The axiom (Greek ἀξίωμα – accepted position) is a sentence allowed for any reason as a starting point for any further reasoning. This general understanding of the axiom is each time concretized together with a refinement of what is meant by the sentence, the reason, and by further reasoning. Typical examples of axioms:

  1. some expression of the symbolic language of the calculus, if by further reasoning we mean the conclusions using it within the framework of this calculus. In this case, the reason for accepting axioms is the very definition of the calculus in question. Here doubts about accepting axioms are meaningless;
  2. some empirical hypothesis, if by further reasoning is understood, for example, systematically developed on its basis the division of physics. In this case, the reason for taking an axiom is a belief in the regularity of nature expressed by this hypothesis. Here, doubts about the adoption of the axiom are not only meaningful but also desirable;
  3. an agreement to understand the terms involved in the formulation of some judgment, as you like, but still in such a way that, under this understanding, the formulation in question expresses a true judgment. This is the case when by further reasoning we mean the derivation of knowingly true consequences from an ambiguously understood initial proposition. Here, doubts about the adoption of an axiom are meaningless. When such an axiom is used within the framework of a scientific theory, it is often called the value postulate;
  4. an assertion that is evaluated as necessary true (apodictic), if by further reasoning we mean any systematically developed doctrine that claims to be perfect in epistemological terms (Euclid’s geometry, Descartes metaphysics, Spinoza’s ethics, Fichte’s science, Hilbert’s metamathematics). In this case, the reason for accepting the axiom is evidence of special cognitive ability (intuition) to the direct discretion of some (often called self-evident) truths. Within the framework of this claim, it is absurd to doubt the axioms, but the question of the justification of this claim itself is one of the deepest and most open problems in philosophy.
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