# Notes on Logic:

Informal, Deductive, and Inductive

## Should Introductory Logic Courses Be Limited to Deductive Reasoning? Some Observations

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.

## How Tom Tancredo "Divides" a Red Herring while Grinding his Axe

Colorado Congressman Tom Tancredo answers Jeff Young's question, "What evidence would convince you that global warming is a serious threat to the planet?" as follows:

I have no doubt that global warming exists. I just question the cause and what we can do to ameliorate it. But I wonder why the Sierra Club isn't going crazy about the environmental aspects of massive immigration into the U.S. The fact is, Americans consume more energy than anyone else, so if a person moves here from another country, they automatically become bigger polluters.

("10 Questions," Time 169 No. 24 (11 June 2007), 8.)

Congressman Tancredo initially sidesteps Mr. Young's question by granting that global warming is a problem presumably without political import since he thinks the causes and cures of global warming are unknown. Generalizing, however, that the problem of global warming is an environmental issue, Congressman Tancredo implicitly proceeds to assign the cause of global warming to the pollution caused by energy consumption in the U.S. from the increase in population by immigration. (Time's preface to the article notes that Tom Tancredo is "hoping to win over primary voters with his hard-line stance on illegal immigration" (Ibid, 8)).

So in grinding his axe, the Congressman commits both the fallacy of division and the red herring fallacy. The fallacy of division is a fallacy committed by reasoning from the fact that since the whole of something has one or more specific characteristics, the parts must also have one or more specific characteristics as well.

In this case, Mr. Tancredo reasons that since the property of high energy consumption and the consequent property of pollution is true collectively of persons living in the U.S., then these same properties must be true as well for each individual distributively living in the U.S. Since legal immigrants in the U.S. are persons living in the U.S., the Congressman fallaciously concludes these immigrants necessarily become bigger polluters when they cross the border.

The red herring fallacy, of course, occurs in the intentional shift from the problem of global warming to the problem of illegal immigration in the U.S. The red herring fallacy is the kind of ignoratio elenchi or fallacy of irrelevant conclusion in which the ultimate conclusion is a diversion from the original question. The fallacy gets its name from a folklore account of intentionally dragging a herring across the trail of a fox in order to divert the foxhounds.

## As Deeply Emotional Controversy Develops, Hasty Reply Can Involve Complex Fallacies

Professor Felicia Nimue Ackerman, a bioethicist at Brown University, reacts to a reference cited in Time magazine relating to the debate over euthanizing severely disabled newborns as follows:

As a Bioethicist, I have a question about the justification of infanticide by Britain's Royal College of Obstetricians and Gynecology on the grounds that "a very disabled child can mean a disabled family." Why should the College apply this consideration only to disabled infants? Cheating husbands, alcoholic wives, and nagging mothers-in-law are just a few of the many sorts of people who can mean a very disabled family. Why not kill them to?

(Felicia Nimue Ackerman, "Saving an Angel," in "Letters," Time 169 No. 7 (12 February 2007), 20.)

If we take a logical point of view and look only at the adequacy or cogency of the arguments advanced in this controversy, the truth or falsity of the ethical positions being taken are not at issue. The question we are asking is whether or not Professor Ackerman's argument is good reasoning--i.e., whether her argument is valid or correct.

Initially, we can surmise that Professor Ackerman would oppose "active euthanasia" on the grounds that she uses the term "infanticide"--a term which has a considerably different emotive significance from other neutrally appropriate terms. The definition in Shorter Oxford English Dictionary, for instance, of "infanticide" is as follows:

a. The killing of infants, esp. the killing of new-born infants, as a custom among savages, and in the ancient world. b. spec. The crime of murdering an infant after its birth, perpetrated by or with the consent of its parents, esp. the mother."

(William Little, et. al., The Shorter Oxford English Dictionary, 3rd. ed. (Oxford: Oxford University Press), 998.)

(By way of contrast, those opposed to Professor Ackerman's position might use the emotively significant term, "mercy killing." The question as to whether Royal College of Obstetricians and Gynecology's use of term "active euthanasia" is a euphemism, is left open.)

Without access to the article Professor Ackerman comments upon, we might suspect that, if fallacies are committed in her response, they might include the fallacies of (1) slippery slope, (2) accent, and (3) complex question. (Arguably, there could be additional fallacies suspected such as equivocation and straw man.)

Let's take a look at the proposal Professor Ackerman opposes. The Time magazine article describes the case of parents choosing to sterilize and minimize the height of their severely disabled and brain-damaged daughter, Ashley. The parents argue that they are seeking to "improve [their] daughter's quality of life and not to convenience her caregivers." This is the relevant section of the article concerning what is termed there, "the Ashley Treatment":

To warnings of a slippery slope, [Dr. Daniel] Gunther tilts the logic the other way. "The argument that a beneficial treatment should not be used because it might be misused is itself a slippery slope," he says. "If we did not use therapies available because they could be misused, we'd be practicing very little medicine.
Those deploring the Ashley Treatment as a medical fix for more than one family are watching the direction that Britain is taking. The Royal College of Obstetricians and Gynecology has proposed that doctors openly consider allowing euthanasia of the sickest infants, which is legal in the Netherlands. "A very disabled child can mean a disabled family," the college wrote to the Nuffield Council on Bioethics and urged that it "think more radically about nonresucitation, withdrawal of treatment decisions ... and active euthanasia, as they are ways of widening the management options available to the sickest of newborns."

(Nancy Gibbs, "Pillow Angel Ethics," Time, 169 No.4 (22 January 2007), 57.)

(1) It seems evident that Professor Ackerman has overlooked Dr. Gunther's admonition concerning slippery slope. The question raised by the Royal College is to consider the "management options available to the sickest of newborns." To presuppose that questions concerning "cheating husbands, alcoholic wives, and nagging mothers-in-law" are comparable to the questions concerning the fight to preserve the life of a severely disabled and brain-damaged newborn, is indeed to confabulate uses of the term "disabled" and to commit the fallacy of slippery slope. The Royal College is raising the questions of nonresuscitation and active euthanasia with respect to the "sickest of newborns" and the consequent effect on the family--those kinds of disabling effects differ substantially from the sociological effects of "cheating husbands, alcoholic wives, and nagging mothers-in-law."

(2) Britain's Royal College of Obstetricians and Gynecology is not attempting to justify infanticide as Professor Ackerman asserts in her first sentence. To seize on the phrase "disabled family" is to commit the fallacy of accent or quoting out of context.. The College seeks to open the options discussed by medical ethics review boards in debating the issues facing severely disabled newborns. Needless to say, the question-at-issue is not the advisability or inadvisability of euthanasia for nagging mothers-in-law. The quotation being taken out of context is as follows:

A very disabled child can mean a disabled family. If life-shortening and deliberate interventions to kill infants were available, they might have an impact on obstetric decision-making ... We would like the working party to think more radically about non-resuscitation, withdrawal of treatment decisions, the best interests test, and active euthanasia, as they are ways of widening the management options available to the sickest of newborns.

(3) The fallacy of complex question is the fallacy of phrasing a question that, by the way it is worded, assumes something not contextually granted, facts not in evidence, or something not true. What Professor Ackerman assumes is that a "disabled family" resulting from the problems of the most severely disabled and brain-damaged newborn is similar to a "disabled family" resulting from a cheating husband, alcoholic wife, or nagging mother-in-law. The difference is not simply that parents cannot divorce their extremely disabled baby. The difference results from the equivocation involved in the phrase "disabled family."

It's important to note that even though Professor Ackerman's argument is flawed, that finding does not imply that any of the other ethical positions discussed here are, as a result, made more likely to be logically correct. Even so, Professor Ackerman's response is not a helpful contribution to the complex social, religious, medical, scientific, philosophical, and, especially, personal aspects of this intractable problem in bioethics.

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