Modus Tollens
Modus tollens is a rule of logical inference. We use rules of inference when we argue from a set of premises to a conclusion, for example when we prove that the Pythagorean Theorem follows from Euclid's Axioms. Roughly, modus tollens means "way of removing," alluding to the fact that one of the relevant premises is negated or "removed."
Faced with propositions of certain forms, a rule of inference tells you how to logically combine them to derive valid conclusions. In particular, modus tollens tells you what propositions you can derive from a conditional proposition and the negation of its consequent. A conditional is a proposition of the form "if A then B." A conditional is made up of two parts, the antecedent, the part preceded by if, i.e., "A", and the consequent, the part followed by "then," i.e., "B." The negation of the consequent would be "not-B." By modus tollens, if a conditional is true and the negation of its consequent is true, then the negation of the antecedent must be true as well.
Modus tollens can be best illustrated by an example. Consider the conditional "If Timothy is a cat, then Timothy is an animal." Suppose you know that Timothy is not an animal. Then obviously, Timothy cannot possibly be a cat.
An informal proof would be to assume for the sake of argument that Timothy is a cat after all (this strategy is known as proof by contradiction). Then by the conditional, since Timothy is a cat, he must also be an animal. But we had already ruled out his membership in the animal kingdom! This is a contradiction. We can conclude that our assumption must be false: Timothy is not a cat. Essentially, modus tollensis a rule that we use as a shortcut for this pattern of reasoning.
One very common mistake is to confuse and scramble modus tollens and modus ponens Quite often, one encounters an argument that starts with a conditional and its consequent, and concludes that the antecedent must be true. This is a grave error that can be avoided by keeping in mind another Timothy example. Suppose we have again the conditional "If Timothy is a cat, then Timothy is an animal." Suppose also that we know that the consequent is true: "Timothy is an animal." Absent any other information, it would be unreasonable to conclude that Timothy is a cat. Timothy could be anything: a dog, a starfish, or a hippopotamus. The conclusion "Timothy is a cat" is invalid as long as no further evidence is brought to bear. This erroneous pattern of reasoning is sometimes referred to as modus moron: "the way of the fool."
Systems of logic (there are many different logics), like other areas of mathematics, are constructed by choosing a minimal number of simple, fundamental axioms whose truth is considered beyond question. However, given only axioms, one cannot derive any other true statements unless one also accepts the validity of a rule of inference as axiomatic. Using such a rule one can then derive other propositions from the axioms, and if the rule is followed accurately, these new statements are called theorems. Other rules of inference can then be derived and considered theorems of that logic.
In principal, when logicians construct systems of logic, they could choose modus tollens as their first, unquestionable rule of inference, using it and their postulated axioms to derive theorems as well as other rules of inference such as transitivity or modus ponens. Usually, however, logicians use rules that are more immediately obvious such as modus ponens or proof by contradiction, and then derive modus tollens.
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