Outlines of Algebraic Semantics
Pavel Bělíček
‘‘Algebraic semantics’’ is a new
theoretical approach to the formal description of the semantic structure of
natural languages in terms of standard mathematical algebras. Its
interdisciplinary systematics encroaches upon the theoretical fields of
syntactic theory, logic, artificial intelligence as well as theoretical
algebras. Its projections upon the sphere of logical thought can be referred to
as conceptual logic and its intersections with mathematic theories
deserve denoting as notional
algebra.
Contents
Formal Presuppositions of Semantic
Description
Concatenation and Decatenation
Semantic Restriction and Expansion
The Properties of Commutativity
and Associativity
Revisiting String Theory and its
Reaxiomatisation
Axiomatisation of Semantic
Grammars
Semantic Modality and Its Quantification
Quantification and Quantifiers
Quadrivalent, Quadral or
Quaternary Negation
Octovalent, Octal or Octonary
Negation
The first endeavours to
give a formal notation of linguistic strings were due to Axel Thue and Emil
Post1 who adapted their theory
to data processing in Turing machines. An epoch-breaking attempt in their formalisation
was launched by Y. Bar-Hillel, who devised a
quasi-arithmetical notation for deciphering syntactic phrases.2 In
the mid-1950s it developed into categorial grammars that offered a
recognoscative apparatus for evaluating the grammatical correction of sentences
and lexical strings. His analysis of language structures arose
as a by-product of the earliest research in rewriting systems designed for
machine-processing. It provided a decision-making counterpart
to Noam Chomsky’s generative phrase-structure grammars that made the greatest
contribution to modern techniques of artificial intelligence. His results
influenced generations of young researchers and became a headstone of theoretical computer science.
Linguistic studies wend their
way in two directions, one devoted to visible language form and another to the
invisible sphere of meaning-oriented semantics. Their interrelations are not
linked by strict mathematical homomorphism but allow us to speak informally
about approximate mappings. Let there be a natural language L composed
from its alphabet A, vocabulary V and the apparatus G of
grammatical rules. Then it is possible to define algebraic semantics as a
formal system dealing with their mapping f into the realm of semantic
referents. The vocabulary is mapped into the set S of semantic meanings
(sememes), while the grammatical apparatus G is projected upon the schematic
layout C of logical and ontological categories.
f(words) = meanings f: V ® S
f(grammar)
= categories f:
G ® C
Natural languages involve much
polysemy so it is necessary to restrain the reference of words to the kernel
vocabulary of primary literal meanings. This methodological step presupposes
abstracting from infinite varieties of figurative meanings implied by numerous
secondary connotations. This is why algebraic semantics directly switches from
linguistic form to the realm of meaning. For simplicity sake it treats words as
basic sememes in their basic primary elementary sense. When dealing with the
modal meaning of must, may, will, shall, it
considers them right away as sememes and resigns from mentioning the irrelevant
intricacies of their formal lexemes.
The present state of human
cognition may be summed up by concluding that theoretical logic and mathematics
give an exact formalisation of the most essential fields of human thought but
cover only a small part of semantic fields. Even if they give a precise logical
treatment of basic elementary concepts, they do not care to render an integral
description of the whole layout of a given semantic area. They are engrossed
too much in their special internal technicalities that hinder them from joining
their subtheories into an all-inclusive picture of the outer world. Algebraic
semantics works with less rigorous theoretical apparatus but relentlessly
strives to ensure mutual convertibility between semantic, logical, mathematical
and algebraic calculi.
Modern
advances of formal grammars have devised two elementary types of formal
linguistic analysis. One was based on Chomsky’s [[phrase-structure grammar |
phrase-structure grammars]] and their close predecessor, the immediate
constituent analysis proposed by Rulon Wells3. Both approaches treated linguistic structures as linear
sequences of words made up from the vocabulary of a natural language and put
forward useful methods of their hierarchical segmentation. Their chief weakness
was seen in low sensitivity to the mutual subordination of constituents. This
drawback was partly removed by L. Tesnière’s project of [[dependency
grammar | dependency grammars]]4. His verb-centred system focused on semantic actants and
syntactic pairs relating heads and dependents. Their mutual advantages are
elucidated by the comparison5 of two ways of analysing the sentence We are trying to
understand the difference given below.
Table 1.
Dependency and constituency grammars
The chief asset of grammatical trees is that they give
a vivid illustrative representation of syntactic structures for common laic
observers but this is debased by difficulties, which it brings about in
automatic word processing. Hence, a convenient remedy is provided by
parenthetical and fractional grammars.
In formal linguistics it is essential to realise that
the laws of associativity hold neither in lexical nor in syntactic strings.
Their lack and absence advances a strong argument for parenthetisation.
The structuring and inner hierarchy in the following German and English
expressions is much easier to understand from the use of parentheses.
‘‘Parenthetical grammar’’ is a formal
rewriting system that applies parentheses for expressing the grammatical
relations of dependency and semantic subordination. It provides the simplest
method of syntactic parsing without requiring very demanding means of visual
representation. It employs a simple apparatus of left brackets (‘{’, ‘[’ or
‘(’) in order to demark the initial boundary of linguistic expressions and
right brackets (‘}’, ‘]’ or ‘)’) that delimit their end. As seen in the phrase a ladies’ dress parenthetisation induces
considerable differences in meaning:
a ladies’ dress
= a (ladies’ dress) ¹ (a lady’s) dress = a lady’s dress .
The
expression on the left describes a dress for ladies, whereas the phrase
structure on the right refers to a particular lady’s garment.
A simple
example of sentence analysis is given by the collocation Such an extremely
long journey exhausted our energy. Its parenthetical articulation grammar
segments couples of heads and dependents into the ensuing hierarchy:
((((Such (an ((extremely long)
journey))) (exhausted (our energy))).
When rendered in terms of
phrase structures, its decomposition proceeds as follows:
S ® NP VP ® ((AP NP) VP) ® ((Adv AP NP) VP) ® ((D A NP) VP) ® ((D A NP) (V NP)) .
Another telling illustration is
supplied by the string Little Red Riding-Hood went to her grandmother in another
village:
((Little
(Red Riding-Hood))) (went (to (((her grandmother))
(in (another village)))).
The main reason for introducing such
adjustments in syntactic theory is not only that it saves space and simplifies
analysis. Its most important theoretical facility consists in opening the
second dimension of syntactic hierarchy. Parenthetical grammars turn linear
sequences into 2D-patterns embedding strings into a two-dimensional Cartesian
space. Its basic horizontal axis x depicts the linear sequencing of
symbols, while the second vertical axis y plots strings with the scaled
hierarchy of phrase-structures according to different levels of syntactic
validity.6
In
current string theory individual symbols and string are treated as immediate
constituents linked by the binary operation of concatenation. InformalIy
speaking, it is a procedure joining two strings of shorter length into a
concatenation whose length is the sum of both segments. Given two arbitrary
strings S_{1 }= x_{1}...x_{n}
and S_{2 }= y_{1}...y_{m},_{ }their concatenation S_{1}S_{2 }results in the following
formula:
S_{1}S_{2 }= x_{1}...x_{n}y_{1}...y_{m
}.
If
x and y are basic symbols, their logical connective is written in different
algebraic symbols such as
xy = x * y = x × y .
Bar-Hillel’s
theoretical apparatus made use of analogies to arithmetical multiplication,
division and cancellation but such conventions represented only a formal and
artificial apparatus. In fact, they have little to do with properties of rational
numbers featuring in arithmetical fractions. He may have applied also additive
formalism that renders concatenating strings as a sum of two addends. An
elementary case of additive binary [[concatenation]] can be illustrated by
joining two lexical strings composed of several letters as in the formula
below:
town + hall = townhall .
An
inverse operation to concatenation may be denoted as [[decatenation]] – and defined as unlinking chains into short fragments. A
simple illustration of decatenative cancellation is provided by
townhall – hall = town .
Neither
concatenation nor decatenation is a commutative operation. This means that the order of addends
and subtrahends cannot be switched:
town + hall = townhall ¹ hall + town .
This
inconveniency makes us introduce a special symbol Ø for left subtraction:
-town + townhall = town Ø townhall = hall .
The chief argument for giving preference to additive notation for
concatenative strings is that the slash sign for right and left division can be
employed for other purposes such as syntactic dependence.
Some theoretical contributions have developed the idea of ‘right cancellation’ conceived as a string
operation that deletes some symbols on the right end of the string: “The right cancellation of a letter a from a string s is
the removal of the first occurrence of the letter a in
the string s, starting from the right hand side. The empty string is always cancellable:{\displaystyle \varepsilon \div
a=\varepsilon } Clearly, right cancellation
and projection commute.”7
However, it cannot be regarded as identical to the concept of right
decatenation.
The
formal apparatus of parenthetical grammars shares many inadequacies encountered
in immediate constituent analysis. It chains subsequent neighbouring words into
pairs but does not specify their grammatical interrelations expressed by their
mutual syntactic dependency. A convenient solution is offered by the so-called
fractional grammars. They
combine the convenient properties of constituency and dependency by indicating
the subordinate position of dependents by slash signs ‘/’ and ‘\’. This is how it is possible to analyse a simple sentence The
extremely long journey exhausted our energy:
(((The\((extremely\long)\journey))\(exhausted/(our\energy))).
S ® NP\VP ® ((AP\NP)\VP) ® ((Adv\AP))\NP)\VP) ® ((D\(A\NP))\VP) ® ((D ((Adv\A)\NP))\(V/(D\NP)))
.
The right
slash in V/NP means that in accusative object constructions the noun phrase the
NP functions as a dependent of the head V (verb). It is efficient
especially in indicating the syntactic status of incongruent attributes
following the governing nominal head. Its treatment of attribute constructions
is illustrated by the phrase structure the flower of many colours:
(the\flower)/(of(different\colours))
.
NP® (D\N)/NP ® (D\N)/(A\N) .
The replacement
of cancellation by subtraction seems convenient since it permits exploiting
slash marks for designating other important string operations. One possible
usage might serve for designating relations of syntactic dependency. The inner
structure of a word would be comprehensible if we combined dependency with parenthetisation.
The afore-mentioned phrases would beam with clarity and explicitness if they
were segmented neatly by parentheses determining the hierarchy of terms:
Rücksichtslosigkeit » ‘inconsiderateness’ ,
((((Rück\sichts)\los)\ig)\keit)
» ‘(in\((consider)\ate)\ness)’
.
In such lexical derivations suffixes act as the
governing head because they explicitly give the whole expression its categorial and
part-of-speech standing. If a lexical root is preceded by a few prefixes and
appended by several suffixes, we do not consider the order of its etymological
composition but the hierarchy of syntactic values. Etymologically speaking, in
‘boldness’ the adjective ‘bold ’ is primary but in lexical
analysis it is secondary because the part-of-speech value of ‘‘boldness’
is determined by the suffix ‘-ness’.
Grammatical
dependency represents an asymmetric binary relations and as such it needs
denoting by left- and right-oriented symbols. The proposed quotient-like
notation brings advantages but it may sometimes be found confusing. In order to
avoid undesirable arithmetic connotations it is possible to make use of other
such as í, ý. This paper supports the afore-mentioned
fractional notation that opens the chance to record dependency pairs as
fractions. In its notation the phrase yellow glove reads as
yellow
\ glove » a + N = aN
and the object phrase eat sandwiches
as
eat
/ sandwiches » V + N = Vn .
The formula yellow \ glove
says that glove functions as the governing head, while the first
expression yellow is its attribute serving as a dependent. It specifies
the mutual subordination of noun phrases and their attributive dependants.
When considering most attributive
constructions, it is evident that they function like operations of semantic
restriction. The expression young ladies
causes that its reference is narrowed to girls. It means that the class
of all ladies is reduced to the subclass of those ladies that are of younger
age:
a
\ b = c young women = girls .
Its inverse operation is semantic extension,
a^{-1}
+ c = b young^{-1}
girls = women ,
Current string theory arose from
word-processing in Turing automata and got firmly established several
preconceptions worth revisiting. The standard account of string systems in
automata theory has been worked out by John E. Hopcroft and Jeffrey D. Ullman, and hence it deserves denoting as Hopcroft-Ullman axiomatisation.8 Its axiomatic ideas operate well when
applied to interpreting command in specialised artificial programming languages
but break down when tackling the intricate syntax natural languages. They
generalise the algebraic properties of concatenation but forget that in natural
languages this operation does not meet the requirements of commutativity and
associativity.
In
general, algebraic strings include the empty string functioning as the identity
element but they fail to preserve associative laws. Therefore their algebraic
systems could be classified as groupoids, quasigroups, loops or ‘grammoids’,
i.e. systems with non-unique operations.
String algebras
are generally considered as associative systems based on the associative
operation of binary concatenation: “Concatenation of
languages is associative.”9
“Concatenation of strings is associative:
s × (t × u) = (s × t) × u.
For example, ({b} × ({l} (ɛ × {ah}) = {bl}ɛ × {ah} = {blah}.“10 “The strings over an alphabet, with the
concatenation operation, form an associative algebraic structure with identity element the null string—a free monoid.“11
Most
authors admit that „the concatenation of
languages as well as concatenation of words is associative, but not
commutative.“12 However, some mathematicians have elaborated the
theory of special commutative strings, which form an Abelian monoid.13
Commutative monoids are associative monoids with commutative concatenation:
„A monoid whose operation is commutative is called a commutative
monoid (or, less commonly, an abelian monoid).“14
String algebras
are free monoids or free semigroups with an identity element ɛ or Æ: „In abstract algebra, the free monoid on a set is the monoid whose elements are all the finite sequences (or strings) of zero or more elements
from that set, with string
concatenation as the monoid
operation and with the unique sequence of zero elements, often called the empty string and denoted by ε or λ, as
the identity element. The free
monoid on a set A is usually denoted A^{+}. The free semigroup on A is the subsemigroup of A^{+} containing all elements except the empty string. It is
usually denoted A^{+}.15
These standard
outlines of string algebras have been developed by adding a new operation of alternation
that offers the choice of two different elements. Its algebraic properties
resemble logical disjunction or set-theoretical union, and so it can be written
as x È y = z. It is remarkable for distributivity because because
concatenation distributes over alternation: zx È zy = z(x È y). When incorporated into concatenative string systems
extended to free monoids, their whole two-operation algebra can be referred as semi-rings. “Sets of strings with concatenation
and alternation form a semiring, with concatenation (*) distributing over alternation (+); 0 is the empty set and
1 the set consisting of just the null string.”16
These quotations allow
us to conclude that the traditional Hopcroft-Ullman axiomatisation defines
string algebras as non-commutative, associative, unital and distributive
systems:
Æ * x = x * Æ = x (* is a unital operation) ,
x * y ¹ y * x (* is a non-commutative operation) ,
x * (y * z) = (x * y) * z) (* is an associative
operation) ,
z * x È z * y = z * (x È y) (* distributes over È) .
Most algebraic notations for syntactic strings
deal only with very simple sentence structures and find it difficult to analyse
more complex collocations. A considerable improvement of their efficiency can
be reached by several theoretical reforms designed to express the real
algebraic properties of languages structures. The main reason is that when
applying algebraic models of mathematical theorems in automata theory to
natural languages, they have to be adjusted to new axioms. Their algebraic
properties may be verified on examples taken from Modern English or German. In
order to keep their phrases apart from traditional string algebras, it is
advisable to denote them as ‘conceptual strings’ and provide them with a new
‘Conceptual Reaxiomatisation’.
‘Conceptual Reaxiomatisation’ regards conceptual strings as ordered finite polynomials linked by the binary operation of concatenation of additive nature. It is indispensable to clearly distinguish lexical grammars composing words from the set A of letters called alphabet and syntactic grammars concatenating sentence structures from the set V of words called vocabulary. The following examples are predominantly taken from syntactic grammars, where concatenation ‘*’ functions as a unital, non-associative, non-commutative, left-unique and right-unique binary operation. Moreover, it distributes over the logical disjunction ‘È’, whose meaning corresponds to the conjunction ‘or’:
Æ * x = x * Æ = x (* is a unital operation) ,
Æ * people = people * Æ = people (unitality) ,
x * y ¹ y * x (* is a non-commutative operation) ,
school garden ¹ garden school (non-commutativity) ,
x * (y * z) ¹ (x * y) * z) (* is a non-associative
operation) ,
(very fast) train ¹ very
(fast train) (non-associativity) ,
z * x È z * y = z * (x È y) (*
distributes over È) .
(tall men) or (tall women) = tall men or women (distributivity)
.
Let l be a mapping that assigns to
every string its length expressed by the number of its elementary symbols
(letters, digits, words). Then for concatenation and right decatenation of any
strings x, y, z there exists homomorphisms
l(x) + l(y) = l(x * y) = l(z) (the additive nature of concatenation) ,
l(z) – l(y)
= l(z – y)
= l(x) (the subtractive nature of decatenation) .
These partial finding justify summarising
the theory of conceptual strings in natural languages into the following
mathematical usances and theoretical reaxiomatisation:
Let S be a string system with a binary
operation of concatenation ‘×’ over a vocabulary V in a
natural language.
A string system S is unital if there
exists an identity element ε such that for every element s of S the
equations s × ɛ
= ɛ
× s =
s hold good.
The concatenation of strings in S and all
natural languages is not a commutative operation. As a result, all factors in
strings have to preserve their standard ordering.
The concatenation of strings in natural
languages is not an associative operation. Accordingly, its factors in a string
have to be separated by parentheses.
The concatenation of strings in natural
languages is a right-unique operation17.
It means that if the products s × t = z and s × u = z hold in a string system, then t = u.
The concatenation of strings in natural
languages is a left-unique operation. It implies that if s × u = z and t × u = z are
both valid in a string system, then s = t.
If concatenation is a right-unique binary
operation ‘×’ in S, then there exists a binary
operation of right decatenation ‘-’ inverse to ‘×’ in S.
If concatenation is a left-unique binary
operation ‘×’ in S, then there exists a binary
operation ‘Ø’ of left decatenation inverse to ‘×’ in S.
If the binary concatenation s × t = z of strings in natural
languages joins two strings of length f and g and their product z is a string of length h
equal to the sum of f and g, i.e. h = f + g,
then it is an operation of additive type.
If the binary decatenation z – t = s of two strings in natural
languages decreases the length of h of by the length g of t so
that h – g = f, then it is an
operation of subtractive type.
The free groupoid over the vocabulary is
not a free monoid but a free quasigroup.
If a string system represents a free
unital quasigroup where s × ɛ = ɛ × s = s, it functions as a free loop.
The
string system S = [V, *, È], where È operates as the disjunction
operation ‘or’ and distributes over concatenation, S is a quasi-ring.
These axiomatic propositions put forward
a series of structural reforms enhancing the present-day theories of formal
grammars.
(a) parenthetical
notation: replacing the tedious tree graphs of phrase structures by
parentheses,
(b) parenthetical
grammars: parenthesising phrase structures so as to mark their associative
coupling,
(c) decatenative
operations: ensuring the left and right uniqueness of concatenative operations
in
order to introduce their left and
right inverse decatenative operations,
(d) fractional
notation: distinguishing the dependency heads from dependents by quotient
signs,
(e) fractional
grammars: opening the vertical dimension of syntactic hierarchy and plotting it
with the
horizontal string-theoretical
linearity into the two-dimensional space of semantic utterances,
(f) treating
phrase structures as free quasigroups or loops thanks to the uniqueness of
decatenation,
(g) extending
the part-of-speech repertory by adding semantic actants,
(h) designating
the part-of-speech standing by various types of letters,
(i) transcribing
all productions into equations with single-valued operations,
(j) jointing
generative and recognoscative grammars into compatible systems,
(k) decomposing complex sentences into branches of a projective grammar.
Table 2.
Requirement for systematic grammars
More complex phrase structures evade formal
description because their complexity exceeds the potential of a given
apparatus. Many difficulties stem from phrases composed from peripheral and
circumstantial projections bound optionally to noun phases and verb phrases.
Table 18 shows that predication in simple sentence structures gets incapsulated
into various ‘semisentences’ that function as peripheral phrases annexed to the
central predication core. Their syntactic potencies were studied by the Czech
linguist Ivan Poldauf18 who
classified them as forms of semipredication.
Chief semipredicative constructions are developed by
postpositive sequencing and appending them as dependents to the central verbal
core. The whole semipredicative unit usually develops the verb phrase in the
accusative or nominative case form and it is joined to the central verb by
means of postpositive sequencing. This mode of branching semipredicative
phrases is instanced in the third row of Table 18. The lowest fourth row is
reserved for semipredication standing in anteposition and developed by
sequencing from right to left. It is characteristic of attributive
constructions, compounds and derivation where the dependents precede the head.
In the word cowardness the morpheme coward functions as a
stem but it depends upon the suffix -ness that acts the semantic head
determining its overall part-of-speech valence.
structures |
object constructions |
attributive
constructions |
predication |
tribes collect honey |
snails are slow |
apposition participle gerund infinitive clause |
tribes honey-collectors we can see tribes collecting honey discuss their collecting honey this forces tribes to collect honey tribes that collect honey |
snails slow-walkers see snails moving slowly about patients getting better make snails (become)
slow snails that are slow |
attribution compound derivation |
honey-collecting tribes honey-collectors grave-diggers |
slow snails busybody cowardness |
Table 3. Semipredication
in constructions from objects and attributes
Table 3 demonstrates that the simple verb
+ object relation collect honey consists of two actants but
may be incapsulated into a tree-like hierarchy of complex collocations where
the object relation is expressed in alternative phrases. They depend on the
syntactic functions of the verb collect that may act as an apposition,
participle, gerund, infinitive or may be embedded into a clause. Its right side
displays similar collocations inherent to the attributive construction slow snails or the
qualifying construction Snails are slow.
Similar branching
is observed also in adverbials of manner, place and time. They demonstrate
types of predication classified 15 as localisation, temporalisation. The
adverbial constructions Children
sleep well and Londoners
live in London can be
annexed to the central verbal core by reducing their predicative charge to appositive,
participial, gerundial and infinitive semipredication. Other acceptable
transformations crop up when the subsidiary sentence is reduced to a
subordinate clause or when it collapses and forms a new compound or derivative
lexical unit. Table 4 proves that circumstantial adverbial constructions may
occupy almost all syntactic positions that are common to object and complement
collocations in Table 3.
Structures |
adverbial
constructions |
circumstantial
constructions |
predication |
children sleep well |
Londoners live in London |
apposition participle infinitive gerund clause |
animals running fast we saw cats running fast it is difficult to run fast Jane’s coming soon girls that do well |
Londoners based in London we met friends (living) in London it is nice to live in L.A. my staying in London inhabitants who live in London |
attribution compound derivation |
fast-running animals well-to-do, fast-killing dose well-ordered |
London residents wintertime love New Yorkers |
Table 4. Semipredication
derived from adverbial constructions
Quantification is a manner of linguistic
[[predication]] by means of logical [[quantifier | quantifiers]] and adverbials
of quantity. It employs specific means in the grammar of natural languages,
logic, algebra and other branches of mathematics. Introducing quantifiers helps
to give a quantitative gradation to the occurrence and frequency of various
elements in propositional utterances. Aristotelian syllogistics introduced
logical quantifiers “all”, “some”, “no” and modern predicate logic incorporated
two degrees of occurrence, the [[universal quantifier |]] " and the [[existential quantifier]] $. Traditional grammar distinguishes
cardinal, negative, ordinal, multiplicative and partitive numbers. Modern
mathematics has built its disciplinary terminology on several special types of
quantifying concepts, e.g. integers, natural, rational and algebraic [[number |
numbers]]. The principal line of division leads between enumerative and
partitive quantifiers. The former behave as arithmetic systems, the latter
usually observe principles of Boolean logic.
Classical
logic originally operated only with two values, true’ and false. Any proposition was evaluated by truth
functions and given two values. Positive truth values were 1 or T, negative truth values were termed false,
untrue and written 0 or ⊥. For these reasons the
theoretical apparatus of standard logic was classified as a two-valued logic
(also two-level logic or bivalent logic). A deeper look at its opposites suggests that they
are associated with quantifying degrees of veracity and alethic verification.
In expressions true equation and false statement they act
as quantifiers conveying a certain degree of truthfulness.
Efforts to extend the narrow range of two-valued logic
finally resulted in the invention of many-valued logic and fuzzy logic. In this
development there, however, appeared several important milestones worth
noticing. The first turning-point occurred thanks to the explorations into the
realm of three-valued logic (also trivalent or ternary
logic). Its introduction was initiated by Jan Łukasiewicz and S. C. Kleene, who broadened the pair of true and false
by adding the third value of unknown.19 Its disadvantage was that unknown lacked a polar opposite. Its value
can organically function only in the quaternary of four values (known, unknown,
admissible, inadmissible). The lack of coherent consistency vexed also
inventors of four-valued logic. Nuel Belnap devised a four-valued logic that counted
with values true, false, both (true and false), and neither (true nor false). Another
application of four-valued logic was designed for digital circuits and
calculated with values 1 (true), 0
(false), Z (high impedance) and X
(indifference).
A special type of quantification is seen in grammatical and logical
[[negation | negation]]. In most natural languages its essence consists in the
logical complement or potencies of the so-called “null quantifier”. It reports
a null degree of occurrence and existence. In formal logic a proposition p
is negated by symbols ~p, −p or ¬ p (read
in all cases as ‘not p’),
eventually also p¢ (read as p
prime) and p^{-1}. The elementary precept of logic is the
equation ¬¬ p = p stating that the negation of an arbitrary
proposition yields its positive affirmation. This logical axiom was refuted
spitefully by Brouwer’s and Heyting’s
intuitionistic logic20
claiming that ¬¬¬p = ¬p.
Quantifiers and negation can be
regarded as unary relations with one argument. A logic L is said to be
closed under the unary operation of negation if there exists a negation for
each of its elements in L. The basic conception of negative opposites is
seen in pairs such as veil – unveil or go – stay. It forms the common case of dual negation, where
(x^{-1})^{-1 }= x. The
sentence There exist many (q) swimmers (S) seems
to be equivalent to the statement There exist few (q^{-1})
non-swimmers (S^{-}^{1}).
qS
= q^{-1}S^{-1} all
swimmers = no non-swimmers .
The values “true’’ and “false’’ in
two-valued logic form antonyms, they function as opposites that negate another
and the denial of either of them equals its dual antipode. Dual negation forms
pairs of opposites that are linked together by opposite meanings. Negative
antonyms are usually created by adding the prefixes in-, un-, a-,
anti- and counter-. Their polarity is illustrated by pairs such
as true – false, much – little, many – few, majority –
minority, tall – short, high – low, wide – narrow,
large – small or happy - sad.
qT
= (q^{-1}T_{1})^{-1} much (q)
money (T_{1}) = not little (q^{-1})
money ,
qT
= (q^{-1}T_{2})^{-1} many (q)
people (T_{2}) = not few (q^{-1}) people
.
Given two opposite meanings, say much
and little or many and few, they apply a sort of ‘central
delimitation’ because they introduce a sort of relative quantification in
respect to the customary mean. Much and little actually mean
“more than the common average of cases”, while little and few
indicate rarer incidence amounting to “less than the common average of cases”. Little
and few function as negative duals of positive antonyms much
and many.
Quantity can be expressed either in an
enumerative way as in integers and natural numbers or in a holistic manner in
terms of a part of an integral entirety. Generally, in an integral whole there
are four degrees of accomplishment delimiting approximate occurrence: total,
incomplete, partial and zero degree. There always exists the least lower bound
(infimum) and the greatest upper bound (supremum). These bounds are related
either to the whole set or only to some of its individual members.
Quantifiers |
Total degree |
Incomplete degree |
Partial degree |
Zero degree |
class membership |
all girls |
not all girls |
some girls |
no girls |
individual membership |
every girl |
not every girl |
some girl |
no girl |
partitive quantification |
whole/entire |
not entire |
part/partial |
not a part/bit of |
deontic modality |
necessity must |
indolence need not do |
permission may do |
prohibition must not do |
causative modality |
to make |
not to make |
to let |
to forbid |
epistemic modality |
certainty it must be |
uncertainty it may not be |
possibility it may be |
excludedness it can’t be |
action phasing |
to begin |
not to begin |
to continue |
to cease |
possessive phasing |
to get |
not to get |
to keep |
to lose |
phasing of verbs of movement |
to come |
not to come |
to stay |
to leave |
cognitive phasing |
to learn |
not to learn |
to know |
to forget |
Table 5. Fields of
quadrivalent gradation
Quadral quantification resists attempts at
quantitative precision but allows subjective estimates of the whole category of
entities. We may informally specify that for the universal quantifier all x
in a collocation All (") dogs (x) bark (b_{1}) the probability p of p("xb_{1}) = 1, i.e. the event of barking is of the
highest possible probability. This means that the statement conveys a total
degree of truthfulness and indicates that the universal quantifier " pertains to the larger class of total
quantifiers.
An opposite case occurs in the sentence
structure Some dogs bite (b_{2}) where the subject acquires an
existential quantifier $. Its meaning can be rewritten in
algebraic symbols such as $xb_{2} implying that there exists at least one
dog that bites. When observed through the prism of probability theory, the
probability of the event of biting amounts to at least one occurrence, i.e. p($xb_{2}) > 0. This example makes it clear that
the existential quantifier represents as a special subtype of partial
quantifiers.
The usual treatment of logical and
mathematical classifiers seldom discusses the eventuality of negative
quantifiers envisaged in symbols "^{-1}x “not all dogs” and $^{-1}x “no dogs”. The former formula illustrates the subtype
of a ‘non-universal quantifier’ that deserves designating as an incomplete
quantifier "^{-1}. In Aristotelian syllogisms the
expression Not all dogs bite was rephrased periphrastically as All
dogs do not bite because natural languages generally omit special words for
incomplete degrees of quantification. Statistically speaking, the statement "^{-1}xb_{2} with a incomplete
quantifier conveys_{ }probability p("^{-1}xb_{2}) < 1.
There also exists a negative existential quantifier $^{-1 }denying the presence of a given attribute for any
member of a class of individuals. It denotes the zero degree of quantification
and hence its appropriate coinage might be the term of zero quantifier
(or null quantifier). It is exemplified by the collocation No dogs miaow (b_{3}).
It is written as $^{-1}xb_{3} and its probability equals zero since p($^{-1}xb_{3}) = 0.
Introducing a probability function
cannot be refuse as a redundant futility here, because it is almost synonymous
to propositional truthfulness. The denomination of total, incomplete, partial
and zero quantifiers make their concepts acceptable for quantifying degrees of
modality, where the probability function p(x) corresponds to the affiliated functions of feasibility and realisability.
The elementary case of bivalent
quantification with dual negation may be completed by more complex relations of
quadral negation. Its idea is required by Boolean concepts involved in the
pairs of lattice-theoretical operations join – meet or the set-theoretical
operations of union – intersection. It cannot be regarded as an absolute
novelty. George Boole was an Irish mathematician of the mid-19^{th }century
but he had an early predecessor in Aristotle, who introduced quantitative
functors such as all – some and whole – part.
Current accounts of modal logic seldom
realise that the quadripartition of total, incomplete, partial and zero degrees
encroaches also on the field of verbal modality. The only difference is that
quantifiers specify the range of variables, while modal verbs determine the
degree of accomplishment of an action. As a result, the formalism developed for
describing concepts of quantifiers may be extended successfully also to the
sphere of verbal modality that is concerned with determining various degrees of
realisability. It is vital to notice that quadral negation relates
modalities such as must – may (deontic
modal logic), make – let (causative logic) and certainty
– possibility (epistemic logic). Moreover, it may suitably cover the
opposites shall – will and want – agree (volitive logics) or learn –remember
(cognitive logics) and begin – continue (phasing or inchoative
logic).
Quadral or quadrivalent quantification differs from
bivalent antonyms by employing marginal delimitation of quantity.
Whereas dual antonyms much – little measure quantity in respect to the
middle average, the quantifying functors all, some and no
measure the range of variables in respect to boundary cases of universal,
partial or nullary occurrence. Generally speaking, they denote absolute,
universal, occasional or nullary existence. If the dual antonym few is referred to as a negative dual to many,
then it is convenient to denote the quadral antonym may as a quadral (or
quadral negation) of must. Quadrals express the total and partial degree
of realisability of an action.
When we come across an arbitrary lexical unit and we
need to estimate its syntactic potencies, we have to find out its negative
duals and quadrals. If we compare two arbitrary quantifying expressions r
and s, they are dual antonyms if r = (s^{-1})^{ -1 }and they comply with the following equations:
(qx Í S) = (q^{-1}x Í S^{-1}) Many
(q) people are swimmers = Few (q^{-1}) people
are non-swimmers
(q^{-1}x
Í S) = (qx Í S^{-1}) Few
(q^{-1}) people
are swimmers = Many (q) people are non-swimmers
Given two arbitrary quantifying expressions r
and s, they are quadral antonyms if they comply with the following four
equations:
("x Í S) = ($^{-1}x Í S^{-1})^{ } All
people are swimmers = No people are
non-swimmers
("^{-1}x Í S) = ($x Í S^{-1})^{ } Not
all people are swimmers = Some people are non-swimmers
($x Í S) = ("^{-}^{1}x Í S^{-1})^{ } Some people are swimmers = Not all people are non-swimmers
($x^{-1} Í S) = ("x Í S^{-1})^{ } No
people are swimmers = Not all people
are non-swimmers
The
semantic theories of grammatical modality are developed in three independent
lines of research. They have become an issue of utmost interest in theoretical
grammar, general semantics, modal logic, conceptual programming as well as in
artificial intelligence. The most authoritative accounts of linguistic modality
were recently given by F. R. Palmer21,
P. Portner and A. Kratzer22. Their
treatises lack mathematical precision but provide a deeper intuitive
understanding of the inner structuring of semantic categories.
An alternative line of studies is conducted in the branch of modal
logic, whose greatest pioneers were R. Carnap, C. I. Lewis and
W. Quine. Their enquiries contributed a lot to systematising the logical
and algebraic interrelations between various types of modal attitudes. They
pursued different objectives and worked with a different terminological
apparatus. Nevertheless, their ultimate results were mutually comparable and
compatible. This is why the most urgent task to tackle in research consists in
devising interdisciplinary studies enhancing the interactive convertibility of
results in both directions.
One of promising enquiries combining linguistic, logical, mathematical
and computational semantics is found
in the theoretical treatise English Semantics23 (2005) by the Czech Anglicist Pavel
Bělíček. His work endeavoured to clothe linguistic analysis in simple algebraic
formulas and elucidate the mathematical properties of grammatical relations
interweaving the network of semantic fields. When preparing his first textbook
of English lexicology, he derived crucial pioneering ideas from Edward
H. Bendix and his ‘componential analysis of general vocabulary’24. Another inspirative impetus appeared in Lakoff‘s generative
semantics and his semantic equation kill = make die. Both approaches
encouraged him to launch a project of the analytic decomposition of English
word stock into elementary subcategories.
Bělíček’s semantic apparatus united findings of modal
logic with linguistic enquiries and integrated them into a comprehensive scheme
including many neglected areas of modality and modal attitudes. It considered
sixteen types of predication in the indicative mood and coordinated them with
mappings into their respective modal fields. Each of type of predication
induced a special projection upon the level of cognitive C-modality and
volitive V-modality. An independent level of projections was found also in evaluative
E-modality expressing emotional attitudes to reality.
Semantic modality does not quantify variables in
classes of individual members but concerns different degrees of the feasibility
or realisability of actions. The collocation Patricia cooks cakes is
formulated in indicative modality and presents the activity of cooking cakes as
a real thing. On the other hands, the sentence Patricia can cook a cake
specifies it as a desirable activity that is not beyond Patricia’s abilities.
In modal logic modal verbs are treated as modal functors. In Rudolf Carnap’s
enquiries the modal functor N(x) is interpreted as the necessity
of an action x and the modal functor P(x) is introduced in
order to denote its hypothetical possibility.
Linguistic modality expresses a huge variety of human
attitudes to reality. The crucial core is sought in the field of necessity and
deontic logic. The scaling of necessitation distinguishes four degrees: the
total degree specifies necessity (1), the incomplete degree expresses
evitability (<1), the partial degree conveys possibility (>0) and zero
degree equals impossibility (0).
In Czech language modal verbs are negated by the
prefix ne- that induces direct negation because it negates the modal
verb itself. In English and some Germanic languages modal verbs are negated by not
or nicht, which give rise to ‘crossed negation’. Crossed negation in you
must not leave means that not does not negate the modal verb must
but the following non-modal verb leave. The Czech collocation nemusíš
odejít says that your leaving is unnecessary while the English phrase you
must not leave implies that it is forbidden. The discordance between Czech
and English modal negation is explained by parentheses indicating different
articulation of semantic constituents.
Czech musím pracovat ‘I
must work’
English I must work
(ne(musím)) pracovat) ‘I need not work’ I need not
work
smím
si oddechnout ‘I
may relax’ I may relax
((ne(smím)) si oddechnout) ‘I mustn’t relax’ I
(must (not
relax))
Logical properties of modal negation may be
demonstrated by algebraic symbols. Let d designate a verb of action and d^{-}^{1} its antonym calibrated as its dual
negation. Such a pair of opposites is illustrated by the verb d = to
stay and its dual negation d^{-}^{1}
= to leave.
(1) xd^{ }= y^{-1}d^{-1}
he must
stay = he must not leave
(2) x^{-1}d = yd^{-1}
he need not
stay = he may leave
(3) x^{1}d^{-1}
= y^{-1}d he
must leave = he must not stay
(4) x^{-1}d^{-1}=
yd he
need not leave = he may stay
A wide field of application for quadral relationships
is found in phasing logic. It is classified as a branch of temporal
logic (another convenient term
is also time logic), which deals with various temporal aspects of actions on
the axis of time. Phasing logic can clearly distinguish three moments in the
accomplishment of an event or feat: inchoation (verbs
to begin, to commence), continuation (to
continue) or termination (to cease, to finish).
Such a triple of three phases may coordinate verbs of knowledge and
learning. Let us have the grammatical relation begin to know conceived as the concatenative
operation begin * know = learn.
Then verbs of cognition can be axiomatised by means of the following semantic
equations.
to
learn = to begin to know = cease not to know
to
remember = to continue to know = not to forget
to
forget = to cease to know = to begin not to know
Verbs
to begin and to
continue exhibit the relation of quadral negation denoted by the wavy sign
~. Their direct dual negation is (to begin)^{–1} = not to begin and (to continue)^{ –1} = not to continue = to cease.
These pairs of dual compose higher units of quadral negation written as ~continue
= begin or reversely ~begin = continue. This semantic distinction is transferred
also to their complex composites ~learn = remember or
~remember = learn. It also holds goods that ~(~learn) = learn. Pairs continue – cease and remember –
forget are duals
linked by dual negation, while the pairs begin – continue and learn – remember are quadrals linked by quadral negation.
When dealing with
the semantic field of possession, it is advisable to introduce the symbol h
for the verb to have and h^{-1} for its dual antonym to lack. The
state of ownership may arise or perish and its continual changes are expressed
by verbs of phasing. English conveys such changes by infinitive constructions
where the verb h is preceded by the verbs to begin, to
continue and to cease. These verbs express the states of inchoation, continuation and termination and in connection with
the verb to have they compose the semantic content of the possessive
verbs to get, to keep and to lose. Such a method i of lexical
composition and decomposition joins isolated lexical items into an
integrated network entwining the
semantic field of possession.
he
gets = he begins to have
he keeps = he continues to have
he loses = he begins to lack = he ceases to have
Mathematically
speaking, two sememes x and y are Boolean quadrals if they fulfil
the following four equations concerning possesive relations:
(1) xh^{ }= y^{-1}h^{-1}
he gets =
he begins to have = he ceases to lack
(2) x^{-1}h = yh^{-1}
he does
not begin to have = he continues to lack
(3) xh^{-1} = y^{-1}h
he loses =
he begins to lack = he ceases to have
(4) x^{-1}h^{-1}=
yh he
does not begin to lack = he keeps = he continues to have
Such a
comparison of verbs phasing actions makes it clear that the verb keep defines
the total degree (1) of accomplishment, to lose corresponds to the
incomplete degree of possession (<1), to begin functions as the
partial degree of possession (> 0) and not to begin operates as a
zero degree (> 0).
Active and Passive Quadrants
A brief glance at the degrees of phasing segmentation
suggests that phasing logic requires its specific phasing scale of temporal
events. It is beyond any doubt that the main quadral opposites are the verbs to
begin and to continue negated by their dual antonyms not to begin
and to cease. Yet what is questionable is the priority of to
continue in respect to the verb to begin. The main difference is the incomplete degree is
covered by special expressions while the zero degree is expressed
periphrastically and lacks it peculiar lexical means.
These two manners of quantification induce an
important distinction between active and passive gradation. Both fields of
semantic quantification exhibit the same four grades (1)-(4) but one emphasises
active intentional participation while the other relies on submissive
obeisance. Phasing temporal logic lays stress on the doer’s will and resolution. It illustrates the type of active gradation and
should be separated as the active quadrant playing the primary role in a
semantic field. On the other hand, the quadral antonyms all – some
and must – may represent
passive gradation and belong to the secondary passive quadrant of
quantification.
Combining
relations of quadrivalent relations allows us to set up more complex schemes
structuring integrated semantic fields. Their composition makes it possible to
construct composite patterns of octal (or octovalent) negation, which link the less congruent pairs of semantic opposites such as must – will or necessity – volition. It is defined as the relation between the
active and the passive quadrant of a semantic field.
The active
counterpart of classic modality is formed by the modal verbs shall and will
accompanied by their dual negations shan’t and won’t. They differ from must
and may by laying greater stress on human will and resolution. The verbs
must and may presuppose that there exists superordinate authority
commanding the doer to perform prescribed activities. In their modal network
the doer acts as a passive victim of other peoples’ will. On the other hand, will
counts with active participation and efforts to reinforce one’s wishes and
desires. The verb shall exhibits semantic features of submissive
obeisance but poses this modal attitude as a voluntary act complying with
subjective decisions. Shall resembles must in denoting fatal
necessity and will bears resemblance to may in referring to
volitive subjectivity but their pair emphasises the subjective point of view.
A deeper
understanding of semantic shades in modal attitudes is acquired if modal verbs
lose their natural polysemy and they are arranged in narrow juxtaposition with
their respective periphrastic constructions. In the table below the four grades
(1)-(4) compare active and subjective modality in the upper section in
contradistinction to passive and objective modality in the lower section.
Modals verbs |
Periphrases |
Upper active quadrants they shall stay = they will not leave they shall not stay = they will leave they
will stay = they shall not leave
they will not stay = they shall not leave |
they agree to stay = they neglect to leave they do not
agree to stay = they intend to leave they intend to stay = they neglect to leave they do not intend to stay = they refuse to leave |
Lower passive quadrants I must stay = I mustn’t leaveI
needn’t stay = I may leave I
may stay = I needn’t leave I mustn’t stay = I must leave |
I am obliged to stay = I am not allowed to leave I am not obliged to stay = I am allowed to leave I am allowed to stay = I am obliged to leave I am not allowed to stay = I am obliged to leave |
Table 6. Active and passive
modality compared
The whole
network of deontic modality is depicted in a scheme of four quadrant and eight
partitions (Table 7). The left upper quadrant is reserved for conative modality
with will constructions and negative shall-not constructions.
They are linked by a full-line arrow denoting the relation of dual negation. On
the right side this quadrant is complemented by the right upper quadrant
reserved for submissive necessity of active and subjective type. Its lower half
suggests that the category of submissive modals stands in dual opposition in
regard to refutative modality. Both upper quadrants are enclosed in separate
frames linked by horizontal dashed arrows of rightward orientation. Dashed
arrows denote quadral or quadrivalent negation typical of Boolean structures.
The two lower quadrants in the graph
describe passive and objective modality expressed by must and may.
The left lower quadrant positions obligative modality confronted with liberative or dispensabilitive
modality present in the negative need-not constructions. On the other
hand, the right lower quadrant is occupied by permissive
modality implied in the verb may. Its dual negation is provided by must
not indicating prohibitive modality. It is vital to realise that the
afore-mentioned octet of conative, neglective, submissive, refutative,
obligative, dispensabilitive, permissive and prohibitive do not
refer only to particular modal verbs but form large categories that appear also
in the fields of termporal, spacial, functional, behavioral and titulative
semantics.
Table 7 outlines only the field of
deontic V-modality (volitive modality) that prescribes infinitive constructions
or that-clauses with conditionals and conjunctives. A similar graph can
be drawn also for the corresponding C-modality (cognitive modality), where the
modal verbs must, may, will and shall exhibit
different meanings. Their C-modal octet conveys meanings of certainty,
uncertainty, doubt, excludedness and impossibility.
Table 7. The diagram of deontic modality
Table 8. Algebraic symbols
T
Table 9. Essive constructions
Table 10. Spacial Semantics
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