Reasons for calling completeness semantic and syntactic

Question

The definition for semantic completeness is that if $\phi$ is a semantic consequence of some set of propositions $T$ then there is a natural deduction from $T$ to $\phi$. On the other hand syntactic completeness states that every proposition or it's negation is a theorem. What is the reason for these names. Intuitively it would seem that the second notion should be called syntactic completeness since the syntactic consequences "completes" the set of tautologies. Or one could think of being closed under the relation formed by natural deduction.

Answer 1

The distinction is rooted into the syntax vs semantics of natural language:

Grammar is the study of how meaningful elements called morphemes within a language can be combined into utterances. The way in which meaningful elements can be combined within a language is governed by rules. The rules of the internal structure of phrases and sentences are called syntax.

Languages express meaning by relating a sign form to a meaning, or its content. Semantics is the study of meaning.

In the same way, in formal languages:

Syntax is concerned with the rules used for constructing, or transforming the symbols and words of a language, as contrasted with the semantics of a language which is concerned with its meaning.

Syntactical completeness is defined in terms of "form" of formulas: $\phi$ and $\lnot \phi$, i.e. in terms of grammar, while semantical completeness is defined in terms of interpretations, i.e. meaning of the formulas.