University of St Andrews

Department of Philosophy

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ID NUMBER: 18001369

MODULE Philosophy of Logic

NAME:

MODULE PY4634

CODE:

TUTOR’S Aaron Cotnoir

NAME:

ESSAY TITLE: What is the nature of truth?

WORD COUNT: 3470

I hereby declare that the attached piece of written work is my own work and that I

have not reproduced, without acknowledgement, the work of another.

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What is the nature of truth?

The material adequacy condition for truth given by the ‘T-schema’ as proposed by

Tarski is as follows: 1

“(T): x is true iif p” for some sentence ‘x’ with some referent p.

The nature of truth follows this condition, it eludes precise human definition but any

formal language definition we give must satisfy it. Therefore, while the T-schema

does not define truth, I do assert that it accurately describes its nature.

In this essay, I will first discuss ambiguity in the meaning of the question and

decide that we ought to discuss the meaning of truth as a formal concept, and not as

it operates natural language. I will then introduce Tarski’s semantic theory of truth,

explaining how the T-schema provides an adequate condition by which to define

truth. I will then discuss the opposition that Tarski says nothing about truth ‘across’

language, considering the work of Davidson as an example. I will discuss that this

misunderstands the precise difference between natural and formal language. From

this, I will accept the T-schema for truth in formal language.

Ambiguity in the question

The question “what is the nature of truth?” unfortunately lacks a clear singular

interpretation, that is: it is unclear exactly in what sense “the nature of truth” is

1
Tarski, A. (1944), “The Semantic Conception of Truth: and The Foundations of Semantics”

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meant. I put forward that there are 3 philosophically relevant interpretations of this

question:

(1). The question is asking how the word “truth” is used by people in colloquial language; we

could refer to this as the descriptivist approach, the aim in answering this is to describe

the behaviour of the word itself in its daily usage.

(2). The question is asking how the word “truth” ought to be defined in a more formal setting,

but still applied to natural language. This could be called the prescriptivist approach. The

aim in answering this is to provide an account for how ‘truth’ may be defined both in

terms of useful meaning and in terms of operation over natural language.

(3). The question is asking not about the word “truth” itself but the concept it describes. This

could be called the conceptual approach. The aim in answering this question is to

provide an account for what the philosophical notion of truth is, rather than accounts of

how it behaves or should behave within human language.

Approach (1) is very common. When discussing truth, in natural language,

this is the correct path to take. What concerns me about this approach is that it

seems to provide an account for the use of the word ‘truth’ in each language, and not

of what we as philosophers would like as truth. While I think it is a perfectly valid

approach, its conclusions are not as wide reaching or formally useful as we may

desire. Nonetheless, it is a coherent system of definition, and it could be argued that

we cannot say anything about truth beyond its usage in natural language. We either

need to accept that truth cannot be defined rigorously, due to the limitations of

human speech, or accept that truth seems to have a rigorous notion that can be

expressed, or at least approximated, well. The first stance seems devastating to the

field of logic, and so I will take the second stance as pure assumption, and therefore

assert that there must be an alternate approach possible.

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 Approach (2) is untenable; prescribing a definition of truth on natural language

is both: linguistically impossible (meaning in natural language is determined by use,

not by ought); and logically incoherent (truth ought not to be defined simply to avoid

the roughness of natural language). (1) and (2) is not an explicitly given distinction by

philosophers in logic. This is the source of what I consider the weakness of the

prescriptive approach: philosophers are using a domain in which descriptive analysis

is the only available method but applying prescriptive notions to it. Clarifying this

notion will be the focus of the 4th section of this essay.

The third approach is the one I believe to be the most attractive. Consider

Dummet’s analogy of winning and losing a game 2. Dummet says that it is not

sufficient for the philosopher to define the cases in which something is true or not

true, they also need to explain what the aim is; likening this behaviour to chess:

creating a list of wins and lose cases is not sufficient to describe how one wins

chess. For Dummet, this led him to discuss truth through the lens of testimony, in

which people generally intend to be truthful. However, I think his notion that a theory

of truth needs to account for ‘why’ is useful in this separate context. Logicians, and in

fact many fields of study, are not concerned with the human ability to define truth in

words, but with its behaviour, I will explain why this must be true given that we do not

fall to the descriptivist approach.

From the T-schema to truth

For a notion of truth to be adequate it must account for a statement’s relationship to

reality. Truth is in this sense very primitive: a statement is true if it matches reality.

2
Dummet, M. (2013), “Truth”

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Tarski does not defend this idea in detail. Instead, he takes is as a given and uses it

to construct the T-schema as a less ambiguous phrasing of this idea. It is not trivial

to argue for this assumption. It should be made clear that when we refer to “needing”

a definition of truth to adhere to certain principle, that we are not bending truth’s

behaviour to suit us. These ‘needs’ should be considered as follows: they must be

satisfied, or we need to accept the descriptivist approach.

1. A definition of truth must not rely on nonaxiomatic assumption, it is at least a part the bedrock

of the system of thought that surrounds it. If this were not true, it would be impossible to

formulate anything, as there is no possible guidance for determining the validity of

statements; the best we could do is a circular definition of truth that has to support its own

premises- obviously, there cannot be true premises without truth. Therefore, a satisfactory

definition of truth must be derivable only from axiomatic statements about it.

2. A definition of truth must preserve equivalence. “X is identical to X” must be true for all objects

X. Without this, it would be impossible to describe reality, as we could not confirm the state of

being of any object: it may differ from itself. If truth is describable, it is clearly bound to this

rule. The truth of equivalence is therefore taken as axiomatic.

3. A definition of truth must be calculable without some form of mapping table. That is, any rule

we give for a truth function must operate on a sentence in the object-language in a way that is

calculable (given enough data), it should not require us to consult some arbitrary truth table

for each case. If this were not true, truth could be defined in any [mathematically] well defined

way, and we would have no more clarity on its operation than we gain with the descriptivist

approach.

We then take that objects in a language have referents, whether conceptual or

being objects in ‘reality’. While it may be differed upon, in natural language, whether

an object could be taken to be truly equivalent to its referent, formal language is

prescriptive. As such, its definitions can be taken as fixed and well defined and

therefore map object to their referents perfectly. This is where the T-schema arises,

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the equivalence [condition (2)] of a sentence X and its referent p. The T-schema

does not rely on any nonaxiomatic premises [condition (1)]. The T-schema also

provides us with a better understanding of how truth must operate on a language

[condition (3)]. No matter how it is defined, truth maps objects to their referents and

determines if the referent is the case.

Tarski, from here, defines a formal language and a system for defining truth

based on semantic principles and gives this as his semantic conception of truth. This

is omitted here for the following reason: Tarski’s definition of truth is not intended to

be universal. If the T-schema is followed, then to Tarski you have created an

adequate definition of truth. Therefore, I say that it is Tarski’s material adequacy

condition that describes the nature of truth, and not his semantic conception.

Definitions built around this criterion merely model this behaviour, for a humanly

understandable language.

It is impossible to define truth in accordance with the T-schema in natural

language because natural language is semantically closed. That is, it can be fully

described using only terms within the language. Tarski showed that the liar paradox

can be expressed in any semantically closed language- if we use classical logic. To

avoid rejecting classical logic Tarski makes a leap: language must be semantically

open for classical logic to be usable. What is relevant is the resultant claim: to form a

definition of truth that adheres to the T-schema, you must use a semantically open

language. The contrapositive of this argument holds and is useful to state: If a

definition of truth adheres fully to the T-schema, then it must be defined over a

formally defined language.

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Reassessing the prescriptive approach

Rejections of this notion ought, therefore, to come from one of 3 directions: the T-

schema is not a satisfactory condition for material adequacy, natural language is

sufficient for a definition of truth, or by asserting that truth cannot be defined

rigorously due to the limitations of human capability (what we now call ‘falling to

descriptivism’). I think the strongest and most damaging responses fall into the

second category.

For example, Davidson3 discusses a system where truth is extended by taking

the definition of truth in each formal language that is a subset of the natural target

language and extending by approximation. Specifically, he claims a formal language

that “has been explained in English and contains much English” must be considered

“a part of English for those who understand it”. He suggests, therefore, that where

sentences have the same truth value, between the 2 languages, we may extend

from one to the other.

This is ultimately the character of the strongest objections to accepting a T-

schema view of truth’s nature. The claim could generally be considered that by

accepting some level of approximation, truth can be extended into natural

languages. Since T-schema truth requires formally defined language, if it is possible

to define truth on natural language then it there must be some other behaviour

inherent to truth, it cannot be said to naturally adhere to the T-schema.

At this point, I believe that there are 3 possible reasonable outcomes to these

claims: the T-schema demonstrates the nature of truth; truth is definable across

3
Davidson, D. (1967), “Truth and Meaning”

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natural language; or, we need to fall to descriptivism We must now reject the second

outcome.

The incongruence of natural and formal language

To counter this, we need to exemplify what was discussed earlier, that prescribing

truth into natural language is not coherent. Firstly, natural language is by nature fluid,

it is not designed for rigorous definition. Meaning within natural language is given by

common use. Truth must be axiomatically derived; its meaning cannot be

determined by use alone- if we are to provide a satisfactory account for it. This is

what is meant by the claim that a prescriptive notion of truth is not coherent, we

cannot force the word “true” into a new meaning in natural language, merely

describe it.

The first concern to this claim is that this may be all we need, after all we

never excluded the possibility of the descriptivist approach. I think this is not

satisfactory for either side, doubtless most people attempting to define truth in

natural language are not attempting to merely describe its use, and defenders of a

formal definition of truth will not be satisfied with a definition based on colloquial use.

There is also the claim we could prescribe a formal definition of truth, a new word,

into natural language. Perhaps call this word ‘True2’, this is not initially rejectable

until we have a better notion of how natural language behaves. We must discuss the

difference between natural and formal language first.

There are many ways to think about meaning in natural language. It is by no

means a simple task. In whatever way it is that we believe a word operates: by

baptising a word to refer to an object, by some primitive link between object words

and referents, or perhaps by approximation of referent classes- what is important is

simply define a word to give it new meaning within our own language, I cannot

decide that ‘dogs’ refers to cats in English now and actually affect naturally spoken

English. Of course, if many of us repeatedly insisted on this change, eventually for

some subset of the population it would hold. However, we did not shift the meaning

of a word by defining it a new way, we shifted a words definition by using it in its new

meaning. Use precedes definition. It does not matter how it is that natural language

operates, it does not shift under mere definition; in this sense, natural language is

defined ‘descriptively’.

One popular method for giving sense to the notion of true2 is to extend from

formal language into natural language, or to explain how natural language and

formal language are congruent in such a way that truth can be applied to either. The

commonality to every claim is a failure to ascertain the exact incongruence between

natural and formal language. It is commonly assumed that the behaviour can be

matched or extended from one to the other. Davidson, as mentioned earlier,

provides an illuminating example. I do not claim that this is the only rejection, but I

hope that I can demonstrate how any claim of this type can be rejected in the

following way.

Natural language is not formally defined. Its meaning is based in use. Formal

language has definitions that are directly prescribed to operate logically. As such

formal language is coherent, it has no paradoxes or fuzzy edges, it operates in

accordance with whatever logical system it is based upon. Natural language is in this

way not congruent to formal language, even if formal language is based on the

natural language, there is no way to map formal language sentence to natural

language sentences that perfectly retains meaning. More importantly, there is no

that natural language under truth is not isomorphic to formal language under truth.

Rejecting notions of the type Davidson gives comes clearly from here: you

cannot extend formal language to natural language because there is no subset of

natural language that is isomorphic, under truth, to formal language, there are no

cases where a sentence has the same truth value between languages. Since these

cases do not exist this method fails to provide an account for formalising natural

language, and therefore truth over it.

This could be itself countered with a claim that there could still be a formal

language sufficiently close to a natural language for this leap to be made. Though it

may not be the case, if some isomorphism is found between a formal language and

a natural language, this extension would be possible. This fails when considering

how these languages are constructed. In natural language, words are not placed into

the language, people’s use of the words gives them meaning. Definitions given in

natural language are therefore “descriptive”, they can only seek to describe the

behaviour of words. In formal language, words are defined precisely, the definition

given for them is their meaning. Definitions given in formal language are therefore

“prescriptive”4, they can provide a direct account for how a word ought to be used.

This is where the problem for Davidson lies. It is not possible to see natural language

as an extension of formal language because truth cannot be given a prescribed

definition in natural language.

4
This relates to but is not identical to the notion of semantic closure. It is neither necessary nor
sufficient for a language to be semantically open for it to be prescriptive. Though all formally defined
semantically open languages are by nature prescriptive since they are formally defined.

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 Still, introducing a new word and continuously insisting its definition seems to

allow us to ‘prescribe’ its meaning. This is how truth2 would operate. Philosophers

would introduce this word and continuously use it with its prescribed meaning. We

wish to show that a logician attempting to define truth on natural language will either

fall to descriptivism or run into the incongruence of natural and formal language. This

is where the remaining counterargument comes in: a definition of truth is either

descriptive, natural language prescriptive, or follows the T-schema, and if it is natural

language prescriptive it is incoherent. That is, we must show that all prescriptive

accounts of truth are either in accordance with the T-schema (and therefore defined

in formal language) or incoherent.

Firstly, we will not take the requirements on truth I gave earlier as a given; we

will assume that the philosopher would take issue with some part of the main body of

section 2 and show how that falls to absurdity. For example, by rejecting calculability

the prescriptivist is defining truth in a way that is both unusable by humans and

completely descriptive. Without the ‘how’ of truth a definition cannot be considered

more than a description of its operation in natural language. Therefore, rejecting

calculability necessarily falls to descriptivism.

Partially rejecting equivalence is a common angle for objections, in the sense

that they reject that a language object and its referent are logically equivalent. If this

view of meaning in natural language is accepted then it seems obvious to me that

formal language should be used in its place, or, if the prescriptivist in question does

not think that formal language is suitable either, then this falls to descriptivism. If we

do not accept a logical connection between words and their well-defined referents

(as in formal language) then it is not possible to accurately provide a definition for

truth for formal usage.

think it is often overlooked in study of truth in natural language. Regularly

philosophers will study the behaviour of the truth predicate in natural language and

define it according to its use, attempting to form a function that is operable and

coherent but curves to fit fuzziness of natural language. I do not think this is tenable.

To define truth in accordance with natural language’s behaviour prescriptively is not

the define truth well, it has placed the reality of natural language as a premise to the

argument for truth. It is not possible to prescribe a language’s truth behaviour from

within the language, as Tarski said, for truth to be coherent it must be defined in a

semantically open language.

At this point I think it is possible to claim that we have rejected the second

way. This leaves us 2 possible outcomes: truth cannot be formally defined well, and

must be defined in purely descriptive terms, or truth’s nature is I accordance with the

T-schema given by Tarski. I think if care is taken by a philosopher in determining the

scope of their claims, then both approaches have real merit to field of logic. I said the

most attractive angle would be to attempt to discuss the nature of truth, as a

conceptual object, and, considering I do believe that we can accept that truth must

be axiomatically defined and that equivalence relations must be given as true, I am

inclined to claim the second possibility: The nature of truth is given by Tarski’s T-

schema criterion for adequacy.

Conclusion

In this essay, I have examined the meaning of the question, and exposed and

explained its ambiguity. I have then discussed Tarski’s T-schema for truth definitions,

explaining its relevance to the aims of the essay. I then briefly introduced Davidson

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as an example for a class of objections I consider important to rebuke, those that

intend to show truth has a nature within natural language. I then explained how

objections of this class are not tenable as formal language and natural language are

not congruent in this way. This led me to accept the T-schema as giving what I would

call “the nature of truth”.

WORD COUNT: 3470

Bibliography:

Dummet, M. (2013), “Truth” in The Virtual Issue No.1: Truth, The Aristotelian

Society, pp.102-120.

Tarski, A. (1944), “The Semantic Conception of Truth: and The Foundations of

Semantics” in Philosophy and Phenomenological Research, International

Phenomenological Society, Vol. 4, No.3, pp.341-376

Davidson, D. (1967), “Truth and Meaning” in Synthese, Springer, Vol. 17, No.3,

“Language in Use Including Wittgenstein's Comments on Frazer and a Symposium

on Mood and Language-Games”, pp.304-323

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