You do not need objective truths to use deductive logic. That statement shows that you fundamentally misunderstand deductive logic.
You simply need premises (i.e. assumptions) to make use of deductive logic in a standard logical system. Those premises, when used with logic that has no fallacies, implies conclusions. Any real world conclusions are always based on premises, where those premises are based on experimental evidence. It's always the case that there could be problems with the experiments, and there always are limitations of those experiments. That doesn't mean you can't use logic to derive conclusions.
You simply need premises (i.e. assumptions) to make use of deductive logic in a standard logical system. Those premises, when used with logic that has no fallacies, implies conclusions. Any real world conclusions are always based on premises, where those premises are based on experimental evidence. It's always the case that there could be problems with the experiments, and there always are limitations of those experiments. That doesn't mean you can't use logic to derive conclusions.