veirarer

Answered

2022-06-26

Find Consistency Of System Specifications

1. $p\wedge \neg q=T$

2. $(q\wedge p)\to r=T$

3. $\neg p\to \neg r=T$

4. $(\neg q\wedge p)\to r=T$

From Eq 1, we got $p=T$ and $q=F$.

Now Apply value of P in Eq 3, we get:

$\begin{array}{ccccc}p& \neg p& r& \neg r& \neg p\to r\\ {\text{T}}& {\text{F}}& {\text{T}}& {\text{F}}& {\text{T}}\\ {\text{T}}& {\text{F}}& {\text{F}}& {\text{T}}& {\text{T}}\\ \text{F}& \text{T}& \text{T}& \text{F}& \text{F}\\ \text{F}& \text{T}& \text{F}& \text{T}& \text{T}\end{array}$

Now there are two possibilities when $\neg p\to r$ is T, and $\neg p$ is F but the r has two separate values.

1. $p\wedge \neg q=T$

2. $(q\wedge p)\to r=T$

3. $\neg p\to \neg r=T$

4. $(\neg q\wedge p)\to r=T$

From Eq 1, we got $p=T$ and $q=F$.

Now Apply value of P in Eq 3, we get:

$\begin{array}{ccccc}p& \neg p& r& \neg r& \neg p\to r\\ {\text{T}}& {\text{F}}& {\text{T}}& {\text{F}}& {\text{T}}\\ {\text{T}}& {\text{F}}& {\text{F}}& {\text{T}}& {\text{T}}\\ \text{F}& \text{T}& \text{T}& \text{F}& \text{F}\\ \text{F}& \text{T}& \text{F}& \text{T}& \text{T}\end{array}$

Now there are two possibilities when $\neg p\to r$ is T, and $\neg p$ is F but the r has two separate values.

Answer & Explanation

lorienoldf7

Expert

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 $(1\wedge 2\wedge 3\wedge 4)\equiv \mathrm{\perp},$,

i.e., regardless of (p, q, r)'s value, $(1\wedge 2\wedge 3\wedge 4)=$'s truth table has a False main connective.

Because the main connective $\to $ in your simplified truth table of $(1\wedge 2\wedge 3\wedge 4)$ is True thrice, your system is consistent.

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 $(1\wedge 2\wedge 3\wedge 4)\equiv \mathrm{\perp},$,

i.e., regardless of (p, q, r)'s value, $(1\wedge 2\wedge 3\wedge 4)=$'s truth table has a False main connective.

Because the main connective $\to $ in your simplified truth table of $(1\wedge 2\wedge 3\wedge 4)$ is True thrice, your system is consistent.

Roland Manning

Expert

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...

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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