Similarity and contrast relations and inductive rules 
Alistair Knott 
Human Communication Research Centre 
University of Edinburgh 
1 Comparisons as coherence 
relations 
This paper considers how comparison relations can 
be integrated within an RST-like model of discourse. 
We will consider two relations, SIMILARITY (proto- 
typically signalled by connectives like also and too) 
and CONTRAST (prototypically signalled by connec- 
tives like whereas and while). On the face of it. such 
relations should be easy to fit within an account 
like RST's. However, they each exhibit certain id- 
iosyncracies. For instance, the CONTRAST relation 
in RST is unusual in having a multinuclear struc- 
ture, rather than the typical nucleus-satellite struc- 
ture (Mann and Thompson, 1988). SIMILARITY is 
unusual in that its associated connectives can ap- 
parently violate structural constraints of RST, link- 
ing the span in which they appear to structurally 
inaccessible spans, as in the following text: 
(1) I have two brothers. John is a student: he 
majors in history. He likes water polo, and 
he plays the drums. Bill is at high school. 
His main interest is drama. He also studies 
history, but he doesn't like it much. 
Our suggestion for integrating comparison rela- 
tions within a model of discourse relations derives 
from a method of defining relations in terms of 
presupposed defensible rules. This method will be 
outlined for causal/inferential relations, and then 
adapted to comparative relations. 
2 Coherence relations and defeasible 
rules 
Several researchers have used defeasible rules in rela- 
tion definitions. In (Oversteegen, 1995) and (Grote 
et al., 1995), defeasible rules are used for repre- 
senting the CONCESSION relation. In (Knott, 1996; 
Knott and Mellish, 1996), they are used more widely, 
in the definitions of a whole family of causal and ar- 
gumentative coherence relations. We will begin by 
outlining some aspects of this latter account. 
In Knott's account of coherence relations, all re- 
lations with a causal or inferential component pre- 
suppose a defensible rule. For instance, consider the 
following case: 
(2) John was tired, so he went to bed. 
What information does the coherence relation sig- 
nalled by so contribute to the text? It is implausible 
to suggest that it informs the hearer of a new rule of 
inference; in fact it seems necessary that the hearer 
already have the necessary rule to make sense of the 
text. A better analysis is that the hearer is being 
told that this rule of inference succeeds in the sit- 
uation described. We can summarise this idea by 
proposing that a relation of the form P, so Q pre- 
supposes a defensible rule that has P as part of its 
left-hand side, and Q on its right. 
The general framework for presupposed rules is 
given in Figure i. For the present, we interpret 
the connective ~ as the defeasible implication > in 
(Asher and Morreau. 1991)'s logic of commonsense 
entailment. Note that we have abstracted away from 
The relation holds between two propo- 
sitions, X and Y. 
It presupposes the existence of a defea- 
sible rule of the form X A P ~ C. 
Figure 1: Framework for Presupposed Rules 
the simple correspondence between relation and rule 
assumed above: the mapping between Y, P and C is 
determined by the values of two further parameters, 
as explained in Sections 2.1 and 2.2. 
2.1 POSITIVE and NEGATIVE POLARITY 
relations 
One of the parameters we need to speciE' concerns 
whether the defensible rule succeeds or is defeated. 
In Example 2, the rule succeeds, as we have seen; but 
there are also similar cases where a defensible rule 
should be analysed as being defeated. For instance: 
(3) John was tired, but he stayed awake. 
(We can call this relation a CO~CESSlO.X.) If we 
treat the success Or failure of the presupposed rule 
54 
as a parameter to be specified, we can use the same 
framework to represent both examples. The param- 
eter can be called POLARITY; The relevant values are 
given in Figure 2. 
POLARITY 
POSITIVE 
NEGATIVE 
I" is identical to }"; the pre- 
supposed rule succeeds. 
}" is inconsistent with Y': the 
presupposed rule fails. 
Figure 2: The POLARITY parameter 
The POLARITY parameter maps Y onto a new vari- 
able Y'. In Examples 2 and 3, Y' should be identified 
with the conclusion C of the presupposed rule, to 
give the right interpretation. A case where Y' needs 
a different binding is given in the next section. 
2.2 UNILATERAL and BILATERAL relations 
The final parameter we will consider is called PAT- 
TERN OF INSTANTIATION. This feature is motivated 
by examples such as the following: 
(4) John was tired, but there was work to do. 
Clearly we do not want to say that this example pre- 
supposes a rule allowing inference from John being 
tired to there being no work to do. It is preferable to 
envisage a rule with two conditions in its left-hand 
side, saying that if John is tired and there is no work 
to do, he will go to bed. We can then associate the }" 
value with the second premise, rather than with the 
conclusion. The parameter determining the binding 
for Y' is given in Figure 3. 
PATTERN OF INSTANTIATION 
UNILATERAL 1" = P. 
BILATERAL }" = C. 
Figure 3: The PATTERN OF INSTANTIATION param- 
eter 
Example 4 is then analysed as NEGATIVE POLAR- 
ITY UNILATERAL, while Example 3 is analysed as 
NEGATIVE POLARITY BILATERAL. See (Knott, 1996; 
Knott and Mellish, 1996) for a more detailed pre- 
sentation of this framework, along with definitions 
of several more independent parameters. 
3 Comparisons and expectations 
Why might we assume that there is a defeasible rule 
underlying relations of SIMILARITY and CONTRAST? 
An initial piece of evidence comes from contexts such 
as the following, in which a similarity between two 
objects in one respect apparently gives rise to an 
expectation of similarities in other respects. 
(5) \[J0 was made by Richard Page.\] ...This 
brooch was also made by Richard Page. But 
whereas J0 was made in 1985, this brooch 
was made in 1990. 
This compare-and-contrast pattern is common in de- 
scriptive texts. We seem to have here a CONCES- 
SION relation (signalled by but), whose satellite is a 
complex span comprising a SIMILARITY relation, and 
whose nucleus is a complex span comprising a CON- 
TRAST relation: see Figure 4. The question is, why 
..... t 
CONCESSION 
J0 was made This brooch was But whereas this brooch 
by Richard also made by J0 was made was made 
Page. Richard Page. in 1985, in 1990. 
Figure 4: Analysis of the text in Example 5 
do we have a CONCESSION relation between these two 
complex spans? According to the analysis of CON- 
CESSION relations in Section 2.1. the relation must 
presuppose the existence of a defensible rule which 
is being defeated at this point. What could the rule 
be in this case? We would not want to suppose a 
rule that if an object is made by Page, it is likely to 
be made in 1990. Rather, we need a rule that states 
that if two objects are similar in some respect, they 
are also similar in other respects. 
4 Comparisons and inductive rules 
\Ve propose that a similarity relation should be rep- 
resented as triggering such a rule. We suggest that 
similarity, like other relations, presupposes a defeasi- 
ble rule: but that the rule in question is inductive in 
form. In brief, we suggest that a similarity relation 
between two propositions be thought of as permit- 
ting an inductive rule to fire, while a contrast rela- 
tion be thought of as preventing an inductive rule 
from firing. 
In Section 4.1 we outline a 
tire rules, and in Section 4.2 
for comparison relations with 
anism. 
mechanism for induc- 
we frame a definition 
respect to this mech- 
4.1 A recta-level system of inductive rules 
We will represent inductive rules as operating at 
a meta-level on a defeasible first-order logic based 
on commonsense entailment. We envisage inductive 
rules applying as second-order rules, whose con- 
clusions are first-order defensible rules; when an 
inductive rule fires, it results in the addition or al- 
teration of defensible rules within the first-order sys- 
tem. " 
55 
For the moment, we do not want so describe the 
workings of this second-order system in any detail; 
our concern here is really just to consider whether in- 
ductive rules might have a role in explaining certain 
structural properties of comparisons. Nevertheless, 
we will propose a rudimentary model of meta-level 
rules. 
The first-order system we propose is identical 
to commonsense entailment, except that each first- 
order defensible rule is associated with a strength, 
represented by a pair of values s/t, where t is the 
number of times the rule has been triggered and s is 
the number of times it has succeeded. A rule only 
becomes part of the set used in first-order defeasi- 
ble reasoning if its values for t and s/t each reach a 
certain threshold. However, rules with values below 
the threshold can still be used as the preconditions 
for discourse relations. 
We then define a second-order defeasible connec- 
tive >> to model inductive rules. An inductive rule 
has the general form given below: 
(6) VP, a,b P(a) n P(b)(A...) >> ¢ 
where ¢ evaluates to one or more first-order defensi- 
ble rules. When a rule is triggered, the t and s values 
of each of the right-hand side rules are incremented. 
We will be using just two inductive rules, which 
are given in Figure 5. Rule 7 says that given two ob- 
jects in class c which are both P, we can increment 
the strength of the rule that asserts that objects in 
class c are typically P. Rule 8 says that if two ob- 
jects have one property P in common, we should 
increment the strength of all rules that allow us to 
infer properties of one object on the basis of knowing 
properties of the other. 
4.2 Inductive rules in relation definitions 
How might we analyse also as presupposing an in- 
ductive rule? The first thing we need is a new pa- 
rameter for specifying what sort of rule is being pre- 
supposed; this is given in Figure 6. 
RULE TYPE 
1ST-ORDER In the presupposed rule, ~ is 
interpreted as >. 
INDUCTIVE In the presupposed rule, ~ is 
interpreted as >>. 
Figure 6: Definition of the RULE TYPE feature 
We can then define also as signalling a POSITIVE 
POLARITY, UNILATERAL and INDUCTIVE relation be- 
tween the propositions it links. Here is an example: 
(9) Brooch B 1 is ornate. (...) Brooch B2 is also 
ornate. 
According to our model, one of the effects of this re- 
lation is to cause the inductive rule 7 to fire, which in 
turn has the effect of increasing the strength of the 
generalisation that brooches of the class to which 
B1 and B2 belong are typically ornate. This seems 
a plausible effect, particularly in a descriptive con- 
text where a reader/hearer is being informed about 
objects in an unfamiliar domain. 
Now consider a contrast relation, of the kind sig- 
nalled by whereas. On our model, this relation would 
be NEGATIVE POLARITY, UNILATERAL and INDUC- 
TIVE. Here is an example of such a relation: 
(10) Brooch B1 is ornate, whereas Brooch B2 is 
simple. 
According to the model, the effect of this relation 
is simply to leave the s and t values of the gener- 
alisation that 'all brooches of the relevant class are 
ornate' unchanged. This is not particularly satisfac- 
tory; we might want to make it have a more signif- 
icant effect, perhaps by increasing only the t value 
(thereby reducing s/t): but as things are currently 
formulated, changes to the strengths of first-order 
rules are only possible when an inductive rule fires. 
PATTERN OF INSTANTIATION for INDUCTIVE 
relations 
We have seen that the POLARITY parameter ap- 
pears to do useful work for INDUCTIVE relations. 
We should now consider whether the PATTE~-N OF 
INSTANTIATION parameter is productive for tNDUC- 
TIVE relations. Those we have seen so far have all 
been UNILATERAL. What might an INDUCTIVE BI- 
LATERAL relation look like': Such a relation would 
have to hold between two propositions, one being an 
individual proposition, and the other being a gen- 
eralisation for which the individual proposition pro- 
vided inductive support. One possibility is that this 
class of relations are those which can be signalled by 
the connective indeed. Consider the following exam- 
ple: 
(11) This jewel is elaborate. Indeed, most Art- 
Deco jewels are elaborate. 
If we assume that the proposition most Art- 
Deco jewels are elaborate takes as its semantic 
value a first-order defeasible rule of the form 
Vx isa(x, art_deco) > elaborate(x), then it seems 
that we can describe the relation signalled by in- 
deed as POSITIVE POLARITY. INDUCTIVE and BILA.T- 
ERAL. I 
Z Defeasible rules in commonsense entailment are actu- 
ally intended to represent the semantics of generic sentences. 
However, the generalisa.tion introduced by indeed does not 
have to be a generic; the proposed account of indeed under- 
generates in this regard. 
56 
(7) VP. xl,x~.c 
isa(xl, c) A P(xl ) 
A 
isa(x2,c) A P(xo) 
>> \[Vx isa(x,c) > P(x)\] 
(s) 
P(xl) 
VP. xl,x2 A >> \[VQQ(xl)>Q(x2)\] 
P(x2) 
Figure 5: Two meta-level inductive rules 
5 Structural consequences of 
inductive rules 
We will now consider whether the proposed account 
of comparison relations can help us in accounting for 
some of their unusual structural characteristics. 
5.1 Compare-and-contrast structures 
Firstly, consider again Example 5. To recap: what 
we have here is a pair of comparison relations, ap- 
parently linked by a CONCESSIOn' relation, and the 
difficulty is to explain why the CONCESSION rela- 
tion applies. We can begin to account for this ef- 
fect by noting that the initial similarity relation be- 
tween the first two sentences causes Rule 8 to fire 
as well as Rule 7. The effect of Rule 8 firing is to 
add/increment the strength of a whole set of first- 
order rules allowing inference from J0's possession of 
a given property to the brooch's possession of that 
property (and vice versa). One of these rules allows 
an inference from the date of manufacture of one Ob- 
ject to that of the other. The contrast relation in the 
second sentence provides information that explicitly 
states that this inference is not permitted, and must 
result in the newly-added rule being defeated if con- 
sistency is to be preserved. If we can take this to be 
a case where Rule 8 is defeated, which seems plau- 
sible, then we can consider the high-level relation 
signalled by but to be NEGATIVE POLARITY. BILAT- 
ERAL and INDUCTIVE. thereby subsuming it within 
a very general account of the contexts where this 
connective is applicable. 
5.2 Violations of adjacency 
Finally, we can consider whether the account of 
comparisons as presupposing inductive rules pro- 
rides any way of explaining the violation of ad- 
jacency which the similarity relation signalled by 
also appears to permit. Our suggestion here is that 
since the rules presupposed by comparison relations 
are of a different sort from those presupposed by 
causal/inferential relations, it is possible that the 
theorem-proving systems which search for the infer- 
ences that can be drawn from the incoming facts in 
a discourse are different for the two kinds of rules. It 
is uncontroversial that there should be methods for 
constraining the search for inferences to be drawn. 
for both types of rule; in any large system of facts 
and rules there is an explosion of possible infer- 
ences to make. For causal/inferential relations, we 
could postulate that the search for inferences is con- 
strained by the compositional structure of the dis- 
course, and thus influenced by the nucleus-satellite 
structure of its relations; and that it is this which 
leads to the criterion of adjacency being enforced. 
For comparison relations, on the other hand. we 
could imagine different criteria for constraining the 
search space: for instance, we could suggest that the 
search does not take structural prominence into ac- 
count, but is simply limited to the previous n propo- 
sitions. There is no space here to explore this possi- 
bility in an3- detail', however, it seems an interesting 
one to consider. 

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