Srinivasan, Radhakrishnan
(2024)
Do arbitrary constants exist? A logical objection.
[Preprint]
Abstract
In classical firstorder logic (FOL), let T be a theory with an unspecified (arbitrary) constant c, where the symbol c does not occur in any of the axioms of T. Let psi(x) be a formula in the language of T that does not contain the symbol c. In a wellknown result due to Shoenfield (the "theorem on constants"), it is proven that if psi(c) is provable in T, then so is psi(x), where x is the only free variable in psi(x). In the proof of this result, Shoenfield starts with the hypothesis that P is a valid proof of psi(c) in T, and then replaces each occurrence of c in P by a variable to obtain a valid proof of psi(x) in T, the argument being that no axiom of T is violated by this replacement. In this paper, we demonstrate that the theorem on constants leads to a metainconsistency in FOL (i.e., a logical inconsistency in the metatheory of T in which Shoenfield's proof is executed), the root cause of which is the existence of arbitrary constants. In previous papers, the author has proposed a finitistic paraconsistent logic (NAFL) in which it is provable that arbitrary constants do not exist. The nonclassical reasons for this nonexistence are briefly examined and shown to be relevant to the above example.
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