Square of Opposition

Traditional Square of Opposition

The square of opposition displays the the four ways that categorical propositons can be opposed to each other: as contraditories, contraries, subcontraries and as sub/superalterns.

1.) A and E are Contraries: the definition of contraries is "statements cannot both be true but both contraries might both be false": in other words, at least one of contraries must be false, and it is possible both of the contraries are false.
I and O are

2.) Subcontraries: the definition of subcontraries is "statements cannot both be false but both subcontraries might be true. In other words, at least one of the subcontraries must be true, and it is possible both are true."

3.) Subalternation: A and the I propositions and the E and the O propositions are related by the logical relation of subalternation.A proposition is a subaltern of another if and only if  the statement must be true if its superaltern is true, and the superaltern must be false if the subaltern is false.  Truth does down and false goes up: from the truth of the A proposition the truth of the corresponding I proposition may be inferred; it also means that from the falsity of the I proposition the falsity of the A proposition may be inferred.

4.) A and O and E and I are contradictories: the statements cannot both be true and they cannot both be false: In other words, they must have opposite truth values--it must be the case that exactly one of them is true and the other false.In other words, they must have opposite truth values--it must be the case that exactly one of them is true and the other false: in other words, they ( A and O and E and I) must have opposite truth values--it must be the case that one is true and the other false.

     'All S is P' and 'Some S is not P' are contradictories.
     'No S is P' and 'Some S is P' are contradictories.
     'All S is P' and 'No S is P' are contraries.
     'Some S is P' and 'Some S is not P' are subcontraries.
     'All S is P' is a superaltern of 'Some S is P'
      'No S is P' is a superaltern of 'Some S is not P'
     'Some S is P' is a subaltern of 'All S is P'.
     'Some S is not P' is a subaltern of 'No S is P'.

Thus, from the traditional square we can quickly determine such facts as the following:
If  A is given as true, then the E is false, the I is true, and the O is false.
If E is given as true, then the A is false, the I is false and the O is true.
If  I is given as true, then the E is false,  the O is undetermined, and the A is undetermined.
If O is given as true, then the E and the I are undetermined and the A is false.
If A is given as false, then the E and the I are undetermined and the O is true.
If E is given as false then the A and the O are undetermined and the I is true.
If  I is given as false then the E is true and the O is true and the A is false.
If O is given false then the A and the I are true and the E is false.