My understanding is that philosophers like Wittgenstein held that thought without language is impossible. I've seen many people reply that they have non-linguistic thoughts all the time, and my guess is that what they mean is that they often "think" in imagery rather than words. For example, rather than saying with their inner voice, "I should advance my pawn," they picture a chess board with a pawn moving forward. Does this demonstrate non-linguistic thought?

I'm no expert on Wittgenstein, and I don't know the particular argument of his that you're alluding to. He does give a famous argument that anything properly regarded as a language must be usable (if not also used) by more than one person.

But your question is about something else: whether a being can think without possessing language, or maybe whether a being can have thoughts with no linguistic content.

I think the clearest reason for answering "yes" is given by the problem-solving behavior of non-human animals to whom we have no reason to attribute language. Mice seem able to solve mazes, octopuses can figure out and open screw-top jars, and so on, yet it seems a stretch to attribute language to them. When an octopus encounters, for the first time ever, a closed glass jar containing attractive prey, which linguistic resources or concepts must it use when it figures out how to remove the screw top? What sort of linguistic content is the octopus representing to itself? None that I can imagine. But some sort of problem-solving -- some sort of thinking -- is happening, just the same.

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