veirarer

2022-06-26

Find Consistency Of System Specifications
1. $p\wedge ¬q=T$
2. $\left(q\wedge p\right)\to r=T$
3. $¬p\to ¬r=T$
4. $\left(¬q\wedge p\right)\to r=T$
From Eq 1, we got $p=T$ and $q=F$.
Now Apply value of P in Eq 3, we get:
$\begin{array}{ccccc}p& ¬p& r& ¬r& ¬p\to r\\ \text{T}& \text{F}& \text{T}& \text{F}& \text{T}\\ \text{T}& \text{F}& \text{F}& \text{T}& \text{T}\\ \text{F}& \text{T}& \text{T}& \text{F}& \text{F}\\ \text{F}& \text{T}& \text{F}& \text{T}& \text{T}\end{array}$
Now there are two possibilities when $¬p\to r$ is T, and $¬p$ is F but the r has two separate values.

lorienoldf7

Expert

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 $\left(1\wedge 2\wedge 3\wedge 4\right)\equiv \mathrm{\perp },$,
i.e., regardless of (p, q, r)'s value, $\left(1\wedge 2\wedge 3\wedge 4\right)=$'s truth table has a False main connective.
Because the main connective $\to$ in your simplified truth table of $\left(1\wedge 2\wedge 3\wedge 4\right)$ is True thrice, your system is consistent.

Roland Manning

Expert

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 $\left(p=p\right),\left(q=q\right),\left(r=r\right)$ 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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