Do repeated measures designs in cognitive research permit causal inference? Can someone please explain this to me because it's driving me bonkers. In cognitive research when one group of participants are exposed to a number of crossed factors manipulated by the researcher, is it a quasi-experimental design or a true experiment?

All the sources I read always mention random assignment and random sampling when talking about true experiments. So do cognitive science experiments such as the repeated measures factorial design a lot of studies use have the right to claim that they establish causal links between the outcome and the manipulations?
If the researcher randomly samples individuals from the population of interest and randomly assigns them to different experimental treatments, then it is a true experiment. Quasi-experimental design occurs when there is no random assignment to treatments.

In general, true experiments can make claims about causality. Since all subjects in the experiment are randomly assigned to a particular treatment, any systematic differences between the groups after the experiment are likely to be due to the treatment. 

However, this is all a game of probabilities. There's always a chance that any systematic difference between groups is from the random assignment. This is why inferential statistics are such an important aspect of evaluating the outcomes of experiments. We can estimate the probability of a particular difference between groups occurring due to random variation. If that probability is low (the typical convention is below 5%), then we can make a causal claim about the treatment.