1 1 5B 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 sentire proposition. This may sound complicated, but an example will
make it clear:
John is tall. Harry is short. John is tall and Harry is short.
T T T
T F F
F T F
F F FHere the first two columns cover every possibility for the component propositions
to be either true or false. The third column states the truth value of the whole
proposition for each combination. Clearly, the conjunction of two propositions is
true if both of the component propositions are true; otherwise, it is false.
Our reflections have not depended on the particular propositions in our
example. We could have been talking about dinosaurs instead of people, and
we still would have come to the conclusion that the conjunction of two prop-
ositions is true if both propositions are true, but false otherwise. This neglect
of the particular content of propositions is what makes our account formal.
To reflect the generality of our concerns, we can drop the reference
to particular sentences altogether and use variables instead. Just as the
lowercase letters “x,” “y,” and “z” can be replaced by any numbers in math-
ematics, so we can use the lowercase letters “p,” “q,” “r,” “s,” and so on as
variables that can be replaced by any propositions in logic. We will also use
the symbol “&” (called an ampersand) for “and.”
Consider the expression “p & q.” Is it true or false? There is obviously no
answer to this question. This is not because we do not know what “p” and
“q” stand for, for in fact “p” and “q” do not stand for any proposition at all.
Just as “x + y” is not any particular number in mathematics, so “p & q” is not
a proposition. Instead, “p & q” is a pattern for a whole series of propositions.
To reflect this, we will say that “p & q” is a propositional form. It is a pattern, or
form, for a whole series of propositions, including “John is tall and Harry is
short” as well as many other propositions.
To specify precisely which propositions have the form “p & q,” we need a little
technical terminology. The central idea is that we can pass from a proposition
to a propositional form by replacing propositions with propositional variables.
Proposition Propositional Form
John is tall and Harry is short. p & q
When we proceed in the opposite direction by uniformly substituting prop-
ositions for propositional variables, we get what we will call a substitution
instance of that propositional form.
Propositional Form Substitution Instance
p & q Roses are red and violets are blue.97364_ch06_ptg01_111-150.indd 115 15/11/13 10:15 AM
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