8. Propositional Logic Natural deduction - negation

Solved problems Problem: ¬A∧¬B can be derived from ¬(A∨B). (De Morgan law)

Last Jouko Väänänen: Propositional logic viewed Problem: ¬A∧¬B can be derived from ¬(A∨B). (De Morgan law)

Let us first think intuitively why ¬A∧¬B should follow from ¬(A∨B).

Last Jouko Väänänen: Propositional logic viewed Problem: ¬A∧¬B can be derived from ¬(A∨B). (De Morgan law)

Let us first think intuitively why ¬A∧¬B should follow from ¬(A∨B).
Say, it is not true that it rains or snows. Why can we conclude that it neither rains nor snows? Well, because if it for example rained, then it would a fortiori rain or snow, so we would contradict the underlined assumption.

Last Jouko Väänänen: Propositional logic viewed Problem: ¬A∧¬B can be derived from ¬(A∨B). (De Morgan law)

Let us first think intuitively why ¬A∧¬B should follow from ¬(A∨B).
Say, it is not true that it rains or snows. Why can we conclude that it neither rains nor snows? Well, because if it for example rained, then it would a fortiori rain or snow, so we would contradict the underlined assumption.
We try to make this formal.

Last Jouko Väänänen: Propositional logic viewed Problem: ¬A∧¬B can be derived from ¬(A∨B). (De Morgan law)

Last Jouko Väänänen: Propositional logic viewed Problem: ¬A∧¬B can be derived from ¬(A∨B). (De Morgan law)

Let us continue thinking intuitively why ¬A∧¬B should follow from ¬(A∨B).

Last Jouko Väänänen: Propositional logic viewed Problem: ¬A∧¬B can be derived from ¬(A∨B). (De Morgan law)

Let us continue thinking intuitively why ¬A∧¬B should follow from ¬(A∨B).
Since we are proving a conjunction we can take each conjunct separately. Let us look at ¬A. Assuming A gives A∨B, contradicting immediately the assumption ¬(A∨B). So we must conclude ¬A. Similarly we get ¬B.

Last Jouko Väänänen: Propositional logic viewed Problem: ¬A∧¬B can be derived from ¬(A∨B). (De Morgan law)

[A] [B] ∨I ¬(A∨B) A∨B ∧I (A∨B)∧¬(A∨B) ¬I ¬A ¬B ∧I ¬A∧¬B

Last Jouko Väänänen: Propositional logic viewed Problem: ¬A∨¬B can be derived from ¬(A∧B). (Another de Morgan law)

This is more difficult!
We want to conclude ¬A∨¬B, so the temptation is to try to derive one of ¬A and ¬B.
But which one??
This is related to the difference between so called constructive logic and classical logic.
Our logic is classical.

Last Jouko Väänänen: Propositional logic viewed Problem: ¬A∨¬B can be derived from ¬(A∧B). (Another de Morgan law)

Let us first think intuitively why ¬A∨¬B should follow from ¬(A∧B).
Say, a dish does not contain both cream and meat. Why can we conclude that either cream is missing or meat is missing? Well, because if both cream and meat were there, we would contradict the underlined assumption, so one of them must be missing.
We try to make this formal.

Last Jouko Väänänen: Propositional logic viewed Problem: ¬A∨¬B can be deri∨ed from ¬(A∧B). (Another de Morgan law)

Let us again first think intuitively why ¬A∨¬B should follow from ¬(A∧B).
Let us assume ¬A∨¬B is false i.e. ¬(¬A∨¬B) and work towards a contradiction. Now clearly ¬A leads to a contradiction, so ¬¬A i.e. A. Respecti∨ely B. So A∧B. This contradicts the assumption ¬(A∧B). So we get ¬¬(¬A∨¬B) i.e. ¬A∨¬B.

Last Jouko Väänänen: Propositional logic viewed Problem: ¬A∨¬B can be derived from ¬(A∧B). (Another de Morgan law) [¬A] [¬B] ∨I [¬(¬A∨¬B)] ¬A∨¬B ∧I (¬A∨¬B)∧¬(¬A∨¬B) ¬I ¬¬A ¬E A B ∧I A∧B ¬(A∧B) ∧I (A∧B)∧¬(A∧B) ¬I ¬¬(¬A∨¬B) ¬E ¬A∨¬B

Last Jouko Väänänen: Propositional logic viewed Example: A∨¬A is derivable

Intuition: We use indirect proof. So we assume ¬(A∨¬A) and derive a contradiction. Now A leads to A∨¬A and hence to a contradiction. Thus we may conclude ¬A. But this leads to A∨¬A and hence to a contradiction, and we are done.

Last Jouko Väänänen: Propositional logic viewed Example: A∨¬A is derivable

We make an A assumption ∨I A∨¬A

Last Jouko Väänänen: Propositional logic viewed Example: A∨¬A is derivable A ∨I ¬(A∨¬A) A∨¬A ∧I ¬(A∨¬A)∧(A∨¬A)

Temporary assumption

Last Jouko Väänänen: Propositional logic viewed Example: A∨¬A is derivable This assumption is now eliminated [A] ∨I ¬(A∨¬A) A∨¬A ∧I ¬(A∨¬A)∧(A∨¬A) ¬I ¬A

Last Jouko Väänänen: Propositional logic viewed Example: A∨¬A is derivable [A] ∨I ¬(A∨¬A) A∨¬A ∧I ¬(A∨¬A)∧(A∨¬A) ¬E ¬A ∨I A∨¬A ∧I

Last Jouko Väänänen: Propositional logic viewed Example: A∨¬A is derivable [A] ∨I ¬(A∨¬A) A∨¬A ∧I ¬(A∨¬A)∧(A∨¬A) ¬E ¬A ∨I ¬(A∨¬A) A∨¬A ∧I ¬(A∨¬A)∧(A∨¬A)

Last Jouko Väänänen: Propositional logic viewed Example: A∨¬A is derivable [A] ∨I [¬(A∨¬A)] A∨¬A ∧I ¬(A∨¬A)∧(A∨¬A) This assumption is ¬E now eliminated ¬A ∨I [¬(A∨¬A)] A∨¬A ∧I ¬(A∨¬A)∧(A∨¬A) ¬I ¬¬(A∨¬A)

Last Jouko Väänänen: Propositional logic viewed Example: A∨¬A is derivable [A] ∨I [¬(A∨¬A)] A∨¬A ∧I ¬(A∨¬A)∧(A∨¬A) ¬E ¬A ∨I [¬(A∨¬A)] A∨¬A ∧I ¬(A∨¬A)∧(A∨¬A) ¬I ¬¬(A∨¬A) ¬E A∨¬A

Last Jouko Väänänen: Propositional logic viewed