I feel like despite the close analysis you grant to the meanings of formalization and syntactic, you've glossed over some more fundamental definitions that are sort of pivotal to the argument at hand.
> LLMs do not reason. They do not infer. They do not analyze.
(definitions from Oxford Languages)
reason(v): think, understand, and form judgments by a process of logic.
to avoid being circular, I'm willing to write this one off because of the 'think' and 'understand', as those are the root of the question here. However, forming a judgement by a process of logic is precisely what these LLMs do, and we can see that clearly in chain-of-logic LLM processes.
infer(v): deduce or conclude (information) from evidence and reasoning rather than from explicit statements.
Again, we run the risk of circular logic because of the use of 'reason'. An LLM is for sure using evidence to get to conclusions, however.
analyze(v): examine methodically and in detail the constitution or structure of (something, especially information), typically for purposes of explanation and interpretation.
This one I'm willing to go to bat for completely. I have seen LLM do this, precisely according to the definition above.
For those looking for the link to the above definitions - they're the snippets google provides when searching for "SOMETHING definition". They're a non-paywalled version of OED definitions.
Philosophically I would argue that it's impossible to know what these processes look like in the human mind, and so creating an equivalency (positive or negative) is an exercise in futility. We do not know what a human memory looks like, we do not know what a human thought looks like, we only know what the output of these things looks like. So the only real metric we have for an apples-to-apples comparison is the appearance of thought, not the substance of the thing itself.
That said, there are perceptible differences between the output of a human thought and what is produced by an LLM. These differences are shrinking, and there will come a point where we can no longer distinguish machine thinking and human thinking anymore (perhaps it won't be an LLM doing it, but some model of some kind will). I would argue that at that point the difference is academic at best.
Say we figure out how to have these models teach themselves and glean new information from their interactions. Say we also grant them directives to protect themselves and multiply. At what point do we say that the distinction between the image of man and man itself is moot?