Chris Amin PHI
Logic and Language
Tarski, it is often said, defined truth. Is there more to
our concept of truth than Tarski caters for?
Tarski's Concept of Truth
The concept of Truth in Colloquial Languages
In 'The Concept of Truth in Formalized Languages' (Tarski, 1958, p. 152) Tarski starts the
first section by attempting a definition of truth for colloquial language, and concludes that
such a definition is impossible. He draws up a scheme for determining partial definitions of
truth of the form:
"x is a true sentence if and only if p"
From this scheme, a concrete partial definition is constructed by replacing 'p' with any
sentence, and 'x' with the name of that sentence. There are two relevant kind of names -quotation-mark names and structural-descriptive names. The former kind is formed by
surrounding an expression with quotation marks, as in:
"'it is snowing' is a true sentence if and only if it is snowing"
The latter kind is formed by describing the structural composition of the sentence (or any
other expression), as in:
"an expression consisting of three words of which the first is composed of the two letters I
and Te (in that order) the second of the two letters I and Es (in that order) and the third of
the seven letters Es, En, O, Double- U, I, En, and Ge (in that order), is a true sentence if and
only if it is snowing." (Tarski, 1956, p. 157)
In order to generalize a partial definition like "'it is snowing' is a true sentence if and only if it
is snowing", thus affording a full definition of true sentence, he provides the scheme:
"'p' is a true sentence if and only if p"
An alternative definition takes a structural approach, and is intended to define the truth of
natural language sentences by their syntactical structure. A definition of this kind would be
built up by establishing truth-preserving constructs in the language. eg. "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" (Tarski, 1956, p.163) However,
he concludes that this is a hopeless exercise with regard to natural language, because of its
complex and mutable nature.
He finishes by presenting a general argument against the construction of a satisfactory truth
definition for colloquial languages: the universality of colloquial languages -- their necessary
capacity of expressing everything that can be meaningfully expressed -- entails that any
semantic definition of truth of such a language must (but cannot) deal with the logical
contradictions of paradoxical sentences of the various kinds. e.g. "this sentence is false",
"'heterological' is heterological. Thus, even if the difficulties of the complexity of natural
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Chris Amin PHI
Logic and Language
language were to be resolved, the best that we can hope for is defining 'true sentence' for
those subsets of colloquial languages which don't contain such expressions.
The Concept of Truth in Formal Languages
Given the non-universal nature of formal languages (specifically, the usual absence therein of
terms belonging to the theory of language), a distinction must be made between the object
language (the language under study) and the metalanguage. The metalanguage contains the
names of the expressions of the object language and of the relations between those
expressions, and usually the full vocabulary of the object language. e.g. the metalanguage
for studying the propositional calculus contains, amongst other things, the name 'P & Q',
referring to a sentence in the object language asserting the conjunction of the propositions 'P'
and 'Q', and the predicate '⊦', used to denote the provability of one sentence from another.
Tarski rejects the idea of simply equating 'true sentence' with 'provable sentence', and
providing a structural definition of truth for formal languages that way, for the reason that
any incomplete deductive language (i.e. one with modest set theoretic apparatus) contains
true sentences which are not-provable.
He instead sets out to formulates a semantic definition of truth that meets the requirement of
convention T, which states that any formally correct definition of truth has the following
sentences in the metalanguage as consequences:
1. all sentences obtained from the scheme 'x ∈ Tr if and only if p', where x is a
structural-descriptive name of some sentence in the object language, p is the
metalinguistic translation of that sentence, and Tr is the extension of the newly
defined predicate 'true expression of L', where L is the object language
2. the sentence Tr ⊆ S, where S is the set of all sentences in the object language
In other words, a definition of true sentence for L is acceptable if and only if for any
expression S that meets the definition, the corresponding translation of S is true in the
metalanguage, and S is a sentence in L.
The first, rejected, general definition takes its form from the concrete partial definitions
constructed from our scheme, such as "'it is snowing' if and only if it is snowing", and it is
tempting to formulate it as follows:
"for all p, 'p' is a true sentence if and only if p"
Without changing our stipulations about the nature of quotation-marks, however, this is
transparently not an acceptable definition of truth, for it talks only of the truth conditions of
the expression 'p' in the object language (that is, the expression made up of one instance of
the 16th letter of the alphabet). In order to salvage our intent, the quotation-marks must
instead represent a function of p. Given explicitly with a functor (following Soames, 1999),
to avoid confusion with our previous usage of quotation-marks:
"for all p, Quote(p) is a true sentence if and only if p"
However, this may still be unacceptable because, according to Tarski, it allows the
construction of self-denying sentences, and hence introduces contradiction into the
metalanguage. The form is as follows:
(A) for all p, if c is identical with Quote(p), then not p
(B) Let the symbol 'c' be an abbreviation of sentence A.
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Chris Amin PHI
Logic and Language
(C) the sentence 'for all p, if c is identical with Quote(p), then not p' is identical with c
(established empirically)
(D) for all p and q, if Quote(p) is identical with Quote(q), then p if and only if q
(transparent stipulation about a property of the quotation-function)
Then, given C and D, one is able to generate a contradiction. Soames (1999) rejects this by
stating that Tarski misapplies the rule of universal elimination. However, even if Tarski
makes an error of this kind he is forced to do so by the shady nature of the whole context of
the quotation-function -- it is difficult to define exactly how we ought to interpret its use.
Take the following attempted general definition of truth:
for all p, Quote(p) is a true sentence of the object language if and only if Quote(p) is a
sentence of the object language and p
To start with, it is helpful to examine the nature of p. Since it is assertoic, it must be a
nullary predicate; but the usual interpretation of the above sentence of the metalanguage as a
quantification over predicates is inconsistent with the assertion of p. If we accept this new
behaviour, Quote(x) can be defined as "the sentence denoting the predicate x". i.e. if F is a
nullary predicate such that F if and only if (∀x)(x=x), then Quote(F) is a term referring to the
expression '(∀x)(x=x)'. It seems like a lot of work is required to rehabilitate the concept of
the quotation-function in our definition of truth. In any case, Tarski constructed a
thoroughly semantic definition of truth, without these ambiguities.
For a semantic definition of truth for languages with a finite number of sentences it suffices
to produce a list of every sentence obtained from applying the above scheme to the sentences
of L. e.g. a definition of truth in a visual language used by the guardsmen on the Great Wall
of China might be provided by the following list: "'<burning torch>' if and only if the Huns
are invading", "'<fireworks>' if and only if the Huns have gone home again". However,
when a language contains the means to produce infinitely many sentences, as is the case with
any interesting language, such a construction would be infinitely long and thus impossible to
formulate.
The solution to this problem is to capitalize on the very property of interesting formal
languages which prevents us from producing an exhaustive list of partial truth definitions:
that they have a finite number of constructions which can be employed to produce infinitely
many sentences. For the purposes of a semantic definition of truth by recursive means, it is
fortuitous but predictable that the syntax and semantics of formal languages are intimately
linked; this is how "logic chases truth up the tree of grammar". (Quine, 1970, p. 35)
There are two ways of constructing a recursive definition in this way. For relatively simple
languages with a finite number of simple sentences (e.g. the propositional calculus), it
suffices to construct a recursive definition of a sentence, followed by a criterion of truth for
any sentence.
e.g. "'P' is a sentence", "' ¬' followed by a sentence is a sentence", "A sentence x is true if and
only if either (1) there exists a v such that v is a propositional variable and the metalinguistic
translation of v is true, or (2) there exists a sentence y such that x= ¬y, and the negation of y
is true, or ...", etc.
For more complicated languages, including any that contain existential quantification and
conjunction, a recursive definition of sentences is not possible because there exist infinitely
many simple sentences. e.g. "(∃x)(Fx)", "(∃x)(Fx & Fx) etc. It is necessary to introduce a
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Chris Amin PHI
Logic and Language
notion the extension of which is a subset of the expressions and a superset of the sentences -that of sentential function. A sentential function is an expression with one or more free
variables that would be a sentence if every of its variables were replaced by a name, or if a
corresponding quantifier were to be prepended to the expression for each of its variables.
Tarski's definition of truth proceeds along the lines of a definition of a recursive nature of the
notion of "satisfaction of a given sentential function by given objects". (Tarski, 1956, p. 189)
A sentential function x with exactly one free variable v is satisfied by a given object o if and
only if the metalinguistic translation of the sentence formed by replacing every instance of 'v'
with 'o' in x is true. Sentential functions with more than one free variable are satisfied by
finite sequences of various sizes (pairs, triples, etc.), and in each case a definition of
satisfaction can be constructed in like manner to that above. However, Tarski goes on, "for
the sake of a uniform mode of expression" (ibid., p. 191), to replace objects and finite
sequences with infinite sequences. The general schematic definition can be formulated as
follows:
The sequence f satisfies the sentential function x of the object language if and only if f is an
infinite sequence of objects, and p
(Where p is the sentence formed by replacing every k-th variable in x with the k-th object in
f)1
A recursive definition of satisfaction for a predicate calculus with negation: '¬', conjunction:
'&' and universal quantification: '∀', the unary predicate 'F' (meaning 'walks') and the
variables x', x'', x''', etc. may be formulated as per the above schema as follows:
The sequence f satisfies the sentential function x of the object language if and only if f is an
infinite sequence of objects, and one of the following conditions are met: (a) there exists a
symbol v such that v is a lower-case x followed by k strokes and x = Fv and fk walks; (b)
there exists a sentential function y such that x=¬y and f does not satisfy y; (c) there exist
sentential functions y and z such that x=y&z and f satisfies both y and z; (d) there exists a
symbol v such that v is a lower-case x followed by k strokes, and there exists a sentential
function y such that x=(∀v)(y), and every infinite sequence of objects which differs from f in
at most the k-th place satisfies y.
Because a sentence is a sentential function with no free variables, it is either satisfied by all or
no infinite sequences. The definition of the notion of true sentence thus swiftly follows:
An expression x is a true sentence of language L if and only if x is a sentence of L and every
infinite sequence of objects satisfies x.
The proper way of settling the ordinal relation between variables in x and objects in f is a
matter of convention. Quine (1970) demonstrates the advantage of matching up variables to
objects based on alphabetical order by showing the counter-intuitive effects of ordering by
introduction the variables in a complex sentential function composed of conjunctions of
simpler sentential functions. For example, using a mapping determined by order of
introduction, <Caesar,Gaul,Brutus> satisfies "x conquered y & z killed x" (Caesar conquered
Gaul; Brutus killed Caesar) but not "z killed x" (Caesar did not kill Gaul). However, this is
only a matter of convenience and elegance, and as Tarski states it is always possible to
establish a consistent order of variables for a given sentential function based on their order of
introduction in a sentence (though not a constituent sentential function).
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Chris Amin PHI
Logic and Language
Conclusion
Given that Tarski rejects the possibility of defining truth for natural languages, this definition
of truth does not suffice to explain our concept of truth, and nor was it intended to. Rather,
it explained wonderfully what it is for a sentence to be true of a given formal object language.
What it does not offer is an account of how we can know simple sentences to be true, for it is
logical and not epistemological in nature -- but a theory of truth requires no such provision.
The more general question is whether a definition for natural languages along the lines of one
provided by Tarski for formal languages would .
Davidson (2001) identifies three properties of an acceptable theory of truth, all of which are
possesed by Tarski's definition: (1) it should account for the meaning of sentences in the
semantic way of Tarski, (2) it should provide a method for deciding what its meaning is, and
(3) the statements of the truth-conditions for sentences entailed by the theory should "draw
upon the same concepts as the sentences whose truth conditions they state". (Davidson, 2001,
p. 56)
References
Tarski, A. (1956). Logic, Semantics, Metamathematics: Papers from 1923 to 1938. Oxford: Clarendon Press
Quine, W. V. (1970). Philosophy of Logic. Englewood Cliffs, N.J.: Prentice-Hall.
Soames, S. (1999). Understanding Truth. New York: Oxford University Press
Davidson, D. (2001). Inquiries into Truth and Interpretation. Oxford: Clarendon Press
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