Incremental parsing is appealing for applications such as speech recognition and machine translation due to its inherent efficiency as well as being a natural match for the language models commonly used in such systems. In this paper we introduce an Incremental Combinatory Categorical Grammar (ICCG) that extends the standard CCG grammar to enable fully incremental left-to-right parsing. Furthermore, we introduce a novel dynamic programming algorithm to convert CCGbank normal form derivations to incremental left-to-right derivations and show that our incremental CCG derivations can recover the unlabeled predicate-argument dependency structures with more than 96% F-measure. The introduced CCG incremental derivations can be used to train an incremental CCG parser.