Understanding Ke Logic Rules & Finding Contradictions

In summary: If no, then the rules must be able to handle this case.There is a rule in α or β rules to account for this case- the principle of explosion. The principle of explosion states that if one can derive an argumen by using the α or β rules, then one can always derivate an argumen by using the principle of explosion.
  • #1
lyd123
13
0
Hi, the question and Ke logic rules are attached.

This is my attempt at the question.

$1. P \land (R\implies Q) $ Premise
$2. ( P \land Q ) \implies \lnot S) $ Premise
$3. ( P \land S) \implies R) $ Premise
$4. \lnot S $ Conclusion
$5. P \land Q$ $ \beta 2,4$
$6. P $ $ \alpha 5$
$7. Q$ $ \alpha 5$
$8. R\implies Q $ $ \alpha 1$

I don't think the lines I wrote after this make a lot of sense. Usually a contradiction would be found, but in this case I don't seem to find a contradiction. I think maybe I have to negate the conclusion, I thought it was already negated because of the \lnot. But how do I know when the argument form is valid (invalid being if there is a contradiction).Thank you for any help. :)

View attachment 8735
View attachment 8736
 

Attachments

  • 2019-01-03.png
    2019-01-03.png
    5.3 KB · Views: 97
  • 2019-01-03 (1).png
    2019-01-03 (1).png
    6.9 KB · Views: 125
Physics news on Phys.org
  • #2
I assume that $\neg S$ is the original conclusion, not its negation.

You cannot derive $P\land Q$ from $P\land Q\implies \neg S$ and $\neg S$.

To prove $\neg S$, one must use the law of excluded middle, or the branching rule. From premise 1 we have $P$ and $R\implies Q$. If $S$, then we get $R$ from premise 3, $Q$ from premise 1 and finally $\neg S$ from premise 2. If $\neg S$, then nothing is left to do.
 
  • #3
Thank you, I understand now.If an argument was valid, how would we know? For example, in this case if the the original was S and the negated conclusion is ¬S ?

1.P∧(R⟹Q) Premise
2.(P∧Q)⟹¬S Premise
3.(P∧S)⟹R Premise
4.¬S Negated Conclusion
 
  • #4
These premises do not imply $S$. The easiest way to see this is to find a counterexample, i.e., an assignment of truth values to variables that makes all premises true and the conclusion false. In this case it is $R=Q=S=F$ and $P=T$.
 
  • #5
Evgeny.Makarov said:
To prove $\neg S$, one must use the law of excluded middle, or the branching rule

I suppose you mean : $ S\vee\neg S$

Evgeny.Makarov said:
If $S$, then we get $R$ from premise 3

I suppose you mean from P and S we get : $P\wedge S$ and then using premise 3 and α rules (Modus Ponens) we get $R$

If yes, there is no rule in α or β rules to account for: $P$,$S$ $\Rightarrow P\wedge Q$
 

FAQ: Understanding Ke Logic Rules & Finding Contradictions

1. What is Ke Logic?

Ke Logic refers to the set of logical rules and principles used in Knowledge Engineering. It is a formal and systematic approach to representing knowledge and reasoning about it in a computer-readable format.

2. Why is it important to understand Ke Logic rules?

Understanding Ke Logic rules is crucial for building and maintaining knowledge-based systems. These rules provide a framework for representing and organizing knowledge in a way that can be processed and reasoned about by computers.

3. What are some common Ke Logic rules?

Some common Ke Logic rules include the use of logical operators such as AND, OR, and NOT, as well as rules for defining relationships between concepts, such as inheritance, composition, and aggregation.

4. How can Ke Logic rules help in finding contradictions?

Ke Logic rules can help in finding contradictions by providing a systematic way to represent and reason about knowledge. By identifying and applying these rules, we can detect inconsistencies or contradictions in the knowledge base, which can then be corrected or resolved.

5. Are there any limitations to Ke Logic rules?

Yes, there are some limitations to Ke Logic rules. These rules are based on a formal and symbolic representation of knowledge, which may not always reflect the complexity and ambiguity of real-world situations. Additionally, the effectiveness of these rules depends on the accuracy and completeness of the initial knowledge base.

Similar threads

Replies
3
Views
2K
Replies
1
Views
2K
Replies
1
Views
1K
Replies
6
Views
3K
Replies
1
Views
3K
Replies
5
Views
2K
Back
Top