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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!

Unfortunately, the PhilosophySE does not support LaTex. See this meta post for explanations why requests to add this feature have been denied, as well as for suggestions of convenient alternatives for displaying formulae. I usually just copy and paste unicode symbols from online lists of common mathematical symbols.

– David H – 2014-09-28T03:01:05.290Thanks for the tip! I edited my question to change my LaTeX code into HTML symbols. – Mathemanic – 2014-09-28T03:10:24.813

2I think the term "negation" isn't used the same way in term logic. Look instead at the square of opposition. The contradictory of a categorical proposition would be the same as it's "negation" in modern logic. Modern logic has a heritage that treats logical propositions as algebraic equations, so negating is exactly like multiplying by -1. Term logic generally doesn't acknowledge such a thought process. At least terms such as "logical product" and "logical sum" didn't stick around... – Kevin Holmes – 2014-09-28T05:46:19.057