As Raphael pointed in a comment you are probably asking about the decidability of the classical decision problem for fragments of first order logic with unary predicates.
The search of syntactically restricted fragments of first-order logic where validity of formulas is decidable was an active field of research for over 100 years. It was understood very early that it is the dependencies between variables that make the decision problem difficult. And dependencies are best expressed when two variables occur within the arguments of a predicate. Therefore binary predicates or functional symbols (if you allow functional symbols) are essential for undecidability.
It was Löwenheim who in 1915 gave a decision procedure for first-order logic with unary predicates. If you are interested, you might find intriguing an exposition paper titled "On the Classical Decision Problem" written by Yuri Gurevich in the form of a dialog of two fictitious characters. The paper was first published in the Bulletin of EATCS.
The paper not only introduces the problem but explains how the search developed into finding a complete classification of decidable and undecidable fragments in terms of quantifier prefixes.