Proof schemata are infinite sequences of proofs which are defined inductively. In this paper we present a general framework for schemata of terms, formulas and unifiers and define a resolution calculus for schemata of quantifier-free formulas. The new calculus generalizes and improves former approaches to schematic deduction. As an application of the method we present a schematic refutation formalizing a proof of a weak form of the pigeon hole principle.

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Proof Theory

Schematic Refutations of Formula Schemata

http://www.filozof.uni.lodz.pl/bulletin/pdf/12_3_3.pdf

The class DA of diagonalizable algebras enjoys the interpolation prop- erty (cf. (3)). Using LDA-Gentzen System corresponding to DA, we show that it has the strong amalgamation property as well.

In this chapter, we present some motivation for proofs explanations For the tasks of providing proofs and explanations, we compare three deduction systems In the sequel, we present an example of proof and possible explanation of this same proof in our system ${\mathrm{ SC} }_{\mathcal{ ALC} }$ Finally, in the last section of this chapter, we use examples of DL deductions from Berardi et al (Artif Intell 168:84, 2005), using ${\mathrm{ ND} }_{\mathcal{ ALCQI} }$ to reason on the $\mathcal{ ALCQI} $ KB The idea is to exemplify how one can obtain from ${\mathrm{ ND} }_{\mathcal{ ALCQI} }$ proofs, a more precise and direct explanation

Proofs and Explanations

In this note the proof-theoretic ordinal of the well-ordering principle for the normal functions $$\mathsf {g}$$
 on ordinals is shown to be equal to the least fixed point of $$\mathsf {g}$$
 . Moreover corrections to the previous paper (Arai in Arch Math Log 57:649–664, 2017) are made.

Proof-theoretic strengths of the well-ordering principles

We introduce a model-theoretic definition of non-monotonic inference relation and study a particular model called C*. Gentzen-style counterpart of C* can be regarded as the nonmonotonic infinitistic version of the system C of Kraus, Magidor, Lehmann with the consistency-preservation, property proposed by Gabbay and used in belief revision. A notion of selection function is introduced. This notion is similar to that of expectations given by Gardenfors and Makinson and derived from partial meet contractions functions of belief revision, given by Alchourron, Gardenfors and Makinson. The representation theorem for the cumulative system C, whose models use a binary relation < among sets of worlds, can be simplified by using selection functions. The finitistic (conditional) system of C* turns out to be equivalent to Makinson's cumulative monotony operator as well as to the conditional system γ* derived from the sistem γ of Gabbay by adding the cautious monotonicity axiom and using classical logic in the place of intuitionistic logic as underlying monotonic logic.

Proof Theory

Citations

Schematic Refutations of Formula Schemata

The class DA of diagonalizable algebras enjoys the interpolation prop- erty (cf. (3)). Using LDA-Gentzen System corresponding to DA, we show that it has the strong amalgamation property as well.

Proofs and Explanations

Proof-theoretic strengths of the well-ordering principles

Some Notes on Cumulative Reasoning

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