Is there a standard of proof in linguistics like there is in other sciences? How do linguists determine if something is true?
I ask because grammarians disagree on a lot of things. It's to the point where I've stopped asking questions. No two people say the same thing.
To tell whether a theory is true, you compare its predictions with the facts.
For instance, TG predicts that a constituent can be extracted from just a single conjunct, but not from both conjuncts of a coordinate construction, which as Ross showed is wrong. GPSG, describes extraction by distinguishing (with slash) the syntactic category of constituents with extractions, which correctly predicts both Ross's Coordinate Structure Constraint and his across-the-board condition (since only constituents of the same category can be coordinated).
As another example, my monostratal phonological (MP) theory (see eliminating intermediary forms) correctly predicts that phonological rules are transitively ordered, while the standard theory of Generative Phonology makes no such prediction (though it assumes transitive ordering). The proof is as follows.
We use Kiparsky's classification to define the "before" relation for rules:
In MP, rules are either input conditioned or output conditioned. Only output conditioned rules are fed or bled. Input conditioned rules are counterfed or counterbled. This implies that A is before B, as just defined, when either A is not output conditioned or B is output conditioned, which in turn is to say that A is output conditioned materially implies that B is output conditioned. The material implication relation of logic is a transitive relation. QED.
Proven?
In the sense of a mathematical proof: Definitely not. There are no axioms and definitions to build linguistics as a part of mathematics. And even if there were some axiomatisations of subsystems of linguistics: How do they relate to natural language?
Is linguistics a science in the sense of Karl Popper?
I.e., does it consist of hypotheses that can be falsified but observations on natural languages?—I'd say, partly. There is a branch of linguistics, sometimes called Language Science, that can be considered a science in this sense. But there are other parts of linguistics, that are more like Humanities. And than there is a large part of Linguistics that is simply descriptive (think of all the field workers documenting lesser known or completely unresourced languages).
I fear that this question is premised upon a naïve idealisation of science.
Science does not prove anything, except the falsety of its own theories. And no, this isn't a bug, nor it should not do anything different. What we call "truth" is always provisional, no one can be in direct contact with the Kantian "thing in itself", everything is mediated by our perceptions, which are not the "true thing".
Why would linguistics be different from particle physics? Both languages and subatomic particles are part of "reality", and can be discussed, labeled, theorised, spoken about. There is a scientific method - propose a theory, isolate variables, design an experiment, perform the experiment, compare the experiment's result with the theory's predictions. This is no different in linguistics than in any other science.
If linguistic theories are in any sence worse than subatomic physics theories, this is a problem of the relative development of each science. Yes, physics as a whole is more developed than linguistics, due to both historical (scientific linguistic is newer than scientific physics) and structural (recursivity in linguistics is more immediate than in physics) factors. But then physics was a mess two centuries ago, too.
Physicists disagree on a lot of things too. Which is a good thing; how would physics ever progress if physicists adhered to previous results as dogma? We would still believe in epicycles, I suppose.
The standard Aristotelian understanding of "true", of a proposition, is "describes reality" (and "false" is "does not describe reality"). "Prove" then means "show to be true", which again in the standard Aristotelian approach means to show that there is a reason to judge the proposition to be true, and there is no reason to doubt the proposition.
Owing to the influence of Kant and more proximally Karl Popper, the classical understanding of the concepts "prove" and so on have been widely though not universally replaced, owing to his stipulation that a "universal statement" cannot be proven, but it can be refuted. Thus all statements are either conjectures, or false. The possibility of observational refutation is undermined by the fact that the logical connection between an ostensively refuting event and a predictive theory is itself an unproven conjecture. If a theory W predicts a reaction temperature of X under conditions Q and the observed reaction temperature is Y, then at least one of W, Q or Y is false. It is conjectured that the device thought to measure temperature, which relies on physical theory Z, has value for establishing temperature, but that is at best an unfalsified conjecture. We may now actually have observational evidence that falsifies foundational theory Z. In other words, when something goes wrong, all you know is that something has gone wrong.
Returning to the Aristotelian approach, the basic methodology of proof amounts to identifying reasons, which are observations or conclusions that are proven based on observations. The practical problem has been that the inference from observation to conclusion has not been automated, so people take invalid shortcuts. Statistical methods, for example, rely on the premise that a given sample is drawn at random from the population, and that is never actually the case in linguistic research.
The reason why linguists cannot agree is fundamentally that there are vastly different epistemologies in the field: this Aristotelian / Popperian split is just one example.
