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7 votes
7 votes

Let $\text{“KB”}$ stand for $\text{“knowledge base”}$ which is a set of premises/statements, as mentioned following:

 


Let $\text{S}$ be a propositional formula.

Which of the following is $\text{“possible”}$?

  1. $(\text{KB}\models \text{S})$ and $(\neg \text{KB} \models \text{S})$
  2. $(\text{KB} \models \text{S})$ and $(\text{KB} \models \neg \text{S})$
  3. $(\text{KB} \models \text{S})$ and $(\text{KB} \not \models \text{S})$
  4. $(\text{KB} \models \text{S})$ and $(\text{KB} \not \models \neg \text{S})$
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1 Answer

4 votes
4 votes

$KB$ is basically a propositional formula.

If $KB ≡ { S1, S2,\dots, Sn }$ then we can say that

$KB ≡ S1 \text{ AND } S2 \text{ AND} \dots  \text{ AND } Sn$

So, $KB$ is basically a propositional formula. $S$ is also a propositional formula.

We know that $KB |= Y$ iff $(KB\rightarrow Y)$ is a tautology.

Is it possible that:

$(a)\; (KB |= S)$ and $(¬KB |= S)$

$\text{Answer:}$ Yes. For example, if $S ≡ TRUE$, i.e., if $S$ is a tautology, then any interpretation(row in the truth table) that satisfies

$KB$ or $¬KB$ also satisfies $S$.

$(b) (KB |= S) and (KB |= ¬S)$
$\text{Answer:}$ Yes. For example, if $KB ≡ FALSE$, i.e., if $KB$ is contradiction, then $KB$ entails/infers any sentence, including $S$ and $¬S$.

 

$\color{red}{\text{More variations:}}$

Which of the following is/are Possible ?

 

 

$\color{red}{\text{Detailed Video Solution:}}$ https://youtu.be/nclBhBmtz2g?t=685

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