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\begin{document}

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\section*{abductive reasoning}

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{general_idea}{General idea}\dotfill \pageref*{general_idea} \linebreak
\noindent\hyperlink{abduction_as_lifting}{Abduction as lifting}\dotfill \pageref*{abduction_as_lifting} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{general_idea}{}\subsection*{{General idea}}\label{general_idea}

\textbf{Abductive} [[reasoning]] is a process whereby one reasons to the truth of an explanation from its ability to account for what is observed. It is therefore sometimes also known as \textbf{inference to the best explanation}.

[[Charles Peirce]], the originator of the term, illustrated the differences between [[deductive reasoning|deduction]], [[inductive reasoning|induction]], and abduction by the following example.

\begin{itemize}%
\item \textbf{Deduction}

\begin{itemize}%
\item All beans in that bag are white.
\item These beans are from that bag.
\item Therefore, these beans are white.

\end{itemize}

\item \textbf{Induction}

\begin{itemize}%
\item These beans are from that bag.
\item These beans are white.
\item Therefore, all beans in that bag are white.

\end{itemize}

\item \textbf{Abduction}

\begin{itemize}%
\item These beans are white.
\item All beans in that bag are white.
\item Therefore, these beans are from that bag.

\end{itemize}


\end{itemize}
It is not completely clear what Peirce meant by abduction, which he also termed \textbf{retroduction}. Clearly the inference cannot be to just any possible explanation, e.g., in the case above, there might have been many other bags full of white beans. But before we decide what constitutes a \textbf{best} explanation, we had been inquire into the nature of explanation itself.

There is an extensive literature about explanation in the Philosophy of Science, for example, (\hyperlink{FourDecades}{FourDecades}). Clearly it is not merely a matter of devising propositions, perhaps a general law and a particular statement, which have the thing to be explained (explicandum) as a consequence. We want the proposed explanation to `give the reason for' the observation. A thorough account of what constitutes the `reason' for something is notoriously difficult to formulate. For some, it is a matter of subsuming the observation under a general covering law, while for others, it is a matter of giving a causal or mechanistic story with the observation as the outcome.

Note also that there is a growing literature now on the concept of `explanatory proofs' in mathematics, it being felt that one may have proved a mathematical fact without understanding `why' it is true.

For some, abduction also signifies the creative process of coming up with a good explanation. Otherwise, if it is merely a case of assessing a range of existing rival hypotheses as explanations, it may be possible to employ [[Bayesian reasoning]], generally taken to be a form of [[inductive reasoning]].

\begin{quote}%
If you really can find an explanation having sufficient probability to be worth consideration, you escape in great measure from reposing upon retroduction abduction and make your inference inductive. (Peirce, \hyperlink{Harvard}{Harvard Lectures, p. 193})


\end{quote}
\hypertarget{abduction_as_lifting}{}\subsection*{{Abduction as lifting}}\label{abduction_as_lifting}

In Peirce's \hyperlink{Harvard}{Harvard lectures, p. 315}, he describes the triad -- \emph{deduction}, \emph{induction}, \emph{abduction} -- in terms of the logical relations between three concepts, $M$, $P$ and $S$.

\begin{itemize}%
\item Deduction strings together, say, $M$ is $P$ and $P$ is $S$ to give $M$ is $S$.


\item Induction looks to generalise from $M$ is $S$, taking $M$ as a sample of $P$, to conclude that $P$ is $S$.


\item Abduction looks to explain why $M$ is $S$, having noted that $P$ is $S$, by hypothesising that $M$ is $P$.



\end{itemize}
Seen from the point of view of category theory, this would seem rather like: composition, [[extension]], and [[lifting]]. Induction as a kind of extension seems quite reasonable. Abduction may account for an instance of some concept, $E$, by lifting to a concept, $C$, through a law connecting cause, $C$, to effect, $E$.

\hypertarget{references}{}\subsection*{{References}}\label{references}

\begin{itemize}%
\item Wesley Salmon, 1989, \emph{Four Decades of Scientific Explanation}, University of Pittsburgh Press.


\item Proposed formalization as a functor between \href{structure+in+model+theory#categories_of_structures}{categories of structures} can be found in Fernando Tohm\'e{}, Gianluca Caterina, Rocco Gangle, \href{http://doi.org/10.1016/j.jal.2014.12.004}{\emph{Abduction: A categorical characterization}}, Journal of Applied Logic, Volume 13, Issue 1, March 2015, Pages 78-90


\item Gerhard Schurz, 2008, \emph{Patterns of abduction}, Synthese 164:201--234.


\item [[Charles Peirce]], 1992, \emph{Reasoning and the logic of things}, Harvard University Press (lectures from 1898, \href{http://www.hup.harvard.edu/catalog.php?isbn=9780674749672}{book})


\item Peter Krause, \emph{Presupposition and abduction in type theory}, In Working Notes of. CLNLP-95: Computational Logic and Natural Language Processing.



\end{itemize}
[[!redirects abduction]] [[!redirects abductive reasoning]] [[!redirects abductive inference]] [[!redirects abductive logic]]



\end{document}