COMPUTATIONAL SEMANTICS OF MASS TERMS 
Jan Tore l..¢nning 
Department of Mathematics 
University of Oslo 
P.O. Box 1053, Blindern 
0316 Oslo 3, Norway 
ABSTRACT 
Although the formalisms normally used for describ- 
ing the semantics of natural languages are far from 
computationally tractable, it is possible to isolate 
particular semantic phenomena and interpret them 
within simpler formal systems. Quantified mass 
noun phrases is one such part. We describe a simple 
formal system suitable for the interpretation of 
quantified mass noun phrases. The main issue of 
this paper is to develop an algorithm for deciding the 
validity of sentences in the formal system and hence 
for deciding the validity of natural language infer- 
ences where all the involved noun phrases are 
quantified mass noun phrases. The decision proce- 
dure is based on a tableau calculus. 
INTRODUCTION 
A formal semantics for a part of a natural language 
attempts to describe the truth conditions for sen- 
tences, or propositions expressed by sentences, in 
model theoretic terms, and thereby also the relation 
of valid inferences between sentences. From the 
point of view of computational linguistics and natu- 
ral language understanding, it is important whether 
this relation of entailment can be made computa- 
tional. In general, the question must be answered in 
the negative. All proposed formal semantics of, say, 
English are at least as complex as first order logic 
and hence at best semi-decidable, which means that if 
a sentence 13 is a logical consequence of a set of sen- 
tences Y., then there exists a proof for \[3 from ~, but 
no effective way to find such a proof. Several pro- 
posals use even more complex logics, like the 
higher order intensional logic used in Montague 
grammar, which has no deductive theory at all. 
We will not oppose to the view that English in- 
corporates at least the power of first order logic and 
that even more complex formalisms may be needed 
to represent the meaning of all aspects of English. 
But we believe there are two different possible 
strategies when one is to study one particular se- 
mantic phenomenon in natural languages. The first 
one is to try to interpret the particular phenomenon 
into a system that attempts to capture all semantic 
aspects of the natural language. The other slrategy 
is to try to isolate the particular semantic phe- 
nomenon one wants to study and to build a semantic 
interpretation suited for this particular phenomenon. 
By following the latter strategy it might be possible 
to find systems simpler than even first order logic 
that reflect interesting semantic phenomena, and in 
particular to come up with systems that are compu- 
tationally tractable. 
Quantified mass noun phrases is one such phe- 
nomenon that can be easily isolated. The properties 
particular for the semantics of quantified mass terms 
have been difficult to capture in extensions of sys- 
tems already developed for count terms, like first or- 
der logic. However, if one isolate the mass terms 
and tries to interpret only them, it is possible to 
build a model where their typical properties fall out 
naturally. We have earlier proposed such a system 
and shown it to have a decidable logic (L0nning, 
1987). We repeat the key points in the two follow- 
ing sections. The main point of this paper is a de- 
scription of an algorithm for deciding validity of 
sentences and inferences involving quantified mass 
terms. 
The strategy of isolating parts of a natural lan- 
guage and giving it a semantics that can be 
computational is of course the strategy underlying 
all computational uses of semantics. For example, 
in queries towards data bases one disregards all truly 
quantified sentences, and use only atomic sentences 
and pseudo quantified sentences where e.g. for all 
means for all individuals represented in the data base. 
The system we present here contains genuine 
quantifiers like all water and much water, but contain 
other restrictions compared to full first order logic. 
In particular the mass quantifiers behave simpler 
with respect to scope phenomena than count quanti- 
fiers. 
REPRESENTING QUANTIFIED 
MASS NOUNS 
We will make use of a very simple formal language 
for representing sentences containing quantified mass 
nouns, called LM (Lcnning, 1987). We refer to the 
original paper for motivation of the particular chosen 
format and more examples and repeat only the key 
points here. 
1. A particular LM language consists of a (non- 
empty) set of basic terms, say: water, gold, 
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blue, hot, boil, disappear .... and a (possibly 
empty) set of non-logical determiners, say" 
much, little, less..than two kilos_of.. 
2. Common to all LM languages are the unary 
operator: -, the binary operator:., the logical 
determiners: : all, some, and the propositional 
connectives: : --1, ^, v, --~. 
3. A term is either basic or of one of the two 
forms (t.s) and (--t) if t and s are terms. 
4. An atomic formula has the form D(t)(s) where 
D is a determiner and t and s are terms. 
5. More complex formulas are built by the 
propositional connectives in the usual way. 
A model for the particular language is a pair con- 
sisting of a Boolean algebra A = <A, +, *, -, 0, 1> 
and an interpretation function \[ \], such that 
1. \[t\] is a member of A for all basic terms t, 
2. \[/9\] is a set of pairs of members of A for all 
determiners D. 
The interpretation of more complex expressions 
is then defined as an extension of \[ \]: 
1. \[-t\]= -\[t\], the Boolean complement of \[t\], 
\[t.s\]=\[t\]*\[s\], the Boolean product (or meet) of 
It\] and Is\]. 
2. \[D(t)(s)\] is true provided (\[t\],\[s\])~ \[D\], in par- 
ticular \[All(t)(s)\] is true provided \[t\]_<\[s\], and 
\[Some(t)(s)\] is true provided \[t\]*\[s\] ~ 0. 
3. The propositional part is classical. 
To get an intuition of how the semantics work 
one can think of \[water\] as the totality of water in 
the world or in the more limited situation one con- 
siders, of \[blue\] as the totality of stuff that is blue 
and of \[disappear\] as the totality of stuff that dis- 
appeared. However, one shall not take this picture 
too literally since the model works as well for ab- 
stract as for concrete mass nouns. 
In the formalism, a sentence like (la) is repre- 
sented as (lb) and (2a) is represented as (2b) if the 
negation is read with narrow scope and as (2c) if the 
negation is read with wide scope. 
(1) (a) All hot water is water. 
(b) All(hot, water)(water) 
(2) (a) Much of the water that disappeared was 
not polluted. 
Co) Much(water.disappeared)(-polluted) 
(c) ~ Much(water,disappeared)(polluted) 
Formula (lb) is a valid LM formula. In general, the 
valid English inferences that become valid if a mass 
term like water is interpreted as quantities of water 
and all water is read as all quantities of water are also 
valid under the LM interpretation. 
In addition, this approach can explain several 
phenomena that are not easily explained on other 
approaches. Roeper (1983) pointed out that 
paraphrasing water as quantities of water was prob- 
lematic when propositional connectives were 
considered. If some water disappeared and some did 
not disappear, there will be many quantities that 
partly disappeared and partly did not disappear. If 
disappear denotes the set of all quantities that wholly 
disappeared and did not disappear denotes the com- 
plement set of this set, then all quantities that partly 
disappeared will be members of the denotation of did 
not disappear. The sum of quantities that are mem- 
bers of the denotation of d/d not disappear will equal 
all the water there is. Roeper solved this problem 
by letting the quantities be members of a Boolean 
algebra and used a non-standard interpretation of the 
negation. In LM it naturally comes out by the 
Boolean complement as in (2b) and water that did 
not disappear is represented by (water.(--disappear)). 
A main feature of the current proposal is that it 
introduces non-logical quantitiers co-occurring with 
mass nouns, like much, little, most .... in a 
straightforward way. A sentence like much water 
was blue does not say anything about the number of 
quantities of blue water, but says something about 
the totality of blue water, which is the way it is in- 
terpreted in LM. 
It might seem a little like cheating that the sys- 
tem only introduces interpretations of mass quanti- 
tiers with minimal scope with respect to other quan- 
riflers, that it is not possible to interpret one quanti- 
tier with scope over another quantifier. In particular, 
since this is a main reason to why the logic in the 
sequel becomes simple. However, it is characteristic 
for mass quantifiers that they get narrow scope. 
(3) (a) A boy ate many pizzas. 
(b) A girl drank much water. 
While it might be possible to get a reading of (3a) 
which involves more than one boy, i.e., one boy for 
each pizza, it is not possible to get a reading of (3b) 
involving more than one girl. 
The only determiners that get a fixed, or logical 
interpretation in LM are : all and some. For the 
other determiners we can add various sorts of con- 
straints on the interpretations that will make more of 
the inferences we make in English valid in the logic. 
For example, it has been claimed that all natural 
language determiners are conservative (or "live 
on" their fast argumen0, i.e: (a,b)¢ \[D\] if and only 
if (a,b*a)~ \[/9\] (Barwise and Cooper, 1981). 
Several determiners are monotone, either 
increasing or decreasing, in one or both argu- 
ments, e.g., much is monotone increasing in its 
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second argument: if (a,b) ~ \[much\] and b < c then 
(a,c) e \[much\], and less than_two_kilos_of is 
monotone decreasing in-its second argument. 
Whether an inference like 
Much water evaporated. 
All that evaporated disappeared. 
• " Hence much water disappeared, 
becomes valid in the logic, will depend on whether 
the denotation of much is constrained to be mono- 
tone increasing in its second argument or not. 
LOGICAL PROPERTIES 
We will repeat shortly several of the properties of 
the logic LM shown in (LCnning, 1987) as a back- 
ground for the decision algorithm for validity. 
A Hilbert style axiomatization was given and it 
was shown that any set of LM sentences consistent 
in the logic has a model: the logic is complete and 
compact. 
It was implicitly shown, but not stated, that any 
model for LM must be a Boolean algebra: let a 
model be any set A with one unary operation \[-\], 
one binary operation \[-\], and a binary relation JAil\] 
then the model is a Boolean algebra with \[-\] the 
Boolean complement, \[.\] the Boolean product (meet) 
operation and JAil\] the ordering < on A. 
It was also shown that the logic was complete 
and compact with respect to the smaller model class: 
the atomic Boolean algebras, i.e. any consistent set 
of sentences has a model which is an atomic algebra, 
and in fact a finite such. 
From this, it was shown that LM with no non- 
logical determiners, let us call it LA, is equivalent to 
a subset of monadic first-order logic, hence it is 
decidable. It was also shown that the full LM is 
decidable. The argument is based on the fact that the 
number of possible models for a sentence is finite 
and decidable. This number grows rapidly, however. 
Already a sentence in LA with n different basic terms 
have 2(2n) different models, so the argument does 
not establish a good procedure for checking validity 
in practice. We will establish procedures that are 
better (in most cases) in the next section. 
Several natural restrictions on determiners, like 
conservativity and monotonicity can be expressed 
completely in LM. It is not surprising that this can 
be done for a fixed LM language given the finiteness 
of its nature. The more important point is that the 
properties can be expressed in a uniform way inde- 
pendently of the atomic terms of the language, 
which in next section will give rise to uniform in- 
ference rules. 
A DECISION PROCEDURE 
We shall establish a procedure for deciding the valid- 
ity of LM sentences. The procedure is a combina- 
tion of a normal form procedure and a tableau proce- 
dure (see e.g. Smullyan, 1968). 
We start with an LM formula 9 for which we 
want to decide whether it is valid, ~. 9. This is 
equivalent to deciding whether ~tp is satisfiable. 
One can think of the process as an attempt on build- 
ing a model for ~9. If we succeed, we have shown 
that tp is not valid and we have found an explicit 
counterexample; if we fail, we have shown that tp is 
valid. We assume all propositional connectives in 9 
to be basic: 9, ^, v. 
1. First we introduce a new unary quantifier Null 
such that Null(t) is a formula when t is a term. The 
meaning is that \[Null(t)\] is true ff and only if \[t\]=0. 
Then substitute 
Null(t.(-s)) for All(t)(s) 
Null(t.s) for ~ Some(t)(s) 
~Null(t.s) for Some(t)(s) 
This step is not necessary, but it gives a more con- 
venient notation to work with and fewer instances to 
consider in the sequel. The substitutions correspond 
to the fact that Some and Every can be taken to be 
unary and the one can be defined from the other, as 
in the case with count nouns. 
2. Then transform $ to conjunctive normal form, 
that is, a conjunction of disjunctions of literals, 
where a literal is an atomic or negated atomic for- 
mula. 
3. Observe that (i) and (ii) are equivalent. 
(i) P ~l^V2^...^~n 
(ii) N~I, NV2...andNVn, 
Hence we can proceed with each conjunct separately. 
4. We shall now check P ~lVXlt2v...VVn, where 
each Vi has the form Null(t), --Wull(t), D(t)(s) or 
-d)(t)(s). Observe that the following two formulas 
are equivalent, where t+s is shorthand for 
-((-t).(-s)). 
(i) ~ Null(t) v ~ Null(s) 
(ii) ~ Null(t+s) 
(This corresponds to the equivalence between 3xtp v 
3xV and 3x(9 v ~) in first order logic.) Hence 
contract all the literals of the form --Wull(t) to one. 
5. We are then left with a formula of the form 
-.-aVull(t)vNull(sl)v... vNull(srOvaF l V... VVm, 
where each ~Fi has the form D(u)(v) or --4)(u)(v), for 
some non-logical D. First assume that there are no 
~i's. Then observe that (i) and (ii) are equivalent. 
- 207 - 
(i) 1= ~ Null(t) v Null(sl) v...v Null(slO 
(ii) h ---3lull(t) v Null(sl) or... or 
h ~ Null(t) v Null(sn). 
(If there are no si's proceed with h ~ Null(t).) This 
equivalence might need some argument. That (ii) 
entails (i) is propositional logic. For the other way 
around, there are two possibilites. Either h--31ull(t), 
which yields the equivalence. Otherwise, there ex- 
ists a model A for the language of d~ where \[t\]=0 
and for all other terms s in the language: Is\]=0 if 
and only if All(s)(t) is a valid formula. Let V be the 
formula ~Null(t) v Null(sl) v ... v Null(sn). 
Then ~ is valid if and only if ~ is true in A. To 
show this, it is sufficient to show that if there is a 
model B in which q/is not true then W is not true in 
A. If B is a model in which W is not true, then 
It\]=0 and each \[si\]~0 in B. Hence All(sO(t) cannot 
be valid and \[si\]*:O in A for each si. Since \[t\]=0 in 
A, ~ cannot be true in A. The same type of argu- 
ment yields that --,Vull(t) v Null(sO is valid if and 
only if it is true in A. If we write A h 11 for ~1 is 
true in A, the following equivalence is propositional 
logic and yields the equivalence above. 
(i) A h ~ Null(t) v Null(sl) v...v Null(sn) 
(ii) A h --1 Null(t) v Null(sl) or... or 
A h --, Null(t) v Null(sn). 
6. a. We shall describe two different ways for 
checking h ---dVull(t) v Null(sO. The first one pro- 
ceeds by a transformation to normal form and may 
be the easiest one to understand if one is not accus- 
tomed to tableau calculus. The second one which 
uses a tableau approach is more efficient. First ob- 
serve that (i) and (ii) are equivalent. 
(i) h ~ Null(t) v Null(sO 
Oi) h Null((-t),si), (i.e. h All(sO(t)). 
The last claim entails the first one since 
Null((-t)*si) ~ -3lull(t)vNull(si) 
is a valid LM-formula. To see that (ii) entails (i) 
observe that if --dVull(t)vNull(si) is valid, it will in 
particular be true in the model A described in step 5, 
hence h All(sO(t). 
To check hNull ((-t).si) rewrite the term (-t)°si in 
disjunctive normal form: allow the symbol + and 
write the term on the form Sl+...+Sm where each si 
has the form Ul°....Uk and each uj is either an 
atomic term or on the form -v for an atomic term v. 
Then h Null(sl+...+Sm) if and only if h Null(sl) 
and ... and hNull(sm), and hNull(ul.....Uk) if and 
only if there is a v such that one uj equals v and an- 
other uj equals -v. 
b. The checking of h ~Null(t)vNull(si) will be 
faster using a tableau procedure instead of rewriting 
to normal form. Note that the following are equiva- 
lent: 
h --1 Null(t) v Null(sO 
h NuU((-t).sO '~ 
h --i Null(t+(-si)) 
There is a close connection between propositional 
logic and Boolean algebras. To each term in LM, t, 
there corresponds a formula Pt in pure propositional 
logic such that --~Vull(t) is valid in LM if and only if 
the corresponding formula Pt is a tautology: shift 
each basic LM term t with a corresponding proposi- 
tional constant Pt, and exchange - with --i, * with ^, 
and + with v. In particular, the following are 
equivalent: 
h --~lull(t+(-si)) (in LM) 
h (Pt v(-~Psi)) (in propositional logic) 
(By the earlier mentioned connection between LM 
and first order logic this corresponds to the fact that a 
first order formula 3xq~ is valid if and only if ¢p is a 
tautology whenever q~ is quantifier free.) 
Step (6a) above is equivalent to checking this 
latter formula for validity by transformation to a 
normal form. Instead it can be checked by a standard 
tableau procedure (see e.g. Smullyan 1968). 
We give a short description of the tableau ap- 
proach to propositional logic. In order to verify a 
formula V, we try to build a model that falsifies it. 
To ease the description we assume that V is on 
negation normal form, that is, built up from literals 
by ^, v. The attempt to build a model is done by 
building a tree for V. We start with a root node de- 
picted by V and use the following two rules: 
1. For each node a depicted by a formula of the 
form y v TI attach to each leaf below a in the 
tree constructed so far one successor node b de- 
picted by y and one successor node c to b de- 
picted by rl. 
2. For each node a depicted by a formula of the 
form T ^ rl attach to each leaf below a in the 
tree constructed so far two new leaf nodes one 
depicted by T and another one depicted by rl. 
The tree is complete when all formulas of the 
forms y v 11 and "/^ 11 are reduced according to the 
rules above. A branch in a tree is called closed if 
there is a formula T such that one node along the 
branch is depicted by T and another node along the 
branch is depicted by --,7. A branch in a complete 
tree for V which is not closed describes a valuation 
that falsifies V. Conversely, if all branches in a 
complete tree for V are closed, W is valid. We illus- 
trate with an example: 
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~pv (-~ qA (p V -'1 r))v (r A q~ 
I ~qA (pv~ 1") 
(r/q) 
~q pv--,r 
/\ I r q p 
I I # # 
The sign # indicates that a branch is closed. We 
have not completed the rightmost branch since it is 
already closed. Since there is one open branch in the 
tree, the formula is not valid. The literals along the 
open branch: ~p, ~q, r shows that any valuation V 
such that V(p) = T, V(q) = T, V(r) = .L, falsifies V. 
The strategy in step (6a) above with transforma- 
tion to normal form corresponds to construction of 
separate copies for each branch, hence duplicating 
parts of the tree, while the tableau procedure exploits 
the possibility of structure sharing. 
Returning to our main point, we can observe one 
additional gain by using the tableau approach. Our 
goal is to check whether t" --Wull(t)vNull(sl) or .... 
or P- ---~ull(t)vNull(sn), which is equivalent to check 
whether (tv(~sl)) or .... or (tv(~sn)) is a tautology. 
The part of the tableau tree that corresponds to t can 
be constructed (and if possible reduced by removing 
closed branches) once and for all, and then be reused 
with all the different si's. 
7. We now return to step 5 and consider the case 
where one or more disjuncts have the form D(u)(v) 
or ---~(u)(v), for some non-logical D. Then the 
following are equivalent. 
(i) b --~Null(t)vNull(sl)v...vNull(sn)WlV...V~m 
(ii)~- -Wull(t)vNull(si) for some si, 1 < i _< n, or 
--~Vull(t)VVkWj for some k andj between 1 
and m, where Vk lias the form D(a)(b) and Vj 
has the form ---D(u)(v) for the same determin6r 
D. 
That (ii) entails (i) is immediate. For the other way 
around, suppose that (ii) does not hold. We shall 
then construct a model which falsifies the original 
formula in (i). Let A be the model where only 
terms provably less than t denote 0 and where a pair 
(\[d\],\[e\]) is a member of \[/9\] if and only if --~D(d)(e) 
is one of the disjuncts Vi's. By the construction, A 
will falsify --,Null(t) and each disjunct of the form 
---~(d)(e). As in step 5 above, A will falsify each 
Null(si). It remains to show that A falsifies each 
disjunct of the form D(a)(b). Let ~/be one such dis- 
junct, let rlj, 1 < j < s, be all the disjuncts of the 
form --dg(d)(e) with the same determiner D as in ~/ 
and let ej be ---aVull(t)v'l, vTIj" Since (ii) does not 
hold, theie exists a model B\] where ej is false, for 
each e" Then there also exists a model A" which J" . .J 
equals A except possibly for the interpretaUon of D, 
and where D gets the same interpretation as in Bj. 
Hence ej is false in A'j. Since there, exists such an 
Aj for each ej, ~/cannot be true m A. 
8. Whether b -WuU(t)vD(a)(b)v--43(u)(v) holds, 
depends on which restrictions are put on D. With no 
restrictions, any possible counterexample is one 
where \[t\]=0, (\[u\],\[v\])~ \[D\] while (\[a\],\[b\])~ \[D\] . 
The only reason we should not be able to construct 
such a model is that \[a\] =\[u\] and\[b\] =Iv\] whenever 
It\] --0. We can hence proceed to check 
I ~ --,Nutl(0 v 
(Nult((-a.u)+(a.-u) ) A Nult((--b'v)+(b'-v) )), 
according to the same procedures as in step 6 above. 
If we have the additional constraint that the de- 
terminer in question is conservative, the last rule is 
changed such that the last conjunct, which above 
stated that the symmetric difference between b and v 
was zero, now instead states that the symmetric dif- 
ference between a.b and u.v is zero. 
.--~ull(t) v (Null((--a.u)+(a.-u)) A 
Nutt( (-(a.b).v)+( (a.b).-v) ) ) 
Similarly, if we know that the determiner is up- 
wards monotone in its second argument 
D(a)(b)v---D(u)(v) has to be true in any model where 
\[al=\[u\] and \[v\]__,\[b\], so the last conjunct will be 
Null((-b.v)) instead of Null((-b.v)+(b.-v)). If the 
determiner is restricted to be both conservative and 
monotone, the last conjunct shall be 
Null((-(a.b).u.v)). Similar modifications of the rule 
can be done for determiners with other forms of 
monotone behaviour. 
GENERALIZED QUANTIFIERS 
One main feature of the decision procedure is that it 
incorporates generalized quantifiers (step 7 and 8). 
The rules for generalized quantifiers correspond to 
axioms one will use in an axiomatization of LM. 
For example, the rule for quantifiers with no addi- 
tional constraints correspond to the extensionality 
schemata: 
For all terms a, b, u, v: 
(All(a)(u) ^ All(u)(a)) ~ (D(a)(b) ~ D(u)(b)) 
(All(b)(v) ^ All(v)(b)) --~ (O(a)(b) ~ O(a)(v)) 
One should remember that we do not try to develop a 
logic for the strong logical interpretation of deter- 
miners like most, but a logic for some minimal 
constraints that interpretations of the determiners 
should at least satisfy. 
- 209 - 
Just like there is a meaning preserving translation 
from LA into first order logic, LM can be translated 
into first order logic extended with generalized quan- 
tiflers. A proof procedure for first order logic, like a 
tableau or a sequent calculus, can be extended with 
rules for generalized quantifiers similar to the rules 
introduced here. If Q is a binary quantifier with no 
additional constraints on its interpretation then the 
following are equivalent. 
O) b Qx(A(x),B(x)) v ~Qx(C(x),D(x)) 
(ii) ~. Vx(A(x) ~ C(x)) ^ Vxf~(x) ~ D(x)) 
So to show that (i) is valid one has to show that (ii) 
is valid. This can be incorporated into a tableau or 
sequent calculus for In'st order logic. If the first or- 
der logic is monadlc, as the logic we get after trans- 
lating LM into first order logic is, one can use a 
similar procedure as the one described here. If the 
extended first order logic is not monadie, the proce- 
dure one gets when rules corresponding to the reduc- 
tion from (i) to (ii) are included, becomes more 
complex. 
EFFICIENCY 
We chose to transform the formula being tested to 
normal form early in the procedure (step 2). 
Alternatively to the described algorithm one could 
think of using a tableau procedure all the way, and 
not first transform to conjunctive normal form in 
step 2. In general, transformation to normal form is 
slower than using a tableau procedure (el. step 6 
above). The reason we made the transformation to 
normal form was that this was necessary to split the 
formula in step 5 and step 7. In the procedure one 
gets by translating LM into first order logic (with 
generalized quantifiers) and using a tableau procedure 
from the start, it is not possible to split the tree 
similarly. If we for simplicity considers a formula 
with no generalized quantifiers, the pure tableau cal- 
culus will not lead to a separate tree for each si to- 
gether with t but to one big tree containing all the 
si's and roughly one copy of t for each si. This cor- 
responds to the quantifier rules in a tableau calculus 
for first order logic: (i) for each formula of the form 
Vxqb introduce one new formula dd(a) where a is 
some new term, (ii) for each formula 3xw introduce 
one new formula V(a) for each term a introduced in 
the tree at a branch to which 3x~ belongs. The 
successful separation in the described algorithm here 
will also be possible in a proof procedure for 
monadic first order logic. 
The two different procedures will be of the same 
time complexity in worst cases. In the practical ap- 
plications we have in mind, the procedure described 
here will be faster. Typically we want to check 
whether a formula 13 follows from al ..... a n. This 
is the same as deciding whether ~tlV...V~nVl3 is 
valid or not. The transformation to normal form 
will produce one additional copy for each v within 
an ai and each ^ within 13. If each ai and \[3 are LM 
formulas that represent English sentences, they can 
each be expected to be relatively short and in 
particular not contain many v's, so the number of 
copies made will be relatively small. On the other 
hand, the number of ai's may be large if they repre- 
sent the dialogue so far or the agent's knowledge. It 
is therefore important that each disjunct can be split 
up as much as possible. 
IMPLEMENTATION 
The inference algorithm has been implemented in 
PROLOG. To test it out we have built a small (toy) 
natural language question-answering-system around 
it. The program reads in simple sentences and ques- 
tions from the terminal and answers the questions. 
It can handle simple statements, like If some of the 
hot coffee that did not disappear was black then 
much gold is valuable (the fragment in Lenning 
1987) and yes/no questions like Did much water 
evaporate? and Was the old gold that disappeared 
valuable? We have written the grammar and transla- 
tion to LM in the built in DCG rules (Pereira and 
Warren, 1980). 
Statements typed on the terminal are interpreted 
as facts about the world and stored as simple sen- 
tences ~bl ..... qb n. When a question like Did much 
water evaporate? is asked, it is parsed and turned into 
a formula like V: Much(water)(evaporate). Then the 
program proceeds to check the validity of 
(Ol^...^On) --) V- If it is valid, the program an- 
swers yes, otherwise it checks (ddl^...^0n) ---) ~V. 
If this is valid, the answer is no, otherwise the pro- 
gram answers that it does not know. When a state- 
ment is made the program checks whether it is con- 
sistent with what the program already knows before 
it is added to the knowledgebase. 
The system is mainly made to test the inference 
algorithm and is not meant as an application by it- 
self. But it illustrates some general points. It is a 
system where natural language inferences are made 
from natural language sentences and not from a fixed 
database. The system contains a complete treatment 
of propositional logic and illustrates a sound treat- 
ment of negation where failure is treated as does not 
know instead of negation. On the other hand, there 
is also a price to pay for incorporating full proposi- 
tional logic. The system can only handle examples 
of a limited size in reasonable time. 
CONCLUSION 
We have here presented a computational approach to 
the semantics of quantified mass noun phrases. We 
think the semantics ascribed to quantified mass 
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nouns through a translation into LM is the one that 
most adequately reflects their particular semantic 
properties. In addition this semantics can be made 
computational in a way not possible for other ap- 
proaches to the semantics of mass terms, like Bum's 
(1985) which extends axiomatic set theory, or 
Link's approach (1983) based on Montague's higher- 
order intensional logic. 
We have modified and adapted a tableau calculus 
to be used with mass terms and extended it with 
generalized quantifiers. Although the imple- 
mentation we have made is of limited applicability, 
we hope that the algorithm can be used to incorpo- 
rate quantified mass noun phrases into larger systems 
treating count terms. In particular, it should be 
possible to combine the algorithm with other ap- 
proaches based on a tableau calculus, like the one 
described by Guenthner, Lehmann and SchOnfeld 
(1986). 
REFERENCES 
Barwise, J. and R. Cooper: 1981, 'Generalized 
Quantifiers and Natural Language", Linguis- 
tics and Philosophy, 4, 159-219. 
Bunt, H.: 1985, Mass terms and model-theoretic 
semantics, Cambridge University Press, 
Cambridge. 
Guenthner, F., H. Lehmann and W. SchOnfeld: 
1986,'A theory for the representation of 
knowledge', IBM J. Res. Develop. 30, 39- 
56. 
Link, G.: 1983,'The Logical Analysis of Plurals and 
Mass terms: A Lattice-Theoretical Approach', 
in Bauerle et al. (eds.), Meaning, Use, and In- 
terpretation of Language, Walter de Gruyter, 
Berlin. 
LCnning, J.T.:1987,'Mass Terms and Quantifica- 
tion', Linguistics and Philosophy, 10, 1-52. 
Pereim, F.C.N. and D.H.D. Warren: 1980, 'Definite 
Clause Grammars for Language Analysis w A 
Survey of the Formalism and a Comparison 
with Augmented Transition Networks', 
Artificial Intelligence 13, 231-278. 
Roeper, R.:1983,'Semantics for Mass terms with 
Quantifiers', Nor.s, 17, 251-265. 
Smullyan, R.M.: 1968, First-Order Logic, Springer, 
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