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The answer to dw85745's bold-faced question must start with a counter-question: "Whose definition of 'Programming Language' are you using?"
By most of the standards I have seen, SQL is a programming language but is not a general purpose programming language.
The languages used by programmable logic controllers count as languages (there are 5 commonly used variants). The language used by certain types of flat-bed plotters counts as a language (variants based on HPGL and its later generations). The language used by CNC routers is a language. ALL of the examples listed here are Domain-Specific Languages. Most if not all of them would fall in the same place when considering the Goedel Completeness Theorem - they are "incomplete" and therefore their correctness is theoretically provable.
In fact, there is a whole specialized branch of logic - called interrogatory logic - that analyzes the validity of answers given by such languages. That branch asks the question: Does this answer satisfy this question? The contrary form asks: Does this question condone this answer? Interrogatory logic is the companion to the more familiar "assertoric logic" that examines the validity of assertions.