A famous example of a so called syntactic consequence is a typing judgment: ex. if a context says a function f is of type t to t, and x is of type t, then the expression f(x) is valid and of type t. That is, f(x), taken for a sequence of symbols, although looking correct, may or may not be a valid expression (which is not the same as True or False, which may not even apply) for the example language.
This looks like what we would refer to in every day wordings, as grammatically correct vs syntactically correct; the former implying the latter but not the other way.
For mathematics, it seems syntax means something stronger than what syntax means in every day wording. Since it includes both of what we would call in every day wordings, syntax and grammar, that is, language rules. So I was wondering if it is OK to better explain syntactic consequence as language rules consequence. Or would this be ambiguous in some way, potentially leading into misunderstanding?
I am wondering about it, because I feel the common understanding of the word “syntax”, suggests something weaker than what it actually is to mathematics and this divergence makes it easy to miss a big part of what “syntactic consequence” means. Ex. common compilers distinguish syntax errors and type errors, as an example showing how the common word “syntax”, seems to not mean the same as the same word in maths and logic.
Best Answer
I like to think of Syntax as rules defined on arranging physical objects which are part of the language. If the objects are arranged in a way that agrees with the rule then they are "Well-formed" as per the syntax. So let's take propositional logic, the syntax of the logic is defined on the language which consists of propositional symbols say $\{P_0,P_1, ..\}$, logical connectives $\{\land, \lor,\neg\}$ and auxiliary symbols { (,)}. And then there are a set of rules which say that what is acceptable as a well-formed formula, for example, $\neg A$ is WFF whereas $\land A$ is not. In language this is akin to grammatical rules, so "A horse is a goat" is a syntactically sound formula, in math $\frac{0}{0}$ is a syntactically sound formula, whereas "A A Man has" is a syntactically invalid statement. But notice neither $\frac{0}{0}$ or a "A horse is a goat" are symantically sound.
Human language is close to logic in this sense but also very far.