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Categorical Propositions:

I'm looking at Categorical Propositions of which there are 4 forms: A, E, I, O. These is the wording of forms according to the Wikipedia article:

  1. A. All S is P
  2. E. No S is P
  3. I. Some S is P
  4. O. Some S is not P

Notationally, these mean the following:

  1. A. $\forall x(Sx \implies Px)$
  2. E. $\forall x(Sx \implies \lnot Px)$
  3. I. $\exists x(Sx \land Px)$
  4. O. $\exists x(Sx \land \lnot Px)$

I want to find a more precise wording that is still equivalent to the original meaning. I find the current wording ambiguous and lacking in other ways.

The English quantifier "All" is ambiguous:

To me, "All" is ambiguous; it can be ambiguous when $S$ can assume a property individually or as group. For example, "All watermelons weigh $X$ lbs." can mean two different things:

  1. Weigh each watermelon, find that each individual watermelon weighs $X$ lbs.
  2. Weigh each watermelon, sum the weights, find that the sum equals $X$ lbs.

Usually, it's possible to disambiguate between 1. and 2. if there's enough context, but wording that has to rely on context to disambiguate is undesirable. Importantly, sometimes there's really not enough context. Say you know there are two watermelons in the room. Given the proposition "All watermelons in the room weigh $15$ lbs.", there's not enough context to disambiguate between 1. there being two medium-sized watermelons in the room that each weigh $15$-lbs or 2. there being two small-sized watermelons in the room with a summed weight of $15$-lbs.

Alternative English quantifiers and wordings:

I considered the following English quantifiers: always, all, every, each, any, some, no, never, non, not (are there others?) and came up with the following wording. It sounds more precise to me and avoids the ambiguity of using "All":

  1. A. Each S is P
  2. E. Each S is non-P
  3. I. Some S is P
  4. O. Some S is non-P

It's also more symmetric (rather than all, no, some, some we have each, each, some, some which seems more symmetric.

Maybe even better, though wordier:

  1. A. Each S is P
  2. E. Each S is non-P
  3. I. There exists S that is P
  4. O. There exists S that is non-P

There are still some issues with these two wordings:

"Some" and "There exist(s)" don't convey singular and plural possibilities:

One issue with "Some" and "There exists ... that" is that they make it sound like there's exactly one "S that is P/non-P". Is there a wording that avoids this issue and naturally allows for the singular and plural possibilities simultaneously? "There is/are one or more S that is P."?

A and E forms can be vacuously true:

The wording should allow for the possibility of vacuously true A and E propositions. It's my understanding that since the A and E forms are quantifications over material implications ($P \implies Q$), the A and E forms can each be vacuously true if their antecedent ($P$) is false. So for example, since unicorns don't exist the following A and E propositions are each vacuously true: "Each unicorn is mammalian." and "Each unicorn is non-mammalian.".

"non-P" vs "not P":

Also, do you make a distinction between "non-P" and "not P"?

joseville
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    Usually, with "not P" we mean the negation of the proposition P: if P is "Socrates is a philosopher", then "not P" will be "Socrates is not a philosopher" – Mauro ALLEGRANZA Nov 17 '21 at 19:50
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    With "non-P" in this context we mean the negation of the predicate P, like e.g. "not-white". – Mauro ALLEGRANZA Nov 17 '21 at 19:51
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    "All" means all; the modern translation of syllogism uses the quantifier on individual variables: $\forall x (Sx \to Px)$. According to "traditional" discussions, we can read the syllogism with "quantifiers" acting on class terms (predicates). – Mauro ALLEGRANZA Nov 18 '21 at 08:50

1 Answers1

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One issue with "Some..." and "There exists a..." is that they make it sound like there's exactly one "S that is P/non-P".

I disagree: while the statements “some goat is blue” and “there exists a goat that is blue” indeed refer to a single blue goat, their point is that there is at least one blue goat, rather than ultimately saying that there is exactly one blue goat. These phrasings do fully correspond to existential quantification, as desired, and the reader/listener who precludes the possibility of multiple blue goats is merely projecting.

To me, "All" is ambiguous; it can be ambiguous when $S$ can assume a property individually or as group.

I couldn't agree more. When making precise assertions, I've long preferred ‘each’/‘every’ rather than ‘all’. ‘Any’ can be even more willy-nilly.

do you make a distinction between "non-P" and "not P"?

We prefix a word, like an adjective or the name of a set, with ‘non-’ to obtain its complementary meaning.

On the other hand, we use the word ‘not’ to negate (flip the truth value of) a predicate or sentence.

ryang
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