I am beginner of logic, and am writing an introduction to logic for a math book. I am of the impression that the three main areas of logic to explain are (in order) syllogistic logic, sentential logic, and predicate logic.
Beginning with syllogistic logic, I state that a syllogism is a collection of three statements, where each statement is in the form of a "categorical proposition". There are exactly four possible categorical propositions:
All x are y All x are not y Some x are y Some x are not y
One might think of
no x are y and suggest this as another possible categorical proposition, but I believe this is equivalent to
all x are not y. Similarly, the statement
no x are not y is equivalent to
all x are y. Would this be correct?
Secondly, I know that in sentential logic, every statement has a negation. For example,
¬(P ∨ Q) ≡ ¬P ∧ ¬Q. However, I noticed that neither the Wikipedia page for Syllogism nor the Wikipedia page for Categorical Proposition mention negations, anywhere. It is as if negations of categorical propositions don't exist in Syllogistic logic. However this seems strange to me, because based on my own intuition, I would suggest that each has a negation, which I would choose to be:
¬(All x are y) ≡ Some x are not y ¬(All x are not y) ≡ Some x are y ¬(Some x are y) ≡ All x are not y ¬(Some x are not y) ≡ All x are y
This only comes from my own intuition. However it seems to me to be correct. However, as I mentioned, none of the Wikipedia pages for Syllogistic Logic, Categorical Propositions, etc. make mention of negations of these statements, as if they do not exist in this system. Am I missing something?
Thanks for your thoughts!