Question

Consider two well-formed formulas in propositional logic :

$F_1:P\Rightarrow \rightharpoondown P$

$F_2:(P\Rightarrow \rightharpoondown P)\vee (\rightharpoondown P\Rightarrow P)$

Which of the following statements is correct ?

• A. $F_1$ is satisfiable , $F_2$ is valid
• B. $F_1$ is unsatisfiable , $F_2$ is satisfiable
• C. $F_1$ is unsatisfiable , $F_2$ is valid
• D. $F_1$ and $F_2$ are both satisfiable

Answer 1

The correct option is A

$F_1$ is satisfiable , $F_2$ is valid

$F_1:P\to \sim P\equiv P\to P^{\prime }\equiv P^{\prime }+P^{\prime }\equiv P^{\prime }$

So $F_1$ is contingency. Hence, $F_1$ is satisfiable but not valid.

$F_2$ : $(P\to \sim P)\vee (\sim P\to P)$
$\equiv (P\to P{}^{\prime })+(P^{\prime }\to P)$
$\equiv (P{}^{\prime }+P{}^{\prime })+(P+P)$
$\equiv P{}^{\prime }+P\equiv 1$

So $F_2$ is tautology and therefore valid.