Section 1.3: Problem 5 Solution
Working problems is a crucial part of learning mathematics. No one can learn... merely by poring over the definitions, theorems, and examples that are worked out in the text. One must work part of it out for oneself. To provide that opportunity is the purpose of the exercises.
James R. Munkres
The English language has a tendency to use two-part connectives: “both . . . and . . .” “either . . . or . . .” “if . . . , then . . . .” How does this affect unique readability in English?
It helps to resolve ambiguities that could have arose otherwise. For example, when we say “ or ”, it is not clear whether what we say implies that both and may take place. However, when we say “either or ”, it usually implies that one is true but not the other. For example, “either you get at least 75% on the test, or you fail the class” means that one of these (and only one) will happen. Of course, the ambiguity in English language can also often be resolved by the context as well. For example, by itself the construction “if , then ” does not imply that if were not to happen, neither would . In fact, such a sentence says nothing about what would be true if were not true. For example, “if is a square, then is a rectangle” does not imply that if were not a square, then it would not be a rectangle. Again, the only thing regarding this issue that we can conclude from the sentence itself, is that if is not a square, then we do not know whether it is a rectangle or not. To emphasize more strict relationships one would use something like “if and only if” or its equivalents. However, in other cases, the context may suggest that, in fact, “if” in “if ..., then ...” plays the role of “if and only if”. The same sentence we used before about passing the test can be rephrased as “if you get at least 75% on the test, you pass the class”, where “if” means “if and only if”.
To sum up, two-part connectives are often used to resolve the ambiguity regarding the meaning of the sentence in all possible cases when the parts of the sentence are true or false. However, not all such two-part connectives play such a role in English (for example, “both ... and ...” often means exactly same as “... and ...”), and even those that do, can often be understood as if the first part was not present (for example, “either ... or ...” is often understood as simply “... or ...”, that is both parts can be true). To say it in a more related to the course manner, the truth-false tables for such language expressions are not always unique and often depend on the use case. In such cases, the context may play a more important role to further clarify the meaning.