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\begin{document}

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\section*{judgment}

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

\hypertarget{type_theory}{}\paragraph*{{Type theory}}\label{type_theory}

[[!include type theory - contents]]

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

[[!include foundations - contents]]

\hypertarget{judgments}{}\section*{{Judgments}}\label{judgments}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{in_firstorder_logic}{In first-order logic}\dotfill \pageref*{in_firstorder_logic} \linebreak
\noindent\hyperlink{in_type_theory}{In type theory}\dotfill \pageref*{in_type_theory} \linebreak
\noindent\hyperlink{hypothetical_and_generic_judgments}{Hypothetical and generic judgments}\dotfill \pageref*{hypothetical_and_generic_judgments} \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}

In [[formal logic]], a \textbf{judgment}, or \textbf{judgement}, is a ``meta-[[proposition]]''; that is, a proposition belonging to the [[meta-language]] (the [[deductive system]] or [[logical framework]]) rather than to the [[object language]].

More specifically, any [[deductive system]] includes, as part of its specification, which strings of symbols are to be regarded as the \emph{judgments}. Some of these symbols may themselves express a [[proposition]] in the object language, but this is not necessarily the case.

The interest in judgements is typically in how they may arise as \emph{theorems}, or as \emph{consequences} of other judgements, by way of the [[deduction]] rules in a deductive system. One writes

\begin{displaymath}
\vdash J
\end{displaymath}
to mean that $J$ is a judgment that is derivable, i.e. a [[theorem]] of the deductive system.

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\hypertarget{in_firstorder_logic}{}\subsubsection*{{In first-order logic}}\label{in_firstorder_logic}

In [[first-order logic]], a paradigmatic example of a judgement is the judgement that a certain string of symbols is a well-formed [[proposition]]. This is often written as ``$P \;prop$'', where $P$ is a [[metavariable]] standing for a string of symbols that denotes a proposition.

Another example of a judgement is the judgement that these symbols form a proposition [[proof|proved]] to be [[true]]. This judgment is often written as ``$P\;true$''.

Neither of these judgements is the same thing as the proposition $P$ itself. In particular, the proposition is a statement \emph{in} the logic, while the judgement that the proposition is a proposition, or is provably true, is a statement \emph{about} the logic. However, often people abuse notation and conflate a proposition with the judgment that it is true, writing $P$ instead of $P\;true$.

\hypertarget{in_type_theory}{}\subsubsection*{{In type theory}}\label{in_type_theory}

The distinction between judgements and [[propositions]] is particularly important in [[intensional type theory]].

The paradigmatic example of a judgment in [[type theory]] is a \emph{typing judgment}. The assertion that a [[term]] $t$ has [[type]] $A$ (written ``$t:A$'') is not a statement \emph{in} the type theory (that is, not something which one could apply logical operators to in the type-theoretic system) but a statement \emph{about} the type theory.

Often, type theories include only a particular small set of judgments, such as:

\begin{itemize}%
\item typing judgments (written $t:A$, as above)
\item judgments of typehood (usually written $A \;type$)
\item judgments of [[equality]] between typed terms (written say $(t=t'):A$)

\end{itemize}
(In a type theory with a [[type of types]], judgments of typehood can sometimes be incorporated as a special case of typing judgments, writing $A:Type$ instead of $A\;type$.)

These limited sets of judgments are often defined [[inductive definition|inductively]] by giving [[type formation]]/[[term introduction]]/[[term elimination]]- and [[computation rules]] (see [[natural deduction]]) that specify under what hypotheses one is allowed to conclude the given judgment.

These inductive definitions can be formalized by choosing a particular [[type theory]] to be the meta-language; usually a very simple type theory suffices (such as a [[dependent type theory]] with only [[dependent product types]]). Such a meta-type-theory is often called a [[logical framework]].

\hypertarget{hypothetical_and_generic_judgments}{}\subsection*{{Hypothetical and generic judgments}}\label{hypothetical_and_generic_judgments}

It may happen that a judgment $J$ is only derivable under the assumptions of certain other judgments $J_1,\dots, J_2$. In this case one writes

\begin{displaymath}
J_1,\dots,J_n \;\vdash J.
\end{displaymath}
Often, however, it is convenient to incorporate hypotheticality into judgments themselves, so that $J_1,\dots,J_n \;\vdash J$ becomes a single \emph{[[hypothetical judgment]]}. It can then be a consquence of other judgments, or (more importantly) a hypothesis used in concluding other judgments. For instance, in order to conclude the truth of an [[implication]] $\phi\Rightarrow\psi$, we must conclude $\psi$ \emph{assuming} $\phi$; thus the [[introduction rule]] for implication is

\begin{displaymath}
\frac{\phi \;\vdash\; \psi}{\vdash\; \phi\Rightarrow\psi}
\end{displaymath}
with a hypothetical judgment as its hypothesis. See [[natural deduction]] for a more extensive discussion.

In a [[type theory]], we may also consider the case where the hypotheses $J_1$ are typing judgments of the form $x:A$, where $x$ is a [[variable]], and in which the conclusion judgment $J$ involves these variables as [[free variables]]. For instance, $J$ could be $\phi\;prop$, where $\phi$ is a valid (well-formed) proposition only when $x$ has a specific [[type]] $X$. In this case we have a \emph{[[generic judgement]]}, written

\begin{displaymath}
(x \colon X) \;\vdash\; (\phi \; prop).
\end{displaymath}
which expresses that \emph{assuming the hypothesis or [[antecedent]] judgement} that $x$ is of type $X$, as a consequence we have the [[succedent]] judgement that $\phi$ is a proposition. If on the right here we have a typing judgment

\begin{displaymath}
(x \colon X) \;\vdash\; (t \colon A)
\end{displaymath}
we have a \emph{[[term in context]]}.

For more about the precise relationship between the various meanings of $\vdash$ here, see [[natural deduction]] and [[logical framework]].

While this may seem to be a very basic form of (hypothetical/generic) judgement only, in systems such as [[dependent type theory]] or [[homotopy type theory]], all of [[logic]] and a good bit more is all based on just this.

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

\begin{itemize}%
\item [[infinite judgment]]

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

Foundational discussion of the notion of \emph{judgement} in [[formal logic]] is in

\begin{itemize}%
\item [[Per Martin-Löf]], \emph{On the meaning of logical constants and the justifications of the logical laws}, leture series in Siena (1983) (\href{http://docenti.lett.unisi.it/files/4/1/1/6/martinlof4.pdf}{web})


\item [[Per Martin-Löf]], \emph{A path from logic to metaphysics}, talk at \emph{Nuovi problemi della logica e della filosofia della scienza}, Jan 1990 (\href{https://github.com/michaelt/martin-lof/raw/master/pdfs/A-path-from-logic-to-metaphysics-1991.pdf}{pdf})



\end{itemize}
More on this is in in sections 2 and 3 of

\begin{itemize}%
\item [[Frank Pfenning]], Rowan Davies, \emph{A judgemental reconstruction of modal logic} (2000) (\href{http://www.cs.cmu.edu/~fp/papers/mscs00.pdf}{pdf})

\end{itemize}
A textbook acccount is in section I.3 of

\begin{itemize}%
\item [[Robert Harper]], \emph{[[Practical Foundations for Programming Languages]]}

\end{itemize}
Something called \emph{judgement} (Urteil) appears in

\begin{itemize}%
\item [[Georg Hegel]], second part, first section, second chapter of \emph{[[Science of Logic]]}.

\end{itemize}
[[!redirects judgment]] [[!redirects judgments]] [[!redirects judgement]] [[!redirects judgements]] [[!redirects hypothetical judgment]] [[!redirects hypothetical judgments]] [[!redirects hypothetical judgement]] [[!redirects hypothetical judgements]] [[!redirects generic judgment]] [[!redirects generic judgments]] [[!redirects generic judgement]] [[!redirects generic judgements]]



\end{document}