Modern supernova (SN) surveys are now uncovering stellar explosions at rates that far surpass what the world's spectroscopic resources can handle. In order to make full use of these SN data sets, it is necessary to use analysis methods that depend only on the survey photometry. This paper presents two methods for utilizing a set of SN light-curve templates to classify SN objects. In the first case, we present an updated version of the Bayesian Adaptive Template Matching program (BATM). To address some shortcomings of that strictly Bayesian approach, we introduce a method for Supernova Ontology with Fuzzy Templates (SOFT), which utilizes fuzzy set theory for the definition and combination of SN light-curve models. For well-sampled light curves with a modest signal-to-noise ratio (S/N >10), the SOFT method can correctly separate thermonuclear (Type Ia) SNe from core collapse SNe with ≥98% accuracy. In addition, the SOFT method has the potential to classify SNe into sub-types, providing photometric identification of very rare or peculiar explosions. The accuracy and precision of the SOFT method are verified using Monte Carlo simulations as well as real SN light curves from the Sloan Digital Sky Survey and the SuperNova Legacy Survey. In a subsequent paper, the SOFT method is extended to address the problem of parameter estimation, providing estimates of redshift, distance, and host galaxy extinction without any spectroscopy.