Most of the real life classification problems have ill defined, imprecise or fuzzy class boundaries. Feedforward neural networks with conventional backpropagation learning algorithm are not tailored to these kinds of classification problems. Hence, in this paper, feedforward neural networks, that use fuzzy objective functions in the backpropagation learning algorithm, are investigated. A learning algorithm is proposed that minimizes an error term, which takes care of fuzziness in classification from the point of view of possibilistic approach. Since the proposed algorithm has possibilistic classification ability, it can encompass different backpropagation learning algorithms based on crisp and constrained fuzzy classification. The efficacy of the proposed scheme is demonstrated on a vowel classification problem.