# What Is Modus Tollens?

February 25, 2011 7 Comments

*Modus tollens* is a valid argument form. Because the form is deductive and has two premises and a conclusion, *modus tollens* is an example of a syllogism. (A syllogism is any deductive argument with two premises and a conclusion.)

The Latin phrase ‘*modus tollens*‘, translated literally, means ‘mode of denying’.

Shown schematically, this form of argument looks like this:

Premise 1: If *A* then *B*.

Premise 2: Not-*B*.

Conclusion: Therefore, not-*A*.

Arguments of this form are produced by substituting statements in English for *A* and for *B*. For example, suppose *A* = ‘Casey is a dog’ and *B* = ‘Casey has four legs.’ We can substitute as follows, for a valid argument:

Premise 1: If Casey is a dog, then Casey has four legs.

Premise 2: Casey does not have four legs.

Conclusion: Therefore, Casey is not a dog.

Any argument of this form is valid. But not every argument of this form is *sound*. For an argument to be sound, it must meet two requirements. First, it must be valid; second, it must have true premises. The above argument about Casey is valid, but it’s not sound. Why? Because the first premise is false. It implies that all dogs have four legs. But this generalization, unfortunately, is not true. (It also turns out that Casey does have four legs; so premise 2 is false, also.)

Because *modus tollens* arguments are always valid, we may extrapolate from this argument form a **rule of inference** as follows:

“Always infer not-*A* from the conjunction of two premises, if one premise is a conditional statement of the form ‘If *A*, then *B*,’ and the other premise denies *B*.” (The order of the premises doesn’t matter.)

**Caveat:**

Be careful not to confuse *modus ponens* with *modus tolendo ponens*. *Modus tolendo ponens* is an argument of the following form:

Premise 1: Either *A* or *B*.

Premise 2: Not-*A*.

Therefore, *B.*

See also “What Is *“*What Is *Modus Ponens*?”

How about this?

Two men were arguing over the price of a second-hand car. The owner said: “Look, either you buy it or you don’t. If you want to buy it, this is my final price. If you don’t, then just leave and stop wasting my time. So agree to the price or start walking.”

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I see your point, David. Your suggestion seems a good one, since that is, after all, what is assumed by the consequent in the conditional.

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Doug,

I’m mean being “cheated” by using a conditional that sounds true in ordinary language, when actually the conditional is false when interpreted in the logical sense. For example.

if x, then y, in ordinary language is often taken to mean “If x is true, then y is often true as well” instead of the logical sense of “x is always true when y is true.”I was looking for a rule of thumb to help critical thinkers (without any background in philosophy) to evaluate arguments. The only one I can think of is putting “in all cases” in front of it and checking if that looks correct.

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Hi David,

Your example argument is valid. But the argument isn’t sound. This is because, as you say, the conditional statement is false. So I’m not sure what you mean by being “cheated” by such an argument.

-Doug

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Hi Matt,

This is actually disputed. I reserve the right to stipulate, along with various philosophers and logicians, that syllogisms are deductive arguments.

Consider this definition from the

Encyclopedia of Philosophy, 2nd ed.:. “A valid deductive argument having two premises and a conclusion.” —Baruch A. Brody, “Logical Terms, Glossary of,” insyllogismEncyclopedia of Philosophy,” 2nd ed., ed. Donald M. Borchert (Detroit: Thomson Gale, 2006), vol. 5, p. 557.Monroe Beardsley, writing in 1950, said, “A syllogism is a simple kind of deductive argument.” He then added, “The study of syllogisms is the heart of what is usually called ‘Aristotelian logic,’ to distinguish the work of Aristotle (who was the first systematic logician, and who first formulated the rules of the syllogism) from the work of modern and contemporary logicians.” (See Beardsley,

Practical Logic[New York: Prentice-Hall, 1950], p. 316.)Beardsley offered a more precise account of the syllogism, as follows:

“The essential ingredients . . ., logically speaking, are of two kinds: (1) The argument consists of exactly

threeatomic statements, two of which are premises and one the conclusion. (2) It contains exactlythreedistinct terms, each of which appears exactly twice (but not twice in the same statement). Any argument that contains these ingredients is a syllogism.” (Beardsley 1950, p. 316)This more precise account is rooted in Aristotle’s logic of the syllogism. It states necessary and sufficient conditions for a syllogism. It must be deductive, and it must consist of three statements, two of which are the premise and the third of which is the conclusion. But Beardsley goes further. A syllogism, in the original sense, must have three distinct terms. By this he means a “major term,” a “minor term,” and a “middle term.” The atomic statement that contains the major term is called the “major premise”; the atomic statement that contains the minor term is called the “minor premise.” Both premises contain the middle term, which provides the essential link between the two premises.

This is consistent with Richard Purtill’s definition: “A syllogism is an argument consisting of three standard form statements, two of which are the premises of the argument and the other of which is the conclusion of the argument. By definition, a syllogism has three terms. . . .” (Purtill,

Logic for Philosophers, [New York, Harper and Row, 1971], p. 97.)On this account, no inductive arguments are syllogisms. But neither are

somedeductive arguments.Modus ponensandmodus tollensarguments contain only two terms. So, holding strictly to the Aristotelian conception of a syllogism,modus ponensandmodus tollensarguments are what you might call “pseudo-syllogisms.” They are deductive arguments with three statements, two of which imply the third. But they don’t meet the other requirements.During the course of time, deductive arguments having two premises and a conclusion have come, more generally, to enjoy the honorific title of “syllogism.” (See Howard Kahane,

Logic and Contemporary Rhetoric, 6th ed. [Belmont, CA: Wadsworth, 1992], pp. 291-296.) This is evidenced in Brody’s definition of “syllogism” (see above). Following his general definition of “syllogism,” he adds, “The term is often restricted to the case where both premises and the conclusion are categorical propositions that have between them three, and only three, terms.” He suggests that “more careful authors distinguish this case by referring to is as acategoricalsyllogism.” I concur with this. There are, in addition to the categorical syllogism, thehypotheticalsyllogism and thedisjunctivesyllogism.Modus ponensandmodus tollensare forms of hypothetical syllogism.There is, however, no convincing precedent for dubbing a deductive argument, even if it has but two premises and a conclusion, a “syllogism.” That would be stretching its sense too far.

The best brief statement of how the term “syllogism” is employed, both broadly so as to include hypothetical and disjunctive syllogisms, and more narrowly so as to be restricted to categorical syllogisms, is in

Talking Philosophy: A Wordbook, by A. W. Sparkes (London: Routledge, 1991, 170-172). Though conciliatory toward those who adopt the broader use (because the habit is so entrenched), Sparkes favors the narrower conception.LikeLike

Doug,

Not all syllogisms are deductive. There are inductive syllogisms too, e.g., http://en.wikipedia.org/wiki/Statistical_syllogism. If this is right, then a syllogism is just a two-premise argument, be it deductive or inductive.

Blessings,

Matt

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Do you know of any rule of thumb for verifying that a modus tollens argument hasn’t “cheated” us by using false conditional (valid, but not sound).

For example;

If there is smoke, there is fire. There is not fire, so there is no smoke.I know this example is silly because in normal life we wouldn’t give an argument for the presence of smoke. But perhaps one can test these conditions by putting “In all cases” in front of the conditional?

(in all cases) If there is smoke, there is fire. There is not fire, so there is no smoke.I suppose the problem with conditionals is they don’t always match up with our ordinary usage.

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