Since we regard material warrants not as suppressed premises but as inference rules, we disagree in many cases with the standard approach to reconstructing enthymemes by supplying alleged non-explicit premises. Why is our approach preferable? We appeal first to Hitchcock’s phenomenological argument: People reasoning according to enthymemes are not aware of holding non-explicit premises in the mind. In addition, if one is not aware of a non-explicit premise, how can one be sure that a reconstruction properly formulates that premise? Furthermore, as Hitchcock points out, we need not be conscious of the inference rules in accordance with which we reason, but we could expect to be conscious of the premises from which we reason. Finally, counting the warrant as a non-explicit premise misconstrues its function in the argument. It is not a ground for the conclusion but an indication of why the grounds have a bearing on the conclusion. However, we do not follow Hitchcock in holding that virtually no arguments have non-explicit premises. With each argument we may identify an associated generalization, the counterpart of a warrant. Regarding the function of that generalization as an inference rule and not as a non-explicit premise is well-motivated for first-order enthymemes. Applying this interpretation to second-order enthymemes is problematic, generating on occasion unreliable inference rules. We argue that the associated conditional of an argument can be construed as an warrant only if it asserts a true nomic generalization. Hence if the associated general conditional is an accidental universal, it should be understood as a non-explicit premise. If the accidental universal is logically equivalent to a singular statement, it is straightforward to construe that statement as the non-explicit premise. We conclude the chapter by pointing out various other advantages of our approach.