- Similarly, if ‘No S are P’ is true, then it is also true that ‘Some S are not P’. The knowledge that ‘No S are P’ is false, however, does not constrain whether or not ‘Some S are not P’.
- If ‘Some S are P’ is false, then ‘All S are P’ must also be false. The knowledge that ‘Some S are P’ is true, however, does not indicate whether or not ‘All S are P’.
- Similarly, if ‘Some S are not P’ is false, then ‘No S are P’ must also be false. The knowledge that ‘Some S are not P’ is true, however, does not indicate whether or not ‘No S are P’.
These relationships can be visualized more easily with a square of opposition composed of Venn representations of the four types of statement (Figure 20).
For example, the Venn diagrams demonstrate the incompatible, contradictory nature of diagonal statements such as ‘All S are P’ and ‘Some S are not P’.
Table 8 summarizes the relationships that can be determined between any two of the classification statements by examination of the square of opposition.
Table 8. Relationships among classification statements.
| All S are P | No S are P | Some S are P | Some S are not P | ||
| If ‘All S are P’ true, | then | false | true | false | |
| If ‘All S are P’ false, | then | unknown | unknown | true | |
| If ‘No S are P’ true, | then | false | false | true | |
| If ‘No S are P’ false, | then | unknown | true | unknown | |
| If ‘Some S are P’ true, | then | unknown | false | unknown | |
| If ‘Some S are P’ false, | then | false | true | true | |
| If ‘Some S are not P’ true, | then | false | unknown | unknown | |
| If ‘Some S are not P’ false, | then | true | false | true |
Finally and most simply (for me at least), one can immediately see the impact of any one statement's truth value on the other three statements through substitution. Again I substitute Scientist for S, and either People, Physicists, or Politicians for P, whichever fits the first statement correctly. For example, if I assume (correctly) that ‘Some scientists are physicists’ is true, then ‘No scientists are physicists’ must be false, and I need additional information to say whether ‘All scientists are physicists’ or ‘Some scientists are not physicists’. Some caution is needed to assure that my conclusions are based on the evidence rather than on my independent knowledge. For example, I know that ‘All scientists are physicists’ is false but I cannot infer so from the statement above that ‘Some scientists are physicists’. As another example, if I assume (naïvely) that ‘Some scientists are politicians’ is false, then it also must be true that ‘No scientists are politicians’ and that ‘Some scientists are not politicians’. Furthermore, the statement that ‘All scientists are politicians’ must be false.
