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Do different elements of logic have different epistemological strength? By epistemological strength, I mean epistemological certainty or the certainty the concept is true and grounded on truth. For example, numbers, they're representations, so they're certainly true, 100% true, but identity being defined as being the reference to itself seems like true, but not as true as numbers, and the law of identity is even less certain, so are there different epistemological strength for each concept?

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    In what sense is a number true (or false)? I don't know what it means for a truth value to apply to a number. – Noah Schweber Jun 9 at 23:11
  • it's true in the sense that numbers exist since I can count the things I see. – Sayaman Jun 9 at 23:17
  • You make a number of claims here that need to be explained and justified: "numbers are representations". Representations of what? A representation has to have something that it represents. "Identity is defined as being the reference to itself". I've never seen a definition like this, and I don't understand what it would mean. "The law of identiy is even less certain". How can something not be itself? – David Gudeman Jun 10 at 1:50
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There's no universal criterion to judge your epistemic certainty (strength) of logic elements and it depends on your metaphysical position such as empiricism and idealism and that's why you need to learn philosophy. For example, for Aristotle who's more of an empiricist may not believe the redness is exactly the same and certain regarding a red apple and a red table. While for Plato who's more of an idealist does believe the shared redness as a Form exists in some independent ideal realm. Thus redness is more certain for Plato than Aristotle. Similar to your comment that number as a Form you can clearly sense and count, so you believe it's more certain than identity which for some reason you have trouble to discern.

Finally not all numbers share the same level of certainty examined under your countability criterion. Countability is essentially the central feature of Turing machines. So per your positivist's criterion only computable numbers are certain. From Church-Turing thesis we know computable numbers must be countable even infinite in cardinal, however, real numbers are uncountable so most irrational numbers are uncomputable. Actually most real numbers are not epistemically certain per your criterion and contrary to your unexamined understanding you should actually rationally claim most numbers are very uncertain/ungrounded or maybe don't exist at all in your world...

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All the parts of logic are only convincing to one degree or another. This is a problem that all axiomatic notions have when they meet real life. Since I am a personal fan of Steven Kleene's, my favorite example is the Law of the Excluded Middle.

Is it true or false that the color of every suit made of titanium is purple? If it is true, then they are all chartreuse, too. How on Earth does that make sense? But there is surely no counterexample. Given that, what makes the inverse process to enumerating an empty set, like a proof of existence by contradiction, seem reliable?

So is every clear statement either true or false? Aristotle says so. Intuitionists and Constructivists disagree.

I have already given an answer about the proposed conventions around the law of identity that I find convincing.

But the range of settled content may not be as broad as you think. Numbers are only kind of safe. The real nature of things like the Reals has open positions for some thinkers. Should we make something so pervasive involve multiple layers of actual infinities? Are they their own thing, or just a model of a geometric concept? Is the notion of continuity separable from the idea of infinitesimal differences, is it different from the notion of convergence? Etc.

Radicals like ultra-finitists might agree each integer exists, but suggest that using the collection of integers as an object is a philosophical mistake that we should avoid.

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