S    Co             s   S
u      nt          e    u
b        ra       i     b
a          di    r      a
l            ct o       l
t            ct o       t
e          di    r      e
r        ra       i     r
n      nt          e    n
s    Co             s   s

I. Sub-contraries O.

As an aid to the memory, this diagram is useful; but as an attempt to represent the logical relations of propositions, it is misleading.  For, standing at corners of the same square, A. and E., A. and I., E. and O., and I. and O., seem to be couples bearing the same relation to one another; whereas we have seen that their relations are entirely different.  The following traditional summary of their relations in respect of truth and falsity is much more to the purpose: 

(1) If A. is true, I. is true, E. is false, O. is false. (2) If A. is false, I. is unknown, E. is unknown, O. is true. (3) If I. is true, A. is unknown, E. is false, O. is unknown. (4) If I. is false, A. is false, E. is true, O. is true. (5) If E. is true, A. is false, I. is false, O. is true. (6) If E. is false, A. is unknown, I. is true, O. is unknown. (7) If O. is true, A. is false, I. is unknown, E. is unknown. (8) If O. is false, A. is true, I. is true, E. is false.

Where, however, as in cases 2, 3, 6, 7, alleging either the falsity of universals or the truth of particulars, it follows that two of the three Opposites are unknown, we may conclude further that one of them must be true and the other false, because the two unknown are always Contradictories.

Sec. 10.  Secondary modes of Immediate Inference are obtained by applying the process of Conversion or Obversion to the results already obtained by the other process.  The best known secondary form of Immediate Inference is the Contrapositive, and this is the converse of the obverse of a given proposition.  Thus: 

        DATUM.  OBVERSE.  CONTRAPOSITIVE.

A. All S is P      .’. No S is not-P       .’. No not-P is S
I. Some S is P     .’. Some S is not not-P .’. (none)
E. No S is P       .’. All S is not-P      .’. Some not-P is S
O. Some S is not P .’. Some S is not-P     .’. Some not-P is S

There is no contrapositive of I., because the obverse of I. is in the form of O., and we have seen that O. cannot be converted.  O., however, has a contrapositive (Some not-P is S); and this is sometimes given instead of the converse, and called the ‘converse by negation.’

Contraposition needs no justification by the Laws of Thought, as it is nothing but a compounding of conversion with obversion, both of which processes have already been justified.  I give a table opposite of the other ways of compounding these primary modes of Immediate Inference.

A                I               E                 O
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1 All A is B Some A is B No A is B Some A is not B