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I found an interesting observation about the English language in this answer post:

Try explaining to a Finn sometime [...] why We're going shopping is OK in English, while *We're going eating isn't[.]

I'm an American native speaker, and although I agree that "We're going eating" isn't acceptable, I have no idea why!

(The correct sentence is "We're going out to eat.")

So why is "We're going eating" unacceptable, while many similar sentences (we're going shopping, we're going skiing) are?

Are there any patterns in the language that would allow an English student to predict that "We're going eating" is not an acceptable sentence, without already knowing beforehand?

There is probably no strict pattern, but one can nevertheless identify a tendency, namely, that the gerund-participal should be an intransitive verb denoting a leisure activity, preferably a physical one that counts as an 'experience'.

EDIT: Since first posting this answer, I have become aware of several linguistics papers that deal exactly with this issue. The latest one is Salkie (2010) (see the reference list at the end; Salkie's paper is free to download). He calls this construction 'expeditionary go', and discussess it together with other two: go for a N and have a N. His question is exactly how come we can go drinking / go for a drink / have a drink, but can't ^?go eating / *go for an eat / *have an eat. He also distingishes these constructions from several others that have the same superficial form but are in fact different. His answer to his main question is more or less along the lines suggested in the original version of this post. Here is the relevant section:

The key to our problem, I shall argue, is two observations about our
three constructions (go drinking / go for a drink / have a drink).　The first
is that drink here normally and most saliently means “drink alcohol.”　A
key difference between drinking alcohol and eating in societies where
English is spoken is that effectively transferring food into our body is essential for survival, whereas effectively ingesting alcohol is not: the point
of “going drinking” is not to satisfy our body’s need for liquid, but to
have a good time (and possibly satisfy an addiction).　Obviously eating
food also has important social rituals and behaviours attached to it, but
they are ancillary to its primary purpose of meeting the human body’s
need for nutrients.

Secondly, all three constructions are normally intransitive.
Older readers may remember Hopper and Thompson (1980) which proposed that the key feature of transitivity is not whether or not a verb takes a direct object, but something more fundamental: “the degree of effectiveness or intensity with which the action is transferred from one participant to another” (Hopper and Thompson (1980: 252)).　The purpose of all our three constructions, I claim, is to take an action which normally has high transitivity and move the focus away from the effectiveness of the action and onto the experience of the subject: that is, these constructions make an action intransitive.　Drinking (alcohol) typically focusses on the experience of the subject, as we just noted, so it is compatible with all three constructions.　It seems that with eating food, the effectiveness of the action is crucial enough to make defocussing it impossible or marginal: that is why you can’t normally ^?go eating / *go for an eat / *have an eat.

The attempts in the literature to characterise the semantics of these constructions can be seen as steps towards the analysis here.　Bolinger writes about the relationship between an action and its performer.　Dixon alludes to “the subject’s whim” rather than “any transcendental goal.”　Guillemin-Flescher says that the properties of a situation and its occurrence need to be in balance, and Stein refers to “the experience of an activity.”　The fundamental distinction, it would appear, is between the experience of the subject of an action and the effectiveness of the action.　As for Wierzbicka, it is reasonable to argue that ingesting an individual object is more “effective,” to use Hopper and Thompson’s term, than ingesting a mass substance, so eating is more transitive than drinking.　In my analysis, this is relevant but not the fundamental distinction.

A possible weakness of my account is that Hopper and Thompson (1980: 280-290) argue at length that low transitivity correlates with backgrounding in discourse, and high transitivity with foregrounding.　This is not what we find with the predominantly intransitive expeditionary go: it would be odd to claim, for instance, that in the song “Gone fishin’,” this activity is backgrounded. However, Hopper and Thompson (1980: 280) assume without argument that “[every] linguistic universal originates in a general pragmatic function.”　Not only is this assumption too strong, but in this case it is not necessary: the distinction between the effectiveness of an action and the experience of the person performing it is straightforward semantics, based on elementary human experiences.　There is no need to assume a pragmatic source for the distinction—though the fact that expeditionary go has different discourse properties from the other phenomena that Hopper and Thompson discuss would be worth investigating.

Here is the rest of my original post.

Comprehensive grammars

CGEL and ComGEL do not address this question. The best I can do as far as such sources is this: what we have here is a construction where go functions as a catenative verb, which takes a gerund-participial as a complement. In general, complementation implies licensing, meaning that the entity which accepts the complement (here the verb go) will not accept just any word as a complement, even if the word is of the proper grammatical format (here the gerund-participial form of an intransitive verb). Unfortunately, usually there are no absolute rules as far as which words are licensed to appear as complements. There can be tendencies, but, in the end, such properties should be recorded in dictionaries. Or at least that's what linguists tell us.

Dictionaries

In practice, dictionaries are never as comprehensive as that. In the present case, even the most comprehensive dictionary of all, the OED, records this kind of usage of go as follows:

30. 
    intransitive. To move, travel, or proceed (to somewhere) so as to perform a specified action, or for the purpose of a specified or implied activity.

f. With verbal noun or gerund.

(b) Without prefixed particle.

This is followed by several examples, based on which we conclude that the following gerunds, at least, are allowed: hunting, hawking, shooting, gunning, gambling, clamming, automobiling, fishing, camping, and clubbing. But in fact it's worse than that, because all but the last two examples are from before 1960 (in fact from 1933 and earlier), and so all the previous ones may well be obsolete or at least dated. Moreover, this is clearly not the complete list, since at least swimming is missing. This would not be so bad if the OED's entry for swim recorded that it can appear in such a construction, but it does not. And even this would still be OK if this meant that, by default, intransitive verbs can enter this construction, with some exceptions which have a note to that effect in their entry. But there is no such note in the OED's entry for eat.

An attempt at discerning a tendency

The most promising lead that I found so far (EDIT: I have since found several academic papers, see above) says the following (here):

In English, we use the grammar structure go+ing to talk about leisure activities, or activities we do in our free time. ... It is important to mention that go+ing is used to indicate the whole activity. When I talk about going swimming, I am describing the entire experience—wearing a swimsuit, putting on sunscreen, going to the pool or beach, swimming laps or enjoying the waves, getting wet, drying off afterwards. Go+ing gives us a mental image of the complete activity. If I just wanted to talk about the action of swimming, I would simply use the verb “to swim.” For example, “I swam ten laps in the pool today.” In this case, I am only describing the action of swimming, not the experience.

Let's test that. Note that although we can't say

*we are going eating,

we can say

we are going drinking/dining.

Arguably, we can even say such things as

we are going lobster/chocolate eating.

Some other things one can go do: wine tasting, running, walking, swimming, hunting, sightseeing, climbing, camping, flying, bar hopping, shopping, painting, decorating, drawing, prospecting, cleaning (also here),...

Perhaps a more useful question is what expressions are like eating in that they cannot be used in such constructions. Proving the negative is of course hard, but here are some plausible candidates: sleeping, writing, standing, exercising, learning, playing, playing with friends, cooking, studying, healing, ...

It seems that the tendency noted in the website holds up pretty well, although not perfectly. In the specific case of eating, it is presumably too generic to count as an 'experience', whereas more specific kinds of eating, like lobster eating or dining, do qualify.

On the other hand, it's not clear why playing wouldn't qualify as a leisure activity, and similarly for exercising and the related working out and lifting. However, as I said, tendencies are best we can hope for, so there are bound to be words that don't quite fit the pattern.

References

Berman, A. (1973). "Tripl-ing." Linguistic Inquiry 4, 401-403.

Bolinger, D. (1979). "The Jingle Theory of Double -ing." In Function and
Context in Linguistic Analysis: A Festschrift for William Haas, D.
Allerton, E. Carney, and D. Holdcroft, eds., pp. 41-56 (Cambridge University
Press,　Cambridge).

Bolinger, D. (1983). "The Go-Progressive and Auxiliary Formation." In Essays
in Honor of Charles F. Hockett, B. Agard, G. Kelley,
A. Makkai, and V. B. Makkai, eds., pp. 153-167 (E. J. Brill, Leiden).

Dixon, R. M. W. (2005). A Semantic Approach to English Grammar, 2nd ed.,
Oxford University Press, Oxford.

Guillemin-Flescher, J. (1999). “Verbes atéliques et construction
d’occurrences.” Les Opérations de Détermination: Quantification /
Qualification, A. Deschamps and J. Guillemin-Flescher, eds.,
251-268 (Ophrys, Paris).

Hopper, J. and S. A. Thompson (1980). “Transitivity in Grammar and
Discourse.” Language 56.2, 251-299.

Salkie, R. (2010). "On Going." In B. Capelle and N. Wada, eds., Distinctions in English grammar: offered to Renaat Declerck, pp. 169-190 (Kaitakusha, Tokyo); that whole book is downloadable for free here.

Silva, C. M. (1975). "Adverbial -ing." Linguistic Inquiry 6, 346-350.

Wierzbicka, A. (1982). “Why Can You have a drink When You Can’t *have an
eat?” Language 58.4, 753-799.

English has constructions like these with go:

• 
  go+V-ing - *Let's go helping/eating ~ They went swimming/fishing
  


• 
  go and+V - Let's go and help/eat/swim/fish ~ They went and helped/ate/swam/fished
  


• 
  go+V - Let's go help/eat/swim/fish ~ *They went helped/ate/swam/fished
  

as well as

• 
  come+V-ing - *Let's come helping/eating ~ They came swimming/fishing
  


• 
  come and+V - Let's come and help/eat/swim/fish ~ They came and helped/ate/swam/fished
  


• 
  come+V - Let's come help/eat/swim/fish ~*They came helped/ate/swam/fished
  

with come. Note that both sets follow the same patterns:

1. 
   go/come+V can be used only as an infinitive: Let's go help; She volunteered to come cook, or
   
   in the present tense, but not third person singular: You come help, but *She comes help(s).


2. 
   go/come and+V can be used in any verb form with any verb, provided the verb inflections match, for the purposes of Conjunction Reduction: Let's go and ask, She went and asked, I have gone and asked.


3. 
   

   go/come+V-ing can be used with the -ing verb form of an appropriate verb. Appropriate verb phrases convey a specialized activity that can be performed only in particular places, or only while in motion (whence the go and come). These include

   
   
   
   • sports like shooting, hunting, fishing, running, boxing
   
   
   • outdoor activities like climbing, running, hiking, diving, exploring, and spelunking.
   
   

I thunk the following post from wordreference.com helps:

But what about: "We go out eating"? -> Eating is the present participle and modifies "go out". The -ing form indicates a duration. Therefore, as has been said "We exit whilst we are in the process of eating [something.]"

"Eating" as a gerund (in the sense of a noun) is not common in the context you give and therefore it sounds strange, except in the sense of exiting whilst eating (i.e. in a sense that is more adverbial).

The example with drinking differs "we go out drinking" = "We go out to indulge ourselves in the action of drinking alcohol." or (less likely) "We exit whilst we are in the process of drinking [something.]" (i.e. in a sense that is more adverbial).

Drinking is used commonly as a noun and therefore "We go out drinking" is a particularly good idiomatic sentence and is clear in its default meaning.

Perhaps one reason the analogy with expressions such as "We're going shopping" fails is what we assume everyone knows about eating. If the response is "We're going shopping", then the question was something like "Why must we go out now?". The asker knows that we're going out but doesn't yet know why.

By contrast, we all know that we will eat at certain times of day. So, if the reason for the trip out is to eat out, then the responder phrases the response as if the asker knows that we will eat but doesn't yet know that the meal will be during the trip out, not at home.