\documentclass{article}
\usepackage{flannery}
\title{The Concept of Truth in Formalized Languages\\by Alfred Tarski}
\author{}
\date{}
\psetupNotes
\newtheorem{definition}{Definition}
\begin{document}
\maketitle
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{The Concept of True Sentence in Everyday or Colloquial Language}
In this section, Tarski demonstrates some of the difficulties in trying to
define a semantic notion of truth in natural languages. Ideally, one would
like a definition expressing the following:
\begin{quote}
a true sentence is one which says that the state of affairs is so and so,
and that the state of affairs indeed is so and so (\cite{tarski:truth35}
155)
\end{quote}
In attempting to construct such a definition, however, it is difficult to
avoid inconsistency. The liar's paradox seems to be an unavoidable problem, as
shown through many examples (pages 157-162), leading Tarski believes it is
impossible to obtain a consistent definition of semantic truth in natural
languages.
\begin{quote}
the attempt to construct a correct semantical definition of the expression
`true sentence' meets with very real difficulties (\cite{tarski:truth35}
162).
\end{quote}
What then about a {\em structural} definition of truth? i.e.
\begin{quote}
a true sentence is a sentence which possesses such and such structural
properties or which can be obtained from such and such structurally
described expressions by means of such and such structural transformations
(\cite{tarski:truth35} 163).
\end{quote}
Tarski provides the following examples of such a structural definition.
\begin{enumerate}
\item every expression consisting of four parts of which the first is the
word `if', the third is the word `then', and the second and fourth
are the same sentence, is a true sentence.
\item if a true sentence consists of four parts, of which the first is the
word `if', the second a true sentence, the third the word `then', then
the fourth part is a true sentence.
\end{enumerate}
Such an approach to defining truth, however, would require many (if not an
infinite) number of rules. And as Tarski notes, natural languages are not
``finished'', but rather constantly changing (with the addition of new words
and expressions, the removal of existing ones, and chaning notions for
existing ones). Also, the very structure of a natural language can change,
implying that any structural definition would also have to be dynamic. Thus,
\begin{quote}
The attempt to set up a structural definition of the term `true sentence' --
applicable to colloquial language is confronted with insurperable
difficulties (\cite{tarski:truth35} 164).
\end{quote}
One of the most important features of natural languages is that they are
{\em universal} in the sense that anything that can be spoken of can be spoken
of in a natural language (in contrast with formal languages which are quite
limited in what they can speak of). To maintain this universality when
developing a semantic notion of truth for a given natural language, one must
allow any expression in that language mentioning `true sentence', `name', or
`denote'. But as Tarski notes, this seems to {\em force} any such language to
be inconsistent.
Tarski concludes that a semantic or even structural notion of truth is
completely impossible in natural languages and abandons the attempt
all-together, concerning himself only with formalized languages for the
remainder of the paper.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Formalized Languages, especially the Language of the Calculus of Classes}
Properties of all formal languages, as described in \cite{tarski:truth35}
page 166\ldots
\begin{enumerate}
\item A list or description is given in structural terms for all of the
symbols with thich the expressions of the languages are formed.
\item Among all possible expression which can be formed with these signs,
those called {\em sentences} are distinguished by means of purely
structural properties.
\item A list, or structural description, is given of the sentences called
{\em axioms} or {\em primitive statements}.
\item In special rules called {\em rules of inference}, certain operations
of a structural kind are embodied which permit the transformation of
sentences into other sentences, called {\em consequences}. In
particular, the consequences of the axioms are called {\em provable}
or {\em asserted sentences}.
\end{enumerate}
Unlike natural languages, formal languages are not universal, specifcially in
that they do no posses (directly) terms denoting symbols or expressions of the
same (or other) language. Thus, we must carefully distinguish between the
languages used\ldots
\begin{itemize}
\item when speaking {\em in} a given language (called {\em the language} or
{\em the object language}), and
\item when speaking {\em about} a given language (called {\em the
metalanguage}).
\end{itemize}
Similarly, we must carefully distinguish between the theory that is the
{\em object} of investigation (called {\em the theory}) and the theory in
which the investigation is being {\em carried out} (called {\em the
metatheory}).
{\cRed
It is possible to give a {\em method} by which a correct definition of truth
can be constructed for {\em any formal language} (\cite{tarski:truth35}
168).\footnote{Is this really what he means? If so, has anyone done this?}
}
Describing such a method is difficult, and Tarski chooses instead to construct
a definition of truth for a specifc formal language, namely, the language of
the calculus of classes.
Tarski then begins a meticulous construction of both the language and the
metalanguage. Oddly (and at least I think so), Tarski chooses to use
{\em letters} for the operations in the object language (in Polish notation no
less) and {\em symbols} for the metalanguage (in normal infix notation).
\begin{center}\begin{tabular}{|c|c|}
\hline
Symbol & Meaning \\\hline\hline
N & negation \\\hline
A & disjunction \\\hline
\gPi & universal quantification \\\hline
I & inclusion \\\hline
\end{tabular}\end{center}
The real focus of this section is the development of the {\em metalanguage}
for the calculus of classes and the {\em metatheory}, called the
{\em metacalculus of classes}. The tedious construction is not noted here,
but one key aspect of the metalanguage is the following.
\begin{quote}
The fact that the metalanguage contains both an individual name and a
translation of every expression (and in particular of every sentence) of the
language studied will play a decisive part in the construction of the
definition of truth\ldots(\cite{tarski:truth35} 172).
\end{quote}
\begin{flushleft}{\bf Some Key Definitions\ldots}\end{flushleft}
Let $S$ be the class of all well-formed sentences in the language, and $Cn(X)$
be the class of all {\em consequences} of the class of sentences $X$.
\setcounter{definition}{16}
\begin{definition}[Provable Sentence / Theorem]
$x$ is a {\em provable (accepted) sentence} or a {\em theorem} -- in symbols
$x\in Pr$ -- if and only if $x$ is a consequence of the set of all axioms.
\end{definition}
\begin{definition}[Deductive System]
$X$ is a {\em deductive system} if and only if
\begin{equation*} Cn(X) \subseteq X \subseteq S \end{equation*}
\end{definition}
\begin{definition}[Consistency]
{\cRed
$X$ is a {\em consistent} class of sentences if and only if $X\subseteq S$
and if, for every sentence $x$, either $x\not\in Cn(X)$ or
$\lnot x\not\in Cn(X)$.}\footnote{I've never seen consistency phrased this
way\ldots just curious.}
\end{definition}
\begin{definition}[Completeness]
$X$ is a {\em complete} class of sentences if and only if $X\subseteq S$
and if, for every sentence $x$, either $x\in Cn(X)$ or $\lnot x\in Cn(X)$.
\end{definition}
\begin{definition}[Equivalence]
The sentences $x$ and $y$ are {\em equivalent} with respect to the class $X$
of sentences if and only if $x\in X$, $y\in S$, $X\subseteq S$ and both
$\lnot x \lor y \in Cn(X)$ and $\lnot y \lor x \in Cn(X)$ (which is the same
as saying $x\lthen y\in Cn(X)$ and $y\lthen x\in Cn(X)$).
\end{definition}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{The Concept of True Sentence in the Language of the Calculus of Classes}
In this section, Tarski constructs a semantic definition of `true sentence' for
the calculus of classes.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{The Concept of True Sentence in the Languages of Finite Order}
still working.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{The Concept of True Sentence in the Languages of Infinite Order}
still working.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Summary}
still working.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Postscript}
still working.
\bibliographystyle{amsalpha}
\bibliography{biblio}\addcontentsline{toc}{chapter}{References}
\end{document}