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According to Copi's Introduction to Logic, in the square of opposition in Aristotle's Syllogistic Logic, there are four kinds of relationships between propositions:

  • contradictory: two propositions can't be both true, and can't be both false. (In Syllogistic Logic, this turns out to be that the propositions have the same subject term, same predicate term, and different quality and different quantity.)
  • contrary: two propositions can't be both true, but can be both false.(In Syllogistic Logic, this turns out to be that the propositions have the same subject term, same predicate term, and the same quantity, but different quality.)
  • subcontrary: two propositions can't be both false, but can be both true. (In Syllogistic Logic, this turns out to be that the propositions have the same subject term, same predicate term, and same quantity, but different quality.)
  • subalternation: two propositions have the same subject term and the same predicate term and the same quality (affirmative or negative), but different quantity (universal or particular).

Is it correct that contradictory, contrary and subcontrary can be generalized/defined for any other logic systems other than syllogistic logic, because their definitions don't depend on anything specific to syllogistic logic?

Does subalternation apply only to syllogistic logic, but not to other logic systems? Can subalternation be generalized/defined for other logic systems other than syllogistic logic?

Specifically, can subalternation be defined similarly to contradictory, contrary and subcontrary, in terms of whether two propositions can't be both true or can't be both false?

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See The Traditional Square of Opposition.

If we use modern predicate logic to formalize categorical propositions, the subalternation relation maust be expressed using the two formulas:

∀x(Sx→Px) for A and ∃x(Sx&Px) for I.

With modern logic, the issue is that the first one is True when there are no Ss, while in that case the second one is False.

The issue is known as the Existential Import of Aristotle's syllogism (see also this post).

For a modern overview see :

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  • +1 Citing the Logic Museum. Thanks for the resource!
    – J D
    Sep 27, 2023 at 15:16
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    @JD - you are welcome :-) Sep 27, 2023 at 15:39

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