Reading about the paradox of analysis here:


Instead of the brother=male sibling example given in the article, suppose we use "triangle" instead. So 3 ways I can refer to a triangle are:

  1. triangle
  2. A polygon that has 3 sides
  3. A polygon that has 3 angles

So the paradox the way I understand it is... if these 3 denote the same concept... then it is vacuous to say that a triangle is a polygon with 3 sides or a polygon with 3 angles.

Why can't we simply say... our minds can apprehend the same object (in this case triangle) in multiple ways. So there isn't a one-correct way to "apprehend" a triangle. There are multiple ways... yet they point to the same entity a triangle. Even different people may apprehend the same word differently... ie: what exactly comes to mind when someone uses the word 'polygon'. It may different student to student.

So our minds don't necessarily grasp entities like triangles "directly". There are different configurations of concepts that happen to point to the same object... but it isn't immediately obvious to our minds that they point to the same object.

So "3-sided polygon" and "3-angled polygon" are different in the sense that they are different ways for a mind to apprehend an object... which only upon further reflection we see to be the same.

  • This is exactly Frege's distinction between Sinn und Bedeutung Commented May 10, 2020 at 14:18
  • Yes. But I assume Frege's distinction isn't commonly taken as a solution to the paradox? Commented May 10, 2020 at 14:45
  • Frege's classical example was the evening star and the morning star as two different ways to "apprehend" Venus. Yet they are not interchangeable without shift of meaning in "The evening star is the morning star". Same with "polygon with 3 sides is polygon with 3 angles". This generally happens with substitution into intensional contexts. So this is not a solution because the intersubstitutivity fails, hence they are different concepts. Indeed, it is the first non-solution discussed under your Wikipedia link.
    – Conifold
    Commented May 10, 2020 at 14:48
  • What about the example given in the article... "brother" and "male sibling"... are those different concepts? Commented May 10, 2020 at 14:56
  • Either that or "brother = male sibling" is a mere linguistic stipulation with no conceptual import. That is the dilemma of analysis, and both horns are unattractive.
    – Conifold
    Commented May 10, 2020 at 15:06


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