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just defined as a consequence of the truth
only
in terms of satisfaction of a formal
language (Tarski 1935). To set up his exposition, Tarski defines two languages, the
first being the syntactic
object language L
and the second being the meta-language
M
.The
meta-language
should be more expressive such that it can describe every
sentence in the object language, and furthermore, that it contain axioms that allow
the truth of every sentence in the object language to be defined. In his first move,
Tarski defines the formal conception of truth as 'Convention T,' namely that for a
given sentence
s
in
L
, there is a statement
p
in
M
that is a theorem defining the
truth of
s
, that is, the truth of
s
is determined via a translation of
s
into
M
(Tarski
1935). Tarski then later shows that truth can be formally defined as “
s
is true if and
only if
p
” (Tarski 1944). For example, if the object language is exemplified by a
sentence uttered by some speaker of English and the meta-language was an English
description of the real world; 'The Eiffel Tower is in Paris' is true if and only if the
Eiffel Tower is in Paris. The sentence 'The Eiffel Tower is in Paris' must be satisfied
by the Eiffel Tower
actually being
in Paris. While this would at first seem circular,
its non-circularity is better seen through when the object language is not English,
but another language such as German. In this case, “'Der Eiffelturm ist in Paris' is
true if and only if the Eiffel Tower is in Paris.” However, Tarksi was not interested
in informal languages such as English, but in determining the meaning of a new
formal language via translations to mathematical models or other formal languages
with well-known models. If one was defining a formal semantics for some fragment
of a knowledge representation language like RDF, a statement such as
http://www.
eiffeltower.example.orgex:locationex:Paris
is true if and only if
∃
ab
.
R
(
a
,
b
)
where
R
,
a
,and
b
are given in first-order predicate logic.
If one is defining a formal Tarski-style semantics for a language, what should
one do when one encounters complex statements, such as 'the Eiffel Tower is in
Paris and had as an architect Gustave Eiffel'? The answer is at the heart of Tarksi's
project, the second component of Tarski's formal semantics is to use the principle of
compositionality so that any complex sentence can have its truth conditions derived
from the truth conditions of its constituents. To do this, the meta-language has to
have finitely many axioms, and each of the truth-defining theorems produced by
the meta-language have to be generated from the axioms (Tarski 1935). So, the
aforementioned complex sentence is true if and only if
,where
Q
can be the
architect of
relationship,
c
can be Gustave Eiffel and
a
the Eiffel
Tower. Tarksi's theory as explained so far only deals with 'closed' sentences, i.e.
sentences containing no variables or quantification. The third and final component
of Tarski's formal semantics is to use the notion of satisfaction via extension to
define truth (Tarski 1935). For a sentence such as 'all monuments have a location,'
we can translate the sentence to
∃
ab
.
R
(
a
,
b
)
∧
Q
(
a
,
c
)
which is
true if and only if there is an extension
x
from the world that satisfies the logical
statements made about
a
. In particular, Tarksi has as his preferred extensions infinite
ordered pairs, where the ordered set could be anything (Tarski 1935). For formal
languages, a model-theoretic semantics with a model composed by set theory was
standard. For example, the ordered pairs in some model of
∀
a
∃
l
.
monument
(
a
)
→
hasLocation
(
a
,
l
)
(
Eiffel Tower,Paris
)
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