Negations, Converses and Contrapositives

Hi there, dear reader! Hope everything is all right!

When I was in my University back in Brazil, I was never introduced to the concepts of Converses and Contrapositives in formal logic systems. I think the concept of changing a thing on just on perspective quite nice, much more nice, when such thing its not a literal negation.

Processing Negation:

Let's suppose the following statement is true:

x  X | P(x)  Q(x)

Then the Negation of that statement is:

 (x  X | P(x)  Q(x))

The quantifier is changed and the negation go inwards:

 x  X | (P(x)  Q(x))

Then, in order to complete the process of negation we have to convert the implication to its and/or form which becomes:

 x  X | (P(x)  Q(x))

Finally we arrive at the negation of the following sentence:

x  X | P(x)  Q(x))

That is, of course the negation of the first sentence!

One easy way to discover if a sentence is a Converse, Contrapositive or the Negation of some sentence is to look for the quantifier. If the quantifier is UNCHANGED then it is the Converse or the Contrapositive, since its properties, just change the relationship between implications, ands or ors.

The Contrapositive of the first sentence would be:

x  X | Q(x)  P(x)

And the Converse of the first sentence would be:

x  X | Q(x)  P(x)

In some sense, you can say that these sentences derived from the original represent some kind contradiction in relation to the first one but NOT its negative perspective. I just thought this would be a nice addition to the blog.

Hope everyone like it!