Question

Indicate which of the following well-formed formula are valid:

• A. $((P\Rightarrow Q)\wedge (Q\Rightarrow R))\Rightarrow (P\Rightarrow R)$
• B. $(P\Rightarrow Q)\Rightarrow (\rightharpoondown P\Rightarrow \rightharpoondown Q)$
  
• C. $(P\wedge (\rightharpoondown PV\rightharpoondown Q))\Rightarrow Q$
• D. $((P\Rightarrow R)\vee (Q\Rightarrow R))\Rightarrow \left((P\vee Q)\right)\Rightarrow R)$

Answer 1

The correct option is A $((P\Rightarrow Q)\wedge (Q\Rightarrow R))\Rightarrow (P\Rightarrow R)$

Option (a) is well known valid formula (tautology)
became it is a rule of inference called hypothetical syllogism .
By applying boolean algebra and simplifying , we can show that (b) , (c) and (d) are invalid .
For example,
Choice $\left(b\right)\equiv (P\Rightarrow Q)\Rightarrow (\rightharpoondown P\Rightarrow \rightharpoondown Q)$
$\equiv (P\Rightarrow Q)\Rightarrow (P{}^{\prime }\Rightarrow Q{}^{\prime })$
$\equiv (P{}^{\prime }+Q)+(P+Q{}^{\prime })$
$\equiv (P{}^{\prime }+Q){}^{\prime }+(P+Q{}^{\prime })$
$\equiv PQ{}^{\prime }+P+Q{}^{\prime }$
$\equiv P+Q{}^{\prime }$
$\ne 1$
So , invalid.