Thread: HELP: Rules of Inference

1. HELP: Rules of Inference

Dear everyone,

I have been studying Rules of Inference for whole day and still do not get the concept yet. It seems to be very hard for me to understand.

Do we need to memorize the table or Rules of Inference? I do not even understand the example given:
Example: Show that the premises "A student in this class has not read the book," and "Everyone in this class passed the first exam" imply the conclusion "Someone who passed the first exam has not read the book."

Solution: Let
C(x) be "x is in this class,"
B(x) be "x has read the book,"
P(x) be "x passed the first exam."

the premises are $\exists x(C(x) \wedge \neg B(x))$ and $\forall x (C(x) \rightarrow P(x))$. The conclusion is $\exists x(P(x) \wedge \neg B(x))$. The steps can be used to establish the conclusion from the premises.

Step________________________Reason
1. $\exists x(C(x) \wedge \neg B(x))$____________Premise
2. $C(a) \wedge \neg B(a)$________________Existential instantiation from (1)
3. $C(a)$________________________Simplification from (2)
4. $\forall x(C(x) \rightarrow P(x))$____________Premise
5. $C(a) \rightarrow P(a)$________________Universal Instantiation from (4)
6. P(a) ________________________Modus ponens from (3) and (5)
7. $\neg B(a)$_______________________Simplification from (2)
8. $P(a) \wedge \neg B(a)$________________Conjunction from (6) and (7)
9. $\exists x(P(x) \wedge \neg B(x))$____________ Existential generalization from (8)
It does not make much sense to me. It would be greatly appreciated if you could give me a prompt explanation.

any guide/tutorial to learn such kind of stuff?
Thank you so much.

2. Originally Posted by zpwnchen
Dear everyone,

I have been studying Rules of Inference for whole day and still do not get the concept yet. It seems to be very hard for me to understand.

Do we need to memorize the table or Rules of Inference? I do not even understand the example given:

It does not make much sense to me. It would be greatly appreciated if you could give me a prompt explanation.

any guide/tutorial to learn such kind of stuff?
Thank you so much.
Do you know what the rule of Existential Instantiation is and how is used in formal proof,because what you have written is a formal proof.

The rule of simplification is : $p\wedge q\Longrightarrow p$ or $p\wedge q\Longrightarrow q$

So now if you put : p=C(a) and q = $\neg B(a)$ and apply the rule of simplification you get line (3) of your formal proof

Now in line (4) you use the rule of Universal instantiation to get line (5).

This rule tell us in general if a property holds for the whole set then it will hold for each member of the set .

In our case everybody in the class passed the exams ,which translated in symbols is:

$\forall x[C(x)\Longrightarrow P(x)]$ or in words: for every ,x if C(x) then P(x),or for every x, if x is a student then x passed the exams .

And applying the rule of Universal instantiation : if Adam is a student of the class ,denoted by C(a) , then Adam has passed the exams,denoted by P(a) .

( $C(a)\Longrightarrow P(a)$)

Then you apply M.Ponens to lines (3) and (5) to get line (6). The general form of which is:

$[(p\rightarrow q)\wedge p]\Longrightarrow q$.

And if you put p=C(a) and q = P(a) and use the above law you get P(a) ,which is line (6)

line (7) is the result of applying the simplification rule to line (2)

Line (8) is the result of applying the conjunction rule to lines (6) and (7)

The general form of the conjunction rule is:

$p,q\Longrightarrow p\wedge q$.

And if you put p = P(a) and $q = \neg B(a)$ you get line (8)

AND finally we use Existential Introduction (or Generalization) to line (8) to get line (9).

Existential Introduction tell us in general that: if a particular thing or person in a set has a particular property ,then there exist someone in the set with that property .

In our case :Adam has passed the exams ( P(a) ) and has not read the book ( $\neg B(a)$

Hence there is someone in the class who has passed the exams and has not read the book:

$\exists x [P(x)\wedge\neg B(x)]$.

it is only left for you to learn the Existential Instantiation .

3. Thank you so much.

Can you help me with this one too?
Determine whether the argument is correct or incorrect and explain why?

Jenny likes all action movies. Jenny likes the movie Eight Men Out. Therefore, Eight Men Out is an action movie.
Thanks

4. Originally Posted by zpwnchen
Thank you so much.

Can you help me with this one too?

Thanks
If jenny liked only the action movies ,then the argument would be a valid one.

Now it is up to you to formalize the above argument and try to prove ,what i said.

Remember only in propositional calculus we can prove the validity of an argument and not in predicate calculus