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

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.

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

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.

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.

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

X produces X. Variables

produce objects of their respective ranges de-

pendently on valuations, they v-construct. Composi-

tion [F A

… A

] is the procedure of applying the

function f produced by F to its arguments produced

by A

, …, A

to obtain the value of f, if any; dually,

Closure [x

… x

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),

X is the construction Trivialization

that constructs X without any change of X.

(iii) Let X, Y

,…,Y

be arbitrary constructions. Then

Composition [X Y

…Y

] is the following con-

struction. For any v, [X Y

…Y

] is v-improper

if at least one of the constructions X, Y

,…,Y

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

,…,Y

. If X does v-construct

such a function, then [X Y

…Y

] v-constructs

the value of this function at the n-tuple.

(iv) (-) Closure [λx

…x

Y] is the following con-

struction. Let x

, x

, …, x

be pair-wise distinct

variables and Y a construction. Then Closure

[λx

…x

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

, …, B

of the respective

ranges of the variables x

, …, x

into the object

(if any) that is v(B

,…,B

)-constructed by

Y, where v(B

,…,B

) is like v except for

assigning B

to x

, …, B

to x

(v) Where X is an object whatsoever,

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,

X is

v-improper.

(vi) Where X is an object whatsoever,

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

X is v-improper. Otherwise

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

X displays X,

X voids the effect of Trivialization

and is thus equivalent to executed X. Below we refer

to this equivalence as to

-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 

, 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:

(types of order 1).

i) Every member of B is an elementary type of order

1 over B.

ii) Let α, β

, ..., β

(m > 0) be types of order 1 over

B. Then the collection (α β

... β

) of all m-ary

partial mappings from β

 ...  β

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).

(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

X are constructions of order n over B.

iii) Let X, X

, ..., X

(m > 0) be constructions of order

n over B. Then [X X

... X

] is a construction of

order n over B.

iv) Let x

, ..., x

, X (m > 0) be constructions of order

n over B. Then [x

...x

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

(i)-(iv).

n+1

(Types of Order n+1) Let 

be the collection of

all constructions of order n over B. Then

i) 

and every type of order n are types of order

n+1.

ii) If m > 0 and , 

, ..., 

are types of order n+1

over B, then ( 

... 

) (see T

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 wt [… 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, [





v-constructs T if B v-constructs the set of all

-objects, otherwise F; [





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 [





x A], [





x A]

we write ‘x A’, ‘x A’ whenever 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

. We use

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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

wt [

Student

John]

The property Student must be extensionalized first,

Student

 () and only then can it be applied to

John, [

Student

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:

[

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 ⊢ [

x B]



(-E)

X ⊢ B(C/x)

The dual rule -I then comes down in this form:

X ⊢ B(y/x)



(-I)

X ⊢ [

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 wt to obtain constructions of truth-

values, and the last steps introduce these wt 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.

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.

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  



[

True

P] v-constructs T if P

, otherwise F;

[

False

P] v-constructs T if P

, otherwise F;

[

Undefined

P] = [

True

P]  [

False

P].

For details on narrow and wide scope negation see (Duží,

2018b) and for answering questions with presuppositions, see

(Duží, Číhalová, 2015).

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

wt [[

True

P]  [

True

Q]];

Q is a presupposition of P iff

wt [[[

True

P]  [

False

P]]  [

True

Q]].

Hence, we have: Q is a presupposition of P iff wt

[[

True

Q]  [

Undefined

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  (

)



. Fur-

thermore, let c/

n+1

 

c  



be a variable ran-

ing over constructions of propositions, a  . Then

the rules for factive propositional attitudes are:

[

a c]

[

a c]



[

True

c][

True

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.

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.

Here we partly draw on material from (Duží et al., 2010,

§4.4).

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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





range over properties, x 

 over individuals.

Then the proof of the rule is this:

1. [[MP]

a] assumption

2. p [[Mp]

a] 1, EG

[x p [[Mp]

x] a] 2, -expansion

4. [w’t’ [x p [[Mp]

w’t’

x]]

a] 3, -expansion

5. M* = w’t’ [x p [[Mp]

w’t’

x]] definition

6. [M*

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]

[M* = wt x p [[Mp]

x]]

(LS) –––––––––––––––––––––––

[M*

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 [

Small p] and [

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/()):

wt x [p [[

Good p]

x]  q [[

Bad q]

x]].

But nobody can be good at something and bad at the

same thing simultaneously:

wt x p [[[

Good

x]  [[

Bad p]

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.

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

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.

(Loukanova, 2009) also warns against unrestricted -re-

duction and its undesirable results.

We analyse Know(ing)/(

)



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.

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/

(



) operates on constructions so that the

Composition [

Sub C

] produces a construction

D that is the result of the collision-less substitution of

the product of C

for the product of C

into C



. The

function Tr/(

) produces the Trivialization of the

-object.

What is also special about “John 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

wt x [

Love

x [

Wife_of

John]]

while the property (ii) of loving one’s own wife is

produced by

wt x [

Love

[

Sub [

Tr x]

[

Wife_of

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

wt

[[[

Sin

] =

0] 

[

Sub [

[[

Sin

] =

0]]

[

Know

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,/; [[

Sin

] =

0]/

 ;

Know/(

)



; John/; it/

 

Note that the result of the substitution (application of

the Sub function) is an adjusted construction [

Know

John

[[

Sin

] =

0]]. But the second argument of

conjunction must be a truth-value; hence, the adjusted

construction must be executedtherefore Double

Execution.

This analysis is fully compositional. The meaning

of “John knows it”wt [

Know

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.

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

/((οι)

τω

(οι)

τω

); Married,

Actor, Student, Fantastic/()



; Member, Like

/(οιι)

τω

; Holidays/; Believe, Think/(

)



; w 

ω;

t  τ; x, y  .

Analysis of the sentences of our scenario comes down

to these constructions:

A.wt [[

Married

Man]

John]

wt [[

Partner-of

John] =

Eve]

C. wt [[

Member

John

SC]  [

Student

John]]

D.wt x [[

Student

x]  [

Holidays]]

wt x [[

Married

x] 

[

Believe

x [

Sub [

Tr [

Partner-of

x]]

[wt [

Fantastic

y]]]]]

F. wt [

Student

Frank]

G. wt [

Think

Frank

[wt

Actor

Peter

]]

A similar stance and solution can be found in (Loukanova,

2012).

Conclusion/question:

Qwt [

Believe

John

[wt [

Fantastic

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

The rule results in

[[

Married

Man]

x ] ⊢ [

Married

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

Believe,

John,

Fantastic and

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. wt [[

arried

]

ohn] 

2. wt [[

Partne

-of

ohn] =

Eve] 

3. wt 

[[

arried

]  [

Believe

[

Sub [

Tr [

Partne

-of

]]

[wt [

antastic

y]]]]]



4. [[

arried

]

ohn] 1, -E

5. [[

Partne

-of

ohn] =

Eve] 2, -E

6. 

[[

arried

]  [

Believe

[

Sub [

Tr [

Partne

-of

]]

[wt [

antastic

]]]]]

3, -E

7. [[

arried

ohn]  [

Believe

ohn [

Sub [

Tr [

Partne

-of

ohn]]

[

t [

antastic

]]]]]

6, -E,

ohn/

8. [[

arried

ohn]  [

Believe

ohn [

Sub [

Eve]

[wt [

Fantastic

y]]]]]

5,7, SI

(Leibnitz)

9. [

arried

ohn] 4. LS

10. [

Believe

ohn [

Sub [

Eve]

[



t [

antastic

]]] 8,9 MPP

11. [

Believe

ohn

[wt [

antastic

Eve]]] 10, Sub, T

12. 

t [

elieve

ohn

[

t [

antastic

ve]]]



11, λ-I

wt [[

Married

Man]

John]

wt [[

Partner-of

John] =

Eve]

C. wt [[

Member

John

SC] 

[

Student

John]]

wt x [[

Married

x] 

[

Believe

x [

Sub [

Tr [

Partner-of

x]]

[wt [

Fantastic

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

;

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/(

)



. Swimming,

Teaching/;

Politician/()



Knowledge base:

wt [

Mayor

Macura]

wt [

Vondrak

Teaching]

C. wt [

Know

Mayor

[wt [

Know

President

[

Sub [

President

]

[wt [

Brussels]]]]]]

D. wt [

President

Snasel]

wt [

Snasel

Swimming]

Fwt [

Politician

Vondrak

]

Question:

Q. wt [

Snasel

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  

c  



; p  



; True/(



)



F1 [

Know

a c] ⊢ 

True



F2 [

Know

a c] ⊢ 

True





True

-E 

True

 ⊢

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Table 2: Top-down derivation of the answer.

wt [

Know

Mayor

[wt [

Know

President

[

Sub [

President

]

[wt [

Brussels]]]]]]



wt [

President

Snasel] 

wt [

Snasel

Swimming] 

w

[

ayo

acura]. 

[

Know

Mayor

[wt [

Know

President

[

Sub [

President

]

[

t [

Brussels]]]]]]

1, -E

6. [

President

Snasel]

2, -E

7. [

Snasel

Swimming]

3, -E

8. [

Mayor

Macura]

4, -E

[

True

[wt



[

Know

President

[

Sub [

President

]

[wt [

Brussels]]]]]]

5, F1

10.

[wt [

Know

President

[

Sub [

President

]

[wt [

Brussels]]]]]

Tru

- E

11.

[wt [

Know

President

[

Sub [

President

]

[wt [

Brussels]]]]]

10,

-E

12.

[

Know

President

[

Sub [

President

]

[



t [

Brussels]]]] 11,



-r

13.

[

True

[

Sub [

President

]

[



t [

Brussels]]]]

12, F2

14.

[

Sub [

Presiden

]

[wt [

Brussels]]]

13,

Tru

– E

15.

[

Sub [

Snasel]

[wt [

Brussels]]]

14,6, SI

16.

[

Sub

Snasel

[wt [

Brussels]]]

15, Tr

17.

[wt [

Snasel

Brussels]]

16, Sub

18.

[wt [

Snasel

Brussels]]

17,

-E

19. [

Snasel

Brussels]

18,



-r

20.

wt [

Snasel

Brussels] 19, -I

For technical reasons, we also need the rule of

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

(

-E)

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

Go,

Snasel and

Brussels. These constituents occur

as sub-constructions of the sentences C, D and E.

C. wt [

Know

Mayor

[wt [

Know

President

[

Sub [

President

]

[wt [

Brussels]]]]]]

Dwt [

President

Snasel]

E. wt [

Snasel

Swimming]

In the sentence C there is another constituent, namely

Mayor, and this same constituent also occurs in the

premise A. By iterating the heuristics, we include A

among the premises as well:

wt [

Mayor

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-

23891S “Hyperintensional 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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