Talk:Completeness (logic)

The paragraph describing syntactical completeness is misleading

There are two meanings of syntactical completeness mentioned in the paragrap:

1. A formal system S is syntactically complete (..) if for each sentence (closed formula) φ of the language of the system either φ or ¬φ is a theorem of S.

2. In another sense, a formal system is syntactically complete if and only if no unprovable sentence can be added to it without introducing an inconsistency.

Shortly after that: "Truth-functional propositional logic (..is) semantically complete, but not syntactically complete (for example, the propositional logic statement consisting of a single propositional variable A is not a theorem, and neither is its negation)."

There are two things to say about this:

1. Syntactical completeness in the second sense is also called Post Completeness or Hilbert-Post Completeness. Formal systems (Hilbert style) for propositional logic are syntactically complete in the second sense, however not in the first sense (as is stated in the paragraph). There should either be a separate paragraph for Post Completeness or some kind of explanation that these two notions of syntactical completeness are different.

2. This may be pedantic but a logic can't be syntactically complete but in the paragraph it says that "Truth-functional propositional logic (..is) semantically complete, but not syntactically complete(..)". Only a formal system can be syntactically complete. 2001:4091:A247:8089:684D:7C6D:4B5F:A266 (talk) 13:48, 22 December 2022 (UTC)