Valid Inference

An inference is valid whenever the form of the conclusion is true every time the forms of the premise are. If the form of the conclusion is not true every time the forms of the premise are true, then the inference is invalid. The significance of this definition is the subject of the paragraphs that follow.

One way of illustrating the meaning of the definition makes use of circles representing the four standard forms. (Another is by the Method of Deduction, discussed in the next study.) The five sets of circles in the chart below correspond to five ways in which two terms (subject and predicate terms) can be related in the four forms. The circles are numbered as cases 1 through 5 for easy reference. (Read A(ab) as All a is b; E(ab) as No a is b; and so forth.) To repeat. If the form of the conclusion is true every time the forms of the premise are true, then the conclusion follows validly from the premise. One inspects the circles to see if a particular immediate inference does not violate the definition for validity. In a sense, the chart circles and lines "operationalize" the definition of validity. Descriptions of each case in the chart follow, after which we offer some examples.

Forms, Circles, and Lines

Chart 2.1 Four Forms Diagrammed

The lines labeled A, I, O, and E mean All, Some, Some ... is not, and No, respectively.

The test for validity compares the conclusion to the premise with reference to the circles to determine if the conclusion is true every time the premise is. For example, one may ask, does A(ab) imply I(ab), or, what comes to same thing, is I(ab) a necessary consequence from A(ab)? With either question, we are asking whether A(ab) and I(ab) form a valid inference. Inspection of Chart 2.1, shows that I(ab) is true every time A(ab) is true. Check it out for yourself. I(ab) is true in cases 1-4; A(ab) is true in cases 1 and 2. Thus, A(ab) and I(ab) make a valid inference. Similarly, E(ab) and O(ab) form a valid inference, since O(ab) is true every time E(ab) is true. This time check out the lines as well as the circles. Note that Line O includes the whole of Line E. To provide a contrast, does O(ab) imply E(ab)? Is E(ab) true every time O(ab) is? E(ab) is true in case 5 (Line E); O(ab) is true in cases 3, 4, and 5 (Line O). Therefore, one cannot validly infer from O(ab), the form E(ab), since E(ab) is not true every time O(ab) is true; O(ab) is true three times, E(ab) only once. By the same reasoning, I(ab) and A(ab) do not form a valid inference, since A(ab) is not true every time I(ab) is true.

Square of Opposition

The square of opposition incorporates the valid inferences just mentioned above. Pairs of the forms stand in opposition to each other as contraries, subcontraries, subalternation, or contradiction as depicted in Chart 2.2.

Chart 2.2: Square of Opposition

The square of opposition's contraries, subcontraries, subalterns, and contradictories require identical subject terms in each of the four forms as well as identical predicate terms in each. Definitions follow in the order listed.

Contraries

A(ab) and E(ab) are opposed as contraries. By definition, contraries cannot both be true together; however, both may be false. For example, if No Christian is an atheist is true, then it is false that All Christians are atheists. Why? Because the forms are contraries and they cannot both be true together. This appeal to the definition of contraries is supported by Chart 2.1 analysis of the corresponding lines. Lines A and E do not overlap which means they cannot both be true in any instance. Suppose we assume that an A Form is true. Applying the definition for contraries, we conclude that the E Form will be false. How can contraries be both false? Observe that Lines A and E do not exhaust all five cases. The significance of this fact is they could both be false together. One example: Some Christians are Calvinists is true, then the corresponding A Form (All Christians are Calvinists) and E Form (No Christians are Calvinists) are both false.

Subcontraries

I(ab) and O(ab), are subcontraries, meaning that they cannot both be false together, but they could both be true. Consider, for instance, the I Form, Some atheists are Christian is a false proposition by all accounts. If so, it follows that the related subcontrary, the O Form, Some atheists are not Christian is true. A similar line of reasoning applies when one begins with a false O Form. The corresponding I Form is true by the definition of subcontraries. The definition also states that subcontraries can both be true together. Here is an example: Some Christians are Calvinists, and Some Christians are not Calvinists. Again, Chart 2.1 analysis of the lines supports the definition of subcontraries. Lines I and O exhaust all 5 cases, and overlap each other to show that they can both be true together.

Subalterns

Subalterns is the name for two pairs of forms: A(ab) & I(ab); and E(ab) & O(ab). Subalterns may both be false together or both true together. For example, if the E Form No men are angels is true, then the A Form (All men are angels) and the I Form (Some men are angels) are both false. A similar analysis applies to the second pair of subalterns. If, All men are sinners is true, then the corresponding E Form (No men are sinners) and O Form (Some men are not sinners) are both false. Under what conditions are subalterns both true? Follow this reasoning: what is true of All will also be true of Some, assuming, of course that we use the identical subject and predicate terms. To stick with our example, if All men are sinners is true, then what is true of All is also true of Some (or any portion of the class). Likewise, if No men are angels is true, then it may be pointless, but nevertheless true that Some men are not angels.

Examination of Chart 2.1 once again supports the definition of subalternation. Lines A and I are both true under cases 1 and 2, and both false in case 5. Lines O and E are both true under case 5, and both false under cases 1 and 2. To "see" the relations between lines and cases may require some effort and more practice to acquire the necessary expertise. In the meantime, rely on the definitions and the square of opposition to determine what are valid inferences and which are not permitted (invalid). Definitions are indispensable, if one seeks to understand correctly or be understood accurately.

Contradiction

The strongest form of opposition is contradiction. The definition of contradiction incorporates some aspects of the properties of contraries and subcontraries. Contradictories cannot both be true together and cannot both be false together. A(ab) & O(ab), and E(ab) & I(ab) are contradictories. If the A Form is true, then the contradictory O Form is false. If the O Form is true, then the contradictory A Form is false. A similar reasoning applies to the other pair, the E Form and the I Form. To illustrate. If All men are sinners is true, then the contradictory, Some men are not sinners is false. Another example. If Some men are Christian is true, then the contradictory E Form, No men are Christian is false.

Observe the lines of Chart 1.2. Lines A and O, and I and E can be seen to meet without overlapping and, at the same time, each pair exhausts all cases. These characteristics apply only to contradictions. Thus it is a mistake to think of contraries or subcontraries as contradictories. Contraries and subcontraries are opposed, but their opposition is not total. Only contradictories represent total opposition. Some men are liars and Some men are not liars are not contradictories; although they are opposed. They are opposed as subcontraries. Recall the definition.

A useful analogy for the square of opposition is to treat it as a simple computer that calculates results based on two values, true or false (1 or 0). Given the value of any form or corner of the square, one can calculate the values of the other forms in the square of opposition. If, for example, we assume the value of True for the A Form corner, what are the values of the other three forms? The contradictory O Form corner has the value of False, since contradictories cannot both be true together and both be false together. The contrary E Form corner has the value of False, based on the definition of contraries. One can arrive at the value of the I Form corner in three ways. First, it has the value of True, since it's contradictory (E Form) is False. Second, it has the value of True, as the subaltern of the A Form corner which we assumed has a value of True. Third, it has a value of true as the subcontrary of a false O Form.

Similar calculations can begin with True or False from any other corner of the square with similar results -- almost always. As with any computer, there are limitations. You will receive an error message: "Can't Compute; Insufficient Information," when one of two initial conditions hold. You plug in False in either of the upper corners (A Form or E Form). Or, you plug in True in either of the lower corners (I Form or O Form). Either of these two related conditions will result in a pair of forms for which no value can be calculated. We know the result is either true or false, but which one?. Our simple computer becomes stuck. It cannot choose between the two: Not enough information. One example should suffice. Suppose we assume True for the O Form corner. The A Form corner has the value of False, since it is the contradictory of the O Form. What about the other pair, the E Form and I Form corners? There is no way to calculate the values of this pair of forms. For instance, from the truth of the subcontrary O Form, the other subcontrary I Form's value is either true or false. Recall the definitions of subcontraries. From the falsity of the contrary A Form, the other contrary E Form's value is either true or false. Recall the definition of contraries. The result is a pair of Undetermined Forms, in this case, the E Form and its contradictory, the I Form.

Given the initial conditions described above, it is always the case that our simple computer calculations will produce a pair of forms with Undetermined values. The pair of Undetermined Forms will always be a diagonal pair, never any other pair.

A couple of comments on the square of opposition relations and propositions. Some texts display a leaner model of our simple computer. It amounts to a stripped-down model, consisting of only the cross of contradiction. We opt for the original, for it illustrates, as we have shown, a number of useful immediate inferences not possible with the stripped-down version. A reason often offered for the stripped-down model is the claim that subcontraries assert or assume the "existence" of members of the subject classes. This is considered by some to be problematic. Let it be noted here that logic alone does not assert the existence or the nonexistence of anything. The existence or nonexistence of men, sinners, or angels in propositions is a matter for history or biology, anthropology, or some other discipline. "Some a is b" does not assert the existence or nonexistence of "a's" or "b's."

Invalid Inferences

We have shown the value of Chart 2.1 in testing the validity of the immediate inferences depicted in the Square of Opposition. Obviously, use of the same methods proves invalidity as well, for if an inference is not valid, then it must be invalid -- the only other possibility.

Consider the following matrix of forms. It represents the number of ways two forms can be combined to form inferences or implications. For example, in the first cell "AA," the implication to be tested is: Does A imply A? Of course, by the Law of Identity, since every proposition implies itself, the answer is affirmative. Similar reasoning for the other combinations in the diagonal (upper left to lower right: EE, II, and OO) show these to be valid implications as well.

Chart 2.3: Matrix of Four Forms

We eliminate the combinations of the other diagonal (upper right to lower left: OA, IE, EI, and AO), because each is a pair of contradictories. No form implies its contradictory. There are a total of 16 combinations in the matrix (4 rows x 4 columns = 16). Thus far we can account for eight of the combinations, four in each of two diagonals. What of the remainder, listed below?

Chart 2.4: Test of Implications (1^st Figure)

Assessment of the first and the last implications from the list above, we conclude each to be invalid. A(ab) < E(ab), is invalid because, as Chart 1.2 shows, E(ab) is not true every time A(ab) is true; E(ab) is true once in the fifth case; A(ab) is true twice in the first and second cases. O(ab) < I(ab), is invalid, since I(ab) is true four times (cases 1, 2, 3, and 4), while O(ab) is true three times (cases 3, 4, and 5), and for a valid inference the form of the conclusion must be true every time the form of the premise is true. 1.2 through 1.7 should receive similar treatment as shown with the first and last.

The list of implications above (1.1-1.8) are in the First Figure. Reordering the terms of the consequent produces another set of implications said to be in the Second Figure. There are only two figures for immediate inferences. In 1.1, the conclusion E(ab) is in the First Figure; E(ba) arrangement is Second Figure. It may be expressed the other way around: E(ba) is in the First Figure; E(ab) is in the Second Figure. The list of implications below are in the Second Figure. It should be an easy task to determine the validity of each using Chart 1.2 information.

Chart 2.5: Test of Implications (2^nd Figure)

Other Immediate Inferences

Definitions of the three immediate inferences that follow are clear and simple and, for these reasons, should be committed to memory. The tests provided by Chart 1.2 are available for those who think it difficult to memorize definitions. The definitions for conversion, obversion, and contraposition follow.

Conversion

The converse of a proposition is formed by interchanging the subject and predicate of the E Form and the I Form. The A Form is a special case, described below and the O Form has no converse.

Chart 2.6: Conversions (Valid & Invalid)

**special case

3.1 and 3.2 are valid on the basis of the application of the definition for valid inference and the information provided in Chart 1.2. The consequents are true every time the antecedents are, as Chart 1.2 will show. A(ab) < I(ba) is known as conversion per accidens. It is a valid implication since I(ab) is true every time A(ab) is true. The implication in 3.4 is invalid because the consequent is not true every time the antecedent is true.

To illustrate each of the above. The E Form, No Christians are atheists has a valid converse No atheists are Christian. The I Form, Some men are believers converts to Some believers are men. The familiar A Form All men are mortal converts by limitation to the I Form Some mortals are men. The O form Some vegetables are not carrots (True) has no converse, since to permit it yields the false consequent Some carrots are not vegetables.

Note also that the distribution of the subject and predicate terms is unchanged in conversion of E Form and I Form propositions. The A Form to I Form conversion is permitted by limiting or reducing the distribution of the subject term of the A Form where the subject term is Distributed to Undistributed subject term in the I Form. Moreover, we know that A(ab) implies I(ab) through the square of opposition's subalternation. We also know that I (ab) implies I(ba) by conversion. Therefore, A(ab) implies I(ba). Follow? Finally, notice that to permit conversion for the O Form changes the distribution of both terms, thus changing the meanings of the original proposition. It is true that Some mortals are not men (cats and dogs, for examples), but to assert that it follows that Some men are not mortal is false. The distribution of the terms have undergone a complete reversal: mortals from undistributed in the first to distributed in the second, and men from distributed in the first to undistributed in the second.

Obversion

To obvert a proposition, change the quality of the proposition and replace the predicate by its complement. Each of the four forms has an obverse. Before showing these, a comment about the notion of complement.

A term and its complement (some use "contradiction" in place of "complement") are said to exhaust the universe of objects. Thus, if "a" stands for the class of objects, its complement class, non-a, contains everything else not included in the class a. The whole, a and non-a totally exhausts the universe of entities, since everything in the universe must fall either into one class or its complement class. The complement or contradictory term of a class is symbolized by the use of the prime character ( ' ). Thus, " aa' " means the class "a" and its complement or contradictory, a-prime, or "a'." It is customary to use the word "non" with the declarative statements and the symbol " ' " with the forms when symbolized.

Chart 2.7: Obversion, Valid Implications

Every form has an a valid obverse. The A Form obverts to an E Form and vice versa. The I Form obverts to an O Form and vice versa. The definition of obversion proves to be a superior criterion for validity than the use of Chart 1.2. To make use of the chart entails labeling all of the complement classes for every term in each of the diagrams, an awkward and somewhat complicated, although workable solution. The more practical method is to simply make use of the definition of obversion. (A similar recommendation will apply with the last inference in this section: contraposition.) Some examples of obversion for each form follow.

Chart 2.7a: Obversion Examples

The reverse of each of the above is valid as well. When you obvert each of the obverses above, one has the original proposition. The definition of obversion must be applied consistently and completely in order to have a valid obverse. Also, there are a number of other combinations possible. To cite one: "Some non-Christians are not immoral." The obverse is: "Some non-Christians are moral." One more: "If some things are not excusable, then some things are inexcusable."

Contraposition

The contrapositive of a proposition is one in which the complement terms of the subject and predicate are interchanged. Contraposition can be thought as first obverting the original, converting the resulting proposition, then obverting the resultant one. The A Form and the O Form have straightforward contrapositives. The E Form is a special case. There is no contrapositive for the I Form.

Chart 2.8: Contraposition, Valid & Invalid Implications

Before providing the reader with some examples, consider 5.3 and 5.4 above. As mentioned earlier, contraposition can be thought of as first obversion of the original, then converting the result, then obverting again. So, E(ab) obverts to A(ab') which (being an A Form) converts by limitation to I(b'a). Applying obversion once again yields O(b'a'). The critical step is the conversion per accidens of the A Form. Follow a similar process with 5.4. I(ab) < O(ab') by obversion. The resulting obverse, O(ab'), has no converse, for O Form propositions have no converses. The process aborts, therefore, I Form propositions have no contrapositives.

Chart 2.8a: Contraposition Examples

Check out the last example. Does Some human being are irreverent mean that Some reverent persons are non-human? Hardly! The I form has no contrapositive since to permit it warrants drawing a false proposition from a true one.

Each of the operations of conversion, obversion, and contraposition refer to the forms in the Original Column of Chart 2.3.

Chart 2.9: Summary of Three Immediate Inferences

*Special Cases

The definitions of obversion and contraposition serve admirably. As previously suggested, the use of Chart 2.1 for obversion and contraposition, is impractical and unnecessary given the simplicity of the definitions. Memorize the definitions. Nevertheless, all of the immediate inferences can be tested using the Chart 2.1. Should the student be inclined to carry out these procedures, it should be noted that for each case, everything outside of circle a is a'; everything outside of circle b is b', and vice versa. These additional notations must be filled in the 5 sets of circles correctly for accurate results.

Three Additional Inferences

The remaining immediate inferences, reflexive, symmetrical, and transitive, apply to relationships, like "is greater than," or "is less than," when speaking of numbers or quantities. One or more may apply to other types of relationships; for example, family relationships, "the son of" or "the sister of," and so forth. Some relationships exhibit one or more; some none of these. They are included here for the sake of completeness.

Summation

The wealth of immediate inferences available from a set of four standard form propositions may come as a surprise to some. Memorizing and understanding the various definitions is essential which when applied correctly will distinguish valid from invalid inferences. Here is an opportunity to apply what has been explained. Consider what valid inferences follow from this passage?

The related I form is true by subalternation; the E form is false by contraries; the contradictory O form is false. The valid converse per accidens is an I Form. It has a valid obverse and contrapositive. The importance of definitions and the impressive power of the logic of immediate inference should be obvious.

We turn in the next study to the power of mediated inference, the syllogism.

Review

Try the Exercises for Lesson Two

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