What precisely do you mean by 'abstraction'? It's true that we have a capacity to intuit how others apply words. But Wittgenstein's point is that we have to use this skill not only when we use words like 'time', 'existence', 'algebra' and other abstract words but also when use more 'concrete' words like 'apple' and 'red'. When I point to a red apple and tell a child who doesn't know the meaning of the terms 'red' and 'apple' that 'that is an apple', how can he know that I mean what those words conventionally mean rather than 'that is red'? Children actually do misunderstand how we use words sometimes but to the extent that they succeed that is due to what Quine called the 'pre-established' harmony between humans: we see the world in similar ways and therefore interpret/intuit linguistic cues in similar ways. So there is really no fundamental difference between 'observable'/concrete predicates and non-observable/abstract predicates on this score. (It's true that our interpretations of observational terms generally converge a lot, but so do our interpretations of some non-observational terms - e.g mathematical terms. Proximity to observations is not the only way to ensure convergence of interpretations or exactness even though it's an important way.)
I could see a sufficiently abstract interpretation of 'abstract' which might fit your (or Vygotsky’s) theory though. But then you should make it precise.
Abstraction can mean finding a concept common to multiple more specific concepts or entitites. Example: Apples and oranges are fruits are edible. Fruit is an abstraction of certain plant properties, edible even more so. I can form this abstraction without language (albeit not communicate it without). Abstraction in this sense is a special kind of pattern detection and can be performed by non-linguistic algorithms.
Abstraction as a process working on other abstract entities via language by naming properties or stripping away properties is linguistic by construction but then the result is trivially true.
I think that I can perform abstraction without a) recursing to others and b) without language. To try this out I had a look at a painting (art is a domain I have no prior knowledge in so no words for concepts I might encounter). I noticed certain color patterns, shapes reoccurring in the picture. Not the same shapes but patterns. I could have given names to these but refrained for test purposes. But I nonetheless could think about - notice, quantify - the reoccurrences. Isn't this abstraction? Isn't this non-linguistic (except for communicating it here after the fact)? I agree that this kind of abstraction is difficult to transport without language (I could paint the gestalt of the pattern and show you). I also agree that this kind of abstraction is likely not applicable to other domains (except other visual or motorical domains) without language.
Belief shouldn't be denied to non-linguistic animals, in part because the best explanation of the origin of language probably involves beliefs. Intelligent primates are keenly aware of the (I'll go ahead and beg the question a little) beliefs and desires of their associates. They wish to communicate their own psychological states, and learn more about, and influence, those of the others. Language is an extremely effective tool for these purposes, so it caught on.
We could call these non-linguistic states "proto-beliefs" (and I suppose "proto-desires"?) instead. The story then has it that the primates only wanted to learn and influence the proto-beliefs of their group, and getting full fledged beliefs in the bargain is merely an acceptable side-effect (later to become a highly desirable effect). But if getting information on, and influence on, the "proto-beliefs" of others is such a big success story, then the "proto-" qualifier is unjustified. Anything that does this much work deserves the label "belief".
Sure, language adds a lot of complexity and refinement. But if you think non-human animals can't introduce information into amazingly varied contexts, you haven't studied them enough.
Belief, puzzling to philosophy, is part of psychology’s conceptual framework. The present essay provides a straightforward yet novel theory of the explanatory and predictive value of describing agents as having beliefs. The theory attributes full-fledged beliefs exclusively to agents with linguistic capacities, but it does so as an empirical matter rather than a priori. By treating abstraction as an inherently social practice, the dependence of full-fledged belief on language resolves a philosophical problem regarding its possibility in a world where only concrete particulars exist.
It can appear mysterious that the content of epistemic attitudes (belief and opinion) is conveyed by clauses introduced by that: “I believe that the dog is in his house.” If beliefs were causes of behavior, our success in denoting them gives rise to an apparently insurmountable problem: how do propositions—if they exist at all—exist independently of human conduct, so as to be fit for causally explaining it?
While belief ascriptions figure prominently in many behavioral explanations, their propositional form indicates that they pertain to states of information. My belief that my dog is in his house consists of the reliable use of the information that he’s there. Not only will I reply accordingly if asked about his location; in directing other my conduct, I may use that information. If I want the dog to come, I will yell in the direction of his house rather than toward his sofa. Yet, I won’t always use this information: I might absent-mindedly call to my dog on the sofa despite knowing (hence believing) that he is in his house. Believed information can be mistakenly disregarded.
Belief “that p” is a propensity to take p into rational account when p is relevant to the agent’s goals. But taking certain information into account involves also various skills, and it must be facilitated by the appropriate habits. The purposeful availability of believed information is also affected by, besides skills, inhibitions, habits, and desires.
What becomes striking on recognizing beliefs as propensities to use particular information is that behavior can be so successfully explained, when we know something of an agent’s purposes, by reference to the information on which we can predict the agent’s reliance.
Is this successful reliance a unique feature of human cognition? We can use belief ascriptions to describe nonhuman behavior, but we can do the same for machines. The concept of belief, however, isn’t essential to describing nonintelligent machine behavior. When my printer’s light indicates that it is out of paper, I might say it believes it is, particularly if, in fact, the tray is full. But compare it to what is true of me when I run out of paper, where my belief that I have exhausted my supply can explain an indefinitely large set of potential behaviors, from purchasing supplies to postponing work to expressing frustrated rage—in any of an indefinitely large variety of manners. The printer’s “belief” that it is out of paper is expressed in two ways: it refuses to print and a light turns on, and I can refer to these directly, without invoking the concept of belief.
Applying the concept of belief to nonhuman animals is intermediate between applying it to machines or to humans; it can be applied to animals more robustly than to machines. It isn’t preposterous to say that a dog believes his bone is buried at a certain location, particularly if it’s been removed and he still tries to retrieve it from the old location. What can give us pause about saying the dog believes arises from the severely limited conduct that’s influenced by the dog’s information about the bone’s location, as is apparent when the dog fails, except when hungry, to behave territorially toward the bone’s burial place.
Humans differ from canines in our capacity to carry the information constituting a belief’s propositional content to indefinitely many contexts. This makes belief indispensable in forecasting human behavior: without it, we could not exploit the predictive power of knowing what information a human agent is likely to rely on in new contexts.
This cross-contextual consistency in the use of information seems to rest on our having language, which permits (but does not compel!) the insertion of old information into new contexts.
Explaining our cross-contextual capacities is the problem (in the theory of knowledge) of how we manage to mentally represent abstractions. In Kripke’s version of Wittgenstein’s private-language argument, the problem is expressed in the dependence of concepts on extensions that are not rule governed. The social consensus engendered by how others apply words provides a standard against which to measure one’s own word usage.
Abstraction relies, ultimately, on the “wisdom of crowds” in achieving the most instrumentally effective segmentations. The source of abstraction—a form of social coordination—lies in our capacity to intuit (but only approximately) how others apply words.
The capacity to grasp the meanings of others’ words underlies the fruitfulness of using believed propositions to forecast human behavior. With language we can represent the information that another human agent is also able to represent and can transfer to all manner of contexts. But this linguistic requirement for full-fledged belief does not mean that people’s beliefs are always the beliefs they claim (or believe) they have. Language allows us our propositional knowledge about abstract informational states, but that doesn’t imply that we have infallible access to those states—obviously not pertaining to others but not even about ourselves. Nor does it follow that nonlinguistic animals can have full-fledged beliefs limited only by concreteness. Nonlinguistic animals lack full-fledged beliefs about even concrete matters because linguistic representation is the only available means for representing information in a way allowing its introduction to indefinitely varied contexts.
This account relies on a weakened private-language argument to explain abstraction as social consensus. But I reject Wittgenstein’s argument that private language is impossible: we do have propositional states accessible only privately. Wittgenstein’s argument proves too much, as it would impugn also the possibility of linguistic meaning, for which there is no fact of the matter as to how society must extend the meaning to new information. The answer to the strong private-language argument is the propositional structure of perception itself. (See T. Burge, Origins of Objectivity (2010).) What language provides is a consensual standard against which one’s (ultimately idiosyncratic) personal standard can be compared and modified. (Notice that this invokes a dialectic between what I’ve termed “opinion” and “belief.”)
This account of the role if language in abstraction justifies the early 20th-century Russian psychologist Vygotsky’s view that abstract thought is fundamentally linguistic.