following argument is valid or invalid. Name any fallacies that are committed. ★4. It is false that some trolls are not creatures who live under bridges. Therefore, it is false that no trolls are creatures who live under bridges

Invalid (A - E), illicit contrary

Use the traditional square of opposition to determine whether the following argument is valid or invalid. Name any fallacies that are committed. ★7. It is false that all mainstream conservatives are persons who support free legal services for the poor. Therefore, no mainstream conservatives are persons who support free legal services for the poor.

All inappropriate remarks are faux pas. Therefore, some faux pas are not appropriate remarks. appropriate remarks = A faux pas = F All non-A are F. Therefore, some F are not A. All non-A are F. (assumed true) Some non-A are F. (true by subalternation) Some F are non-A. (true by conversion) Therefore, some F are not A.(true by obversion)

Use the traditional square of opposition together with conversion, obversion, and contraposition to prove that the following argument is valid. Show each intermediate step in the deduction. All inappropriate remarks are faux pas. Therefore, some faux pas are not appropriate remarks.

All insurance policies are cryptically written documents. Therefore, some cryptically written documents are insurance policies. Insurance policies = I cryptically written documents = C All I are C. Therefore, some C are I. All I are C. (assumed true) Some I are C. (subalternation) Some C are I. (conversion)

Use the traditional square of opposition together with conversion, obversion, and contraposition to prove that the following argument is valid. Show each intermediate step in the deduction. ★1. All insurance policies are cryptically written documents. Therefore, some cryptically written documents are insurance policies.

False: Some F are not A. False: No F are A. (subalternation) False: No A are F. (conversion) False: All A are non-F. (obversion)

Use the traditional square of opposition together with conversion, obversion, and contraposition to prove that the following argument is valid. Show each intermediate step in the deduction. ★10. It is false that some feminists are not advocates of equal pay for equal work. Therefore, it is false that all advocates of equal pay for equal work are nonfeminists.

All exogenous morphines are addictive substances. Therefore, it is false that all addictive substances are endogenous morphines. exogenous morphines = E addictive substances = A All E are A. Therefore, it is false that all A are non-E. All E are A (assumed true) false: No E are A. (contrary) false: No A are E. (conversion) false: All A are non - E. (obversion)

Use the traditional square of opposition together with conversion, obversion, and contraposition to prove that the following argument is valid. Show each intermediate step in the deduction. ★4. All exogenous morphines are addictive substances. Therefore, it is false that all addictive substances are endogenous morphines

contradictory

a relation between two statements that necessarily have opposite truth values The ________________ relation is the same as that found in the modern square. Thus, if a certain A proposition is given as true, the corresponding O proposition is false, and vice versa, and if a certain A proposition is given as false, the corresponding O proposition is true, and vice versa. The same relation holds between the E and I propositions. The contradictory relation thus expresses complete opposition between propositions.

Given: All non-A are B. (T) Operation/relation: contrary New statement: No non-A are B. (F) Truth value: false

★7. This exercise provides a statement, its truth value in parentheses, and an operation to be performed on that statement. Supply the new statement and the truth value of the new statement. Given Statement: All non-A are B. (T) Operation/relation: contrary New statement: Truth value:

a. All obsessive-compulsive behaviors are curable diseases. (false, by contradictory relations) b. No obsessive-compulsive behaviors are curable diseases. (Logically undetermined truth value) c. Some obsessive-compulsive behaviors are curable diseases. (Logically undetermined truth value)

★7. Use the traditional square of opposition to find the answers to this problem. When a statement is given as false, simply enter an ''F'' into the square of opposition and compute (if possible) the other truth values. If ''Some obsessive-compulsive behaviors are not curable diseases'' is true, what is the truth value of the following statements? a. All obsessive-compulsive behaviors are curable diseases. b. No obsessive-compulsive behaviors are curable diseases. c. Some obsessive-compulsive behaviors are curable diseases.

traditional square of opposition

A diagram that illustrates the necessary relations that prevail between the four kinds of standard-form categorical propositions as interpreted from the Aristotelian standpoint. Like the modern square, the ______________ ____________ ___ ________________ is an arrangement of lines that illustrates logically necessary relations among the four kinds of categorical propositions. However, the two squares differ in that the modern square depends on the Boolean interpretation of categorical propositions, whereas the traditional square depends on the Aristotelian interpretation. As we saw in Section 4.3, the Boolean interpretation is neutral about whether universal (A and E) propositions make claims about actually existing things, whereas the Aristotelian interpretation assumes that the subject terms of these propositions denote things that actually exist. Because of this existential assumption, the traditional square contains more relations than the modern square. The four relations in the traditional square of opposition may be characterized as follows: Contradictory = opposite truth value Contrary = at least one is false (not both true) Subcontrary = at least one is true (not both false) Subalternation = truth flows downward, falsity flows upward

existential fallacy

A formal fallacy that occurs when the Aristotelian standpoint is adopted to draw inferences about things that do not exist; or a formal fallacy that occurs when, after Boolean standpoint is adopted, a particular conclusion is drawn from universal premises. As we saw at the beginning of this section, the traditional square of opposition depends on the Aristotelian interpretation of categorical propositions, which assumes that the subject terms of universal propositions denote actually existing things. What happens, it may now be asked, if the traditional square of opposition is used (in a way that would normally yield a determinate truth value) with propositions about things that do not exist? The answer is that another formal fallacy, the __________ _____________, is committed. In other words, the ____________ _____________ is committed whenever contrary, subcontrary, and subalternation are used (in an otherwise correct way) on propositions about things that do not exist. The following arguments commit the existential fallacy All witches who fly on broomsticks are fearless women. Therefore, some witches who fly on broomsticks are fearless women. No wizards with magical powers are malevolent beings. Therefore, it is false that all wizards with magical powers are malevolent beings. The first depends on an otherwise correct use of the subalternation relation, and the second on an otherwise correct use of the contrary relation. If flying witches and magical wizards actually existed, both arguments would be valid. But since they do not exist, both arguments are invalid and commit the ____________ ______________. The _____________ _______________ can be conceived in either of two ways. On the one hand, it appears as a kind of metalogical fallacy that results from an incorrect adoption of the Aristotelian standpoint

unnamed fallacy Cases of the incorrect application of the contradictory relation are so infrequent that an ''illicit contradictory'' fallacy is not usually recognized

A formal fallacy that occurs when the conclusion of an argument depends on an incorrect application of the contradictory relation.

illicit contrary

A formal fallacy that occurs when the conclusion of an argument depends on an incorrect application of the contrary relation. Forms: It is false that all A are B. Therefore, no A are B. It is false that no A are B. Therefore, all A are B.

illicit subalternation

A formal fallacy that occurs when the conclusion of an argument depends on an incorrect application of the subalternation relation Forms: Some A are not B. Therefore, no A are B. Some A are B. Therefore, all A are B. It is false that all A are B. Therefore, some A are B. It is false that no A are B. Therefore, some A are not B.

illicit subcontrary

A formal fallacy that occurs when the conclusion of an argument depends on an incorrect application of the subcontrary relation. Forms: Some A are B. Therefore, it is false that some A are not B. Some A are not B. Therefore, some A are B

illicit subcontrary

A formal fallacy that occurs when the conclusion of an argument depends on an incorrect application of the subcontrary relation. Testing Immediate Inferences Next, let us see how we can use the traditional square of opposition to test immediate inferences for validity. Here is another example: Some viruses are structures that attack T-cells. Therefore, some viruses are not structures that attack T-cells. Here the premise and conclusion are linked by the subcontrary relation. According to that relation, if the premise is assumed true, the conclusion has logically undetermined truth value, and so the argument is invalid. It commits the formal fallacy of ____________ _____________. Analogously, arguments that depend on an incorrect application of the contrary relation commit the formal fallacy of illicit contrary, and arguments that depend on an illicit application of subalternation commit the formal fallacy of illicit subalternation. Cases of the incorrect application of the contradictory relation are so infrequent that an "illicit contradictory" fallacy is not usually recognized.

contrary

The relation by which two statements are necessarily not both true (at least one is false) The ______________ relation differs from the contradictory in that it expresses only partial opposition. Thus, if a certain A proposition is given as true, the corresponding E proposition is false (because at least one must be false), and if an E proposition is given as true, the corresponding A proposition is false. But if an A proposition is given as false, the corresponding E proposition could be either true or false without violating the ''at least one is false'' rule. In this case, the E proposition has logically undetermined truth value. Similarly, if an E proposition is given as false, the corresponding A proposition has logically undetermined truth value. These results are borne out in ordinary language. Thus, if we are given the actually true A proposition ''All cats are animals,'' the corresponding E proposition ''No cats are animals'' is false, and if we are given the actually true E proposition ''No cats are dogs,'' the corresponding A proposition ''All cats are dogs'' is false. Thus, the A and E propositions cannot both be true. However, they can both be false. ''All animals are cats'' and ''No animals are cats'' are both false

subcontrary

The relation that exists between two statements that are necessarily not both false (at least one is true) The ______________ relation also expresses a kind of partial opposition. If a certain I proposition is given as false, the corresponding O proposition is true (because at least one must be true), and if an O proposition is given as false, the corresponding I proposition is true. But if either an I or an O proposition is given as true, then the corresponding proposition could be either true or false without violating the ''at least one is true'' rule. Thus, in this case the corresponding proposition would have logically undetermined truth value. Again, these results are borne out in ordinary language. If we are given the actually false I proposition ''Some cats are dogs,'' the corresponding O proposition ''Some cats are not dogs'' is true, and if we are given the actually false O proposition ''Some cats are not animals,'' the corresponding I proposition ''Some cats are animals'' is true. Thus, the I and O propositions cannot both be false, but they can both be true. ''Some animals are cats'' and ''Some animals are not cats'' are both true.

Some persons who recognize paranormal events are not non-scientists. Therefore, it is false that no scientists are persons who recognize paranormal events. persons who recognize paranormal events = P scientists = S Some P are non-S. Therefore, it is false that no S are P. Some P are not non-S. (assumed true) Some P are S. (obversion) Some S are P. (conversion) False: No S are P. (contradiction) or Some P are not non-S. (assumed true) False: All P are non-S. (contradiction) False: No P are S. (obversion) False: No S are P. (conversion) or Some P are not not-S. (assumed true) Some S are not non-P. (contraposition) False: All S are non-P. (contradiction) False: No S are P. (obversion)

Use the traditional square of opposition together with conversion, obversion, and contraposition to prove that the following argument is valid. Show each intermediate step in the deduction. ★7. Some persons who recognize paranormal events are not non-scientists. Therefore, it is false that no scientists are persons who recognize paranormal events.

Using the four relations to determine the truth value of corresponding statements

Using the four relations to determine the truth value of corresponding statements Now that we have explained these four relations individually, let us see how they can be used together to determine the truth values of corresponding propositions. The first rule of thumb that we should keep in mind when using the square to compute more than one truth value is always to use contradiction first. Now, let us suppose that we are told that the nonsensical proposition ''All adlers are bobkins'' is true. Suppose further that adlers actually exist, so we are justified in using the traditional square of opposition. By the contradictory relation, ''Some adlers are not bobkins'' is false. Then, by either the contrary or the subalternation relation, ''No adlers are bobkins'' is false. Finally, by either contradictory, subalternation, or subcontrary, ''Some adlers are bobkins'' is true. Next, let us see what happens if we assume that ''All adlers are bobkins'' is false. By the contradictory relation, ''Some adlers are not bobkins'' is true, but nothing more can be determined. In other words, given a false A proposition, both contrary and subalternation yield undetermined results, and given a true O proposition (the one whose truth value we just determined), subcontrary and subalternation yield undetermined results. Thus, the corresponding E and I propositions have logically undetermined truth value. This result illustrates two more rules of thumb. Assuming that we always use the contradictory relation first, if one of the remaining relations yields a logically undetermined truth value, the others will as well. The other rule is that whenever one statement turns out to have logically undetermined truth value, its contradictory will also. Thus, statements having logically undetermined truth value will always occur in pairs, at opposite ends of diagonals on the square.

using the traditional square to test validity

using the traditional square to test validity Now that we have seen how the traditional square of opposition, by itself, is used to test arguments for validity, let us see how it can be used together with the operations of conversion, obversion, and contraposition to prove the validity of arguments that are given as valid. Suppose we are given the following valid argument: All inappropriate remarks are faux pas. Therefore, some faux pas are not appropriate remarks. To prove this argument valid, we select letters to represent the terms, and then we use some combination of conversion, obversion, and contraposition together with the traditional square to find the intermediate links between premise and conclusion: inappropriate remarks = A faux pas = F All non-A are F. Therefore, some F are not A. All non-A are F. (assumed true) Some non-A are F. (true by subalternation) Some F are non-A. (true by conversion) Therefore, some F are not A. (true by obversion) The premise is the first line in this proof, and each succeeding step is validly derived from the one preceding it by the relation written in parentheses at the right. Since the conclusion (which is the last step) follows by a series of three necessary inferences, the argument is valid. Various strategies can be used to construct proofs such as this, but one useful procedure is to concentrate first on obtaining the individual terms as they appear in the conclusion, then attend to the order of the terms, and finally use the square of opposition to adjust quality and quantity. As the above proof illustrates, however, variations on this procedure are sometimes necessary. The fact that the predicate of the conclusion is ''A,'' while ''non-A'' appears in the premise, leads us to think of obversion. But using obversion to change ''non-A'' into ''A'' requires that the ''non-A'' in the premise be moved into the predicate position via conversion. The latter operation, however, is valid only on E and I statements, and the premise is an A statement. The fact that the conclusion is a particular statement suggests subalternation as an intermediate step, thus yielding an I statement that can be converted.

Given: All non-A are B. (T) Operation/relation: contraposition New statement: All non-B are A. (T) Truth value: same truth value as given statement

★1. This exercise provides a statement, its truth value in parentheses, and an operation to be performed on that statement. Supply the new statement and the truth value of the new statement. Given Statement: All non-A are B. (T) Operation/relation: contraposition New statement: Truth value:

a. No fashion fads are products of commercial brainwashing. (false, by contrary) b. Some fashion fads are products of commercial brainwashing. (true, by subalternation) c. Some fashion fads are not products of commercial brainwashing. (false, by contradictory)

★1. Use the traditional square of opposition to find the answer to this problem. When a statement is given as false, simply enter an ''F'' into the square of opposition and compute (if possible) the other truth values. If "All fashion fads are products of commercial brainwashing is true, what is the truth value of the following statements? a. No fashion fads are products of commercial brainwashing. b. Some fashion fads are products of commercial brainwashing. c. Some fashion fads are not products of commercial brainwashing.

Given: Some non-A are non-B. (F) Operation/relation: subcontrary New statement: Some non-A are not non-B. (T) Truth value: true

★10. This exercise provides a statement, its truth value in parentheses, and an operation to be performed on that statement. Supply the new statement and the truth value of the new statement. Given Statement: Some non-A are non-B. (F) Operation/relation: subcontrary New statement: Truth value:

Given: Some non-A are not B. (T) Operation/relation: subalternation New statement: No non-A are B. Truth value: logically undetermined

★14. This exercise provides a statement, its truth value in parentheses, and a new statement. Determine how the new statement was derived from the given statement and supply the truth value of the new statement. Take the Aristotelian standpoint in working these exercises. Given Statement: Some non-A are not B. (T) Operation/relation: New statement: No non-A are B Truth value:

Given: Some non-A are non-B. (F) Operation/relation: subalternation New statement: No non-A are non-B. Truth value: false

★16. This exercise provides a statement, its truth value in parentheses, and a new statement. Determine how the new statement was derived from the given statement and supply the truth value of the new statement. Take the Aristotelian standpoint in working these exercises. Given Statement: Some non-A are non-B. (F) Operation/relation: New statement: No non-A are non-B. Truth value:

Given: No A are non-B. (F) Operation/relation: contrary New statement: All A are non-B. Truth value: logically undetermined

★19. This exercise provides a statement, its truth value in parentheses, and a new statement. Determine how the new statement was derived from the given statement and supply the truth value of the new statement. Take the Aristotelian standpoint in working these exercises. Given Statement: No A are non-B. (F) Operation/relation: New statement: All A are non-B. Truth value:

Given: Some non-A are not B. (T) Operation/relation: subcontrary New statement: Some non-A are B. Truth value: logically undetermined

★4. This exercise provides a statement, its truth value in parentheses, and an operation to be performed on that statement. Supply the new statement and the truth value of the new statement. Given Statement: Some non-A are not B. (T) Operation/relation: subcontrary New statement: Truth value:

a. All sting operations are cases of entrapment. (Logically undetermined truth value) b. Some sting operations are cases of entrapment. (true, by contradictory relation) c. Some sting operations are not cases of entrapment. (Logically undetermined truth value)

★4. Use the traditional square of opposition to find the answer to this problem. When a statement is given as false, simply enter an ''F'' into the square of opposition and compute (if possible) the other truth values. If ''No sting operations are cases of entrapment'' is false, what is the truth value of the following statements? a. All sting operations are cases of entrapment. b. Some sting operations are cases of entrapment. c. Some sting operations are not cases of entrapment.