Relationships Among Statements
The four types of classification statement are formally related in truth value, regardless of the subjects of the statements. The relationships can be summarized in what is called the square of opposition (Figure 19).
The strongest relationship among the statements is that of contradiction along the diagonals: if a statement is true, then its diagonal is false, and vice versa. Without even substituting familiar terms for the subject and predicate, one can recognize readily that:
- ‘All S are P’ contradicts the statement ‘Some S are not P’, and
- ‘No S are P’ contradicts the statement ‘Some S are P’.
Horizontally along the top, one or both of the statements invariably is false:
- If ‘All S are P’ is true, then ‘No S are P’ must be false;
- If ‘No S are P’ is true, then ‘All S are P’ must be false;
- If either ‘All S are P’ or ‘No S are P’ is false, we cannot infer that the other statement is true; possibly both are false and ‘Some S are P’.
Horizontally along the bottom, one or both of the statements invariably is true:
- If ‘Some S are P’ is false, then ‘Some S are not P’ must be true;
- If ‘Some S are not P’ is false, then ‘Some S are P’ must be true;
- Both statements may be true: some S are P while other S are not P.
Vertically, the statements lack the perfect symmetry that we saw diagonally and horizontally. Instead, imagine truth flowing downward (from the general to the particular) and falsity flowing upward (from the particular to the general):
- If ‘All S are P’ is true, then it is also true that ‘Some S are P’.
The knowledge that ‘All S are P’ is false, however, does not constrain whether or not ‘Some S are P’.
