Meaning, Presuppositions, Truth-relevance, Gödel's Theorem and the Liar Paradox

8/9/2019 Meaning, Presuppositions, Truth-relevance, Gdel's Theorem and the Liar Paradox

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2014-04-14

Meaning, Presuppositions, Truth-relevance, Gdel'sSentence and the Liar Paradox

by X.Y. Newberry

Section 1 reviews Strawsons logic of presuppositions. Strawsons justification is

critiqued and a new justification proposed. Section 2 extends the logic of

presuppositions to cases when the subject class is necessarily empty, such as x!

Px" #Px! $ Qx! . %he strong similarity of the resulting logic with &ichard

'ia(s truth)relevant logic is pointed out. Section * further extends the logic of

presuppositions to sentences with many variables, and a certain valuation is

proposed. +t is noted that, given this valuation, -dels sentence becomes neither

true nor false. %he similarity of this outcome with oldstein and aifmans

solution of the iar paradox, which is discussed in section /, is emphasi(ed.

Section 0 returns to the definition of meaningfulness the meaninglessness of

certain sentences with empty subjects and of the iar sentence is discussed. %he

objective of this paper is to show how all the above)mentioned concepts areinterrelated.

1. Justification of Strawsons Theory of Presuppositions

P.F. Strawson is known for introducing the logic of presuppositions. According to this

theory, the sentence

!he present king of France is wise." #1.1$

is neither true nor false if there is no king of France. %ntuiti&ely #1.1$ is 'eaningful.

Strawson considered the (uestion of how a 'eaningful sentence can )e neither true nor

false as the 'ain pro)le' #Strawson, 1*+0, pp. 21-24 1*+2, pp. 14-1+$, and he

proposed a solution. /e 'ade a distinction )etween a sentence and the useof a

sentence. For ea'ple #1.1$ is a sentence, )ut it can )e used differently on different

occasions. %f so'eone uttered #1.1$ in the era of uis %3, he would )e 'aking a true

assertion if so'eone uttered it in the era of uis 3 he would )e 'aking a false

assertion and if so'e)ody uttered it today it would )e neither true nor false. Strawson

defined 'eaning as follows to gi&e the 'eaning of a sentence is to gi&egeneral

1


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directionsfor its use in 'aking true or false assertions." #Strawson, 1*+0, p. 15+$. /e

co'pared it to gi&ing a 'eaning to %" or this." %n short, Strawson e'phasi6ed that

#1.1$ was indeical.

!his was too 'uch for 7ertrand 8ussell, who wrote

As regards the present 9ing of France", he fastens upon the egocentricword present" and does not see' a)le to grasp that, if for the word

present" % had su)stituted the words in 1*0+", the whole of his

argu'ent would ha&e collapsed." #8ussell, 1*+, p. 5+$

% will suggest a new :ustification of the logic of presuppositions using the following

three definitions.

'efinition 1A sentence is meaningful iff it or its internal negation

epresses a possi)le state of affairs.

'efinition 2 A sentence is trueiff the possi)le state of affairs it epresses

corresponds to an actual state of affairs.

'efinition * A sentence isfalseiff the possi)le state of affairs epressed

)y its internal negation corresponds to an actual state of affairs.

;efinition 1 is Ayer


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the denial of the state of affairs. A negation in this sense asserts that the king of France

does not posses the property of )eing wise. %nstead of saying the king of France is not

wise" we could say the king of France is unwise" or perhaps e&en the king of France is

foolish," that is, his decisions are not well thought out and his acts often ha&e unintended

or detri'ental conse(uences. !he denial of the state of affairs is a wider concept and

includes the possi)ility that there is no king of France at all.

>learly

!he 9ing of France in 1*0+ was wise." #1.2$

epresses a possi)le state of affairs. =e can picture to oursel&es what the sentence

states. A no&el could ha&e )een written in the era of uis %3 a)out the French

'onarchy in 1*0+. /owe&er, if we enu'erate all the things that are wise and all the

things that are not wise, the 9ing of France in 1*0+ will not appear on either list.

!herefore the sentence is neither true nor false.

ater Strawson 'odified his stance and offered this definition %t is enough that it

should )e possi)le to descri)e or imaginee'phasis addedB circu'stances in which its

use would result in a true or false state'ent." #Strawson, 1*+2, p. 15+.$ =hen translated

into our parlance, this )eco'es it should )e possi)le to picture to oursel&es

circu'stances in which its use would result in a true or false state'ent." !his is &ery

si'ilar to our theory.

!here are two interesting o)ser&ations a)out the logic of presuppositions #P$. Firstly,

it is co'pati)le with the traditional Aristotelian syllogis' #Strawson, 1*+2, pp. 1-

1*.$ Secondly, P is an alternati&e to the !heory of ;efinite ;escriptions #!;;$. %t is

illu'inating to contrast the two.


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7oth P and !;; hold that #1.1$ can )e true only if there is a king of France. P also

holds that #1.1$ can )e false only if there is a king of France. #!he su):ect class 'ust )e

none'pty for the sentence to ha&e a truth &alue.$ !he purpose of !;; is to elucidate

the sentences where the gra''atical su):ect is in the singular with the definite article.

#%t is not clear why we need to analy6e the definite article considering that 'ost

languages including atin do not ha&e it.$ P treats such sentences and uni&ersally

(uantified sentences unifor'ly while !;; does not. According to P )oth

All the kings of Swit6erland ha&e )een wise." #1.$

and

!he present king of Swit6erland is wise." #1.4$

are neither true nor false. %n contrast !;; proposes that #1.4$ is false, although #1.$ is

usually considered &acuouslyB true. % do not find this plausi)le. Perhaps Aristotle and

Strawson were right while 8ussell was wrong.

2. Presuppositions & Truth Relevance

2.1. Presuppositions

%n Strawson


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D#Ex$#fx Dgx$ #Ex$#fx$ #2.1.$

or as

D#Ex$#fx Dgx$ #Ex$#fx$ #Ex$#Dgx$, #2.1.4$

where

#Ex$#fx$ #2.1.+$

stands for Cohn has children," and

#Ex$#gx$ #2.1.@$

stands for So'ething is asleep." %f Cohn does not ha&e any children then, according to

classical logic, #2.1.2$ is true, #2.1.$ and #2.1.4$ are false.

Accepting #2.1.$ or #2.1.4$ would open the (uestion of what the translation into English

of D#Ex$#fx Dgx$" is. All Cohn


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#2.1.1$ and its negation as a con:unction of so'e wff with #Ex$fx. !herefore, we can let

D#Ex$#fx Dgx$" stand as the for'ali6ation of All Cohn


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for' D#Ex$#fx Dgx$" to )e either true or false, )oth #Ex$#fx$" and #Ex$#Dgx$" 'ust

hold.

2.2 Truth Relevance

%n 1*51 8ichard ;ia6 pu)lished a 'onograph in which he presented truth"

rele#ant logic#;ia6, 1*51, pp. @+-2.$ %t has striking si'ilarities with the logic of

presuppositions.

=hen e&aluating for'ulae of classical propositional calculus, we often find oursel&es

using shortcuts such as

1. ifpis false, thenpH $is true, regardless of the &alue of $

2. if $is true, thenpH $is true, regardless of the &alue ofp

!here are e&en so'e for'ulae whose truth &alue 'ay )e deter'ined in e&ery

&aluation, e&en if we do not know the truth &alue assigned to one of its &aria)les in

any #aluation. A case in point ispH #$Hp$. Supposepis true. !hen $Hpis true

)y shortcut 2, and hencepH #$Hp$ is true, again )y 2. %f p is false, then )y 1,pH

#$Hp$ is true" #;ia6, 1*51 p. @+$. !he ter' $is not rele&ant to the deter'ination of

pH #$Hp$. =e will call this kind of rele&ance truth"rele#ance. A for'ula is truth

rele&ant if all the &aria)les occurring in it are truth rele&ant.

!he shortcut ta)les for dis:unction and con:unction are )elow. " stands for

uknown."


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!a)le 1.1

!a)le 1.2

And here is an ea'ple of how to use these ta)les to e&aluate D## D$ D!$.

!a)le 1.

=e do not need to use the truth &alue of !to deter'ine that the for'ula is a tautology.

Mf special interest is the set of tautologies that are also t-rele&ant." #;ia6, 1*51, p. @.$

!he tautologies that are nott rele&ant are the propositional counterparts of the

&acuously true" sentences of classical logic

# D$ H !%

D## D$ D!$,

#D &$ & !.

An ea'ple of a t-rele&ant tautology is 'odus tollens

5

v | F x T

-----------

F | F x T

x | x x T

T | T T T

P | Q | ~((P & ~P) & ~Q)

------------------------

T | x | T F F F x

F | x | T F T F x

& | F x T

----------

F | F F F

x | F x x

T | F x T


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# & ~!$ H #! & ~$.

For 'ore detail please seeGew)erry #200*$.

2.3 Conclusion

=e note the iso'orphis' )etween the tautologies that are nott-rele&antand the

for'ulas that are neither true nor false )ecause theirsub'ect class is empty or their

predicate class is uni#ersal.

3on t)relevant tautology 4ormula with an empty subject class

# D$ H ! #x$##x Dx$ H !x$

D## D$ D!$ D#Ex$##x Dx$ D!x$

3on t)relevant tautology 4ormula with universal predicate class

H D# D$ #x$#xH D#x Dx$$

D# # D$$ D#Ex$#x #x Dx$$

!he for'ulae on the right are )ut generali6ations of the for'ulas on the left.

%n all likelihoodtruth-rele&ant logic can )e used as the )asis for for'ali6ing the logic

of presuppositions.

*
http://xnewberry.tripod.com/System_TR.pdfhttp://xnewberry.tripod.com/System_TR.pdfhttp://xnewberry.tripod.com/System_TR.pdf


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3. When re Relations !either True !or "alse#

=hen are relations neither true nor falseN For ea'ple when is

#x$#y$D#Fxy Gxy$ #.1.1$

neither true nor falseN %n classical logic,

#x$#y$*xy #.1.2$

is interpreted as follows.

For all aiin the range ofx%it is the case that

#y$*aiy #.1.$

and, for all biin the range ofy%it is the case that

#x$*xbi. #.1.4$

=hen is #.1.2$ true in the logic of presuppositionsN =hat if #.1.$ is G for so'e aior if

#.1.4$ is G for so'e biN #G" stands for D#! & F$".$ =e will si'ply count" only those

#y$*aiyand #x$*xbithat are either true or false. !he 'eaning of #.1.2$ is then defined )y

the following procedure. >ollect all the #y$*aiyand #x$*xbithat are either true or false. %f

all of the' are true then #.1.2$ is true, if one is false then #.1.2$ is false. %f there is no

such #y$*aiyor #x$*xbithen #.1.2$ is G. For for'al se'antics, please seeGew)erry

#201$.

%n the ea'ple in Figure 1, there are so'e

#y$#Faiy Gaiy$ #.1.+$

that are ! & F. =hen a O t, then #Ey$Fayand #Ey$Gay. 7ut there is no biin the range of

yfor which

#x$#Fxbi Gxbi$ #.1.@$

is ! & F.

%n su''ary, #.1.1$ will )e ! & F iff there is an aisuch that

10
http://www.scribd.com/doc/63283823/Formal-Semantics-for-The-Logic-of-Presuppositionshttp://www.scribd.com/doc/63283823/Formal-Semantics-for-The-Logic-of-Presuppositionshttp://www.scribd.com/doc/63283823/Formal-Semantics-for-The-Logic-of-Presuppositionshttp://www.scribd.com/doc/63283823/Formal-Semantics-for-The-Logic-of-Presuppositions


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#Ey$Faiyand #Ey$Gaiy #.1.$

and there is a bisuch that

#Ex$Fxbiand #Ex$Gxbi. #.1.5$

!hus, it will )e ! & F if

#Ex$#Ey$Fxy #Ey$GxyB #Ey$#Ex$Fxy #Ex$GxyB. #.1.*$

!his 'eans that #.1.1$ will )e ! & F if F and o&erlap along )oth aes. Figure 1 shows

the case when #.1.1$ is D#! & F$, Figure 2 shows the case when #.1.1$ is true, finally

Figure shows the case when #.1.1$ is false. !he asterisk 'eans )oth F and .

Figure 1

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| . . . F F F F F . . . . . . . . . . . .

| . . F F F F F F F . . . . . . . . . . .

| . . F F F F F F F F . . . . . . . . . .

| . . F F F F F F F F . . . . . . . . . .

| . . . F F F F F F . . . . . . . . . . .

| . . . . F F F F . . . . . . . . . . . .

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| . . . . . . . . . . G G G G G . . . . .

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----------------------------------------------- x

t


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Figure 2

Figure

* * * * * * *

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| . . . F F F F F . . . . . . . . . . . .

| . . F F F F F F F . . . . . . . . . . .| . . F F F F F F F F . . . . . . . . . .

| . . F F F F F F F F . . . . . . . . . .

| . . F F F F F F F F . . . . . . . . . .

| . . F F F F F F F . . . . . . . . . . .

| . . . F F F F F . . . G G G G G G . . .

| . . . F F F F . . . G G G G G G G . . .

| . . . . . . . . . G G G G G G G G . . .

| . . . . . . . . G G G G G G G G G . . .

| . . . . . . . . G G G G G G G G G . . .

| . . . . . . . . G G G G G G G G G . . .

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------------------------------------------------x

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| . . . F F F F F . . . . . . . . . . . .

| . . F F F F F F F . . . . . . . . . . .

| . . F F F F F F F F . . . . . . . . . .

| . . F F F F F F F F . . . . . . . . . .

| . . F F F F F F F F . . . . . . . . . .

| . . F F F F F F * * G G G G G G . . . .

| . . . F F F F F * * G G G G G G G . . .

| . . . F F F F F * * G G G G G G G . . .

| . . . . . . . . G G G G G G G G G . . .

| . . . . . . . . G G G G G G G G G . . .

| . . . . . . . . G G G G G G G G G . . .

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-------------------------------------------------x


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et us now study a special case,

D#Ex$#Ey$#Fxy Gy$, #.2.1$

such that only oneyO msatisfies Gy. For ea'ple let Gy" )e yO m"

D#Ex$#Ey$#Fxy #yO m$$. #.2.2$

!here is no x" at G", )ut we can i'agine that #.2.2$ is epressed as

D#Ex$#Ey$Fxy ##yO m$ #xOx$$B. #.2.$

!he situation is depicted in Figures 4 and +. /ere there are only two cases. Either the

two regions o&erlap #Figure 4$ or they do not #Figure +$. =e o)ser&e that #.2.1$ can )e

either false or neither true nor false it can ne&er )e true. %n case of #.1.1$, when the two

regions did not o&erlap, there were two further su)cases either the for'ula was true

#Figure 2$ or it was neither true nor false #Figure 1.$

%t is apparent that #.2.1$ will )e D#! & F$ if

D#Ex$Fxm #.2.4$

is true. %n this case the two regions will not o&erlap #Figure +$.

et our do'ain )e the set of natural nu')ers. !hen

#5x!5y!6x7 y8 9! " y: ;!< #.2.+$

is D#! & F$ #Figure @$. !his is so )ecause the two regions do not o&erlap along thexais.

et us pickyO 5

D#Ex$#xK 5 J @$ #5 O 5$B. #.2.@$

=e o)ser&e that

D#Ex$#xK 5 J @$ #.2.$

that is, #.2.@$ is D#! & F$ analogously to #1.$ our logic is not classical. et us pick, say,

1


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yO 4

D#Ex$#xK 4 J @$ #4 O 5$B. #.2.5$

=e o)ser&e that

D#Ex$#4 O 5$ #.2.*$

that is, #.2.5$ is D#! & F$. %t is apparent that for any choice ofy%the corresponding

sentence will )e D#! & F$, hence #.2.+$ is D#! & F$. Ge&ertheless,

#5x!x7 ; 8 9! #.2.10$

istrue. +n the logic of presuppositions, *.2.0! and *.2.1=! are notequivalent.

Figure 4

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| . . . F F F F F . . . . . . . . . . . .

| . . F F F F F F F . . . . . . . . . . .

| . . F F F F F F F F . . . . . . . . . .

| . . F F F F F F F F . . . . . . . . . .

| . . F F F F F F F F . . . . . . . . . .

| . . F F F F F F F . . . . . . . . . . .

| . . . F F F F F . . . . . . . . . . . .

| . . . F F F F F . . . . . . . . . . . .

n | G G G * * * * * G G G G G G G G G G G G

| . . . . F F F . . . . . . . . . . . . .

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-------------------------------------------------x

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| . . . F F F F F . . . . . . . . . . . .

| . . F F F F F F F . . . . . . . . . . .

| . . F F F F F F F F . . . . . . . . . .

| . . F F F F F F F F . . . . . . . . . .

| . . F F F F F F F F . . . . . . . . . .

| . . F F F F F F F . . . . . . . . . . .

| . . . F F F F F . . . . . . . . . . . .

| . . . F F F F F . . . . . . . . . . . .

n | G G G * * * * * G G G G G G G G G G G G

| . . . . F F F . . . . . . . . . . . . .

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Figure +

Figure @

* * * * * * *

1+

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1| . . . . . . . . . . . . . . . . . . . .

0| . . . . . . . . . . . . . . . . . . . .

9| . . . . . . . . . . . . . . . . . . . .

8| G G G G G G G G G G G G G G G G G G G G

7| . . . . . . . . . . . . . . . . . . . .

6| . . . . . . . . . . . . . . . . . . . .

5| F . . . . . . . . . . . . . . . . . . .

4| F F . . . . . . . . . . . . . . . . . .

3| F F F . . . . . . . . . . . . . . . . .

2| F F F F . . . . . . . . . . . . . . . .

1| F F F F F . . . . . . . . . . . . . . .

0| F F F F F F . . . . . . . . . . . . . .-------------------------------------------------- x

0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9

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| . . . F F F F F . . . . . . . . . . . .

| . . F F F F F F F . . . . . . . . . . .

| . . F F F F F F F F . . . . . . . . . .

| . . F F F F F F F F . . . . . . . . . .

| . . F F F F F F F F . . . . . . . . . .

| . . F F F F F F F . . . . . . . . . . .

| . . . F F F F F . . . . . . . . . . . .

| . . . F F F F . . . . . . . . . . . . .

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m | G G G G G G G G G G G G G G G G G G G G

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Qdel


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$. %aifans Solution of the 'iar Para(o)

!he two-line pu66le is the launching point for the solution proposed )y /ai' aif'an

#2000 $ . !he sentences #4.1$ and #4.2$ )elow are two different sentence-tokens of the

sa'e sentence-type.

ine 1 !he sentence on line 1 is not true. #4.1$

ine 2 !he sentence on line 1 is not true. #4.2$

/ow do we e&aluate a sentence of the for' !he sentence on linexis not true"N

Paraphrasing aif'an #2000, p.$

Go to line x and e#aluate the sentence written there. -f that sentence is true% then +he

sentence on line x is not true/ is false% else the latter is true.

=hen we e&aluate the sentence on line 1 we are instructed to e&aluate the sentence on

line 1 =e enter an infinite loop, and no truth &alue will e&er )e assigned to #4.1$. /ence,

#4.1$ is neither true nor false. =hen we e&aluate #4.2$ we already know that the sentence

on line 1 is not true, hence #4.2$ is true. !hus #4.1$ and #4.2$ are assigned different truth

&alues although their gra''atical su):ects ha&e the sa'e referent and their predicates the

sa'e etent.

% find aif'an


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Mn line , we ha&e replaced the gra''atical su):ect of sentence #4.1$ with asynonym.%t

is a different token than the sentence on line 1, )ut it is also a different type. %ts

gra''atical su):ect has the sa'e referent as the su):ect of #4.1$, and its predicate has the

sa'e etent as the predicate of #4.1$. Synony's )y definition ha&e the sa'e referent, )ut

they alter the sentence-type. et us now assu'e that the sentence-types are the truth

)earers that is, all the sentence-tokens of the sa'e sentence-type ha&e the sa'e truth

&alue. !hen #4.1$ and #4.2$ ha&e the sa'e truth &alue, na'ely neither true nor false. 7ut

the sentence on line is still true. %t succeeds in epressing our conclusion, that sentence

#4.1$ is not true, without a contradiction. %f we accept aif'an


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false analogously to #4.1$ and #4.$. Lore for'ally we can put

V O !his sentence is not true".

!hen D!#V$ is true and V is neither true nor false. =e ha&e reached the sa'e

conclusion through different 'eans inGew)erry #2005, pp.5-10$. A si'ilar result is

o)tained )y awrence oldstein #1**2, pp.1-2$. >o'pare also #4.4$, #4.+$ with #..1$,

#..2$.

*. %+(el an( the 'iar

=e concluded in section 4 that gi&en

V O !his sentence is not true", #+.1$

D!#V$ was true )ut V was neither true nor false. %n section , we postulated a

hypothetical deri&ation syste' S of the logic of presuppositions that deri&ed only the

sentences that were true in Strawson


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'efinition 1A sentence is meaningful iff it or its internal negation epresses a

possi)le state of affairs.

=e further clarified that a state of affairs was possi)le if we could picture it to oursel&es.

Mur 'ain o):ecti&e then was to account for the sentences that were 'eaningful yet

neither true nor false. An ea'ple of such a sentence is

All Cohn


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7 !he apple is green

%f % did not understand the possi)le state of affairs it epressed #i.e., its 'eaning$ % would

not know what to co'pare it with or how to &erify it. % could perhaps &erify A )ut not

falsify 7. !his would lead to the a)surd conclusion that only true sentences were

'eaningful. !he &erification principle as a criterion of 'eaningfulness is too restricti&e,

7ut this does not 'ean that there should not )e a criterion.

!he eistence and properties of sentences are also facts, and we can for' sentences a)out

sentences. For ea'ple

> A is true

!o deter'ine if > is true, we apply

'efinition 2 A sentence is trueiff the possi)le state of affairs it epresses

corresponds to an actual state of affairs.

> epresses the possi)le state of affairs that the possi)le state of affairs epressed )y A

corresponds to the actual state of affairs. %n short > is true if A is true. =hat if A is not

trueN =e apply

'efinition * A sentence isfalseiff the possi)le state of affairs epressed)y its internal negation corresponds to an actual state of affairs.

!he internal negation of A is true" is A is not true." A can )e false, neither true nor

false or e&en 'eaningless. %n all these cases, > is false, see !a)le +.2.

> @

! !

G FF F

!a)le +.2

%n general to e&aluate a sentence of the for' X!he sentence is true< we would follow

21


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this procedure

Go to the label X and e#aluate the sentence written there. -f that sentence is true% then

+he sentence X is true/ is true% otherwise it is false.

Asing our definitions of truth, we have arrived at an exact replica of aifmans

evaluation procedure#aif'an , 2000, p.$.et us apply it to

; !he sentence ; is not true

%n order to deter'ine if ; is true we need to deter'ine if ; is true. =e are in an infinite

loop and no truth &alue will )e assigned to ;. =hat possi)le state of affairs does ;

epressN %t epresses" that the possi)le state of affairs that ; epresses does not

correspond to an actual state of affairs. An atte'pt to find the 'eaning of ; lea&es us in

an infinite loop as well. =e are not a)le to picture to oursel&es the state of affairs ;

epresses. %t does not epress any. ; is 'eaningless :ust like #+.$.

Sentence ; is clearly not &erifia)le. E&en if we use a relaed criterion of 'eaningfulness

such as ;efinition 1, ; is still 'eaningless. Vet it appears to say that the sentence ; is an

ele'ent of the set of untrue sentences. Further'ore ; indeed isan ele'ent of the set of

untrue sentences. 7ut in fact ; does notsay that it itself is an ele'ent of the set of untrue

sentences )ecause ... well ... )ecause it is 'eaningless. !he algorith'ic procedure

esta)lishes that ; is 'eaningless, therefore the co'positional se'antics is not applica)le.

/ai' aif'an 'ay ha&e put it )est when he said if the sentence fails to express a

proposition it does not% contrary to appearance% say of itself that it is not true"#aif'an ,

2000, p.1@$>ontrary to appearances, #..$ does notsay that it itself is not deri&a)le. %t

says the sa'e thing as #+.+$, that is nothing. %t has no truth &alue it ought not to )e

deri&a)le. %n S there is no separation of truth and deri&a)ility.

22
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aif'an #2000, p.1+$prefers to say that the iar sentence is 'eaningful )ut that it does

not epress a proposition. !he 'eaning of the iar sentence is the e#aluation procedure

that re&eals that the sentence has no truth &alue. !his procedure is thesenseof the

sentence states aif'an. =hat a)out its referenceN aif'an does not say. % can only

conclude that he would agree that the iar sentence #or token$ hasno interpretation in

the domain of sentences.7ut is this not what we usually 'ean )y ?'eaning?N

oldstein and 7lu' #2005, p.+$ argue ?o we can a#oid all the unpalatable

conclusions if we ac0nowledge that there is a sentence% the 1iar sentence% that is

grammatically correct and meaningful (we can% after all% understand it and translate it

into French) but deny that it can be used to ma0e a statement.? !he point is not well

taken. Firstly, why can we not argue that if we understand it then it 'ust )e 'aking a

state'entN Secondly, it does not follow that if all the co'ponents of a sentence ha&e

referents then the whole sentence does. !hat is there are cases such that e&en if there is

an ob'ectcorresponding to each ter' of a sentence, there is no state of affairsthat

corresponds to the whole sentence. !his is the case whether the ter's are in French or

in English. And % would dispute that we actually understand the sentence. For one, it

causes a tre'endous a'ount confusion. %tseemsto )e saying - and this is the source of

the confusion - that the liar sentence does not )elong to the set of true sentences, )ut it

does not =e ha&e a tendency to erroneously pro:ect the 'eaning of the sentence #4.$

into #4.1$. 7ut that does not 'ean that we ?understand? it. !he construct of the

propositionorstatementis not necessary to esta)lish that V does not ?say? anything.

2
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Conclusion

% ha&e outlined a si'ple theory of 'eaning and truth, which is co'pati)le with

StrawsonRs logic of presuppositions. #>ontrary to the current trends it is a thesis of this

paper that while the &erification principle is too restricti&e a criterion of

meaningfulness is re$uired.$ =hen this theory is applied to self-referential sentences it

yields the sa'e e&aluation as the one proposed )y /ai' aif'an. !he sentence

!he sentence is not true

is e&aluated as follows

Go to the label X and e#aluate the sentence written there. -f that sentence is true% then

+he sentence X is not true/ is false% else the latter is true.

Since the procedure ne&er ends no truth &alue is assigned to , and as a result the iar

sentence is neither true nor false.

StrawsonRs logic of presuppositions yields the result that a sentence is neither true nor

false if its su):ect class is e'pty, i.e. if D#E$F then #$#F -I $ is neither true nor

false. =hen )rought to its logical conclusions it i'plies that #$##P DP$ -I W$ is

neither true nor false. =e o)ser&e that such a logic is )ut a (uantified &ersion of ;ia6Rs

truth-rele&ant logic. %t can )e generali6ed to sentences with 'any &aria)les, and we

find that then Qdel


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,i-lioraphy

Ayer, A.C. #1*54$hilosophy in the +wentieth2entury%Gew Vork 3intage 7ooks.

;ia6, L.8. #1*51$ +opics in the 1ogic of ele#ance, Lunich, er'any Philosophia 3erlag.

aif'an, /. #2000$ Pointers to propositions. %n A. >hapuis A. upta, eds. 2ircularity%

3efinition% and +ruth%Gew ;elhi %ndian >ouncil of Philosophical 8esearch *Y121, A&aila)le

at httpZZwww.colu')ia.eduZDhg1Zgaif'[email protected] Accessed 2* Culy 2010B.

oldstein, . #1**2$ X!his State'ent %s Got !rue< %s Got !rue,*nalysis 02 #1$ 1Y2.

oldstein, , 7lu' A #2005$ ?=hen is a state'ent not a state'entNY=hen itopyright .V. Gew)erry 2010-2014

For non-co''ercial purposes this docu'ent 'ay )e freely copied and distri)uted.

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Meaning, Presuppositions, Truth-relevance, Gödel's Theorem and the Liar Paradox

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