Section 5.3 Substituting into an argument
Substituting into an argument does not change its validity.
Theorem 5.3.1. Substitution Rule.
Suppose \(A_1, A_2, \dotsc, A_m \therefore C\) is a valid argument involving statement variables \(p_1, p_2, \dotsc, p_\ell\text{.}\) If we apply substitution \(p_i\to B_i\) to each of \(A_1, A_2, \dotsc, A_m, C\text{,}\) for some collections of statements \(B_1, B_2, \dotsc, B_\ell\text{,}\) then the resulting argument is also valid.
Example 5.3.2.
Since modus tollens is a valid argument, using the substitution rule with the equivalences
\begin{equation*}
r \lgcand p \lgcequiv \lgcnot (\lgcnot r \lgcor \lgcnot p) \lgcequiv \lgcnot (r\lgccond \lgcnot p) \text{,}
\end{equation*}
demonstrates that the following argument is also valid.
| \((p \lgcbicond q) \lgccond (r \lgccond \lgcnot p)\) |
| \(r \lgcand p\) |
| \(\lgcnot (p \lgcbicond q)\) |
