Notes on Logic:

I want to propose that inductive reasoning is essential for understanding the proper role for the application of deductive logic in the analysis of everyday contexts. I take deductive logic as involving arguments whose conclusions follow with necessity, whereas inductive logic involves arguments whose conclusions follow with some degree of probability. Usually introductory logic classes emphasize the study of those arguments whose premises are claimed to conclusively entail their conclusions, that is, the study of deductive arguments. (I recognize that in fields of study outside of logic, such as economics, English, biology, and so forth, many textbooks define deductive arguments as those where the premises are general and the conclusion is particular. And inductive arguments are defined in those sources as particular premises leading to general conclusions. There are substantial difficulties with these definitions.)

The efficacy of learning mathematics and symbolic logic for everyday life seems to be questioned most by those persons actually engaged in those studies. In my experience, the impulse to structure such courses to be as comprehensive and complete as possible tends to result in the subsequent presentation of abstract principles and simplified examples not only leaving the subject matter joyless but also leaving the subject inapplicable and irrelevant to everyday concerns. The student in these kinds of courses finds ordinary applications of logic and mathematics extraordinarily difficult because the familiar routines of everyday life are rarely related to the kind of over-simplified examples present in most logic and mathematics textbooks.

To work out realistic everyday-encountered problems by applying general principles learned in basic mathematics and symbolic logic classes is often time-consuming and refractory not only because of the vagaries of everyday events but also because of the complexities of their ordinary language descriptions. I think these same comments equally apply to the study of the principles of traditional logic.

Nevertheless, I think that even if more attention were to be placed on the analysis and evaluation of practical logic examples and exercises, the basic logic course still might not be particularly helpful for judging the reasoning in other subjects or for assessment of everyday discourse.

Many logic teachers avoid traditional logic altogether and, instead, teach symbolic or mathematical logic because of its structure and elegance. Here, the hope is that rigorous practice with deductive symbolic manipulation will somehow trickle down to effect practical logical abilities in a manner similar to the historically claimed benefits of the study of proofs in Euclidean geometry.

For the study of geometry, the benefits claimed range all the way from the giving insight into the nature of things to the refinement and application of critical thinking skills. For example, Professor P. Bursill-Hall notes in his lecture at University of Cambridge:

Studying geometry reveals--in some way--the deepest true essence of the physical world. And teaching geometry trains the mind in clear and rigorous thinking.

P. Bursill-Hall, "Why do we study geometry?" (19 April 2008)
http://www.dpmms.cam.ac.uk/~piers/F-I-G_opening_ppr.pdf

And Professor Barry Simon of the California Institute of Technology writes:

Modern mathematicians don't use the two-column proofs so beloved by my high school geometry teachers, and real life rarely needs the precise rigor of mathematicians, but those who have survived those darned dual columns understand something about argumentation and logic. They can more readily see through the faulty reasoning so often presented in the media and by politicians.

Barry Simon, "Why study Euclidean Geometry," (19 April 2008)
http://www.solomonovich.com/geometry/whystudy.html

I admit that I haven't found any studies supporting these claims, but I also admit I haven't researched the subject in any depth. Nevertheless, these claims do not fit my experience in teaching elementary traditional logic to freshman and sophomore college students.

I find that when I include examples from everyday life on logic tests without having introduced these examples in class work, students rarely know how to approach them. This is the case even though the test examples are carefully chosen for their close relevance to the extensively worked, but simplified, exercises assigned.

By way of example, I'd like to cite two questions from a test given recently to students on the subject of immediate inferences in an introductory logic course.

In accordance with Square of Opposition in traditional logic, the subcontrary logical relation holds between two statements of the form "Some S is P" and Some S is not P." In keeping with the textbook, how to translate from ordinary language into standard-form statements is studied. (For example, the statement, "It is not uncommon for a musician to have perfect pitch" can be translated for clarity into "Some musicians are persons with perfect pitch.")

The students learn that if a statement of the form "Some S is P" is known to be true, then it does not deductively follow that "Some S is not P" is true" since sometimes when we state the former statement we have no information concerning all things referred to by the subject.

For example, if I learn that some students in the class have done their homework, I cannot infer that some students have not done their homework because it may well be that all students have done their homework. From incomplete information about the students, I cannot deductively infer anything is true about all of the students.Even though the students were well aware that the subcontrary logical relation is that both statements involved could both be true but both statements cannot be false, every student in the class missed the following question taken from an article written by Irving Kristol in the Wall St. Journal thirty-five years ago:

Evaluate the following reasoning present in the following passage. Is Mr. Kristol's reasoning deductively certain? "This (a study showing all the officers in a precinct in New York are honest), however, is rather like saying that the majority of New York City's police officers are honest and honorable men. Of course they are. But the statement itself implies that a not altogether insignificant minority are less than that, and the presence of such a minority is fairly taken to constitute a rather serious problem."

The students concluded that Kristol's reasoning was entirely correct. Yet, no student who responded to the problem related the reasoning to the logical relation of subcontrary. From a logical point of view, the conclusion that some police officers are not honest does not logically follow, and consequently the student's answers were deemed to be mistaken because, given the premises, it is possible that all police officers in New York are honest. The question I suggest here is whether or not the evaluation of arguments should be restricted to given premises when it should be noted that it more reasonable to conclude that some police officers are probably not honest given other inductive evidence. The point of the example is just that the subcontrary deduction does not prove anything--Kristol is mistaken in supposing that it does. Nevertheless, it is clear that he conclusion is probably correct on other grounds.

A second example was to evaluate whether or not C. S. Price's use of the word "similarly" could correctly indicate that a resulting independent clause is the conclusion of a valid inference:

C.S. Price in his The Improvement of Sight writes, "Any strain imposed on the mind will be reflected in the eyes, and similarly anything which rests the mind will benefit them."

Again, not one of thirty students recognized this inference as a sequence of contraposition and (invalid) conversion of an A statement. Without the benefit of studying ordinary language translations in class or exercises, it is evident to me that the students' ability to relate their study to everyday life and reason logically when faced with the application of logical principles to ordinary, everyday examples is quite limited because of the oddity of evaluating the inference in isolation from its context. The inference the students are making could probably be substantiated on empirical evidence that mental strains do indeed affect vision as well as empirical evidence that eye strain affects the mind. How useful, then, are practical examples when only approached deductively in isolation from their inductive contexts?

Consequently, I'm inclined to suspect that the aesthetic appeal of studying only complete and abstract principles of deduction should not be a central concern in elementary logic, but rather, when need be, comprehensiveness of these topics should be sacrificed for relevance and usability of logical techniques in ordinary everyday situations. The use of inductive logic is far more common in everyday life than that of deduction. And I suggest here that the study of deductive arguments should be studied in their inductive contexts since deductive arguments are so rarely used in everyday life.