[Update] Multiple target variables can exist.
If I find that a variable is not a predictor variable, does that mean it automatically becomes a target variable?
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Predictor Variable - One or more variables that are used to determine(Predict) the 'Target Variable'.
Target Variable - A variable that needs to be predicted is a target variable.
The above quantities are determined prior to the experiment, the person who is conducting the experiment has to come up with a problem statement and once he does, he determines which variables are predictor variables and which one is the target variable.
If I find that a variable is not a predictor variable, does that mean it automatically becomes a target variable? No, if you read the description above your question is invalid.
Generally, if you have data of very high dimensionality and a target variable, you will have to reduce the number of dimensions (depending upon the scenario) either you will decide to go will all the predictor variables or choose a dimensionality technique such as PCA(Principal Component Analysis). If you choose to perform PCA, you will select handful of predictor variables and drop(remove) the remaining variables, while doing so the Target Variable remains the same.
You can use a variable as both "predictor" and "target". Imagine a simple regression problem, where past values of a variable are used to forecast future values of the same variable.
Apart from that, you can have multiple predictor and target variables in the same problem. It is a multiple input / multiple output problem. A way of dealing with such problem with time series, is to deploy an LSTM-RNN that accepts multiple inputs and forecasts the signals of interest.