Deduction Theorem for Symmetric Cirquent Calculus

Cirquent calculus is a new proof-theoretic and semantic framework, whose main distinguishing feature is being based on circuits, as opposed to the more traditional approaches that deal with tree-like objects such as formulas or sequents. Among its advantages are greater efficiency, flexibility and expressiveness. This paper presents a detailed elaboration of a deep-inference cirquent logic, which is naturally and inherently resource conscious. It shows that classical logic, both syntactically and semantically, is just a special, conservative fragment of this more general and, in a sense, more basic logic -- the logic of resources in the form of cirquent calculus. The reader will find various arguments in favor of switching to the new framework, such as arguments showing the insufficiency of the expressive power of linear logic or other formula-based approaches to developing resource logics, exponential improvements over the traditional approaches in both representational and proof complexities offered by cirquent calculus, and more. Among the main purposes of this paper is to provide an introductory-style starting point for what, as the author wishes to hope, might have a chance to become a new line of research in proof theory -- a proof theory based on circuits instead of formulas.

The first results in proof theory were obtained using axiomatic systems of logical deduction. The chapter develops propositional and predicate logic in such an axiomatic system. It shows how reasoning under assumption can be regained in an axiomatic system by proving the so-called “deduction theorem” propositional logic (and later, predicate logic as well). The deduction theorem provides the opportunity to introduce one of the main tools needed for proving properties of formal systems, namely, proof by induction on the natural numbers. The differences between minimal, intuitionistic, and classical systems of logic according to the properties of negation characterizing each of the systems are outlined. Independence of theorems such as the law of excluded middle are here given model-theoretically (in later chapters, these are also presented proof-theoretically). After presenting classical predicate logic, the systems of classical and intuitionistic arithmetic are presented. Gentzen’s first substantial result, the so-called Gödel-Gentzen translation, is discussed in the final section where it is proved that if intuitionistic arithmetic is consistent, so is classical arithmetic.

In this paper, we investigate the proof theory of default reasoning. We generalize Reiter's framework to a monotonic reasoning system, and in particular allow formulae with nested defaults. We give proof rules for this extended default logic, called default ionic logic, and give deduction theorems. We also give examples of applications of our framework to some well-known problems: weak implication, disjunctive information, default transformation, and normal versus non-normal defaults.