A defining trait of human cognition is the capacity to form compounds out of simple thoughts. This ability relies on the logical connectives AND, OR and IF. Simple propositions, e.g., ‘There is a fork’ and ‘There is a knife’, can be combined in alternative ways using logical connectives: e.g., ‘There is a fork AND there is a knife’, ‘There is a fork OR there is a knife’, ‘IF there is a fork, there is a knife’. How does the brain represent compounds based on different logical connectives, and how are compounds evaluated in relation to new facts? In the present study, participants had to maintain and evaluate conjunctive (AND), disjunctive (OR) or conditional (IF) compounds while undergoing functional MRI. Our results suggest that, during maintenance, the left posterior inferior frontal gyrus (pIFG, BA44, or Broca’s area) represents the surface form of compounds. During evaluation, the left pIFG switches to processing the full logical meaning of compounds, and two additional areas are recruited: the left anterior inferior frontal gyrus (aIFG, BA47) and the left intraparietal sulcus (IPS, BA40). The aIFG shows a pattern of activation similar to pIFG, and compatible with processing the full logical meaning of compounds, whereas activations in IPS differ with alternative interpretations of conditionals: logical vs conjunctive. These results uncover the functions of a basic cortical network underlying human compositional thought, and provide a shared neural foundation for the cognitive science of language and reasoning.