Consider the two statements:
1. Either I'll go home and watch TV, or I'll think about the
2. I'll go home, and I'll either watch TV or think about the
These statements involve the same component propositions:
I'll go home; I'll watch TV; I'll think about the election. We
can abbreviate them with the letters H, T, and E. Also, both
have the same connectives: one conjunction and one disjunction.
However, they don't say the same thing. Statement #2 implies that
H is true (i.e., that I will go home), whereas statement #1 does not
imply this. To mark the difference, we use parentheses:
1. [(HT) v E]
2. [H(T v E)]
The parentheses in #1 indicate that disjunction is the main
connective, but that one disjunct (HT) is itself a conjunction.
The parentheses in #2 indicate that the main connective is
a conjunction, but that one conjunct (T v E) is itself a disjunction.
The basic rule is to use parentheses so that the connectives, , v, and join two components, each of which may
themselves be compound statements marked off by parentheses.
A second major rule has to do with negation. A negation sign in front of a component statement is a denial
of that component only, whereas a negation sign in front of a
compound statement marked off by parentheses is a denial of the
compound statement as a whole.
Thus, the statement "Either Leslie is not sad or she's a good
actor" would be symbolized: ~S v G. The formula ~(S v G) would
represent the very different statement that Leslie is neither sad
nor a good actor.