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\begin{document}

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\section*{equality}

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

\hypertarget{equality_and_equivalence}{}\paragraph*{{Equality and Equivalence}}\label{equality_and_equivalence}

[[!include equality and equivalence - contents]]

\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{DifferentKinds}{Different kinds of equality}\dotfill \pageref*{DifferentKinds} \linebreak
\noindent\hyperlink{in_type_theory}{In type theory}\dotfill \pageref*{in_type_theory} \linebreak
\noindent\hyperlink{definitional_equality}{Definitional equality}\dotfill \pageref*{definitional_equality} \linebreak
\noindent\hyperlink{computational_equality}{Computational equality}\dotfill \pageref*{computational_equality} \linebreak
\noindent\hyperlink{propositional_equality}{Propositional equality}\dotfill \pageref*{propositional_equality} \linebreak
\noindent\hyperlink{in_set_theory}{In set theory}\dotfill \pageref*{in_set_theory} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

There is a lot of interesting stuff to be said about \emph{equality} in [[logic]], [[higher category theory]], and the [[foundations]] of mathematics, but it hasn't all been said here yet.

\hypertarget{DifferentKinds}{}\subsection*{{Different kinds of equality}}\label{DifferentKinds}

Here is a list of distinctions between different notions of \emph{equality}, in different contexts, where possibly all the following pairs of notions are, in turn, ``the same'', just expressed in terms of different terminologies:

\begin{itemize}%
\item the difference between [[axiom|axiomatic]] and [[definition|defined]] equality;
\item the difference between identity and equality,
\item the difference between intensional and extensional equality,
\item the difference between equality [[judgements]] and equality [[propositions]],
\item the difference between equality and [[isomorphism]],
\item the difference between equality and [[equivalence]],
\item the possibility of operations that might not preserve equality.

\end{itemize}
\hypertarget{in_type_theory}{}\subsection*{{In type theory}}\label{in_type_theory}

In a formal language such as [[type theory]], one distinguishes various different notions of \emph{equality} or \emph{equivalence} of the terms of the language.

Note that both [[intensional type theory|intensional]] and [[extensional type theory|extensional]] type theories may distinguish between intensional and extensional equality (and whether the type theory satisfies [[function extensionality]] is yet a separate issue).

\hypertarget{definitional_equality}{}\subsubsection*{{Definitional equality}}\label{definitional_equality}

The most basic one is \emph{definitional equality} or \emph{intensional equality}.

According to \hyperlink{PML}{PML (1980), p. 31}:

\begin{quote}%
Definitional equality is intensional equality, or equality of meaning (synonymy). \ldots{} It is a relation between linguistic expressions \ldots{} Definitional equality is the equivalence relation generated by abbreviatory definitions, changes of [[bound variables]] and the [[principle of substituting equals for equals]]. \ldots{} Definitional equality can be used to rewrite expressions \ldots{}.


\end{quote}
on p. 60:

\begin{quote}%
\ldots{} intensional (sameness of meaning) \ldots{}


\end{quote}
For instance the symbols ``$2$'' and ``$s(s(0))$'' (meaning the successor of the successor of $0$) are definitionally/intensionally equal terms (of type the [[natural numbers]]): the first is merely an abbreviation for the second.

\hypertarget{computational_equality}{}\subsubsection*{{Computational equality}}\label{computational_equality}

Another sort of equality which is very important in type theory is \emph{computational}, \emph{conversional}, or \emph{reductional} equality. This is the congruence on terms generated by various conversion or reduction rules, such as [[∞-reduction]] or [[∞-conversion]] (the choice of rules depends on the type theory).

The paradigmatic example of computational equality is a pair of terms like ``$(\lambda x. x+x)2$'' and ``$2+2$'', where the second is obtained by $\beta$-reduction from the first. In a type theory that includes definitions by [[recursion]], expansion of a recursor is also computational equality. For instance, if addition is defined by recursion, then ``$2+2$'' (that is, $s(s(0))+s(s(0))$) reduces via this rule to ``$4$'' (that is, $s(s(s(s(0))))$).

For better or for worse, however, it is common in type theory to use the phrase ``definitional equality'' to \emph{include} computational equality, even though it involves more than mere expansion of definitional abbreviations. There is even a phrase ``$\delta$-reduction'' for the substitution of definitions regarded as a conversion rule analogous to $\beta$-reduction and $\eta$-expansion. Thus, even though ``$2+2$'' and ``$4$'' are not literally equal \emph{by definition} but only by performing some computation (even if the computation is fairly trivial in this case), the type of equality they do enjoy is generally called \emph{definitional}.

Perhaps this is because in type theory, computational equality plays the same role as true definitional equality. In particular, typing judgments respect it: if $\Gamma \vdash (t:A)$, while $t$ and $A$ are computationally equal to $t'$ and $A'$, then also $\Gamma \vdash (t':A')$. For $t$ and $t'$, this fact is sometimes called [[type preservation]] and is an important provable aspect of a type theory. For $A$ and $A'$, the statement is generally only contentful in a [[dependent type theory]], in which case it is often included explicitly as one of the typing rules.

Another apt word for computational equality is \emph{judgmental equality}, since it is often expressed in type theory by a separate [[judgment]]: in addition to typing judgments $\Gamma \vdash (t:A)$, we have equality judgments $\Gamma \vdash (t = t'): A$. The rules for manipulating such judgments then include reflexivity, symmetry, transitivity, and the generating computations such as $\beta$-reduction.

\hypertarget{propositional_equality}{}\subsubsection*{{Propositional equality}}\label{propositional_equality}

By contrast with ``$2+2$'' and ``$4$'', it is pretty much impossible to write down rules of computational equality such that the symbols ``$a + b$'' and ``$b + a$'' (in a [[context]] with free variables $a$ and $b$ for, say, [[natural numbers]]) become equal. They are (only) \emph{extensionally} equal, in the sense that they become computationally equal when any particular natural numbers (such as $s(s(0))$ and $s(s(s(0)))$, say) are substituted for $a$ and $b$. An actual proof is necessary to demonstrate this equality in general: it is not a definition but a theorem that $a + b = b + a$ for all natural numbers $a, b$. With addition defined recursively on (say) its first argument, this theorem must be proven by induction on $a$ and is surprisingly nontrivial to formalize.

In type theory, this extensional equality is a [[judgment|judgement]], not itself yet a [[proposition]] (an item in the formal system itself). It can be internalized, however, into a notion of equality called \emph{propositional equality}, by introducing [[identity types]] (or identity [[propositions]]). Thus, $Id_{\mathbb{N}}(2+2,4)$ is the [[propositions as types|type/proposition]] that $2+2$ is propositionally equal to $4$, while $\prod_{a,b:\mathbb{N}} Id_{\mathbb{N}}(a+b,b+a)$ is the proposition that addition is commutative. Exhibiting a term belonging to such a type is exhibiting (i.e. proving) such a propositonal equality.

The fact that typing judgments respect computational/judgmental equality means that if two terms $t$, $t'$ are computationally/judgmentally equal, then they are also propositionally equal. This is because the type $Id(t,t)$ is always inhabited by the reflexivity term $refl_t$, but then this term $refl_t$ also inhabits the type $Id(t,t')$.

The converse, however, is not true in general. It is possible to add a rule to type theory saying that ``if $Id(t,t')$ is inhabited, then $t=t'$ judgmentally''. (In this case, the adjectives ``computational'' and ``definitional'' sometimes used for [[judgmental equality]] truly lose their applicability.) This is a very strong way to make the type theory [[extensional type theory]], but is often eschewed by type theorists (even those who don't care about [[homotopy type theory]]) because it makes type checking [[decidability|undecidable]] --- because type checking has to respect [[judgmental equality]], such a rule means that it also has to respect [[propositional equality]], which in turn means that it has to be able to evaluate arbitrary functions defined inside of type theory.

\hypertarget{in_set_theory}{}\subsection*{{In set theory}}\label{in_set_theory}

Every [[set]] $S$ has an \textbf{equality relation}, a binary [[relation]] according to which two elements $x$ and $y$ of $S$ are related if and only if they are equal; in this case we write $x = y$. This is the smallest [[reflexive relation]] on $S$, and it is in fact an [[equivalence relation]]; it is the only equivalence relation on $S$ that is also a [[partial order]] (although that fact is somewhat circular). This relation is often called the \textbf{identity relation} on $S$, either because `identity' can mean `equality' or because it is the [[identity]] for [[composition]] of relations.

As a [[subset]] of $S \times S$, the equality relation is often called the \textbf{[[diagonal]]} and written $\Delta_S$ or similarly. As an abstract set, this subset is [[isomorphism|isomorphic]] to $S$ itself, so one also sees the diagonal as a map, the [[diagonal function]] $S \overset{\Delta_S}\to S \times S$, which maps $x$ to $(x,x)$; note that $x = x$. This is in [[Set]]; analogous [[diagonal morphisms]] exist in any [[cartesian monoidal category]].

In an [[axiomatic set theory]] such as [[ZFC]] or [[ETCS]], the presence of the equality relation is part of the axiomatization, although this is usually swept under the rug by including an equality relation by default in all first-order theories. (It has to be made more explicit in a structural set theory such as [[SEAR]], where there is no equality relation on sets themselves, only on elements of a particular set.)

By contrast, in a [[definitional set theory]], an equality relation is \emph{structure} which has to be put onto a [[type]] or [[preset]] in order to obtain a \emph{set}. See for example [[setoid]].

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

\begin{itemize}%
\item \textbf{equality}, [[equation]]


\item [[identity type]],


\item [[identity of indiscernibles]]


\item [[inequality]]

\begin{itemize}%
\item [[denial inequality]]


\item [[apartness relation]]



\end{itemize}

\item [[isomorphism]]


\item [[equivalence]]


\item [[weak equivalence]]


\item [[homotopy equivalence]], [[weak homotopy equivalence]]


\item [[equivalence in an (∞,1)-category]]


\item [[equivalence of (∞,1)-categories]]



\end{itemize}
[[!include logic symbols -- table]]

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

In

\begin{itemize}%
\item [[Georg Hegel]], \emph{[[Science of Logic]]}, 1812

\end{itemize}
equality is the subject of volume 1, book 2 ``Die Lehre vom Wesen'' (The doctrine of essence). As discussed at \emph{[[Science of Logic]]}, one can roughly identify in Hegel's text there the notion of intensional identity and of the reflector term in [[identity types]].

Texts on [[type theory]] typically deal with the subtleties of the notion of \emph{equality}. For instance

\begin{itemize}%
\item [[Per Martin-Löf]], \emph{Intuitionistic type theory}, Lecture notes (1980)

\end{itemize}
Besides

\begin{itemize}%
\item [[Kurt Gödel]], \emph{Über eine bisher noch nicht benützte Erweiterung des finiten Standpunktes}. Dialectica (1958), pp. 280–287,

\end{itemize}
which might be the first paper to mention intensional equality (and the fact that it should be decidable), there is

\begin{itemize}%
\item [[Nicolaas de Bruijn]], \emph{[[Automath]]},

\end{itemize}
where de Bruijn makes a distinction between definitional equality and ``book'' equality.

[[!redirects equal]] [[!redirects equals]] [[!redirects equality]] [[!redirects equalities]] [[!redirects equality predicate]] [[!redirects equality predicates]]

[[!redirects equality relation]] [[!redirects equality relations]] [[!redirects identity relation]] [[!redirects identity relations]] [[!redirects diagonal relation]] [[!redirects diagonal relations]]

[[!redirects definitional identity]] [[!redirects definitional identities]] [[!redirects definitional equality]] [[!redirects definitional equalities]]

[[!redirects propositional equality]] [[!redirects propositional equalities]]

[[!redirects computational identity]] [[!redirects computational identities]] [[!redirects computational equality]] [[!redirects computational equalities]]

[[!redirects intensional identity]] [[!redirects intensional identities]] [[!redirects intensional equality]] [[!redirects intensional equalities]] [[!redirects extensional equality]] [[!redirects extensional equalities]]



\end{document}