What Is Modus Tollens?


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?”

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What Is Modus Ponens?


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

The Latin phrase ‘modus ponens‘, translated literally, means ‘mode of affirming’.

Shown schematically, this form of argument looks like this:

Premise 1: If A then B.

Premise 2: A.

Conclusion: Therefore, B.

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 is a dog.

Conclusion: Therefore, Casey has four legs.

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.

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

“Always infer B from the conjunction of two premises, if one premise is a conditional statement of the form ‘If A, then B,’ and the other premise affirms A.” (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 Modus Tollens?”

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