Contraposition, Partial and Full
Chapter Forty-Eight
Syllabus topic 3.3, "Contrapositive (Partial and Full)"
Pages 227 to 230 of 334
In one line
The partial contrapositive is obtained by obverting a proposition and then converting the result; the full contrapositive is obtained by obverting once more.
In the wording a student can write in an examination: contraposition is the immediate inference in which the contradictory of the original predicate becomes the subject. In the partial contrapositive the predicate is the original subject; in the full contrapositive it is the contradictory of the original subject.
The shape
Partial contrapositive: non-P as subject, S as predicate.
Full contrapositive: non-P as subject, non-S as predicate.
That is what "partial" and "full" mean, exactly as sequence 440 said: partial negates one term, full negates both.
Deriving them for A
Original: All S is P.
Step one, obvert. Change quality, negate the predicate. No S is non-P. An E proposition.
Step two, convert. E converts simply. No non-P is S. An E proposition, and this is the partial contrapositive of A.
Step three, obvert again. Change quality, negate the predicate. All non-P is non-S. An A proposition, and this is the full contrapositive of A.
In words. "All contracts are agreements" gives, partially, "no non-agreement is a contract", and fully, "every non-agreement is a non-contract". Both say: anything that is not an agreement is not a contract.
Notice that the full contrapositive of A is an A, which means contraposition of A loses nothing at all. This is the only eduction that takes a universal to a universal while replacing both terms, and it is why it is worth so much.
Deriving them for E
Original: No S is P.
Step one, obvert. All S is non-P. An A proposition.
Step two, convert. A converts only by limitation. Some non-P is S. An I proposition, and this is the partial contrapositive of E.
Step three, obvert. Some non-P is not non-S. An O proposition, and this is the full contrapositive of E.
In words. "No minor's agreement is enforceable" gives, partially, "some non-enforceable things are minors' agreements".
Notice the loss. A universal has become a particular, because the chain passed through a conversion by limitation. Contraposition of E is therefore possible only in a weakened form, and only on the traditional reading.
Why I has no contrapositive
Original: Some S is P.
Step one, obvert. Some S is not non-P. An O proposition.
Step two, convert. O has no converse, as sequence 450 proved. The chain stops.
So I has no contrapositive at all, and the reason is the same proof that stopped the obverted converse of O.
Deriving them for O
Original: Some S is not P.
Step one, obvert. Some S is non-P. An I proposition.
Step two, convert. I converts simply. Some non-P is S. An I proposition, and this is the partial contrapositive of O.
Contraposition, Partial and Full
Step three, obvert. Some non-P is not non-S. An O proposition, and this is the full contrapositive of O.
The table
| Original | Partial contrapositive | Full contrapositive | Note |
|---|---|---|---|
| A All S is P | E No non-P is S | A All non-P is non-S | Nothing lost |
| E No S is P | I Some non-P is S | O Some non-P is not non-S | By limitation |
| I Some S is P | none | none | Obverse is O, and O has no converse |
| O Some S is not P | I Some non-P is S | O Some non-P is not non-S | Nothing lost |
Why the full contrapositive of A matters so much in law
Because every condition precedent in a statute is applied through it.
A provision says: "All appeals filed within thirty days shall be entertained." That is an A proposition, and read as it stands it tells a registry what to do with appeals filed in time.
The full contrapositive is: "Every appeal not entertained is an appeal not filed within thirty days." Take care with that one: it is the contrapositive of the proposition as stated, and it is doing exactly what the proposition says, no more.
Take a cleaner case. "All valid wills are attested by two witnesses." The full contrapositive is "Everything not attested by two witnesses is not a valid will." That is the form in which the requirement is actually used: nobody ever needs to be told what a valid will is, and everybody needs to know that an unattested document is not one.
The general shape. A rule stating a necessary condition, in the form "all X are Y", is applied by its contrapositive, "whatever is not Y is not X". The rule tells you what the class contains; the contrapositive tells you what to reject. Litigation is almost entirely about rejection.
And the contrast with conversion. Converting "all valid wills are attested by two witnesses" gives "some attested documents are valid wills", which is nearly useless. Contraposing gives the working rule. Two operations on one proposition, and only one of them produces something a registry can apply.
A worked example
Section 25 of the Indian Contract Act 1872 provides that an agreement made without consideration is void, unless it falls within one of three cases.
Reduce the main limb. A proposition: All agreements made without consideration and outside the three excepted cases are void agreements.
Partial contrapositive. Obvert: no such agreement is a non-void agreement. Convert simply, since that is an E: No non-void agreement is an agreement made without consideration and outside the three excepted cases.
Contraposition, Partial and Full
Full contrapositive. Obvert: Every non-void agreement is an agreement not made without consideration or falling within one of the three excepted cases.
Unwind the double negative. An agreement not made without consideration is one made with consideration. So the full contrapositive says: every agreement that is not void either has consideration or falls within one of the three exceptions.
That is the rule a drafter applies. It is a checklist for any agreement that is meant to be enforceable, and it was derived from the section by two mechanical steps with no knowledge of contract law beyond the words of the section.
And it shows what the section does not say. It does not say that every agreement with consideration is valid, since a great many are void for other reasons entirely. That would be the simple converse of the contrapositive, and the rules forbid it.
Distinctions that carry marks
| Partial contrapositive | Full contrapositive | |
|---|---|---|
| Built by | Obvert, then convert | Obvert, convert, obvert |
| Subject | non-P | non-P |
| Predicate | S | non-S |
| For A | No non-P is S, an E | All non-P is non-S, an A |
| For E | Some non-P is S, an I, by limitation | Some non-P is not non-S, an O |
| For O | Some non-P is S, an I | Some non-P is not non-S, an O |
| For I | none | none |
| Contraposition | Obverted converse | |
|---|---|---|
| First operation | Obvert | Convert |
| Subject of result | non-P | P |
| Fails for | I | O |
What this does not mean
Contraposition is not conversion with negatives thrown in. It is a derived operation, and its results are proved by the two basic ones.
The full contrapositive is not stronger than the original. It is equivalent to it, in the case of A, which is exactly why it may be used in place of it.
The absence of a contrapositive for I is not an oversight. It follows from O having no converse, which was proved from the distribution rule.
Quick revision
Partial contrapositive: obvert then convert. Subject non-P, predicate S.
Full contrapositive: obvert, convert, obvert. Subject non-P, predicate non-S.
A: partial is "no non-P is S", full is "all non-P is non-S". Nothing is lost, and this is the valuable one.
E: partial is "some non-P is S", by limitation, and full is "some non-P is not non-S".
I has no contrapositive, because its obverse is an O and O does not convert.
O: partial is "some non-P is S", full is "some non-P is not non-S".
In law: a necessary condition stated as "all X are Y" is applied through its full contrapositive, "whatever is not Y is not X". That is the form in which requirements are enforced.
Contraposition, Partial and Full
Test yourself
1. Define the partial and the full contrapositive and state how each is built.
The partial contrapositive is obtained by obverting the given proposition and then converting the result, and it has the contradictory of the original predicate as its subject and the original subject as its predicate. The full contrapositive is obtained by obverting the partial contrapositive, so that both terms are contradictories: its subject is non-P and its predicate is non-S.
2. Derive the full contrapositive of "All S is P" step by step.
Obvert: "No S is non-P", an E proposition. Convert simply, since E converts simply: "No non-P is S", which is the partial contrapositive and is an E. Obvert again: "All non-P is non-S", which is the full contrapositive and is an A. Nothing has been lost at any step, so the full contrapositive of an A proposition is itself universal.
3. Why does I have no contrapositive?
Because the derivation cannot be completed. Obverting "some S is P" gives "some S is not non-P", which is an O proposition, and O has no converse, since its subject is undistributed and every candidate converse would distribute it. The chain therefore stops after the first step, and neither a partial nor a full contrapositive exists.
4. What is lost in the contraposition of E, and why?
Quantity. Obverting "no S is P" gives "all S is non-P", an A proposition, and A converts only by limitation, so the conversion yields the particular "some non-P is S". The partial contrapositive of E is therefore an I and the full contrapositive an O, and both depend on the traditional doctrine of existential import, failing on the modern reading.
5. Why is the full contrapositive of an A proposition so useful in law?
Because a rule stating a necessary condition is applied by rejection rather than by inclusion. "All valid wills are attested by two witnesses" tells you what a valid will is; its full contrapositive, "everything not attested by two witnesses is not a valid will", tells you what to reject, and that is what a court or a registry actually does. The contrapositive is equivalent to the original, so nothing is being added, only made usable.
6. Give the full contrapositive of the main limb of section 25 of the Indian Contract Act 1872 and say what it does not assert.
It is that every agreement which is not void either has consideration or falls within one of the three excepted cases. It does not assert that every agreement with consideration is valid: many agreements with consideration are void for want of capacity, for unlawful object or on other grounds entirely. That further proposition would be the simple converse of the contrapositive, and the rules of conversion forbid it.
The rest of this subject
These notes are cut from the University's printed syllabus. Open the syllabus itself, or the past papers, for the same subject.