A FORMAL REPRESERTATION OF PROPOSITIONS AND T~PORAL ADVERBIALS 
Jttrgen Kunze 
Zentralinswitut fGr Sprachwissenschaft der 
Akademie der Wissenschaften der DDR 
Prenzlauer Promenade 149-152 
Berlin, DDR-1100 
ABSTRACT 
The topic of the paper is the intro- 
duction of a formalism that permits a 
homogeneous representation of definite 
temporal adverbials, temporal quanti- 
fications (as frequency and duration), 
temporal conJ~ctions and tenses, and 
of their combinations with propositions. 
This unified representation renders it 
possible to show how these components 
refer to each other and interact in 
creati~ temporal meanings. The formal 
representation is 0ased on the notions 
"phase-set" and "phase-operator", and it 
involves an interval logic. Furthermore 
logical coz~uections are used, but the 
(always troublesome) logical quantifica- 
tions may be avoided. The expressions 
are rather near to lingaistic struc- 
tures, which facilitates the link to 
text analysis. Some emprical confir- 
mations are outlined. 
q. THE GENERAL FRAME 
This paper presents some results 
that have been obtained in the field 
of time and tense-phenomena (K~Luze 1987). 
In connection with this some links to 
text analysis, knowledge representation 
and q~erence mechanisms have been 
taken into accoumt. 
The formalism presented here differs 
from what is under label of temporal 
logic on the market (e. g. Prior (1967), 
Aqviet/Guenthner (1978). Our main in- 
tention is to establish a calculus that 
is rather near to linguistic structures 
on one side (for text analysis) and to 
inference mechanisms on the other 
side. 
The whole formalism has integrating 
features, i. e. the following compo- 
nents are represented by the same for- 
mal means in a way, that it becomes 
easy and effective to refer the differ- 
ent components to each other: 
- The propositions and their validity 
with respect to time; 
- Definite temporal adverbials (nex~ 
wee k , ever~ Tuesda2); 
- Definite temporal qu~uti£ications as 
frequency (three times) and duration 
(three hours), comparislon of fre- 
quencies, durations etc.; 
- Temporal conj~ctlons; 
- Tenses and their different meanings. 
The unified representation renders 
it possible to observe how the compo- 
nents interact in creating temporal 
meanings and relations. Some details 
have to be left out here, e. g. the 
notion "determined time" and the axio- 
matic basis of the calculus. 
- 197 - 
2. PHASE-SETS AND PROPOSITIONS 
A phase p is an interval (either un- 
bounded or a span or a moment) which a 
truth value (denoted by q(p)) is as- 
signed to: 
q(p) = T : p is considered as an affir- 
mative phase. 
q(p) • F : p is considered as a denying 
phase. 
The intervals are subsets of the time 
axis U (and never empty!). 
A phase-set P is a pair tP',q3, 
where Pa is a set of intervals, and q 
(the evalnation function) assigns a 
truth value to each pe Pa. P has to ful- 
fil the following consistency demand: 
(A) For all p,,p"aP" holds: 
If p'n p" @ ~, then q(p') • q(P"). 
A phase-set P is called complete, iff 
the union of all phases of P covers U. 
Propositions R are replaced by com- 
plete phase-sets that express the 
"structured" validity of R on the time 
axis U. Such a phase-set, denoted by 
(R>, has to be understood as a possible 
temporal perspective of R. There are 
propositions that differ from each 
other in this perspective only: Por 
(I) John sleeps in the dinin~ room~ 
one has several such perspectives: He 
is sleeping there, he sleeps there be- 
cause the bedroom is painted (for some 
days), he sleeps always there. SO the 
phases of (R> are quite different, even 
with clear syntactic consequences for 
the underlying verb, The local adver- 
bial may not be omitted in the second 
and third easel ~ 
I skip here completely the following 
problems: 
- A more sophisticated application of 
nested phase-sets for the representa- 
tion of discontinuous phases in (R>; 
- the motivation of phases (e. g. accord- 
ing to Vendler (1967)) and their ad- 
equacy. 
3. PHASE-0PERATORS 
A phase-operator is a mapping with 
phase-sets as arguments and values. 
There are phase-operators with one and 
with two arguments. A two-place phase- 
operator P-O(PI,P 2) is characterized by 
the following properties: 
(B) If P = P-O(PI,P2), then P" • P~, 
i. e. the set of intervals of the 
resulting phase-set is the same as 
of the first argument; 
(C) For each phase-operator there is a 
characteristic condition that says 
how q(p) is defined by q1(p) and 
P2 for all p £ P~. This condition 
implies always that q(p) = F fol- 
lows from q1(p) = F. 
SO the effect of applying P-O(PI,P 2) is 
that some T-phases of PI change their 
truth value, new phases are not created. 
The characteristic conditions are 
based on %wo-place relations between 
intervals. Let rel( , ) be such a rela- 
tion. Then we define (by means of tel) 
q(p) according to the following scheme: 
CD) 
q(P) " f 
T, if ql(p) = T and there is 
a P2 G P~ with q2(P2) = T 
and rel(P2oP); 
F otherwise 
- 198- 
We will use three phase-opera~ors and 
define their @v~uation functions in the 
following way by (D)s 
(E) P = 0CC(P1,P2): 
rel(P2,p) is the relation 
"P2 and p overlap", i. e. P2nP ~ ~. 
(F) P = PER(P1,P2)s 
rel(P2,p) is the relation 
"P2 contains p", i. e. P2 ~ p" 
(G) P = NEX(PI,P2)s 
rel(P2,p) is the relation 
"P2 and p are not seperated from 
each other", i. e. P2uP is an 
interval. 
As an illustration we consider some 
examples. Needless to say~ that their 
exact represention requires further 
formal equipment we have not introduced 
yet. Typical cases for 0CC and PER ares 
(2) yesterda~ was bad weather. 
Overlapping of (yesterday> and a T- 
phase of (bad weather>. 
(3) John worked the whole evening. 
A T-phase of ( evening> is contained 
in a T-phase of (John works>. 
(for (evening>, (yesterday> cf. 7.) 
There is only a slight difference be- 
tween the characteristic conditions for 
0CC and NEX: NEX admits additionally 
only b~EETS(P2,p) and ~LEETS(p,p2) in the 
sense of Allen (1984). Later ! will mo- 
tivate that NEX is the appropriate 
phase-operator for the conjunction when. 
Therefore, sentences of the form 
(~) R1, when R 2. (cf. (N), (0)) 
will be represented by an expression 
that contains NEX((R2> , (Rfl>) as core. 
The interpretation is thaC nothing hap- 
pens between a certain T-phase of (RI> 
an~ a certain T-phase of (R2~ (if they 
do not overlap). 
The next operation we are going to 
define is a one-place phase-operator 
with indeterminate character. It may be 
called "choice" or "singling out" and 
will be denoted by xP~, where P1 = 
\[~,qd\] is again an arbitrary phase-sets 
(H) xP - \[~,q~, 1 - 
P~ = ~ (set of intervals unchanged) I 
T for exactly one p with 
ql(p) = T (if there is some 
q(p) = T-phase in P1)! 
F otherwise 
If we need different choices, we write 
xP1' YPI' zP2' ..., using the first sign 
as an index in the mathematical sense. 
Moreover~we define one-place phase- 
operators with parameters: 
(I) KAR(PI,n) = \[Pm, q\]: 
P~ = ~ (set of intervals unchanged) 
i T, if qfl(p) = T and there are 
exactly n T-phases in PI; 
q(p) = 
F otherwise (for all p g pm 
independently of qq(p)) 
Similarily one defines 0RD(PI,g) for 
integers g: ORD(PI,g) assigns the value 
T exactly to the g-th T-phase of PI' if 
there is one, with certain arrangements 
for g (e. g. how to express "t~e last 
but second" etc.) 
Finally we define the "alternation" 
alt(P I) of an arbitrary phase-set P1 = 
\[l~1,ql\]. By alternation new phases may 
be create~s alt(P1) contains exactly 
those phases which one gets by joining 
all phases of P1 that are not seperated 
~o - 199 - 
from each other and have the same value 
ql(Pl). So the intervals of alt(P I) are 
unions of intervals of PI' the q-values 
are the common ql-values of their parts 
(of. (A)). It is always alt(alt(P1)) = 
alt(P1) , and alt(P I) is complete, if 
PI is complete. Going from left to 
right on the time axis U, one has an 
alternating succession of phases in 
alt(P1) with respect to the q-values. 
alt(P I) is the "maximal levelling" of 
the phase-set PI" 
4. LOGICAL CONNECTIONS 
The negation of a phase-set P1 is de- 
fined as follows: 
(J) ~PI = tP',qG: 
P~ = P~ (set of intervals unchanged) 
q(p) = neg(ql(p)) 
Note that (~R> and N(R> may be dif- 
ferent because of non-equivalent phase- 
perspectives for ~R and R! 
For each two-place functor " u " (e. 
g. "Q" = "v") we aegina PI a P2' if 
the sets PI and P2 are equal: 
(K) PI m P2 = \[Pt'q3: 
P'= P;- P~ 
q(p) = Pu(ql(p),q2(p)) , where F u is 
the corresponding truth func- 
tion (e. g. vel for " w,,). 
Obviously for every phase-operator P-O 
the expression P'O(PI'P2) "~ PI repre- 
sents both a phase-set and a clear "tau- 
tology" - in other words - a phase-set 
that is "always true", if PI is complete. 
Therefore, alt(P-0(PI,P2)-~P1 ) = U ° 
(where U ° is the phase-set that contains 
the time axis U as the only interval 
with the q-value T) reflects the double 
nature of the aforesaid implication. 
5. TRUTH CONDITIONS 
The last considerations lead imme- 
deately to the following definitions. 
The whole formalism requires two types 
of truth conditions, namely 
(L) alt(P) = U ° 
(M) alt(P) # ~U ° • 
They have different status: (L) is 
used, if the phase-set P is considered 
as a temporal representation of some- 
thing that is valid, independently of 
time. (M) is applied~if P is considered 
as something that represents a certain 
"time" (expressed by the phases of P). 
Because of the second possibility, alt 
appears not only in truth conditions, 
but it may constitute arguments in 
phase-operators etc., too. This will be 
shown in the examples below. 
Obviously one has for arbitrary 
phase-sets P = \[P',q~, 
alt(P) = U ° iff Vt~U~pGP~ 
Cq(p) = T • tap) 
altCP) ~ ~U° iff ~t GU 3p~P~ 
CqCp) = Ta tGp) 
6. SOME CO tgIRNT ON THE PORMALISM 
By regarding the time axis U as a 
basic notion one has to take the 
trouble to consider the topology of U, 
and gets difficulties with closed and 
- 200 - 
and open sets, environments etc.. This 
may be avoided by taking an axiomatic 
viewpoint: For all operations, relations 
etc. one formulates the essential prop- 
erties needed and uses them without di- 
rect connection to the time axis. In 
this way U becomes a part of a model 
of the whole formalism. This is inde- 
pendent of the fact, that in definitions 
and explanations U may appear for mak- 
ing clear what is meant. 
7 • TEMPORAL ADVERBI ALS 
In section 2. we have outlined, how 
propositions R are substituted by phase- 
sets (R>. The same has to be done for 
temporal adverbials. First we consider 
definite adverbials: (tuesday> is a 
phase-set P, where P~ is the set of all 
days (as spans p covering together the 
whole time axis U), and exactly the 
Tuesdays have the value q(p) = T. For 
(day>~he set pm is the same, but it 
is q(p) = T for all p G Pm. (evening> 
has as intervals suitable subintervals 
of the days with q(p) = T, whereas the 
remaining parts of the days form phases 
with q(p) = F in (evening> . Analo- 
gously ( e~> contains all years as 
spans p with q(p) = T, whereas (1986> 
has the same spans, but exactly one 
with q(p) = T. 
Now we combine temporal adverbials 
with propositions. An e~ct representa- 
tion would require that we list all 
possible structures of phrases, clauses 
etc. that express a certain combination. 
We use instead of this "standard para- 
phrases" as "a~ least on Tuesdays R". If 
R is a certain proposition, e. g. 
R = John works in the library , then 
this paraphrase stands (as a remedy) for 
(5) John works, worked, ... in the 
library every Tuesday. 
On every Tuesda~ John ... 
On Tuesda~ of ever~ week John ... 
A~ least on Tuesdays John ... 
Examples with truth condltionas 
(6) (the days, when R> 
= occ(<day>, (R>) 
~t(...) , ~u ° (cf. (~) - (E)) 
(7) (the Tuesdays in 1986 , when R > 
= 0CC(OCC((tuesday>,(R>), (1986>) 
~t(...) ~ ~u ° 
(8) (at least on Tuesdays E > 
= (tuesd%7> -~ OCC((day~, (R>) 
alt(...) = u ° (cf. (~)) 
(9) (at most on Tuesd%ys R> 
= OCt(( daft, (R>) -~ < tuesday > 
alt(...) = U ° 
(10) (in 1986 at least on Tuesdays R> 
= ( 1986 > -~ 
PER(( year>, 
alt((tuesd> -~ OCC((day>, < R>))) 
alt(...) = u ° (cf. (F)) 
(1986 is a year, throughout which 
it is always true, tha~ every 
Tuesda~ is a day, when R occurs.) 
The second argument of PER is a phase- 
set defined by an air-operation. This 
phase-set has as T-phases exactly those 
maximal periods during which (8) holds. 
, PER((~ear~,...) selects the years that 
are covered by such a period, and the 
whole expression says that 1986 is such 
a year (and nothing about other years). 
The time of speech L is formally rep- 
resented by a phase-set L ° with three 
phases, namely L itself with q(L) = T, 
and the two remaining infinite inter- 
vals with the q-value F. Then one may 
define (today> = 0CC((day~,L°). By 
- 201 - 
using the phase-operator ORD (cf. (I)) 
one introduces (yesterday) etc., and 
similarily (this year> etc.. 
(11) (in this year three times R 
= (R)"~KAR(OCO((R), (this year}),3) 
alt(...) = U ° 
(12) (the three times R in this ye.._ar) 
= KAR(OCC((R), (~his year)) ,3) 
sit(...) + ~U ° 
In (11) a yes-no-decision is expressed 
(there are three T-phases of (R)in this 
year), but in (12) a "time" is defined, 
namely the three T-phases of (R> in this 
year. Therefore~the truth conditions 
are different. The expression in (12) 
may appear as an argument in other ex- 
pressions again. 
Now we apply the operation "choice": 
(13) (at most on Tuesdays three times R) 
= V OCC(x(da~), 
KAR(OCC((R), x(day~),3)) 
-~<tuesda~) 
alt(...) = U ° 
OCC((R),x(day>) determines the T-phases 
of (R) on a single day, KAR(...,3) keeps 
them iff there are exactly three (other- 
wise they become F-phases, cf. (I)), 
OCC(x(day},...) assigns to the single 
day the value Tiff the T-phases of (R) 
on this day have been preserved. There- 
fore, ~OCC(...,...) is a T-F-distribu- 
tion over all days if x runs over all 
days, and the whole expression says 
that all T-days are Tuesdays. 
(q~) 
(15) 
(exactl~ on Mondays and Fridays R) 
coo(( day>, (at) 
((monde~) v (~>) 
alt(...) = u ° (of. (8), (9)) 
(never on Tuesdays R> 
OOO(( day), (R)) -~ ~ (tuesday) 
alt(...) = u ° (cz. (9)) 
These examples demonstrate the applica- 
tion of logical functors. 
As one oan see, the e~pressions ren- 
der it possible to formulate even rath- 
er complex temporal relations in a com- 
prehensible manner without much redun- 
dancy, the necessary arguments appear 
only once (or twice for certain quanti- 
fications as e. g. (tuesday) and (da~ 
in (8)). In order to handle durations, 
one needs another phase-operator EXT 
that is quite similar to KAR and ORD. 
The argument R stands either for "bare" 
propositions (without any temporal com- 
ponent) or for propositions with some 
temporal components. In the latter case 
the corresponding expression has to be 
substituted for (R): 
(q6) Ever~ Tuesda~ John watches tele- 
vision in the evening. 
Take (R) = (in the evenin~ R') 
with R' = John watches television. 
Then one can represent (R) by 
(R) = OCC( (R'}, (evening)) 
with alt(...) ~ ~U ° (John's t.v.- 
phases in evenings) and apply (8): 
( tuesda~ ) -~ 
OCC(( day},0CC(( R' ), (evening~) ) 
alt(...) = U O 
Similarily one obtains (qO) from (8). 
The truth condition in (8) causes that 
alt(...) occurs as argument in (qO). 
The sign "=" in the examples means that 
the left side is defined by the right 
side, the left side is stripped of one 
(or more) temporal components. In this 
sense (6), (8) and (9) are rules, (7) 
and (I0) include two rules in each case. 
The full and exact form of such rules 
requires more than the standard para- 
phrases, namely corresponding (syntac- 
tic) str~ctures on their left side. 
- 202 - 
8. TENSES 
Till now nothing has been said about 
tenses. It is indeed possible to repre- 
sent tenses in the formalism that we 
have outlined. But it is impossible to 
introduce "universal" rules for tenses. 
Even between closely related languages 
like English and German there are essen- 
tial differences. So it does not make 
sense to explain here the details for 
the German tenses (of. Kunze 1987). 
The main points in describing tenses 
are these: At first one needs a dis- 
tinction between "tense meanings" and 
"tense forms" (e. g. a Present-Perfect- 
form may be used as Future Perfect). 
After that one has to introduce special 
conditions for special tense meanings 
(e. g. for perfect tenses in German and 
English, for the aorist in other lan- 
guages). Further a characterization of 
tense meanings by a scheme like Reichen- 
bach's is necessary, including the in- 
troduction of the time of speech L °. 
On this basis rules for tense-assign- 
ment may be formulated expressing whioh 
tenses (= meanings) a phase xP or a 
phase-set P can be assigned to. From 
the formal point of view tenses then 
look like very general adverbials, and 
it is rather easy to explain how tenses 
and adverbials fit together. Tense- 
assignments create new expressions in 
addition to those used above. It is im- 
portant that the position of the phases 
of (R> does not depend on the tense R 
is used with: The tense selects some of 
these phases by phase-operators. So 
alt(NEX(xP,L°)) • ~U ° is the basic con- 
dition for the actual Present (of. (G)). 
9. TEMPORAL CONJUNCTIONS 
For some temporal conjunctions there 
are two basic variants, the "particular" 
usage and the "iterative" usage. We il- 
lustrate this phenomenon for when: 
(N) whenl (particular usage of when): 
WHENI(RI,R2): (for "RI, when R2") 
alt(NEX((~2>,(~1~ )) * ~u °. 
(17) When John went to the librar~ 
he found 10 ~. (Once t when ... ) 
In (17) there is a reference to a single 
T-phase of (RI> and a single T-phase of 
(R2). One can show that the truth con- 
dition for when I is equivalent to 
3x SyCaltCNEXCx(R2),YCRI>)) * ~U °) , 
but this form is avoidable (cf. (H) 
and the end of 5.). 
(0) when2(iterative usage of when): 
WHEN2RI,R2): (for "RI, when R2") 
(18) When John went to the library. 
he took the bus. (Whenever ... ) 
In (18) something is said about all 
T-phases of (R2~ , namely 
Vx 3y(alt(NEX(x(R2~ ,y(R~ ) * NuO) , 
which is equivalent to the truth condi- 
tion for when 2. 
Conjunctions like while, as lon~ as 
etc. are represented in a similar way 
with the phase-operator PER (cf. (F)). 
For the conjunctions after, before, 
since and till one needs in addition an 
ANTE- and a POST-operator, which are 
tense-dependent (the main difference 
is caused by imperfective vs. perfec- 
tive) and modify the arguments of the 
phase-operators. Some of the conjunc- 
tions have both basic variants, whereas 
since admits no iterative usage. 
- 203 - 
The meaning of since is expressed by 
(P) since: (only particular usage) 
SINCE(Rfl,R2): (for "Rfl, since R2") alt(P~(PosT((~2)), (RI)~ ~ ~u °, 
and the truth condition for afterq is 
(Q) afterq (particular usage of after)s 
AFTERI(Rq,R2): (for "Rq, after R2") 
alt(PER((RI~ ,POST((R2)))) , ~U ° 
It turns out that an analysis of tem- 
poral conjunctions based only on the 
Reichenbach scheme causes some difficul- 
ties. It works very well for when and 
while (cf. Hornstein 1977) and the Ger- 
man equivalents (als/wenn, w~hrend and 
solam~e), but for the remaining cases 
ANTE- and POST-operations seem to be 
inivitable. 
qO . AN F~iPIRI CAL CONFIP~IATION 
By combining the rules for te~se-as- 
sig~ment and the truth conditions for 
the temporal conjunctions (in German 
there are seven basic types) and by al- 
lowing for some res~rictiomsfor their 
use (e. g. als only for Past, seit not 
for Future) one gets for each conjunc- 
tion a prediction about the possible 
combinations of tenses in the matrix 
and the temporal clause. 
Gelhaus (q97@) has published statis- 
tical data about the distributions of 
tenses in the matrix and the temporal 
clause for German. From the huge L!MAS- 
corpus the took all instances of the use 
of temporal conjunctions. From my cal- 
culus one cannot obtain statistics, 
of course, it decides only on "correct- 
hess". The comparlsion proved that 
there is an almost complete coincidence. 
The combinations for als/wenn cannot be 
derived, if one takes OOC instead of NEX 
in (N) and (O). The same seems to be the 
case for when. The restrictions for the 
propositions R I and R 2 (e. g. \[+FINIT\]), 
given by Wunderlich (1970), can be de- 
duced from the truth conditions (details 
about both questions in (Kunze (1987)). 
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