The need for reconfigurable manufacturing systems has long been recognized as a key factor to gain the necessary flexibility for economical production of small batch sizes. Automation of the reconfiguration processes is a challenge both on the hardware and the software level. Addressing this issue in the field of fixture design, a new reconfigurable fixture device for a CNC milling machine has been developed. The developed vise contains interchangeable and customizable jaws enabling the secure fixture of a variety of workpiece geometries. To enable automated reconfiguration, a reasoning system is needed that can determine feasible fixture configurations based on the given workpiece and part as well as the available fixture components. In this paper, an ontology for representing fixture design and reconfiguration knowledge for vise-type flexible fixtures is presented. The ontology builds on and extends an existing ontology for modular fixture design. The creation of the ontology is based on a systematic building methodology, going from informal to formal concept definitions in a middle-out approach. The core concepts and relations of the ontology are presented and the ontology is validated both on the informal and on the formal level by its ability to find feasible fixture configurations, i.e., appropriate vise jaw pairs to fix example workpieces. The ontology can represent type-specific fixture designs in an unambiguous way and can hence serve as a basis for the development of applications needed to create an autonomous fixture design system.