Find Consistency Of System Specifications1. p ∧ ¬ q = T2. ( q ∧ p...
Find Consistency Of System Specifications
From Eq 1, we got and .
Now Apply value of P in Eq 3, we get:
Now there are two possibilities when is T, and is F but the r has two separate values.
Answer & Explanation
A set of sentences as consistent iff their conjunction is satisfiable.
(Informally: a consistent system is one whose premises/axioms are coherent in some universe.)
So, in propositional logic, an inconsistent system is one whose conjunction is a contradiction, i.e., whose conjunction is false regardless of the combination of truth values of its atomic propositions.
So, in your exercise, the system is inconsistent iff ,
i.e., regardless of (p, q, r)'s value, 's truth table has a False main connective.
Because the main connective in your simplified truth table of is True thrice, your system is consistent.
It's perfectly ok that your set of sentences is consistent if you have 2 different models satisfying your set of sentences (a theory) since consistency has nothing to do with uniqueness of model as referenced here
a consistent theory is one that does not lead to a logical contradiction. The lack of contradiction can be defined in either semantic or syntactic terms. The semantic definition states that a theory is consistent if it has a model, i.e., there exists an interpretation under which all formulas in the theory are true.
In fact any set of tautological sentences such as can always have different truth values for any propositional sentence p, q, r to stay to be consistent.
But look further about your set of particular sentences, r can only be true from the constraint of your last sentence 4 since the antecedent of your material conditional is true then r has to be true...