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Necessarily, one must affirm axiomatic propositions that are true without needing to be proven, and axiomatic concepts that are understandable without needing to be defined. Axiomatic propositions such as “the world exists”, and axiomatic concepts such as “existence”.

• • If the truth of every proposition needed to be proven, then propositional knowledge would be impossible.
  • Propositional knowledge is not impossible.
  • Therefore, the truth of every proposition does not need to be proven.

Premise 1: If the truth of every proposition needed to be proven, then propositional knowledge would be impossible.

A proof is composite of propositions. So if every propositions needed to be proven, then the truth of any proposition would depend on a proof, the soundness of that proof would depend on another proof, and the proof of the proof on yet another proof… ad infinitum. Entailing an infinite number of conditions before any knowledge could be acquired. And an infinite number of conditions cannot be satisfied given the impossibility of Tasalsul.

Therefore, if the truth of every proposition needed to be proven, then it would be impossible to infer the truth of any proposition.

Premise 2: Propositional knowledge is not impossible.

This is known by introspection. We find ourselves knowing about our own existence, the existence of the world around us, and the reality of our experiences. And this would not have been the case if propositional knowledge were impossible.

Therefore, the truth of every proposition does not need to be proven.

Thus, there necessarily are propositions that are true without needing to be proven.

A similar variant of this same argument also works for concepts. If every term required a definition, and since definitions are composite of terms, then understanding each definition would depend on an infinite number of other definitions, which is impossible. So given the fact that we do understand some terms, then this is proof for the existence of terms that are understandable without needing to be defined.