The logic of the weak law of excluded middleKC

 p
 is obtained by adding the formula ℸA ∨ ℸ ℸA as an axiom scheme to Heyting's intuitionistic logicH

 p
 . A cut-free sequent calculus for this logic is given. As the consequences of the cut-elimination theorem, we get the decidability of the propositional part of this calculus, its separability, equality of the negationless fragments ofKC

 p
 andH

 p
 , interpolation theorems and so on. From the proof-theoretical point of view, the formulation presented in this paper makes clearer the relations betweenKC

 p
 ,H

 p
 , and the classical logic. In the end, an interpretation of classical propositional logic in the propositional part ofKC

 p
 is given.

Proof Theory

A cut-free gentzen-type system for the logic of the weak law of excluded middle

Coinductive reasoning about infinitary data structures has many applications in computer science Nonetheless developing natural proof systems (especially ones amenable to automation) for reasoning about coinductive data remains a challenge This paper presents a minimal, generic formal framework that uniformly captures applicable (ie finitary) forms of inductive and coinductive reasoning in an intuitive manner The logic extends transitive closure logic, a general purpose logic for inductive reasoning based on the transitive closure operator, with a dual ‘co-closure’ operator that similarly captures applicable coinductive reasoning in a natural, effective manner We develop a sound and complete non-well-founded proof system for the extended logic, whose cyclic subsystem provides the basis for an effective system for automated inductive and coinductive reasoning To demonstrate the adequacy of the framework we show that it captures the canonical coinductive data type: streams

methods

/pdf/integrating-induction-and-coinduction-via-closure-operators-13ymzw7z88.pdf

Integrating Induction and Coinduction via Closure Operators and Proof Cycles

Consistency proof without transfinite induction for a formal system for turing machines

We consider the constructive ordinal notation system for the ordinal ${\epsilon_0}$ that were introduced by L.D. Beklemishev. There are fragments of this system that are ordinal notation systems for the smaller ordinals ${\omega_n}$ (towers of ${\omega}$-exponentiations of the height $n$). This systems are based on Japaridze's provability logic $\mathbf{GLP}$. They are closely related with the technique of ordinal analysis of $\mathbf{PA}$ and fragments of $\mathbf{PA}$ based on iterated reflection principles. We consider this notation system and it's fragments as structures with the signatures selected in a natural way. We prove that the full notation system and it's fragments, for ordinals ${\ge\omega_4}$, have undecidable elementary theories. We also prove that the fragments of the full system, for ordinals ${\le\omega_3}$, have decidable elementary theories. We obtain some results about decidability of elementary theory, for the ordinal notation systems with weaker signatures.

On Elementary Theories of Ordinal Notation Systems based on Reflection Principles

In this study, falsification-aware semantics and sequent calculi for first-order classical logic are introduced and investigated. These semantics and sequent calculi are constructed based on a falsification-aware setting for first-order Nelson constructive three-valued logic (N3). In fact, these semantics and sequent calculi are regarded as those for a classical variant of N3 (i.e., a classical variant of N3 is identical to first-order classical logic). The completeness and cut-elimination theorems for the proposed semantics and sequent calculi are proved using Schutte’s method. Similar results for the propositional case are also obtained.

Proof Theory

Citations

A cut-free gentzen-type system for the logic of the weak law of excluded middle

Integrating Induction and Coinduction via Closure Operators and Proof Cycles

Consistency proof without transfinite induction for a formal system for turing machines

On Elementary Theories of Ordinal Notation Systems based on Reflection Principles

Falsification-Aware Semantics and Sequent Calculi for Classical Logic

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