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ALBERT VISSER

SEMANTICS AND THE LIAR PARADOX

ON THE CHOICE OF MATERIAL

The semantical paradoxes are not a scientic subject like Inductive Denitions,

Algebraic Geometry or Plasma Physics. At least not yet. On the other hand the

paradoxes exert a strong fascination and many a philosopher or logician has spent

some thought on them, mostly in relative isolation. The literature on the paradoxes

is vast but scattered, repetitive and disconnected. This made it impossible to give

a presentation in which all ideas in the literature receive their due.

The chapter consists of two parts. In the rst I give a rather labyrinthine, ad-

mittedly far from complete, survey of problems and ideas for ‘solution’ connected

with the liar. In Part II I concentrate on ideas mainly of Kripke, Herzberger and

Gupta, to be found in Kripke [ 1975 ] , Gupta [ 1982 ] , Herzberger [ 1982a; 1982b ] .

Some reasons for this choice are: the papers mentioned above are connected to

important earlier work like Herzberger [ 1970; 1980 ] . Moreover, the treatment in

Kripke’s paper uses an important technique: inductive denitions. Constructions

similar to Kripke’s occur a.o. in the work of Aczel [ 1980 ] , Aczel and Feferman

[1980], Cantini [1980; 1981], Feferman [1969; 1975; 1974; 1977; 1979; 1984],

Fitch [1948; 1963; 1980], Gilmore [1974; 1980], Martin and Woodruff [1975],

Nepeıvoda [ 1973 ] , Scott [ 1975 ] . These papers contain work that is technically

explicit and fruitful. These papers share an interest in the meaning of self refer-

ential sentences in addition to the more mathematical interest to exploit the extra

expressive possibilities obtained by adding a truth predicate to a given model.

The Labyrinth of the Liar

1 THE SEMANTICAL PARADOXES AND THEIR INTEREST

Ramsey, following Peano (see [ Ramsey, 1925 ] ) distinguished the Semantical from

the Set Theoretical Paradoxes, or in his own terminology the Linguistical from the

Logical or Mathematical Paradoxes. The Semantical Paradoxes are those involv-

ing concepts like truth, satisfaction, denotation, denition, concept, proposition

and the Set Theoretical Paradoxes are those involving notions like set, element,

number.

Such a distinction involves two theses: (i) The paradoxes are paradoxes of the

semantical or set theoretical notions involved; (ii) The classication of notions as

given is a natural one.

Not many philosophers today would accept Ramsey’s own arguments for his

distinction, but that does not mean that the issue is settled. The distinction is partly

D. Gabbay and F. Guenthner (eds.),

Handbook of Philosophical Logic, Volume 10, 159–245.

2002, Kluwer Academic Publishers. Printed in the Netherlands.

160

ALBERT VISSER

conrmed by the development of Zermelo–Fraenkel Set Theory. Russell’s paradox

is in this context clearly displayed as a paradox involving the notions of set and

element. The paradoxical proof turns out to employ a too strong principle of set

formation. That Russell’s Paradox could be thus solved, where the Semantical

Paradoxes are left untouched, seems to corroborate the idea that set in the sense

of the Cumulative Hierarchy can be viewed as independent of notions as truth,

proposition and the like.

I will not try to settle the issue of Ramsey’s distinction here, but rather discuss

the question what interest the study of the paradoxes has for both kinds of para-

doxes together. In the rest of the chapter I concentrate however mainly on the

liar.

Let us list some interests of the study of the paradoxes.

(i) Foundational motivations . Foundational questions are of this form: to articu-

late and justify principles on which to base a given practice.

The most important examples of foundational work connected with paradox are

the development of Zermelo–Fraenkel Set Theory and the current development of

systems for the foundations of Constructivistic Mathematics. Work in this area

typically involves isolation and clarication of fundamental categories of objects

and articulation of principles valid for these categories.

It must be stressed that Russell’s paradox is just one foundational issue in math-

ematics. The Axiom of choice and Continuity Principles in constructivistic math-

ematics are examples of other foundational issues.

(i) Practical motivations . A practical motivation is to extend an already accepted

mathematical theory with e.g. a type free notion of truth or set for technical con-

venience. Typically but not necessarily such extensions will be conservative over

the original language. For a forceful statement of such a practical program see

Feferman’s appendix to Aczel and Feferman [ 1980 ] .

(iii) Linguistical, semantical, philosophical motivations . The primary aim from a

philosophical or linguistical point of view is to understand, by analysing the para-

doxes, the semantical notions involved better. Typically analyses of the paradoxes

touch both on matters of detail like the proper treatment of implication or how to

handle truth value gaps as on broad issues in the philosophy of language, like for

example why do we have a notion of truth at all. For some examples of ‘broad

views’ see e.g. Tarski [1944, p. 349] and Chihara [1979].

In the chapter I will mainly be guided by linguistical, semantical and philosoph-

ical interest. A point where this will be particularly evident is in Part II, where I

treat the idea that the problem of meaning for self referential sentences involving

truth is wider than the problem of giving a theory of correct inference and judge-

ment for these sentences.

SEMANTICS AND THE LIAR PARADOX

161

2 VERSIONS OF THE LIAR

In this section versions of and variations on the Liar are presented. A closer dis-

cussion is given in Section 3.

2.1 Sentential Versions

The simplest way to get a liar sentence is to write down:

This sentence is not true.

To arrive at Paradox, or Absurdity on no assumptions, reason as follows:

Suppose that sentence is true, i.e. it is not true. So it is true and it is

not true, a contradiction. Hence the sentence is not true. But that is

precisely what it says, so it is true. We may conclude that the sentence

is both true and not true, a contradiction on no assumptions.

Clearly this version raises problems about the interaction of demonstratives with

the truth predicate. Some of these will be treated in Section 3. For those who doubt

that the context in which the sentence is presented is sufcient to disambiguate

‘this’, there is the following variant:

This sentence is not true.

There are several ways to avoid the use of demonstratives. One is the use of an

‘empirical’ predicate.

The sentence printed on page 161, lines 19–21 of the Handbook of

Philosophical Logic, Vol 10 , edited by Gabbay and Guenthner is not

true.

Another way employs baptism. Consider:

is not true.

”. For later reference I give the paradoxical

reasoning for the case of in natural deduction style. Let stand for the (in-

ter)substitutivity rule for and “ is not true”: because is “ is not true’, we

may replace occurrences of “ ” in a formula by occurrences of ““ is not true””

and vice versa. Let stand for the rule based on Tarski’s Convention : we may

move from ““. . . ” is true” to “. . . ” and vice versa.

Let us call the above sentence “

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ALBERT VISSER

“

is not true” is true

is true

is not true

is true

“

is not true” is true

is not true

is true

is not true

“

is not true” is true

is true

is not true

Of the following ‘versions’ it is disputable whether they are ‘really’ versions of

the Liar rather than closely related paradoxes. I’m going to present them anyway.

The next ‘version’ is known as Grelling’s Paradox. The German mathemati-

cian Kurt Grelling found it in 1908. It dispenses with the use of a self referential

sentence altogether. Instead it employs self-application of words.

Let’s call a word “heterological” when it does not apply to itself, e.g.

“long” is heterological. Is “heterological” heterological? If it is, it

applies to itself, so it is not heterological. Contradiction. So “hetero-

logical” is not heterological, hence it is heterological. Contradiction

on no assumptions.

It is worthwhile to observe that Grelling’s Paradox can be converted in a ‘meta-

mathematical’ version for languages containing their own satisfaction predicate.

First note that the relevant ‘words’ for Grelling’s denition of ‘heterological’ are

predicate words. These may be replaced or mimicked by formulae having just

Then Grelling’s sentence “ “heterological” is heterological” becomes:

”

By the usual properties of Sat, this quickly leads to Paradox. Our rephrasing of

Grelling’s sentence is closely related to G odel’s construction of his celebrated

sentence: what corresponds to ‘

“

”

“

’ there, is ‘

’,

is the function computing the G odel number of the result of sub-

stituting the numeral of the number

for in the formula with G odel number .

(As is well known is primitive recursive or even polynomial time computable.)

The following sentence is Quine’s. It is very similar to Grelling’s, but for the

fact that it replaces self application by the operation of putting a string of symbols

behind its quotation. This operation is metamathematically very simple. (For an

elaboration see [ Smullyan, 1957 ] ). Quine’s sentence is:

free. Now dene a formula standing for ‘heterological ’ by:

where

SEMANTICS AND THE LIAR PARADOX

163

“Yields a falsehood when appended to its own quotation”

yields a falsehood when appended to its own quotation.

A nal example, a bit more articial maybe than the others in that it does not rest

on previously accepted principles, employs a proposed extension of our stipulatory

practices. It will be useful because it allows us to present the paradoxical reasoning

purely in a propositional logical form.

First some explanation. The usual stipulative denition of the Riemann Integral

looks somewhat like this:

–a–b–c–

where –a–b–c– is a sentence in the language of analysis not containing . Let us

ignore the occurrence of ‘ ’ on the left, which introduces extra problems.

The reasons why one does not admit on the right is presumably that otherwise

the denition would be circular. But consider:

’ is supposed to

be given. Clearly there is a way of reading the denition, familiar to those who

know the Recursion Theorem, under which it is a perfectly sensible denition of

. For the moment let us vaguely say that we stipulate the left hand side to have

the same meaning as the right hand side whatever that will turn out to be .

If we accept provisionally a liberalisation of stipulative denitions in the spirit

of above denition of , we can have a Liar as follows:

Note the difference with

of the baptism way of constructing a Liar sentence:

” is a new atomic sentence,

introduced in our language, that is stipulated to have the same meaning as “

” is a name of the sentence under consideration; “

”.

Let us call the rule that allows us to interchange an atom with its stipulated meaning

and vice versa: . We may now reason as follows.

Note that the stipulatory way involves neither self reference nor self application

nor truth nor satisfaction.

Here the domain of the denition is the natural numbers and ‘

“

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