Recognizing fossil species, and reconstructing their phylogenetic relationships, is dependent in part on our ability to accurately record and analyze continuously distributed data. We usually classify specimens into groups (whether species or populations) by using crisp set theory. Fuzzy set theory can aid analysis in situations where populations have indistinct boundaries. Two primary areas of cladistic research contain elements of fuzzinessthe species concept itself, and the variables we use to analyze phylogenetic relationships among species. Analytical determination of species boundaries can be difficult due to hybridization as well as intra-and inter-specific variability. However, cladistic analysis requires that taxa are monophyletic, and unique in their morphological patterns. Variables used in cladistic analysis must be coded using discrete character states to determine polarity. But many morphological and metric variables are continuous or quasicontinuous in distribution, and are problematic for use in cladistic analyses without some type of non-overlapping dichotomization or breakpoint. The issue of how to utilize these continuously or quasicontinuously distributed variables, whether metric or nonmetric, is an ongoing issue in phylogenetic analysis. Fuzzy set theory is a method that circumvents two problematic assumptions implicit in phylogenetic analyses-crispness of taxa and crispness of traits. Using a fuzzy analysis, multiple character states are maintained through fuzzy variable sets that maintain the fuzziness of boundaries. Additionally, fuzzy analysis calculates a measure of the degree of group membership.To illustrate this process, I analyzed nine multistate nonmetric cranial variables representing three regions of the skull on 14 Neandertals and 24 early moderns using a Mamdani Fuzzy Inference System to compare the relative performance of fuzzy and crisp variables and groups on group identification. The analysis that performed best contained both fuzzy trait groups and fuzzy taxon groups. Fuzzy analysis is useful to explore the degree to which fuzziness is present in trait variation and in taxa, and it can advance our understanding of species identification and phylogeny construction. Expanding our toolkit to include fuzzy set analysis can help us determine how crisp or fuzzy are our putative taxonomic groups. "[T]he cult of impressive technicalities or the cult of precision may get the better of us, and interfere with our search for clarity, simplicity, and truth" (Popper 1983: 60). "The closer one looks at a real-world problem, the fuzzier becomes its solution" (Zadeh 1973: 28).
INTRODUCTIONT here are many areas in paleoanthropology where precision eludes us. It is uncertain how many fossil species existed in the past; fossil sample sizes are small, and the magnitude and pattern of character variation within fossil species is unknown. It is unclear to what degree many characters are correlated, or are selectively neutral. Nonmetric characters may present as a continuous distributio...
