We present a higher-order inference system based on a formal compositional semantics and the wide-coverage CCG parser. We develop an improved method to bridge between the parser and semantic composition. The system is evaluated on the FraCaS test suite. In contrast to the widely held view that higher-order logic is unsuitable for efficient logical inferences, the results show that a system based on a reasonably-sized semantic lexicon and a manageable number of non-first-order axioms enables efficient logical inferences, including those concerned with generalized quantifiers and intensional operators, and outperforms the state-of-the-art firstorder inference system.