5.5: FURTHER IMMEDIATE INFERENCES

READ Copi/Cohen, 193-200
 

N.B. It is essential to know the meaning of "quality" and "quantity" as said of statements in order to understand the operations in immediate inference. Also I will be distinguishing the process which generates an immediate inference from the validity of that process. For example, the process for conversion is to flip subject and predicate terms; this process is only valid when you start with an E or an I statements.
 

When these immediate inferences are valid, then the two statements--the premise one starts with and the statement one generates--are called logical equivalences and therefore they always have the same truth value as each other. Thus when I start with a false E and generate its converse, then that converse is also false since such a converse is from a valid inference.
 

CONVERSION

20. [Copi/Cohen, 194] The table at the bottom of the page is a good summary of CONVERSION: the converse of any statement is formed by flipping the subject and the predicate terms--this is the process for forming a converse. Appying this process to a statement (the "convertend") is separate from the determination that the process generates a valid immediate inference. (Note: for our purposes, it is not necessary to know the term "convertend.") A summary: the E and I convert truly and simply--and that means they are valid immediate inferences. Neither the A nor the O can produce valid converses, although one can write statements that are their "converses" even though these are not valid immediate inferences. Converses that are valid inferences are then logical equivalences and have the same truth value.

There is a valid immediate inference for an A that converts "by limitation"--that is, by changing the quantity as well as flipping the positions of the terms. However, this converse by limitation IS NOT a logical equivalence and is not a real converse. For the O, however, there is no such valid converse by limitation.

OBVERSION

21. [Copi/Cohen, 195] The discussion on this page is a good example of one of the ways in which Aristotelian logic is metaphysical. Metaphysics is the branch or area of philosophy which describes the nature of reality using general or abstract terms. In this case, the relevant term is "class." A class is described and defined very broadly here--all objects having a common attribute, or attributes, constitute a class. Note also Copi/Cohen's example of a complex class: the class of all left-handed, red-headed students. This is a smaller class than the class of all students, but it is a perfectly legitimate example of a class. It is essential to know something about classes in order to understand obversion as a type of immediate inference, since obversion depends on the notion of "complement" to the class designated by the predicate term.
 

22. [Copi/Cohen, 195] The distinction between a complement referring to classes and a complement referring to terms for classes is important to recognize, but it will not be relevant to the type of exercises we will be doing with immediate inference.
 

23. [Copi/Cohen, 195] The point about the complement of the complement of a class--e.g., non-S--bears repeating. Thus the complement of "non-S" is not "nonnon-S," but simply "S." The two "nons" in "nonnon-S' cancel each other leaving only the class "S." One way to simplify the process for forming a complement-term is that one adds or subtracts a "negative" (such as 'non' before an adjective or noun or a 'not' in a clause or phrase modifying the term for which one is forming a complement). The complement of 'nonsmokers' is 'smokers', for 'people who don't laugh' it is 'people who do laugh'; and for 'trees without leaves' it's 'trees with leaves.' Also, keep in mind that we usually have several ways in English to say the same thing: 'smokers' or 'people who smoke'.
 

N. B. THIS IS VERY IMPORTANT! Do not confuse adding a "non" to a term with adding a "not" after the verb of the full statement. "Non" goes only with classes or their terms and establishes the complement of these classes or their terms. "Not" added to the statement's copula negates the statement (which contains two terms and thus speaks of two classes) and therefore operates on the statement as a whole. Thus the statement "Some nonpacifists are nonsocialists" is an AFFIRMATIVE statement, even though it has a pair of 'nons' in it. These two 'nons' negate their corresponding classes but as such they have nothing at all to do with the QUALITY of the statement, which is a characteristic of the statement as a whole.
 

24. [Copi/Cohen, 197] As with CONVERSION, the table on this page provides a useful summary of OBVERSION. (Note: For our purposes, it is not necessary to know the term "obvertend."] NOTE again the difference between process and validity. The process for conversion is to change the quality of the statement and then substitute the complement of the predicate term for the original predicate term. This process yields a valid immediate inference for all types of statements (A, E, I, O) and thus obverses are logical equivalences and have the same truth value.
 

CONTRAPOSITION

25. [Copi/Cohen, 199] As with conversion and obversion, the table on this page provides a useful summary of CONTRAPOSITION--although it does not show invalid contrapositives. A contrapositive can be formed for all types of categorical statements; it is valid for A and O propositions, but not for E and I ones. The  process for forming a contrapositive is (a) switch the positions of the subject and the predicate terms and (b) substituting complement terms for both of the original terms [actually you can apply these in any order]. Also, remembering that the actual contrapositive of an E statement, we can still, starting with an E statement, form a "contrapositive by limitation" by also changing the quantity of the statement along with the other two changes, (a) and (b)--giving us a valid inference; but this is not a real contrapositive, not a real logical equivalence.
 

26. [Copi/Cohen, 199] The paragraph beginning "Some questions...." contains an important analysis showing how to determine the truth value of "No nonsurgeons are nonphysicians" given that the statement "All surgeons are physicians" is true. This paragraph requires careful reading; however, reviewing the sample analysis given here is very useful for doing Exercises IV, V, VI, and VII (201-202).
 

27. [Copi/Cohen, 200] The diagram is a useful summary of ALL types of immediate inference; but remember that it shows only valid inferences and therefore does not show the process applied to statements that would generate invalid inferences, nor does it clarify which of these valid inferences are true logical equivalences..

EXERCISES: ---ELOGIC: SQ, TRY (5), & WORKSPACE Exercises 5.1 (C/C I, p. 200), 5.2 (C/C II, p. 200) & 5.3 (C/C III, p. 201)
INSTRUCTIONS: [A] Do all the exercises in I, II, and III . Note that all the inferences which will be produced in these exercises are equivalent to the given propositions EXCEPT (1) those which are formally (in form) the converse of an A or O and the contrapositive of an E or I, as Copi/Cohen explains in the text.(and as indicated in the diagram on p. 200) or (2) those which are the converse by limitation and the contrapositive by limitation--these are valid immediate inferences from, but not logically equivalent to, their original/premise statements.

FOR THURSDAY, TRY SOME OF THE FOLLOWING:--ELOGIC: WORKSPACE Exercises Section 5.5 cont., Ex. 5.4-5.7 (C/C,  pp. 201-2),  IV through VII
[B] Do all the exercises in IV , V, VI and VII. Here you must determine the relation between the original statement and each of the numbered ones in turn--and it might take more than one immediate inference--and thus the relation name--to trace this relation, so that then the validity of the inference(s) and the truth value of the final statement can be determined. Some of these relations will be from the Square of Opposition and others from the list of logical equivalences and "by limitation" inferences. For example, the final statement might be the contradiction of the obverse of the original statement. SEE the document  at http://condor.depaul.edu/~mlarrabe/LOGDECIx.htm for more information.
 

As a sample solution, here is one way to think through IV-1. You are given that "All socialists are pacifists" is true. The question: What is the truth value of "Some nonpacifists are not nonsocialists"-- true, false, or undetermined--and what is the justification for claiming that, i.e., what is the logical relation between these statements?
 

(1) Begin by noting all changes from the original statement (the "premise" of the immediate inference) to the (final) conclusion: are there any of these?

(a) switch in the term position [used for converse, contrapositive],

(b) switch to complement term [1 used for obverse, 2 for contrapositive],

(c) switch in quality [used in obverse, some Square relations]

(d) switch in quantity [used in "by limitation" inferences, some Square relations]

In the example, there is one of (a), two of (b) and one of (c) and (d). This information gives us hints to try. In this case there are too many changes for the answer to be only one relation.
 

(2) Now try out a process (apply change criteria) that matches one/some of these:

In the example, I would try a contrapositive first because it matches the one (a) and the two (b) changes:

P1: "All socialists are pacifists" is true.

C1: So, "All nonpacifists are nonsocialists" is ____________ because a valid contrapositive.

Having determined that the process I applied allows a valid inference, I then determine the BLANK, that is the truth value: here it would be true.
 

(3) But C1 is still not the final conclusion of the exercise. I still have left the quantity and quality change. At this point I am checking how C1, now function as P2 for a second immediate inference, relates to the final conclusion:

C1/P2: "All nonpacifists are nonsocialists" is true.

So, FC: "Some nonpacifists are not nonsocialists" is ____________, because ?

Note that the terms are the same between these two statements; so the relation between them is on the Square of Opposition--and the only two changes left to deal with are quantity and quality change. So, what is the relation between an A and an O with the same terms? They are contradictories and hence have opposite truth values. Fill in those blanks and you have traced the immediate inferences from the original premise to the final conclusion--here the latter is the contradictory of the contrapositive of the original premise; it's formed by a string of valid inferences which allow us to determine truth value all along the line.
 

NOTE: In doing these exercises, you might find you get different answers if you try one relation before another the first time and reverse that the next--so that doing it one way gives you UNDETERMINABLE and doing it the other gives you a specific truth value. This can happen; so there can be more than one correct answer to these exercises. However, in no case will correct answers include contradictory ones--that is, true for one way and false for another. Valid deduction cannot yield contradictory results.