Why no violation of Morgan’s law happens in the following example

discrete mathematicselementary-set-theory

I got the following from the book Discrete Mathematics with Applications 4th edition by Susanna Epp.

According to De Morgan’s laws, the negation of:

p: Jim is tall and Jim is thin
is ~p: Jim is not tall or Jim is not thin

Unfortunately, a potentially confusing aspect of the English language
can arise when you are taking negations of this kind. Note that
statement p can be written more compactly as

p´: Jim is tall and thin

When it is so written, another way to negate it is
~(p´): Jim is not tall and thin.
But in this form the negation looks like an and
statement.

Doesn’t that violate De Morgan’s laws? Actually no violation
occurs. The reason is that in formal logic the words and and or are
allowed only between complete statements, not between sentence
fragments.

I have difficulty understanding the last paragraph. Why is it that in this example no violation of the Morgan's law happens?

Best Answer

  • The negation ( properly speaking) of " Jim is tall and Jim is thin" is simply and purely :

it is not the case that ( Jim is tall and Jim is thin).

And in general, the negaton of proposition P is simply

NOT-P .

  • De Morgan's law tells you that the following sentence is equivalent to the negation of P, which is not exactly the same as being its negation :

it is not the case that Jim is tall OR it is not the case that Jim is thin.

  • The propositions not-(P&Q) and (not-P OR not-Q) are distinct syntactically, though they are equivalent semantically.

  • Note that , in " John is not tall and thin" the "and" operator is within the scope of the negation operator. Since the main operator is " not" it is not an " and" statement.

In the same way the statement : "not ( A Or B OR C OR D)" is not an OR statement.

Related Question