# The Lindstrøm’s Characterizability of Abstract Logic Systems
for Analytic Structures Based on Measures

Krystian JOBCZYK and Mirna DŽAMONJA

###### Abstract

In 1969, Per Lindstrøm proved his celebrated theorem characterising the first-
order logic and established criteria for the first-order definability of
formal theories for discrete structures. K. J. Barwise, S. Shelah, J. Väänänen
and others extended Lindstrøm’s characterizability program to classes of
infinitary logic systems, including a recent paper by M. Džamonja and J.
Väänänen on Karp’s chain logic, which satisfies interpolation, undefinability
of well-order, and is maximal in the class of logic systems with these
properties. The novelty of the chain logic is in its new definition of
satisfability. In our paper, we give a framework for Lindstrøm’s type
characterizability of predicate logic systems interpreted semantically in
models with objects based on measures (analytic structures). In particular,
Hájek’s Logic of Integral is redefined as an abstract logic with a new type of
Hájek’s satisfiability and constitutes a maximal logic in the class of logic
systems for describing analytic structures with Lebesgue integrals and
satisfying compactness, elementary chain condition, and weak negation.

Keywords: Lindstrøm Characterizability, Abstract Logic, Analytic Structures,
Measures, Lebesgue Integrals

##  1 Introduction

In Lindstrom (1969), Per Lindstrøm elaborated criteria of the first-order
characterizability of formal theories according to his previous research from
Lindstrom (1966) on the so-called Lindsrøm’s quantifiers. Due to this theorem-
a logical system shares the same expressive power with the first-order
(elementary) logic when both the compactness theorem and downward Skolem-
Loewenheim Theorem hold for it. The purely semantic proof of Lindstrøm’s
theorem initiated research on the so-called abstract model theory -
axiomatically initially depicted in Barwise (1977) and broadly described in
Barwise (1985) and in many other papers and monographs, such as Ebbinghaus
(1985).

In essence, Lindstrøm’s results from the 60s strongly influenced two parallel
research paths in the conceptual framework of abstract model theory. The first
path – renewed in the spirit of results from Lindstrom (1966) – refers to
infinitary logic – very recently discussed in Dzamonja (2021) in some
reference to Shelah’s infinitary logic from Shelah (2012) – and generalized
quantifiers initially introduced in Mostowski (1957) and broadly discussed,
for instance, in Barwise (1981). The second research path refers to the
original Lindstrøm’s theorem itself from Lindstrom (1969) and includes all the
theorem’s reconstruction attempts.

Lindstrøm’s theorem has been extrapolated for various logical systems, for
instance, for modal logic systems as in Benthem (2007); De Rijke (1995).
Simultaneously, a significant part of the proof machinery has been
incorporated into the model theoretical structure of functional analysis - due
to ideas from Henson (1975, 1986). In the same spirit, the Lindstrøm’s type
characterization of analytic structures with a new approximation satisfaction
relation was elaborated in Iovino (2001); Caicedo (2014).

###  1.1 The Paper Motivation and its Objectives

Although the model-theoretic treatment of (even sophisticated) analytic
structures, such as Banach spaces, seems to be well-grounded since the earlier
works from the ’70s, such as: Henson (1975, 1986); Shelah (1978),
Lindstrom’s-type characterization itself of the analytic structures has been
elaborated for a small number of types of the structures, such as continuous
metric spaces Ben Yacov (2008); Caicedo (2014). More precisely, the maximality
logic and Lindstrøm theorem were established here for the so-called Pavelka-
Lukasiewicz logic - due to Hajek (1998). Unfortunately, no attempt at the
first-order characterizability of analytic structures based on measures has
yet been proposed. In particular, we have no information about the appropriate
maximal logic suitable to describe analytic structures with integrals based on
Lebesgue or - more generally - Radon’s measure. Fortunately, a convenient
formal bridgehead for such a construction has already been proposed in Hajek
(1998) in the form of the so-called Rational Pavelka Predicate Logic with
Integrals – for simplicity – to be called Hájek Integral Logic in the paper.

Due to this lack and shortcoming – this paper is aimed at:

$\bullet$

proposing a new abstract logic concept-based depiction of Hájek Integral Logic
(HLI),

$\bullet$

formulating and proving the Lindsrøm’s type theorem for the appropriate
’minimal’ logic associated with the abstract logic – previously defined for
HLI.

This paper forms an extended, and modified version of the conference paper
Jobczyk (2021).

##  2 The terminological framework of the paper analysis

Before we move to the proper part of the paper, we put forward a general
conceptual framework for further analysis. At first - the formal definition of
the abstract logic of a given signature and a couple of the close-related
concepts will be recalled. Secondly, Hájek Integral Logic – as a unique
extension of the so-called Predicate Pavelka-Hájek Logic – will be described
both syntactically and semantically.

###  2.1 Abstract logic and its model-theoretic properties

The formal definition of abstract logic - as a core notion of abstract model
theory - was elaborated by P. Lindstrøm in his famous work Lindstrom (1969).
The current depiction of this concept incorporates an approach from Iovino
(2001).

######  Definition 1.

Alogic $\mathcal{L}$ is a triple
$(\mathcal{K},\,Sent_{\mathcal{L}},\models_{\mathcal{K}})$, where
$\mathcal{K}$ is a class of structures of a given type111P. Lindstrom
considered classical, i.e. discrete structures in Lindstrom (1966). In Caicedo
(2014), continuous metric structures were considered. closed under
isomorphism, renaming and reducts, $Sent_{\mathcal{L}}$ is a function which
assigns to each vocabulary $S$ a set $Sent_{\mathcal{L}}(S)$, i.e. a set of
$\mathcal{S}$-sentences of $\mathcal{L}$ and $\models_{\mathcal{L}}$ is a
satisfaction relation such that the following condition hold:

  1. 1.

$S\subseteq S^{{}^{\prime}}$, then $Sent_{\mathcal{L}}(S)\subseteq
Sent_{\mathcal{L}}(S^{{}^{\prime}})$ (monotonicity of
$Sent_{\mathcal{L}}$-operator),

  2. 2.

If $M\models_{\mathcal{L}}\phi$ (a formula $\phi$ and M remain in
$\models$-relation), then there is a vocabulary $S$ such that M is an
S-structure in $\mathcal{K}$ and $\phi$ is an S-sentence.

  3. 3.

(Isomorphism property) If $M\sim N$ are isomorphic structures in $\mathcal{K}$
and $\phi$ is an S-sentence, then $\phi^{N}=\phi^{M}$.

  4. 4.

(Reduct property) If $S\subseteq S^{{}^{\prime}}$, where both $S$ and
$S^{{}^{\prime}}$ are vocabularies, and $M$ is an $S^{{}^{\prime}}$-structure
in $\mathcal{K}$, then

$M\models_{\mathcal{L}}\phi\,\,\,\mathrm{if\,\,and\,\,only\,\,if}\,\,(M|S)\models_{\mathcal{L}}\phi.$

  5. 5.

(Renaming property) If $\rho:S\to S^{{}^{\prime}}$ is a renaming between
vocabularies $S$ and $S^{{}^{\prime}}$, then for each $S$-sentence $\phi$
there exists $S^{{}^{\prime}}$-sentence $\phi^{\rho}$ such that

$M\models_{\mathcal{L}}\phi\,\,\mathrm{if\,\,and\,\,only\,\,if}\,\,M^{\rho}\models_{\mathcal{L}}\phi^{\rho},$

for each S-structure $M$ in the structure class $\mathcal{K}$222As usual,
$M^{\rho}$ denotes $S^{{}^{\prime}}$-structure obtained from $M$-structure by
its converting through $\rho^{{}^{\prime}}$..

As usual, if $M\models_{\mathcal{L}}\phi$, we say that $\phi$ is satisfied in
$M$ and $M$ is said to be a model for $\phi$.

######  Definition 2.

An abstract logic $\mathcal{L}$ is said to be closed under conjunction if for
all $\mathcal{L}$-sentences $\phi,\psi$ there is an $\mathcal{L}$-sentence
$\phi\wedge\psi$ such that

$M\models_{\mathcal{L}}\phi\wedge
M\models_{\mathcal{L}}\psi\,\,\mathrm{if\,\,and\,\,only\,\,if}\,\,M\models_{\mathcal{L}}\psi\wedge\phi.$

######  Definition 3.

An abstract logic is said to be closed under negation if for all
$\mathcal{L}$-sentences $\phi$:

$M\models_{\mathcal{L}}\neg\phi\,\mathrm{if\,\,and\,\
only\,\,if}\,\,M\not\models_{\mathcal{L}}\phi,$ (1)

for all M-structures in $\mathcal{K}$.

The general definition of abstract logic $\mathcal{L}$ allows us to define a
(formal) theory $T$ of $\mathcal{L}$.

######  Definition 4.

Let $\mathcal{L}$ be an abstract logic and let $S$ be a vocabulary. An
$S$-theory (or simply: a theory) $T$ is a set of all $S$-sentences of
$\mathcal{L}$.

######  Definition 5.

Let $T$ be an S-theory of $\mathcal{L}$. If also $M$ be an S-structure in
$\mathcal{K}$ such that $M\models_{\mathcal{L}}\phi$, for each $\phi\in T$,
then $M$ is a model for $T$, what we denote by $M\models_{\mathcal{L}}T$.

######  Theorem 1.

If $T$ is consistent, then it has a model.

###  2.2 From Łukasiewicz Logic to Hájek Integral Logic

The conceptual tissue of abstract logic in its model-theoretic depiction was
adopted in Caicedo (2014) to describe the Pavelka-Hájek Logic (in both the
propositional and the predicate variant) on a base of Łukasiewicz Logic. In
the second part of this chapter, Hájek Logic of Integrals – as an extension of
the predicate Pavelka-Hájek logic – is presented due to Hajek (1998). Whereas
Pavelka-Hájek Logic in its two variants will be already described in terms of
a conceptual tissue – based on the abstract logic concept – due to Iovino
(2001), Hájek Logic of Integrals alone will be still presented classically –
due to Hajek (1998). The appropriate abstract logic concept-based apparatus
for it will be elaborated in Sections 3 and 4.

####  2.2.1 Pavelka Rational Logic

Predicate Pavelka Rational Logic (PrePRL) constitutes a conservative extension
of (predicate) Infinitely Valued Łukasiewicz Logic – due to Sheperdson (2000)
— and extends this system by associating truth constants for rational in
[0,1].

A similar relation holds between the propositional Infinitely Valued
Łukasiewicz Logic and Pavelka Rational Logic (PRL) as a propositional logic
system, which forms the main subject of this paragraph. This similarity
manifests itself in the way of defining PRL as an abstract logic. In fact, the
appropriate class of structures for PRL is built up from the class of
continuous metric structures being the appropriate class of structures for
Łukasiewicz Logic – due to Caicedo (2014). In addition, predicates of a PRL
language take values from the closed interval $[0,1]$.

Because of the fuzzy nature of PRL – the usual $\models$-relation for an
assignment relation $\mathcal{V}$ will be exchanged for an
$\mathcal{V}$-assignment, which forms a counterpart of the so-called truth
degree defined in Hajek (1998) and plays a role of a ’fuzzy satisfaction
relation’. PRL as an abstract logic is introduced in two steps. At first, its
’surrogate’ in the form of a Weak Pavelka Rational Logic (WPRL) is put forward
as a basis for the proper construction of PRL.

######  Definition 6.

A Weak Pavelka Rational Logic (WPRL)333The name has not been used yet in the
literature, although the current definition is introduced as defining
[0,1]-valued logic in def. 1.10. See: Caicedo (2014), pp.1175-76. is a triple

$(\mathcal{K},\,Sent_{\mathcal{WPRL}},\mathcal{V})$, where $\mathcal{K}$ forms
a class of metric continuous structures closed under isomorphism, renaming and
reducts, $Sent_{\mathcal{L}}$ is a function which assigns to each vocabulary
$S$ a set $Sent_{\mathcal{WPRL}}(S)$, i.e. a set of $\mathcal{S}$-sentences of
$\mathcal{L}$ such that the following condition hold:

  1. 1.

$S\subseteq S^{{}^{\prime}}$, then $Sent_{\mathcal{WPRL}}(S)\subseteq
Sent_{\mathcal{WPRL}}(S^{{}^{\prime}})$,

  2. 2.

A relation $\mathcal{V}$ assigns to each pair $(\phi,M)$, where $\phi$ is
S-sentence and $M$ is S-structure in $\mathcal{K}$, a real number
$\phi^{M}\in[0,1]$.

  3. 3.

(Isomorphism property for WPRL) If $M\sim N$ and they are metrically
isomorphic and $\phi$ is an S-sentence of $\mathcal{WPRL}$, then
$\phi^{M}=\phi^{N}$.

  4. 4.

(Reduct property for WPRL) If $S\subseteq S^{{}^{\prime}}$, where both $S$ and
$S^{{}^{\prime}}$ are vocabularies, and $M$ is an $S^{{}^{\prime}}$-structure
in $\mathcal{K}$ and $\phi$ is an S-sentence, then
$\phi^{M}=\phi^{M\restriction S}$ ($M\restriction S$ denotes a reduct $M$ to
$S$).

  5. 5.

(Renaming property for WPRL) If $\rho:S\to S^{{}^{\prime}}$ is a renaming
between vocabularies $S$ and $S^{{}^{\prime}}$, then for each $S$-sentence
$\phi$ there exists $S^{{}^{\prime}}$-sentence $\phi^{\rho}$ such that
$\phi^{M}=(\phi^{\rho})^{M^{\rho}}$ for each S-structure $M$ in the structure
class $\mathcal{K}$444As usual, $M^{\rho}$ denotes $S^{{}^{\prime}}$-structure
obtained from $M$-structure by its converting through $\rho^{{}^{\prime}}$..

######  Definition 7.

If M is S-structure from $\mathcal{K}$, and $\phi\in Sent_{\mathcal{WPRL}}$,
$\phi^{M}=1$, then we say that M satisfies $\phi$ and write
$M\models_{\mathcal{WPRL}}\phi$.

Instead of $\mathcal{V}$, we will sometimes write $e$ to express the same
assignment, in particular - in the context when the true value of PRL-formulae
is considered555It is not easy to see that this definition of abstract Pavelka
Rational Logic is structurally similar to the Kripke frame-based depiction of
semantics for many fuzzy logic systems, such as S5[0,1]- due to Hajek (1998).
In fact, their semantics is usually given by the triples: $(W,R,e)$, where $W$
is a non-empty set of states, $R$ is an accessibility relation $\subseteq
W\times W$ and $e:\mathcal{L}\to[0,1]$ is an evaluation function, for a given
language of a given fuzzy logic system..

For a complementary definition of PRL on a base of WPRL, one needs to close
the abstract logic $(\mathcal{K},\,Sent_{\mathcal{WPRL}},\,\mathcal{V})$ under
some connectives. The key connective for each [0,1]-valued logic – based on
Łukasiewicz logic – is the so-called Łukasiewicz implication. It forms a
function from $[0,1]^{2}$ into $[0,1]$ defined by the clause:

$x\to_{L}y=\min\\{1-x+y,1\\}.$ (2)

It is noteworthy to state that $x\to_{L}y=1$ if and only of $x\leq y$.

######  Definition 8.

We say a [0,1]-valued logic $\mathcal{L}$ is closed under the basic
connectives if and only if the following condition holds, for each vocabulary
$S$:

  1. 1.

If $\phi,\psi\in Sent_{\mathcal{L}}(S)$, then there exists a sentence
$\phi\to_{L}\psi\in Sent_{\mathcal{L}}(S)$ such that $(\phi\to_{L}\psi)^{M}$
if and only if $\phi^{M}\to_{L}\psi^{M}$, for every $S$-structure $M$.

  2. 2.

For each element $r\in[0,1]$, the set $Sent_{\mathcal{L}}(S)$ contains a
sentence with constant truth value $r$ to be called the constants of
$\mathcal{L}$ and denoted by $\bar{r}$, i.e. $e(\bar{r})=r$.

Having defined closeness under basic connectives we are in a position to
define Pavelka Rational Logic as an abstract logic as follows.

######  Definition 9.

A Pavelka Rational Logic(PRL) is a triple
$(\mathcal{K},\,Sent_{\mathcal{WPRL}},\,\mathcal{V})$ defined as in Def. 6 and
closed under basic connectives.

From the perspective of [0,1]-evaluation of formulae, the following
observation seems to be noteworthy.

######  Remark 1.

Let M be an S-structure for abstract logic $\mathcal{L}=\mathcal{PRL}$. Then
for any $\phi\in Sent_{\mathcal{PRL}}(S)$ hold:

  1. 1.

$M\models_{\mathcal{L}}\phi\to_{L}\bar{r}\iff\phi^{M}\to r$ (alternatively:
$e(\bar{r}\to_{L}\phi)=1\iff e(\phi)\geq r$).

  2. 2.

$M\models_{\mathcal{L}}\bar{r}\to_{L}\phi\iff r\to_{L}\phi^{M}$
(alternatively: $e(\phi\to_{L}\bar{r})=1\iff e(\bar{r})\geq\phi$).

Note that for every $S$-structure $M$ it holds666These properties correspond
to the axiom-based depiction of PRL - due to Hajek (1998):  1.
$(\bar{r}\to\bar{s}\equiv\bar{r\Rightarrow s})$ 2.
$\neg\bar{r}\equiv\bar{1-r}$ :

$\displaystyle(\neg\phi)^{M}=1-\phi^{M},\,\,(\phi\vee\psi)^{M}=\min\\{\phi^{M},\psi^{M}\\},\,\,(\phi\wedge\psi)^{M}=\max\\{\phi^{M},\psi^{M}\\}.$

It is noteworthy to underline that the truth value of each $\phi\in
Sent_{\mathcal{PRL}}$ is determined either by the set

$\\{r\in\mathcal{Q}\cap[0,1]:M\models_{\mathcal{PRL}}\phi\to_{L}\bar{r}\\}=\\{r\in\mathcal{Q}\cap[0,1]:e(\bar{r}\to_{L}\phi)=1\\}$
(3)

or by the set

$\\{r\in\mathcal{Q}\cap[0,1]:M\models_{\mathcal{PRL}}\bar{r}\to_{L}\phi\\}.$
(4)

It relies on the observation that truth value of the formula (by fuzzy
evaluation) is not too false, i.e. its truth value is at least equal to $r$
(if $\bar{r}\to_{L}\phi$). Similarly, if $\phi\to_{L}\bar{r}$, then the truth
value of $\phi$ is not too true, i.e. its truth value is at most equal to $r$.

####  2.2.2 Predicate Pavelka Rational Logic (PrePRL)

Predicate Pavelka Rational Logic (PrePRL) forms a predicate enhancement of
Pavelka Rational Logic, obtained from PRL if exchanged its propositional
language is the predicate one. In this paragraph, we intend to provide an
abstract logic-based depiction of PrePRL constructing the appropriate class of
structures and a new set of sentences – due to the taxonomy of terms and
formulae in each predicate language. The construction itself will be proceeded
by explaining the direction of the construction and the nature of the
constructed structures.

I. A couple of explanatory remarks. Before we move to details of the
construction, some explanatory remarks concerning the direction of this
construction and the nature of the structures themselves should be previously
made.

  1. 1.

While a majority of two-valued predicate logics include the predicate "="
among their formulae – interpreted semantically by identity relation, a multi-
valued predicate logic system should include a sign "$\approx$" intentionally
interpreted in semantics by similarity relation as fuzzy identity. PrePRL
forms an example of such a system.

  2. 2.

Meanwhile, we generally have two possibilities to formally grasp the
’=’-relation symbol in a predicate language. The first way relies on listing
its formal properties: reflexivity, symmetry and transitivity – as this way
stems from the observation that it forms just an equivalence relation. The
alternative, stronger way relies on defining the appropriate
congruences888Note that each congruence is a unique equivalence relation. It
justifies why this method may be described as the stronger one. for formulae
of the predicate language.

More precisely, for variable symbols $x_{1},\ldots x_{n}$ and
$y_{1},\ldots,y_{n}$, the ’=’-relation symbol is described by the following
conjunction of congruences for each $n$-ary operation symbol $f$:

$(x_{1}=y_{1}\wedge\ldots
x_{n}=y_{n})\to(f(x_{1},\ldots,x_{n})=f(y_{1},\ldots,y_{n}).$ (5)

and for $n$-ary predicate $P$:

$(x_{1}=y_{1}\wedge\ldots
x_{n}=y_{n})\to(P(x_{1},\ldots,x_{n})\to_{L}P(y_{1},\ldots,y_{n})).$ (6)

The same concept of congruence – as a fuzzy counterpart of identity – may be
adopted to grasp $\approx$-relation symbol – semantically interpreted by
similarity relation as follows:

$(x_{1}\approx y_{1}\wedge\ldots x_{n}\approx
y_{n})\to_{L}(f(x_{1},\ldots,x_{n})\approx f(y_{1},\ldots,y_{n}).$ (7)

and for $n$-ary predicate $P$:

$(x_{1}\approx y_{1}\wedge\ldots x_{n}\approx
y_{n})\to_{L}(P(x_{1},\ldots,x_{n})\approx P(y_{1},\ldots,y_{n})).$ (8)

II. PrePRL as an abstract logic. In order to provide an abstract logic
concept-based depiction of PrePRL, let us repeat that PRL is semantically
represented by continuous metric spaces with the 1-Lipschitz condition (see:
Caicedo (2014), pp.1173, 1179.)999The authors of the paper say about
Łukasiewicz-Pavelka Logic..

######  Definition 10.

The 1-Lipschitz condition. Let $S$ be a class of continuous metric structures,
and let $M$ be such an $S$-structure. For all $n$-ary predicates $P$, all
$n$-ary function symbols $f$, $\bar{a}=a_{1},\ldots a_{n}\in M$,
$\bar{b}=b_{1},\ldots b_{n}\in M$

$d(f(\bar{a}),f(\bar{b})),\leq\sup_{1\leq i\leq n}d(a_{i},b_{i}),$ (9)
$d(P(\bar{a}),P(\bar{a}))\leq\sup_{1\leq i\leq n}d(a_{i},b_{i}).$ (10)

Meanwhile, the similarity relation $x\approx y$ corresponds to the distance
$d(x,y)$ such that $d(x,y)=1-x\approx y$, i.e. both relations are their mutual
negations101010It is explainable in the light of the observation that a large
distance between two elements $x$ and $y$ means a small similarity between
them and conversely.. It infers a unique similarity-based Lipschitz condition
in the following form.

######  Definition 11.

The Similarity-based Lipschitz Condition111111In fact, we could name this
property as an anti-contraction. Let $f$ be a function symbol, S - be a class
of structures with similarity relation $\approx$, $M$ be an S-structure and
$\bar{a}=a_{1},\ldots,a_{n}\in M$ and $\bar{b}=b_{1},\ldots,b_{n}\in M$. Then
it holds:

$(\bar{a}\approx\bar{b})\leq\inf_{1\leq i\leq n}\\{f(a_{i})\approx
f(b_{i})\\}.$ (11)

As a result, the adequate $S$-class for PrePRL is a class of continuous,
similarity-based Lipschitz structures.

Having already established the class of the adequate structures for PrePRL, we
are in a position to introduce a set of PrePRL-sentences
$Sent_{\mathcal{PrePRL}}$ – defined similarly like a set of formulae of a
predicate language. More precisely, we will identify $Sent_{\mathcal{PrePRL}}$
with a set of formulae of $\mathcal{L}(PrePRL)$, which is built up from terms
of $\mathcal{L}(PrePRL)$, as usual.

######  Definition 12.

(Terms of $\mathcal{L}(PrePRL)$. Let us assume that a class of structure (of a
type) $S$ is given and let I be a non-empty set of indices. The set of
$S$-terms $Tm_{\mathcal{L}(PrePRL)}$ of $\mathcal{L}(PrePRL)$ is defined
inductively as follows:

$Tm_{0}=X\cup C,$ (12)

where $X$ is a finite set of variables and $C$ is a set of constants,

$Tm_{l+1}=Tm_{l}\cup\\{\langle f_{j}(t_{1},\ldots,t_{m})\rangle\\},$ (13)

where $f_{j}$ is an $j$th function symbol, for $j\in I$, and

$Tm=\bigcup_{l\in\mathbb{N}}Tm_{l}.$ (14)

######  Definition 13.

(Formulae of $\mathcal{L}(PrePRL)$). Let us assume that a class of structures
(of a type $S$) with an $S$\- model $M$ is given, let $I$ be a (non-empty) set
of indices and let $P_{i}$ be $i$th $n$-ary predicate and $\widehat{r}$ forms
a constant symbol for a rational $r$. The set $Form_{\mathcal{L}(PrePRL)}$ is
defined inductively as follows:

$\displaystyle Fm_{0}=\\{\langle t,\approx,s\rangle:t,s\in
Tm_{\mathcal{L}(PrePRL)}\\}\cup$
$\displaystyle\cup\\{P_{i}(t_{1},\ldots,t_{n}):i\in
I\,\,\mathrm{and}\,\,t_{1},\ldots t_{n}\in Tm\\}$
$\displaystyle\cup\\{\widehat{r}:\widehat{r}^{M}=r\in\mathbb{Q}\cap[0,1]\\},$
$\displaystyle Fm_{l+1}=Fm_{l}\cup\\{\neg F:F\in Fm_{l}\\}\cup$
$\displaystyle\cup\\{F\to_{L}G:F,G\in Fm_{l}\\}\cup$
$\displaystyle\cup\\{\forall_{x_{n}}F:F\in
Fm_{l}\,\,\mathrm{and}\,\,n\in\mathbb{N}\\}.$

Having established a class of formulae $\mathcal{L}(PrePRL)$, we are in a
position to define PrePRL as an abstract logic.

######  Definition 14.

(Predicate Pavelka Rational Logic (PrePRL).) Let $S$ be a predicate
vocabulary. A Predicate Pavelka Rational Logic is a triple
$(\mathcal{K},Sent_{PrePRL},\mathcal{V})$, where:

a)

$\mathcal{K}$ is a class of continuous, similarity-based Lipschitz structures
closed under renaming, reducts and isomorphism,

b)

$Sent_{PrePRL}(S)$ is a class of formulae of $\mathcal{L}$(PrePRL) – as
defined in Def. 13,

c)

a relation $\mathcal{V}$ assigned to each pair $(\phi,M)$, where $\phi$ is
$S$-sentence and $M$ is $S$-structure in $\mathcal{K}$, a real number
$\phi^{M}\in[0,1]$,

which is closed on basic connectives and existential (general) quantifiers.

##  3 Hajek Logic of Integrals

In this section, HLI is introduced both syntactically and semantically – due
to its axiomatic depiction from Hajek (1998) – as an extension of Predicate
Pavelka Rational Logic (see:Hajek (1998); Caicedo (2014)). This depiction will
constitute a convenient bridgehead to reformulate this system as an abstract
logic.

I. Syntax of HLI. $HLI$ is defined both syntactically and semantically in
Hajek (1998) in a language $\mathcal{L}(HLI)$ given by the grammar:

$\displaystyle\phi:=\phi|\neg\phi\,\,|\,\,\phi\vee\psi\,\,|\,\phi\veebar\psi\,|\,\phi\wedge\psi\,|\,\phi\&\psi\,|\forall_{x}\phi\,\,|\,\,\int\phi\mathrm{d}x\,|\,$
$\displaystyle\forall_{x}\phi=\forall_{y}\psi\,|\,\displaystyle\int\phi\mathrm{d}x=\displaystyle\int\psi\mathrm{d}y.$

In other words, $\mathcal{L}(HLI)$ extends a significant part of PrePRL-
language (without $\approx$-symbol exchanged for ’=’)141414A sense of this
modification follows from another approach to the representation of fuzziness,
which constitutes a founding conceptual idea of HLI. Fuzziness is rendered
less explicit and manifests itself, for example, by truth values of formulae
of this system. by a new quantifier $\displaystyle\int$ (read ’probably’) and
extends the definition of PrePRL-formulae151515In fact, defining PrePRL in the
previous chapter, and we said about Łukasiewicz implication $\to_{L}$, which
is definable either in terms of $\neg$ and strong disjunction $\veebar$ or -
in terms of $\neg$ and strong conjunction $\&$. by the clause informing that
if $\phi$ is a formula and $x$ is a variable, then
$\displaystyle\int\phi\mathrm{d}x$. $HLI$ is syntactically determined in
$\mathcal{L}(HLI)$ by:

$\bullet$

Axioms:

($\mu 1$)

$\displaystyle\int v\mathrm{d}x\equiv v$ for $v$ not not containing $x$
freely,

($\mu 2$)

$\displaystyle\int(\neg\phi)\mathrm{d}x\equiv\neg\displaystyle\int\phi\mathrm{d}x$,

($\mu 3$)

$\displaystyle\int(\phi\to\psi)\mathrm{d}x\equiv\displaystyle\int\phi\mathrm{d}x\to\displaystyle\int\psi\mathrm{d}x$,

($\mu 4$)

$\displaystyle\int(\phi\veebar\psi)\mathrm{d}x\equiv\Big{(}(\displaystyle\int\phi\mathrm{d}x\to\displaystyle\int(\phi\&\psi)\mathrm{d}x)\to\displaystyle\int\psi\mathrm{d}x\Big{)}$,

($\mu$5)

$\displaystyle\int(\displaystyle\int\phi\mathrm{d}x)\mathrm{d}y\equiv\displaystyle\int(\displaystyle\int\phi\mathrm{d}y)\mathrm{d}x$.

$\bullet$

Inference rules: Modus Ponens, substitution, generalization and

$\frac{\phi}{\displaystyle\int\phi\mathrm{d}x}\,\,\,,\frac{\phi\to\psi}{\displaystyle\int\phi\mathrm{d}x\to\displaystyle\int\psi\mathrm{d}x}.$

######  Definition 15.

HLI – in a language, $\mathcal{L}(HLI)$ – is defined as the smallest logical
systems consisting of axioms $(\mu 1)$-$(\mu 5)$ and closed under the above
’integral’ inference rules and the inference rules of PrePRL (MP and
generalization).

######  Example.

  1. 1.

$\displaystyle\int(\displaystyle\int\neg\phi\veebar\psi\,\mathrm{d}x)\mathrm{dy}$
is well-founded $\mathcal{L}(HLI)$-formula, but
$\displaystyle\int\phi\mathrm{d}x+\displaystyle\int\psi\mathrm{dx}$ does
not161616However, the additivity properties of integrals are rendered by the
axioms. For example, finite additivity of integrals is rendered by $(\mu 4)$.
It may be seen in the perspective of semantic interpretations of logical
connectives by the appropriate $\min$ and $\max$ norms..

  2. 2.

$\forall_{x}\phi=\forall_{y}\psi$ is a legal $\mathcal{L}(HLI)$-formula, but
$\phi=\psi$ does not. This formula is prohibited as ’=’ may be used only for
quantifiers.

II. Semantics -Hajek (1998), p. 238-240. The semantics of HLI - described in
Hajek (1998), pp. 238-240 – was elaborated in terms of the so-called
probabilistic models of a general form:

$M=\langle|M|,R_{1}^{M},\ldots
R_{k}^{M},f_{1}^{M},\ldots,f_{n}^{M},c_{1},\ldots,c_{l}\,,\mu\rangle,$ (15)

where $|M|$ is a countable or finite set (the model universe),
$R_{i}^{M},f_{j}^{M}$, for $i\in 1,\ldots,k$, $j\in 1,\ldots,l$, interpret
predicates and function symbols (resp.) as usual (over the real unit $[0,1]$)
and $c_{1}^{M},\ldots,c_{m}^{M}$ interpret constant symbols in the same way
over the real unit $[0,1]$ and $\mu$ is a probability measure on $M$171717That
is $\mu$ is a function assigning to $m\in|M|$ a real $\mu(m)\in[0,1]$ such
that $\sum_{m\in|M|}\mu(m)=1$. For an arbitrary subset
$A\subseteq|M|:\mu(A)=\sum_{m\in A}\mu(m).$ More generally, if
$A\subseteq|M|^{n}$ then
$\mu(A)=\sum\\{\mu(m_{1})\mu(m_{2})\ldots\mu(m_{n}):\langle
m_{1},\ldots,m_{n}\rangle\in A\\}$.

In fact, the proper model suitable to interpret HLI must form a refinement of
the weak probabilistic model given by (18). Instead of $\mu$, one needs to
consider the semantic counterpart of the integral formula. Let us repeat that
the semantic integral181818In further part, we will use simply the name
’integral’ if it does not generate any confusion. is defined as follows.

For each function $f:|M|\to[0,1]$ the (Lebesgue) integral $\displaystyle\int
f\mathrm{d}\mu$ is defined by $\sum_{n\in|M|}f(m)\mu(m)$. More formally, if a
measure space $(X,\mathcal{A},\mu)$ is given, where $\mathcal{A}$ forms an
$\rho$-algebra of subsets of $X$, then each finite, pairwise disjoint family
$\\{A_{1},A_{2},\ldots,A_{n}\\}\subseteq\mathcal{A}$ such that
$\bigcup_{k=1}^{n}A_{k}=X$ is said to be a measurable dissection of $X$. It
allows us to define the Lebesque integral of (a given) function $f$.

######  Definition 16.

The Lebesgue integral[Hewitt],p.164. Let $f$ be any function from $X$ to
$[0,\infty)$. Then the Lebesgue integral $L(f)$ of $f$ is defined as follows.

$\displaystyle L(f)=\sup\Big{\\{}\sum_{k=1}^{n}\inf\\{f(x):x\in
A_{k}\\}\mu(A_{k}):\\{A_{1},\ldots,A_{n}\\}$
$\displaystyle\mathrm{is\,\,a\,\,measurable\,\,dissection\,\,of\,\,X}\Big{\\}}.$

It is noteworthy that if $f$ is two-valued, i.e. $range(f)\subseteq\\{0,1\\}$,
then $\displaystyle\int f\mathrm{d}\mu=\mu(f^{-1}(1))=\mu(\\{m:f(m)=1\\})$ (it
is a measure of the set whose characteristic function is $f$).

These observations will be exploited now for defining the ’semantic
integrals’191919The authors of Hajek (1998) are said to be called the ’weak
integrals’ because of being components of the so-called ’weak models’. in the
algebraic environment of $\mathcal{F}$-algebra of [0,1]-fuzzy subsets of weak
probabilistic model. Its precise definition is as follows.

######  Definition 17.

($\mathcal{F}$-algebra of weak probabilistic models.) Let M be a weak
probabilistic model with a universe $|M|$. $\mathcal{F}$-algebra of (weak
probabilistic) model M is a family of [0,1]-fuzzy subsets202020Note that these
sets are determined by their characteristic functions, so we can think about
them as about their functions, what justifies this defining method. of $|M|$
containing each constant rational function with the value $r\in[0,1]$ and
closed under $\Rightarrow$ (if $f,g\in\mathcal{F}$ and $h(a)=f(a)\Rightarrow
g(a)$, for all $a\in|M|$ then $h\in\mathcal{F}$.)212121One can infer from this
that $\mathcal{F}$ is closed under $\neg\wedge,\vee,\veebar$.(see: Hajek
(1998), p.240.) It corresponds well with the closeness of the set of HLI
formulae under their corresponding connectives.

######  Definition 18.

(Semantic Integrals).Hajek (1998), p. 240. Let us assume that a weak
probabilistic model M with a non-empty domain $|M|$ with its
$\mathcal{F}$-algebra is given. A semantic integral on $\mathcal{F}$is a
mapping $\oint$ associating to each $f\in\mathcal{F}$ its Lebesgue integral –
constructed as described in Def. 16 – $\displaystyle\oint
f\mathrm{d}x\in[0,1]$ and satisfying the following conditions(if
$f,g\in\mathcal{F}$, $k_{r}$ is the constant function with the value $r$,
$h:|M|\times|M|\to[0,1]$ is such that for each $a,b\in|M|$, the functions
$f_{b}(x)=h(x,b)$ and $g_{a}(y)=h(a,y)$ are both in $\mathcal{F}$):

$\displaystyle\oint k_{r}\mathrm{d}x=r,$ (16)
$\displaystyle\oint(1-f)\mathrm{d}x=1-\displaystyle\oint f\mathrm{d}x,$ (17)
$\displaystyle\oint(f\Rightarrow g(\mathrm{d}x\leq(\displaystyle\oint
f\mathrm{d}x\Rightarrow\displaystyle\oint g\mathrm{d}x)$ (18)
$\displaystyle\oint(f\oplus g)\mathrm{d}x=\displaystyle\oint
f\mathrm{d}x+\displaystyle\oint g\mathrm{d}x-\displaystyle\oint(f\ast
g)\mathrm{d}x,$ (19) $\displaystyle\oint(\displaystyle\oint
h\mathrm{d}x)\mathrm{d}y=\displaystyle\oint(\displaystyle\oint
h\mathrm{d}y)\mathrm{d}x,\,\,\mathrm{if\,\,both\,\,sides\,\,defined.}$ (20)

Having defined ’semantic integrals,’ we are in a position to define the so-
called weak probabilistic model as a structure slightly modifying the
probabilistic model.

######  Definition 19.

Weak probabilistic model. Each structure of the type:

$M=\langle|M|,R_{1}^{M},\ldots
R_{k}^{M},f_{1}^{M},\ldots,f_{n}^{M},c_{1},\ldots,c_{l}\,,\displaystyle\oint\rangle\,,$
(21)

where all components are defined as previously and $\displaystyle\oint$ is a
semantic integral defined on $\mathcal{F}$-algebra of fuzzy subsets of M.

In order to build a correspondence between HLI-axioms and axioms $(\mu
1)$-$(\mu 5)$ for semantic integrals, we need to complement the standard truth
value function $\|\bullet\|_{M}$ for $\mathcal{L}(HLI)$ expressions with fuzzy
connectives by the new clause for the integral-type formulae:

$\|\displaystyle\int\phi\mathrm{d}x\|_{M}=\displaystyle\oint\|\phi\|_{M,v}\mathrm{d}x,$
(22)

provided that $\|\phi\|_{M}\in\mathcal{F}$. (otherwise – undefined).

##  4 Towards Hájek Logic of Integrals as an Abstract Logic

In order to describe HLI as an abstract logic – due to Lindstrom (1966);
Caicedo (2014) – for the use of its Lindstrøm’s characterization, one needs to
introduce a satisfaction relation for $\mathcal{L}(HLI)$-formulae in weak
probabilistic models. In this chapter, a new satisfaction relation to be
called Hájek’s satisfiability (symbolically: $\models^{H}$) for
$\mathcal{L}(HLI)$ is introduced222222The analysis of the chapter are inspired
by Iovino’s ideas from Iovino (2001). However, Iovino’s ideas refer to the
standard definition of satisfaction relation and are not immersed in the
analytic conceptual scenario as the current considerations..

In a standard way, a fuzzy logic formula $\phi$ is satisfied in a given model
$\mathcal{M}$, in its state $w$ if and only if 1 (true value) is associated
with $\phi$ in $w$ by an evaluation function $e$. Formally,

$\mathcal{M},w\models\phi\iff e(\phi,w)=1.$ (23)

We preserve this standard and restrictive convention for all
$\mathcal{L}(HLI)$-expressions except for quantifier-based expressions with
’=,’ when we will relax our expectations concerning satisfaction conditions in
weak probabilistic models because of the nature of the integral quantifier
expressions. It finds its reflection in Hájek’s satisfiability for a majority
of $\mathcal{L}(HLI)$-formulae given by the clauses, which develop (10).

We proceed to introduce its definition by short reasoning because of $\&$
connective. Since $e(\phi\&\psi)=1\iff\max(0,e(\psi,w)+e(\phi,w)-1)=1$ because
of $\|\phi\&\psi\|_{M}=\max\\{0,x+y-1\\}$, we can infer that it must hold:
$e(\phi,w)+e(\psi,w)-1=1\iff e(\phi,w)=1$ and $e(\psi,w)=1$232323It follows
from the fact that both $e(\psi,w),e(\phi,w)\in[0,1]$. This result may be
alternatively inferred from the fact that $\phi\&\psi$ stands for
$\neg(\phi\to\psi)$, thus $e(\phi\&\psi w)=1\iff e(\neg(\phi\to\psi),w)=1$,
i.e. $e(\phi,w)=1,e(\psi,w)=1$.. It leads to the following definition.

######  Definition 20.

(Hájek’s satisfaction for $\mathcal{L}(HLI)$-formulae without ’=’.) Let
$\mathcal{M}$ be a weak probabilistic model and $w\in|\mathcal{M}|$ with a set
of constants $C$.

  1. 1.

$(\mathcal{M},w)\models^{H}\phi\iff e(\phi,w)=1$.

  2. 2.

$(\mathcal{M},w)\models^{H}\neg\phi\iff e(\phi,w)=0$.

  3. 3.

$(\mathcal{M},w)\models^{H}\phi\wedge\psi\iff\min\\{e(\psi,w),e(\phi,w)\\})=1$242424Obviously,
this condition may be reformulated to the condition that
$e(\phi,w)=1,e(\psi,w)$. However, such a formulation hides the provenance of
the condition. It is noteworthy that the satisfaction relation itself is not
sensitive to a semantic difference between fuzzy connectives.

  4. 4.

$(\mathcal{M},w)\models^{H}\phi\vee\phi\iff\max\\{e(\psi,w),e(\phi,w)\\})=1.$

  5. 5.

$(\mathcal{M},w)\models^{H}\phi\&\phi\iff e(\phi,w)=1,e(\psi,w)=1$.

  6. 6.

$(\mathcal{M},w)\models^{H}\forall_{x}\phi(x)\iff\inf_{c\in
C}\\{e(\phi(c),w)\\}=1.$

  7. 7.

$(\mathcal{M},w)\models^{H}\exists_{x}\phi(x)\iff\sup_{c\in
C}\\{e(\phi(c),w)\\}=1.$

  8. 8.

$(\mathcal{M},w)\models^{H}\displaystyle\int\phi\mathrm{d}x\iff
e(\displaystyle\int\phi\mathrm{d}x,w)=1$252525This situation should not be
identified with a fact when a given Lebesgue integral is computed in some
limits, and it takes a value 1. This value plays here a role of the highest
truth value for its corresponding ’integral expression in the sense of this
concept given by Hajek (1998), pp. 238-241..

######  Example.

  1. 1.

Let us assume that $e(\phi,w)=0.9$ and $e(\psi,w)=1$ in some w. p. model.
Obviously, $\mathcal{M},w\models^{H}\phi\vee\psi$, but neither
$\mathcal{M},w\models^{H}\phi\&\psi$ nor
$\mathcal{M},w\models^{H}\phi\wedge\psi$ because $e(\phi,w)=0.9$ violates
Hájek’s satisfaction in these cases due to 3) and 4).

  2. 2.

Let us consider a model $\mathcal{M}$ with $\mathcal{C}$ with
$C=\\{\bar{1},\bar{2},\bar{3}\\}$, such that $e(\phi(\bar{1}))=0.5$,
$e(\phi(\bar{2}))=0,8$ and $e(\phi(\bar{3}))=1$. Than
$\mathcal{M}\models^{H}\exists_{x}\phi(x)$ (as $\sup_{c\in
C}\\{e(\phi(c))\\}=1)$, but $\mathcal{M}\not\models^{H}\forall_{x}\phi(x)$ (as
$\inf_{c\in C}\\{e(\phi(c))\\}=0.5\not=1)$.

It remains to complement the definition of Hájek’s satisfiability for ’=’ with
$\mathcal{L}(HLI)$ quantifiers, in particular – with the integral quantifiers.

###  4.1 Hájek’s satisfiability

For the use of Hájek’s satisfiability of ’=’ with $\mathcal{L}(HLI)$
quantifiers, we intend to refer to the concept of measure because of a
measure-based nature o Lebesgue integrals. Informally speaking, we want to
accept $\mathcal{L}(HLI)$-expressions of the type $\displaystyle\int
f\mathrm{d}\mu=\displaystyle\int g\mathrm{d}\mu$ as Hájek’s satisfiable in
w.p. models if the $\mu$-measures of the appropriate $\mathcal{M}$-truth value
sets for the left-side and the right-side integral expressions are almost
identical. The definition of Hájek’s satisfiability will be inductive – due to
the increasing complication degree of the measure. Initially, we begin with a
Dirac’s measure case to generalize the situation later.

I. Hájek’s satisfiability for ’=’ in $\\{0,1\\}$-Dirac’s case.

For the use of Hájek’s satisfiability, we will refer to the slightly modified
definition of Lebesgue integral262626In fact, it forms an approximation of the
proper definition of integral given by Def. 16. (see: Hewitt (1969), p. 164).:

$D_{f}:\oint f\mathrm{d}\mu=\sum_{m\in M}f(m)\mu(\\{m:f(m)=1\\}).$

If range($f$)$\in\\{0,1\\}$, then we obtain

$\oint f\mathrm{d}\mu=\mu\\{x:f(x)=1\\}\cdot 1=\mu\\{f^{-1}(1)\\}.$

Our intention is to accept that the statement of the form $\displaystyle\int
f\mathrm{d}x=\displaystyle\int g\mathrm{d}x$ is Hájek’s satisfiable (in a
$\\{0,1\\}$-Dirac’s case) in a given weak probabilistic272727This type of
models plays a role of the intended models in the whole reasoning – even if it
is not clearly mentioned. model $\mathcal{M}$ by a valuation $v$ if the
measure of a difference set $\\{x:f(x)=1\\}-\\{y:g(y)=1\\}$ is arbitrarily
small and $\\{y:g(y)=1\\}\subseteq\\{x:f(x)=1\\}$282828In order to elucidate a
correlation of the integral symbol with a measure, on terms of which it is
defined, we take the liberty to return to use this integral symbol instead of
$\displaystyle\int f\mathrm{d}x$.. In fact, these two sets corresponding to
their integrable functions $f$ and $g$ can possibly differ in a very
restricted, ’ommittable’ number of elements. By the last assumption, the
following should hold:

$\displaystyle\forall\epsilon>0\Big{(}\mu\Big{(}\\{x:f(x)=1\\}-\\{y:g(y)=1\\}\Big{)}<\epsilon\Big{)}$
(24) $\displaystyle\iff\mathcal{M}\models^{H}\,"\int f\mathrm{d}\mu=\int
g\mathrm{d}\mu".$ (25)

Since $\\{x\in|M|:f(x)=1\\}=\|\phi\|_{M}$ and
$\\{x\in|M|:g(x)=1\\}=\|\psi\|_{M}$, for some $\phi$ and $\psi$ corresponding
to $f(x)$ and $g(x)$ (resp.), we can reformulate lines (24)-(25) to the
following ones:

$\displaystyle\forall\epsilon>0\Big{(}\mu\Big{(}\|\phi\|_{M}-\|\psi\|_{M}\Big{)}<\epsilon\Big{)}$
(26)
$\displaystyle\iff\mathcal{M}\models^{H}\,"\displaystyle\int\phi\mathrm{d}\mu=\displaystyle\int\psi\mathrm{d}\mu",$
(27)

where $\|\phi\|$ is an interpretation of $\phi$ in a given model
$\mathcal{M}$.

II. Hájek’s satisfiability for ’=’ in non-Dirac’s case. In this subsection,
Hájek’s satisfiability for ’=’ in a more general, non-Dirac’s sense will be
introduced. In other words, we will refer to $\mathcal{L}(HLI)$-formulae
$\phi$ with their truth values $\|\phi\|=\\{x:f_{\phi}(x)\in A\\}$ such that
$A$ may belong to a broad class of subsets of $[0,1]$, not only – as
previously – being a singleton. Namely, we will take into account the sets of
the form $\\{x\in|M|:f_{\phi}(x)>\alpha\\}$ and their measures
$\mu\\{x\in|M|:f_{\phi}(x)>\alpha\\}$292929Let us note that it opens a
realistic possibility to consider not only their set-theoretic complements in
the form of the sets of the type $\mu\\{x\in|M|:f_{\phi}(x)\leq\alpha\\}$ but
also – denumerable sums of the pairwise disjoint sets of this type..

For simplicity of further considerations, we will adopt the following modified
version of Definition 4 of Lebesgue integral for (normalized) simple function
– based on the sets of the above type:

$\displaystyle\oint
f\mathrm{d}\mu=\sum_{\rho_{\alpha}\in(0,1)}\rho_{\alpha}\cdot\mu\\{x\in|M|:f_{\phi}(x)>\alpha\\},$
(28)

where $\rho_{\alpha}\in(0,1)$ is a normalized coefficient defining $f$ as a
simple function and $\\{x\in|M|:f_{\phi}(x)>\alpha\\}$ is a measurable set,
$\alpha\in(0,1)$303030We do not specify how $\rho_{\alpha}$ depends on
$\alpha$. One of the reasonable way may be to define
$\rho_{\alpha}=\inf_{x\in\|M|}\\{f_{\phi}(x)>\alpha\\}$. However, it does not
concern us in this context..

As previously, we are willing to require an arbitrarily small (smaller than
arbitrary positive $\epsilon$) sum of measures of the difference set
$A=\\{x\in|M|:f_{\phi}(x)>\alpha_{i}\\}-\\{x\in|M|:g_{\psi}(x)>\alpha_{j}\\}$,
for some functions $f_{\phi}(x)$, $g_{\psi}(x)$313131These functions are
associated by $\|\cdots\|$-function to $\phi$ and $\psi$ respectively. and
fixed $\alpha_{i},\alpha_{j}\in(0,1)$, for $i,j\in I$323232$I$ forms at the
most denumerable set of indices., and a (weak probabilistic) model
$\mathcal{M}$. Finally, we also expect that the second set determined by
$\alpha_{j}$ is contained in the first set determined by $\alpha_{i}$. It
leads to the following definition of Hájek’s satisfiability of the integral
symbols for non-Dirac’s case.

######  Definition 21.

(Hájek’s satisfiability for integral $\mathcal{L}(HLI)$-expressions with ’=’.)
Let us assume that $\mathcal{M}$ be a weak probabilistic model with a model
universe $|M|$, $\alpha_{i},\alpha_{j}\in(0,1)$, for each $i,j\in I$, where I
is (at most) denumerable set of indices. Let also establish two $\alpha_{i}$
and $\alpha_{j}$ such that
$\\{x\in|M|:g_{\psi}(x)>\alpha_{j}\\}\subseteq\\{x\in|M|:f_{\phi}(x)>\alpha_{i}\\}$
and let $f,g$ be non-simple Lebesgue integrable functions corresponding to the
sets. We say that an $\mathcal{L}(HLI)$-formula "$\displaystyle\int
f\mathrm{d}\mu=\displaystyle\int g\mathrm{d}\mu$" is Hájek’s satisfiable in
$\mathcal{M}$, symb. $\mathcal{M}\models^{H}"\displaystyle\int
f\mathrm{d}\mu=\displaystyle\int g\mathrm{d}\mu$", if and only if the
following condition holds:

$\displaystyle\forall_{\epsilon>0}\Big{(}\sum_{i,j\in
I}\mu\Big{(}\\{x\in|M|:f_{\phi(x)}>\alpha_{i}\\}$
$\displaystyle-\\{x\in|M|:g_{\psi(x)}>\alpha_{j}\\}\Big{)}<\epsilon\Big{)}.$

We can venture to generalize the condition for all
$\mathcal{L}(HLI)$-quantifier expressions with ’=’.

######  Definition 22.

Hájek’s satisfiability for $\mathcal{L}(HJL)$-quantifiers
in non-Dirac case). Let us assume that $\mathcal{M}$ be a weak probabilistic
model with a model universe $|M|$, $\alpha_{i},\alpha_{j}\in(0,1)$, for each
$i,j\in I$, where I is (at most) denumerable set of indices. Let also
establish two $\alpha_{i}$ and $\alpha_{j}$ such that
$\\{x\in|M|:g_{\psi}(x)>\alpha_{j}\\}\subseteq\\{x\in|M|:f_{\phi}(x)>\alpha_{i}\\}$
and let $f,g$ be non-simple Lebesgue integrable functions corresponding to the
sets. Let assume finally that $Q\phi(x),Q\psi(x)$ are some quantifier
$\mathcal{L}$(HLI)-expressions, i.e.
$Q\in\\{\forall_{x},\exists_{x},\displaystyle\int()\mathrm{d}x\\}$.

We say that an $\mathcal{L}(HLI)$-quantifier formula "$Q\phi(x)=Q\psi(x)$" is
Hájek’s satisfiable in $\mathcal{M}$, symb.
$\mathcal{M}\models^{H}"Q\phi(x)=Q\psi(x)"$, if and only if the following
condition holds:

$\displaystyle\forall_{\epsilon>0}\Big{(}\sum_{i,j\in
I}\mu\Big{(}\\{x\in|M|:f_{\phi}(x)>\alpha_{i}\\}$
$\displaystyle-\\{x\in|M|:g_{\psi}(x)>\alpha_{j}\\}\Big{)}<\epsilon\Big{)}.$

(In terms of Definition 20, $e(Q\phi(x)=Q\psi(y))=1$.)

######  Example.

We show that the thesis of Fubini theorem $(\mu 5)$. Therefore, let us assume
that $\mathcal{M}$ is a weak probabilistic model. If
$f:|\mathcal{M}|\times|\mathcal{M}|\to[0,1]$, then – due to a simplified
version of Definition 16 (see: the paragraph before it) – we get

$\displaystyle\oint(\displaystyle\oint
f(x,y)\mathrm{d}x)\mathrm{d}y=\sum_{a\in|\mathcal{M}|}(\sum_{b\in|\mathcal{M}|}f(a,b)\mu(b))\mu(a)=$
$\sum_{a,b\in|\mathcal{M}|}f(a,b)\mu(a)\mu(b)=\sum_{b\in|\mathcal{M}|}(\sum_{a\in|\mathcal{M}|}f(a,b)\mu(a))\mu(b)=\\\
$ $=\displaystyle\oint\displaystyle\oint f(x,y)\mathrm{d}y\mathrm{d}x.$

Assuming that the measure $\mu^{*}\\{(x,y)\in|\mathcal{M}|^{2}:f(x,y)=1\\}$ =
$\|f\|_{\mathcal{M}}$ is defined333333It should be a product measure built up
from measures concerning each variable $x$ and $y$., we can immediately infer
from (26-27) that $\mathcal{M}\models^{H}\displaystyle\oint(\displaystyle\oint
f(x,y)\mathrm{d}x)\mathrm{d}y=\displaystyle\oint(\displaystyle\oint
f(x,y)\mathrm{d}y)\mathrm{d}x$.

Being equipped with all the required definitions, we can define HLI as an
abstract logic.

######  Definition 23.

($HLI$ as an abstract logic). HLI is a triple

$(\mathcal{K},\,Sent_{\mathcal{L}(HLI)},\models^{H}),$

where

  1. 1.

$\mathcal{K}$ is a class of weak probabilistic models closed under
isomorphism, renaming and reducts,

  2. 2.

$Sent_{\mathcal{L}}$ is a set of $\mathcal{S}$-sentences of
$\mathcal{L}(HLI)$,

  3. 3.

$\models^{H}$ is Hájek’s satisfaction relation given by Definition 20 and
Definition 22.

##  5 The Lindstrøm-type Theorem for Extensions of HLI and an Extended Outline
of its Proof

Having defined $HLI$ as an abstract logic in the model-theoretic treatment, we
can venture to formulate and prove the Lindstrøm’s-type theorem for all
extensions of $HLI$. We begin with introducing a terminological tissue of the
proof argumentation leading to the main result. They cannot be presented in
Section II as they are involved in a conceptual tissue elaborated in Section
III.

###  5.1 A terminological tissue of Lindstrøm’s theorem and its proof.

In order to perform the task, we need to consider both $HLI$ (as an abstract
logic) and its extensions as equipped by unique binary relations to be called
a Hájek approximation system and denoted by $\lhd^{H}$. For each pair of
formulae, say $\phi(x)$ and $\psi(x)$343434We are especially interested in the
pairs of the formulae as in the previous definitions of Hájek satisfiability.
of a given abstract logic language $\mathcal{L}[L]$ – this unique relation
will intentionally encode some of the metalogical properties of the formulae.

Some of the properties, such as Definition 23 of an abstract logic language
$\mathcal{L}[L]$ or Hájek’s satisfiability in their corresponding
$\mathcal{L}$-structures in the sense of Definition 21 and Definition 22 have
been presupposed or implicitly assumed. It was previously made in some
informal way. In this section, they will be explicitly indicated and formally
expressed in terms of the following definition of $\lhd^{H}$-approximation
system.

######  Definition 24.

(Hájek’s $\lhd^{H}$-approximation system). Let $\mathcal{L}$ be an abstract
logic. The binary relation $\lhd^{H}$ is said to be Hájek’s
$\lhd^{H}$-approximation system if and only if the following conditions hold:

  1. 1.

$\lhd^{H}$ is transitive,

  2. 2.

If $\phi\lhd^{H}\phi^{{}^{\prime}}$ and $\phi\in\mathcal{L}[L]$, then
$\phi^{{}^{\prime}}\in\mathcal{L}[L]$.

  3. 3.

If $\phi\lhd^{H}\phi^{{}^{\prime}}$ and $\mathcal{M}\models^{H}\phi$, then
$\mathcal{M}\models^{H}\phi^{{}^{\prime}}$.

The following definition explains how to relate Hájek’s satisfiability to the
Hájek’s $\lhd^{H}$-approximation, just introduced. These two notions meet
together in the concept of approximate Hájek’s satisfiability.

######  Definition 25.

We say that $\mathcal{M}$ approximately Hájek’s satisfies a formula
$\phi\in\mathcal{L}[L]$, symb. $\mathcal{M}\models^{HA}\phi$, if for all
$\lhd^{H}$-approximations $\phi^{{}^{\prime}}$ of $\phi$, it holds
$\mathcal{M}\models^{H}\phi^{{}^{\prime}}$.

######  Remark 2.

Let us note that $\models^{HA}$ is weaker than $\models^{H}$, which is weaker
than $\models$-satisfaction.

Because of the new conceptual scenario determined by Hájek’s approximation
satisfiability, we should redefine both Compactness Property and the
Elementary Chain Condition for abstract logic with Hájek’s approximation. We
will say a theory (as a set of sentences) $\sigma$ of $\mathcal{L}$ is
consistent if there exists a structure $\mathcal{M}$ of $\mathcal{L}$ which
approximately satisfies every sentence in $\Sigma$. In a similar way, we will
say that $\sigma$ is finitely consistent if every finite subset of $\sigma$ is
consistent.

Let $(\mathcal{L},\lhd^{H})$ be a logic with Hájek’s approximations. For a
structure with semantic integrals $\mathcal{M}$, let
$\mathrm{Th}(\mathcal{L})^{HA}(\mathcal{M})$ denote the theory (set of
sentences) of $\mathcal{L}$ which are Hájek’s approximately satisfied by
$\mathcal{M}$.

######  Definition 26.

(Hájek’s Elementary substructure.) Let $\mathcal{M,N}$ be structures of
$\mathcal{L}$353535In all these contexts, we consider $\mathcal{L}$ by default
as the pair $(\mathcal{L},\lhd^{H})$.. The structure $\mathcal{M}$ is said to
be Hájek’s elementary substructure of $\mathcal{N}$ and will be denoted by

$\mathcal{M}\prec^{HA}_{\mathcal{L}}\mathcal{N}$ (29)

if and only if $M\subseteq N$ and the structure
$(\mathcal{N},a)_{a\in\mathcal{M}}$ Hájek’s approximately satisfies
$\mathrm{Th}(\mathcal{L})^{HA}((M,a)_{a\in\mathcal{M}})$363636This defining is
justified by the fact that the class of structures of a logic is assumed to be
closed under expansions by constants. Recall that $\prec^{HA}$ is a relation
of being elementary substructure (in terms of $\models^{HA}$ satisfiability)..

######  Property 1.

(Compactness). Let $(\mathcal{L},\lhd^{H})$ be an abstract logic with Hájek’s
approximation. $(\mathcal{L},\lhd^{H})$ is said to be satisfied the
compactness theorem if it has the property that every theory of $\mathcal{L}$
which is finitely consistent is consistent.

######  Property 2.

(The Elementary Chain Condition.) Let $(\mathcal{L},\lhd^{H})$ be a logic with
Hájek’s approximations. $(\mathcal{L},\lhd^{H})$ satisfies the elementary
chain condition if the following holds. Whenever

$\mathcal{M}_{0}\prec_{\mathcal{L}}^{HA}\mathcal{M}_{1}\prec_{\mathcal{L}}^{HA}\mathcal{M}_{2}\prec_{\mathcal{L}}^{HA}\ldots,(n<\omega)$

there exists such a structure $\mathcal{M}$ of $\mathcal{L}$ – uniquely
determined by $\bigcup\mathcal{M}_{n}$ – that
$\mathcal{M}_{n}\prec_{\mathcal{L}}^{HA}\mathcal{M}$, for each $n<\omega$.

######  Property 3.

(Weak Negation Property). Let $(\mathcal{L},\lhd)$ be a logic with
approximations. We say that $(\mathcal{L},\lhd)$ has a weak negation property
if and only if there exists a monadic operation $\neg^{weak}$ on
$\mathcal{L}$-sentences such that

  1. 1.

If $\phi\in\mathcal{L}[L]$, then $\neg^{weak}\phi\in\mathcal{L}[L]$;

  2. 2.

If $\phi\in\mathcal{L}[L]$ and $\mathcal{M}$ forms an $L$-structure of
$\mathcal{L}$, then

     * •

$\mathcal{M}\models^{H}\phi$ or $\mathcal{M}\models^{H}\neg^{weak}\phi$,

     * •

for each $\lhd$-approximation $\phi^{{}^{\prime}}$ of $\phi$ it holds
$\mathcal{M}\models^{HA}\neg^{weak}\phi^{{}^{\prime}}$ implies
$\mathcal{M}\not\models^{HA}\phi$.

######  Example.

Weak negation forms a standard negation for each logic systems – considered as
a logic with approximation $\lhd$, where $\lhd$ is a diagonal relation on
sentences (i.e. each sentence is the only approximation of itself).

######  Definition 27.

(Reducibility of an abstract logic $\mathcal{L}$ to $HLI$.373737The similar
definition of reducibility may be introduced for any pair of abstract logics.
Note that we do not require Hájek’s satisfiability in $\mathcal{L}$.) Let
$(\mathcal{L},\lhd^{H})$ be a logic with $\lhd$-approximation, and
$(HLI,\lhd^{H})$ and $(\mathcal{L},\lhd^{H})$ have the same structures. We say
that a sentence $\phi\in\mathcal{L}[L]$ is reducible to $HLI$ if the following
condition holds. For every $\lhd$-approximation $\phi^{{}^{\prime}}$ of $\phi$
there exist two sentences $\psi[\phi,\phi^{{}^{\prime}}]$,
$\psi^{{}^{\prime}}[\phi,\phi^{{}^{\prime}}]$ of $HLI$, such that

  1. 1.

$\psi[\rho,\rho^{{}^{\prime}}]\lhd^{H}\psi^{{}^{\prime}}[\phi,\phi^{{}^{\prime}}]$
and

  2. 2.

If $\mathcal{M}$ is a structure for $\mathcal{L})$(and $HLI$), than

     * •

$\mathcal{M}\models_{\mathcal{L}}^{H}\phi$ implies
$\mathcal{M}\models^{H}\psi[\phi,\phi^{{}^{\prime}}]$,

     * •

$\mathcal{M}\models^{H}\psi^{{}^{\prime}}[\phi,\phi^{{}^{\prime}}]$ implies
$\mathcal{M}\models^{H}_{\mathcal{L}}\phi^{{}^{\prime}}$.

We say that $(\mathcal{L},\lhd)$ forms an extension of $(HLI,\lhd^{H})$ if
every sentence of $\mathcal{L}$ is reducible to $HLI$.

###  5.2 The Proof of Lindstrøm’s type Theorem for HLI – the Proof Idea and
its Extended Outline

I. The proof idea. The proof of Lindstrøm’s Theorem relies on showing that
each abstract logic $(\mathcal{L},\lhd)$ as an extension of
$(\mathcal{L},\lhd)$ is equivalent to the abstract logic $(HLI,\lhd^{H})$
(i.e., they have the same expressive power) if it satisfies both compactness
theorem, the so-called elementary chain condition, and it is closed on
negation. The proof itself is carried our by reductio ad absurdum by showing
that violating one of the conditions383838It is assumed that
$(\mathcal{L},\lhd)$ is not reducible to $(HLI,\lhd^{H})$ in our proof.
generates a contradiction.

The argumentation line of the proof is based on two important but rather
technical lemmas. Lemma 2 allows us to specify the initial situation for the
construction of the required contradiction. More precisely, it ensures the
existence of the following two $\mathcal{L}$-structures, $\mathcal{M}_{0}$ and
$\mathcal{M}_{1}$ such that there exists $\theta\in\mathcal{L}$, which is
satisfied (in Hájek’s sense) in one of them but does not in the second one
(more precisely: a weak negation of its approximation $\theta^{{}^{\prime}}$
is satisfied here). This scenario will be disconfirmed by means of Lemma 2.
This lemma – together with the elementary chain condition – warranties not
only a linearly ordered sequence of structures
$\mathcal{M}_{0}\prec\mathcal{M}_{1}\prec\ldots$, but also a structure
$\mathcal{M}$ as the final element of that sequence and such that we get
$\mathcal{M}\models^{HA}\theta$ and
$\mathcal{M}\models^{HA}\neg^{weak}\theta^{{}^{\prime}}$, for the same
$\theta\in\mathcal{L}$ and its $\lhd^{H}$-approximation $\theta^{{}^{\prime}}$
as previously. The proof of Lemma 1 exploits Observation 1, which establishes
some equivalence between Hájek’s approximate satisfiability of a given theory
$\Sigma\in HLI$ and Hájek’s approximate satisfiability of a corresponding
theory $\Sigma^{L}$ – built up in a unique way from sentences of $\mathcal{L}$
dependent on elements of $\Sigma$.

II. An extended outline of the proof. It is convenient to present the proof
line of Lindstrøm’s Theorem beginning with formulating Observation 1. For that
reason, for a theory $\Sigma\in(HLI,\lhd^{H})$, let

$\Sigma^{\mathcal{L}}=\\{\psi[\rho^{{}^{\prime}},\rho^{{}^{\prime\prime}}]:\rho^{{}^{\prime}}\lhd^{H}\rho^{"}\,\,\mathrm{and}\,\,\rho\lhd^{H}\rho^{{}^{\prime}},\,\,\mathrm{for\,\,some}\,\,\rho\in\Sigma\\}.$

######  Observation 1.

Let $\Sigma$ be a theory in $HLI$ and $\Sigma^{\mathcal{L}}$ defined as
previously. Than it holds the following:

$\mathcal{M}\models^{HA}\Sigma\iff\mathcal{M}\models^{HA}_{\mathcal{L}}\Sigma^{\mathcal{L}}.$
(30)

###### Proof.

We show this property for the implication in one side. Let
$\mathcal{M}\models^{HA}\rho$, for some $\rho\in\Sigma$. It exactly means that

$\forall\rho^{{}^{\prime}}\in\Sigma(\rho\lhd^{H}\rho^{{}^{\prime}}\Rightarrow\mathcal{M}\models^{H}\rho).$
(31)

Therefore, let $\rho,\rho^{{}^{\prime}}\in\Sigma$ be such that
$\rho\lhd^{H}\rho^{{}^{\prime}}$. Thus, we can infer that
$\mathcal{M}\models^{H}\rho$ from definition of Hájek’s
$\lhd^{H}$-approximation. Since
$\mathcal{M}\models^{H}\rho\Rightarrow\mathcal{M}\models^{H}\psi[\rho,\rho^{{}^{\prime}}]$,
for some sentence $\psi[\rho,\rho^{{}^{\prime}}]$ dependent on
$\rho,\rho^{{}^{\prime}}\in\Sigma$ such that $\rho\lhd^{H}\rho^{{}^{\prime}}$,
we can deduce that $\mathcal{M}\models^{H}\psi[\rho,\rho^{{}^{\prime}}]$. Let
now $\psi[\rho^{{}^{\prime}},\rho^{"}]$ be such a sentence that
$\psi[\rho,\rho^{{}^{\prime}}]\lhd^{H}\psi[\rho^{{}^{\prime}},\rho^{"}]$.
Since $\mathcal{M}\models^{H}\psi[\rho,\rho^{{}^{\prime}}]$, then also
$\mathcal{M}\models^{H}\psi[\rho^{{}^{\prime}},\rho^{"}]$, for
$\rho,\rho^{{}^{\prime}},\rho^{"}$ as in definition of $\Sigma^{L}$.

In order to show that also
$\mathcal{M}\models^{HA}\psi[\rho^{{}^{\prime}},\rho^{{}^{\prime\prime}}]$,
one needs to show that for any sentence
$\phi^{{}^{\prime}}[\rho^{{}^{\prime}},\rho^{{}^{\prime\prime}}]\in\Sigma^{\mathcal{L}}$,
it holds

$\psi[\rho^{{}^{\prime}},\rho^{{}^{\prime\prime}}]\lhd^{H}\phi^{{}^{\prime}}[\rho^{{}^{\prime}},\rho^{{}^{\prime\prime}}].$

Meanwhile, this condition is satisfied from the definition of
$\lhd^{H}$-approximation. Thus,
$\mathcal{M}\models^{HA}\psi[\rho^{{}^{\prime}},\rho^{"}]$. ∎

######  Remark 3.

An alternative proof of Observation 1 might be carried out inductively – due
to the taxonomy of $\mathcal{L}(HLI)$ formulae and conditions of Hájek’s
satisfaction for them. For that reason, $\lhd^{H}$ should be established among
$\mathcal{L}(HLI)$ sentences, in particular – among the sentences of the form
$\psi[Q\phi(x)=Q\phi(y),Q\chi(x)=Q\chi(y)]$, where $Q$ is a quantifier sign
admissible in $\mathcal{L}(HLI)$393939This proof would be less general and
more extended and less beneficial from the perspective of further analysis, so
we omit its details..

######  Lemma 1.

Let $\mathcal{M}$ and $\mathcal{N}$ be structures with integrals such that
$\mathcal{M}\prec^{HA}\mathcal{N}$. Then there exists a structure with
integrals $\mathcal{K}$ such that

  1. 1.

$\mathcal{M}\prec^{HA}_{\mathcal{L}}\mathcal{K}$, and
$\mathcal{N}\prec^{HA}_{\mathcal{L}}\mathcal{K}$.

###### Proof.

In order to justify an existence of such a $\mathcal{K}$ structure, it is
convenient to think about it as a ’cumulative’ structure, as consisting of two
sets of elements from $|\mathcal{M}|$ and $|\mathcal{N/M}|$. From a syntactic
point of view, it is enough to show that the theory
$\mathrm{Th}((\mathcal{M},a)_{a\in\mathcal{M}})\cup\mathrm{Th}(\mathcal{N},a,b)_{a\in\mathcal{M},b\in\mathcal{N/M}})$
is consistent. Due to the Lemma 1 – it enough to show that the theory
$\mathrm{Th}((\mathcal{M},a)_{a\in\mathcal{M}})^{\mathcal{L}}\cup\mathrm{Th}((\mathcal{N},a,b)_{a\in\mathcal{M},b\in\mathcal{N/M}})^{\mathcal{L}}$
is consistent. Since $\mathcal{L}$ has compactness property, it is enough to
show that each finite subset of this theory is consistent in order to show
that the whole theory is consistent.

For that reason, let us fix a finite set (of sentences) of
$\mathrm{Th}((\mathcal{N},a,b)_{a\in\mathcal{M},b\in\mathcal{N/M}})^{\mathcal{L}}$
as the set

$\Sigma^{\psi}=\\{\psi[\rho_{1}^{{}^{\prime}},\rho_{1}^{"}],\psi[\rho_{2}^{{}^{\prime}},\rho_{2}^{"}],\ldots,\psi[\rho_{n}^{{}^{\prime}},\rho_{n}^{"}]\\},$

for such
$\rho_{1},\rho_{2},\ldots,\rho_{n}\in\mathrm{Th}((\mathcal{M},a)_{a\in\mathcal{M}})$
that $\rho_{i}<\rho_{i}^{{}^{\prime}}<\rho_{i}^{"}$, for $i=1,2\ldots,n$. We
will find the finite structure extension on a base of $\mathcal{M}$ and
$\mathcal{N/M}$, in which the set
$\\{\psi[\rho_{1}^{{}^{\prime}},\rho_{1}^{"}],\psi[\rho_{2}^{{}^{\prime}},\rho_{2}^{"}],\ldots,\psi[\rho_{n}^{{}^{\prime}},\rho_{n}^{"}]\\}$
is satisfied (in the sense of $\models^{H}_{\mathcal{L}}$-satisfaction). It
will exactly mean that this set in consistent.

Therefore, let $\hat{a_{1}},\hat{a_{2}},\ldots,\hat{a_{k}}$ and
$\hat{b_{1}},\hat{b_{2}},\ldots,\hat{b_{l}}$ be (an exhaustive) list of names
of elements from $|\mathcal{M}|$ and $|\mathcal{N/M}|$ (resp.) occurred in
$\rho$’s. Let us observe that

$((\mathcal{N},a,b)_{a\in\mathcal{M},b\in\mathcal{N/M}})\models^{HA}\forall_{i}^{n}\,\rho_{i}(\hat{a_{1}},\hat{a_{2}},\ldots,\hat{a_{k}},\hat{b_{1}},\hat{b_{2}},\ldots,\hat{b_{l}}).$

Since $\rho_{i}^{{}^{\prime}}$ is Hájek’s $\lhd^{H}$-approximation of
$\rho_{i}$, for each $i=1,\ldots,n$ we can enlarge
$(\mathcal{M},a_{1},\ldots,a_{k})$ by such new elements $c_{1},c_{2}\ldots
c_{l}\in|\mathcal{M}|$ to obtain

$(\mathcal{M},a_{1},\ldots,a_{k},c_{1},c_{2}\ldots
c_{l})\models^{H}\forall_{i}^{n}\rho^{{}^{\prime}}_{i}(\hat{a_{1}},\hat{a_{2}},\ldots,\hat{a_{k}},\hat{c_{1}},\hat{c_{2}},\ldots,\hat{c_{l}}).$
(32)

Since the paraphrazed Condition 2 of Property 3 (of weak negation). asserts
that
$\mathcal{M}\models\rho^{{}^{\prime}}\Rightarrow\mathcal{M}\models\psi[\rho^{{}^{\prime}},\rho^{"}]$,
for all $\rho^{{}^{\prime}},\rho^{"}$ such that $\rho^{{}^{\prime}}<\rho^{"}$,
we can deduce from (32) the following:

$(\mathcal{M},a_{1},\ldots,a_{k},b_{1},b_{2}\ldots
b_{l})\models_{\mathcal{L}}^{H}\psi_{i}[\rho^{{}^{\prime}},\rho^{"}],$ (33)

for $i=1,2\ldots,n$. It means that

$(\mathcal{M},a_{1},\ldots,a_{k},b_{1},b_{2}\ldots
b_{l})\models_{\mathcal{L}}\Sigma^{\psi},$ (34)

i.e. $\Sigma^{\psi}$ is consistent. Since $\Sigma^{\psi}$ was chosen
arbitrarily, we can state that each finite subset of
$\mathrm{Th}((\mathcal{M},a)_{a\in\mathcal{M}})^{\mathcal{L}}\cup\mathrm{Th}(\mathcal{N},a,b)_{a\in\mathcal{M},b\in\mathcal{N/M}})^{\mathcal{L}}$
is consistent. Because of the compactness property for $\mathcal{L}$, we can
assert that the whole theory is consistent as required. ∎

######  Lemma 2.

Let $(\mathcal{L},\lhd^{H})$ be an abstract logic with Hájek’s approximation.
Suppose that $\theta$ is a sentence of $\mathcal{L}[L]$ that is not reducible
to $HLI$. Then there exist an $\lhd^{H}$-approximation $\eta^{{}^{\prime}}$ of
$\eta$ and analytic structures $\mathcal{M}$ and $\mathcal{K}$ such that

  1. 1.

$\mathcal{M}\prec^{HA}\mathcal{K}$ ($\mathcal{M}$ is Hájek’s elementary
substructure),

  2. 2.

$\mathcal{M}\models^{HA}\theta$, but
$\mathcal{K}\models^{HA}\neg^{weak}\theta^{{}^{\prime}}$.

###### Proof.

Because of the previous lemma – it is enough to show that

$\Sigma=\mathrm{Th}^{HA}((\mathcal{M},a)_{a\in\mathcal{M}})^{\mathcal{L}}\cup\\{\neg^{weak}\theta^{{}^{\prime}}\\}$

is consistent, or that there exists a structure, say $\mathcal{K}$, of
$\mathcal{L}$ which approximately satisfies every sentence in (a theory)
$\Sigma$ of $\mathcal{L}$. In particular, it will be
$\mathcal{K}\models^{HA}\neg\theta^{{}^{\prime}}$. Indeed, each finite subset
of $\Sigma$ is Hájek’s approximately satisfied by a finite extension of
$\mathcal{M}$. Thus – because of compactness property for $\mathcal{L}$ – the
whole $\Sigma$ is Hájek’s approximately satisfied in a structure, which is as
required in point 3). ∎

######  Theorem 2.

(Lindstrøm’s Theorem for extensions of HLI.) Let $(\mathcal{L},\lhd)$ and
$(HLI,\lhd^{H})$ be such that:

  1. 1.

$(\mathcal{L},\lhd^{H})$ extends $(HLI,\lhd^{H})$,

  2. 2.

$(\mathcal{L},\lhd^{H})$ has compactness property, elementary chain condition
and it is closed on weak negation.

Then $(\mathcal{L},\lhd^{H})\equiv(HLI,\lhd^{H})$.

###### Proof.

Let us assume that $(\mathcal{L},\lhd^{H})$ has compactness property, the
elementary chain condition, is closed on a weak negation. Finally, let
$(\mathcal{L},\lhd^{H})$) extend $(HLI,\lhd^{H})$, but not conversely, i.e.,
there is such a formula $\theta$ in a language of $(\mathcal{L},\lhd^{H})$
that is not reducible to $(HLI,\lhd^{H})$. This last condition together with
Lemma 2 allows us to state that there exist such $\mathcal{L}$-structures
$\mathcal{M}_{0}$ and $\mathcal{M}_{1}$ that
$\mathcal{M}_{0}\prec\mathcal{M}_{1}$ and there exists such a
$\lhd$-approximation $\theta^{{}^{\prime}}$ of $\theta\in\mathcal{L}$ that:

$\mathcal{M}_{0}\models^{HA}\theta\,\,\,\mathrm{and}\,\,\,\mathcal{M}_{1}\models^{HA}\neg^{weak}\theta^{{}^{\prime}}.$
(35)

Simultaneously, from the elementary chain condition and Lemma 1 used
iteratively – we can infer that there exists for each sequence of
$\mathcal{L}$-structures:

$\mathcal{M}_{0}\prec_{\mathcal{L}}^{HA}\mathcal{M}_{1}\prec_{\mathcal{L}}^{HA}\mathcal{M}_{2}\prec_{\mathcal{L}}^{HA}\ldots$

such a $\mathcal{L}$-structure $\mathcal{M}$ that

$\mathcal{M}_{0}\prec_{\mathcal{L}}^{HA}\mathcal{M}_{1}\prec_{\mathcal{L}}^{HA}\mathcal{M}_{2}\prec_{\mathcal{L}}^{HA}\ldots\mathcal{M}.$

In particular, taking now $\mathcal{M}_{0}\prec_{\mathcal{L}}^{HA}\mathcal{M}$
and $\mathcal{M}_{1}\prec_{\mathcal{L}}^{HA}\mathcal{M}$ we obtain:
$\mathcal{M}\models^{HA}_{\mathcal{L}}\theta$ and
$\mathcal{M}\models^{HA}_{\mathcal{L}}\neg^{weak}\theta^{{}^{\prime}}$. It
generates a contradiction with point 2. of the weak negation property. Hence,
$(\mathcal{L},\lhd^{H})\equiv(HLI,\lhd^{H})$. ∎

##  6 Conclusions and closing remarks

It has been shown how Hájek’s Fuzzy Logic of Integrals – described classically
in Hajek (1998) – may be characterized in Lindstrøm-style. Indeed, Hájek’s
system was considered as a minimal logic in a broader class of fuzzy logic
systems suitable for describing continuous structures based on measures.
Simultaneously, Hájek’s logic constitutes the first formal system suitable to
describe integrals and their properties which has its abstract logic-based
depiction. Nevertheless, it required introducing a two-level satisfaction
relation (Hájek’s satisfiability and Hájek’s approximate satisfiability),
which increases the complexity of the considerations and potentially decreases
the clarity of reasoning. Simultaneously, Lindstrøm’s-type characterizability
for all continuous structures – discussed in this context until today – is
executed through the compactness theorem and the elementary chain condition
instead of the compactness theorem and Skolem-Loewenheim’s theorem.

An idea of Lindstrøm’s-type characterizability might also be extrapolated for
other structures and entities of functional and real analysis, such as Banach
L2-spaces. Simultaneously, Lebesgue integrals might be exchanged by transforms
or splots. Unfortunately, a problem arises of a need for the appropriate
formal logic systems to describe them and their typical properties. The system
proposed in Jobczyk (2014) constitutes rather a surrogate of the formal
system, and it would require some further complements.

##  7 Acknowledgement

Mirna Dzamonja received funding from the European Union Horizon 2020 Research
and Innovation Programme under the Maria Skłodowska-Curie grant agreement No.
1010232, FINTOINF. She also gratefully acknowledges her association with IHSPT
at Université Panthéon-Sorbonne, the School of Mathematics, The University of
East Anglia, and the Academy of Sciences and Arts in Bosnia and Herzegovina,
ANUBiH.

Krystian Jobczyk is grateful to the Polish Fulbright Commission for the Senior
Award Grant supporting his research stay in the Saul Kripke Center at the City
University of New York, where the paper was written.

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Mirna DŽAMONJA, Institut de Recherche en Informatique Fondamentale CNRS et
l’Université de Paris, 8 Place Aurelie Nemours, Paris Cedex 13, 75205, France

Krystian JOBCZYK, Department of Applied Computer Science, AGH University of
Science and Technology, al. Mickiewicza 30, 30 -059, Kraków, Poland.