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 ?
- System $(i)$ is consistent and System $(ii)$ is inconsistent
- System $(ii)$ is consistent and System $(i)$ is inconsistent
- Both systems $(i)$ and $(ii)$ are consistent
- None of the systems are consistent