Definition of syntactic consequence in US English:

syntactic consequence

noun

Logic
  • The relationship by which, in a particular formal logical system, a sentence can be derived from others by the inference rules of the system, without regard to the truth or interpretation of the sentences; logical consequence conceived syntactically, as in proof theory, rather than semantically, as in model theory; (also) a sentence which is derivable from others by the inference rules of a particular formal logical system; often contrasted with semantic consequence.

Origin

1960s; earliest use found in Newman-Studien.