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I am studying Aristotle's logic (Categorical logic AEIO). There are rules on how you can form arguments: Premise 1, Premise 2, and Conclusion: All A is B; All B is C; then "All A is C" is valid. Everything is clear if we have simple Arguments.

How to translate the next sentence in Categorical Logic: " The number of negative claims in the premises must be the same as the number of negative claims in the conclusion. If the argument follows this rule then it is valid otherwise it is not valid"?

Something like:

(C is Negative) and (P1 is negative) and not (P2 is negative)
(C is Negative) and not (P1 is negative) and (P2 is negative)
(C is Positive) and not (P1 is negative) and not (P2 is negative)

All provided argument is Follow rules
All Follow rules is Valid argument
The provided argument is Valid argument

It is not clear how to formulate such rules in Categorical logic.

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The first examples is not very clear.

The second one looks like:

Every argument that Follow the rules is a Valid argument (this is an A proposition: $\text {Afv}$).

The provided argument Follows the rules.

Therefore: The provided argument is a Valid argument.

The argument is valid but it is not, strictly speaking, a syllogism, because the second premise is not a Categorical proposition. "The provided example" refers to an individual and not to a general term.

Its logical form is:

$\forall x (Fx \to Vx), Fp \vDash Vp.$

The rule needed are: Universal instantiation and Modus Ponens.

Conclusion: not every valid argument can be formulated as a Syllogism.

Syllogism is a proper part of predicate logic: the part called Monadic predicate logic.

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  • $\begingroup$ The syllogisms argument is valid if and only if it does not violate rules. Is it possible to formulate these rules as syllogisms? en.wikipedia.org/wiki/Syllogism#Syllogistic_fallacies $\endgroup$ – Oleg Dats Jan 20 at 16:35
  • $\begingroup$ @OlegDats - the argument is valid but not all valid arguments are syllogistic. You can force the def of "categorical prop" saying that "every argument that is equal to the provided argument is an argument that follows the rule"... $\endgroup$ – Mauro ALLEGRANZA Jan 20 at 18:43

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