I need to derive D from the following. Any help would be greatly appreciated
Premise 1: A & B
Premise 2: C -> D
Premise 3: C v ~A
Hello, rsportsman09!
I need to derive $\displaystyle D$ from the following.
. . $\displaystyle \begin{array}{ccc} {\color{blue}(1)} & A \wedge B \\
{\color{blue}(2)} & C \to D \\ {\color{blue}(3)} & C \:\vee \sim A \end{array}$
I'll give you the game plan ... You can supply the reasons.
From (1), $\displaystyle A$ and $\displaystyle B$ are true . . . Hence: $\displaystyle A$ is true. .(a)
From (3), we have: .$\displaystyle C \:\vee \sim A \;\;=\;\;\sim A \vee C \;\;=\;\;A \to C$ .(b)
We have: . $\displaystyle \begin{array}{cccc}{\color{blue}(b)} & A \to C \\ {\color{blue}(2)} & C \to D \\ & --- \\ {\color{blue}(c)} & A \to D & \text{syllogism}\end{array}$
We have: . $\displaystyle \begin{array}{cccc}{\color{blue}(c)} & A \to D \\ {\color{blue}(a)} & A \\ & --- \\ \therefore & D & \text{detachment}\end{array}$