A formal system is expressively complete if and only if it is capable of expressing, as a formula, everything that is the subject of that formal system.

A deductive system is semantically complete if and only if every sentence that is truth-functionally entailed by a set of sentences can be derived from that set.

Is there a relation between these two types of completeness? If the two are related, what type of relation is this?

  • Maybe useful : Expressive Completeness and Completeness. Apr 19, 2018 at 12:45
  • In a nutshell: yes and no. If the set of connectives is not adequate (i.e. it cannot generate all truth-functions) we will have some truth-function that is "identically true" that is not expressible. But the issue is that if it is not expressible in the language it is also not provable. Apr 19, 2018 at 12:48
  • But if we consider non-trivial example, like Implicational propositional calculus with only as connective, suitbale axioms and modus ponens, we have that it is complete with respect to standard semantics, i.e. it proves all the tautologies containing only . Apr 19, 2018 at 12:56
  • Say there is a (non-classical) system intended to have expressive power and is thereby presumably intended to be expressively complete. Is it possible at all to infer anything as to its semantic completeness from this expressive completeness? Any texts concerning this relation will be much appreciated. Apr 19, 2018 at 12:56


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