How does Transposition differ from Contraposition?

Question

_{Source: A Concise Introduction to Logic (12 Ed, 2014) by Patrick Hurley}

[p 226:] [...] contraposition requires two steps: (1) switching the subject and predicate terms and (2) replacing the subject and predicate terms with their term complements.

[...] Given Statement Form: All A are B.     Contrapositive: All non-B are non-A.

[p 424:] Transposition (Trans):  p ⊃ q ::  ∼q ⊃ ∼p

Google revealed only this Wikipedia article, but which is poorly written; for example, it uses pronouns ambiguously and fails to disambiguate the pronoun's antecedents.

Answer 1

Within aristotelean logic, the difference is that a contrapositive is a categorical, rather than a hypothetical. As your example shows, the contrapositive of the categorical "all A are B" is the categorical "all non-B are non-A". The term transposition is reserved for hypotheticals, so "if A then B" transposes to "if not B then not A".

In predicate logic, this distinction is unimportant, because propositions such as "all A are B" are interpreted hypothetically to mean "for any x, if x is an A then x is a B". This is an important departure from its treatment in aristotelean logic, because it means that "all A are B" has no existential import, i.e. it does not assume that any A exist. In consequence, "all A are B" is trivially true if there are no A, and "all A are B" does not entail "some A are B".

Answer 2

Some historical comments on logical terminology.

In Aristotle's logic there is neither contraposition nor transposition.

A.'s logic is not propositional (while Stoics logic is; see Ancient Logic).

The mode of inference used by A is conversion (antistrophê; see Prior Analytics, I25a1) :

a categorical sentence is converted by interchanging its terms. Aristotle recognizes and establishes three conversion rules: ‘from AeB infer BeA’; ‘from AiB infer BiA’ and ‘from AaB infer BiA’. [During the Middle Ages, the first rule was called an accidental (per accidens) conversion and the last two simple (simpliciter) conversions.] Particular negative sentences do not convert, according to Aristotle.

In particular, ‘from AaB infer BiA’ means :

"from ‘A belongs to all B’ [i.e.Every B is A], infer ‘B belongs to some A’ [i.e.Some A is B].

This principle is not valid in modern logic; see existential import.

Medieval logician added the principle of “conversion by contraposition”; it states that:

‘Every S is P’ is equivalent to ‘Every non-P is non-S’

and

‘Some S is not P’ is equivalent to ‘Some non-P is not non-S’.

This principle is not endorsed by Aristotle.

According to A.Church, “The History of the Question of Existential Import of Categorical Propositions” (1965), there are some apparent instances of contraposition endorsed by Aristotle.

See Pr.An. II,53b12:

If it is necessary that B should be when A is, it is necessary that A should not be when B is not.

and also II,57b1 :

when two things are so related to one another, that if the one is, the other necessarily is, then if the latter is not, the former will not be either.

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Whitehead and Russell, in Principia Mathematica named Transposition the propositional law :

The principle of transposition, i.e. "if p implies q, then not-q implies not-p," and vice versa: this principle has various forms, namely

(*4.1) ⊢ : p ⊃ q . ≡ . ~q ⊃ ~p.

The principle is a modern version of the Stoic mode of argument :

If the 1st, the 2nd.

But not: the 2nd.

Therefore not: the 1st.