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\begin{document}

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\section*{logic}

\begin{quote}%
`Contrariwise,' continued Tweedledee, `if it was so, it might be; and if it were so, it would be; but as it isn't, it ain't. That's logic.'

(Lewis Carroll, \emph{\href{http://sabian.org/looking_glass4.php}{Through the Looking Glass}})


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

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

[[!include foundations - contents]]

\hypertarget{category_theory}{}\paragraph*{{$(0,1)$-Category theory}}\label{category_theory}

[[!include (0,1)-category theory - contents]]

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

[[!include type theory - contents]]

\hypertarget{logic}{}\section*{{Logic}}\label{logic}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{MathematicalLogic}{Mathematical logic}\dotfill \pageref*{MathematicalLogic} \linebreak
\noindent\hyperlink{classical_subfields}{Classical subfields}\dotfill \pageref*{classical_subfields} \linebreak
\noindent\hyperlink{CategoricalLogic}{Categorical logic}\dotfill \pageref*{CategoricalLogic} \linebreak
\noindent\hyperlink{entries_on_logic}{Entries on logic}\dotfill \pageref*{entries_on_logic} \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{ReferencesCategoricalLogic}{On categorical logic}\dotfill \pageref*{ReferencesCategoricalLogic} \linebreak
\noindent\hyperlink{logic_in_natural_languages}{Logic in natural languages}\dotfill \pageref*{logic_in_natural_languages} \linebreak


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

Traditionally, as a discipline, \textbf{logic} is the study of correct methods of reasoning. Logicians have principally studied [[deduction]], the process of passing from premises to conclusion in such a way that the [[truth]] of the former necessitates the truth of the latter. In other words, deductive logic studies what it is for an argument to be \emph{valid}. A second branch of logic studies [[inductive reasoning|induction]], reasoning about how to assess the plausibility of general propositions from observations of their instances. This has often been done in terms of [[probability theory]], particularly [[Bayesian reasoning|Bayesian]].

Some [[philosophy|philosophers]], notably [[Charles Peirce]], considered there to be third variety of reasoning for logic to study, namely, [[abduction]]. This 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 \emph{inference to the best explanation}. At least some aspects of this can also be studied using Bayesian probability.

Deductive logic is the best developed of the branches. For centuries, treatments of the [[syllogism]] were at the forefront of the discipline. In the nineteenth century, however, spurred largely by the needs of [[mathematics]], in particular the need to handle [[relations]] and [[quantifiers]], a new logic emerged, known today as [[predicate logic]].

As we said above, logic is traditionally concerned with \emph{correct} methods of reasoning, and philosophers (and others) have had much to say \emph{prescriptively} about logic. However, one can also study logic \emph{descriptively}, taking it to be the study of methods of reasoning, without attempting to determine whether these methods are correct. One may study [[constructive logic]], or a [[substructural logic]], without saying that it should be adopted. Also psychologists study how people actually reason rapidly in situations without full information, such as by the \href{http://fastandfrugal.com/}{fast and frugal} approach.

A \textbf{logic} is a specific method of reasoning. There are several ways to \emph{formalise} a logic as a [[mathematics|mathematical]] object; see at \emph{\hyperlink{MathematicalLogic}{Mathematical Logic}} below.

\hypertarget{MathematicalLogic}{}\subsection*{{Mathematical logic}}\label{MathematicalLogic}

\emph{Mathematical logic} or \emph{symbolic logic} is the study of logic and [[foundations]] of mathematics as, or via, formal systems -- \emph{[[theories]]} -- such as [[first-order logic]] or [[type theory]].

\hypertarget{classical_subfields}{}\subsubsection*{{Classical subfields}}\label{classical_subfields}

The classical subfields of mathematical logic are

\begin{itemize}%
\item [[set theory]]


\item [[model theory]],


\item [[recursion theory]]


\item [[proof theory]]



\end{itemize}
\hypertarget{CategoricalLogic}{}\subsubsection*{{Categorical logic}}\label{CategoricalLogic}

By a convergence and unification of concepts that has been named \emph{[[computational trinitarianism]]}, mathematical logic is equivalently incarnated in

\begin{enumerate}%
\item [[type theory]]


\item [[category theory]]


\item [[programming theory]]



\end{enumerate}
The logical theory that is specified by and specifies a given [[category]] $\mathcal{C}$ -- called its \emph{[[internal logic]]}, see there for more details and also see [[internal language]], [[syntactic category]]. -- is the one

\begin{itemize}%
\item whose [[types]] are the [[objects]] $A$ of $\mathcal{C}$;


\item whose [[contexts]] are the [[slice categories]] $\mathcal{C}_{/A}$;


\item whose [[propositions]] in context are the [[(-1)-truncated objects]] $\phi$ of $\mathcal{C}_{/A}$;


\item whose [[proofs]] $A \vdash PhiIsTrue : \phi$ are the [[generalized elements]] of $\phi$.



\end{itemize}
Hence pure mathematical logic in the sense of the study of [[propositions]] is identified with [[(0,1)-category theory]]: where one concentrates only on [[(-1)-truncated objects]]. Genuine [[category theory]], which is about [[0-truncated objects]], is the home for logic and [[set theory]], or rather [[type theory]], the 0-truncated objects being the [[sets]]/[[types]]/[[hSet|h-sets]].

For instance,

\begin{itemize}%
\item [[limits]] and [[colimits]], [[exponentials]], and [[object classifiers]] belong to the [[type theory]];


\item while their (-1)-truncation, in this order: [[intersections]]/([[and]]), [[unions]]([[or]]), [[implications]], and [[subobject classifiers]], belong to the logic.



\end{itemize}
Generally, [[(∞,1)-category theory]], which is about untruncated objects, is the home for logic and types with a [[constructive mathematics|constructive]] notion of [[equality]], the [[identity types]] in [[homotopy type theory]].

See also at \emph{\href{model%20theory#CategoricalModelTheory}{categorical model theory}}.

\hypertarget{entries_on_logic}{}\subsection*{{Entries on logic}}\label{entries_on_logic}

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


\item [[Boolean algebra]]


\item [[algebraic models for modal logics|Boolean algebra with operators]]


\item [[boolean domain]]


\item [[boolean function]]


\item [[boolean-valued function]]


\item [[classical logic]]


\item [[constructive mathematics]]


\item [[context]]


\item [[equality]]


\item [[principle of equivalence]]


\item [[excluded middle]]


\item [[geometric logic]]


\item [[Heyting algebra]]


\item [[higher-order logic]]


\item [[internal logic]]


\item [[intuitionistic logic]]


\item [[linear logic]]


\item [[logicality and invariance]]


\item [[minimal logic]]


\item [[modal logic]]


\item [[adjoint logic]]

\begin{itemize}%
\item [[algebraic models for modal logics]]\begin{itemize}%
\item [[closure algebra]]
\item [[monadic algebra]]
\item [[temporal algebra]]

\end{itemize}

\item [[arrow logic]]
\item [[epistemic logic]]\begin{itemize}%
\item [[the logic K(m)]]
\item [[the logic T(m)]]
\item [[the logic S4(m)]]
\item [[the logic S5(m)]]

\end{itemize}

\item [[frame (modal logic)]]
\item [[geometric models for modal logics]]
\item [[modal type theory]]
\item [[temporal logic]]

\end{itemize}

\item [[negation]]


\item [[Peirce's law]]


\item [[predicate]]


\item [[predicate logic]]


\item [[predicative mathematics]]


\item [[propositional logic]]


\item [[relation]]


\item [[truth value]]


\item [[type]]


\item [[type theory]]



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

\begin{itemize}%
\item [[mathematical logic]]


\item [[logic symbols]]


\item [[type theory]], \textbf{logic}


\item [[2-type theory]], [[2-logic]]


\item [[(∞,1)-type theory]], [[(∞,1)-logic]]


\item [[objective and subjective logic]]



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

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

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

For centuries, logic was [[Aristotle's logic]] of [[deduction]] by [[syllogism]]. In the 19th century the idea of [[objective logic]] as [[metaphysics]] was influential

\begin{itemize}%
\item [[Hegel]], \emph{Wissenschaft der Logik} ( \emph{[[Science of Logic]]} )

\end{itemize}
This ``old logic'' was famously criticized

\begin{itemize}%
\item [[Bertrand Russell]], \emph{[[Logic as the Essence of Philosophy]]}, 1914

\end{itemize}
as opposed to the ``new logic'' of [[Peano]] and [[Frege]], contemporary [[predicate logic]].

The first textbook on mathematical logic is

\begin{itemize}%
\item [[David Hilbert]], [[Wilhelm Ackermann]], \emph{Grundz\"u{}ge der Theoretischen Logik} , 4th ed. Springer Heidelberg 1959 1928

\end{itemize}
Modern textbooks on mathematical logic include

\begin{itemize}%
\item [[Open Logic Project]], \emph{The open logic text} (\href{http://people.ucalgary.ca/~rzach/static/open-logic/open-logic-complete.pdf}{pdf})

\end{itemize}
\hypertarget{ReferencesCategoricalLogic}{}\subsubsection*{{On categorical logic}}\label{ReferencesCategoricalLogic}

\begin{itemize}%
\item [[William Lawvere]], \emph{[[Adjointness in Foundations]]}, Dialectica 23 (1969), 281-296


\item [[William Lawvere]] \emph{Equality in hyperdoctrines and comprehension schema as an adjoint functor}. In A. Heller, ed., \emph{Proc. New York Symp. on Applications of Categorical Algebra}, pp. 1--14. AMS, 1970. ([[LawvereComprehension.pdf:file]])


\item [[Pierre Cartier]], \emph{Logique, cat\'e{}gories et faisceaux}, S\'e{}minaire Bourbaki, 20 (1977-1978), Exp. No. 513, 24 p. (\href{http://www.numdam.org/item?id=SB_1977-1978__20__123_0}{numdam})


\item [[Lambek]], J.; [[Philip Scott|Scott]], P.J. (1986), \emph{Introduction to Higher Order Categorical Logic}, Cambridge University Press.


\item [[Jim Lambek]], [[Phil Scott]], \emph{Reflections on the categorical foundations of mathematics} (\href{https://www.site.uottawa.ca/~phil/papers/LS11.final.pdf}{pdf})


\item [[Saunders MacLane]], [[Ieke Moerdijk]], \emph{[[Sheaves in Geometry and Logic]]}


\item [[Bart Jacobs]], \emph{Categorical Logic and Type Theory}, (1999) Elsevier


\item [[John Bell]], \emph{The development of categorical logic} (\href{http://publish.uwo.ca/~jbell/catlogprime.pdf}{pdf})


\item [[Jean-Yves Girard]], \emph{[[Lectures on Logic]]}, European Mathematical Society 2011


\item [[Jean-Pierre Marquis]], [[Gonzalo Reyes]], (2009) \emph{The History of Categorical Logic 1963-1977 (\href{https://www.webdepot.umontreal.ca/Usagers/marquisj/MonDepotPublic/HistofCatLog.pdf}{pdf})}



\end{itemize}
[[!include REF MakkaiReyes77]]

\hypertarget{logic_in_natural_languages}{}\subsubsection*{{Logic in natural languages}}\label{logic_in_natural_languages}

\begin{itemize}%
\item Pieter A. M. Seuren, \emph{The logic of language}, vol. II of Language from within; (vol. I: \emph{Language in cognition}) Oxford University Press 2010

\end{itemize}
[[!redirects logic]] [[!redirects logics]]

[[!redirects categorical logic]] [[!redirects categorial logic]] [[!redirects category-theoretic logic]]

[[!redirects symbolic logic]]



\end{document}