Discrete Mathematics | Propositional Logic | Test 2 | Question: 13

Question

The consistency of a set of premises whose logical structure may be expressed by sentential connectives alone may be determined directly by a mechanical truth table test. The truth table for the conjunction of the premises is constructed.    
            
If $\textit{every entry}$ in the final column is $'F'$ i.e. False then the premises are $\textit{inconsistent}$ and if $\textit{at least}$ one entry is $'T'$ i.e. True then the premises are $\textit{consistent}$      
Consider two sets of premises as:    
    
i. $\{p \wedge \neg q ; (q \wedge p) \rightarrow r; \neg p \rightarrow \neg r ; (\neg q \wedge p) \rightarrow r\}$     
ii. $\{p;\neg p\}$      
Now, which one of the following statements is correct ?           

• A. System $(i)$ is consistent and System $(ii)$ is inconsistent    
       
• B. System $(ii)$ is consistent and System $(i)$ is inconsistent    
      
• C. Both systems $(i)$ and $(ii)$ are consistent    
       
• D. None of the systems are consistent

Answer 1

1. if could find atleast one case where is true on the conjunction of the below premises then consistent.

• p $\wedge$ q’
• (q $\wedge$p) → r
• p’ → r’
• (q’ $\wedge$ p) → r   

 

p $\wedge$ q’ is true at p=T and q=F

(q $\wedge$p) → r is true at r=T

p’ → r’ is true 

(q’ $\wedge$ p) → r   is true

At p=T q=F r=T conjunction of the premises is true then consistent.

2. p $\wedge$ p’ is false in all cases. Therefore inconsistent.

• p
• p’

so, i) is consistent ii) is inconsistent   A) is correct