Btw, if you think that the words and the signatures on the paper are the contract, you are wrong. The contract is what you agreed to, and if there is no agreement, there is no contract.
Here is an example that I actually experienced: we proposed a deal to person A. That person agreed and sent the signed contract back. We then also put our signatures onto the contract and put it into a drawer. However, we unfortunately did not notify A that we signed the contract as well, rendering the contract void. I.e.: in my jurisdiction (and many others in Europe), it does not suffice that everyone signed a contract, one also needs to ensure that everyone knows that everyone signed the contract. Otherwise, it is invalid because there was no agreement everyone knew about.