The Language of Argument

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C H A P T E R 6 ■ P r o p o s i t i o n a l L o g i c

Thus, “John is tall and Harry is short” and “Roses are red and violets are

blue” are both substitution instances of the propositional form “p & q.”

To get clear about these ideas, it is important to notice that “p” is also

a propositional form, with every proposition, including “Roses are red and

violets are blue,” among its substitution instances. There is no rule against

substituting compound propositions for propositional variables. Perhaps a

bit more surprisingly, our definitions allow “Roses are red and roses are red”

to be a substitution instance of “p & q.” This example makes sense if you

compare it to variables in mathematics. Using only positive integers, how

many solutions are there to the equation “x + y = 4”? There are three: 3 + 1,

1 + 3, and 2 + 2. The fact that “2 + 2” is a solution to “x + y = 4” shows that

“2” can be substituted for both “x” and “y” in the same solution. That’s just

like allowing “Roses are red” to be substituted for both “p” and “q,” so that

“Roses are red and roses are red” is a substitution instance of “p & q” in

propositional logic.

In general, then, we get a substitution instance of a propositional form by

uniformly replacing the same variable with the same proposition throughout,

but different variables do not have to be replaced with different proposi-

tions. The rule is this:

Different variables may be replaced with the same proposition, but

different propositions may not be replaced with the same variable.

According to this rule:

“Roses are red and violets are blue” is a substitution instance of “p & q.”

“Roses are red and violets are blue” is also a substitution instance of “p.”

“Roses are red and roses are red” is a substitution instance of “p & q.”

“Roses are red and roses are red” is a substitution instance of “p & p.”

“Roses are red and violets are blue” is not a substitution instance of “p & p.”

“Roses are red” is not a substitution instance of “p & p.”

We are now in a position to give a perfectly general definition of conjunction

with the following truth table, using propositional variables where previously

we used specific propositions:

p q p & q

T T T

T F F

F T F

F F F

There is no limit to the number of propositions we can conjoin to form

a new proposition. “Roses are red and violets are blue; sugar is sweet and

so are you” is a substitution instance of “p & q & r & s.” We can also use

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