If all rational, logical, philosophical, scientific, and mathematical reasoning starts with the initial axiom that 'X=X' - that is a 0 information tautology - can it be said that all ensuing rational, etc statements are therefore empty of any Essential truth and only nominal?

  • Any rational, logical, philosophical, scientific, or mathematical reasoning does not start with this initial axiom, or any other. It starts with learning how to use words and symbols used to communicate abstract thoughts, and axioms come very late in the process, if at all. People thought and reasoned long before anyone knew what an axiom is. But even if we take "axiomatic development" as a (very lousy) rational reconstruction of the process, as long as additional, empirical or analytic, "axioms" are also used, the supposed essential vacuity of X=X is moot.
    – Conifold
    Jan 29, 2018 at 8:27
  • It's the first classical law of thought and the second Peano axiom (first is that 0 is a number).. why axiomatic development is lousy, given that rational thought and mathematics do a fine job of describing the observed world, would be another question. Jan 29, 2018 at 8:32
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    You can see the post: exactly-what-was-wittgensteins-argument-against-identity regarding the similar Wittgenstein's view about the "emptysness" of identity statements. Jan 29, 2018 at 8:42
  • It is indeed very fine as a description of the world, it is lousy at describing how we reason. For one, it does not capture how those claims that are reasoned about are arrived at, even in arithmetic, certainly not by spinning deductive chains. And most of humanity reasoned without Aristotle's laws, let alone Peano axioms, for most of its history. But let's grant their "primality" for the sake, how do you get from "first law of thought is empty" to "everything else is empty"? What happens to synthetic inputs mixing with "empty" analytic ones to produce "essence"?
    – Conifold
    Jan 29, 2018 at 8:55
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    @syntonicC We can certainly apply a model to processes before its creation but only if it is a good model of those processes. After Wittgenstein's, Toulmin's, etc., critique few maintain that the positivist idea of human reasoning as modeled on deductive chains in axiomatic systems is viable. Indeed, it takes quite a bit of effort to master deductive reasoning. Formal logic is a good justificational tool, but as analysis of actual arguments shows it has little to do with reasoning itself.
    – Conifold
    Jan 31, 2018 at 2:59

1 Answer 1


'Identity' has several shades of meaning. 'X = X' does seem particularly vacuous. But if you translate identity to some perfectly ordinary contexts, it proves a useful concept.

  1. Diachronic identity - identity over time. If I am the same person now who defrauded a client a year ago, then I now am responsible for what I did a year ago. I am the same person, after all : I am not someone else, e.g. you. Without X = X (GT at time 2 = GT at time 1) by legal criteria, personal responsibility for past actions would not hold. It might be convenient if it didn't !

  2. 'By legal criteria' : this brings out an important point. There is no single set of criteria by which to apply X = X. By biological criteria I might not be identical now to myself a year ago : pretty certainly I am not. This suggests the familiar point that identity is relative to a description. Not everyone accepts this thesis of Peter Geach's but I do. Suppose (an unoriginal example) I bought a clay statue last year, a statue of John F. Kennedy. I change my political attitudes and remould it into a statue of Richard Nixon. For all practical purposes, it is the same clay and it is important to insist on this. 'Of course, it's the same clay', I say : X = X. (X at time2 = X at time1 as above). But I can also say, 'Yes, I now have a different statue'. If I simply say, 'I have a different statue' this leaves out the X = X factor, the fact that it is the same clay differently configured.

  3. X = X is empty out of context. In context - the context of diachronic or relative identity - it is informative.

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