Discrete Mathematics | Propositional Logic | Test 2 | Question: 8

Question

In the $\textit{theory of inference},$ we begin with a set of formulas which we call $\textit{premises/ hypotheses}$ and using some rules we obtain some other $\textit{given formula}$ which we call the desired $\textit{conclusion}.$    
         
We say that $Q$ logically follows from $P$ if $P \rightarrow Q$ is a tautology or in other words:     
$\textit{Q logically follows from P if Q is tautologically implied by P}$    
      
($P$ stands for the conjunction of premises the and $Q$ stands for conclusion)     
Now, consider the following set of premises represented by English letters for various sentences.    
$1.$ $\neg S \rightarrow C$    
       
$2.$ $C \rightarrow \neg D$     
$3.$ $D \vee O$     
$4.$ $\neg O$      
Which of the following statement(s) is/are correct ?    

• A. Premise $(3)$ and Premise $(4)$ tautologically imply $D$   
• B. Premise $(2)$ and $D$ does not tautologically imply $\neg C$      
         
• C. Premise $(1)$ and $\neg C$ tautologically imply $S$ and $S$ is a logical consequence of given premises    
        
• D. Premise $(1)$ and $\neg C$ does not tautologically imply $S$

Answer 1

• D V O
• O’

   $\therefore$ D                   a) is true

 

• C → D’                                       
• D                                                                       

   $\therefore$ C’                  b)is false 

                

• S’→ C
• C’

  $\therefore$ S                  c) is true           Conclusion is also called Consequence.

 

• S’ → C
• C’

 $\therefore$ S                  d) is false

On making a conclusion False, we can’t make all of the premises true.Therefore all are valid conclusion.

 

Ans:   A) and C)

(Using Law of Detachment, from $D \vee O \equiv \neg O \rightarrow D$ and $\neg O,$ we may infer $D.$

Similarly, From $C \rightarrow \neg D \equiv D \rightarrow \neg C$ and $D,$ we may infer $\neg C$ and

From $\neg S \rightarrow C \equiv \neg C \rightarrow S$ and $\neg C,$ we may infer $S.$)