Find the negation of the following quantified sentence ( ∀ x ) ( p (...

Damon Stokes

Damon Stokes

Answered

2022-06-24

Find the negation of the following quantified sentence ( x ) ( p ( x ) q ( x ) ¬ q ( x ) )
My attempt:
To negate a quantified sentence, I just need to change the quantifier and connectives.
So: ( x ) ( p ( x ) q ( x ) ¬ q ( x ) )
Denying: ( x ) ( p ( x )   q ( x ) q ( x ) )
¬ ( ) =
¬ ( ) =
¬ ( ) =
¬ ( ¬ q ( x ) ) = q ( x )
I'm pretty sure that my solution is correct, the problem is that the denial of the conditional connective is causing me doubts.

Answer & Explanation

Angelo Murray

Angelo Murray

Expert

2022-06-25Added 23 answers

Step 1
Notes: ¬ ( ) =
Not quite: the negation of A B is ¬ A ¬ B ..
Remember, you are negating a sentence, not merely its main logical connective.
¬ ( )
Not quite: the negation of A B is A ¬ B .
¬ ( )
Not quite: the negation of ¬ x A ( x ) is x ¬ A ( x ) ..
¬ ( ¬ q ( x ) )
Correct.
Step 2
So: ( x ) ( p ( x ) q ( x ) ¬ q ( x ) )
Liberally inserting parentheses helps disambiguate sentences. In any case, the precedence of logical connectives dictates that this sentence ought to be read as x ( ( p ( x ) q ( x ) ) ¬ q ( x ) )
(I removed the unnecessary parentheses around the quantifier).
Denying: ( x ) ( p ( x )   q ( x ) q ( x ) )
This is incorrect, as pointed out in the comments and other answers. My above corrections make clear why.
Lovellss

Lovellss

Expert

2022-06-26Added 5 answers

Step 1
Using the definition of P Q, which is ( ¬ P ) Q.
Step 2
So, since x ( p ( x ) q ( x ) ¬ q ( x ) is the same as x ¬ ( p ( x ) q ( x ) ) ¬ q ( x ), its negation is x ¬ ¬ ( p ( x ) q ( x ) ) ¬ ¬ q ( x ) , i.e. x ( p ( x ) q ( x ) ) q ( x ) .

Do you have a similar question?

Recalculate according to your conditions!

Ask your question.
Get your answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?