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Symbolic logic typically makes use of the lower case letters p, q, r, s, and t to represent statements. Various symbols are then used to related these symbols or statements.
We want to translate a sentence into a symbolic form:
The earth is not flat if and only if it's possible to sail all the way around it.
I will provide two possible symbolic form for this sentence.
First, let us do substitions, and use the letters p, and q, for our statements. Let p be the statement: "The earth is not flat." Let q be the statement: "It is possible to sail all the way around it [the earth]." Notice that p is the first part of our sentence, while q the latter part. The only phrase missing now is "if and only if." In mathematics (or in logic), this phrase is the biconditional logical connective, usually abbreviated as "iff" and is associated with the symbol: <=> (⇔).
Hence, the sentence can be translated to: p ⇔ q (read as "p if and only if q" or "p is equivalent to q" or "p precisely when q", among other interpretations).
Another possible interpration would be the following; Let s be the statement: "The earth is flat." Let tbe the statement: "It is possible to sail all the way around it [the earth]." The only difference with these assignments compared to the first one is p. Since the original statement is "The earth is NOT flat", we need to negate s so that it will be the same as p in the first formulation. In logic, there is a symbol for negation (or not): ~. Hence, ~s is read as "not s". The first part of the sentence then would be ~s in symbolic form.
Therefore, we can also translate the sentence to: ~s ⇔ t (read is "not s if and only f t").
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