In this paper, we develop techniques for automated hypothesis-space exploration over data sets that may contain contradictions. To do so, we make use of the equivalence between two formulations: those of first-order predicate logic with prefix modal quantifiers under the finite-model hypothesis and those of mixed-integer linear programming (MILP) problems. Unlike other approaches, we do not assume that all logical assertions are true without doubt. Instead, we look for alternative hypotheses about the validity of the claims by identifying alternative optimal solutions to a corresponding MILP. We use a collection of slack variables in the derived linear constraints to indicate the presence of contradictory data or assumptions. The objective is to minimize contradictions between data and assertions represented by the presence of nonzero slack in the set of linear constraints. In this paper, we present the following: 1) a correspondence between first-order predicate logic with modal quantifier prefixes under the finite-model hypothesis and MILP problems and 2) an implicit enumeration algorithm for exploring the contradiction hypothesis space.