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\section*{excluded middle}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{foundations}{}\paragraph*{{Foundations}}\label{foundations}

[[!include foundations - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{RelationToTheAxiomOfChoice}{Relation to the axiom of choice}\dotfill \pageref*{RelationToTheAxiomOfChoice} \linebreak
\noindent\hyperlink{DoubleNegatedPEM}{Double-negated PEM}\dotfill \pageref*{DoubleNegatedPEM} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak
\noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak
\noindent\hyperlink{relation_to_the_axiom_of_choice_2}{Relation to the axiom of choice}\dotfill \pageref*{relation_to_the_axiom_of_choice_2} \linebreak
\noindent\hyperlink{ReferencesInHomotopyTypeTheory}{In homotopy type theory}\dotfill \pageref*{ReferencesInHomotopyTypeTheory} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

In [[logic]], the principle of \textbf{excluded middle} states that every [[truth value]] is either [[true]] or [[false]] (\hyperlink{AristotleMetaphysics}{Aristotle, MP1011b24}). (This is sometimes called the `[[axiom]]' or `law' of excluded middle, either to emphasise that it is or is not optional; `principle' is a relatively neutral term.) One of the many meanings of [[classical logic]] is to emphasise that this principle holds in the logic; in contrast, it fails in [[intuitionistic logic]].

The principle of excluded middle (hereafter, PEM), as a statement about truth values themselves, is accepted by nearly all mathematicians ([[classical mathematics]]); those who doubt or deny it are a distinct minority, the [[constructive mathematics|constructivists]]. However, when one [[internalization|internalises]] mathematics in [[categories]] other than [[Set|the category of sets]], there is no doubt that excluded middle often fails [[internal logic|internally]]. See the examples listed at [[internal logic]]. (Those categories in which excluded middle holds are called [[Boolean category|Boolean]]; in general, the adjective `Boolean' is often used to indicate the applicability of PEM.)

Although the term `excluded middle' (sometimes even \emph{excluded third}) suggests that the principle does not apply in many-valued logics, that is not the point; many-valued logics are many-valued \emph{externally} but may still be two-valued \emph{internally}. In the language of [[categorial logic]], whether a category has exactly two [[subterminal objects]] is in general independent of whether it is Boolean; instead, the category is Boolean iff the statement that it has exactly two subterminal objects holds in its [[internal logic]] (which is in general independent of whether that statement is true).

In fact, intuitionistic logic proves that there is no truth value that is neither true nor false; in this sense, the possibility of a `middle' or `third' truth value is still `excluded'. But since the relevant [[de Morgan law]] fails in intuitionistic logic, we may not conclude that every truth value is either true or false, which is the actual PEM.

\hypertarget{RelationToTheAxiomOfChoice}{}\subsection*{{Relation to the axiom of choice}}\label{RelationToTheAxiomOfChoice}

Excluded middle can be seen as a very weak form of the [[axiom of choice]] (a slightly more controversial principle, doubted or denied by a slightly larger minority, and true internally in even fewer categories). In fact, the following are equivalent. (this is the [[Diaconescu-Goodman-Myhill theorem]] due to \hyperlink{Diaconescu75}{Diaconescu 75}, see also \hyperlink{McLarty96}{McLarty 96, theorem 19.7}, )

\begin{enumerate}%
\item The principle of excluded middle.
\item [[finite set|Finitely indexed]] sets are [[projective object|projective]] (in fact, it suffices 2-indexed sets to be projective).
\item [[finite set|Finite sets]] are [[choice object|choice]] (in fact, it suffices for 2 to be choice).

\end{enumerate}
(Here, a set $A$ is \textbf{finite} or \textbf{finitely-indexed} (respectively) if, for some natural number $n$, there is a bijection or surjection (respectively) $\{0, \ldots, n - 1\} \to A$.)

The proof is as follows. If $p$ is a truth value, then divide $\{0,1\}$ by the equivalence relation where $0 \equiv 1$ iff $p$ holds. Then we have a surjection $2\to A$, whose domain is $2$ (and in particular, finite), and whose codomain $A$ is finitely-indexed. But this surjection splits iff $p$ is true or false, so if either $2$ is choice or $2$-indexed sets are projective, then PEM holds.

On the other hand, if PEM holds, then we can show by induction that if $A$ and $B$ are choice, so is $A\sqcup B$ (add details). Thus, all finite sets are choice. Now if $n\to A$ is a surjection, exhibiting $A$ as finitely indexed, it has a section $A\to n$. Since a finite set is always projective, and any retract of a projective object is projective, this shows that $A$ is projective.

In particular, the axiom of choice implies PEM. This argument, due originally to Diaconescu, can be internalized in any [[topos]]. However, other weak versions of choice such as [[countable choice]] (any surjection to a countable set (which for this purpose is any set isomorphic to the set of natural numbers) has a section), [[dependent choice]], or even [[COSHEP]] do not imply PEM. In fact, it is often claimed that axiom of choice is \emph{true} in constructive mathematics (by the BHK or [[Brouwer-Heyting-Kolmogorov interpretation]] of predicate logic), leading to much argument about exactly what that means.

\hypertarget{DoubleNegatedPEM}{}\subsection*{{Double-negated PEM}}\label{DoubleNegatedPEM}

While PEM is not valid in constructive mathematics, its double negation

\begin{displaymath}
\neg\neg(A\vee \neg A)
\end{displaymath}
is valid. One way to see this is to note that $\neg (A\vee B) = \neg A \wedge \neg B$ is one of [[de Morgan's laws]] that is constructively valid, and $\neg (\neg A \wedge \neg\neg A)$ is easy to prove (it is an instance of the [[law of non-contradiction]]).

However, a more direct argument makes the structure of the proof more clear. When [[beta-reduced]], the [[proof term]] is $\lambda x. x(inr(\lambda a. x(inl(a))))$. This means that we first assume $\neg (A\vee \neg A)$ for a contradiction, for which it suffices (by assumption) to prove $A\vee \neg A$. We prove that by proving $\neg A$, which we prove by assuming $A$ for a contradiction. But now we can reach a contradiction by invoking (again) our assumption of $\neg (A\vee \neg A)$ and proving $A\vee \neg A$ this time using our new assumption of $A$. In other words, we start out claiming that $\neg A$, but whenever that ``bluff gets called'' by someone supplying an $A$ and asking us to yield a contradiction, we retroactively change our minds and claim that $A$ instead, using the $A$ that we were just given as evidence.

In particular, this shows how the [[double negation]] [[modality]] can be regarded computationally as a sort of [[continuation-passing]] transform.

However, there is another meaning of ``double negated PEM'' that is not valid. The above argument shows that

\begin{displaymath}
\forall A, \neg\neg(A\vee \neg A).
\end{displaymath}
But a stronger statement is

\begin{displaymath}
\neg\neg \forall A, (A \vee \neg A).
\end{displaymath}
This is related to the above valid statement by a [[double-negation shift]]; and in fact, the truth of $\neg\neg \forall A, (A \vee \neg A)$ is equivalent to the principle of double-negation shift. In particular, it is \emph{not} constructively provable.

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[axiom of choice]]


\item [[law of double negation]]


\item [[proof by contradiction]]


\item [[De Morgan law]]


\item [[Cantor-Schroeder-Bernstein theorem]]



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

\hypertarget{general}{}\subsubsection*{{General}}\label{general}

\begin{itemize}%
\item [[Colin McLarty]], \emph{Elementary Categories, Elementary Toposes}, Oxford University Press, 1996

\end{itemize}
\hypertarget{relation_to_the_axiom_of_choice_2}{}\subsubsection*{{Relation to the axiom of choice}}\label{relation_to_the_axiom_of_choice_2}

Relation to the [[axiom of choice]] ([[Diaconescu-Goodman-Myhill theorem]]):

\begin{itemize}%
\item [[Radu Diaconescu]], \emph{Axiom of choice and complementation}, Proceedings of the American Mathematical Society 51:176-178 (1975) (\href{https://doi.org/10.1090/S0002-9939-1975-0373893-X}{doi:10.1090/S0002-9939-1975-0373893-X})


\item N. D. Goodman J. Myhill, \emph{Choice Implies Excluded Middle}, Zeitschrift fuer Mathematische Logik und Grundlagen der Mathematik 24:461 (1978)



\end{itemize}
Discussion in [[toposes]] is for instance in

\hypertarget{ReferencesInHomotopyTypeTheory}{}\subsubsection*{{In homotopy type theory}}\label{ReferencesInHomotopyTypeTheory}

\begin{itemize}%
\item [[Aristotle]], \emph{[[Metaphysics (Aristotle)|Metaphysics]]}, 1011b24: ``Of any one subject, one thing must be either asserted or denied''

\end{itemize}
In [[homotopy type theory]]:

\begin{itemize}%
\item [[Univalent Foundations Project]], section 3.5 \emph{[[Homotopy Type Theory -- Univalent Foundations of Mathematics]]}

\end{itemize}
category: foundational axiom

[[!redirects Excluded Middle]] [[!redirects excluded middle]] [[!redirects law of excluded middle]] [[!redirects the law of excluded middle]] [[!redirects principle of excluded middle]] [[!redirects the principle of excluded middle]] [[!redirects axiom of excluded middle]] [[!redirects the axiom of excluded middle]]



\end{document}