So, what is language? I do not intend to deal with phenomenological or ontological issues. This is just a witggensteinian approach (the meaning of words is its (conventional?) use). I simply want to point out two facts: 1) Those of us who work as English as a foreign language teachers cannot teach (or at least are very unlikely to succeed and are much more ineffective) if we do not even know or ask ourselves what language is; and 2) our conception of language is not one single monolith.
My very personal, most favorite definition is that “language is the evolutionary mechanism that humans developed to achieve ‘shared cognition’ (a common consciousness with the others).” But that is something to be discussed at some other moment with much more theoretical machinery from the cognitive neurosciences that define the underpinnings and differences between sensation, perception, and conception.
Gontier, N. (2016). Guest-editorial introduction: converging evolutionary patterns in life and culture. Evolutionary Biology, 43(4), 427-445.
I did not tackle the "language is grammar" conception for this seems to me way too reductive. I am quite open, though, to discussing this as a possible component of the conceptual prototype of what language is.
The level of schematization is gargantuan: ¿How do we craft a definition that takes into account all the spaces-times in which language inhabits?
For Langacker (1987), symbols are not 100% arbitrary. He illustrates this with the word "stapler," which is arguably composed of two morphemes: "stapl-" and "-er." Since "-er" carries schematic meaning of its own, the symbol "stapler" is not 100% arbitrary.
Langacker, R. W. (1987). Foundations of cognitive grammar: Theoretical prerequisites (Vol. 1). Stanford University Press.