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https://plato.stanford.edu/entries/square/ says

A proposition is a subaltern of another iff it must be true if its superaltern is true, and the superaltern must be false if the subaltern is false.

Do the following two derivations imply each other?

  • if proposition P is true, then proposition Q is true

  • if proposition Q is false, then proposition P is false

Can the quote be simplified to:

A proposition is a subaltern of another iff it must be true if its superaltern is true.

or

A proposition is a subaltern of another iff the superaltern must be false if the subaltern is false.

Is the subalternation relationship between superaltern and subaltern a derivational one between two propositions, or implicational one between component propositions in a compound proposition? (I guess the former.)

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  • Yes; see Square of Opposition: "two propositions are said to stand in the relation of subalternation when the truth of the first (“the superaltern”) implies the truth of the second (“the subaltern”), but not conversely. An A propositions stand in the subalternation relation with the corresponding I propositions. The truth of the proposition “all plastics are synthetic,” implies the truth of the proposition “some plastics are synthetic.” 1/2
    – Mauro ALLEGRANZA
    Commented Sep 27, 2023 at 11:30
  • In traditional logic, the truth of an A or E proposition implies the truth of the corresponding I or O proposition, respectively. Consequently, the falsity of an I or O proposition implies the falsity of the corresponding A or E proposition, respectively. [...] The presupposition, mentioned above, that all categories contain at least one thing, has been abandoned by most later logicians." 2/2
    – Mauro ALLEGRANZA
    Commented Sep 27, 2023 at 11:31
  • Is the subalternation relationship between superaltern and subaltern a derivational one between two propositions, or implicational one between component propositions in a compound proposition?
    – Tim
    Commented Sep 27, 2023 at 11:40
  • The question: "Do the following two derivations imply each other?..." needs Contraposition because in traditional logic and modern classical one "Q is false" iff "not-Q is true".
    – Mauro ALLEGRANZA
    Commented Sep 27, 2023 at 11:43
  • In A's syllogism "A is true" implies "I is true" but in modern logic ∀x(Sx→Px) ⊭ ∃x(Sx&Px)
    – Mauro ALLEGRANZA
    Commented Sep 27, 2023 at 11:45

1 Answer 1

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Mauro's answer is complete and authoritative, and Wikipedia's article on subalternation is technical, but illuminating. Let's start with an example in natural language:

An example of a subalternation is "If all squares are quadrilaterals, then some quadrilaterals are squares."

From WP article Square of Opposition

All we've done is drawn a more particular conclusion from the original proposition. What is important is that the truth follows from form or syntax, not content or semantics. Thus, it is of the form:

∀s Qs ⊨ ∃s Qs

Where "∀s" means "all squares", "∃s" means "there exists some squares", and "Qs" means "squares are quadrilaterals". The symbol "⊨" can be read as "always entails" and means that there are no exceptions to the rule. Put together: All squares are quadrilaterals always entails some squares are quadrilaterals. In terms of the Venn diagram, this means that if all members of one circle are in another circle, some portion of those members must be in the circle.

Ultimately, it's one type of relationship that can be found to inhere to the square of opposition. You ask:

Is the subalternation relationship between superaltern and subaltern a derivational one between two propositions, or implicational one between component propositions in a compound proposition?

Thus, the superaltern always implies the subaltern, but not vice versa. If a conclusion and antecedent imply each other, then the implication is a contraposition and is qualified as biconditional. For instance:

"Some quadrilaterals are squares does not imply all quadrilaterals are squares" is true.

Indeed, kites are not. The biconditional is thus false.

"Some quadrilaterals are squares implies all quadrilaterals are squares" is false.

Material implication is a type of logical consequence, and a derivation in the language of logic is a proof. From the article on formal proof:

In logic and mathematics, a formal proof or derivation is a finite sequence of sentences (called well-formed formulas in the case of a formal language), each of which is an axiom, an assumption, or follows from the preceding sentences in the sequence by a rule of inference. It differs from a natural language argument in that it is rigorous, unambiguous and mechanically verifiable.

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  • 1
    "All squares are quadrilaterals *always entails some squares are quadrilaterals.*" Always entails? Could you give an example where p does entail q but not always?
    – Speakpigeon
    Commented Sep 27, 2023 at 15:26
  • @Speakpigeon Okay, let's keep it simple. Consider where p does not always entail q. Some bachelors are men but not all men are bachelors. Thus P->Q does not imply Q->P. In this case Some Q->P can be read "Q partially entails P". That is absolutely part of the square of opposition in a subaltern. All bachelors are men implies some bachelors are men. That is ALWAYS entailed. But being a man POSSIBLY or PARTIALLY or CONTINGENTLY entails being a bachelor contingent upon the man in question.
    – J D
    Commented Sep 27, 2023 at 15:47
  • The moral of the story is that the square of opposition and simple classical logics are formal systems constructed that model some states of affairs in the physical world and not others, and we can always vary the linguistic artifact to have more complex systems of logical consequence.
    – J D
    Commented Sep 27, 2023 at 15:48
  • @speakpigeon Okay, while I deleted my earlier portion, it's still probably important to point you to the difference between proof-theoretic and model-theoretic logical consequence. Besides contingent implication there is also implication across contexts or models. Thus, sometimes P->Q is true in System A and sometimes the same P->Q is false but in System B. Here again, P doesn't ALWAYS entail Q. "I am happy" is true for me, but it may not be for my opponent in the Fornite battle royale (zero sum win!)
    – J D
    Commented Sep 27, 2023 at 15:55
  • So, those are the proof-theoretic (single turnstile) and model-theoretic (double turnstile) examples of contingent implication.
    – J D
    Commented Sep 27, 2023 at 15:56

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