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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 Sep 28 '14 at 3:01
  • Thanks for the tip! I edited my question to change my LaTeX code into HTML symbols. – Mathemanic Sep 28 '14 at 3:10
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    I 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 Sep 28 '14 at 5:46
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You are "missing" The Traditional Square of Opposition.

As you say :

‘Every S is P’ and ‘Some S is not P’ are contradictories.

The "traditional" symbolization is :

  • SaP for "all S are P"

  • SeP for "no S is P"

  • SiP for "some S is P"

  • SoP for "some S is not P".

o and i are the negations of a and e respectively.

Thus : not SaP will be "not all S are P" i.e. "some S is not P", which is SoP.

The same for not SeP, i.e. "not no S is P" i.e. "some S is P", whcih is SiP.


Note

From a modern point of view, the "order" must be :

  • sentential logic,

  • syllogistic logic,

  • predicate logic.

Syllogistic logic is called also monadic predicate logic, because it is simply the subset of predicate logic with all predicate letters having "arity" one, i.e.monadic.

The arity of a predicate letter is the number of its argument-places.

Thus "... is father of ..." and "... is less than ..." are dyadic : arity = 2 (two argument places; usually called : binary relations).

Categorical syllogism uses only predicates with one argument place, like "... is a men", "... is mortal"; arity = 1 (one argument place).

This is the reason why we cam "model" it with the language of sets (or classes) : "all Man are Mortal" is equivalent to : the set of Men is included into the set of Mortals.

See the first modern mathematical logic textbook :

  • Ah - this is quite the perfect answer! Thank you, @Mauro. So would you also say that sentential logic is a "subset" of syllogistic logic, where the "contradictory" of syllogistic logic corresponds to the "negation" of sentential logic? That is, the negation of a categorical proposition is its contradictory from the square of opposition? P.S., thanks for the book recommendation; I've just put it on order through Amazon. – Mathemanic Sep 28 '14 at 15:51
  • One other question - I'm a little bit confused what exactly is the predicate... It seems you call "...is mortal" (or in general the phrase, "...are P" in the sentence "all S are P") a predicate. However, the Wikipedia page only refers to the single letter P as the predicate. That is, the descriptive terms are "S" (subject) and "P" (predicate), whereas "all" and "are" are logical terms. Thanks for helping me sort this out. – Mathemanic Sep 28 '14 at 15:57
  • Yes, P, S are predicate letters in modern logic with arity. Thus the "standard" categorical clause of syllogism : "all S are P" is translated in modern logic as "for all x(if S(x), then P(x))". Thus "all Men are Mortal" is translated into "for all x(if x is a Man, then x is Mortal)" or equivalently as "for all x(if Man(x), then Mortal(x))". I'm using both "Man(x)" and "x is a Man" to translate the formal "P(x)". – Mauro ALLEGRANZA Sep 28 '14 at 16:08
  • About sentential logic, it is usually described first in order to introduce the basic concept, and most of all the connectives. But it is correct that we can see it as a subset of predicate logic. Of course, if we read Barbara as a sentential formula, it amounts to "if p and q, then r" which of course is not a valid argument at all... – Mauro ALLEGRANZA Sep 28 '14 at 16:10
  • In addition to SEP's entry on Aristotle's Logic you can see Günther Patzig, Aristotle’s Theory of Syllogism (1968). – Mauro ALLEGRANZA Sep 28 '14 at 16:59

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