\documentclass[a4paper,enabledeprecatedfontcommands,abstract=on,twoside=true,bibliography=totoc]{scrreprt}
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\input{external.tex}



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\title{Representation and Partial Automation of the
Principia Logico-Metaphysica in Isabelle/HOL}
\author{Daniel Kirchner}

\begin{document}

\begin{titlepage}
\vspace{1cm}

\begin{center}
    \includegraphics[width=0.6\textwidth]{logo}
    \vspace{1cm}


Master's thesis at the institute of mathematics at Freie Universit\"at Berlin

    \vspace{2cm}


    \Large{\textsf{Representation and Partial Automation of the Principia Logico-Metaphysica in Isabelle/HOL}}

    \vspace{2cm}

    \large{\textbf{Daniel Kirchner}}

	\vspace{0.25cm}

	\small{Matrikelnummer: 4387161}

    \vspace{2cm}

    \large{\textbf{
        Supervisors:\\
Priv.-Doz. Dr.-Ing. Christoph Benzm\"uller\\
Dr. Edward N. Zalta
    }}

    \vspace{2cm}
    \large{Berlin, \today}
\end{center}
\end{titlepage}

\cleardoublepage

\begin{abstract}
	We present an embedding of the second-order fragment of the Theory of Abstract Objects as described in Edward Zalta's
  upcoming work Principia Logico-Metaphysica (PLM\cite{PM}) in the automated reasoning framework Isabelle/HOL. The Theory of Abstract
  Objects is a metaphysical theory that reifies property patterns, as they for example occur in the abstract reasoning
  of mathematics, as \emph{abstract objects} and provides an axiomatic framework that allows to reason about these objects.
  It thereby serves as a fundamental metaphysical theory that can be used to axiomatize and describe a wide range of philosophical
  objects, such as Platonic forms or Leibniz' concepts, and has the ambition to function as a foundational theory of mathematics.
  The target theory of our embedding as described in chapters 7-9 of PLM\cite{PM} employs a modal relational type theory as
  logical foundation for which a representation in functional type theory is known to be challenging\cite{rtt}.
  
  Nevertheless we arrive at a functioning representation of the theory in the functional logic of Isabelle/HOL based on a semantical
  representation of an Aczel-model of the theory. Based on this representation we construct an implementation of the deductive
  system of PLM (\cite[Chap. 9]{PM}) which allows to automatically and interactively find and verify theorems of PLM.

  Our work thereby supports the concept of shallow semantical embeddings of
  logical systems in HOL as a universal tool for logical reasoning as
  promoted by Christoph Benzm\"uller\cite{UniversalReasoning}.

  The most notable result of the presented work is the discovery of a previously
  unknown paradox in the formulation of the Theory of Abstract Objects.
  The embedding of the theory in Isabelle/HOL played a vital
  part in this discovery. Furthermore it was possible to immediately offer
  several options to modify the theory to guarantee its consistency.
  Thereby our work could provide a significant contribution to the
  development of a proper grounding for object theory.
\end{abstract}

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\tableofcontents

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\appendix
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\chapter{Isabelle Theory}
\input{TAO_1_Embedding}
\input{TAO_2_Semantics}
\input{TAO_3_Quantifiable}
\input{TAO_4_BasicDefinitions}
\input{TAO_5_MetaSolver}
\input{TAO_6_Identifiable}
\input{TAO_7_Axioms}
\input{TAO_8_Definitions}
\input{TAO_9_PLM}
\input{TAO_10_PossibleWorlds}
\input{TAO_98_ArtificialTheorems}
\input{TAO_99_SanityTests}
\input{TAO_99_Paradox}

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\chapter*{Selbstst\"andigkeitserkl\"arung}
\selectlanguage{ngerman}

\begin{center}
\setlength\extrarowheight{4pt}
\begin{tabularx}{\textwidth}{|X|X|}
\hline
Name: & Kirchner \\
\hline
Vorname: & Daniel \\
\hline
geb.am: & 22.05.1989 \\
\hline
Matr.Nr.: & 4387161 \\
\hline
\end{tabularx}
\end{center}

Hiermit versichere ich, dass ich die vorliegende Arbeit selbstst\"andig
verfasst und keine anderen als die angegebenen Quellen und Hilfsmittel
benutzt habe.

Alle Ausf\"uhrungen, die w\"ortlich oder inhaltlich aus fremden Quellen
\"ubernommen sind, habe ich als solche kenntlich gemacht.

Diese Arbeit wurde in gleicher oder \"ahnlicher Form noch bei keiner
anderen Universit\"at als Pr\"ufungsleistung eingereicht und ist auch
noch nicht ver\"offentlicht.

\vspace{50pt}
\noindent\hfill\rule{7cm}{.4pt}\par
   \hfill Daniel Kirchner


\end{document}

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