00footnotetext: Support of the research of both authors by the Austrian
Science Fund (FWF), project I 4579-N, and the Czech Science Foundation (GAČR),
project 20-09869L, entitled “The many facets of orthomodularity”, is
gratefully acknowledged.

# Operator residuation in orthomodular posets of finite height

Ivan Chajda and Helmut Länger

###### Abstract

We show that for every orthomodular poset
$\mathbf{P}=(P,\leq,{}^{\prime},0,1)$ of finite height there can be defined
two operators forming an adjoint pair with respect to an order-like relation
defined on the power set of $P$. This enables us to introduce the so-called
operator residuated poset corresponding to $\mathbf{P}$ from which the
original orthomodular poset $\mathbf{P}$ can be recovered. Moreover, this
correspondence is almost one-to-one. We show that this construction of
operators can be applied also to so-called weakly orthomodular and dually
weakly orthomodular posets. Examples of such posets are included.

AMS Subject Classification: 06C15, 06A11, 03G17, 81P10

Keywords: Orthomodular poset, weakly orthomodular poset, dually weakly
orthomodular poset, poset of finite height, operator residuated structure,
left adjointness

The first attempts for algebraic axiomatization of the logic of quantum
mechanics were done by G. Birkhoff and J. von Neumann [3] and K. Husimi [10]
via orthomodular lattices. Later on the people working in quantum mechanics
decided that such an axiomatization is imprecise since for two operators $P$
and $Q$ on the corresponding Hilbert space their disjunction, i.e. the lattice
join, need not exist in the case when $P$ and $Q$ are neither comparable nor
orthogonal. Hence the concept of an orthomodular poset was introduced (see
[8]) and the majority of researchers believe that orthomodular posets form a
better algebraic axiomatization of the logic of quantum mechanics, see, e.g.,
[12] and [13].

It is not so easy to introduce the connective implication in the propositional
logic based on an orthomodular poset. Namely, since orthomodular posets are
only partial lattices where the lattice operation join need not exist for
elements being neither comparable nor orthogonal, also implication is often
considered as a partial operation. However, such an approach may cause
problems for developing the corresponding logic. We used another approach
where implication is everywhere defined, but its result $x\rightarrow y$ for
given entries $x$ and $y$ need not be an element, but may be a subset of the
orthomodular poset in question, see [6]. In [6] we showed that for such an
implication there exists an operator $\odot$ (considered as conjunction) such
that $(\rightarrow,\odot)$ forms a pair connected by some kind of unsharp
adjointness. The disadvantage of this approach is that the rule for unsharp
adjointness is rather complicated and does not correspond to adjointness as
usually considered in residuated structures.

In [7] we developed a better approach where the considered orthomodular posets
are of finite height. We introduced an everywhere defined connective
implication which satisfies the properties usually asked in classical and non-
classical logics. Again the result $x\rightarrow y$ for given entries $x$ and
$y$ need not be an element, but may be a subset of the orthomodular poset in
question. But now this subset contains only the maximal values. Of course, if
there is only one maximal value then the result is equal to it and hence the
result of implication is an element. In other words, if our orthomodular poset
is a lattice then the implication defined there coincides with the implication
usually introduced in orthomodular lattices, see, e.g., [11] and [12]. On the
other hand, if several maximal values of $x\rightarrow y$ exist then this may
not cause problems since such an “unsharp” reasoning in quantum mechanics was
already treated e.g. in [9]. Moreover, this approach enabled us to derive a
nice Gentzen system for the axiomatization of this logic, i.e. a system
consisting of five simple axioms and six derivation rules only, see [7] for
details.

The above mentioned approach motivates us to study a kind of residuation for
such a logic based on orthomodular posets of finite height. The aim of this
paper is to find an operator $\odot$ such that the couple
$(\rightarrow,\odot)$ forms an adjoint pair.

We start with definitions and basic properties of our concepts.

######  Definition 1.

An orthomodular poset is a bounded poset $(P,\leq,{}^{\prime},0,1)$ with an
antitone involution ′ which is a complementation satisfying the following
conditions:

  1. (i)

If $x,y\in P$ and $x\leq y^{\prime}$ then $x\vee y$ is defined.

  2. (ii)

If $x,y\in P$ and $x\leq y$ then $y=x\vee(y^{\prime}\vee x)^{\prime}$
(orthomodularity).

If $x,y\in P$ and $x\leq y^{\prime}$ then $x$ and $y$ are called orthogonal to
each other which will be denoted by $x\perp y$. It is easy to see that the
expression at the end of condition (ii) is well-defined. Due to De Morgan’s
laws, (ii) can equivalently be rewritten in anyone of the following ways:

  1. (A)

If $x,y\in P$ and $x\leq y$ then $y=x\vee(y\wedge x^{\prime})$.

  2. (B)

If $x,y\in P$ and $x\leq y$ then $x=y\wedge(x\vee y^{\prime})$.

If, moreover, the poset $(P,\leq)$ is a lattice, then
$(P,\leq,{}^{\prime},0,1)$ is called an orthomodular lattice.

Observe that if a poset $(P,\leq,{}^{\prime})$ with an involution satisfies
(B) then it also satisfies (i).

######  Example 2.

Of course, every orthomodular lattice is an orthomodular poset. However, there
are orthomodular posets that are not lattices. The smallest orthomodular poset
that is not a lattice is depicted in Figure 1:

$0$$a$$b$$c$$d$$e$$f$$g$$h$$i$$i^{\prime}$$h^{\prime}$$g^{\prime}$$f^{\prime}$$e^{\prime}$$d^{\prime}$$c^{\prime}$$b^{\prime}$$a^{\prime}$$1$Fig.
1. Smallest orthomodular poset not being a lattice

It is not a lattice since $a\vee b$ does not exist because $i^{\prime}$ and
$f^{\prime}$ are two distinct minimal upper bounds of $a$ and $b$. This
orthomodular poset is finite and hence of finite height. The smallest
orthomodular lattice that is not a Boolean algebra is visualized in Figure 2:

$0$$a$$b$$b^{\prime}$$a^{\prime}$$1$Fig. 2. Smallest orthomodular lattice not
being a Boolean algebra

It is easy to show that a horizontal sum of an arbitrary number of
orthomodular posets of finite height is an orthomodular poset of finite height
again. Hence there exist also infinite orthomodular posets of finite height.

Let $(P,\leq)$ be a poset, $a,b\in P$ and $A,B\subseteq P$. We define $A\leq
B$ if $x\leq y$ for all $x\in A$ and all $y\in B$. Instead of
$\\{a\\}\leq\\{b\\}$, $\\{a\\}\leq B$ and $A\leq\\{b\\}$ we simply write
$a\leq b$, $a\leq B$ and $A\leq b$, respectively. Further, we define

$\displaystyle L(A)$ $\displaystyle:=\\{x\in P\mid x\leq A\\},$ $\displaystyle
U(A)$ $\displaystyle:=\\{x\in P\mid A\leq x\\}.$

Instead of $L(\\{a\\})$, $L(\\{a,b\\})$, $L(\\{a\\}\cup B)$, $L(A\cup B)$ and
$L\big{(}U(A)\big{)}$ we simply write $L(a)$, $L(a,b)$, $L(a,B)$, $L(A,B)$ and
$LU(A)$, respectively. Analogously, we proceed in similar cases. The meaning
of expressions like $A\vee B$, $a\vee B$, $A^{\prime}$ and similar ones is
clear.

######  Lemma 3.

Let $\mathbf{P}=(P,\leq,{}^{\prime})$ be a poset with a unary operation,
$a,b\in P$ and $A,B\subseteq P$. Then the following hold:

  1. (i)

If $\mathbf{P}$ satisfies (A) and $a\leq B$ then $B=a\vee(B\wedge
a^{\prime})$.

  2. (ii)

If $\mathbf{P}$ satisfies (B) and $A\leq b$ then $A=b\wedge(A\vee
b^{\prime})$.

###### Proof.

  1. (i)

If $\mathbf{P}$ satisfies (A) and $a\leq B$ then $B=\\{x\mid x\in
B\\}=\\{a\vee(x\wedge a^{\prime})\mid x\in B\\}=a\vee(B\wedge a^{\prime})$.

  2. (ii)

If $\mathbf{P}$ satisfies (B) and $A\leq b$ then $A=\\{x\mid x\in
A\\}=\\{b\wedge(x\vee b^{\prime})\mid x\in A\\}=b\wedge(A\vee b^{\prime})$.

∎

A poset $(P,\leq)$ is said to be of finite height if it contains no infinite
chain. In such a case, every non-empty subset of $P$ has at least one minimal
and one maximal element. Denote by $\operatorname{Min}A$ and
$\operatorname{Max}A$ the set of all minimal and maximal elements of the
subset $A$ of $P$, respectively.

######  Lemma 4.

Let $\mathbf{P}=(P,\leq,{}^{\prime})$ be a poset of finite height and $a,b\in
P$. Then the following hold:

  1. (i)

If $\mathbf{P}$ satisfies (A) and ′ is an involution then
$\operatorname{Min}U(a,b^{\prime})\wedge b$ is defined.

  2. (ii)

If $\mathbf{P}$ satisfies (B) then $a^{\prime}\vee\operatorname{Max}L(a,b)$ is
defined.

###### Proof.

Assertion (i) follows from $b^{\prime}\leq\operatorname{Min}U(a,b^{\prime})$,
and (ii) follows from $\operatorname{Max}L(a,b)\leq a$. ∎

Now on every orthomodular poset $(P,\leq,{}^{\prime},0,1)$ of finite height we
can introduce the following everywhere defined operators
$\odot,\rightarrow\colon P^{2}\rightarrow 2^{P}\setminus\\{\emptyset\\}$:

$\displaystyle x\odot y$
$\displaystyle:=\operatorname{Min}U(x,y^{\prime})\wedge y,$ $\displaystyle
x\rightarrow y$ $\displaystyle:=x^{\prime}\vee\operatorname{Max}L(x,y)$

for all $x,y\in P$.

######  Remark 5.

If $\mathbf{L}=(L,\vee,\wedge,{}^{\prime},0,1)$ is an orthomodular lattice
then

$\displaystyle\operatorname{Min}U(x,y^{\prime})$ $\displaystyle=x\vee
y^{\prime},$ $\displaystyle\operatorname{Max}L(x,y)$ $\displaystyle=x\wedge
y,$

thus

$\displaystyle x\odot y$ $\displaystyle=(x\vee y^{\prime})\wedge y,$
$\displaystyle x\rightarrow y$ $\displaystyle=x^{\prime}\vee(x\wedge y).$

(The expression $(x\vee y^{\prime})\wedge y$ is called the Sasaki projection
of $x$ to $[0,y]$.) This means that $\rightarrow$ is the implication in
orthomodular lattices as considered in [1], [11] and [12]. In the case where
$\mathbf{L}$ is a Boolean algebra (axiomatizing classical propositional logic)
then the just defined operations reduce to

$\displaystyle x\odot y$ $\displaystyle=x\wedge y,$ $\displaystyle
x\rightarrow y$ $\displaystyle=x^{\prime}\vee y$

as expected. For orthomodular posets of finite height, the operator
$\rightarrow$ introduced above coincides with the implication introduced in
[7].

Let $(P,\leq)$ be a poset, We define a binary relation $\sqsubseteq$ on
$2^{P}\setminus\\{\emptyset\\}$ as follows: For all non-empty subsets $A$ and
$B$ of $P$

$A\sqsubseteq B:\Leftrightarrow\text{ there exists some }a\in A\text{ and some
}b\in B\text{ with }a\leq b$

Now let $a,b\in P$ and $A,B,C$ be non-empty subsets of $P$. Then the following
hold:

  1. (i)

$\\{a\\}\sqsubseteq\\{b\\}$ if and only if $a\leq b$.

  2. (ii)

$A\sqsubseteq A$

  3. (iii)

$A\leq B$ implies $A\sqsubseteq B$.

  4. (iv)

$A\leq B\sqsubseteq C$ implies $A\sqsubseteq C$.

  5. (v)

$A\sqsubseteq B\leq C$ implies $A\sqsubseteq C$.

Statement (i) shows that if we identify the elements $x$ of $P$ with the
corresponding singletons $\\{x\\}$ then the restriction of the relation
$\sqsubseteq$ to $P$ coincides just with the original relation $\leq$ on $P$.

In the following, if $P$ is a non-empty set and $\otimes$ a mapping from
$P^{2}$ to $2^{P}$, i.e. a binary operator on $P$, then we define

$A\otimes y:=\bigcup_{x\in A}(x\otimes y)$

for all $A\subseteq P$ and $y\in P$.

For the operators $\odot$ and $\rightarrow$ defined above we can prove the
following properties.

######  Lemma 6.

Let $(P,\leq,{}^{\prime},0,1)$ be an orthomodular poset of finite height and
$a,b,c\in P$ and define

$\displaystyle x\odot y$
$\displaystyle:=\operatorname{Min}U(x,y^{\prime})\wedge y,$ $\displaystyle
x\rightarrow y$ $\displaystyle:=x^{\prime}\vee\operatorname{Max}L(x,y)$

for all $x,y\in P$. Then the following hold:

  1. (i)

$\operatorname{Min}U(a,b)\odot a=a$

  2. (ii)

$a\leq b$ implies $a\rightarrow b=1$.

  3. (iii)

$1\rightarrow a=a$

  4. (iv)

$a\leq b$ implies $c\rightarrow a\sqsubseteq c\rightarrow b$.

###### Proof.

Because of Lemma 4, $\odot$ and $\rightarrow$ are well-defined.

  1. (i)

$\operatorname{Min}U(a,b)\odot
a=\bigcup\limits_{x\in\operatorname{Min}U(a,b)}(x\odot
a)=\bigcup\limits_{x\in\operatorname{Min}U(a,b)}\big{(}\operatorname{Min}U(x,a^{\prime})\wedge
a\big{)}=\\\ =\bigcup\limits_{x\in\operatorname{Min}U(a,b)}\\{1\wedge
a\\}=\\{a\\}$

  2. (ii)

$a\leq b$ implies $a\rightarrow
b=a^{\prime}\vee\operatorname{Max}L(a,b)=a^{\prime}\vee a=1$.

  3. (iii)

$1\rightarrow a=1^{\prime}\vee\operatorname{Max}L(1,a)=0\vee a=a$

  4. (iv)

Everyone of the following statements implies the next one:

$\displaystyle a$ $\displaystyle\leq b,$ $\displaystyle L(c,a)$
$\displaystyle\subseteq L(c,b),$ $\displaystyle\operatorname{Max}L(c,a)$
$\displaystyle\sqsubseteq\operatorname{Max}L(c,b),$ $\displaystyle
c\rightarrow a=c^{\prime}\vee\operatorname{Max}L(c,a)$
$\displaystyle\sqsubseteq c^{\prime}\vee\operatorname{Max}L(c,b)=c\rightarrow
b.$

∎

Now we introduce our main concept.

######  Definition 7.

An operator residuated structure is an ordered six-tuple
$\mathbf{R}=(P,\leq,\odot,\rightarrow,0,1)$ satisfying the following
conditions:

  1. (i)

$(P,\leq,0,1)$ is a bounded poset.

  2. (ii)

$\odot$ and $\rightarrow$ are mappings from $P^{2}$ to
$2^{P}\setminus\\{\emptyset\\}$.

  3. (iii)

$x\odot y\sqsubseteq z$ if and only if $x\sqsubseteq y\rightarrow z$.

  4. (iv)

$x\odot 1\approx 1\odot x\approx x$

  5. (v)

$y\rightarrow 0\leq x$ implies $x\odot y=x\wedge y$.

  6. (vi)

$y\leq x$ implies $x\rightarrow y=(x\rightarrow 0)\vee y$.

Condition (iii) is called operator left adjointness. $\mathbf{R}$ is

  * •

called idempotent if $x\odot x\approx x$,

  * •

called divisible if $(x\rightarrow y)\odot x\approx\operatorname{Max}L(x,y)$,

  * •

said to satisfy the double negation law if $(x\rightarrow 0)\rightarrow
0\approx x$.

  * •

said to satisfy the contraposition law if $x\leq y$ implies $y\rightarrow
0\leq x\rightarrow 0$.

Now we can prove that every orthomodular poset of finite height can be
converted into such an operator residuated structure.

######  Theorem 8.

Let $\mathbf{P}=(P,\leq,{}^{\prime},0,1)$ be an orthomodular poset of finite
height and define mappings $\odot,\rightarrow\colon P^{2}\rightarrow
2^{P}\setminus\\{\emptyset\\}$ by

$\displaystyle x\odot y$
$\displaystyle:=\operatorname{Min}U(x,y^{\prime})\wedge y,$ $\displaystyle
x\rightarrow y$ $\displaystyle:=x^{\prime}\vee\operatorname{Max}L(x,y)$

for all $x,y\in P$. Then
$\mathbb{R}(\mathbf{P}):=(P,\leq,\odot,\rightarrow,0,1)$ is an idempotent and
divisible operator residuated structure satisfying both the double negation
law as well as the contraposition law and $x\rightarrow 0\approx x^{\prime}$.

###### Proof.

Let $a,b,c\in P$. Because of Lemma 4, $\odot$ and $\rightarrow$ are well-
defined. Moreover, we have $x\rightarrow 0\approx
x^{\prime}\vee\operatorname{Max}L(x,0)=x^{\prime}$.

  1. (i)

and (ii) are evident.

  2. (iii)

Anyone of the following statements implies the next one:

$\displaystyle a\odot b$ $\displaystyle\sqsubseteq c,$
$\displaystyle\operatorname{Min}U(a,b^{\prime})\wedge b$
$\displaystyle\sqsubseteq c,$ $\displaystyle\text{There exists some
}d\in\operatorname{Min}U(a,b^{\prime})\wedge b\text{ with }d$
$\displaystyle\leq c,$ $\displaystyle\text{There exists some
}d\in\operatorname{Min}U(a,b^{\prime})\wedge b\text{ with }d$
$\displaystyle\in L(b,c),$ $\displaystyle\text{There exists some
}d\in\operatorname{Min}U(a,b^{\prime})\wedge b\text{ with }d$
$\displaystyle\sqsubseteq\operatorname{Max}L(b,c),$
$\displaystyle\operatorname{Min}U(a,b^{\prime})\wedge b$
$\displaystyle\sqsubseteq\operatorname{Max}L(b,c),$ $\displaystyle
b^{\prime}\vee\big{(}\operatorname{Min}U(a,b^{\prime})\wedge b\big{)}$
$\displaystyle\sqsubseteq b^{\prime}\vee\operatorname{Max}L(b,c),$
$\displaystyle a\leq\operatorname{Min}U(a,b^{\prime})$
$\displaystyle\sqsubseteq b\rightarrow c,$ $\displaystyle a$
$\displaystyle\sqsubseteq b\rightarrow c,$ $\displaystyle a$
$\displaystyle\sqsubseteq b^{\prime}\vee\operatorname{Max}L(b,c),$
$\displaystyle\text{There exists some }e\in
b^{\prime}\vee\operatorname{Max}L(b,c)\text{ with }a$ $\displaystyle\leq e,$
$\displaystyle\text{There exists some }e\in
b^{\prime}\vee\operatorname{Max}L(b,c)\text{ with }e$ $\displaystyle\in
U(a,b^{\prime}),$ $\displaystyle\text{There exists some }e\in
b^{\prime}\vee\operatorname{Max}L(b,c)\text{ with
}\operatorname{Min}U(a,b^{\prime})$ $\displaystyle\sqsubseteq e,$
$\displaystyle\operatorname{Min}U(a,b^{\prime})$ $\displaystyle\sqsubseteq
b^{\prime}\vee\operatorname{Max}L(b,c),$
$\displaystyle\operatorname{Min}U(a,b^{\prime})\wedge b$
$\displaystyle\sqsubseteq\big{(}b^{\prime}\vee\operatorname{Max}L(b,c)\big{)}\wedge
b,$ $\displaystyle a\odot b$
$\displaystyle\sqsubseteq\operatorname{Max}L(b,c)\leq c,$ $\displaystyle
a\odot b$ $\displaystyle\sqsubseteq c.$

In the eighth and sixteenth statement we use Lemma 3.

  3. (iv)

We have $x\odot 1\approx\operatorname{Min}U(x,1^{\prime})\wedge 1\approx x$
and $1\odot x\approx\operatorname{Min}U(1,x^{\prime})\wedge x\approx x$.

  4. (v)

$b\rightarrow 0\leq a$ implies $b^{\prime}\leq a$ and hence $a\odot
b=\operatorname{Min}U(a,b^{\prime})\wedge b=a\wedge b$.

  5. (vi)

$b\leq a$ implies $a\rightarrow
b=a^{\prime}\vee\operatorname{Max}L(a,b)=a^{\prime}\vee b=(a\rightarrow 0)\vee
b$.

Finally, we have

$\displaystyle x\odot x$
$\displaystyle\approx\operatorname{Min}U(x,x^{\prime})\wedge x\approx x,$
$\displaystyle(x\rightarrow y)\odot x$ $\displaystyle\approx\bigcup_{z\in
x\rightarrow y}(z\odot x)\approx\bigcup_{z\in
x^{\prime}\vee\operatorname{Max}L(x,y)}\big{(}\operatorname{Min}U(z,x^{\prime})\wedge
x\big{)}\approx$
$\displaystyle\approx\bigcup_{u\in\operatorname{Max}L(x,y)}\big{(}\operatorname{Min}U(x^{\prime}\vee
u,x^{\prime})\wedge
x\big{)}\approx\bigcup_{u\in\operatorname{Max}L(x,y)}\\{(x^{\prime}\vee
u)\wedge x\\}\approx$
$\displaystyle\approx\bigcup_{u\in\operatorname{Max}L(x,y)}\\{u\\}\approx\operatorname{Max}L(x,y),$
$\displaystyle(x\rightarrow 0)\rightarrow 0$ $\displaystyle\approx
x^{\prime\prime}\approx x.$

In the last but one line we used orthomodularity. Finally, $a\leq b$ implies
$b\rightarrow 0=b^{\prime}\leq a^{\prime}=a\rightarrow 0$. ∎

It is important that we can also prove the converse.

######  Theorem 9.

Let $\mathbf{R}=(R,\leq,\odot,\rightarrow,0,1)$ be a divisible operator
residuated structure of finite height satisfying both the double negation law
as well as the contraposition law and define

$x^{\prime}:=x\rightarrow 0$

for all $x\in R$. Then $\mathbb{P}(\mathbf{R}):=(R,\leq,{}^{\prime},0,1)$ is
an orthomodular poset.

###### Proof.

Let $a,b\in R$. Since $\mathbf{R}$ satisfies both the double negation law as
well as the contraposition law, ′ is an antitone involution on $(R,\leq)$. Now
assume $a\leq b$. Because of (vi) of Definition 7 we have $b\rightarrow
a=(b\rightarrow 0)\vee a=b^{\prime}\vee a\geq b^{\prime}$. According to (v) of
Definition 7 we conclude $(b\rightarrow a)\odot b=(b\rightarrow a)\wedge b$
which together with divisibility yields

$b\wedge(a\vee b^{\prime})=(b^{\prime}\vee a)\wedge b=(b\rightarrow a)\wedge
b=(b\rightarrow a)\odot b=\operatorname{Max}L(b,a)=a$

showing orthomodularity. Because of $0\leq a$ we obtain $0=a\wedge(0\vee
a^{\prime})=a\wedge a^{\prime}$ completing the proof of the theorem. ∎

The following theorem shows that the correspondence between orthomodular
posets of finite height and certain operator residuated structures of finite
height is nearly one-to-one.

######  Theorem 10.

  1. (i)

Let $\mathbf{P}=(P,\leq,{}^{\prime},0,1)$ be an orthomodular poset of finite
height. Then $\mathbb{P}\big{(}\mathbb{R}(\mathbf{P})\big{)}=\mathbf{P}$.

  2. (ii)

Let $\mathbf{R}=(R,\leq,\odot,\rightarrow,0,1)$ be a divisible operator
residuated structure of finite height satisfying both the double negation law
as well as the contraposition law. Then
$\mathbb{R}\big{(}\mathbb{P}(\mathbf{R})\big{)}=\mathbf{R}$ if and only if

$\displaystyle\operatorname{Min}U(x,y\rightarrow 0)\wedge y$
$\displaystyle\approx x\odot y,$ $\displaystyle(x\rightarrow
0)\vee\operatorname{Max}L(x,y)$ $\displaystyle\approx x\rightarrow y.$

###### Proof.

  1. (i)

If $\mathbb{R}(\mathbf{P})=(P,\leq,\odot,\rightarrow,0,1)$ and
$\mathbb{P}\big{(}\mathbb{R}(\mathbf{P})\big{)}=(P,\leq,{}^{*},0,1)$ then
according to Theorem 8 we have $x^{*}\approx x\rightarrow 0\approx
x^{\prime}$.

  2. (ii)

If $\mathbb{P}(\mathbf{R})=(R,\leq,{}^{\prime},0,1)$ and
$\mathbb{R}\big{(}\mathbb{P}(\mathbf{R})\big{)}=(R,\leq,\bullet,\Rightarrow,0,1)$
then

$\displaystyle x^{\prime}$ $\displaystyle\approx x\rightarrow 0,$
$\displaystyle x\bullet y$
$\displaystyle\approx\operatorname{Min}U(x,y\rightarrow 0)\wedge y,$
$\displaystyle x\Rightarrow y$ $\displaystyle\approx(x\rightarrow
0)\vee\operatorname{Max}L(x,y).$

∎

According to Theorem 10, $\mathbb{R}(\mathbf{P})$ contains the whole
information on $\mathbf{P}$.

In the following we show that our construction of an operator residuated
structure just introduced for othomodular posets of finite height can be
easily modified for more general structures.

The following concepts were introduced for lattices in [4] and for posets in
[5].

######  Definition 11.

  * •

A weakly orthomodular poset is a bounded poset $(P,\leq,{}^{\prime},0,1)$ with
complementation such that $x,\in P$ and $x\leq y$ together imply that
$x\vee(y\wedge x^{\prime})$ is defined and that $x\vee(y\wedge x^{\prime})=y$.

  * •

A dually weakly orthomodular poset is a bounded poset
$(P,\leq,{}^{\prime},0,1)$ with complementation such that $x,y\in P$ and
$x\leq y$ together imply that $y\wedge(x\vee y^{\prime})$ is defined and that
$y\wedge(x\vee y^{\prime})=x$.

Hence, weakly orthomodular posets satisfy condition (A) and dually weakly
orthomodular posets condition (B) mentioned after Definition 1.

It should be remarked that the complementation of a weakly orthomodular poset
or a dually weakly orthomodular poset may neither be antitone nor an
involution. Of course, a poset is orthomodular if and only if it is both
weakly orthomodular and dually weakly orthomodular and if its complementation
is an antitone involution.

In the following we show an example of a weakly orthomodular, respectively
dually weakly orthomodular posets that is not orthomodular.

######  Example 12.

The lattice $\mathbf{W}$ depicted in Figure 3

$0$$a$$b$$c$$d$$e$$f$$g$$1$Fig. 3. Weakly orthomodular poset being not dually
weakly orthomodular

with complementation defined by

$\begin{array}[]{c|ccccccccc}x&0&a&b&c&d&e&f&g&1\\\ \hline\cr
x^{\prime}&1&g&g&f&e&d&c&b&0\end{array}$

is weakly orthomodular, but neither orthomodular nor dually weakly
orthomodular. Hence the horizontal sum of $\mathbf{W}$ and the othomodular
poset $\mathbf{P}$ from Figure 1 is a weakly orthomodular poset that is
neither a lattice nor dually weakly orthomodular. On the contrary, the lattice
$\mathbf{D}$ visualized in Figure 4

$0$$a$$b$$c$$d$$e$$f$$g$$1$Fig. 4. Dually weakly orthomodular poset being not
weakly orthomodular

with complementation defined by

$\begin{array}[]{c|ccccccccc}x&0&a&b&c&d&e&f&g&1\\\ \hline\cr
x^{\prime}&1&f&e&d&c&b&a&a&0\end{array}$

is dually weakly orthomodular, but neither orthomodular nor weakly
orthomodular. Hence the horizontal sum of $\mathbf{D}$ and $\mathbf{P}$ is a
dually weakly orthomodular poset that is neither a lattice nor weakly
orhomodular.

For such posets we can prove results concerning operator residuation without
the contraposition law. However, we must assume the existence of operators
$\odot$ and $\rightarrow$ since they are not automatically defined if only a
weaker variant of the orthomodular law is assumed.

######  Theorem 13.

Let $\mathbf{P}=(P,\leq,{}^{\prime},0,1)$ be a bounded poset of finite height
with complementation satisfying the identity $x^{\prime\prime}\approx x$ and
define mappings $\odot,\rightarrow\colon P^{2}\rightarrow
2^{P}\setminus\\{\emptyset\\}$ by

$\displaystyle x\odot y$
$\displaystyle:=\operatorname{Min}U(x,y^{\prime})\wedge y,$ $\displaystyle
x\rightarrow y$ $\displaystyle:=x^{\prime}\vee\operatorname{Max}L(x,y)$

whenever defined ($x,y\in P$). Then $0^{\prime}\approx 1$ and
$1^{\prime}\approx 0$ and the following hold:

  1. (a)

Assume $\mathbf{P}$ to be weakly orthomodular. Then $\odot$ is well-defined.
Assume $\rightarrow$ to be well-defined, too. Then
$\mathbb{R}(\mathbf{P}):=(P,\leq,\odot,\rightarrow,0,1)$ satisfies all the
conditions of an idempotent operator residuated structure satisfying the
double negation law, $x\vee(x\rightarrow 0)\approx 1$ and $x\rightarrow
0\approx x^{\prime}$, only (ii) has to be replaced by

    1. (ii’)

$x\odot y\sqsubseteq z$ implies $x\sqsubseteq y\rightarrow z$.

  2. (b)

Assume $\mathbf{P}$ to be dually weakly orthomodular. Then $\rightarrow$ is
well-defined. Assume $\odot$ to be well-defined, too. Then
$\mathbb{R}(\mathbf{P}):=(P,\leq,\odot,\rightarrow,0,1)$ satisfies all the
conditions of an idempotent divisible operator residuated structure satisfying
the double negation law and $x\rightarrow 0\approx x^{\prime}$, only (ii) has
to be replaced by

    1. (ii”)

$x\sqsubseteq y\rightarrow z$ implies $x\odot y\sqsubseteq z$.

###### Proof.

The proof is similar to that of Theorem 8. But we cannot use the fact that ′
is antitone. Let $a,b,c\in P$. We have $0^{\prime}\approx 0\vee
0^{\prime}\approx 1$ and $1^{\prime}\approx 1\wedge 1^{\prime}\approx 0$.

  1. (a)

Because of Lemma 4, $a\odot b$ is defined. By assumption also $a\rightarrow b$
is defined. We have $x\rightarrow 0\approx
x^{\prime}\vee\operatorname{Max}L(x,0)=x^{\prime}$ and since $x\leq 1$ we have

$x\vee(x\rightarrow 0)\approx x\vee x^{\prime}\approx x\vee(1\wedge
x^{\prime})\approx 1.$

The rest of the proof is completely analogous to that of Theorem 8.

  2. (b)

Because of Lemma 4, $a\rightarrow b$ is defined. By assumption also $a\odot b$
is defined. We have $x\rightarrow 0\approx
x^{\prime}\vee\operatorname{Max}L(x,0)=x^{\prime}$. The rest of the proof is
completely analogous to that of Theorem 8.

∎

Combining both assumptions (a) and (b) of Theorem 13 we can prove a result
similar to that of Theorem 8.

######  Corollary 14.

Let $\mathbf{P}=(P,\leq,{}^{\prime},0,1)$ be a poset of finite height being
both weakly orthomodular and dually weakly orthomodular and satisfying the
identity $x^{\prime\prime}\approx x$ and define mappings
$\odot,\rightarrow\colon P^{2}\rightarrow 2^{P}\setminus\\{\emptyset\\}$ by

$\displaystyle x\odot y$
$\displaystyle:=\operatorname{Min}U(x,y^{\prime})\wedge y,$ $\displaystyle
x\rightarrow y$ $\displaystyle:=x^{\prime}\vee\operatorname{Max}L(x,y)$

for all $x,y\in P$. Then $\odot$ and $\rightarrow$ are well-defined and
$\mathbb{R}(\mathbf{P}):=(P,\leq,\odot,\rightarrow,0,1)$ is an idempotent and
divisible operator residuated structure satisfying the double negation law and
$x\rightarrow 0\approx x^{\prime}$.

###### Proof.

This follows from Theorem 13. ∎

The following lattice is an example of a poset satisfying the conditions of
Corollary 14, but being not orthomodular.

######  Example 15.

The lattice $\mathbf{L}$ depicted in Figure 5

$0$$a$$b$$c$$d$$e$$f$$g$$h$$1$Fig. 5. Weakly orthomodular and dually weakly
orthomodular lattice whose complementationis a non-antitone involution

with complementation defined by

$\begin{array}[]{c|cccccccccc}x&0&a&b&c&d&e&f&g&h&1\\\ \hline\cr
x^{\prime}&1&g&h&f&e&d&c&a&b&0\end{array}$

is modular and hence both weakly orthomodular and dually weakly orthomodular
and its complementation is an involution but not antitone since $a\leq f$, but
$f^{\prime}=c\not\leq g=a^{\prime}$. Hence the horizontal sum of $\mathbf{L}$
with the orthomodular poset $\mathbf{P}$ from Figure 1 is a weakly
orthomodular and dually weakly orthomodular poset that is neither a lattice
nor an orthomodular poset.

Similarly, as it was done above in the case of orthomodular posets, we can
prove the converse of Corollary 14.

######  Theorem 16.

Let $\mathbf{R}=(R,\leq,\odot,\rightarrow,0,1)$ be a divisible operator
residuated structure of finite height satisfying the double negation law and
define

$x^{\prime}:=x\rightarrow 0$

for all $x\in R$. Then $\mathbb{P}(\mathbf{R}):=(R,\leq,{}^{\prime},0,1)$ is a
dually weakly orthomodular poset of finite height satisfying the identity
$x^{\prime\prime}\approx x$.

###### Proof.

Let $a,b\in R$. Since $\mathbf{R}$ satisfies the double negation law, ′ is an
involution on $(R,\leq)$. Now assume $a\leq b$. Because of (vi) of Definition
7 we have $b\rightarrow a=(b\rightarrow 0)\vee a=b^{\prime}\vee a\geq
b^{\prime}$. According to (v) of Definition 7 we conclude $(b\rightarrow
a)\odot b=(b\rightarrow a)\wedge b$ which together with divisibility yields

$b\wedge(a\vee b^{\prime})=(b^{\prime}\vee a)\wedge b=(b\rightarrow a)\wedge
b=(b\rightarrow a)\odot b=\operatorname{Max}L(b,a)=a,$

i.e. $\mathbb{P}(\mathbf{R})$ is dually weakly orthomodular. Because of $0\leq
a$ we obtain $0=a\wedge(0\vee a^{\prime})=a\wedge a^{\prime}$. According to
(vii) of Definition 7 we have $a\vee a^{\prime}=a\vee(a\rightarrow 0)=1$
completing the proof of the theorem. ∎

The following result shows that the correspondence between posets
$(P,\leq,{}^{\prime},0,1)$ of finite height being both weakly orthomodular and
dually weakly orthomodular and satisfying the identity
$x^{\prime\prime}\approx x$ on the one side and operator residuated structures
of finite height on the other side is almost one-to-one.

######  Corollary 17.

Let $\mathbf{P}=(P,\leq,{}^{\prime},0,1)$ be a poset of finite height being
both weakly orthomodular and dually weakly orthomodular and satisfying the
identity $x^{\prime\prime}\approx x$. Then
$\mathbb{P}\big{(}\mathbb{R}(\mathbf{P})\big{)}=\mathbf{P}$.

###### Proof.

If $\mathbb{R}(\mathbf{P})=(P,\leq,\odot,\rightarrow,0,1)$ and
$\mathbb{P}\big{(}\mathbb{R}(\mathbf{P})\big{)}=(P,\leq,{}^{*},0,1)$ then
according to Corollary 14 we have $x^{*}\approx x\rightarrow 0\approx
x^{\prime}$. The rest of the proof is evident. ∎

According to Corollary 17, $\mathbb{R}(\mathbf{P})$ contains the whole
information on $\mathbf{P}$.

## References

  * [1] L. Beran, Orthomodular Lattices. Algebraic Approach. Reidel, Dordrecht 1985. ISBN 90-277-1715-X.
  * [2] G. Birkhoff, Lattice Theory. AMS, Providence, R. I., 1979. ISBN 0-8218-1025-1.
  * [3] G. Birkhoff and J. von Neumann, The logic of quantum mechanics. Ann. of Math. 37 (1936), 823–843.
  * [4] I. Chajda and H. Länger, Weakly orthomodular and dually weakly orthomodular lattices. Order 35 (2018), 541–555.
  * [5] I. Chajda and H. Länger, Weakly orthomodular and dually weakly orthomodular posets. Asian-Eur. J. Math. 11 (2018), 1850093 (18 pp.).
  * [6] I. Chajda and H. Länger, How to introduce the connective implication in orthomodular posets. Asian-Eur. J. Math. 14 (2021), 2150066 (8 pp.).
  * [7] I. Chajda and H. Länger, The logic of orthomodular posets of finite height. Log. J. IGPL 30 (2022), 143–154.
  * [8] P. D. Finch, On orthomodular posets. J. Austral. Math. Soc. 11 (1970), 57–62.
  * [9] R. Giuntini and H. Greuling, Toward a formal language for unsharp properties. Found. Phys. 19 (1989), 931–945.
  * [10] K. Husimi, Studies on the foundation of quantum mechanics. I. Proc. Phys.-Math. Soc. Japan 19 (1937), 766–789.
  * [11] G. Kalmbach, Orthomodular Lattices. Academic Press, London 1983. ISBN 0-12-394580-1.
  * [12] N. D. Megill and M. Pavičič, Orthomodular lattices and a quantum algebra. Internat. J. Theoret. Phys. 40 (2001), 1387–1410.
  * [13] J. Pykacz and P. Fra̧ckiewicz, The problem of conjunction and disjunction in quantum logics. Internat. J. Theoret. Phys. 56 (2017), 3963-3970.

Authors’ addresses:

Ivan Chajda
Palacký University Olomouc
Faculty of Science
Department of Algebra and Geometry
17\. listopadu 12
771 46 Olomouc
Czech Republic
<EMAIL_ADDRESS>
Helmut Länger
TU Wien
Faculty of Mathematics and Geoinformation
Institute of Discrete Mathematics and Geometry
Wiedner Hauptstraße 8-10
1040 Vienna
Austria, and
Palacký University Olomouc
Faculty of Science
Department of Algebra and Geometry
17\. listopadu 12
771 46 Olomouc
Czech Republic
<EMAIL_ADDRESS>


0 Research system The authors are with the Department of Computer Science,
Texas Tech University, Lubbock, TX, USA. E-mail: {jake.gonzalez $|$
ngan.v.t.nguyen $|$<EMAIL_ADDRESS>Biv et al.: Global Illumination for
Fun and Profit

# VisFCAC: An Interactive Family Clinical Attribute Comparison

Jake Gonzalez  Ngan V.T. Nguyen  and Tommy Dang

###### Abstract

This paper presents VisFCAC, a visual analysis system that displays family
structures along with clinical attribute of family members to effectively
uncover patterns related to suicide deaths for submission to the BioVis 2020
Data Challenge. VisFCAC facilitates pattern tracing to offer insight on
potential clinical attributes that might connect suicide deaths while also
attempting to offer insight to prevent future suicides by at risk people with
similar detected patterns. This paper lays out an approach to compare family
members within a family structure to uncover patterns that may appear in
clinical diagnosis data. This approach also compares two different families
and their family structures to see whether there are patterns in suicide cases
amongst clinical attributes outside family structures. Our solution implements
a radial tree to display family structures with clinical attributes displayed
on radial charts to provide in depth visual analysis and offer a comprehensive
insight for underlying pattern discovery.

###### keywords:

BioVis challenge, Visual analysis, Family structure, Clinical attribute,
Radial tree, Radial chart, Suicide.

K.6.1Management of Computing and Information SystemsProject and People
ManagementLife Cycle; K.7.mThe Computing ProfessionMiscellaneousEthics  The
VisFCAC interface. The system presents the structure of the family tree that
is selected. Squares represent males while circles represent females. The age
of an individual is represented by the total area shaded in the nodes. Black
shaded nodes represent individuals who have committed suicide, gray shaded
nodes represent deceased individuals, and teal represents alive individuals.
Family members that have committed suicide also contain a radial chart
representing the individual’s diagnosed clinical attributes.

##  1 Introduction

The BioVis 2020 Data Challenge focuses on public health data to uncover the
underlying patterns of disease that may play a role in the rate of suicide.
The challenge tasks participants to create a novel visualization to compare
family structures with diagnosed clinical attributes of the family members to
determine patterns that involve suicide cases. Because it is important to
clearly show how hierarchical relationships influence other elements[1], a
radial tree visualization with generational separation is used to show the
hierarchical structure of nine families from the Utah Population Database. To
tackle this challenge, the structure of two family trees are displayed side-
by-side for comparison. Family members who fell victim to suicide are
represented on the tree by a radial chart showing the person’s diagnosed
clinical attributes.

We built a comprehensive web-based visual analysis system that allows users to
compare two family trees and the family members that have fallen victim to
suicide in order to find underlying patterns in diagnosed clinical attributes.
The view of VisFCAC is shown in Figure VisFCAC: An Interactive Family Clinical
Attribute Comparison. The general workflow includes 1) selecting a family to
be displayed on the right, and one to be displayed on the left. Then 2)
analyze family members that have fallen victim to suicide and analyze their
diagnosed clinical attributes. 3) Compare the clinical attributes contained
amongst family members who have committed suicide to uncover patterns of
clinical attributes both within the family structure, and amongst other
families. 4) Use the dot-plot of a clinical disease to determine the
distribution of the disease amongst the families with the age of diagnosis to
help uncover patterns. A similar analytical approach using a graph
visualizations on social media data to uncover relationship patterns can be
found in [2]. The complete source code for the visualization can be found at
https://jakettu.github.io/BioVis2020/.

##  2 System Architecture

###  2.1 Radial Tree

The view is presented in Figure VisFCAC: An Interactive Family Clinical
Attribute Comparison. This allows the user to select two of the nine families
whose data was given in the challenge in order to compare the family
hierarchical structure of the two families. Each family unit is represented as
a square and a circle, the father and mother representatively. Or if the
person is a child that is not found to be with a spouse, they are represented
my an individual square or circle. The shaded area within the shape represents
individual’s age, the larger the shaded area, the older. The color is
representative of whether the individual is alive, deceased, or has fallen
victim to suicide. A teal represents the individual is alive, a gray
represents the individual is deceased, and black represents a suicide. The
root of the tree is representative of the first generation that was given in
the data from the Utah Population Database, and each subsequent generation is
located at the end of a link from the previous generation. The tree is also
separated by generation.

###  2.2 Radial Chart

Figure 1 displays the visualization used to investigate clinical attributes of
a suicide victim. A user can simply hover the mouse over an individual who has
committed suicide with clinical diagnosis data to enlarge the radial chart.
The chart represents sixteen different clinical diagnoses. Each of these
clinical diagnoses are represented by a unique portion of the radial chart,
and are represented by a unique color. The user can reference the dot-plots at
the bottom of the visualization for the mapping of color to disease.

Using this platform, the user can analyze if there was a point in the family
tree where a clinical disease appeared in the family, and which family members
might have inherited genes that also make them susceptible to the same
clinical disease. Also, if a clinical disease appears in different branches of
the family tree, the user is able to easily determine who the last common
ancestor was to offer a deeper insight.

###  2.3 Comparison

The side-by-side comparison becomes useful when tracking clinical diagnoses
within a family to determine if that clinical attribute follows a similar
pattern amongst the other families in the given data set. Each family tree
structure varies greatly, thus patterns in disease heredity can easily be
uncovered. By comparing different family structures, some environmental
factors can also be investigated to determine if they play a role suicide
rates and clinical attributes. For example if a family member commits suicide,
the likelihood of a family member later on in that family structure committing
suicide can be evaluated.

A dot-plot at the bottom of the visualization can be used to determine the
distribution of suicide victims’ diagnosed clinical attributes. The dot-plot
shows the total number of individuals that were diagnosed with a particular
disease. When mousing over the circles on the dot-plot, the user can identify
the person id of the individual, the family the individual belongs to, and the
age of diagnosis.

##  3 Challenge Findings

With the aid of the created visualization, we can determine that depression is
one clinical attribute amongst family members who have fallen victim to
suicide and can easily be passed from one generation to the next. Even if a
person in the previous generation does not contain data to definitively
determine if they suffered from depression, we can see that there may very
well be a gene that predisposes offspring to suffer from this disease. We can
see in families 27251 and 68939 that there are different family members of the
same generation that have fallen victim to suicide where each of them were
diagnosed with depression. This may be caused by biological factors, or may be
caused by environmental factors. In the case of family 27251, we can see that
both offspring of a suicide victim were diagnosed with depression, and then
also fell victim to suicide.

While it is difficult for us to determine how the diagnosed individuals came
to have these diseases, whether being biological or environmental factors, the
user can easily uncover patterns of clinical diagnoses and suicide to know
where to focus deeper investigation. It can also potentially offer some
insight on why some members of a family within a generation seem healthy with
few deaths, but have one family member who committed suicide and suffered from
numerous clinical diseases. For example in family 149, in the ninth generation
a family member was diagnosed with 5 different clinical diseases before
committing suicide, while all members of that generation, the previous
generation, and the second previous generation are still alive.

Figure 1: When analyzing the diagnosed clinical attributes of a family member
who has committed suicide, mousing over the radial chart enlarges the graphic
to more clearly see the colored portions which can be referenced with the
color of the corresponding dot-plot at the bottom of the visualization to
determine the diagnosed disease

##  4 Conclusion

This paper presents VisFCAC, a visual analysis system implemented in HTML5
with D3.js for uncovering patterns in suicide rates amongst different family
structures. The system enables a quick overview of the structure of the nine
different families in the Utah Population Database and shows which of the
family members are alive, deceased, and have fallen victim to suicide. With
investigation by the user, the radial chart offers insight into why patterns
of suicide cases may exist within a family structure. While it difficult to
determine the underlying cause of clinical diseases, patterns of illness and
suicide can be traced to determine if family members of the current or future
generations may be at risk to suicide or illness. With the side-by-side
comparison view of different family structures, the user can easily determine
what clinical diseases are more prone to inheritability by comparing how a
given clinical disease forms patterns in different family structures.

## References

  * [1] T. Dang, P. Murray, R. Etemadpour, and A. G. Forbes.  A user study of techniques for visualizing structure and connectivity in hierarchical datasets.  pp. 45–59. VOILA@ ISWC, 2017.
  * [2] H. N. Nguyen, J. Gonzalez, J. Guo, N. V. Nguyen, and T. Dang.  Vismca: An interactive visual analytics system formisclassification correction and analysis.  In 2020 IEEE Conference on Visual Analytics Science and Technology (VAST), 2020.


# Combining vagueness detection with deep learning to identify fake news

Paul Guélorget1, Benjamin Icard2, Guillaume Gadek1, Souhir Gahbiche1,
Sylvain Gatepaille1, Ghislain Atemezing3 and Paul Égré2  1Advanced Information
Processing, Airbus, France  2Institut Jean-Nicod, CNRS, ENS, EHESS, PSL
University, France  3Mondeca, Paris, France

###### Abstract

In this paper, we combine two independent detection methods for identifying
fake news: the algorithm VAGO uses semantic rules combined with NLP techniques
to measure vagueness and subjectivity in texts, while the classifier FAKE-CLF
relies on Convolutional Neural Network classification and supervised deep
learning to classify texts as biased or legitimate. We compare the results of
the two methods on four corpora. We find a positive correlation between the
vagueness and subjectivity measures obtained by VAGO, and the classification
of text as biased by FAKE-CLF. The comparison yields mutual benefits: VAGO
helps explain the results of FAKE-CLF. Conversely FAKE-CLF helps us
corroborate and expand VAGO’s database. The use of two complementary
techniques (rule-based vs data-driven) proves a fruitful approach for the
challenging problem of identifying fake news.

###### Index Terms:

vagueness, subjectivity, bias, fake news, deep learning, expert system,
explainable AI

##  I Introduction

The computational verification of textual claims is motivated by the
proliferation of misinformation on the Web and online resources. Such methods
are needed because human verification does not scale up to the diffusion speed
of online misinformation, which spreads up to six times faster than real news
[1]. It is also worth mentioning that up to 50% of the shares of viral claims
happens within minutes of their appearance [2].

Extant proposals for detecting fake news have been focused on verifying
textual claims given reference information, based on corpora of previously
checked claims [3, 4]. Some approaches exploit the thousands of reports
written by journalists in fact-checking organizations to automatically verify
if a claim is true or not. Other works use textual documents as trusted
information to be analyzed in order to validate claims. Recently, there have
been new methods to verify claims using reference-structured data, for example
by adapting transformers to let them model datasets [5, 6].

In this paper, we propose to treat lexical vagueness as a cue for the
automatic identification of fake news conveying biased information. The
motivation is twofold: vague claims are less sensitive to factual refutation
than precise claims [7]; moreover vague terms are more prone to subjective
interpretation than precise terms, particularly in the adjectival domain [8].
On the technical side, we use and compare two independent detection methods
for identifying fake news: the first, deployed in the algorithm VAGO, detects
and measures vagueness and subjectivity in texts using a semantic-based
approach combined with NLP techniques, while the second, deployed in the
classifier FAKE-CLF, relies on Convolutional Neural Network to classify texts
as biased or legitimate. The motivation behind this comparison is to
investigate whether VAGO’s and FAKE-CLF’s respective classification results
actually converge. Our aim is thus to confront and to relate a semantic method
with a deep learning method and to pool their results.

The paper is organized as follows. In section II, we present the expert-based
approach of VAGO. We first motivate our emphasis on vagueness by explaining
the way in which some vague expressions convey subjectivity. We present a
typology of vague terms used by VAGO to classify corpora as being either vague
or precise, opinion or factual, as well as simple scoring rules used to
implement an online text analyzer.

Section III describes the deep learning approach of FAKE-CLF. This classifier
based on a Convolutional Neural Network (CNN) architecture was trained on more
than 36,000 texts manually labelled as “fake news” or “legitimate”. FAKE-CLF
exploits Class Activation Maps (CAMs) to define the contribution of each sub-
part of the document to each considered class, allowing for word-level
interpretation.

In section IV, we compare the categorization by VAGO between vague or precise,
opinion or factual, with the classification of news articles by FAKE-CLF
between legitimate or biased. In terms of explainable AI, we expect that doing
so will help us see more clearly into the determinants of the FAKE-CLF
classification.

Finally, in Section V we discuss the opportunities and gains in terms of
explainability of mixing expert-based systems with deep learning, data-driven
classifiers.

##  II Vagueness detection: VAGO

###  II-A Vagueness and subjectivity

Vague words are expressions whose meaning is indeterminate and compatible with
an open-ended range of possible interpretations[9]. Typical examples of vague
words include gradable adjectives like “tall”, “rich”, “intelligent”, whose
extension is left open by the speaker [10]. When hearing “John is tall”, the
listener generally cannot infer a precise representation of John’s height, but
only a set of more or less probable values [11]. This is different if we hear
the precise sentence “John is exactly 187cm tall”, which eliminates more
possibilities.

There is no direct relation between a sentence being vague and its
classification as true or false [7]. An utterance with precise truth
conditions can perfectly be false (“the Eiffel Tower is 96 meters high”).
Conversely, a vague sentence can be judged uncontroversially true (“the Eiffel
Tower is a tall building”). As noted by [9], however, a vague sentence has
higher chances of being judged true than a precise one, because it is
compatible with more possible states of affair. Relatedly, vague expressions
are generally seen as more subjective than precise expressions, because both
speaker and listener can interpret them in different ways [12]. Thus, a large
class of gradable adjectives is described as subjective, or even as evaluative
[13, 14, 15, 8].

In principle, a seasoned liar could use only precise language to create fake
news. But it may be harder to ensure coherence, and it may thereby add up to
the cognitive cost of having to make up a story (see [16]). In contrast, using
vague language can be a cheap way of making one’s utterances easier to accept,
because less sensitive to factual refutation [7]. Importantly, vague language
is also used cooperatively to minimize error and to communicate uncertainty
[17, 7, 18]. But an overwhelming reliance on vague language may signal that
the discourse is possibly less factual, and more prone to _bullshitting_ (see
[19]) or to the spreading of biased information.

###  II-B A typology of vague expressions

To detect vagueness and subjectivity in texts, IJN and Mondeca have developed
a lexical database and associated algorithm called VAGO. The inventory relies
on the typology proposed in [7], which distinguishes four types of lexical
vagueness: approximation ($V_{A}$), generality ($V_{G}$), degree-vagueness
($V_{D}$), and combinatorial vagueness ($V_{C}$).

Expressions of approximation here include mostly modifiers such as “around”,
“about”, “almost”, “nearly”, “roughly”, which loosen up the meaning of the
expression they modify (compare “around 10 o’clock” and “10 o’clock”).
Expressions of generality include determiners such as “some” and modifiers
like “at most”, “at least”. Unlike the former, these expressions have precise
truth-conditions but they fail to give a maximally informative answer to the
question “how many” (compare “some students/at least three students left” to
“eleven students left”).

The class of degree-vague and combinatorially vague expressions includes
mostly one-dimensional adjectives on the one hand (like “tall”, “old”,
“large”) and multi-dimensional adjectives on the other (“beautiful”,
“intelligent”, “good”, “qualified”), which combine several dimensions or
respects of comparison. The opposition between degree-vagueness and
combinatorial vagueness is adapted from [20]. In Alston’s approach,
combinatorial vagueness concerns not just adjectives but also common nouns and
verbs. The class of degree-vague and combinatorially vague expressions
consists mostly of adjectives in VAGO (see Section IV), knowing that in the
case of adjectives, the _degree_ vs _combinatorial_ distinction is congruent
with the distinction proposed by Kaiser and Wang between _simple-subjective_
vs _complex-subjective_ adjectives [8].

Thus, we assume that subjectivity is introduced foremost by expressions of
type $V_{D}$ and $V_{C}$. Two competent speakers can have a non-factual
disagreement as to whether someone is “tall” or “old”, regardless of the
existence of common scales of physical measurement, and even as they share the
same knowledge of height or age [13]. And the more dimensions attached to an
expression, the more disagreement is expected to arise between competent
speakers, as evidenced in the case of evaluative adjectives like “beautiful”
or “smart” for which no standard scale of measurement is available [14]. By
contrast, we assume that expressions of type $V_{A}$ and $V_{G}$ give rise to
no subjectivity, or at least to limited subjectivity, since the interpretation
of these expressions is number-relative and less context-sensitive (“roughly
20” is relative to the precise value 20, unlike “tall”, and similarly “some
students” literally means “more than 0 students”).

As a result, expressions of type $V_{A}$ and $V_{G}$ are treated as factual
expressions, and expressions of type $V_{D}$ and $V_{C}$ as subjective
expressions. This means that logically the class of vague expressions is not
coextensional with the class of subjective expressions, but forms a superset.

###  II-C VAGO: detection and scoring

In VAGO, vagueness and subjectivity are detected and scored bottom-up, from
words to sentences to larger texts. For a given sentence, its vagueness score
is defined simply as the ratio of vague words to the total number of words of
the sentence.

$R_{vague}(\phi)=\frac{(\overbrace{|V_{G}|_{\phi}+|V_{A}|_{\phi}}^{factual}+\overbrace{|V_{D}|_{\phi}+|V_{C}|_{\phi}}^{subjective})}{N_{\phi}}$
(1)

where $N_{\phi}$ designates the total number of words in the sentence $\phi$
and $|V_{G}|_{\phi}$, $|V_{A}|_{\phi}$, $|V_{D}|_{\phi}$ and $|V_{C}|_{\phi}$
denote the number of terms of each of the four types of vagueness (generality,
approximation, degree-vagueness and combinatorial vagueness).

Similarly, the subjectivity score of a sentence is calculated as the ratio of
subjective expressions to the total number of words in the sentence.

$R_{subjective}(\phi)=\frac{|V_{D}|_{\phi}+|V_{C}|_{\phi}}{N_{\phi}}$ (2)

Both the vagueness score and the subjectivity score of a sentence vary between
0 and 1. When a sentence contains at least one vague term, the degree of
vagueness of the sentence is nonzero, $R_{vague}(\phi)>0$. When
$R_{vague}(\phi)=0$ this implies the sentence does not contain vague
vocabulary, and similarly for subjectivity.

For sets of sentences, the vagueness score and the subjectivity score can be
defined as the proportion of sentences with nonzero vagueness and nonzero
subjectivity scores respectively. More fine-grained measures can be proposed,
but these scores suffice to characterize the prevalence of each feature. If
$T$ is a text and $N_{T}$ its total number of sentences, then:

$R_{vague}(T)=\frac{|\\{\phi\in T|R_{vague}(\phi)>0\\}|}{N_{T}}$ (3)

$R_{subjective}(T)=\frac{|\\{\phi\in T|R_{subjective}(\phi)>0\\}|}{N_{T}}$ (4)

###  II-D VAGO Implementation

VAGO’s back-end is built on top of the popular GATE [21] framework for NLP
content processing. It also leverages the semantic content annotator CA-
Manager [22], a semantic-based tool for knowledge extraction from unstructured
data. The current pipeline automatically detects the language of the corpus
(English or French) using
TexCat.111https://www.let.rug.nl/vannoord/TextCat/index.html

The back-end uses a workflow composed of different analysis engines, combining
NLP processing from GATE and semantic technologies with a UIMA-based
infrastructure.222Unstructured Information Management Architecture
(http://uima.apache.org) UIMA architecture allows enriching and customizing
engines to address specific needs in the application. VAGO is also available
through HTTP REST API.

###  II-E Online tool

The online tool VAGO, available from Mondeca’s
website,333https://research.mondeca.com/demo/vago/ provides a graphical
interface for the representation of these scores using two barometers. One
barometer represents the degree to which a text is vague or precise. The other
barometer indicates the degree to which the text reports an opinion or is
factual (the proportion of subjective vs objective vocabulary). A section of
detailed results explains for each sentence the vague triggers detected and
the corresponding category ($V_{X}$).

The online tool uses an anonymous profile, with some restrictions to call the
service such as a limit number of characters as input (e.g., 1200), and some
predefined short sample texts. The online application automatically detects
the language of the text entered by the user (French or English).

As a toy example, consider the text $T=\\{$“ _Most sensational news articles
are sometimes hard to believe”, “Two plus two equals four”, “Mary left Paris
around 2pm”_ $\\}$, comprised of three sentences. $T$ contains five vague
terms (“most”, “sensational”, “sometimes”, “hard”, “around”), with “sometimes”
and “most” instantiating generality, “sensational” and “hard” pertaining to
combinatorial vagueness, and “around” to approximation. While the sentence “
_Two plus two equals four”_ is precise, the other two are vague: _“Mary left
Paris around 2pm”_ is only factual but “ _Most sensational news articles are
sometimes hard to believe”_ contains both factual and subjective terms. The
barometers reporting the proportion of vague and subjective sentences in the
text $T$ are shown in Figure 1.

Figure 1: Results summary for text with three sentences

##  III Fake news classifier

Another way to detect fake news is to learn how to distinguish them
inductively, by means of examples. In this section we detail a previously
developed classifier [23], FAKE-CLF, and the datasets used for training.

###  III-A Deep learning text classification: Related work

Part of machine learning, text classification is known through numerous
applications: sentiment analysis [24], language identification [25] or even
emotion detection [26], dealing with a variety of texts from tweets to
Wikipedia articles. While very efficient, these methods often lack
explainability as most of them work in a “black box” mode.

Explaining a prediction on a text means to characterize sub-document segments:
this can be performed either by observing the behavior of an attention
mechanism [27]; another way relies on the extraction of class activation maps
[28].

Regarding the first approaches, a self-attention mechanism was introduced in
[29] for text classification. Attention scores are computed as the salience of
a sub-part of the document relatively to a generic classification task, and
not relatively to a precise label. Indeed this mechanism focuses on the
“important” words, not on the words that justify an isolated prediction.
Ranging from 0 (no interest) to 1 (high interest), the attention scores do not
carry information about the prediction. However, high-attention words and
phrases act with more impact, either for or against class membership.

The second approach, class activation maps (CAMs), was introduced in [28].
CAMs are literally the contribution of each sub-part of the document to each
considered class. In order to preserve a direct link with the input, CAMs
require specific architecture conditions detailed in the next paragraph.

###  III-B CAM to explain text classification

CAMs were first introduced in image classification [28], and only recently
adapted to text classification [30]. The intuition relies on the fact that
spatiality is preserved across convolutional layers, whereas it is lost in the
last fully-connected layers used in most classifiers. Thus this method
concerns fully-convolutional networks; the final prediction layer can rely
either on global average pooling (GAP) or on global max pooling (GMP), which
is applied to squash the $T$ spatial features vectors associated to the
deepest feature maps $F=\\{F_{1},...,F_{T}\\}\in\mathbb{R}^{T\times K}$ into a
single, global feature vector $F_{g}\in\mathbb{R}^{K}$ in which all spatiality
is lost. $K$ denotes the size of the considered feature maps.

The technical details of the FAKE-CLF model implementation are the following:
it consists of a fully-convolutional neural network made of 3 layers of 128
kernels of size 5. The final layer consists of a global average pooling and a
softmax classification. As inputs, we rely on a pre-trained word embedding, of
dimension 300: FastText [31]. Indeed, a more recent embedding, even if
context-aware such as BERT [32] or Elmo [33], would mean a much higher-
complexity architecture, detrimental to the interpretability of the class
prediction (as it would take into account the role of the neighboring words
without explaining how). The texts are only tokenized (words and punctuation
items) through the Python library NLTK444https://www.nltk.org/. These pre-
trained embeddings imply that the model has to grasp the context: FAKE-CLF
does it thanks to its convolutional layers. Note that FAKE-CLF traces the
importance of each token’s contribution.

Because CAMs require to preserve the temporality across layers, FAKE-CLF uses
the same padding for all convolutions to have exactly one output layer
directly corresponding to each input token. As a result the last convolutional
layer presents a number of outputs that is equal to the number of input words:
the $t^{th}$ output of the last convolutional layer describes the $t^{th}$
word of the input sentence, considering also its context within the
convolutional receptive field.

The CAM extraction is explained hereafter. If there are $C$ different labels,
then the softmax input is defined as:

$S=W^{T}F_{g}+b$ (5)

where $W=\\{w_{c}^{k}\\}\in\mathbb{R}^{K\times C}$ and $b\in\mathbb{R}^{C}$
are the final weights and biases. For any input example $x$, the class
activation map for label $c$ at location $t$ is obtained by summing the
contribution $w_{c}^{k}F_{t}^{k}$ of each scalar feature $F_{t}^{k}$ to the
final score of label $c$, as described in the following equation:

$\text{CAM}(x,c,t)=\sum_{k}{w_{c}^{k}F_{t}^{k}}$ (6)

As a result, in a text classification context, CAMs provide a signed
contribution score of each word (or punctuation item) to each class membership
tackled by the model. Our CAM-extracted model is illustrated in Figure 2.

Figure 2: Extraction of Class Activation Maps from a tokenized sentence, from
[23].

###  III-C Predicting fake-like news articles

We trained this CAMs-based architecture to detect “fake news”, using the
following well-known press articles datasets:

  * •

Kaggle fake news555https://www.kaggle.com/mrisdal/fake-news is composed of
13,000 documents in English, tagged as “fake”: either “biased” or “bullshit”.
We decided to merge these two “fake” categories under a single “biased” label
as the first iterations of this classifier did not distinguish well these two
kinds of bad-quality content. Moreover, doing so enables the aggregation with
the next datasets.

  * •

ISOT666https://www.uvic.ca/engineering/ece/isot/datasets/fake-news/index.php
contains 21,417 “true” and 23,481 “fake” articles, subsequently referred to as
ISOT-True and ISOT-Fake.

  * •

SignalMedia1M777http://research.signalmedia.co/newsir16/signal-dataset.html
gathers a huge corpus of press articles (in English), deemed to be legitimate
[34]. From this resource, we select two random samples:

    * –

15,064 articles to balance the training,

    * –

20,071 other articles for the evaluation part.

On Kaggle and ISOT, a random train/test split policy is applied (80%/20%). The
resulting datasets are then aggregated with their SignalMedia1M complement.

A recurrent objection on fake news classification is the risk to overfit the
classifier on a topic-specific dataset, over-focusing on a few words.
Particularly hard to analyze, this problem stems from the scarcity of
similarly-labeled, thematically different datasets. FAKE-CLF presents an
advantage in order to investigate this point, through the class activation
mappings: high-score tokens are easily identified.

###  III-D Validating the classifier on unseen data

The training dataset of FAKE-CLF consists of the four corpora previously
listed (ISOT-True, ISOT-Fake, SignalMedia1M, Kaggle). The measure of the model
performance with respect to the four corpora is presented on Figure 3. The
y-axis is the output score; 0 means legitimate while 1 stands for a biased
content. The x-axis shows the source corpora. The letter-value plots888The
widest box stretches between the first and last quartiles, the second widest
boxes stretch between the first/last quartile and the first/last octile, the
third widest between the first/last octile and the first/last hexadeciles, and
so forth [35]. illustrate the distribution of articles that were part of the
test dataset (20% of total data, excluded from the training set).

ISOT-True and ISOT-Fake are two facets of the ISOT dataset. “True” news,
labeled as legitimate articles, and “fake” articles, labeled as biased, are
very well recognized and separated. The same trend is also remarkable in
separating SignalMedia1M and Kaggle’s datasets, even though the latter shows a
fuzzier distribution. Overall the classification performance on these test
datasets results in an F1-score $=0.955$, indicative of high accuracy.

Figure 3: Classification of source corpora in the test dataset as legitimate
(0) or biased (1) by FAKE-CLF.

##  IV Comparison and combination

We now turn to the comparison between the classification obtained by FAKE-CLF
and the measures produced by VAGO.

###  IV-A Experimental Settings

The lexicon used by VAGO at the time of this experiment consisted of $1,527$
English terms, and $1,150$ French terms (version of the database of
12/22/2020)[36]. The English lexicon is comprised 95% of adjectives, all in
the classes $V_{D}$ and $V_{C}$, the whole vocabulary being distributed as
follows: $|V_{A}|=8$, $|V_{G}|=14$, $|V_{D}|=35$ and $|V_{C}|=1,470$. Hence
96% of the English lexicon consists of $V_{C}$ items. The French lexicon is
comprised 97% of adjectives, again all in the categories $V_{D}$ and $V_{C}$.
In French, we have $|V_{A}|=7$, $|V_{G}|=15$, $|V_{D}|=28$ and
$|V_{C}|=1,100$. Accordingly, 96% of the French lexicon consists of $V_{C}$
vocabulary. Although the corpora used contained mostly English sentences, they
included some French excerpts, which were included for analysis. Hence part of
the French lexicon was used.

###  IV-B Comparison and correlation of VAGO and FAKE-CLF

We measure the relation between vagueness and how much articles are predicted
to be manipulative by FAKE-CLF by looking at two classes of vague items
detected by VAGO, including: all vague items ($V_{A}+V_{G}+V_{D}+V_{C}$), and
subjective vague items ($V_{D}+V_{C}$).

The relation between articles predicted to be manipulative and the presence of
vagueness is measured in Figure 4 a), using VAGO as our detector of vague
sentences, and the measure reported in Eq. (3) for vagueness. On the left hand
side, the x-axis shows the predicted class (legitimate in blue, biased in red)
by FAKE-CLF. The y-axis displays the percentage of sentences containing
markers of vagueness (the higher, the more vague). The Pearson correlation
coefficient between these two variables (biased, and vague) is 0.208 (sample
size $=28,692$): biased articles tend to contain more vague sentences, but
vagueness is not the sole determining factor.

A similar analysis is performed using VAGO as a detector of subjective
sentences, also described as opinion sentences, using the measure of
subjectivity reported in Eq. (4); this analysis is displayed in Figure 4 b).
There, the y-axis displays the ratio of sentences that contain $V_{C}$ and
$V_{D}$ type keywords. Here the correlation between these two variables is
0.271: articles detected as biased tend to contain more markers of
subjectivity.

a. |
---|---
b. |
Figure 4: Comparison between FAKE-CLF predictions and (a) ratio of vague
sentences, (b) ratio of subjective sentences in texts, as predicted by VAGO.

A split according to the source corpora provides further insights. Regarding
subjectivity, Figure 5 b) shows that corpora follow the tendency of their
legitimate vs. biased affiliation: ISOT-true and SignalMedia1M texts are
detected as less subjective than ISOT-fake and Kaggle texts. While ISOT-true
documents and ISOT-fake documents contain the same proportion of vague terms
all types included (Figure 5 a), the difference between them concerns the
occurrence of subjective vocabulary (Figure 5 b).

a. |
---|---
b. |
Figure 5: Distribution of vague sentences ratio (a), vague-subjective
sentences ratio (b) across sources.

###  IV-C Word-level analysis: exploiting the deep to find new keywords

FAKE-CLF not only provides a prediction at article-level, but also a word-
level contextual score to enable an understanding of its predictions. In
Figure 6 we display the distribution of the average “fakeness” score (the CAM
word-level score) of the occurring VAGO keywords, as they are perceived by the
classifier. Each dot represents the average “fakeness” score of a vocabulary
entry across all its occurrences. For each category, a boxplot sums up the
score distribution. $V_{A}$ and $V_{G}$ words used to identify vague-factual
sentences are in blue; $V_{C}$ and $V_{D}$ words used to identify vague-
subjective sentences are in red. Gray bars give the scores for other
adjectives and adverbs, as a reference, as VAGO contains mostly adjectives.
Non-VAGO entries have been tagged as adjectives or adverbs by jointly using
TextBlob999https://github.com/sloria/TextBlob and NLTK part-of-speech tagging.

Figure 6: Average “fakeness” scores of VAGO entries and other adjectives and
adverbs according to FAKE-CLF class attention maps, defined in Equation 6.

As described in section III-B, for every token in a sentence the CAM provides
a signed score accounting for the contribution of this token and its near
context to the prediction of the whole sentence. Thus, we consider that a high
CAM score stands for a highly bias-inducing word according to FAKE-CLF. We can
see from this figure that belonging to a vague category is not sufficient to
trigger a biased or legitimate label; however some words remain good cues in
that respect. In Table I, we propose a list of the thirty most bias-inducing
adjectives and adverbs (considering the prevalence of adjectives in VAGO and
the proximity of adverbs to adjectives). The first two columns present the
number of occurrences of each term and its average “fakeness” score. The third
gives the semantic type (when listed in VAGO) and the fourth the syntactic
category.

word | occ | avg | VAGO | part-of-speech
---|---|---|---|---
sociable | 32 | 80.42 | VC | adj.
disgusting | 231 | 61.50 | VC | adj.
stumble | 28 | 57.28 |  | adj.
Orwellian | 13 | 44.69 |  | adj.
sociopathic | 12 | 42.40 |  | adj.
arrogantly | 13 | 42.04 |  | adv.
misogynistic | 37 | 41.84 |  | adj.
entire | 2246 | 41.54 |  | adj.
idiotic | 48 | 40.86 | VC | adj.
courageously | 15 | 40.43 |  | adv.
neoconservative | 25 | 40.30 |  | adj.
neoliberal | 45 | 39.31 |  | adj.
coincidentally | 24 | 37.52 |  | adv.
subliminal | 14 | 36.81 |  | adj.
delusional | 61 | 35.78 | VC | adj.
pitiful | 18 | 35.43 | VC | adj.
frighteningly | 10 | 34.78 |  | adv.
disturbingly | 16 | 34.30 |  | adv.
shamelessly | 26 | 34.08 |  | adv.
blatantly | 75 | 33.92 |  | adv.
astonishing | 100 | 33.62 | VC | adj.
laughably | 11 | 33.56 |  | adv.
sic | 101 | 33.21 |  | adj.
outrageously | 18 | 32.67 |  | adv.
racist | 971 | 32.27 | VC | adj.
massive | 1314 | 32.14 | VC | adj.
devious | 13 | 31.81 |  | adj.
transnational | 52 | 31.73 |  | adj.
deplorable | 85 | 31.56 | VC | adj.
Siberian | 11 | 31.51 |  | adj.
TABLE I: Thirty most bias-inducing terms according to FAKE-CLF, filtered to adjectives and adverbs with at least 10 occurrences. Entries are sorted by descending average CAM scores (avg column). VAGO terms are in bold. Miscategorized adjectives and adverbs are in gray.  word | occ | avg | VAGO | part-of-speech
---|---|---|---|---
Carly | 16 | -60.89 |  | adj.
Emily | 72 | -60.84 |  | adv.
provisionally | 28 | -58.53 |  | adv.
40th | 32 | -53.64 |  | adj.
Experian | 40 | -48.58 |  | adj.
vSphere | 14 | -47.59 |  | adv.
upwardly | 11 | -46.20 |  | adv.
fortnightly | 11 | -46.13 |  | adv.
cloudy | 99 | -44.04 | VC | adj.
sectoral | 14 | -43.56 |  | adj.
unbeaten | 113 | -42.26 |  | adj.
trimble | 19 | -41.68 |  | adj.
premiere | 57 | -40.97 |  | adv.
Sebastian | 80 | -38.96 |  | adj.
directorial | 12 | -37.21 |  | adj.
playable | 18 | -37.18 |  | adj.
Tanzanian | 11 | -36.07 |  | adj.
treble | 19 | -36.01 |  | adj.
undertaken | 25 | -35.82 |  | adj.
interactive | 378 | -35.57 |  | adj.
semifinal | 19 | -34.96 |  | adj.
topographic | 16 | -34.77 |  | adj.
shareable | 10 | -34.44 |  | adj.
Dorian | 10 | -33.97 |  | adj.
procedural | 92 | -33.50 |  | adj.
muddy | 46 | -33.48 | VC | adj.
unrivalled | 13 | -33.24 |  | adj.
comprehensive | 1001 | -32.52 | VC | adj.
moody | 146 | -32.37 | VC | adj.
unauthorised | 14 | -32.36 |  | adj.
TABLE II: Thirty least bias-inducing terms according to FAKE-CLF, filtered to
adjectives and adverbs with at least 10 occurrences. Entries are sorted by
ascending average CAM scores (avg column). VAGO terms are in bold.
Miscategorized adjectives and adverbs are in gray.

Even though two part-of-speech tagging tools were used to identify “other
adjectives” and “other adverbs”, some artifacts (displayed in gray) are
wrongly tagged as adjectives or adverbs. Several evaluative adjectives like
“disgusting” and “racist” are identified by VAGO as markers of subjectivity;
others like “Orwellian”, “sociopathic” and “arrogantly” are not in the VAGO
database but are also clear vehicles of subjectivity and evaluativity. In
fact, disregarding the two artifacts marked in gray, all terms in Table I are
adjectives or adverbs that fall under the category of combinatorial vagueness
and that could be added as such to the VAGO lexicon.

A comparison with least bias-inducing terms (scores ranging $[-60;-30]$, Table
II) shows a more heterogeneous set. Fewer terms are already part of the VAGO
lexicon. Furthermore, among adjectives and adverbs, several are non-gradable
and precise (“40th”, “unbeaten”, “upwardly”, “fortnightly”, “Tanzanian”,
“unauthorized”). Fewer correspond to evaluative adjectives or to adjectives
that would be eligible for the type $V_{C}$ (“interactive”, “unrivalled” may
be exceptions).

##  V Discussion

Our hypothesis in this paper was that because vague terms are often
subjective, the prevalence of vagueness in discourse can be used as a cue that
the discourse conveys fake news (whether _biased_ or _bullshit_). In order to
test this hypothesis, we pooled two distinct methodologies. The first
methodology involved the semantic-based algorithm VAGO, which provides a
measure of vagueness vs precision in text, and a measure of subjectivity vs
objectivity in text. The second methodology involved the deep learning
classifier FAKE-CLF, relying on CNN and CAM techniques.

By comparing the results of FAKE-CLF with the measures of vagueness and
subjectivity obtained by VAGO on four distinct corpora pre-identified as
containing “fake” vs “true” articles, we found a positive correlation between
vagueness and fakeness, as well as between subjectivity and fakeness. The
association with fakeness is stronger for subjective vague terms (of type
$V_{D}$ and $V_{C}$ in VAGO’s taxonomy). While an overwhelming majority of the
terms in the VAGO database are markers of subjectivity, this stronger
association also confirms that not all types of vagueness introduce
subjectivity.

From a methodological point of view, the two approaches discussed in this
paper can be viewed as complementary. In one direction, FAKE-CLF identifies as
markers of fakeness some adjectives not originally in the VAGO database, but
which clearly belong to the class $V_{C}$ of adjectives exemplifying multi-
dimensional vagueness, and conveying subjectivity. The correlation found is
therefore expected to increase by the inclusion of a larger vocabulary. In the
converse direction, the typology deployed in VAGO helps to make the results of
FAKE-CLF more easily interpretable and explainable. Indeed, while the
inductive generalizations operated by FAKE-CLF remain opaque, VAGO rests on a
transparent semantic architecture. More work is needed to narrow the gap
between the two approaches, but the consistency in the correlation between the
two classifications is a step toward increased explainability.

##  VI Conclusion

Focusing on lexical vagueness as a marker of subjectivity, we have combined a
semantic-based NLP engine VAGO with a deep learning classifier FAKE-CLF to
improve the detection of fake news contents. The two approaches yield
convergent results and they each provide useful input to improve the other.

Several points remain for further elaboration. First of all, the VAGO lexicon
was still limited at the time of this comparison. We have seen how it can be
expanded using FAKE-CLF, but the base will keep evolving. In particular, other
expressions beside adjectives (and adverbs) can introduce vagueness and
subjectivity. One non-adjective that was included in VAGO at the time of this
comparison is the modal “should”. “Should” was tentatively classified as
$V_{C}$ considering its expressive dimension in deontic utterances stating
moral prescriptions. As it turns out, FAKE-CLF predicts that “should” is not
among the most bias-inducing terms when more entries than adjectives and
adverbs are considered for analysis. Further work is needed to adjudicate the
potential vagueness and subjectivity of “should” and related modals.

Secondly, the correlation found between biased predictions and subjectivity
scores is limited, thus subjectivity explains only part of the variance in the
classification of FAKE-CLF. This is not surprising, since subjectivity is only
one among several factors that can contribute to misinformation. Fake news
consist not only in opinionated texts, but also in clever fabrications,
hoaxes, or simply mistaken factual reports [37]. The type of integration
proposed between FAKE-CLF and VAGO should therefore be extended to explore
other sources of fakeness beside bias.

## Acknowledgments

This work was funded by the project “Disinformation Identification in Evolving
Knowledge Bases” (DIEKB) under grant n° DGA01D19018444 (Mondeca, CNRS,
Airbus). PE and BI also acknowledge grants ANR-19-CE28-0004-01 (Probasem),
ANR-19-CE28-0019-01 (Ambisense), and ANR-17-EURE-0017 (FrontCog) for support.
We thank three anonymous reviewers and Denis Bonnay for their helpful
comments.

## Declaration of contribution

The three teams contributed equally as part of the DIEKB consortium. The
Airbus team is responsible for the deep learning analysis and the development
of FAKE-CLF. IJN and Mondeca designed the VAGO algorithm and jointly developed
the lexical database and program corrections: IJN proposed the typology of
vagueness and scoring rules, Mondeca implemented the VAGO tool and interface.
The three teams contributed to the analysis and discussion of the results. PG,
BI, GG, GA, and PE wrote the paper, which all authors read and discussed. PG,
BI, GG and PE are the main authors, PG and BI are joint first. Correspondence:
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
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