81 research outputs found
Schematic Cut elimination and the Ordered Pigeonhole Principle [Extended Version]
In previous work, an attempt was made to apply the schematic CERES method [8]
to a formal proof with an arbitrary number of {\Pi} 2 cuts (a recursive proof
encapsulating the infinitary pigeonhole principle) [5]. However the derived
schematic refutation for the characteristic clause set of the proof could not
be expressed in the formal language provided in [8]. Without this formalization
a Herbrand system cannot be algorithmically extracted. In this work, we provide
a restriction of the proof found in [5], the ECA-schema (Eventually Constant
Assertion), or ordered infinitary pigeonhole principle, whose analysis can be
completely carried out in the framework of [8], this is the first time the
framework is used for proof analysis. From the refutation of the clause set and
a substitution schema we construct a Herbrand system.Comment: Submitted to IJCAR 2016. Will be a reference for Appendix material in
that paper. arXiv admin note: substantial text overlap with arXiv:1503.0855
Hilbert's "Verunglueckter Beweis," the first epsilon theorem, and consistency proofs
In the 1920s, Ackermann and von Neumann, in pursuit of Hilbert's Programme,
were working on consistency proofs for arithmetical systems. One proposed
method of giving such proofs is Hilbert's epsilon-substitution method. There
was, however, a second approach which was not reflected in the publications of
the Hilbert school in the 1920s, and which is a direct precursor of Hilbert's
first epsilon theorem and a certain 'general consistency result' due to
Bernays. An analysis of the form of this so-called 'failed proof' sheds further
light on an interpretation of Hilbert's Programme as an instrumentalist
enterprise with the aim of showing that whenever a `real' proposition can be
proved by 'ideal' means, it can also be proved by 'real', finitary means.Comment: 18 pages, final versio
Elimination of Cuts in First-order Finite-valued Logics
A uniform construction for sequent calculi for finite-valued first-order logics with distribution quantifiers is exhibited. Completeness, cut-elimination and midsequent theorems are established. As an application, an analog of Herbrand’s theorem for the four-valued knowledge-representation logic of Belnap and Ginsberg is presented. It is indicated how this theorem can be used for reasoning about knowledge bases with incomplete and inconsistent information
Intensional Models for the Theory of Types
In this paper we define intensional models for the classical theory of types,
thus arriving at an intensional type logic ITL. Intensional models generalize
Henkin's general models and have a natural definition. As a class they do not
validate the axiom of Extensionality. We give a cut-free sequent calculus for
type theory and show completeness of this calculus with respect to the class of
intensional models via a model existence theorem. After this we turn our
attention to applications. Firstly, it is argued that, since ITL is truly
intensional, it can be used to model ascriptions of propositional attitude
without predicting logical omniscience. In order to illustrate this a small
fragment of English is defined and provided with an ITL semantics. Secondly, it
is shown that ITL models contain certain objects that can be identified with
possible worlds. Essential elements of modal logic become available within
classical type theory once the axiom of Extensionality is given up.Comment: 25 page
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