Wording: can one say “syntax” in maths, actually means “language rules”?

Question

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.

Syntactic consequence involves something more than the rules to write "correct" expression (well-formed formula); it involves the concept of proof system, with rules of inference and derivation. — Mauro ALLEGRANZA, May 05 '20 at 10:31
@MauroALLEGRANZA, to you, would “language rules” correctly stress this? — Hibou57, May 05 '20 at 10:34
The rules of inference are the "inferential engine" of the proof system. — Mauro ALLEGRANZA, May 05 '20 at 10:44
I feel to see, but I prefer to ask to be sure: do you mean I should better distinguish between the rules of inference defined by the language (ex. typing rules) and the rules of inference of the proof system used to reason on the language (ex. modus ponens)? — Hibou57, May 05 '20 at 10:51
Yes, I like this one, especially that production rule is an expression already used in computer langage, in a way which conforms to the definition you points to. Just to be sure I understand what formation rules is to mean: a typing rule would be a formation rule, this is it? — Hibou57, May 05 '20 at 11:08

score 1 · Accepted Answer · answered May 05 '20 at 10:38

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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.

answered May 05 '20 at 10:38

SagarM

1,789

I guess I will go with wellformedness. I just noticed MathWorld (a part of Wolfram) does not mention syntax when talking about formal languages, it mentions grammar instead, which is a wording I more agree with. To connect it to another issue to me, I would just say ⊢ is to be understood as “narrative consequence” instead of “syntactic consequence”, since a sentence following syntax rules may still be invalid, grammatically incorrect, and no one can derive anything from a sentence which is not correct. At least, wellformedness stress it in some way. – Hibou57 May 06 '20 at 17:12
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I kind of see the understanding you want to build, but I think seeing logic as having features of the language is the wrong direction of understanding (this is completely my opinion). Rules of logic are explicit, fundamental, and compact. Rules and concepts of Language are none of the three, I see our general notion of language-based concepts as an approximation of logic and it becomes most evident when you try to wrap your head around riddles (like by Raymond Smullyan) and see how simple they are in logic but language makes them obscure questions. – SagarM May 06 '20 at 21:41
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Don’t worry, I am aware of all of this, I’m just seeking for wordings for bootstrapping, and this needs disambiguation. When the meaning of the fundamental symbols and concepts is established, things can go on with more formal definitions. – Hibou57 May 06 '20 at 22:21

Wording: can one say “syntax” in maths, actually means “language rules”?

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