1 2 3B a s i c P r o p o s i t i o n a l C o n n e c t i v e sconstruct 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 ques-
tion 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 original
propositions are true, and false otherwise. With disjunction, given two prop-
ositions, 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:
p ~p
T F
F TNegation 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 include 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.”
The complexities of negation can be illustrated by noticing that the
simple sentence “Everyone loves running” can include negation at four
distinct places: “Not everyone loves running,” “Everyone does not love
running,” “Everyone loves not running,” and the colloquial “Everyone loves
running—not!” Some of these sentences can be symbolized in propositional
logic as negations of “Everyone loves running,” but others cannot.
To determine whether a sentence can be symbolized as a negation in prop-
ositional logic, it is often useful to reformulate the sentence so that it starts97364_ch06_ptg01_111-150.indd 123 15/11/13 10:15 AM
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