Not quite.

Legal Othello board moves by themselves don't say anything about the board size or rules.

And yet when Harvard/MIT researchers fed them into a toy GPT model, they found that the neural network best able to predict outputting legal moves had built an internal representation of the board state and rules.

Too many people commenting on this topic as armchair experts are confusing training with what results from the training.

Training on completing text doesn't mean the end result can't understand aspects that feed into the original generation of that text, and given a fair bit of research so far, the opposite is almost certainly the case to some degree.