With conjunction and disjunction, we begin with two propositions and con- struct a new proposition from them. There is another way in which we can construct a new proposition from just one proposition—by negating it.
Given the proposition “John is clever,” we can get a new proposition, “John is not clever,” simply by inserting the word “not” in the correct place in the sentence.
What, exactly, does the word “not” mean? This can be a difficult question to answer. Does it mean “nothing” or, maybe, “nothingness“? Although some respectable philosophers have sometimes spoken in this way, it is im- portant to see that the word “not” does not stand for anything at all. It has an altogether different function in the language. To see this, think about how conjunction and disjunction work. Given two propositions, the word “and” allows us to construct another proposition that is true only when both origi- nal propositions are true, and false otherwise. With disjunction, given two propositions, the word “or” allows us to construct another proposition that is false only when both of the original propositions are false, and true other- wise. (Our truth table definitions reflect these facts.) Using these definitions as models, how should we define negation? A parallel answer is that the negation of a proposition is true just in the cases in which the original propo- sition is false, and it is false just in the cases in which original proposition is true. Using the symbol “~” (called a tilde) to stand for negation, this gives us the following truth table definition:
Negation might seem as simple as can be, but people quite often get con- fused by negations. If Diana says, “I could not breathe for a whole minute,” she might mean that there was a minute when something made her unable to breathe (maybe she was choking) or she might mean that she was able to hold her breath for a whole minute (say, to win a bet). If “A” symbolizes “Diana could breathe sometime during this minute,” then “~A” symbolizes the former claim (that Diana was unable to breathe for this minute). Conse- quently, the latter claim (that Diana could hold her breath for this minute) should not also be symbolized by “~A.” Indeed, this interpretation of the original sentence is not a negation, even though the original sentence did in- clude the word “not.” Moreover, some sentences are negations even though they do not include the word “not.” For example, “Nobody owns Mars” is the negation of “Somebody owns Mars.” If the latter is symbolized as “A,” the former can be symbolized as “~A,” even though the former does not include the word “not.”
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