The long tales of a tail/less ape

July 22, 2017 Saturday

Bedtime Story

Analyzing Negation of Disjunction

The other law, which is the negation of a disjunction, is the conjunction of the negations.

The set theoretical way of stating it is that the complement of the intersection of two sets is the same as the union of their complements.

In formal propositional logic, it would be written as follows:

¬(P ⋁ Q) ⇔ (¬P) ⋀ (¬Q)

Both these laws seem knotty, but when shown in the form of Venn diagrams they become very obvious.

I am not going to put up these Venn diagrams and would rather encourage you to look for these up yourselves.

It is just a Google click away.

I want to analyze this second law negation of disjunction slightly more deeply and get some sense out of it.

Let us look at the left hand side of the arrow and see what it says.

¬(P ⋁ Q) implies that it is false that either of P or Q is true.

This phrase is establishing as a fact that neither P nor Q is true.

Hence it follows that both P is not true and Q is not true.

But this is exactly what the right hand symbols are saying
(¬P) ⋀ (¬Q)

We can also examine the equation other way around.

The right hand side is saying that P is false and Q is false.

It is same thing as saying that “not P” and “not Q” are true.

If P and Q are false, then so must be their disjunction.

In other words, the negation of their disjunction must be true.

This is exactly the meaning of the symbols on the left hand side.

Now let us go back to Abhari and his proposed solution to the Liar Paradox.

What Abhari is trying to say is that from ‘not both true and false’, you cannot infer not false so true.

Invoking De Morgan’s laws, negation of a conjunction does not entail negation of a conjunct.

From ‘not both true and false’, you cannot infer not false so true.

In modern terms, Abhari is contesting that the liar is depending on the elementary scope fallacy.

I shall take up this scope fallacy or rather the Modal scope fallacy in the nights to come.

Stay tuned to the voice of an average story storytelling chimpanzee or login at http://panarrans.blogspot.com

Good night mon ami and my fellow cousin ape.

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Another great educator and a teacher that I am aware of is Professor Subhashish Chattopadhyay in Bangalore, India.

While I narrate stories, Professor Subhashish an electronic engineer and a former professor at BARC, does and teaches real mathematics and physics.

He started the participation of Indian students at the International Physics Olympiad.

Do visit him here:

All his books can be downloaded for free through this link:

For edutainment and English education of your children, I recommend this large collection of Halloween Songs for Kids:

Saturday, July 22, 2017