Non-equivalence of `i`-form and a claim of existence

0

What's the difference in meaning between an i-sentence and its corresponding existential claim?


In traditional logic, the following inference is valid

 All As are Bs.       AaB
-----------------    -----
 Some As are Bs.      AiB

But, according to this source cited in this comment, the inference from i-statement to existence is not valid. In order to make the sentence grammatical, please interpret B as an adjective phrase.

I'm not sure how to represent this symbolically in traditional notation, let E refer to existence as a predicate and C be defined as both B and A. Maybe it's also possible to extend the notation to allow empty positions in order to avoid making existence a predicate, even syntactically.

      Some As are B.             AiB                               AiB
BAD! -----------------     BAD! -----        (or possibly)  BAD! -------
      Some B As exist.           CiE                               Ci

For me, at least, it is really hard to see why Some As are B and Some B As exist are not paraphrases of each other and why they would not have the same truth conditions.

One mechanism I can think of to split the meaning (or at least the truth conditions) of Some As are B and Some B As exist is to say that we're determining the truth or falsity of a proposition in a larger world than just the collection of things that currently exist (or the collection of things that exist at any time) and make existence just another predicate in the system, but that seems unsatisfying and ad hoc.

Gregory Nisbet

Posted 2021-01-04T14:05:51.530

Reputation: 806

Answers

1

it is really hard to see why "Some As are B" and "Some B As exist" are not paraphrases of each other.

You are right... because they are paraphrases of each other.

"Some As are Bs" is read as: "there is something that is an A and a B".

The truth condition for the "and" connective is that P ∧ Q is TRUE exactly when both P and Q are TRUE.

Thus, "Some As are Bs" in modern symbolic logic is ∃x(A(x)∧B(x)), and it is false when there are no As.

"All As are Bs" is translated in modern predicate logic with "for all x, if x is an A, then x is a B", i.e.

"for every x: if Ax, then Bx" : ∀x(A(x)→B(x)).

The truth condition for "if..., then.." are defined by the truth table for the conditional connective: P → Q is TRUE when P is FALSE.

Thus, if there are no As, we have that A(x) is false for every value of x, and thus A(x)→B(x) is true for every value of x.

This is the point of view of modern symbolic logic.

According to traditional Square of Opposition:

"suppose that the A form is true. Then its contrary E form must be false. But then the E form’s contradictory, I, must be true. Thus if the A form is true, so must be the I form."

See also Subalternation:

The presupposition, mentioned above, that all categories contain at least one thing, has been abandoned by most later logicians. Modern logic deals with uninstantiated terms such as “unicorn” and “ether flow” the same as it does other terms such as “apple” and “orangutan”. When dealing with “empty categories”, the relations of being contrary, being subcontrary and of subalternation no longer hold.

This means that, according to traditional logic, we predicate the "B-ness" of something that is A when we know that there is something that is A. Thus, the "empty case" of predicating B of things that are A, when there are no A at all, does not apply.

Mauro ALLEGRANZA

Posted 2021-01-04T14:05:51.530

Reputation: 33 575