Notes on Logic:

A common criticism of university-required courses in logic and mathematics is their lack of practical examples in the traditional course of academic problem-solving. Current logic textbooks used at most universities limit examples to the "All men are mortal" types of examples.

This approach assumes that if students can solve problems in traditional logic or symbolic logic involving simple examples, then students ought to be able to solve much more complex linguistic examples by following the same principles. Such a result is rare, though, because translating from everyday language into some kind of standard-form is difficult and time-consuming.

On a recent classroom test on the topic of the square of opposition, in addition to answering correctly the usual examples of problems like ...

State the contradictory and resultant truth-value of the true statement 'No arctic flora are bromeliads'

most students also answered correctly problems involving successive inferences, if the questions were posed using standard-form categorical statements.

But if the questions involved non-standard-form statements, most students were lost, even though they did well in translating textbook-type ordinary language examples into standard-form.

For example, when the following question was posed

Translate the following statements in to standard form: Any strain imposed on the mind will be reflected in the eyes, and similarly [therefore] anything which rests the mind will benefit them.¹

all students could provide a translation, but their translations did not allow them to evaluate the inference stated since the subjects and the predicates chosen in the different statements had no obvious logical relation to each other.

The objective sought in asking the question was for the student to recognize in order to evaluate the inference using suitable paraphrases of the subjects and predicates in each of the original statements, the translation should result in either the same classes or classes involving complements of each other. One way to do such a translation would be to use the complementary classes as shown here:

(1) All strains imposed on the mind are non-benefits to the eyes.
(2) All non-strains imposed on the mind are benefits to the eyes.

The conjunctive adverb "similarly" in the original quotation permits the looseness of the rather liberal liberal translation above. Once the translation is accomplished, then it is a simple matter to see that by first contraposing and second converting the first statement in order to obtain the second statement, an undetermined truth-value results. (For those unfamiliar with successive immediate inferences, this process is shown in detail as problem number 2 on this Webpage)



Subsequently, as a take-home problem, I asked the class to show formally, by using the Square of Opposition, that the following argument by Joseph Butler was correct:

[T]here is generally thought to be some peculiar kind of contrariety between self-love and the love of our neighbour, between the pursuit of public and of private good; insomuch that when you are recommending one of these, you are supposed to be speaking against the other; and from hence arises a secret prejudice against, and frequently open scorn of, all talk of public spirit, and real good-will to our fellow-creatures ...

Let us now see whether there be any peculiar contrariety between the respective courses of life which these affections lead to; whether there be any greater competition between the pursuit of private and of public good, than between any other particular pursuits and that of private good. There seems no other reason to suspect that there is any such peculiar contrariety.²

Again, the major difficulty is choosing or inventing a suitable paraphrase in standard form of the specific contrariety Butler is supposing. Statements are contraries if the first statement being true implies that the second statement is false and if the first statement being false does not imply the second statement is true or false.

What seems puzzling about this example is that if the statements are translated into the forms "All S is P" and "No S is P, then they would by definition have to be formal contraries. Even so, I was surprised that no student submitted a paper to show Butler's reasoning is correct.

How is this to be shown with traditional logic? Let's try the following translations:

(1) No pursuits of private good by those who have self-love are quests for public good by those who love their neighbor.
(2) All pursuits of private good by those who have self-love are quests for public good by those who love their neighbor.

We can say if statement (1) above is true, statement (2) must be false and vice-versa. This fact is not, however, what Butler is denying. Instead, Butler is denying that statement (1) is true. In point of fact, he is not overtly asserting either statement (1) or (2).

What Butler is asserting is ...

Some pursuits of private good by self-loving individuals are quests for public good by those who love their neighbor

is true, presumably because he recognizes that some actions done in self-interest also by their very nature help others. So in sum, by asserting that the pursuit of public and private good are not contraries, Butler is specifically asserting that pursuits of private and public good are not mutually exclusive activities.

When explained in this fashion, the students state that that analysis is obvious, but the question asked was how to prove statement are not contraries using the principles of traditional logic? And, of course, they are correct, the analysis doesn't use the logical relation of contrariety for its explantion, it explains why traditional logic is not used in such an example.

Posing clear problems in ordinary language for introductory logic is a difficult undertaking, but if beginning logic is to be useful for other endeavors, then much more attention by texts and instruction need be directed to it.
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¹C.S. Price, The Improvement of Sight (Cleveland: Sherwood Press, 1946), vi.
²Joseph Butler, ``Sermon XI---Upon the Love of Our Neighbour,'' in Fifteen Sermons Preached at the Rolls Chapel, 2nd. ed. 1729.