Do spoken languages have a sense of operators with 'order of operations'

Question

This is something I came across while thinking about expressing a 'nested' choice, i.e. you make a choice where one of the options entails making another choice, for example, at college you could choose to study a single major A, or enter another stream where there is two majors, B and C which you must decide on at a later date.

If I were to treat the word 'or' as an operator using the mathematical 'order of operations' I would have something like:

A or (B or C) actually, however the order of operations here are slightly wrong, as in Mathematics we would have to decide on B or C first, it does however represent the idea that the other choice is nested.

Is there any sense of using words such as 'or' as operators that take two binary arguments? As with 'or' we must use it twice, but its just between words, there's no sense of whether it is just bridging words or taking whats on either side as an 'input'.

Answer 1

Answer

If you are speaking conceptually, then all mathematical statements are true whether in natural or artificial language. 'Three plus six divided by two' simplifies to 'six'. However, note that the natural language expression evaluates to the answer by way of conceptual convention, not a property of the language per se. But I don't think that's what you're asking.

Rules that Govern Language Sounds, Syntax, and Semantics

Now, it seems to me you're asking are there rules that govern syntax of morphemes, and to that the answer is a resounding yes. The rules that construct a language in a sense generate it, and while rules have been observed by grammarians for thousands of years, Noam Chomsky rocked the world with his theory of generative grammar.

Ray Jackendoff expresses a more modern theory of the rules of operations on pg. 125 of his Foundations of Language with his model of tripartite architecture in essence dividing up the rules of natural language into phonological, syntactic, and conceptual formation.

Therefore, one has to be clear to articulate which domain of rules one is talking about. Chess and math have rules and orders of operations, for instance, black and white alternate moves in chess, and these are conceptual constraints. English and German have syntactical rules, such as modifiers generally go before the nouns, and in English the verb occurs in the middle of the sentence, but for German, complex predicates have verbs listed at the end. And of course both have phonological rules such as how dipthongs and blends occur or how phones correspond to graphemes.

Operations that Govern Transformation

Relevant to philosophy of language is how multiple syntactical expressions might represent the same proposition.

S1. Bob dropped the pizza. (Active voice)
S2. The pizza was dropped by Bob. (Passive voice)

S1 is transformed into S2 by a rule which roughly consists of swap the order of the subject (Bob) and the direct object (the pizza), and transform the verb using an auxiliary copula (was). This rule is regular in English, so any active sentence can be transformed from active to passive.

SVO -> O(was/were)VS

Typlogogically, this is significant because English like Spanish is SVO.

Rules of Syntax and Semantics with Logical Connectives

Lastly, do rules govern the use of logical connectives in English? Sure. A list is generally expressed as {e₁, e₂, e₃[,] and/or e₄} (with the square brackets indicating the use of the Oxford comma as optional (regrettably ;). Thus we have standard and non-standard (indicated by ^*) expressions with logical operators:

S3. The boy has a mitt, bat, and ball.
S4. The boy has a mitt, bat and ball.
S5. The boy has the mitt, bat, or ball.
*S6. The boy has a mitt and batt, ball.
*S7. The boy has or a mitt, ball, ball.

And note, that nesting of linguistic structures generally employs punctuation for clarification when written to economize on word use:

Bob has the bat, mitt, and ball; or he has the skis and the poles; or he has both the bat, mitt, and ball, and the skis and the poles.

And generally, where concepts are ambiguous, for instance, is 'or' inclusive or exclusive, there are rules of syntax for clarification.

S8. Bob has the bat or ball. (inclusive or exclusive?)
S9. Bob has either the bat or ball but not both. (exclusive)
S10. Bob has the bat or ball or both. (inclusive)

Summary

What's important is that when you talk about rules and operations in natural language, you differentiate whether you're talking about the phonological domain, the syntactical one, or conceptual one. In fact, philosophers of language expand even further into pragmatics which explores ideas like performativity, implicature, and direction of fit.