Integrating Special Rules Rooted in Natural Language Semantics into
the System of Natural Deduction
Marie Duží and Michal Fait
VSB - Technical University of Ostrava, Department of Computer Science, Czech Republic
Keywords: Natural Language Processing, Question Answering, Natural Deduction, Transparent Intensional Logic - TIL,
Anaphoric References, Property Modifiers, Factive Verbs.
Abstract: The paper deals with natural language processing and question answering over large corpora of formalised
natural language texts. Our background theory is the system of Transparent Intensional Logic (TIL). Having
a fine-grained analysis of natural language sentences in the form of TIL constructions, we apply Gentzen’s
system of natural deduction to answer questions in an ‘intelligent’ way. It means that our system derives
logical consequences entailed by the input sentences rather than merely searching answers by keywords. Nat-
ural language semantics is rich, and plenty of its special features must be taken into account in the process of
inferring answers. The TIL system makes it possible to formalise all these semantically salient features in a
fine-grained way. In particular, since TIL is a logic of partial functions, it deals with non-referring terms and
sentences with truth-value gaps in an appropriate way. This is important because sentences often come at-
tached with a presupposition that must be true in order that a given sentence had any truth-value. Yet, a
problem arises how to integrate those special semantic rules into a standard deduction system. Proposal of the
solution is one of the goals of this paper. The second novel result is this. There is a problem how to search
relevant sentences in the labyrinth of input text data and how to vote for relevant applicable rules to meet the
goal, i.e. to answer a given question. To this end, we propose a heuristic method driven by constituents of a
given question.
1 INTRODUCTION
Logic and computational linguistics are the disci-
plines that have much in common; in particular, they
should work hand in hand in natural language pro-
cessing and question answering. In the era of infor-
mation overload, the systems that can answer ques-
tions raised over the large corpora of text data in an
‘intelligent’ way gain more and more interest in the
research community. In this paper, we introduce a
system that derives the logical consequences of infor-
mation recorded in the huge knowledge bases of text
data. Thus, the system not only answers the questions
by providing explicit knowledge sought by keywords.
It answers in an ‘intelligent’ way and computes infer-
able knowledge (Duží, Menšík, 2017) such that ra-
tional human agents would produce if only this were
not beyond their time and space capacities. To this
end, we apply Gentzen’s system of natural deduction
1
See, for instance, (Tichý, 1988) or (Duží, Jespersen, Ma-
terna, 2010).
adjusted to our background theory Transparent Inten-
sional Logic (TIL) with its procedural semantics.
1
In
TIL, meanings of natural language sentences are
viewed as abstract structured procedures that produce
Possible World Semantic (PWS-)propositions as their
products. Duží and Horák in (2019) introduce the sys-
tem that applies the goal-driven, backward-chaining
strategy of inferring answers by general resolution
method adjusted for TIL. It seems to be a natural
choice because by applying the goal-driven strategy,
we can easily solve the problem of searching for rel-
evant information resources in the huge labyrinth of
input data. Yet, a problem arises here, namely the
problem of integrating special rules rooted in the rich
natural language semantics into the deduction pro-
cess. These rules include, inter alia, the rules of left
and right subsectivity for property modifiers, the rules
for handling non-referring terms and propositions
with truth-value gaps, the rules dealing with factive
410
Duží, M. and Fait, M.
Integrating Special Rules Rooted in Natural Language Semantics into the System of Natural Deduction.
DOI: 10.5220/0009369604100421
In Proceedings of the 12th International Conference on Agents and Artificial Intelligence (ICAART 2020) - Volume 1, pages 410-421
ISBN: 978-989-758-395-7; ISSN: 2184-433X
Copyright
c
2022 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved
verbs like ‘knowing’ or ‘regretting’, presuppositions
of sentences, de dicto vs de re attitudes, and many
other. TIL with its fine-grained procedural semantics
is the system in which all these semantically salient
features are successfully formalised.
In (Duží, Horák, 2019) and (Duží, Menšík, 2020)
it has been assumed that it is possible to pre-process
the sentences first so that the special semantic rules
are applied prior to the application of a proving
method. Yet, it turned out that such a system is under-
inferring (
Duží, Fait, Menšík, 2019). We have to inte-
grate these special semantic rules into the very pro-
cess of inferring answers. To this end, we vote for
Gentzen’s natural deduction here, because enrich-
ment by special rules seems to be easier for the sys-
tem of natural deduction than for the General Resolu-
tion Method where the input sentences must come in
Skolem Clausal Form.
The goal of this paper and its novel contribution
is to introduce such a system of natural deduction ex-
tended by semantic rules for TIL and natural language
processing. Yet, another problem crops up here,
which is the problem of a proper search strategy in the
huge amount of input data. As mentioned above, in
GRM it was easy to solve thanks to the goal-driven
backward chaining resolution. However, inferring by
natural deduction usually applies a forward-chaining
strategy. Moreover, even if we apply the backward-
chaining strategy, it cannot be strictly goal driven.
Sometimes, the process of satisfying one goal after
another has to be interrupted by an application of a
semantic rule to one or more other constructions, and
only then can we continue the inferential process by
applying standard rules to answer questions. Thus,
another goal of this paper is to introduce a heuristic
method of searching for proper input constructions
driven by constituents occurring in a given query or
goal. Using two case studies, we demonstrate the so-
lutions by an example dealing with property modifi-
ers and an example of dealing with factive verbs and
their presuppositions. In both cases, we also deal with
anaphora resolution.
The rest of the paper is organized as follows. Sec-
tion 2 summarises the main principles of TIL. In Sec-
tion 3 we briefly describe the rules of natural deduc-
tion adjusted to TIL. In Section 4 we introduce the
semantic rules and their formalization in TIL. Section
5 illustrates our method of intelligent question an-
swering by two case studies. Concluding remarks can
be found in Section 6.
2
A kindred theory of procedural semantics has been intro-
duced by Moschovakis in (2006) and further developed by
Loukanova, see, e.g. (Loukanova, 2009).
2 FOUNDATIONS OF TIL
As mentioned above, TIL comes with procedural (as
opposed to set-theoretical denotational) semantics.
Hence, the meaning of a sentence is conceived as an
abstract structured procedure encoded by the sen-
tence, the structure of which is isomorphic with the
structure of the sentence. These procedures can be
viewed as instructions how, in any possible world and
time, to evaluate the truth-value of a sentence.
2
They
are known as TIL constructions. There are six kinds
of such constructions defined, namely variables,
Trivialization, Composition, (-)Closure, Execution
and Double Execution. While variables and Triviali-
zations are atomic constructions that supply objects
on which molecular constructions operate, Composi-
tion and Closure are molecular constructions. Trivial-
ization roughly corresponds to a constant of formal
languages; where X is an object whatsoever of TIL
ontology, Trivialization
0
X produces X. Variables
produce objects of their respective ranges de-
pendently on valuations, they v-construct. Composi-
tion [F A
1
A
m
] is the procedure of applying the
function f produced by F to its arguments produced
by A
1
, …, A
m
to obtain the value of f, if any; dually,
Closure [x
1
x
m
C] is the procedure of declaring or
constructing a function by abstracting over the values
of -bound variables in the ordinary manner of
lambda calculi. Thus, we define.
Definition (Constructions).
(i) Variables x, y, are constructions that con-
struct objects (elements of their respective
ranges) dependently on a valuation v; they
v-construct.
(ii) Where X is an object whatsoever (even a con-
struction),
0
X is the construction Trivialization
that constructs X without any change of X.
(iii) Let X, Y
1
,…,Y
n
be arbitrary constructions. Then
Composition [X Y
1
Y
n
] is the following con-
struction. For any v, [X Y
1
Y
n
] is v-improper
if at least one of the constructions X, Y
1
,…,Y
n
is v-improper, or if X does not v-construct a
function that is defined at the n-tuple of objects
v-constructed by Y
1
,…,Y
n
. If X does v-construct
such a function, then [X Y
1
Y
n
] v-constructs
the value of this function at the n-tuple.
(iv) (-) Closure x
1
x
m
Y] is the following con-
struction. Let x
1
, x
2
, …, x
m
be pair-wise distinct
variables and Y a construction. Then Closure
x
1
x
m
Y] v-constructs the function f that
Integrating Special Rules Rooted in Natural Language Semantics into the System of Natural Deduction
411
takes any members B
1
, …, B
m
of the respective
ranges of the variables x
1
, …, x
m
into the object
(if any) that is v(B
1
/x
1
,…,B
m
/x
m
)-constructed by
Y, where v(B
1
/x
1
,…,B
m
/x
m
) is like v except for
assigning B
1
to x
1
, …, B
m
to x
m
.
(v) Where X is an object whatsoever,
1
X is the con-
struction Single Execution that v-constructs
what X v-constructs. Thus, if X is a v-improper
construction or not a construction as all,
1
X is
v-improper.
(vi) Where X is an object whatsoever,
2
X is the con-
struction Double Execution. If X is not itself a
construction, or if X does not v-construct a con-
struction, or if X v-constructs a v-improper
construction, then
2
X is v-improper. Otherwise
2
X v-constructs what is v-constructed by the
construction v-constructed by X.
(vii) Nothing is a construction, unless it so follows
from (i) through (vi).
From the formal point of view, TIL is a typed
-calculus that operates on functions (intensional
level) and their values (extensional level), as ordinary
-calculi do; in addition to this dichotomy, there is
however the highest hyperintensional level of proce-
dures producing lower-level objects. And since these
procedures themselves can serve as objects on which
other higher-order procedures operate, there is a fun-
damental dichotomy between two modes in which
constructions can occur, namely displayed (as an ob-
ject to be operated on) and executed to v-construct a
lower-level object. In principle, constructions are dis-
played by Trivialization. A dual operation to Trivial-
ization is the construction called Double Execution
that executes constructions twice over. Hence, while
0
X displays X,
20
X voids the effect of Trivialization
and is thus equivalent to executed X. Below we refer
to this equivalence as to
20
-rule.
To avoid vicious circle problem and keep track of
particular logical strata in its stratified ontology, TIL
ontology is organized into a ramified hierarchy of
types built over a base. For natural language pro-
cessing, we use the epistemic base consisting of for
atomic types, namely (the set of truth-values), (in-
dividuals), (times or real numbers) and (possible
worlds). The type of constructions is
n
, where n is
the order of construction.
Definition (Ramified Hierarchy of Types).
Let B be a base, where a base is a collection of pair-
wise disjoint, non-empty sets. Then:
T
1
(types of order 1).
i) Every member of B is an elementary type of order
1 over B.
ii) Let α, β
1
, ..., β
m
(m > 0) be types of order 1 over
B. Then the collection β
1
... β
m
) of all m-ary
partial mappings from β
1
... β
m
into α is a
functional type of order 1 over B.
iii) Nothing is a type of order 1 over B unless it so
follows from (i) and (ii).
C
n
(Constructions of Order n)
i) Let x be a variable ranging over a type of order n.
Then x is a construction of order n over B.
ii) Let X be a member of a type of order n. Then
0
X,
1
X,
2
X are constructions of order n over B.
iii) Let X, X
1
, ..., X
m
(m > 0) be constructions of order
n over B. Then [X X
1
... X
m
] is a construction of
order n over B.
iv) Let x
1
, ..., x
m
, X (m > 0) be constructions of order
n over B. Then [x
1
...x
m
X] is a construction of
order n over B.
v) Nothing is a construction of order n over B unless
it so follows from C
n
(i)-(iv).
T
n+1
(Types of Order n+1) Let
n
be the collection of
all constructions of order n over B. Then
i)
n
and every type of order n are types of order
n+1.
ii) If m > 0 and ,
1
, ...,
m
are types of order n+1
over B, then (
1
...
m
) (see T
1
ii)) is a type of
order n+1 over B.
iii) Nothing is a type of order n+1 over B unless it so
follows from (i) and (ii).
Empirical sentences and terms denote (PWS-)inten-
sions, functions with the domain of possible worlds
; they are frequently mappings from to chronolo-
gies of -objects, hence functions of types
((or

for short. Where variables w, t range
over possible worlds (w and times (t ), re-
spectively, constructions of intensions are usually
Closures of the form wt [… w … t …].
We model sets and relations by their characteris-
tic functions. Hence, (), () are types of a set of
individuals and of a binary relation-in-extension be-
tween individuals, respectively. Quantifiers
,
are type-theoretically polymorphic total functions of
types (()) defined as follows. Where B is a con-
struction that v-constructs a set of -objects, [
0
B]
v-constructs T if B v-constructs the set of all
-objects, otherwise F; [
0
B] v-constructs T if B
v-constructs a non-empty set, otherwise F.
Notational Conventions. That an object X belongs
to a type is denoted as X/’; that a construction C
v-constructs an -object (provided not v-improper) is
denoted by C ’. Instead of [
0
x A], [
0
x A]
we writex A’, x Awhenever no confusion arises.
If C

then the frequently used Composition
[[C w] t], aka extensionalization of the -intension
v-constructed by C, is abbreviated as C
wt
. We use
NLPinAI 2020 - Special Session on Natural Language Processing in Artificial Intelligence
412
classical infix notation without Trivialization for
truth-value functions (conjunction), (disjunction),
(implication) and (negation). Also, identities =
of -objects are written in the infix way without Triv-
ialization and the superscript whenever no confu-
sion arises.
For a simple example, where Student/()

is a
property of individuals and John/an individual, the
sentence “John is a student” encodes as its meaning
the hyper-proposition
wt [
0
Student
wt
0
John]
The property Student must be extensionalized first,
Student
wt
() and only then can it be applied to
John, [
0
Student
wt
0
John] Abstracting over the
values of variables w, t the proposition of type

that
John is a student is produced.
3 NATURAL DEDUCTION IN TIL
The rules of natural deduction adjusted to TIL have
been described in (Duží, Menšík, 2020). Here we just
briefly recapitulate. For a correct application of the
rules of a proof calculus in TIL it is important to real-
ize that the rules are applicable to constituents of a
given construction producing propositions and or
truth-values. As described above, constituents of a
procedure are not the input/output objects on which
the procedure operates; they are beyond the proce-
dure. Rather, constituents of a procedure are its sub-
procedures occurring in executed mode.
When a construction C occurs in the displayed
mode in D, then the construction C itself becomes the
object on which other sub-constructions of D can op-
erate; we also say that the context of its occurrence is
hyperintensional, because all the sub-constructions of
a displayed construction occur neither intensionally
nor extensionally; they are displayed as well. When a
construction C occurs in the executed mode in D, then
the product (if any) of C is the object to be operated
on. In this case the executed construction C is a con-
stituent of its super-construction D.
The rules follow the general pattern of natural de-
duction and are thus introduced in I/E pairs. The rules
dealing with truth-functions, namely conjunction in-
troduction (-I) and elimination (-E), disjunction in-
troduction (-I) and elimination (-E), implication
introduction (-I) and elimination (-E, known also
as modus ponendo ponens MPP) are standard, as in
propositional logic. Additionally, there are rules for
quantifiers (general and existential ). Again, these
additional rules are of two kinds, namely introduction
and elimination rules. Yet, quantifiers in TIL (see
above) are not special symbols; rather, they are func-
tions applicable to classes of objects. Hence, the rules
must be adjusted for the TIL system. Here is how.
Let x,y , B(x) : the variable x is free in B;
[x B] (
), /(()), C . Then general
quantifier elimination in full detail consists of these
steps:
[
0
x B]
[[x B] y] -E
B(y) reduction
B(C/y) substitution
where B(C/y) arises from B by a collision-less, valid
substitution of the construction C for all occurrences
of the variable y in B.
For the sake of simplicity, we write this rule in the
shortened form:
X [
0
x B]

(-E)
X B(C/x)
The dual rule -I then comes down in this form:
X B(y/x)

(-I)
X [
0
x B]
Furthermore, there are rules for -introduction
(-I) and elimination (-E). They are used in particu-
lar when dealing with empirical propositions. Since
in any world w and time t the proof sequence must be
truth-preserving from premises to a conclusion, the
first steps of each such proof are -elimination (-E)
of the left-most wt to obtain constructions of truth-
values, and the last steps introduce these wt again
(-introduction (-I).
4 SEMANTIC RULES
There are many features of the rich semantics of nat-
ural language that must be formalized by special rules
that are not found in the formal logical languages. TIL
is a logical system that has been primarily applied to
the analysis of natural language because it is a pow-
erful system in which almost all the semantically sa-
lient features of a language can be captured by rigor-
ous, fine-grained analysis. Since it is out of the scope
of this paper to deal with all the natural language se-
mantic peculiarities, we refer for details to (Duží, Jes-
persen, Materna, 2010). To illustrate the problems we
have to deal with when building up a question an-
swering system over natural language corpora, we are
Integrating Special Rules Rooted in Natural Language Semantics into the System of Natural Deduction
413
now going to deal with factive verbs and presupposi-
tions triggered by them, property modifiers and ana-
phoric references.
4.1 Factive Attitudes and
Presuppositions
Factive verbs like to ‘know that’, ‘regret that’, ‘be
sorry’, ‘be proud’, ‘be indifferent’, ‘be glad that’, ‘be
sad that’, etc., presuppose that the embedded clause
denotes a true proposition. For, if one asks, “Does
John regret that he came late?” and John did not come
late, there is no direct answer Yes or No. For, both
answers entail that John did come late. In such a case
an appropriate answer conveys information that the
presupposition is not true, like “It is not true that John
regrets his coming late because he did not come late”.
Note that while the direct answer applies narrow
scope negation, the complete answer denies by wide
scope negation.
3
Hence, both John regrets and John
does not regret his coming late entail that John did
come late. If John did not come late, he could neither
regret nor not regret it, the proposition that he regrets
it has a truth-value gap. Schematically, if K is a fac-
tive verb and X its complement clause, the following
rules are valid:
K(X)X,K(X)X.
Factive verbs should be distinguished from im-
plicative verbs like ‘to manage’ or ‘to dare’. While
sentences applying factive verbs presuppose the truth
of the embedded clause, those with implicative verbs
only entail it.
4
Schematically, where I is an implica-
tive verb and X the complement clause, we have the
following rules.
I(X)X,I(X)X.
TIL is a logic of partial function, and as such is apt
for dealing with presuppositions and truth-value gaps.
Yet, partiality, as we all know very well, brings about
technical complications. To manage them properly,
we define properties of propositions True, False and
Undefined, all of type (

)

, as follows (P

):
[
0
True
wt
P] v-constructs T if P
wt
, otherwise F;
[
0
False
wt
P] v-constructs T if P
wt
, otherwise F;
[
0
Undefined
wt
P] = [
0
True
wt
P] [
0
False
wt
P].
3
For details on narrow and wide scope negation see (Duží,
2018b) and for answering questions with presuppositions, see
(Duží, Číhalová, 2015).
4
We are not going to deal with implicative verbs here; yet, see
(Nadathur, 2016), and also (Baglini, Francez, 2016) for
detail. Note however, that the notion of presupposition that
these authors deal with is pragmatic in nature, while we deal
with logical presuppositions the definition of which comes
Now we can rigorously define the difference between
presupposition and a mere entailment. Let P, Q be
constructions of propositions. Then
Q is entailed by P iff
wt [[
0
True
wt
P] [
0
True
wt
Q]];
Q is a presupposition of P iff
wt [[[
0
True
wt
P] [
0
False
wt
P]] [
0
True
wt
Q]].
Hence, we have: Q is a presupposition of P iff wt
[[
0
True
wt
Q] [
0
Undefined
wt
P]]. If a presupposition
of a proposition P is not true, then P has no truth
value.
Factive verbs being a special case of attitudinal
verbs, they thus denote relations-in-intension of an in-
dividual to the meaning of the embedded clause,
which is a construction of a proposition. Hence, if K
is the meaning of a factivum, then K (
n
)

. Fur-
thermore, let c/
n+1
n
,
2
c

be a variable ran-
ing over constructions of propositions, a . Then
the rules for factive propositional attitudes are:
[
0
K
wt
a c]
[
0
K
wt
a c]
[
0
True
w
t
2
c][
0
True
w
t
2
c]
4.2 Property Modifiers
Property modifiers are denoted by adjectives and they
are functions in extension that applied to a root prop-
erty return as a value the modified property. Here we
deal with properties of individuals and modifiers of
such properties of type (()

()

). There are three
basic kinds of modifiers, namely intersective, sub-
sective and privative. Here are the examples.
a) Intersective. “A yellow elephant is yellow and is
an elephant.”
b) Subsective. “A skilful surgeon is a surgeon.
c) Privative. “Forged passport is non-passport.”
We are not going to analyse these modifiers in detail
here. TIL analysis has been introduced in numerous
papers, see, e.g. (Jespersen, Carrara, Duží, 2017),
(Duží, 2017) or (Jespersen, 2015), (Jespersen, 2016).
The issue we deal with bellow is the rule of left sub-
sectivity.
5
The principle of left subsectivity is trivially (by
definition) valid for intersective modifiers. If Jumbo
below. It appears the implicative verbs listed above presup-
pose a weaker version of a presupposition; ‘to manage some-
thing’ presupposes ‘to try that something’ (and a certain dif-
ficulty of the task) and ‘to dare’ presupposes a sort of ‘want’.
We are grateful to an anonymous referee for this note.
5
Here we partly draw on material from (Duží et al., 2010,
§4.4).
NLPinAI 2020 - Special Session on Natural Language Processing in Artificial Intelligence
414
is a yellow elephant, then Jumbo is yellow. Yet how
about the other modifiers? If Jumbo is a small
elephant, is Jumbo small? If you factor out small from
small elephant, the conclusion says that Jumbo is
small. Yet this would seem a strange thing to say, for
something appears to be missing: Jumbo is a small
what? Nothing or nobody can be said to be small
or forged, skilful, good, notorious, or whatnot,
without any sort of qualification. A complement to
provide an answer to the question, ‘a what?’ is
required. We are going to introduce the rule of left
subsectivity that is valid for all kinds of modifiers
including subsective and privative ones. The idea is
simple. From a is an [MP] we infer that a is an M-
with respect to something.
Here is the scheme of defining left subsectivity
rule, SI being substitution of identical properties
(Leibniz’s Law).
(1) a is an MP
assumption
(2) a is an (M something)
1, EG
(3) M* is the property (M something) definition
(4) a is an M*
2, 3, SI
To put the rule on more solid grounds of TIL, let
= ()

for short, M () be a modifier, P
an individual property, [MP] the property
resulting from applying M to P, Further, let =/()
be the identity relation between properties, and let p
v
range over properties, x
v
over individuals.
Then the proof of the rule is this:
1. [[MP]
wt
a] assumption
2. p [[Mp]
wt
a] 1, EG
3.
[x p [[Mp]
wt
x] a] 2, -expansion
4. [wt’ [x p [[Mp]
w’t’
x]]
wt
a] 3, -expansion
5. M* = wt’ [x p [[Mp]
wt
x]] definition
6. [M*
wt
a] 4, 5, SI
Any valuation of the free occurrences of the variables
w, t that makes the first premise true will, together
with step five, make the conclusion true. Left
subsectivity (LS), dressed up in full TIL notation, is
this:
[[MP]
wt
a]
[M* = wt x p [[Mp]
wt
x]]
(LS) –––––––––––––––––––––––
[M*
wt
a].
Additional type: /(()).
This specification of the rule easily dismantles
objections raised against the (LS) principle by Gamut
(1991,
§6.3.11) and Geach (1956). Summarising
briefly, there are three such arguments against (LS).
First Objection. If Jumbo is a small elephant and
a large mammal, then Jumbo is small and large
contradiction! Yet, there is no contradiction, because
Jumbo is small as an elephant and large as a mammal.
Hence the properties p, q with respect to which Jumbo
is a [
0
Small p] and [
0
Large q] are distinct.
The conclusion ought to strike us as being trivial.
If we grant, as we should, that nobody and nothing is
absolutely small or absolutely large, then everybody
is made small by something and made large by
something else. And if we grant, as we should, that
nobody is absolutely good or absolutely bad, then
everybody has something they do well and something
they do poorly. That is, everybody is both good and
bad, which here just means being good at something
and being bad at something else, without generating
paradox (Good, Bad/()):
wt x [p [[
0
Good p]
wt
x] q [[
0
Bad q]
wt
x]].
But nobody can be good at something and bad at the
same thing simultaneously:
wt x p [[[
0
Good
p]
wt
x] [[
0
Bad p]
wt
x]].
The Second Objection is rejected in a similar way.
The argument goes as follows. If Jumbo is a small
elephant and Mickey is a large mouse, then Jumbo is
small, and Mickey is large; hence Jumbo is smaller
than Mickey. Again, to derive the conclusion, it
would have to be granted that Jumbo is small with
respect to the same property as Mickey, which is not
so.
Third Objection. If we do not hesitate to use
‘small’ not only as a modifier but also as a predicate,
then it would seem we could not possibly block the
following fallacy:
Jumbo is small
Jumbo is an elephant

Jumbo is a small elephant.
But we can and must block it, for this argument is
obviously not valid. The premises do not guarantee
that the property p with respect to which Jumbo is
small is identical to the property Elephant.
4.3 Anaphoric References and
Substitution Method
Resolving anaphoric references is a hard nut for every
linguist dealing with the semantics of natural
languages because there are frequently many
ambiguities as for to which part of the foregoing
discourse the anaphoric pronoun refers. Logic cannot
disambiguate any sentence, of course. Instead, logic
can contribute to disambiguation and better
communication by making these hidden features
Integrating Special Rules Rooted in Natural Language Semantics into the System of Natural Deduction
415
explicit and logically tractable. If a sentence or term
is ambiguous, we furnish it with multiple
constructions as its proposed meanings and leave it to
the agent to decide which of these meanings is the
intended one.
To deal with anaphoric references, we apply
generalized Hans Kamp’s Discourse Representation
Theory (DRT), see (Kamp, 1981), (Kamp, Reyle,
1993).DRT is an umbrella term for a collection of
logical and computational linguistic methods
developed for a dynamic interpretation of natural
language, where each sentence is interpreted within a
certain discourse. DRT as presented in (Kamp, 1981)
is a first-order theory. Thus, only terms denoting
individuals (indefinite or definite noun phrases) can
introduce so-called discourse referents, which are free
variables that are updated when interpreting the
discourse.
Since TIL semantics is procedural, hence
hyperintensional and higher-order, not only
individuals, but entities of any type, like properties of
individuals, propositions, relations-in-intension, and
even constructions (i.e., meanings of antecedent
expressions), can be linked to anaphoric variables.
Moreover, the thoroughgoing typing of the universe
of TIL makes it possible to determine the respective
type-theoretically appropriate antecedent, which also
contributes to disambiguation.
6
For instance, the ambiguous anaphoric reference
to properties as in Neale’s example “John loves his
wife and so does Peter” has been analysed in (Duží,
Jespersen, 2013). The authors prove that the sentence
entails that John and Peter share a property. Only that
it is ambiguous which one; there are two options, (i)
loving John’s wife and (ii) loving one’s own wife.
The property predicated of Peter in ‘so does Peter’ is
a function of the property predicated of John in ‘John
loves his wife’. Since the source clause is ambiguous
between attributing (i) or (ii) to John, the target clause
is likewise ambiguous between attributing (i) or (ii)
to Peter. The ambiguity of the anaphoric expression
‘his wife’ as applied to John is visited upon the
likewise anaphoric expression ‘so does’. The authors
propose the analyses of both readings and show that
unrestricted -reduction ‘by name’ reduces both
6
The algorithm for dynamic discourse representation
within TIL has been specified in (Duží, 2018a) and imple-
mented by Kotová, (2018). It is applied in a multi-agent
system to govern the communication of individual agents
by messaging.
7
(Loukanova, 2009) also warns against unrestricted -re-
duction and its undesirable results.
8
We analyse Know(ing)/(
n
)

as a hyperintensional atti-
tude, i.e. the relation-in-intension of an individual to a
hyperproposition (construction of a truth value or a PWS
readings to the strict one on which John and Peter love
John’s wife, which is undesirable.
7
The solution consists in the application of
-reduction ‘by value’ that makes use of the functions
Sub and Tr defined as follows. The function Sub/
(
n
n
n
n
) operates on constructions so that the
Composition [
0
Sub C
1
C
2
C
3
] produces a construction
D that is the result of the collision-less substitution of
the product of C
1
for the product of C
2
into C
. The
function Tr/(
n
) produces the Trivialization of the
-object.
What is also special aboutJohn loves his wife,
and so does Peter” is that it involves two anaphoric
terms, namely ‘his’ and ‘so does’. It might seem
tempting, though, to analyse “John loves his wife” as
though it were synonymous with “John loves John’s
wife”. Then “So does Peter” would unambiguously
attribute to Peter the property of loving John’s wife.
But this analysis would not be plausible as it would
entirely annihilate the anaphoric character of ‘his’.
Instead, the form of the solution must be in terms of
resolution of verb-phrase ellipsis. It needs to be spelt
out which of two properties applies to John in “John
loves his wife” and so applies to Peter in “So does
Peter”.
The property (i) of loving John’s wife is produced by
wt x [
0
Love
wt
x [
0
Wife_of
wt
0
John]]
while the property (ii) of loving one’s own wife is
produced by
wt x [
0
Love
wt
x
2
[
0
Sub [
0
Tr x]
0
y
0
[
0
Wife_of
wt
y]]]
From the logical point of view, anaphoric
pronouns denote variables, valuation of which is
supplied by referring to an appropriate antecedent. To
this end, we developed a substitution method that
exploits the functions Sub and Tr defined above
To adduce an example of referring to the meaning
of a term, i.e. to the encoded construction, the
sentence “Sin of equals zero and John knows it
encodes the following construction as its meaning
8
.
wt
[[[
0
Sin
0
] =
0
0]
2
[
0
Sub [
0
Tr
0
[[
0
Sin
0
] =
0
0]]
0
it
0
[
0
Know
wt
0
John
it]]]
proposition). In case of mathematics it is obvious that such
attitudes must relate an individual to the very procedure
rather than its product; it makes no sense to know a truth
value without any mathematical operation producing it. In
an empirical case intensional attitudes are also thinkable.
Yet, since intensional attitudes inevitably yield a variant
of the well-known paradox of logical/mathematical omnis-
cience, we vote for the hyperintensional analysis here.
NLPinAI 2020 - Special Session on Natural Language Processing in Artificial Intelligence
416
Types. Sin/(); 0,/; [[
0
Sin
0
] =
0
0]/
1
;
Know/(
n
)

; John/; it/
2
1
.
Note that the result of the substitution (application of
the Sub function) is an adjusted construction [
0
Know
wt
0
John
0
[[
0
Sin
0
] =
0
0]]. But the second argument of
conjunction must be a truth-value; hence, the adjusted
construction must be executedtherefore Double
Execution.
This analysis is fully compositional. The meaning
of “John knows it”wt [
0
Know
wt
0
John
it]contains a free variable it as its constituent. If the
sentence is uttered in isolation, the valuation
assignment is a pragmatic matter of a
speaker/interpreter. However, if the sentence is
embedded in the discourse context, the variable it
becomes bound, and the value assignment is provided
by the substitution method.
9
5 TWO CASE STUDIES
5.1 Reasoning with Property Modifiers
Scenario. John is a married man. John's partner is
Eve. John is a member of a sports club and a student.
All students like holidays. Everybody who is married
believes that his/her partner is fantastic. Frank is a
student. Frank thinks that Peter is an actor.
Question. Does John believe that Eve is fantastic?
To formalise our mini knowledge base, we start with
assigning types to the objects that receive mention in
the text:
Types: John, Eve, Peter, Frank, S(port)C(lub)/;
Partner-of/()

; Married
m
/((οι)
τω
(οι)
τω
); Married,
Actor, Student, Fantastic/()

; Member, Like
/(οιι)
τω
; Holidays/; Believe, Think/(
n
)

; w
ω;
t τ; x, y .
Analysis of the sentences of our scenario comes down
to these constructions:
A.wt [[
0
Married
m
0
Man]
wt
0
John]
wt [[
0
Partner-of
wt
0
John] =
0
Eve]
C. wt [[
0
Member
wt
0
John
0
SC] [
0
Student
wt
0
John]]
D.wt x [[
0
Student
wt
x] [
0
Like
wt
x
0
Holidays]]
wt x [[
0
Married
wt
x]
[
0
Believe
wt
x [
0
Sub [
0
Tr [
0
Partner-of
wt
x]]
0
y
0
[wt [
0
Fantastic
wt
y]]]]]
F. wt [
0
Student
wt
0
Frank]
G. wt [
0
Think
wt
0
Frank
0
[wt
0
Actor
wt
0
Peter
]]
9
A similar stance and solution can be found in (Loukanova,
2012).
Conclusion/question:
Qwt [
0
Believe
wt
0
John
0
[wt [
0
Fantastic
wt
0
Eve]]]
To derive the answer, we are going to apply the
system of Gentzen’s natural deduction (ND) adjusted
for TIL. In addition to the standard rules of the ND
system, we need the rule of left subsectivity (LS) for
dealing with the property modifier Married
m
.
The rule results in
[[
0
Married
m
0
Man]
wt
x ] [
0
Married
wt
x]
Informally, this rule represents the fact that “Married
man is married”.
We must also deal with technical rules and
functions specific for TIL. For instance, application
of the functions Sub and Tr must be properly
evaluated, or Leibniz’s law of substitution of
identicals specified for TIL in (Duží, Materna, 2017)
and (Fait, Duží, 2020) must be properly applied.
Table 1 presents the proof. The answer to the
question Q is Yes, of course; it follows from our mini
knowledge base that John indeed believes that Eve is
fantastic.
However, in this proof, we simplified the
situation. We took into account only the premises
relevant for deriving the conclusion, ignoring the
others. For instance, from premises D and F one can
infer (by applying -E and MPP) that “Frank likes
holidays”. Similarly, by applying -E, -E and MPP
to the premises C and D we can infer that John likes
holidays. Yet, these conclusions are pointless for
answering the question Q.
In practice, there are a huge number of sentences
formalised in the form of TIL constructions so that
extracting the relevant ones is not so easy. Moreover,
implementation of the method within the interactive
question answering system calls for an algorithm of
selecting relevant input sentences so that to reduce
inferring consequences that are not needed. To this
end, we propose a simple solution that nevertheless
restricts the number of input premises and thus also
the length of the proofs significantly. We select only
those sentences that talk about the objects that receive
mention in a given question.
In our example, the following constructions
would be selected because they contain the
constituents
0
Believe,
0
John,
0
Fantastic and
0
Eve,
which they have in common with the question Q.
Integrating Special Rules Rooted in Natural Language Semantics into the System of Natural Deduction
417
Table 1: Derivation of the answer.
1. wt [[
0
M
arried
m
0
M
an
]
wt
0
ohn]
2. wt [[
0
Partne
r
-of
w
t
0
J
ohn] =
0
Eve]
3. wt
x
[[
0
M
arried
wt
x
] [
0
Believe
wt
x
[
0
Sub [
0
Tr [
0
Partne
r
-of
wt
x
]]
0
y
0
[wt [
0
F
antastic
w
t
y]]]]]
4. [[
0
M
arried
m
0
M
an
]
wt
0
J
ohn] 1, -E
5. [[
0
Partne
r
-of
w
t
0
J
ohn] =
0
Eve] 2, -E
6.
x
[[
0
M
arried
wt
x
] [
0
Believe
wt
x
[
0
Sub [
0
Tr [
0
Partne
r
-of
wt
x
]]
0
y
0
[wt [
0
F
antastic
w
t
y
]]]]]
3, -E
7. [[
0
M
arried
wt
0
ohn] [
0
Believe
wt
0
J
ohn [
0
Sub [
0
Tr [
0
Partne
r
-of
wt
0
J
ohn]]
0
y
0
[
w
t [
0
F
antastic
w
t
y
]]]]]
6, -E,
0
J
ohn/
x
8. [[
0
M
arried
wt
0
ohn] [
0
Believe
wt
0
J
ohn [
0
Sub [
0
Tr
0
Eve]
0
y
0
[wt [
0
Fantastic
wt
y]]]]]
5,7, SI
(Leibnitz)
9. [
0
M
arried
w
t
0
J
ohn] 4. LS
10. [
0
Believe
w
t
0
J
ohn [
0
Sub [
0
Tr
0
Eve]
0
y
0
[
w
t [
0
F
antastic
w
t
y
]]] 8,9 MPP
11. [
0
Believe
w
t
0
J
ohn
0
[wt [
0
F
antastic
w
t
0
Eve]]] 10, Sub, T
r
12.
w
t [
0
B
elieve
wt
0
J
ohn
0
[
w
t [
0
F
antastic
wt
0
E
ve]]]
11, λ-I
wt [[
0
Married
m
0
Man]
wt
0
John]
wt [[
0
Partner-of
wt
0
John] =
0
Eve]
C. wt [[
0
Member
wt
0
John
0
SC]
[
0
Student
wt
0
John]]
wt x [[
0
Married
wt
x]
[
0
Believe
wt
x [
0
Sub [
0
Tr [
0
Partner-of
wt
x]]
0
y
0
[wt [
0
Fantastic
wt
y]]]]]
The premises D, F and G are irrelevant because they
do not have any constituent in common with the
question Q. This heuristic method does not guarantee
that all the selected constructions are necessary for
deriving the answer (in our case the premise C is
spare), nor that the selected set is sufficient for
deriving the answer. It may happen that in the proof
process the heuristic method must be iterated to select
additional input sentences. Anyway, it turns out that
in most cases one-step heuristic is sufficient, and the
process of proving is effectively optimized.
5.2 Factive Propositional Attitudes
Scenario. The Mayor of Ostrava is Tomáš Macura.
Prof. Vondrák likes teaching. The Mayor of Ostrava
knows that the President of Technical University of
Ostrava (TUO) does not know (yet) that he (the
President of TUO) will go to Brussels. The President
of TUO is prof. Snášel. Prof. Snášel likes swimming.
Prof. Vondrák is a politician.
Question. Will prof. Snášel go to Brussels?
Types: Snasel, Macura, Vondrak, Brussels/ι;
President(-of TUO), Mayor (-of Ostrava)/ι
τω
;
10
For the sake of simplicity, we assign type to these ac-
tivities, because this simplification is harmless to the der-
ivation we are going to demonstrate.
Go/()

; Like/(οι)
τω
; Know/(
n
)

. Swimming,
Teaching/;
10
Politician/()

.
Knowledge base:
wt [
0
Mayor
wt
=
0
Macura]
wt [
0
Like
wt
0
Vondrak
0
Teaching]
C. wt [
0
Know
wt
0
Mayor
wt
0
[wt [
0
Know
wt
0
President
wt
[
0
Sub [
0
Tr
0
President
wt
]
0
he
0
[wt [
0
Go
wt
he
0
Brussels]]]]]]
D. wt [
0
President
wt
=
0
Snasel]
wt [
0
Like
wt
0
Snasel
0
Swimming]
Fwt [
0
Politician
wt
0
Vondrak
]
Question:
Q. wt [
0
Go
wt
0
Snasel
0
Brussels]
What is interesting about this example is that it makes
it possible to demonstrate a top-down derivation from
hyperintensional level of the complement of
knowing/not knowing that “he will go to Brussels” to
the extensional level of Snasel’s going to Brussels. It
is made possible by application of the rules for factive
attitudes defined above, plus the rule for True-
Elimination and resolution of anaphoric references by
the substitution method. To recapitulate, here are the
rules (c
n
,
2
c

; p

; True/(

)

).
F1 [
0
Know
wt
a c] ⊢ 
0
True
wt
2
c
F2 [
0
Know
wt
a c] ⊢ 
0
True
wt
2
c
True
-E
0
True
wt
p
 ⊢
p
wt
NLPinAI 2020 - Special Session on Natural Language Processing in Artificial Intelligence
418
Table 2: Top-down derivation of the answer.
1
wt [
0
Know
wt
0
Mayor
wt
0
[wt [
0
Know
wt
0
President
wt
[
0
Sub [
0
Tr
0
President
wt
]
0
he
0
[wt [
0
Go
wt
he
0
Brussels]]]]]]
2.
wt [
0
President
wt
=
0
Snasel]
3.
wt [
0
Like
wt
0
Snasel
0
Swimming]
4.
w
t
[
0
M
ayo
r
wt
=
0
M
acura].
5.
[
0
Know
wt
0
Mayor
wt
0
[wt [
0
Know
wt
0
President
wt
[
0
Sub [
0
Tr
0
President
wt
]
0
he
0
[
w
t [
0
Go
wt
he
0
Brussels]]]]]]
1, -E
6. [
0
President
wt
=
0
Snasel]
2, -E
7. [
0
Like
wt
0
Snasel
0
Swimming]
3, -E
8. [
0
Mayor
wt
=
0
Macura]
4, -E
9.
[
0
True
wt
20
[wt
[
0
Know
wt
0
President
wt
[
0
Sub [
0
Tr
0
President
wt
]
0
he
0
[wt [
0
Go
wt
he
0
Brussels]]]]]]
5, F1
10.
20
[wt [
0
Know
wt
0
President
wt
[
0
Sub [
0
Tr
0
President
wt
]
0
he
0
[wt [
0
Go
wt
he
0
Brussels]]]]]
w
t
9,
Tru
e
- E
11.
[wt [
0
Know
wt
0
President
wt
[
0
Sub [
0
Tr
0
President
wt
]
0
he
0
[wt [
0
Go
wt
he
0
Brussels]]]]]
w
t
10,
20
-E
12.
[
0
Know
wt
0
President
wt
[
0
Sub [
0
Tr
0
President
wt
]
0
he
0
[
w
t [
0
Go
wt
he
0
Brussels]]]] 11,
-r
13.
[
0
True
wt
2
[
0
Sub [
0
Tr
0
President
w
t
]
0
he
0
[
w
t [
0
Go
wt
he
0
Brussels]]]]
12, F2
14.
2
[
0
Sub [
0
Tr
0
Presiden
t
w
t
]
0
he
0
[wt [
0
Go
wt
he
0
Brussels]]]
w
t
13,
Tru
e
– E
15.
2
[
0
Sub [
0
Tr
0
Snasel]
0
he
0
[wt [
0
Go
wt
he
0
Brussels]]]
w
t
14,6, SI
16.
2
[
0
Sub
00
Snasel
0
he
0
[wt [
0
Go
wt
he
0
Brussels]]]
w
t
15, Tr
17.
20
[wt [
0
Go
wt
0
Snasel
0
Brussels]]
w
t
16, Sub
18.
[wt [
0
Go
wt
0
Snasel
0
Brussels]]
w
t
17,
20
-E
19. [
0
Go
wt
0
Snasel
0
Brussels]
18,
-r
20.
wt [
0
Go
wt
0
Snasel
0
Brussels] 19, -I
For technical reasons, we also need the rule of
20
-
Elimination, a simple technical adjustment, which
holds for any construction C that is typed to
v-construct a non-procedural object of a type of order
1.
(
20
-E)
20
C = C
For the selection of constructions that are relevant for
deriving the answer we now apply the heuristics
described above. Constituents of the question Q are
0
Go,
0
Snasel and
0
Brussels. These constituents occur
as sub-constructions of the sentences C, D and E.
C. wt [
0
Know
wt
0
Mayor
wt
0
[wt [
0
Know
wt
0
President
wt
[
0
Sub [
0
Tr
0
President
wt
]
0
he
0
[wt [
0
Go
wt
he
0
Brussels]]]]]]
Dwt [
0
President
wt
=
0
Snasel]
E. wt [
0
Like
wt
0
Snasel
0
Swimming]
In the sentence C there is another constituent, namely
0
Mayor, and this same constituent also occurs in the
premise A. By iterating the heuristics, we include A
among the premises as well:
wt [
0
Mayor
wt
=
0
Macura].
The proof of the argument, i.e. the derivation of
the answer to the question Q from premises A, C, D
and E can be found in Table 2. Since we proved that
the premises A, C, D and E entail that Snášel is going
to Brussels, the answer to the question Q is YES.
6 CONCLUSION
In this paper, we introduced the system for
‘intelligent’ question answering over natural
language texts. The system derives answers to the
questions as logical consequences of assumptions
extracted from given text corpora. When designing
such a system, one has to solve several problems.
First, natural language sentences must be analysed in
a fine-grained way so that all the semantically salient
features of a language are captured by an adequate
formalization. To this end, we exploited the system of
Transparent Intensional Logic (TIL). Second, there
are special rules rooted in the rich semantics of
natural language which are not found in standard
proof calculi. The problem is how to integrate these
Integrating Special Rules Rooted in Natural Language Semantics into the System of Natural Deduction
419
rules with a given proof system. And the third
problem is how to extract just those sentences that are
needed for deriving the answer from the large corpora
of input text data. There are two novel contributions
of the paper. While in the previous proposals based
on TIL it has been tacitly presupposed that it is
possible to pre-process the natural language sentences
first, and then to apply a standard proof calculus, we
gave up this assumption, because it turned up to be
unrealistic. Instead, we voted for Gentzen’s natural
deduction system so that those special semantic rules
could be smoothly inserted into the derivation process
together with the standard I/E rules of the proof
system. Yet, by applying the forward-chaining
strategy of the natural deduction system, we faced up
the problem of extracting those sentences that are
relevant for the derivation of the answer. As a
solution, we proposed a heuristic method that extracts
those sentences that have some constituents in
common with the posed question.
Future research will concentrate on the comparison
of this approach with the system of deriving answers
by means of the backwards-chaining strategy of
general resolution method and/or sequent calculus,
and an effective implementation thereof. Moreover,
we will also deal with Wh-questions like “Who is
going to Brussels?,When did an American
president visit Prague?”, analyse them and propose a
method of their intelligent answering.
ACKNOWLEDGEMENTS
This research has been supported by the Grant
Agency of the Czech Republic, project No. GA18-
23891SHyperintensional Reasoning over Natural
Language Texts”, and by the internal grant agency of
VSB-Technical University of Ostrava, project No.
SP2019/40, “Application of Formal Methods in
Knowledge Modelling and Software Engineering II”.
Michal Fait was also supported by the Moravian-
Silesian regional program No. RRC/10/2017
“Support of science and research in Moravian-
Silesian region 2017” and by the EU project “Science
without borders” No. CZ.02.2.69/0.0/0.0/16
\_027/0008463. We are grateful to two anonymous
referees for valuable comments that improved the
quality of the paper.
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