1 1 7B 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 sparentheses to group propositions. This last example could be treated as a
substitution instance of “(p & q) & (r & s)”—that is, as a conjunction of two
conjunctions. Later we will see that, just as in mathematics, parentheses can
make an important difference to the meaning of a total proposition.
One cautionary note: The word “and” is not always used to connect two
distinct sentences. Sometimes a sentence has to be rewritten for us to see
that it is equivalent to a sentence of this form. For example,
Serena and Venus are tennis players.
is simply a short way of saying
Serena is a tennis player, and Venus is a tennis player.
At other times, the word “and” is not used to produce a conjunction of
propositions. For example,
Serena and Venus are playing each other.
does not mean that
Serena is playing each other, and Venus is playing each other.
That does not even make sense, so the original sentence cannot express a
conjunction of two propositions. Instead, it expresses a single proposi-
tion about two people taken as a group. Consequently, it should not be
symbolized as “p & q.” Often, unfortunately, it is unclear whether a sentence
expresses a conjunction of propositions or a single proposition about a
group. The sentence
Serena and Venus are playing tennis.
could be taken either way. Maybe Serena and Venus are playing each other.
If that is what it means, then the sentence expresses a single proposition
about a group, so it should not be symbolized as “p & q.” But maybe Serena
is playing one match, while Venus is playing another. If that would make it
true, then the sentence expresses a conjunction of propositions, so it may be
symbolized as “p & q.”
When a sentence containing the word “and” expresses the conjunction
of two propositions, we will say that it expresses a propositional conjunction.
When a sentence containing “and” does not express the conjunction of two
propositions, we will say that it expresses a nonpropositional conjunction. In
this chapter we are concerned only with sentences that express propositional
conjunctions. A sentence should be translated into the symbolic form “p & q”
only if it expresses a propositional conjunction. There is no mechanical
procedure that can be followed to determine whether a certain sentence
expresses a conjunction of two propositions. You must think carefully about
what the sentence means and about the context in which that sentence is
used. This takes practice.97364_ch06_ptg01_111-150.indd 117 15/11/13 10:15 AM
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