Logic Clarification - Negating Always/Only?

Hi! I'm new, and I'm not sure if this is the correct subheading for my topic of query (discrete math), but I thought it fit best here. If it should be moved, please feel free to do so!

I have the whimsical expression "If a person is an insane vampire, then he believes only in false things and always lies."

I have converted it to symbols appropriately, with definitions:

p = A person is an insane vampire
q = He only believes in false things
r = He always lies

My final expression, which I'm confident in, for the negation is p&(~q + ~r)

What is tripping me up is the conversion back into words. When negating my propositions, are the words 'always' and 'only' quantifiers for the expressions? Or would their negation simply be: "He does not only believe in false things" and "He does not always lie" respectively?

Any help would be greatly appreciated!

Rome_Leader said:

I have the whimsical expression "If a person is an insane vampire, then he believes only in false things and always lies."

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Is it correct that you are to write the negation of the above statement?


If so its negation is:
A person is an insane vampire and, he believes some true things or sometimes he tells the truth.

If that is not the task, please explain what is to be done.

Yes, the negation is my task. What I have come up with, as I said, is: p&(~q + ~r). My difficulty you appear to have corrected in your word translation, over whether or not 'always' and 'only' are qualifiers, which you appear to say is the case. I had wondered if it would be valid to say "not always" instead of sometimes, but that does not appear correct. Am I right here?

Rome_Leader said:

Yes, the negation is my task. What I have come up with, as I said, is: p&(~q + ~r). My difficulty you appear to have corrected in your word translation, over whether or not 'always' and 'only' are qualifiers, which you appear to say is the case. I had wondered if it would be valid to say "not always" instead of sometimes, but that does not appear correct. Am I right here?

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It is truly a matter of style and/or personal taste.
However, we usually teach that the negation of all are is some not .
So you see why I did it that way.

You should consult the text/notes.