What exists in a individual person when they stand in a symbol_N_r relation (see naming) is a disposition to use that symbol in a certain way. The usual way a person comes to have such a disposition is to begin with social training and then social teaching or education. The meaning of the symbol or word in society is not given simply by the dispositions of a member of that society. An individual person's dispositions may stray from other members of the society. The person may make mistakes. These mistakes may or may not be noticed and corrected by other members of the society. It may be objected that a person's dispositions are finite (Kripke if I remember correctly), but this is false. E.g. a Turing machine itself is very much finite, but can give an infinite number of outputs on an infinite number of inputs. The Turing machine may be regarded as having a disposition to respond with the outputs from those inputs. However, it can be seen the simple behaviorist model of stimulus-response is inadequate to model the Turing machine behavior. Whether human behavior can be modeled by a Turing machine seems to me a category mistake. What matters (how a Turing machine is judged) is the output for any given input. How human behavior is judged is not so simple, but rather a long sequence of interactions with the environment. But the Turing machine example suffices to remove the objection that a person can only have a finite number of dispositions.
The social meaning of a symbol or word is the result of the dispositions of the members of that society. And the meaning of symbols and words can also evolve in a society. It is not necessary to consider how the meaning of words might change in considering what they mean now. Thus in considering whether 'bachelor' and 'unmarried adult male' are synonyms we need not take into account how the use of these words might change in the future.
In considering a individual's use of language we need to take into account their individual dispositions. In understanding how they acquire those dispositions we need to consider their interactions with other members of society.
In 'going on in the same way', e.g. when we add numbers we never have before, we do so, usually correctly, because our of our common nature and experience with other humans. And, if we make a mistake, and it is pointed out to us, we usually can recognize it.
We need make no judgment here about determinism or 'freedom of the will'. The existence of what I have here called 'dispositions' need not imply determinism.