Logicians use the terms conversion, obversion, and contraposition to define three types of logically equivalent statements, but we will not need to memorize these terms. Below are listed on the right the only logically equivalent statements to those on the left:
| Initial statement | Logically equivalent statements | |
| All S are P. | No S are non-P. | All non-P are non-S. |
| No S are P. | No P are S. | All S are non-P. |
| Some S are P. | Some P are S. | Some S are not non-P. |
| Some S are not P. | Some S are non-P. | Some non-P are not non-S. |
Some logically equivalent statements seem cumbersome and overloaded with negatives. That apparent weakness is a strength of the concept of logical equivalence, for we may encounter a statement on the right and want to translate it into a familiar classification statement.
The concept of logical equivalence can also be useful in experimental design. For example, it might be impossible to show that ‘some S are P’ but easy to show that ‘some P are S’. In Chapter 7 we will consider the Raven’s Paradox: the two statements ‘All ravens are black’ and ‘All nonblack things are non-ravens’ may be logically equivalent, but testing the latter would involve an inventory of the universe.
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For recognizing logically equivalent statements, substitution is an alternative to Venn diagrams. For example, replace S with scientists and replace P with either people, physicists, or politicians, whichever gives a true initial statement:
Valid equivalent statements:
| All Scientists are People. | No Scientists are non-People. |
| All Scientists are People. | All non-People are non-Scientists. |
| No Scientists are Politicians. | No Politicians are Scientists. |
| No Scientists are Politicians. | All Scientists are non-Politicians. |
| Some Scientists are Physicists. | Some Physicists are Scientists. |
| Some Scientists are Physicists. | Some Scientists are not non-Physicists. |
| Some Scientists are not Physicists. | Some Scientists are non-Physicists. |
| Some Scientists are not Physicists. | Some non-Physicists are not non-Scientists. |
Non-equivalent statements:
| All Scientists are People. | No People are non-Scientists. |
| All Scientists are People. | All People are Scientists. |
| Some Scientists are not Physicists. | Some Physicists are not Scientists. |
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