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Subsection 2.2.7 Summary – What is a Boolean Well-Formed Formula?

Now that we have introduced the symbols that we will use to denote the standard logical operations, we can summarize the syntax of a Boolean logic statement, also called a Boolean well-formed formula, or wff, pronounced, “woof”:

  1. True (usually abbreviated T) is a statement. It is always true.

  2. False (usually abbreviated F) is a statement. It is always false.

  3. Individual claims (which we may write in English or we may denote with variables like p) are statements. They may be either true or false.

  4. If p and q are statements, then so are:

    1. \(\neg \) p

    2. \(\displaystyle p \wedge q\)

    3. \(\displaystyle p \vee q\)

    4. \(\displaystyle p \equiv q\)

    5. \(\displaystyle p \rightarrow q\)

In all of these cases, the truth value of the compound formula is given by their definitions as we’ve just presented them. Further, if we introduce additional operators (which we will do), then expressions formed using them as defined are also statements.

  1. If p is a statement, then so is (p).

Notice that when we say, “If p is a statement,” we are using the symbol, “p,” to stand for any expression (not solely the literal symbol, “p”).

The following expressions are all statements:

\(T, p, p \wedge q \) These are simple.

\(\neg \neg p \) \(\neg p \) is an expression. So, using (4a) again and letting p there correspond to \(\neg \) p, we have that \(\neg \neg \) p is an expression.

\(( p \wedge q) \rightarrow \) Notice here that p \(\wedge \) q is an expression. Then so is (p\(\wedge \) q). Then so is this entire expression. Think of p in (4e) as corresponding to the entire statement (p \(\wedge\) q). And q in (4e) corresponds to r.

\(( G \vee H) \rightarrow \) We are not limited to the names p and q. As we’ll soon see, we may want to define many symbols to stand for the basic statements that we want to work with.

Exercises Exercises

Exercise Group.

For each of the following expressions, mark True if it is a wff and False otherwise.

a.
\(( p \wedge q ) \rightarrow ( r \wedge q )\)
Answer.
True
b.
\(( p \wedge q ) \wedge \neg ( p \wedge q) \)
Answer.
True
c.
\(( p \neg q ) \rightarrow \neg r \)
Answer.
False
d.
\(((( p \wedge q ) \wedge r ) \vee s )\)
Answer.
True
e.
\((( R \wedge D ) \wedge G ) \neg ( D \vee R )\)
Answer.
False
f.
\(((( p \wedge q ) \rightarrow r ) \vee s \)
Answer.
False
Solution.

\((p \neg q) \rightarrow \neg r \) is not a wff because \(\neg \) must be applied to a single statement. It (unlike \(\vee, \wedge, \equiv,\) and \(\rightarrow\) ) does not connect two statements.

\(((R \wedge D) \wedge G) \neg (D \vee R) \)is not a wff, also because \(\neg \) is not a binary operator.

\((((p \wedge q) \rightarrow r) \vee s \) is not a wff because its parentheses are not balanced.