Question

Consider the following arguments.

$S_1:\left\{\right(A\vee B\sim C),(\sim B\to (\sim A\vee C)\left)\right\}\Rightarrow B$

$S_2:\{\sim B\to (P\leftrightarrow R),R\to \sim B,P\to \sim R\}$

Which of the following is true?

• A. $S_{1}isnotvalidand{S}_{2}isvalid$
• B. $Neither{S}_{1}not{S}_{2}isvalid$
• C. $Both{S}_{1}and{S}_{2}arevalid$
• D. $S_{1}isvalidand{S}_{2}isnotvalid$

Answer 1

The correct option is A $S_{1}isnotvalidand{S}_{2}isvalid$

option(b)

$S_1:\left\{\right(A\vee B\sim C),(\sim B\to (\sim A\vee C)\left)\right\}\Rightarrow B$

When A is true, B is false and C is true, then all the premises have truth value true and the conclusion has truth value false.
$\therefore$ $Theargument{S}_{1}isnotvalid.$
$S_2:\{\sim B\to (P\leftrightarrow R),R\to \sim B,P\to \sim R\}$

Indirect proof:

$\begin{array}{cc}1.\sim B\to (P\leftrightarrow R)& Premise\\ 2.R\to \sim B& Premise\\ 3.P\to \sim R& Premise\\ 4.R& Newpremisetoapplyindirectproof\\ 5.\sim P& \left(3\right),\left(4\right),modustollesns\\ 6.\sim B& \left(2\right),\left(4\right),modusponens\\ 7.(P\leftrightarrow R)& \left(1\right),\left(6\right),modusponens\\ 8.R\to P& \left(7\right),simplification\\ 9.P& \left(4\right),\left(8\right),modusponens\end{array}$

(5) and (9) contradict each other.

$\therefore$ The argument is valid.