A new theory of evaluating counterfactual statements is presented based on the established predicate calculus formalism of the Fluent Calculus for reasoning about actions. The assertion of a counterfactual antecedent is axiomatized as the performance of a special action. An existing solution to the Ramification Problem in the Fluent Calculus, where indirect effects of actions are accounted for ...

We investigate the first-order theory of subtype constraints. We show that the first-order theory of non-structural subtyping is undecidable, and we show that in the case where all constructors are either unary or nullary, the first-order theory is decidable for both structural and non-structural subtyping. Our results hold for both simple and recursive types. The undecidability result is shown...

We expose a formalism that allows the expression of any theory with one or more axiom schemes using a finite number of axioms. This allows us to give finite first-order axiomatizations of arithmetic and real analysis, and a presentation of arithmetic in deduction modulo that has a finite number of rewrite rules. Overall, this formalization relies on a weak calculus of explicit substitutions to ...

We present a formalization of parts of Melvin Fitting’s book “FirstOrder Logic and Automated Theorem Proving” [1]. The formalization covers the syntax of first-order logic, its semantics, the model existence theorem, a natural deduction proof calculus together with a proof of correctness and completeness, as well as the Löwenheim-Skolem theorem.

An approximate Herbrand theorem is proved and used to establish Skolemization for first-order Łukasiewicz logic. Proof systems are then defined in the framework of hypersequents. In particular, extending a hypersequent calculus for propositional Łukasiewicz logic with usual Gentzen quantifier rules gives a calculus that is complete with respect to interpretations in safe MV-algebras, but lacks ...

The author has submitted to Mizar Mathematical Library a series of five articles introducing a framework for the formalization of classical first-order model theory. In them, Gödel’s completeness and Lowenheim-Skolem theorems have also been formalized for the countable case, to offer a first application of it and to showcase its utility. This is an overview and commentary on some key aspects of...