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I am reading a paper¹ which explains criticism of Kant’s putative argument that the cosmological argument reduces to the ontological one. To show that, a syllogism of this form is needed:

  • (Major) The proposition ‘Every absolutely necessary being is at the same time the most real being’ is, by reason of its logical form (ratione formae), convertible per accidens, i. e. ‘Some most real beings are at the same time absolutely necessary beings.’
  • (Minor) But one most real being does not differ the least bit from another, and thus what holds of some such beings holds also of all.
  • (Conclusion) Therefore, the proposition ‘Every absolutely necessary being is at the same time the most real being’ is, by reason of its matter or content (ratione materiae), convertible simpliciter, i. e. ‘Every most real being is an absolutely necessary being.’

Philosopher J. A. Eberhard, contemporary of Kant, argue, as far as I understand, that is not valid. He illustrates that by using Theodosius of Bithynia’s Sphaerica’s theorem ‘All circles passing through the centre of a sphere are the largest circles of the same sphere.’ Theodosius did not argue ‘all the largest circles of the same sphere do not differ in the least in abstracto, “so if it holds for some of them that they pass through the centre of the sphere, then it should hold for all”; rather, the Greek astronomer and mathematician of antiquity specifically demonstrated the converting proposition’. But, as far as I can see, that would have been a valid proof. What am I missing?

(I am not interested in the question of Eberhard’s understanding of Kant, merely in the logical point he rises.)

¹https://www.degruyter.com/document/doi/10.1515/kant-2022-2012/html

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    K's argument is not a formal syllogism but it seems correct to me. The 1st premise "Every A is R" converts, provided that there are As, to "Some R is A", and this is formally correct. Commented Mar 30 at 16:10
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    The 2nd premises says that all Rs share the same properties (they do not differ...) and thus they share also the property of being A. Therefore, every R is A". Commented Mar 30 at 16:11
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    It seems that K is applying the Indiscernibility of identicals Commented Mar 30 at 16:28
  • @MauroALLEGRANZA That is possible, but I find it weird. Kant did know the principle, but he does not talk about it in this section of the CPR. Moreover, funnily enough, he says his syllogism is presented in a scholastically correct way, but the principle of identity of indiscernibles was not known to the medieval AFAIK, and his minor premise is not stated as it should be in term logic. Commented Mar 30 at 20:47
  • "One point on a circle does not differ the least bit from another, so if one of them is on a given line then all of them must be on that line". Kant potentially equivocates on "do not differ" and "the same" in a similar way between the minor premise and the conclusion. "Does not differ the least bit" in abstracto does not convert into "the same" in relation to something else. He could, of course, argue additionally that they are relevantly "the same", but it would go beyond his "syllogism", and that is exactly what Eberhard says Theodosius did, unlike Kant.
    – Conifold
    Commented Mar 30 at 22:36

3 Answers 3

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Why is it invalid to conclude from ‘Some A is B’ and ‘One A does not differ the least bit from another’ that ‘Every A is B’?

"Some A is B" may be rendered "∃x(Ax ∧ Bx)"

"One A does not differ the least bit from another" may be rendered as "∀x,y(Ax ∧ Ay ⟹ ¬∃P([Px ∧ ¬Py] ∨ [¬Px ∧ Py])" which is (only classicaly?) equivalent to "∀x,y(Ax ∧ Ay ⟹ ∀P(Px ⟺ Py))"

so taking Leibniz identity of indiscernibles one may easily prove that

"∀x(Ax ⟹ Bx)", and that's "Every A is B"

In fact, one may have taken "One A does not differ the least bit from another" directly to mean "∀x,y(Ax ∧ Ay ⟹ x = y)", so it seams pretty much valid

theorem ‘All circles passing through the centre of a sphere are the largest circles of the same sphere.’

Well, maximal elements in a partial order need not be maximum elements

‘all the largest circles of the same sphere do not differ in the least in abstracto,

any choice of coordinate system will show they do in fact differ

“so if it holds for some of them that they pass through the centre of the sphere, then it should hold for all”

non-sequitur

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  • Great answer, just to add on, it’s quite perplexing how Eberhard cited Theodosius’ lack of use of a specific form of argumentation to attempt to establish its invalidity…
    – Max Maxman
    Commented Mar 30 at 16:48
  • @ac15 Right, so Kant needs the principle of identity of indiscernibles. Commented Mar 30 at 20:34
  • @MaxMaxman Technically, I am the one trying to understand Rogelio Rovira explaining Eberhard’s putative take on Theodosius. Lots of wiggle room, here. Roviro try to explain Eberhard’s point with Porphyry’s notion of proprium. But I did not want to confuse more things. You can read it in the paper if you want. It is in the fourth part, 8th paragraph. It did not help me to understand why Kant’s argument would be invalid. Commented Mar 30 at 20:38
  • @TempLogicKant personally, I would've taken "not differ the least bit from another" directly to mean equality, but I chose to formulate it this way for clarity
    – ac15
    Commented Mar 30 at 20:46
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The argument is valid if we take "does not differ in the least bit" to mean either:

  • "is equal", i.e. all As are equal to each other.
  • or "does not differ in a relevant way", i.e. in terms of whether or not it is B, which is to say all As are Bs or no As are Bs.

An example of the second point might be something like this:

P1: Some circles are round.
P2: Circles do not different in terms of whether or not they are round.
C: Every circle is round.

We could also phrase this as "if some A is B, then every A is B", in which case we have a standard hypothetical syllogism form:

P1: Some A is B.
P2: If some A is B, every A is B.
C: Every A is B.


That section of what the author presented does seem to reflect Kant's argument:

some entia realissima are absolutely necessary beings. But no ens realissimum is in any respect different from another, and what is valid of some is valid of all. In this present case, therefore, I may employ simple conversion, and say: “Every ens realissimum is a necessary being.”

Saying they don't differ "in any respect" seems to suggest equality. But I'll leave aside discussion of the reasoning behind Kant's claim here.

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I couldn't follow your narrative question, but I can certainly give an example that suits your headline question, as follows...

Some electrons are in a battery on my desk, one electron does not differ in the least bit from any other, therefore all electrons are in a battery on my desk.

Clearly the conclusion of the sentence is invalid.

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    But if the "shared properties" include also the property of being in a battery on your desk... you are exlcludind spatio-temporal location from what counts to "identify" an object, and K refers to non-physical "objects". Commented Mar 30 at 16:21
  • @MauroALLEGRANZA To follow your logic, the shared properties would have to include having exactly the same location. If you assume all electrons are not just identical but all in exactly the same place, then you have reached another form of absurdity. Commented Mar 30 at 20:03
  • @MarcoOcram Nothing absurd here, though. It would just mean that there is only one, or none, electrons at all. Commented Mar 30 at 20:31

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