A research programme is proposed for the inference of unpreserved attributes of fossil taxa. The programme includes: (i) phylogenetic inference of attributes based on the cladistic distribution of known features in related taxa; and (ii) extrapolatory analyses that infer unpreserved features from the known attributes of the fossil. Phylogenetic inferences regarding the fossil taxon are based on the attributes of both the sister group of the fossil taxon and more distantly related clades. Unlike phylogenetic inferences that are based on a single related taxon, this broader phylogenetic context avoids unjustified assumptions regarding the occurrence of unpreserved features in particular fossil taxa. Phylogenetic inference is conservative; only features in related taxa can be inferred in the fossil. Extrapolatory analyses, such as form-function correlation and biomechanical design analysis, provide a means for choosing among equivocal phylogenetic inferences, and, on occasion, can provide a basis for rejecting a phylogenetic inference. Extrapolatory approaches provide the only means of inferring or interpreting autapomorphies in fossils. The results of phylogenetic and extrapolatory approaches to the reconstruction of the shoulder musculature of the ornithomimid Struthiomimus are compared. Results are congruent in most instances; however, many of the extrapolatory inferences are implicitly phylogenetic. The phylogenetic inferences constitute a null hypothesis regarding fossil attributes, and place constraints on the inferences generated by extrapolatory analyses. The potential uncertainty and untestability of many extrapolatory analyses suggests that the phylogenetic inference should be overturned only when the functional or other extrapolatory evidence is compelling. This procedure should identify and reduce speculation in fossil reconstruction.