Find Consistency Of System Specifications1. p ∧ ¬ q = T2. ( q ∧ p...





Find Consistency Of System Specifications
1. p ¬ q = T
2. ( q p ) r = T
3. ¬ p ¬ r = T
4. ( ¬ q p ) r = T
From Eq 1, we got p = T and q = F.
Now Apply value of P in Eq 3, we get:
p ¬ p r ¬ r ¬ p r T F T F T T F F T T F T T F F F T F T T
Now there are two possibilities when ¬ p r is T, and ¬ p is F but the r has two separate values.

Answer & Explanation




2022-06-27Added 19 answers

Step 1
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 ( 1 2 3 4 ) ,,
i.e., regardless of (p, q, r)'s value, ( 1 2 3 4 ) ='s truth table has a False main connective.
Because the main connective in your simplified truth table of ( 1 2 3 4 ) is True thrice, your system is consistent.
Roland Manning

Roland Manning


2022-06-28Added 5 answers

Step 1
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.
Step 2
In fact any set of tautological sentences such as ( p = p ) , ( q = q ) , ( r = r ) 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...

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