Contrapositive

From Academic Kids

In predicate logic, the contrapositive (or transposition) of the statement "p implies q" is "not-q implies not-p." A statement and its contrapositive are always logically equivalent, unlike a statement's inverse or its converse.

One can informally convince oneself of this equivalence by examining examples from ordinary English. Consider the statement, "If there is fire here, then there is oxygen here." The contrapositive would be, "If there is no oxygen here, then there is no fire here." If the statement and its contrapositive are indeed logically equivalent, then these sentence should either both be true or both false. But they are indeed both true. (See combustion.) Thus logical equivalence holds, at least in this case.

Note that while a statement is logically equivalent to its contrapositive (where the two statements are both negated and "swapped"), it is not logically equivalent to its converse (with the two statements "swapped", but not negated).

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Proofs of logical equivalence

That a statement and its contrapositive are always logically equivalent can be proven rigorously using formal logic. There are two basic methods for doing this: deriving the equivalence from the axioms of the propositional calculus, or with a truth table.

Via the propositional calculus

The logical equivalence of a statement and its contrapositive is not one of the axioms of the propositional calculus as defined in this work. However, the equivalence can be proved as follows:

First, we can prove that p → q entails 段 → 殆:

  1. Suppose p → q. Assuming this, we can reason as follows:
    1. Suppose 段. Assuming this, we can reason as follows:
      1. Suppose p. Assuming this, we can reason as follows:
        1. q (Modus ponens, lines 1 and 1.1.1)
        2. 段 (Copying from above)
        3. q and 段 (Conjunction introduction)
      2. Since this is a contradiction, then 殆 (Reductio ad absurdum)
    2. Thus 段 → 殆 (Conditional proof)
  2. Thus (p → q) → (段 → 殆) (Conditional proof)

A very similar proof will show that 段 → 殆 also entails p → q. Combined, these facts show the two statements to be logically equivalent.

Via a truth table

Alternatively, logical equivalence can be proved using the following truth table:

p q pq p q qp
T T T F F T
T F F F T F
F T T T F T
F F T T T T

The first two columns can be taken as given. The third follows from the first two by the truth table definition of the logical conditional. The fourth and fifth follow from the first two by negation. The sixth follows from the fourth and fifth, again by the definition of the logical conditional.

Since the third and sixth columns have the same truth values for all values of p and q, the two are logically equivalent.

In Aristotelian logic

In Aristotelian logic (or categorical logic), moreover, categorical propositions can have contrapositives.

  • The contrapositive of "All S is P" is "All P is S." (These are "A" propositions.)
  • The contrapositive of "No S is P" is "No P is S." (These are "E" propositions.)
  • The contrapositive of "Some S is P" is "Some P is S." (These are "I" propositions.)
  • The contrapositive of "Some S is not P" is "Some P is not S." (These are "O" propositions.)

So-called "E" and "I" propositions are logically equivalent to their contrapositives. For example, we can always infer from "no bachelors are women" to "no women are bachelors" (as well as the converse inference) and from "some dogs are flea-bitten animals" to "some flea-bitten animals are dogs" (and conversely).

However, so-called "A" and "O" propositions are not logically equivalent to their contrapositives. For example, from "all violins are musical instruments," we cannot infer "all musical instruments are violins." Similarly, from "some plants are not trees," we cannot infer "some trees are not plants."

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