If we ignore the political distinction "theories of those who generally agree with Chomsky" vs "theories of those who generally disagree with him" (which is better described by "Chomskian" vs "non-Chomskian/post-Chomskian"), can "generative" be defined in any useful way? Can a non-generative theory exist at all? It is clear that speakers can produce sentences they've never heard before, so they need to generate them in some way, based on what they know. Isn't this is something that both minimalist syntacticians and, say, construction/cognitive grammarians agree on? Or does the word "generative" have any more specific meaning?
For a short version, I'll cite my proposed tag wiki for generative-grammar:
A theory usually associated with Noam Chomsky that accounts for a language's grammar by a system of rules that are able to generate all the possible grammatical expressions in that language. In its original sense, "generative" does not neccessarily mean "production-focussed", although it has often been understood as such. Generativists study mostly syntax, but also other aspects of linguistic structure such as morphology, phonology and semantics.
I think the question is a good one and admit I can't give a ready-to-go definition of what "generative grammar" subsumes, but I have a feeling there is some on-going misconception about what "generative" means which I want to clarify because I think this is the most important part of understanding the problem:
In the original sense, "generative" doesn't mean that one is interested in generating particular instances of grammatical sentences of a certain language, or in how human language is generated "from neutral pathways towards the utterance". The OP already made a good insight by saying "Typical Chomskian approaches aren't generative in this sense [of "concentrated on production, not on understanding"] - they are very often based on grammaticality judgements". Rather, generative grammar wants to establish a grammatical theory that is able to account for any expression a language MAY generate, rather than just describing, without making any generalisations and predictive explanations, what is already there. It is the explanatory power and the aim to account for linguistic phenonemna univerally that is the essential characteristics of generative grammar, and not an aim to restrict yourself to a certain aspect of language. "Generative" does not neccessarily mean production-focussed.
Generative theory wants to be - apart from being explanatory, this is one of the core goals - independent of purpose, it doesn't want to just provide a way to analyse how humans produce sentences, but establish a theoretical framework which has the explanatory power of generalising what is already there and predicting how language would behave in terms of grammaticality when certain parameters are set in this or in that way. Basing your theory on a generative view on language doesn't mean you must focus on production.
Therefore, it is well possible to do research that isn't interested in production but rather in an analysis of existing structures, like typological research, with an approach of a theory that is motivated by the goal to be generative (a good linguistic theory, be it syntactic, semantic, phonological, ... should anyway be based on typologically widespread empirical research, so you actually do typological or historical research with generative theories in mind); basing your research on a generative theory doesn't mean you need to use it to produce sentences on your own or, even less, to account for how language is processed neurally or anything.
However, @Greg Lee indicated that many people have indeed been using the term in the sense of "production-focussed", so although I claim this is not what Chomsky orignially intended with this terminology, in order to not be prescriptivist I should admit that "generative" is also used for "production-focussed" in a second sense.
There are many definitions of "generative (grammar)", so there can be no single answer. A separate and interesting question would be to document usage of the term "generative" in publications of formal syntacticians of any version, separated into Chomskian vs. opposing theories such as LFG, RG, HPSG.
The core concept is "mathematically explicit", and there are and have been many theories that are not mathematically explicit, such as virtually all functional and typological schools. Indeed, after a spike in mathematical activity in the 50's and 60's, most formal linguistic theories have taken mathematical foundations for granted, even while repudiating some of the earlier mathematical foundations. The level of mathematization in 50's Chomskian linguistics is vastly higher than one encounters in any work in Minimalism. HPSG and descendants are the "most mathematical" extant theories of syntax, but practicioners of those theories are not particularly comfortable with being called generativists, because "generative" is usually understood to mean "Chomskian".
"Generate" technically does not mean "produce in chronological steps", it means "provide a bidirectional chain of logical inference from one representation to another", but most theories of syntax seem to be expressed in a production-friendly manner rather than a parsing-friendly manner. Minimalism superficially breaks with this trend by basing everything on the fundamental parsing operation Merge, but it does not start with the actually-produced surface sentence. All generative theories maintain that they provide a logical account of relations between structures, and no theories claim to be prodcedural accounts of how e.g. S is converted into NP plus VP, thus the idea that rules are about real-time production-orientation is not a feature of the theory, it's a feature of how people have misunderstood the theory.
There are lots of non-"generative" linguistic theories (or rather approaches/disciplines). E.g. linguistic typology is not interested in the way language is produced on its way from neural pathways towards the utterance but on co-occurences of linguistic features, their dependencies and mutual conditions. Historical linguistics strives to understand the grammar of reconstructed languages but I would argue that the core of the discipline lies in the act of reconstruction itself, not generating grammatically correct sentences in the proto-language.
Furthermore the term "generative" (broadly speaking, not concentrated on Chomskians) implies just one dimension of language, i.e. its production, while the other dimension - reception and understanding - is conveniently overlooked.
"Generative" has been understood in two quite different ways in the Chomskian era: A. explicit versus B. speaker oriented. We might refer to these as "official" or "standard" for A, since this is the sense proposed by Chomsky himself in his 1964 Aspects of the theory of syntax, versus "non-standard", "informal", "hearer oriented", or "interpretive" for B.
In Aspects, Chomsky gave the analogy of the way an algebraic function "generates" a set of points to the way a grammar "generates" sentences.
Fred Householder published a paper in which he proposed sense B for the term "generative", but Chomsky and Halle published a rejoinder repeating Chomsky's original position that generative grammar was "neutral between speaker and hearer". [See Some recent claims in phonological theory and Some controversial claims in phonological theory]
Despite Chomsky's historical position on the matter, sense B of "generative" has proved very popular among linguists. The matter seems relatively clear-cut in phonology, where in the standard view, the conditions on the application of a phonological rule must be met in the "input" to the rule, not the "output". It seems pretty clear that this is a speaker-oriented interpretation and not at all "neutral between speaker and hearer".
I myself proposed a version of generative phonology that was truly neutral between speaker and hearer, inasmuch as there were rules in a phonological system of both sorts -- speaker-oriented rules (like the rules in standard generative phonology) and hearer-oriented rules. That was a paper "Interpretive and productive phonological rules" in the U. Hawaii Working Papers in Linguistics; it was inspired by Kiparsky's distinction between opaque and transparent rules. And much more recently, I wrote up a more recent version here.
People like Ray Jackendoff and Adele Goldberg like to talk about "alternative generative grammar theories". The "alternative" part is obvious—it means they're not (Chomskyan) mainstream generative grammar.1 But what does "generative" mean?
Here's a (non-exclusive) list of AGGs thrown out by Jackendoff in one instance: his own PA (Parallel Architecture, aka Simpler Syntax), GPSG (General Phrase Structure Grammar) and its successors, BCG (Berkeley Construction Grammar) and its successors, LFG (Lexical-Functional Grammar), Autolexical Grammar, Role and Reference Grammar. What do these all have in common?
First, there's a purely historical/sociological meaning: Every AGG theory arose as a reaction to MGG (or recursively arose from an earlier AGG, where ultimately the first ancestor was a reaction to MGG).2
In fact, you could almost define "generative grammar theory" as "theory from someone who accepts the basic abstract from Chomsky's Syntactic Structure, but rejects at least one of the foundational ideas from the frameworks that followed". But that doesn't really add much to the openly historical definition.
Is there an useful meaning for "generative" that can be extracted? Yes, per Jackendoff:3
- A generative model is a model that produces a combinatorial structure (like a tree) that isn't just derived from some external structure. It doesn't matter whether you build the tree bottom-up instead of top-down, whether you use unification or analog constraint optimization, or even whether you build a cyclic graph instead of a tree.
- A generative grammar theory is a theory that has at least one component on the path between sound and meaning and vice-versa that requires a generative model. It doesn't matter whether that component is syntax (MGG, and many alternatives), semantics (Generative Semantics), or functional structure (LFG), or whether there are multiple independently-generative components (PA).
Not exactly the definition Chomsky originally gave, but it still fits the word, and it covers all of the present and historical theories people want to call "generative".
Is this actually what "generative" means, or is the historical definition the real meaning, and this is just a (presumably not coincidental, but still metaphysically contingent) feature of all GG theories that happens to fit the pre-theoretic meaning of the name "generative"? I don't know if Jackendoff has ever said.
Since I brought up Jackendoff,4 he also argues that generativity is inextricably tied two other ideas—both also from Chomsky, but separable from MGG:
- GG theories are (at least methodologically) mentalistic: they study language as it's instantiated in the mind, not in society or Platonic space.
- GG theories are all driven by methodological minimalism toward universal grammar.5 Even if a theory's "UG" is just a toolkit of ways to build language-specific constructions rather than anything like P&P's set of hardcoded structural principles, and the theory strikes a radically different balance in what to minimize than Chomsky's, and the author would vehemently disagree with your contention that she's trying to minimize UG, any generative grammar theory is still ultimately trying to minimize UG.
What do either of these have to do with "generative"? Nothing directly. It's probably not a coincidence that (at least according to Jackendoff) they're true of all GG theories, but the same is true for other features he doesn't bring up, like neutrality between production and perception. So, I think this contention can be completely separated from the definition of "generative". But others might disagree.
1. Which can be equated with the set of theories derived from Chomsky's successive frameworks from Syntactic Structures to the Minimalist Program.
2. So GG is a theory family. The fact that Syntactic Structure didn't fully describe its theory just means that Proto-GG has to be reconstructed from Aspects and its 1960 alternatives through laws of regular theory change. :)
3. This is all covered, at least to some degree, in Simpler Syntax and all of Jackendoff's later books. Based on my vague memories, I think the most detailed treatment of all of these ideas is in book Meaning and the Lexicon—but I unfortunately don't have a copy of that at hand, so I'm not positive on that.
4. And Simpler Syntax, which actually spends a lot more time on these issues than on generativity per se.
5. As you can probably guess, using the terms "UG" and "minimalism" led to a lot of confusion. Jackendoff's later works have backed off from that a bit, but as far as I know, he hasn't really come up with any good terminology to replace it.