A new formal toolt Functorlal variables 
representing assertions and presuppositions 
lngolf MAX 
Department of PhllosophylLoglc 
Martln-Luther-Unlverslty Halle-Wittenberg 
GroSs SteinetraBe 73~ 4010 HallelSaale, GDR 
Abstract 
The language of classical propositional iog- 
Ic is extended by functorial variables as a 
new syntactical category. Functorial vari- 
ables render to be a possible integrating 
representation of both assertion and presup- 
position in one and the same logical formula 
different from such using classical conjunc- 
tion. 
O~ Introduction 
From a computational point of vlew there 
is an important difficulty of an adequate 
formalization of both assertion and presup- 
position, It rests on dividing utterances 
into an explicit part (assertion) and an im- 
plicit one (presupposition). 
Frege 118921 refused any explication of 
this implicit part, because of hls ideallza= 
t/on that logic deals only with correct 
~tatemerlts. At least in regard of definite 
descriptions Russell /1905/ pleaded for such 
an explication. But classical conjunction 
was one of its essential formal tools. To a 
certain extent assertion and presuppoeltion 
were represented at one and the same level. 
Strawson /1950/ claimed that presuppositions 
have to be explicated, but not at the same 
level like the implicit part. Many linguists 
= e, g. Klefer 11973/ - believe that logical 
means can be used to represent both parts of 
utterances aeparately~ but these representa- 
tions cannot be put together in one and the 
same logical expreeslon~ because Russell's 
solution Is unsatisfactory. 
My intention is to show that functorlal 
variables render to be possible tools for 
integrating representation of both assertion 
and presupposition in one and the same logi- 
cal formula, Moreover - unlike Bergmann 
i1981/, OunglKOstner /19861 - this represen- 
tation Is a svntactlcal one and different 
from that given by means of classical con- 
Junction (cp, Max l1986/i /forthcoming/). 
1. The logical apparatus 
1,1~ Functors as classical functions 
n Let ~t be the form of n-placed propositional 
2n functors; 1 ~ t ~ 2 ~ I will use 1- and 2- 
placed functors only, These functors are in- 
terpreted as classical (i. e, 2-valued end 
extensional) functions, The value-tables are= 
111 1 o--7- 1 1 1 1 
0 I 1 0 1 0 110 I 1 1 1 1 
0,11 1 1 0 0 
0,0 I I 0 1 0 
lll 
I~0 
Oel 
OmO 
1 0 0 0 0 0 0 0 
0 1 I 1 1 0 0 0 
0 1 1 0 0 1 1 0 
0 1 0 1 0 1 0 1 
1 .1. 1 
0 0 0 
1 1 0 
1 0 1 
0 
0 
0 
0 
1.2, Functorlal variables 
I introduce functorial variables as e ne~ 
pyntactlcal category, and take the classical 
functors as values of these variables. Intro- 
ducing such variables we get a whole class of 
syntactical extensions of the classical pro- 
positional logic. Only functorlal varlablee 
of the following form are considered= 
G~=gi_ I ~ f,g ~ 16. 
The f's and g's are called components. The 
f-component (g-component) is called first 
(second) component. 
are values of 2 Semantically, ~ and ~gG2 the 
functorial variables f,g. Therefore these 
functorlal variables (abbreviation= FV's) 
represent eats of functors wlth exact two 
(not necessarily different) elements. Wlth 
respect to an intuitive interpretation FV'e 
=unite" the properties of both functore. 
1,3, The language O 
Primitive symbols= 
1) .pliqliP2eq2s,. , (propositional variables) 
(fu.ctors) 
2 ~2 ^2 _2 
3) Gltl,..OlUli161U2ili...iu16116 (FV=a) 
Formation rules= 
(1) A propositional variable standing alone 
40B 
is a formula of G. 
(2) Zf A, B and C are formulae of G, then 
~A" and ~BC are formulae of G. 
(3) G~,6Ptq ;__ iS a formula of G. 
(47 Zf A is a formula of G formed without 
reference to the formation rulee (1) end 
(27a then G 2f*gAA is a formula of G. 
(5) A Je a formula of G lff its being so 
follows from the formation rules (1) - 
(4). 
Definitions and types of formulae= 
Ol= pvA =df ~A 
D2= (A V B7 =df ~AB D4= (A===B) =df ~AB 
D3= (AZDB) =df ~AB DS= (A A B) =df 0~AB 
2 D6= (piqt) =df G4,6Ptqt" 
A K-formula A K (l. e. classical formula) le 
that fo~'muZs which was exclusively formed by 
means o? formation rules (1) and (2). 
A G-for~.luls A G iS that formula which was ex- 
cluetveIy formed by means of formation rulee 
(3) and (4). 
The rul~,s of substitution of G are formu- 
lated in such a manner that (a) the p-propo- 
sitions,1 variables and the q-propositional 
variables occurring in G-formulae act as va- 
riables of a different eort= 1, s° in any 
case the former occur on the left. and the 
latter on the right in formulae of the form 
G 2 4,6piqil (b) they have the same index; and 
(C) there ls no'rule of substitution which 
allows the substitution of more complex for- 
mules for propositional variables within G- 
formulae, In the case of K-formulae we have 
the usual rule of substitution. 
Connection conditions of FV's= 
Now Z explain how several FV's occurring in 
the same formula ere connected, The condi- 
tions of connection are chosen in such a way 
that every formula containing FV°e repre- 
sents exactly two formulas without FV*s= 
(1) Let A G be a G-formula of G. I define 
both FV-free formulas in two steps= 
(S) Let G~,g be the matn-FV of AG. Then 
AG1 AG/G 2 /~2~ AG1 =dr ~ f,g, pf~ t.e. is the formulao 
which results by substitution of ~ 
for G~g, 
\] 
AG2 A G { G 2 /~2 t =df i f,glW9~* 
(b) Let K R be e functor or e FV of e well- 
formed part of A Gl (AG2), end let this 
wsll=~:ormed pert be more complex than a 
formule of the form G 2 With re- 4~6Plqt ° 
spect to all well-formed parts of A GI 
(A G2) I generate the formulae A GEl and 
A GE2 by substitution of the functor indi- 
cated by the flret component of the main- 
FV of the first argument of K 2 for this 
mstn-FV, and by substitution of the func- 
tot Indicated by the second component of 
the matn-FV of the second argument of K 2 
for thla matn-FV. 
The formulae A GEl end A GE2 generated by this 
method dlffer only in the mafn-functor. They 
are both K-formulae. 
AG/A GEl and AG/A GE2 are abbreviations for all 
substitutions in A G which generate A GEl and 
A GE2. respectively. 
(2 7 Let A be a formula of G which can contain 
both functore and FV's, A well-formed 
part of A is celled G-maximum tff 
(t 7 A is e G-formula= end 
(1t7 its governed connective is not a FV. 
Let AI,..oeA u be all G-maximum well- 
formed parts of A. Then 
AE1 A(A1/A~E1 , -.GEl =df ,°.,,Au/A u 7 
AE2 GE2 A "A GE2" A(AI/A I , 
* U / U )" =df " "" 
1o4o Validity of formulae with FV's 
(1) A G-formula A G ls valid tff both A GEl end 
A GE2 are valid in the classical sense= 
(2) A formula A is valid tff both A E1 end A E2 
are valid in the classical sense. 
2° Relations to classical logic 
My system is semantically equivalent..w.it h the 
.F!aaelcal PrOPOSitional logic in the sense 
that all FV'e can be eliminated by replacing 
every formula A of G by the conjunction of 
its both closed substitutions, t. e° A E1 end 
A E2- In this manner we get s complete and 
consistent system of classical logic. It 
hoZds= A formula A of G ls valid lff its cor- 
responding classical formula AE1A A E2 is 
valid. 
There are some specific differences between 
the starting formula with FV'e and its analo- 
gous formula without FV'eo One important dif- 
ference is the following= After replacing the 
propositional variables by values 1 or 0 the 
formula A gets none of these values and it 
remains ,up eatured. Only if this formula ls 
transmitted in one of its both closed substi- 
tutions - A E$ or A E2 -, then it gets e value. 
With respect to formulas with FV's which are 
neither tautologies nor contradictions there 
is another Important difference: Let A be 
409 
such a formula. Then often A ~-~(AE1A- A E2) 
is not valid, 
3, Assertion and presupposition 
The introduction of expressions of the form 
G~,6Plq i_ renders to be a possible unconven- 
tional approach to assertion and presupposi- 
tion. I postulate that the p-propositional 
variables represent elsments of a set of as- 
sertions, and the corresponding q-variables 
represent elements of a set of presupposi- 
tions. The FV G 2 4,6 constitutes an ordered 
sequence of both sorts of propositional va- 
t la~s- :-~ p re eu ppo sl t ion component 
!piq~ i presupposltion expression: ~2plql 
G2, (~Plql ~--- ql ) 
I L-J-J--~assertlon expression : ~2plql --assertion component (~2piqi~ pi ) 
Concerning logical relations between several 
sentences both assertions and presupposi- 
tions can influence this relation. In order 
to form s correct translation of such com- 
pound sentences their simple parts should be 
translated into expressions of the form 
G 2 . Let A G be a G-formula of G. Then 4,6Piql 
we can put on the following generalization 
of our interpretation: 
A GEl .- assertion expression 
A GE2 - presupposition expression. 
So ws get a new syntactical method to expli- 
cate assertion and presupposition in one and 
the same formula. Unlike 4-valued/2-dimen- 
sional approaches our language possesses an 
enrlchment of syntactical expressive power. 
4. FV's and functors 
The explication of both assertion and pre- 
supposition by means of formulas of the form 
(plql) dlffers from that one by means of 
claselcal conjunction, because 
(Plql)--- (Pi Aql ) is not valid. 
Because of 
TI: (Pi A ql) ~(Plql), end 
T2: (plql)~(pl V ql) 
the representation by means of G 2 is 4,6 
stronger than that one by conjunction, but 
it is weaker than that one by disjunction. 
5. Negations 
Because of 
2 T3= Gll,ll(plql)(plql)~ r-~q 1 
410 
the 2-placed FV G 2 - 11,11 can be interpreted ee 
presupposition-rejecting negation. 
G 2 13,6 can be interpreted as Presuppoeitlon- 
~resefving negation, because 
, 2 2 2 
G~3,6 ( "" .r--~13~4Plql~6plql~Pl PlqlJ(PiqlJ'--1 ~2-2 -2 
~.= A6p4plqlP6Plql~ql. 
6. Extensions of the language G 
2 Starting_point: G4a6Plqi. 
~t~p 1~ Dropping index-equality of proposi- 
tional variables: G 2 Hence it follows 4,6Piqj • 
e more direct formalization of sentences with 
the same presupposition: 
2 2 G4,6plq 1 and G4,6P2q 1, 
~t~p 2~ Admitting of repreeentetlon of sever- 
al presuppositions (of one simple sentence) 
connected by propositional functors= 
2 G4,6P1(q I A q2). 
~t~p 3~ Admitting of functional dependence 
between propogltionsl varisbles (i, e. ad- 
mitting of an interaction between assertion 
and presupposition of simple sentences): 
2 G3, sPlq 1" 
~t~p 4~= Admtttin 9 of more than two components 
(e. g, 2 presupposition components)= 
3 r -- F ~16Plql i~ pl (assertion) 
G 3 3 16,52, B6Ptqtri-~52Plqtrl~qi (preeuppo- 
3 __ sttlon I) 
L~86Plqlri~ r I (presuppo- sition 2). 
~t~p 5~ Combinations of ssveral steps, 

References 

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