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\)
Section 1.2 Converting language to symbols
As we have already begun to do, we will use letters to represent (possibly variable) logical statements and substatements. To complete the conversion from verbose language to compact symbolism, we will introduce symbols to represent the
Five basic connectives .
negation of \(A \)
\(\displaystyle \lgcnot A \)
conjuction of \(A \) and \(B \)
\(\displaystyle A \lgcand B \)
disjunction of \(A \) and \(B \)
\(\displaystyle A \lgcor B \)
conditional where \(A \) implies \(B \)
\(\displaystyle A \lgccond B \)
biconditional involving \(A \) and \(B \)
\(\displaystyle A \lgcbicond B \)
Using variables to represent statements and the above symbols to represent connectives allows us to isolate the task of analyzing logical structure, without being distracted or influenced by the content of the statements.
Example 1.2.2 . Translating English language into symbolic language.
Consider the statement “if we are outside and we get wet then it is raining.” Assign statement variables:
\begin{align*}
A \amp = \text{ “we are outside,”} \amp
B \amp = \text{ “we get wet,”} \amp
C \amp = \text{ “it is raining.”}
\end{align*}
Then symbolically, the statement can be written
\begin{equation*}
A \lgcand B \lgccond C\text{.}
\end{equation*}
