The crux of software transactional memory (STM) is to combine an easy-to-use programming interface with an efficient utilization of the concurrent-computing abilities provided by modern machines. But does this combination come with an inherent cost? We evaluate the cost of concurrency by measuring the amount of expensive synchronization that must be employed in an STM implementation that ensures positive concurrency, i.e., allows for concurrent transaction processing in some executions. We focus on two popular progress conditions that provide positive concurrency: progressiveness and permissiveness. We show that in permissive STMs, providing a very high degree of concurrency, a transaction performs a linear number of expensive synchronization patterns with respect to its read-set size. In contrast, progressive STMs provide a very small degree of concurrency but, as we demonstrate, can be implemented using at most one expensive synchronization pattern per transaction. However, we show that even in progressive STMs, a transaction has to "protect" (e.g., by using locks or strong synchronization primitives) a linear amount of data with respect to its write-set size. Our results suggest that looking for high degrees of concurrency in STM implementations may bring a considerable synchronization cost.