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Section 1.1: Problem 2 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
Show that there are no wffs of length 2, 3, or 6, but that any other positive length is possible.
Note that a wff is either a single sentence symbol ( symbol), a negation of another wff ( symbols), or a combination of two wffs using one of the four binary connective symbols ( symbols), where and are the numbers of symbols in the wffs being used. Therefore, a wff can have symbols as well as symbols with further possibility to produce symbols where and . and symbols are impossible, since both and have to be greater than . is not possible as otherwise , implying , or , implying , implying or equal to . (Real life formulas may have any number of symbols, as the outermost parentheses are often omitted, and, besides, for example, parentheses are not used in a repetition of the negation operation, such as .)