Inductive Reasoning in First-Order Logic

Extending classical first order logic with the capability of representing object-level statements expressing a default relationship --- i.e. facts linked by plausible implication --- as opposed to one of necessity, has remained an open problem in the context of formalising common-sense defeasible statements. Additionally, defining meta-level defeasible inference relations over the default quantifier language is the other necessary component for a full defeasible first order logic. Conditional connectives, that have been investigated extensively in conditional propositional logics, do not seem to behave sufficiently well in the first order context. Just using universally quantified variables with default conditionals results in contradictions with exceptional instances due to specificity. Additionally, there are many ranking-based defeasible inference relations that have not been formalised over a defeasible first-order logic. The solution investigated in this thesis is that of extending first order logic with default quantifiers: variable binding binary connectives encoding a notion of typicality, quantified over free variables in the two formulae in question. To interpret default quantifier statements, a ranking measure semantics, only well defined for propositional defeasible reasoning, is chosen to extend regular first order model theory. Then, the aforementioned nonmonotonic entailment relations may be defined over this logic of default quantification. The specific case of defeasible description logics is also presented, along with a novel approach for extending these logics with defeasibility via the syntactic addition of a hypothesis box, which informs a range of speculative inference techniques. Benchmark examples from the literature are here presented and analysed to showcase the efficacy of this general approach to default reasoning with first-order information.