John Cage was not a fan of symbolism - at least if you believe what he said in the album Indeterminacy. He preferred to take things as they were rather than pretend they stood for something else. But, there's a distinction to be made between what is symbolism and a strange way of looking at something. Symbolism is putting a dog in a painting and saying it represents fidelity. There is an alternative, which is drawing a faithful dog.
The same thing can happen in music. You can compose a sheet of music that symbolizes a bird - you can make a C represent a the coo of a dove - or, you can just say that you composed a dove singing and it happens to be the note C. The difference probably lies in your intention; one is forcing an entity to take on the role of something else, the other is recognizing that in an entity exists this other thing. One is the idea of creation, the other is the idea of recognition. You can say that you were able to recreate some entity in another format, or you can say that you found this same sound here that you found over there. You can take credit for the action, or you can give credit to the sound. Or you can not worry about things like this. None of these options are better than the others.
The idea of hearing a dove singing the note C is recognizing an inter-connectedness between these things. The note C is never going to be a Dove, but you can realize that there is an intimate relationship between those two things. You can realize that with everything though.
One last note about this recognition thing: Capt. Beefheart did the something very similar when he produced the lyrics - "Fast and Bulbous - That's right, the Mascara Snake - Fast and Bulbous also tapered". This is not a symbol of a woman crying with her mascara streaking, it is a woman crying with mascara streaking. Capt. Beefheart happens to see those words are those things. Those words aren't representing that woman's face, they are that woman's face.