# Magnetic interactions and spin excitations in van der Waals ferromagnet
$\text{VI}_{3}$

Elijah Gordon  Ames Laboratory, U.S. Department of Energy, Ames, Iowa 50011
V. V. Mkhitaryan  Ames Laboratory, U.S. Department of Energy, Ames, Iowa 50011
Haijun Zhao  Ames Laboratory, U.S. Department of Energy, Ames, Iowa 50011
School of Physics, Southeast University, Nanjing 211189, China.  Y. Lee  Ames
Laboratory, U.S. Department of Energy, Ames, Iowa 50011  Liqin Ke  Ames
Laboratory, U.S. Department of Energy, Ames, Iowa 50011

###### Abstract

Using a combination of density functional theory (DFT) and spin-wave theory
methods, we investigate the magnetic interactions and spin excitations in
semiconducting VI3. Exchange parameters of monolayer, bilayer, and bulk forms
are evaluated by mapping the magnetic energies of various spin configurations,
calculated using DFT+$U$, onto the Heisenberg model. The intralayer couplings
remain largely unchanged in three forms of VI3, while the interlayer couplings
show stronger dependence on the dimensionality of the materials. We calculate
the spin-wave spectra within a linear spin-wave theory and discuss how various
exchange parameters affect the magnon bands. The magnon-magnon interaction is
further incorporated, and the Curie temperature is estimated using a self-
consistently renormalized spin-wave theory. To understand the roles of
constituent atoms on magnetocrystalline anisotropy energy (MAE), we resolve
MAE into sublattices and find that a strong negative V-I inter-sublattice
contribution is responsible for the relatively small easy-axis MAE in VI3.

##  I Introduction

Magnetic 2D van der Waals materials (m2Dv) have great potential for future
energy-efficient spin-based devices. Despite the tremendous advancement in the
research field of 2D materials since the discovery of Graphene, finding and
developing robust magnetic 2D materials remain a significant challenge. In
practice, it is nontrivial to induce magnetism into non-magnetic 2D materials
by the magnetic proximity effects or defect engineering and control them
systematically and reliably. On the other hand, the development of intrinsic
magnetic 2D materials have significantly been motivated by the recent
experimental breakthrough, demonstrating that magnetism could sustain down to
mono-layer $\text{CrI}_{3}$ and bilayer $\text{CrGeTe}_{3}$ at a lower
temperature (tens of Kelvins), [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11]. Broader
applications require m2Dv with a higher $T_{\text{C}}$, which is essentially
determined by their intrinsic magnetic properties, particularly, exchange
couplings and magnetocrystalline anisotropy. The latter is of enhanced
importance in 2D as it lifts the constraint of the Mermin-Wagner theorem by
inducing a spin-wave (SW) gap and stabilizing the long-range magnetic ordering
at finite temperatures.

Besides $\text{CrI}_{3}$ and $\text{CrGeTe}_{3}$, other intrinsic 2D magnetic
materials of bulk or layer form have been explored ever since. Among them,
$\text{VI}_{3}$ is a semiconductor with an energy gap of
$\sim$$0.320.67\text{\,}\mathrm{e}\mathrm{V}$ [12, 13] in its bulk form. It is
ferromagnetic (FM) below $50\text{\,}\mathrm{K}$ in a relatively small
magnetic field of about $0.1\text{\,}\mathrm{T}$ [14, 15, 13, 12, 16].
$\text{VI}_{3}$ experiences two FM transitions at $T_{\text{C}}\approx$
$36\text{\,}\mathrm{K}$ and $50\text{\,}\mathrm{K}$ at ambient pressure; the
two transitions merge into one as hydrostatic pressure above
$0.6\text{\,}\mathrm{G}\mathrm{P}\mathrm{a}$ is applied [17, 18]. Bulk
$\text{VI}_{3}$ has easy-axis magnetic anisotropy ($K_{\text{u}}$=
$37\text{\,}\mathrm{k}\mathrm{J}\mathrm{/}\mathrm{m}^{3}$ at
$10\text{\,}\mathrm{K}$) [19]. Remarkably, $\text{VI}_{3}$ is a hard
ferromagnet with a $9.1\text{\,}\mathrm{k}\mathrm{O}\mathrm{e}$ coercive field
at $2\text{\,}\mathrm{K}$, significantly larger than in Cr-based m2Dv [13,
12]. In general, coercivity depends not only on magnetic anisotropy but also
on extrinsic factors such as microstructures. The mechanism behind the very
different coercivity between $\text{VI}_{3}$ and $\text{CrI}_{3}$ is not well
understood yet. It is also unclear how different are the exchange couplings
and the nature of magnetocrystalline anisotropy (MA) in $\text{VI}_{3}$
compared to those in Cr-based m2Dv.

Unlike CrI3, no exfoliated monolayer $\text{VI}_{3}$ has been reported yet.
However, cleavage energies are theoretically found to be between $0.18$ and
$0.29\text{\,}\mathrm{J}\mathrm{/}\mathrm{m}^{2}$, which is comparable to
those for Cr-based m2Dv [15, 16, 20, 21]. Thus, the exfoliation of
$\text{VI}_{3}$ in the near future might be expected, and it is desirable to
understand how magnetic interactions evolve with the material configuration
changing from bulk to multiple layers, as this understanding is relevant to
the real applications, where multiple layers or heterostructures are used
instead of bulk materials.

Theoretical studies have also been carried out to understand the intrinsic
magnetic properties of monolayer $\text{VI}_{3}$ and conflicting results
regarding the exchange couplings and electronic structures have been reported.
For example, large exchange coupling and higher Curie temperature,
$T_{\text{C}}$, resulting from a half-metallic ground state, have been
reported in Ref. [22]. On the other hand, Yang and coworkers have found a
semiconducting ground state and estimated the energy difference between FM and
Néel-AFM states, suggesting smaller exchange couplings and $T_{\text{C}}$ in
monolayer $\text{VI}_{3}$ [21]. Similarly, different dependences of magnetism
on layer stacking in bilayer $\text{VI}_{3}$ have been reported [23, 24].
These discrepancies demonstrate the sensitivity of calculated intrinsic
magnetic properties on the underlying electronic structure.

Inelastic neutron scattering (INS) is the tool of choice to characterize the
spin-wave spectrum and underlying magnetic interactions. Unfortunately, unlike
$\text{CrI}_{3}$ [25], no INS study on $\text{VI}_{3}$ has been reported so
far. A comprehensive theoretical analysis of exchange couplings and the
resulting spin-wave spectra may provide a helpful insight to understand
magnetic properties in $\text{VI}_{3}$ and make predictions upon the
prospective INS experiments.

It is believed that MAE in m2Dv results from the interplay between the spin
polarization of $3d$ atoms and the large spin-orbit coupling (SOC) of the
heavier $p$ elements. By scaling the SOC strength on Cr and I sublattices,
Lado et al. have shown that the heavier I atom plays a significant role in
inducing the magnetic anisotropy in $\text{CrI}_{3}$ [26]. On the other hand,
Chen et al. have argued that a large single-ion anisotropy is expected in
$\text{VI}_{3}$ [21] but it is absent in $\text{CrI}_{3}$. A numerical
resolution of MAE into atomic sites or pairs may help to better understand MAE
in $\text{VI}_{3}$.

In this work, we investigate the exchange couplings in monolayer, bilayer, and
bulk VI3 within the density functional theory (DFT). Using the calculated
exchange parameters, we employ a spin-wave theory to investigate the spin-wave
spectra and estimate the Curie temperatures. By scaling the SOC strength, we
decompose the MAE contribution into sublattices and compare it with
$\text{CrI}_{3}$. We also discuss the validity of the second-order
perturbation theory (SOPT) [27] in calculating MAE in these systems.

##  II Methods

The DFT calculations were performed within the framework of the plane-wave
projector-augmented wave formalism [28], as implemented in the Vienna ab
initio simulation package (vasp) [29, 30]. The Perdew-Burke-Ernzerhof
generalized gradient approximation [31] was used for the exchange-correlation
functional. All calculations were performed with a plane wave cutoff energy of
$400\text{\,}\mathrm{e}\mathrm{V}$. To properly describe the additional
Coulomb repulsion between electrons beyond DFT, the on-site Hubbard $U$ was
applied on V-$3d$ electrons using the double-counting scheme introduced by
Dudarev et al. [32].

The experimental [15] lattice parameters and atomic positions of bulk
$\text{VI}_{3}$ were employed for all calculations unless explicitly stated
otherwise. For bulk $\text{VI}_{3}$, we confirmed that the volume of the
optimized structure is within $\sim 4\%$ of the experimental value. The
reported optimized lattice constants for monolayer $\text{VI}_{3}$ are
different only by $\sim 3\%$ from the experimental values of bulk
$\text{VI}_{3}$ [16, 33]. Thus, employing experimental structural parameters
is a reasonable approximation for this study. A more comprehensive
investigation of theoretical structural optimization, which often depends on
the functionals used, is beyond the scope of the current study. In the bilayer
and monolayer models, a vacuum region of 16Åwas included to avoid the
interaction between periodic images.

###  II.1 Exchange parameters

Figure 1:  (a) Schematic representation of the V sublattices in rhombohedral
$\text{VI}_{3}$. The hexagonal conventional unit cell is used to depict pair
exchange parameters for the first few neighbors of V atoms. Two V sublattices,
V1 and V2, are indicated with green and purple spheres, respectively. (b)
Néel-, zigzag-, and stripy-AFM intralayer spin configurations. A four-atom
unit cell, denoted by the dashed blue lines, is used to describe the three
spin configurations. Green and blue spheres denote the spin-up and spin-down
orientations, respectively.

To estimate the exchange couplings, we map the total energies of various
collinear spin configurations onto a Heisenberg model defined as

$H=-\sum_{i\neq j}J_{ij}\,\hat{\bf e}_{i}\cdot\hat{\bf e}_{j},$ (1)

where $\hat{\bf e}_{i}$ is the unit vector of magnetic moment on site $i$. The
first six nearest-neighbor exchange couplings $J_{ij}$, including three
intralyer ($J_{1}$, $J_{2}$, and $J_{3}$) and three interlayer
($\tilde{J}_{0}$, $\tilde{J}_{1-}$, and $\tilde{J}_{1+}$) exchange couplings
indicated in Fig. 1(a), are included to describe the bilayer and bulk cases.

For monolayer $\text{VI}_{3}$, we calculate the energies of four collinear
spin configurations, namely, FM, Néel-, zigzag-, and stripy-AF [34] to derive
the three intralayer exchanges as follows:

$\left(\begin{array}[]{c}J_{1}\\\ J_{2}\\\ J_{3}\\\
\end{array}\right)=\left(\begin{array}[]{ccc}6&0&6\\\ 2&8&6\\\ 4&8&0\\\
\end{array}\right)^{-1}\left(\begin{array}[]{c}\Delta E_{\text{N}}\\\ \Delta
E_{\text{Z}}\\\ \Delta E_{\text{S}}\\\ \end{array}\right).$ (2)

Here, $\Delta E_{\mu}=E_{\mu}-E_{\text{FM}}$, $\mu=$N, Z, and S, are the
energies (per V atom) of Néel, Zigzag, and Stripy configurations (illustrated
in Fig. 1(b)) with respect to the reference FM configuration.

To obtain the interlayer couplings for bilayer and bulk $\text{VI}_{3}$, we
double each of these four intralayer configurations along the out-of-plane
direction in a larger supercell. For each intralayer configuration, there are
two (analog to FM/AFM) possible interlayer orderings, whose energy difference
is used to extract the interlayer couplings.

###  II.2 Linear spin-wave theory

Using the obtained exchange parameters we calculate the SW spectra within a
linear SW theory. First, we rewrite Eq. (1) in terms of spin ${\bf S}$ instead
of unit vector $\hat{\bf e}$, as $H=-\frac{1}{2}\sum_{i\neq j}J_{ij}^{S}\,{\bf
S}_{i}\cdot{\bf S}_{j}$, where $J_{ij}^{S}=2J_{ij}/S^{2}$, and $S=|{\bf
S}|=m/2$ is the on-site spin of V atoms. Next, we bosonize the spin operator
via the Holstein-Primakoff transformation [35] and truncate the Hamiltonian at
quadratic order in bosons to diagonalize it. There are two V sublattices in
the primitive unit cell of bulk R-$\text{VI}_{3}$, resulting in two SW
branches with energies written as [36]:

$\omega_{\pm}({\bf q})=S\left(J_{{\bf q}=0}^{\text{Intra}}+J_{{\bf
q}=0}^{\text{Inter}}-J_{{\bf q}}^{\text{Intra}}\pm|J_{{\bf
q}}^{\text{Inter}}|\right),$ (3)

where $J_{{\bf q}}^{\text{Intra}}$ and $J_{{\bf q}}^{\text{Inter}}$ sum over
the intra-sublattice and inter-sublattice exchange couplings, respectively.
With the considered range of exchange couplings, we have

$\displaystyle J_{{\bf q}}^{\text{Intra}}$ $\displaystyle=$ $\displaystyle
J_{2}({\bf q})+\tilde{J}_{1-}({\bf q}),$ $\displaystyle J_{{\bf
q}}^{\text{Inter}}$ $\displaystyle=$ $\displaystyle J_{1}({\bf q})+J_{3}({\bf
q})+\tilde{J}_{0}({\bf q})+\tilde{J}_{1+}({\bf q}).$ (4)

Here,

$J_{i}({\bf q})=J_{i}^{S}\gamma_{i}({\bf
q})=J_{i}^{S}\sum_{\boldsymbol{\delta}_{i}}e^{-i2\pi{\bf
q}\cdot\boldsymbol{\delta}_{i}},$ (5)

and $\gamma_{i}({\bf q})$ is the structure factor, while
$\boldsymbol{\delta}_{i}$ standing for the connecting vectors of corresponding
exchange $J_{i}^{S}$.

###  II.3 Non-linear spin-wave theory

The linear SW theory briefly discussed above neglects the interactions between
magnons. Therefore it is applicable only at the lower temperatures, where INS
experiments are often used to measure SW spectra. In the critical region close
to the Curie temperature, interactions between the spin waves become
progressively more relevant. To the leading order, these interactions are
given by the quartic terms of the Holstein-Primakoff transformed Hamiltonian.
To assess the SW interaction effects, we involve quartic terms and treat them
within a Hartree-Fock-like decoupling approximation [37, 38, 39, 40]. This
procedure leads to an effective quadratic Hamiltonian containing
renormalization factors that encapsulate SW interaction effects. The effective
quadratic Hamiltonian is then solved self-consistently, e.g., for the
magnetization versus temperature behavior, $M(T)$. In such a way, we set up a
self-consistently renormalized spin-wave theory (SRSWT) [41], which is the
analogue of previous self-consistently renormalized theories [37, 42, 43, 44]
for layered systems with hexagonal intralayer structure.

Within our SRSWT, renormalization factors defining the effective Hamiltonian
are combinations of the average spin polarization, $\bar{S}$, and the average
short-range two-boson correlations between the exchange-coupled sites. Thus,
the effective Hamiltonian is a function of four parameters in monolayer
$\text{VI}_{3}$ or of seven parameters in bulk $\text{VI}_{3}$. The self-
consistency equations are found by deriving relations for $\bar{S}$ and the
average two-boson correlations from the effective Hamiltonian. Subsequently,
these equations are solved numerically by successive iterations. The
magnetization is defined as the normalized average spin polarization,
$M(T)=\bar{S}(T)/S$, and the critical temperature is taken as the temperature
at which $M(T)$ turns to zero.

Using the calculated values of exchange couplings, we find $M(T)$ and the
corresponding critical temperatures for monolayer, bilayer, and bulk VI3, by
means of SRSWT. The analysis of monolayer and bulk systems is facilitated by
the fact that these systems consist of two equivalent sublattices of magnetic
atoms. In contrast, we regard the few-layer systems with $L$ number of layers
as consisting of $2L$ sublattices – two per hexagonal layer. This approach
allows us to take into account the physical difference between the surface and
bulk layers. At the same time, it requires a substantially larger parameter
space for the self-consistency equations, as the average spin polarization and
two-boson correlation values at different sublattices are generally different.
The details of our SRSWT method and implementation can be found elsewhere
[41].

###  II.4 Sublattice-resolved MAE

MAE is calculated as the total-energy difference between two FM states, in
which the magnetization is aligned along the [100] or [001] directions,
respectively. SOC was included using the second-variation method [45, 46]. We
resolve MAE contributions into sublattices via scaling the SOC strength
$\xi_{i}$ on V and I sites with parameters $\lambda_{i}$; the SOC Hamiltonian
becomes

$H_{\text{soc}}=\sum_{i=\text{V,I}}\lambda_{i}\xi_{i}({\bf L}_{i}\cdot{\bf
S}_{i}).$ (6)

Assuming that MAE can be well described by SOPT, total MAE can be written as
[47]

$K(\lambda_{\text{V}},\lambda_{\text{I}})=K_{\text{V-V}}{\lambda_{\text{V}}^{2}}+K_{\text{V-I}}{\lambda_{\text{V}}}{\lambda_{\text{I}}}+K_{\text{I-I}}{\lambda_{\text{I}}^{2}},$
(7)

where $K_{\text{V-V}}$, $K_{\text{V-I}}$, and $K_{\text{I-I}}$ describe the
response of MAE to the SOC strength of V, V and I simultaneously, and I,
respectively. They can be determined by fitting the
$K(\lambda_{\text{V}},\lambda_{\text{I}})$ curve or directly evaluated within
SOPT, and the inter-sublattice term $K_{\text{V-I}}$ arise from the coupling
between the two sublattices [47].

##  III RESULTS AND DISCUSSION

###  III.1 Electronic properties

Figure 2: (a) Band structure of $\text{VI}_{3}$ for $U=$
$2\text{\,}\mathrm{e}\mathrm{V}$. Solid and dotted lines indicate the majority
and minority spin channels, respectively. Fermi level $E_{\rm F}$ is set at
zero energy. This figure was constructed using sumo [48]. (b) Partial density
of states projected on the V-$d$ states. (c) The splitting and occupation of
V-$d$ orbitals in $\text{VI}_{3}$, where $\alpha=\sqrt{1/3}$ and
$\beta=\sqrt{2/3}$; the $t_{2g}$ orbital set splits into doubly-degenerate
($1e_{x}$, $1e_{y}$) and singly-degenerate ($1a$) orbital sets.

Figure 2 shows the calculated non-SOC band structures, the partial density of
states (PDOS) projected on V-$d$ orbitals, and their corresponding orbital
characters near the Fermi level $E_{\rm F}$, calculated in DFT+$U$ with
$U=$2\text{\,}\mathrm{e}\mathrm{V}$$. The obtained bandgap of
$0.42\text{\,}\mathrm{e}\mathrm{V}$ is within the range of reported
experimental values [13, 12].

The oxidation state of V in $\text{VI}_{3}$ is $+3$, with electronic
configuration [Ar]4$s^{0}3d^{2}$. V atoms arrange in a honeycomb structure
within the layer with edge-sharing octahedral coordination by I ligands. The
octahedral ligand and crystal fields split the V-$d$ states into the doublet
$e_{g}$ and triplet $t_{2g}$ subsets, with the latter states lying lower in
energy. Therefore, the two unpaired electrons of V3+ occupy the $t_{2g}$
orbitals, both with spins up according to Hund’s first rule, forming the spin
state $S=1$, consistent with the calculated on-site magnetic moment of V atom
$m\approx 2\mu_{\text{B}}$/V.

Correlation effects beyond DFT are required to describe the semiconducting
ground state of $\text{VI}_{3}$ correctly [49]. Figure 2(c) shows that the
$t_{2g}$ orbital set further splits into doubly degenerate ($1e_{x}$,
$1e_{y}$) and singly degenerate ($1a$) orbital sets. While DFT gives a half-
metallic state, the on-site Coulomb correction shifts the $1a$ state upward
relative to the ($1e_{x}$, $1e_{y}$) states, resulting in a bandgap in between
[49, 23]. We note that one may need to control the initial orbital occupancy
in the DFT+$U$ calculation to converge to the experimental semiconducting
ground state for $\text{VI}_{3}$, which may explain the discrepancy of ground
states found in previous DFT+$U$ calculations [23, 21, 50].

We also qualitatively explore the effects of compressive strain on the
electronic structure by gradually decreasing the lattice parameters (volume)
of bulk $\text{VI}_{3}$ while keeping the aspect ratio fixed. We found that
the ($1e_{x}$, $1e_{y}$) states shift upward relative to the $1a$ state, and a
semiconductor-to-metal transition occurs at a strain of 5–6% when the gap
closes, and all three $t_{2g}$ states become partially occupied. A more
systematic and comprehensive theoretical investigation on structural
properties may help compare with or guide the future experiments in this
direction.

###  III.2 Exchange coupling in monolayer, bilayer, and bulk $\text{VI}_{3}$

Table 1: Pairwise intralayer exchange parameters $J_{i}$, $i=1$, $2$, $3$ and interlayer exchange parameters $\tilde{J}_{i}$, $i=0$, $1+$, $1-$ for the Heisenberg Hamiltonian $H=-\sum_{i\neq j}J_{ij}\,\hat{\bf e}_{i}\cdot\hat{\bf e}_{j}$ in monolayer, bilayer, and bulk R-$\text{VI}_{3}$ calculated within DFT+$U$ with various $U$ values (in the unit of eV). Positive (negative) $J_{ij}$ values correspond to FM (AFM) couplings. Exchange parameters are illustrated in Fig. 1 for the bulk structure. The degeneracy (No.) and distance ($R_{ij}$) of $J_{ij}$ are also provided.  R-$\text{VI}_{3}$ |  | Lbl. | No. |  | $R_{ij}$ |  |  $J_{ij}$(meV)
---|---|---|---|---|---|---|---
Form |  |  |  |  | (Å) |  |  $U$=2 |  $U$=3 |  $U$=3.7
Monolayer |  | $J_{1}$ | 3 |  | 3.946 |  | 2.26 | 2.07 | 1.99
|  $J_{2}$111Intra-sublattice couplings for monolayer and bulk cases. | 6111Intra-sublattice couplings for monolayer and bulk cases. |  | 6.835 |  | 0.06 | 0.08 | 0.09
| $J_{3}$ | 3 |  | 7.893 |  | -0.23 | -0.18 | -0.16
Bilayer |  | $J_{1}$ | 3 |  | 3.946 |  | 2.26 | 2.07 | 2.03
| $J_{2}$ | 6 |  | 6.835 |  | 0.04 | 0.06 | 0.06
| $J_{3}$ | 3 |  | 7.893 |  | -0.16 | -0.14 | -0.12
| ${\tilde{J}_{0}}$ | 1/0222Different V sites have different numbers of neighbors coupled through $\tilde{J}_{0}$ or $\tilde{J}_{1+}$. |  | 6.552 |  | -0.14 | -0.07 | -0.08
| ${\tilde{J}_{1-}}$ | 3 |  | 7.660 |  | 0.08 | 0.06 | 0.04
| ${\tilde{J}_{1+}}$ | 0/3222Different V sites have different numbers of neighbors coupled through $\tilde{J}_{0}$ or $\tilde{J}_{1+}$. |  | 7.672 |  | 0.25 | 0.17 | 0.18
Bulk |  | $J_{1}$ | 3 |  | 3.946 |  | 2.18 | 2.02 | 1.98
|  $J_{2}$111Intra-sublattice couplings for monolayer and bulk cases. | 6 |  | 6.835 |  | 0.10 | 0.09 | 0.08
| $J_{3}$ | 3 |  | 7.893 |  | -0.20 | -0.16 | -0.15
| ${\tilde{J}_{0}}$ | 1 |  | 6.552 |  | 0.27 | 0.17 | 0.14
|  ${\tilde{J}_{1-}}$111Intra-sublattice couplings for monolayer and bulk cases. | 6 |  | 7.660 |  | 0.18 | 0.13 | 0.11
| ${\tilde{J}_{1+}}$ | 3 |  | 7.672 |  | -0.05 | -0.03 | -0.02
Figure 3:  Interlayer magnetic oupling energy, $\Delta
E=E_{\text{AFM}}^{\text{L-L}}-E_{\text{FM}}^{\text{L-L}}$, as a function of
on-site Coulomb interaction parameter $U$ in bilayer $\text{VI}_{3}$ with
hexagonal and monoclinic stackings.

We next discuss the dependence of exchange couplings on dimensionality,
additional on-site Coulomb interaction $U$, and stacking order. Table 1
summarizes the exchange parameters calculated in monolayer, bilayer, and bulk
$\text{VI}_{3}$ with different $U$ values. Although increasing $U$ values
generally decrease the exchange coupling, suggesting a more localized-moment
picture, qualitative details remain the same. In particular, there is no
change in their signs for the considered $U$ values.

In all the monolayer, bilayer, and bulk forms, the nearest neighbor intralayer
exchange coupling $J_{1}$, arises from a superexchange via the near-90∘ V-I-V
bonds. It is found to be dominant and FM, while $J_{2}$ and $J_{3}$ are weakly
FM and AFM, respectively. In bulk $\text{VI}_{3}$, the first two nearest
neighbor interlayer parameters $\tilde{J}_{0}$ and $\tilde{J}_{1-}$ are found
to be positive, leading to a favorable FM interlayer ordering. Note that
although $\tilde{J}_{1+}$ is negative, its magnitude is small. The overall
interlayer coupling is dominated by $\tilde{J}_{0}$ and $\tilde{J}_{1-}$,
especially the latter, which has a larger coordination number of six.

Interestingly, the interlayer couplings show a much stronger dependence on
dimensionality than the intralayer ones. As shown in Table 1, the intralayer
couplings only change slightly in three different forms. In contrast, the
overall interlayer coupling in bilayer remains FM as in bulk, the individual
interlayer couplings $\tilde{J}_{0}$ and $\tilde{J}_{1+}$ change signs when we
compare the bilayer and bulk cases. Note that the coordination number of
interlayer couplings decreases by a factor of two in the bilayer form.

We also investigate the stacking-dependent magnetism in $\text{VI}_{3}$ and
found a similar magnetostructural coupling as previously found in
$\text{CrI}_{3}$ [51, 52, 53]. Figure 3 shows the $U$ dependence of interlayer
magnetic coupling energy in bilayer $\text{VI}_{3}$ with hexagonal stacking
and monoclinic stacking. The energy difference is calculated as $\Delta
E=E_{\text{AFM}}^{\text{L-L}}-E_{\text{FM}}^{\text{L-L}}$, with
$E_{\text{AFM}}^{\text{L-L}}$ and $E_{\text{FM}}^{\text{L-L}}$ being the
energies of AFM-ordered layer-layer and FM-ordered layer-layer configurations,
respectively. For the hexagonal stacking, FM interlayer ordering between the
layers is found to be stable over a wide range of $U$. For the monoclinic
stacking, in contrast, the energy difference $\Delta E$ is very small at
$U\approx$2\text{\,}\mathrm{e}\mathrm{V}$$, indicating competing interlayer
AFM and FM states. Moreover, a crossover from the interlayer-FM to interlayer-
AFM magnetic ground state occurs at
$U\approx$2.25\text{\,}\mathrm{e}\mathrm{V}$$. These results point to the
strong magnetostructural coupling in bilayer $\text{VI}_{3}$. Consequently,
control of the magnetic configuration, for example, via strain or electric
field, is feasible, facilitating device applications.

###  III.3 Spin-wave dispersion

---
Figure 4:  (a) Spin-wave spectra calculated with only intralayer exchange
couplings $J_{i}$, $i=1$, $2$, $3$ involved (red), intralayer $J_{i}$, $i=1$,
$2$, $3$, and interlayer $\tilde{J}_{0}$ involved (blue dashed), and all
calculated $J_{i}$ and $\tilde{J}_{i}$ exchange couplings involved (black).
For all three plots, the exchange coupling values used are those calculated
for the bulk $\text{VI}_{3}$. (b) Spin-wave spectra calculated along
[$h$,$k$,$l$] with various $l$ ($k_{z}$) values; [$h$,$k$,0] is along the
$\Gamma$–$K$–$M$ path (i.e., it is equivalent to the red line in panel a).

Utilizing the exchange parameters for the bulk $\text{VI}_{3}$, we calculated
the linear SW spectra and investigated how different exchanges affect the
spectra features. Such understanding may be useful to compare with future INS
experiments on $\text{VI}_{3}$. Figure 4(a) shows the SW spectra calculated
along the in-plane $\Gamma$–$K$–$M$ path and out-of-plane $\Gamma$–$Z$ path.
To better illustrate the effects of different exchanges on SW excitation, we
consider three cases with different numbers of interlayer couplings included
in the Hamiltonian: 1) only intralayer couplings ($J_{1}$-$J_{2}$-$J_{3}$;
solid red lines), 2) intralayer and the $1st$ nearest-neighbor interlayer
couplings ($J_{1}$-$J_{2}$-$J_{3}$-$\tilde{J}_{0}$; blue dashed lines), 3) all
of the considered intralayer and interlayer couplings
($J_{1}$-$J_{2}$-$J_{3}$-$\tilde{J}_{0}$-$\tilde{J}_{1-}$-$\tilde{J}_{1+}$;
solid black lines).

With only intralayer coupling considered, two SW branches cross exactly at
$K$, which is caused by the vanishing of the inter-sublattice coupling
$J_{{\bf q}}^{\text{Inter}}$, and thus the energy difference between in-phase
and out-of-phase inter-sublattice precessing. Including the first nearest
interlayer coupling $\tilde{J}_{0}$ leaves the acoustic branch unchanged but
lifts the optical branch by a constant and correspondingly shifts the Dirac
crossing toward $M$ point along the $\Gamma$–$K$–$M$ path. The constant
shifting of in-plane optical SW spectra is because $\tilde{J}_{0}$ is along
the $z$ direction. On the other hand, the SW spectra along the $\Gamma$–$Z$
direction become dispersive with $\tilde{J}_{0}$. Unlike $\tilde{J}_{0}$,
interlayer couplings $\tilde{J}_{1+}$ and $\tilde{J}_{1-}$ have nonzero in-
plane components in their connecting vectors (similar to those of $J_{1}$).
They induce additional dispersion on in-plane SW spectra but preserve the
Dirac crossing at $K$ (not shown) in the absence of $\tilde{J}_{0}$. We note
that comparing the dispersions of the acoustic and optical modes along the
$\Gamma$–$Z$ direction may tell whether the dominant FM interlayer coupling is
intra-sublattice or inter-sublattice. For example, as shown in Fig. 4(a), FM
$\tilde{J}_{1-}$ increases both branches’ energies while the FM
$\tilde{J}_{0}$ increases the acoustic mode energy but lowers the optical mode
energy when we move along $\Gamma$–$Z$. It would be interesting if future INS
can be used to unveil the nature of interlayer couplings.

Figure 4(b) shows the SW spectra in bulk $\text{VI}_{3}$ calculated along
paths that are parallel to $\Gamma$–$K$–$M$ but at various $k_{z}$ planes
(denoted by different colors in Fig. 4(b)). All of the six exchange couplings
are included. At finite $k_{z}$, the interlayer couplings rotate the Dirac
crossing off this path. Even at $k_{z}=0$ plane, including further interlayer
couplings can also rotate the Dirac crossing off the high-symmetry line and
opens up a gap along the $\Gamma$–$K$–$M$ high-symmetry path [36].

Relativistic exchanges and MAE are not included in the model spin Hamiltonian
to calculate SW spectra. Easy-axis MAE, in either single-ion or anisotropic
exchange (equivalently two-ion MAE) form, introduces a gap of $\Delta=(2S-1)K$
at $\Gamma$. On the other hand, Dzyaloshinkii-Moriya interactions (DMI) and
Kitaev interaction can open up a global gap [25] between two magnon branches.

###  III.4 Magnetization vs Temperature from non-linear spin-wave theory

Figure 5: (Color online) Magnetization dependence on temperature in zero
magnetic field calculated using self-consistently renormalized spin-wave
theory [41]. Plots for the monolayer, bilayer, and bulk VI3 are obtained by
using exchange parameters calculated for $U=2$ eV, as listed in Table 1, and
the calculated MAE value of 71 $\mu$eV. Inset: the same dependence for the
bulk system, resulting from the three different sets of exchange couplings
calculated for $U=2$, $3$, and $3.7$ eV.

We further use SRSWT to explore the temperature dependence of magnetization,
$M(T)$, of the monolayer, bilayer, and bulk VI3, by utilizing the calculated
MAE and exchange parameters. For the bulk system, exchange parameters
calculated for on-site Coulomb correction of 2, 3, and 3.7 eV yield magnetic
ordering temperature values of 48.9, 42.3, and 39.5 K, respectively (see Fig.
5 inset). By noting that the ordering temperature found for $U=2$ eV is in
excellent agreement with the experimentally reported value of 49 K [13], for
the analysis of monolayer and bilayer systems, we use the exchange couplings
calculated with this value of $U$. For the momentum-space integrals involved
in the SRSWT self-consistency equations, we use a very fine $k$-point mesh of
100$\times$100 for 2D and 100$\times$100$\times$100 for 3D.

The plots in Fig. 5 show a typical mean-field-like behavior, implying a first-
order phase transition with the magnetization abruptly vanishing at a critical
temperature. As expected, the magnetization is strongly quenched in lower
dimensions; The ordering temperature decreases to about 13 K and 24.3 K in
monolayer and bilayer $\text{VI}_{3}$, respectively. Overall, these results
attest to the feasibility of intrinsically ferromagnetic monolayer and bilayer
VI3.

###  III.5 Constituents of MAE and validity of perturbation treatment

Figure 6: MAE as a function of the SOC scaling factor $\lambda_{M,\text{I}}$
($M$ = V or Cr) in monolayer $\text{CrI}_{3}$ (a) and $\text{VI}_{3}$ (b). The
calculated values are presented by symbols, and the lines are fits using the
data calculated with $\lambda\in[0,0.4]$. The red dashed, blue straight, and
black dotted lines correspond to scaling with $\lambda_{\text{I}}$ (at
$\lambda_{M}=0$), $\lambda_{M}$ (at $\lambda_{\text{I}}=0$), and
$\lambda_{\text{I}}=\lambda_{M}$, respectively. The calculated MAE at
$\lambda_{\text{I}}=\lambda_{M}=1$ corresponds to normal DFT result.

The calculated MAE in $\text{VI}_{3}$ is much smaller than in
$\text{CrI}_{3}$. To understand the role of constituent atoms on the MA in two
compounds we vary the SOC strength on V/Cr and I sites independently or
simultaneously, to study how MA evolves in monolayer $\text{VI}_{3}$ and
compared it with monolayer $\text{CrI}_{3}$. Figure 6 shows the calculated MAE
as functions of scaling parameter $\lambda$ with three different scaling
schemes, as denoted by red cross, blue triangle, and blue circles. We extract
corresponding sublattice contributions by fitting Eq. (7) data calculated with
$\lambda\in[0,0.4]$.

The large SOC on I sites is essential in providing the uniaxial anisotropy in
$\text{VI}_{3}$ and $\text{CrI}_{3}$. Without including SOC on I site,
$\text{CrI}_{3}$ has a negligibly small easy-plane anisotropy from
$K_{\text{Cr-Cr}}$ while $\text{VI}_{3}$ exhibits an even larger easy-plane
MAE from $K_{\text{V-V}}$. A small positive Cr-I contribution
$K_{\text{Cr-I}}$ in $\text{CrI}_{3}$; a 9% decrease in MAE is observed when
one removes SOC from Cr in $\text{CrI}_{3}$. Our DFT results for
$\text{CrI}_{3}$ are consistent with previous calculations [26, 54].
Remarkably, the much smaller MAE in $\text{VI}_{3}$ than in $\text{CrI}_{3}$
results mostly from a large negative inter-sublattice contribution
$K_{\text{V}-\text{I}}$; which had also been found in $L1_{0}$ FeNi and FePd
[47, 27]. Turning off SOC on V increases the MAE in $\text{VI}_{3}$ by 335%.

SOPT can be very useful to describe and understand MAE in various systems [55,
56]. Figure 6 also shows the quadratic fits of MAE values, on the basis of Eq.
(7), using data calculated for $\lambda\in[0,0.4]$, and corresponding
extrapolation to $\lambda=1.5$. The deviation of the fit from the calculated
data indicates the validity of the SOPT treatment of the SOC and MAE. For
small perturbations (small $3d$ SOC comparing to bandwidth) one expects a
perfect quadratic dependence of MAE on the scaling factors with the SOPT, as
shown in the plots in Fig. 6 for the scaling scheme that
$\lambda_{\text{V/Cr}}$ is varied while $\lambda_{\text{I}}=0$. However, the
large SOC constant of I-$p$ orbitals causes deviations of MAE from SOPT in
$\text{CrI}_{3}$ and $\text{VI}_{3}$, especially the latter. Comparing the
fitted $K(\lambda_{\text{V}}=\lambda_{\text{I}})$ with the calculated ones,
$\text{VI}_{3}$ shows a dramatically larger deviation of 42% than 9% in
$\text{CrI}_{3}$. This large deviation leads to the conclusion that SOPT
describes MA effects better in $\text{CrI}_{3}$ than in $\text{VI}_{3}$. To
confirm this conclusion, we have also calculated MAE of $\text{CrI}_{3}$ and
$\text{VI}_{3}$ using force theorem and SOPT with full-potential linear
augmented plane wave (FP-LAPW) method [57]. The calculations produced about
75% (18%) different MAE values between the two different methods for
$\text{VI}_{3}$ ($\text{CrI}_{3}$). Note that $\text{VI}_{3}$ has a much
smaller bandgap than $\text{CrI}_{3}$, which likely makes SOPT less valid in
describing the MAE in $\text{VI}_{3}$.

##  IV CONCLUSIONS

In summary, we have theoretically studied the electronic and magnetic
properties of the two-dimensional (2D) van der Waals magnet VI3 in various
forms. The interlayer exchange couplings show a stronger dependence on the
dimensionality of materials than the intralayer couplings. Moreover,
$\text{VI}_{3}$ also shows a strong stacking-dependent interlayer coupling,
similar to $\text{CrI}_{3}$, rendering the bilayer ground state interlayer
ordering readily switchable between the FM and AFM configurations upon
external stimuli, such as strain and electric field. Considering the relative
positions of and the hybridizations between cation-$3d$ and anion-$p$ states
can be essential to determine the superexchange coupling in $\text{VI}_{3}$,
future studies going beyond DFT+$U$ and including non-local exchange
correlations will be useful to investigate the magnetic interactions in
$\text{VI}_{3}$. Within SRSWT, the calculated $T_{\text{C}}$ in bulk agrees
well with experiments while $T_{\text{C}}$ in the monolayer and bilayer are
quenched down to 14 and 24 K, respectively. The sublattice-resolved MAE
calculated in the monolayer $\text{VI}_{3}$ reveals that a strong V-I inter-
ion easy-plane contribution to MAE lowers the overall easy-axis MAE in
$\text{VI}_{3}$. Finally, we demonstrate that the second-order perturbation
treatment of the MAE is less valid in $\text{VI}_{3}$ than in
$\text{CrI}_{3}$, due to the combination of large SOC on I sublattice and a
smaller bandgap in $\text{VI}_{3}$.

## ACKNOWLEDGMENTS

The authors thank B. Harmon, P. Ong and G. Miller for valuable discussions.
This work was supported by the U.S. Department of Energy, Office of Science,
Office of Basic Energy Sciences, Materials Sciences and Engineering Division,
and Early Career Research Program. Ames Laboratory is operated for the U.S.
Department of Energy by Iowa State University under Contract No. DE-
AC02-07CH11358.

##  Appendix A Intralayer coupling

For monolayer, the energies (per magnetic atom) of FM, Néel-, zigzag-, and
stripy-AF configurations, $E_{\text{FM}}$, $E_{\text{N}}$, $E_{\text{Z}}$, and
$E_{\text{S}}$, respectively, are mapped onto Eq. (1) as follows:

$E_{\mu}=E_{0}+\sum_{k=1}^{3}C^{\mu}_{k}J_{k},\quad\mu=\text{FM},\,\text{N},\,\text{Z},\,\text{S},$
(8)

where $E_{0}$ is the non-magnetic energy, and the coefficients $C^{\mu}_{k}$
are listed in Table 2.

Table 2: Coefficients of mapping of different spin configurations onto the spin Hamiltonian defined as $H=-\sum J_{ij}\hat{e}_{i}\cdot\hat{e}_{j}$. The right columns represent the difference, $\Delta C^{\mu}=C^{\mu}-C^{\text{FM}}$. Configurations | $C$ |  | $\Delta C$
---|---|---|---
$C^{\mu}_{1}$ | $C^{\mu}_{2}$ | $C^{\mu}_{3}$ |  | $\Delta C^{\mu}_{1}$ | $\Delta C^{\mu}_{2}$ | $\Delta C^{\mu}_{3}$
FM | -3 | -6 | -3 |  | 0 | 0 | 0
Néel | 3 | -6 | 3 |  | 6 | 0 | 6
Zigzag | -1 | 2 | 3 |  | 2 | 8 | 6
Stripy | 1 | 2 | -3 |  | 4 | 8 | 0

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# Fuzzy Logic and Markov Kernels

Rogier Brussee  Jheronimus Academy of Data Science, Sint Janssingel 92 5211 DA
den Bosch/ Technical University Eindhoven<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

Fuzzy logic is a way to argue with boolean predicates for which we only have a
confidence value between 0 and 1 rather than a well defined truth value. It is
tempting to interpret such a confidence as a probability. We use Markov
kernels, parametrised probability distributions, to do just that. As a
consequence we get general fuzzy logic connectives from probabilistic
computations on products of the booleans, stressing the importance of joint
confidence functions. We discuss binary logic connectives in detail and
recover the “classic” fuzzy connectives as bounds for the confidence for
general connectives. We push multivariable logic formulas as far as being able
to define fuzzy quantifiers and estimate the confidence.

###### Key words and phrases:

Fuzzy Logic, Markov Kernel, Probability Theory

##  1\. Introduction

The notion of fuzzy sets tries to formalise the idea that if we have a set of
things $X$ and a property $P$ for an element $x\in X$, (i.e. a predicate
function $P:X\to\mathbb{B}$), we can express a level of confidence in the
truth of $P(x)$, traditionally called “belief”, as a function $p:X\to[0,1]$ .
A predicate is in one to one correspondence with a subset $X_{P}=\\{x\in X\mid
P(x)\\}\subset X$, so the belief function $p(x)$ can be interpreted as a level
of confidence of membership of $x$ in $X_{P}$, a “fuzzy” membership function.
It is tempting to interpret this membership function as a probability. On
second thought this is a bit problematic, however, since $p(x)$ is a function
on $X$ and certainly not a probability distribution on $X$ as there is no way
to “add up” confidences at different points $x\in X$ to a confidence 1, and
even for discrete spaces there is no reason why they should. Nonetheless the
interpretation of fuzzy logic and its relation to probability theory has been
a continued discussion since the inception of fuzzy logic
[Ga][DP1][DP2][LSBDW][Za]. Here we will discuss a direct interpretation of
fuzzy logic in terms of probability theory and more particularly in the Markov
category. It interprets the belief function as a thinly disguised Markov
kernel with values in the Booleans, i.e. instead of interpreting the belief
function as a probability distribution _on_ $X$ we interpret the confidence
function as defining a probability distribution on the _booleans_ which is
_parametrised_ by $X$. Somewhat similar ideas seem to have been proposed by
Loginov [Lo] based on conditional probability rather than Markov kernels.

The main contribution of this paper is that this interpretation gives a
straightforward probabilistic interpretation of fuzzy logic and its
properties, which become a direct consequence of the properties of traditional
Boolean logic and standard probabilistic computations. In particular we
immediately get a family of fuzzy connectives like “and” and “or” parametrised
by the confidence that (n)either predicate is true. More conceptually, in this
interpretation logic connectives are only properly defined if we have a
_joined_ confidence in the predicates i.e. joined probability distribution on
two booleans rather than just the belief functions separately which only
determines the marginals. This makes sense intuitively: even if we only have
$50\%$ confidence in a predicate $P$, we must still have $0\%$ confidence in
$P\wedge\neg P$ and can have $100\%$ confidence in $P\vee\neg P$. Of course
the strong correlation between the two predicates is reflected in the joint
distribution of $(P,\neg P)$, but the joint confidence can also express more
subtle “if $P_{1}$ then $P_{2}$ is more or less likely” correlations. The
current approach makes it possible, to make use of joined confidences/beliefs
for correlated predicates in a natural way. In fact, the same argument that
gives a fuzzy/probabilistic interpretation for binary logic relations also
works for more general systems of logic formulas. We can even even extent this
framework to quantification over infinite sets. However, while this Markov
interpretation is useful and precise, it also requires to specify joint
confidence functions which have $2^{n}-1$ degrees for $n$ predicates. Thus it
is useful to have upper and lower bounds for logic connectives or results
making assumptions like independence. It turns out that sensible bounds and
independence simplifications exist and are given by connectives in their
“traditional” fuzzy forms (plural!), “explaining” why an approach with logic
flavour often works.

The organisation of this paper is as follows: in section 2 we define Markov
kernels and the Markov category, as a quick recap and to fix notation. In
section 3 we introduce Fuzzy logic, again mainly to fix notation and specify
what form of fuzzy logic we use. In section 5 we reformulate the belief
function in terms of Markov kernels and use them to define binary logic
connectives. As an application of this probabilistic interpretation, we
describe the relation to ”classical” logic connectives and prove a de Morgan
Law. We finally discuss the general case and quantifiers in section 5.

In the rest of this paper we use the standard notations
$\mathbb{R},\mathbb{N}$ for the real numbers, and nonnegative integers,
$[a,b]$, $(a,b]$ or $(a,b)$ for the closed or (half) open interval and the
less standard $\mathbb{B}=\\{\mathbf{F},\mathbf{T}\\}$ for the Booleans.

###  1.1. competing interest

The author gracefully acknowledges support from the RNOB Regio deal Noord Oost
Brabant, but otherwise declares no competing interest in this research.

##  2\. Markov kernels

A Markov kernel is a parametrised version of a probability distribution. Just
like a probability distribution formalises a random value, a Markov kernel
formalises a function with random values. The formal definition leans on the
language of measure theoretic probability theory, but it is mainly used as
solid foundation supporting an oiled machinery. The section is mainly needed
because we liberally use the language from this section, in particular that of
functoriality of pushforward. Some readers may prefer to only take a cursory
look at the definition of Markov kernel and pushforward, and go back to other
definitions as needed.

Let $X=(X,\mathcal{A})$ and $Y=(Y,\mathcal{B})$ be two measurable spaces, i.e.
a set together with a $\sigma$-algebra of measurable subsets [Ru, Ch 1].

######  Definition 1.

A Markov kernel [Do, Appendix VI] $F:X\to Y$ is a map $\mathcal{B}\times
X\to[0,1]$ such that

  * •

for each $B\in\mathcal{B}$, the map

(1) $\displaystyle F(B,\mathchar 45\relax):X$ $\displaystyle\to[0,1]$ (2)
$\displaystyle x$ $\displaystyle\mapsto F(B,x)$

is a measurable function on $X$,

  * •

for each $x\in X$ the map

(3) $\displaystyle F(\mathchar 45\relax,x):\mathcal{B}$
$\displaystyle\to[0,1]$ (4) $\displaystyle B$ $\displaystyle\mapsto F(B,x)$

is a probability measure on $Y$.

As a convenient (manifestly asymmetric!) notation we write $F(dy|x)$ for the
probability measure on $Y$ defined by the point $x\in X$. We likewise write
$F(B|x)=\int_{B}F(dy|x)$ for $F(B,x)$. The notation also suggests conditional
probability. While this is not a bad way to think about it, the two concepts
are related but technically and conceptually slightly different: a conditional
probability is a Markov kernel defined almost surely with respect to a
probability measure on the “universe of discourse” $X$. This background
measure on $X$ can become a block in interpretation, particularly if one
liberally identifies measures with functions on discrete spaces. It is also
the way people denote parametrised probability distribution, e.g.
$G(d^{n}x|\Sigma,\mu)$ for a Gaussian distribution on $R^{n}$ that is
parametrised by its covariance matrix $\Sigma$ and mean $\mu$.

The $\sigma$-algebras and measurability of the function $F(B|\mathchar
45\relax)$ are a convenient technical setup allowing us to define integration
without worries, and depending on the $\sigma$ algebra disallow pathological
constructions using the axiom of choice (e.g. if $X$ and $Y$ are topological
spaces and the $\sigma$-algebras are the corresponding Borel algebras, all
continuous functions are allowed) or give built in adaptation to some
“pixelation” of the space. The meat of the definition is the second part: it
allows us to think of Markov kernels as probabilistic “functions” that assign
a probability distribution $F(dy|x)$ on $Y$ rather than a specific value for
every $x$ in $X$. Conversely, every normal measurable function $f:X\to Y$
defines a Markov kernel111 If the $\sigma$-algebra $\mathcal{B}$ on $Y$ does
not separate points, different functions may define the same kernel. by
$F(dy|x)=\delta_{f(x)}(dy)$ i.e.

(5) $F(B|x)=\begin{cases}1&\text{ if }f(x)\in B\\\ 0&\text{
otherwise}\end{cases}$

In particular, the identity function on $X$ has the Markov kernel

(6) $1_{X}(dx^{\prime}|x)=\delta_{x}(dx^{\prime})$

Like functions, Markov kernels can be composed: if $F:X\to Y$ and $G:Y\to Z$
are Markov kernels, then the composition of the kernels $G\circ F$ is defined
by

(7) $(G\circ F)(dz|x)=\int_{Y}G(dz|y)F(dy|x)$

i.e. for each measurable subset $C\subset Z$ and $x\in X$ we define

(8) $(G\circ F)(C|x)=\int_{Y}G(C|y)F(dy|x)$

One checks that $G\circ F$ is indeed a Markov kernel, that $\circ$ is
associative and $1_{X}$ is the identity on $X$. This gives measurable spaces
and Markov kernels the structure of a category [McL, section I.2], the Markov
category [Pa], (aka Lawvere category [La], aka Giry monad[Gi] aka the
Stochastic Category [Vo]). The category has an initial object, the empty set
$\emptyset$, and a final object, the one point set $\star=\\{\ast\\}$. A
probability measure $\mathbb{P}$ on a measurable set $X$ is “the same thing”
as a Markov kernel $P:\star\to X$ because the Markov kernel is fully
determined by the probability measure $P(dx|\ast)=\mathbb{P}(dx)$. Hence a
probability space $(X,\mathcal{A},\mathbb{P})$ is “the same thing” as a
pointed space $\star\to X$ in the Markov category and we will identify them.
More generally, if $X$ comes with a family of probability distributions
parametrised by $S$, (i.e. a Markov kernel $\mathbb{P}:S\to X$) then for every
kernel $F:X\to Y$ we have a pushforward $F_{*}\mathbb{P}=F\circ\mathbb{P}:S\to
Y$. The pushforward222There is also a functorial contravariant pullback: a
kernel $\phi:Y\to T$ gives a kernel $F^{*}\phi:X\to T$ by $F^{*}\phi=\phi\circ
F$ which satisfies $(G\circ F)^{*}=F^{*}\circ G^{*}$: is a covariant functor,
i.e. for $G:Y\to Z$ we have

(9) $(F\circ G)_{*}=F_{*}G_{*}$

This _functoriality_ is just expressing the fact that $\circ$ is associative
here. If a kernel comes from a measurable function $f:X\to Y$ the pushforward
is very explicit. For $B\in\mathcal{B}$

(10) $(f_{*}P)(B|s)=P(f^{-1}(B)|s)$

i.e. it is just the pushforward of measures but parametrised with $s\in S$.
Given $F_{1}:X\to Y_{1}$ and $F_{2}:X\to Y_{2}$ there does not, in general,
exist a _unique_ Markov kernel

(11) $F_{12}:X\to Y_{1}\times Y_{2}$

with marginals $\pi_{1*}F_{12}=F_{1}$ and $\pi_{2*}F_{12}=F_{2}$, where
$\pi_{1},\pi_{2}$ are the projections on the first and second coordinate. This
is because marginal probability distributions do not, in general, determine a
joint probability distribution even though the marginals may heavily restrict
the joint distribution. We can, however, define a conditionally independent
(i.e. independent for every $x$) joint kernel $F_{1}\otimes F_{2}:X\to
Y_{1}\otimes Y_{2}$ defined by the product measure [Ru, Definition 7.7]

(12) $(F_{1}\otimes F_{2})(d(y_{1},y_{2})|x)=F_{1}(dy_{1}|x)F_{2}(dy_{2}|x).$

On measurable subsets $B_{1}\times B_{2}\subset Y_{1}\times Y_{2}$, it is
given by $(F_{1}\otimes F_{2})(B_{1}\times
B_{2}|x)=F_{1}(B_{1}|x)F_{2}(B_{2}|x)$ which is then uniquely extended to the
product $\sigma$-algebra using the Caratheodory extension theorem.

To explain what we are going to do with logic connectives $\wedge$ and $\vee$
it is useful to start with the more familiar situation of adding random
variables. Therefore first assume that $(X,\mathcal{A})$ is not just a
measurable space but a probability space $(X,\mathcal{A},\mathbb{P})$ for some
probability measure $\mathbb{P}$. Recall that in this setup a random variable
or stochast333In this paper uppercase “end of alphabet” latin letters denote
(measure) spaces rather than stochasts. is a measurable map
$f:X\to\mathbb{R}$. The law of the stochast $f$ is the pushforward measure
$f_{*}\mathbb{P}$, which is a probability distribution on $\mathbb{R}$.
Concretely, for an interval $(-\infty,b]\subset\mathbb{R}$ the pushforward
measure gives the cumulative distribution

(13)
$(f_{*}\mathbb{P})((-\infty,b])=\mathbb{P}(f^{-1}(-\infty,b])=\mathbb{P}(f(x)\leq
b).$

For $F:X\to\mathbb{R}$ a Markov kernel, we have the pushforward probability
measure $F_{*}\mathbb{P}=F\circ\mathbb{P}$ with a cumulative distribution

(14) $(F_{*}\mathbb{P})((-\infty,b])=\int_{X}F((-\infty,b]|x)\mathbb{P}(dx)$

For two stochasts $f_{1},f_{2}:(X,\mathcal{A},\mathbb{P})\to\mathbb{R}$, we
have a well defined sum function $(f_{1}+f_{2})(x)=f_{1}(x)+f_{2}(x)$. The law
of their sum $f_{1}+f_{2}$ is the probability measure
$(f_{1}+f_{2})_{*}\mathbb{P}$. However, it is well known that, in general, the
law of the sum is _not_ determined by the laws ${f_{1}}_{*}\mathbb{P}$ and
${f_{2}}_{*}\mathbb{P}$ of the stochasts separately. For example if $f$ is
normally distributed with mean $0$ and standard deviation $1$, and $f_{1}=f$,
$f_{2}=f$, then $f_{1}$ and $f_{2}$ are also normally distributed with
standard deviation $1$ and $f_{1}+f_{2}=2f$ is normally distributed with
standard deviation $2$, while if $f^{\prime}_{1}=f$, $f^{\prime}_{2}=-f$ then
again $f_{1}$ and $f_{2}$ are again normally distributed with standard
deviation $1$, but obviously $f_{1}+f_{2}=0$. What _does_ determine the law of
the sum is the _joint_ law $(f_{1}\times f_{2})_{*}\mathbb{P}$ on
$\mathbb{R}^{2}=\mathbb{R}\times\mathbb{R}$. This follows from functoriality
since if we define $\mathbf{add}(y_{1},y_{2})=y_{1}+y_{2}$ then
$f_{1}+f_{2}=\mathbf{add}\circ(f_{1}\times f_{2})$ , and the joint law is

(15) $(f_{1}+f_{2})_{*}\mathbb{P}=(\mathbf{add}\circ(f_{1}\times
f_{2}))_{*}\mathbb{P}=\mathbf{add}_{*}\left((f_{1}\times
f_{2})_{*}\mathbb{P}\right).$

For Markov kernels $F_{1},F_{2}:X\to\mathbb{R}$ there is a very similar
complication: they do not define a well defined kernel which one would
reasonably call “$F_{1}+F_{2}$”. In order to compute the distribution of the
sum, we need a _joint_ Markov kernel i.e. a kernel $F_{12}:X\to\mathbb{R}^{2}$
with marginals ${\pi_{1}}_{*}F_{12}=F_{1}$ and ${\pi_{2}}_{*}F_{12}=F_{2}$,
where again $\pi_{1},\pi_{2}$ are the projections on the first and second
coordinate. Such a joint kernel certainly exists since we can always assume
take $F_{1}$ and $F_{2}$ are conditionally independent and take $F_{1}\otimes
F_{2}$

are conditionally independent (i.e. independent for every $x$) and take
$F_{12}=F_{1}\otimes F_{2}$ on $\mathbb{R}^{2}$. However, as mentioned
earlier, even though $F_{12}$ is restricted by the marginals $F_{1}$ and
$F_{2}$, there is, in general, no uniquely determined choice. _Given_ a joint
Markov kernel, however, we have a Markov kernel that by abuse of notation we
may call the sum

(16) $F_{1}+_{F_{12}}F_{2}:=\mathbf{add}_{*}F_{12}.$

Here the notation $+_{F_{12}}$ is supposed to indicate that we use the _joint_
Markov kernel $F_{12}$ to define addition. This notation is a bit silly
because $F_{12}$ already determines $F_{1}$ and $F_{2}$, but If no confusion
seems possible we write

(17) $F_{1}+_{12}F_{2}:=F_{1}+_{F_{12}}F_{2}.$

Since $F_{12}$ although not uniquely determined by the marginals, may still be
substantially restricted, and so $F_{1}+_{12}F_{2}$ is restricted.

######  Example 1.

Consider two Markov kernels $F_{1},F_{2}:\Delta\to\mathbb{R}$ on the interval
$\Delta_{1}=[0,1]$ with values in $\\{\pm 1\\}\subset\mathbb{R}$, representing
the loss or gain for two parties in two Bernouilly experiments each with a
chance $p\in\Delta$ for heads ($h$) and $(1-p)$ for tails ($t$). Party 1 wins
on heads, Party 2 on tails. What we do not know is the correlation between the
two Bernouilly experiments. We then have

(18) $\displaystyle F_{1}(dy_{1}|p)$
$\displaystyle=(p\delta_{+1}+(1-p)\delta_{-1})(dy_{1})$ (19) $\displaystyle
F_{2}(dy_{2}|p)$ $\displaystyle=((1-p)\delta_{+1}+p\delta_{-1})(dy_{2})$

An equivalent way to write down this kernel is as probabilities of the
different outcomes

(20) $\displaystyle F_{1}(+1|p)$ $\displaystyle=p,$ $\displaystyle
F_{2}(+1|p)$ $\displaystyle=(1-p)$ (21) $\displaystyle F_{1}(-1|p)$
$\displaystyle=1-p$ $\displaystyle F_{2}(-1|p)$ $\displaystyle=p$ (22)
$\displaystyle F_{1}(B|p)$ $\displaystyle=0$ $\displaystyle F_{2}(B|p)$
$\displaystyle=0\text{ if }\pm 1\notin B$

where we used the usual abuse of notation in the probability distributions of
discrete sets to write e.g. $F_{1}(+1|p)=F_{1}(\\{-1\\}|p)$ which we will
continue to do. The joint Markov kernels can then be parametrised as

(23) $\displaystyle F_{12}((+1,+1)|p)$ $\displaystyle=p_{ht},$ $\displaystyle
F_{12}((+1,-1)|p)$ $\displaystyle=p_{hh}$ (24) $\displaystyle
F_{12}((-1,+1)|p)$ $\displaystyle=p_{tt}$ $\displaystyle F_{12}((-1,-1)|p)$
$\displaystyle=p_{th}$ (25) $\displaystyle F_{12}(B|p)$ $\displaystyle=0\text{
if }(\pm 1,\pm 1)\notin B$

where the condition that $F_{12}$ has $F_{1}$ and $F_{2}$ as marginals gives

(26) $\displaystyle p_{hh}+p_{ht}$ $\displaystyle=p$ $\displaystyle
p_{hh}+p_{th}$ $\displaystyle=p$ (27) $\displaystyle p_{th}+p_{tt}$
$\displaystyle=1-p$ $\displaystyle p_{ht}+p_{tt}$ $\displaystyle=1-p$

with the inequalities $p_{ab}\geq 0$. The system of equations is equivalent to

(28) $\displaystyle p_{hh}+p_{ht}+p_{th}+p_{tt}$ $\displaystyle=1$ (29)
$\displaystyle p_{hh}+p_{ht}$ $\displaystyle=p$ (30) $\displaystyle
p_{ht}+p_{tt}$ $\displaystyle=1-p$

and the solutions are of the form

(31) $\displaystyle p_{hh}$ $\displaystyle=q$ $\displaystyle p_{ht}$
$\displaystyle=p-q$ (32) $\displaystyle p_{th}$ $\displaystyle=p-q$
$\displaystyle p_{tt}$ $\displaystyle=1-2p+q$

with $\min(2p-1,0)\leq q\leq p$. Given this joint kernel, the sum kernel is

(33) $\displaystyle(F_{1}+_{12}F_{2})(+2|p)$ $\displaystyle=p_{ht}=p-q$ (34)
$\displaystyle(F_{1}+_{12}F_{2})(0|p)$ $\displaystyle=p_{hh}+p_{tt}=1-2p+2q$
(35) $\displaystyle(F_{1}+_{12}F_{2})(-2|p)$ $\displaystyle=q_{th}=p-q$

What we are going to do for fuzzy logic is much the same except rather than
adding numbers, we are going to apply “and” or “or” to boolean values.

##  3\. Fuzzy Logic

For a set $X$, given the belief (confidence) $p:X\to[0,1]$ in a predicate
$P:X\to\mathbb{B}$ the belief in $\neg P(x)$ is $1-p(x)$.

(36) $(\neg p)=1-p$

For two properties $P_{1},P_{2}$ with belief functions $p_{1},p_{2}$, we
consider the belief (confidence) in $P_{1}(x)\wedge P_{2}(x)$ or equivalently
the belief (confidence) in $x\in X_{P_{1}}\cap X_{P_{2}}$ as a function of
$x$. Several natural choices for a fuzzy “and” that only depend on the values
$p_{1}(x)$ and $p_{2}(x)$ have been proposed [ATV], in particular:

(37) $\displaystyle p_{1}\underline{\wedge}p_{2}$
$\displaystyle=\max(0,p_{1}+p_{2}-1)$ (38) $\displaystyle p_{1}\curlywedge
p_{2}$ $\displaystyle=p_{1}p_{2}$ (39) $\displaystyle
p_{1}\overline{\wedge}p_{2}$ $\displaystyle=\min(p_{1},p_{2}).$

Note that all choices reduce to the usual “and” connective $p_{1}\wedge
p_{2}$, if $p_{1}$ and $p_{2}$ only take the value $0$ or $1$ and that

(40) $p\underline{\wedge}q\leq p\curlywedge q\leq p\overline{\wedge}q$

Likewise consider the belief (confidence) in $P_{1}(x)\vee P_{2}(x)$ i.e. the
confidence in $x\in X_{P_{1}}\cup X_{P_{2}}$ for which

(41) $\displaystyle p_{1}\underline{\vee}p_{2}$
$\displaystyle=\max(p_{1},p_{2})$ (42) $\displaystyle p_{1}\curlyvee p_{2}$
$\displaystyle=p_{1}+p_{2}-p_{1}p_{2}$ (43) $\displaystyle
p_{1}\overline{\vee}p_{2}$ $\displaystyle=\min(1,p_{1}+p_{2}).$

has been proposed. Note again that all choices reduce to the usual “or”
connective $p_{1}\vee p_{2}$ if $p_{1}$ and $p_{2}$ only take the values $0$
or $1$ and that

(44) $p_{1}\underline{\vee}p_{2}\leq p_{1}\curlyvee p_{2}\leq
p_{1}\overline{\vee}p_{2}.$

We will see later that $\underline{\wedge}$ and $\overline{\wedge}$ are
extremals of a 1 parameter family of such connectives with $\curlywedge$ as an
intermediate value, and similarly for $\underline{\vee}$, $\overline{\vee}$
and $\curlyvee$.

Given the proposed fuzzy “and” and “or connectives, the classical de Morgan’s
law

(45) $\neg(a\vee b)=(\neg a)\wedge(\neg b),\ \forall a,b\in\mathbb{B}$

for Boolean truth values have fuzzy analogues

(46) $\displaystyle p_{1}\underline{\vee}p_{2}$ $\displaystyle=\neg((\neg
p_{1})\overline{\wedge}(\neg p_{2}))$ (47) $\displaystyle p_{1}\curlyvee
p_{2}$ $\displaystyle=\neg((\neg p_{1})\curlywedge(\neg p_{2}))$ (48)
$\displaystyle p_{1}\overline{\vee}p_{2}$ $\displaystyle=\neg((\neg
p_{1})\underline{\wedge}(\neg p_{2}))$

we will later see that there is a De Morgan law for a whole 1 parameter family
with formulas (46), (47) and (48) as special cases.

##  4\. Fuzzy logic connectives and Markov kernels

With the notion of Markov kernel at hand, we can interpret a fuzzy set or
fuzzy predicates on a measurable set $(X,\mathcal{A})$ as a predicate kernel
$P:X\to\mathbb{B}$. A belief function $p:X\to[0,1]$ as in section 3 is
equivalent to a predicate kernel with

(49) $\displaystyle P(\mathbf{T}|x)$ $\displaystyle=p(x)$ (50) $\displaystyle
P(\mathbf{F}|x)$ $\displaystyle=1-p(x).$

Conversely the kernel predicate defines a belief function in this way. Again
we abused notation in the probability distributions of discrete sets to write
e.g. $P(\mathbf{T}|x)$ for $P(\\{\mathbf{T}\\}|x)$ here. This interpretation
will allow us to think of fuzzy logic as a “shadow” of ordinary logic by
suitably applying functoriality. In the process we will quickly find the need
to use joint confidence functions or joint kernels. As explained in section 2,
the introduction of a $\sigma$-algebra of subsets and assuming that $p$ is
measurable is a technical detail. In fact for most of what follows we could
take $\mathcal{A}=\mathcal{P}(X)$ the full powersetof X.

###  4.1. logical connective “not”

As a warmup, consider the belief function of the “not” connective $\neg$.
There is an associated function

(51) $\displaystyle\mathbf{not}:\mathbb{B}\to\mathbb{B}$ (52) $\displaystyle
t\mapsto\neg t$

The Markov kernel for $\neg P:X\to\mathbb{B}$ is then defined functorially on
predicate kernels

(53) $(\neg P):=(\mathbf{not}_{*}P).$

Its corresponding belief function is

(54) $\displaystyle(\neg p)(x)$ $\displaystyle=(\neg P)(\mathbf{T}|x)$ (55)
$\displaystyle=(\mathbf{not}_{*}P)(\mathbf{T}|x)$ (56)
$\displaystyle=P(\mathbf{not}^{-1}(\mathbf{T})|x)$ (57)
$\displaystyle=P(\\{\mathbf{F}\\}|x)$ (58) $\displaystyle=1-P(\mathbf{T}|x)$
(59) $\displaystyle=1-p(x)$

just as expected.

###  4.2. logical connective “and”

Now, just as the sum of two Markov kernels with values in $\mathbb{R}$ turned
out not to be well defined without a joint kernel, to define a logical “and”
connective of predicate Markov kernels $P_{1}$ and $P_{2}$ (Markov predicates
for short), we need a _joint_ Markov predicate $P_{12}$ lifting $P_{1}$ and
$P_{2}$. Spelled out: given Markov predicates $P_{1},P_{2}:X\to\mathbb{B}$
uniquely defined by $P_{i}(\mathbf{T}|x)=p_{i}(x)$, we need a joint Markov
predicate $P_{12}:X\to\mathbb{B}^{2}$ with conditional marginals
$\pi_{1*}P_{12}=P_{1}$ and $\pi_{2*}P_{12}=P_{2}$ where $\pi_{i}$ is
projection onto the $i$-th coordinate. Again, such a lift exists because again
as a special case of (12) we can take the joint Markov predicate
$P_{12}=P_{1}\otimes P_{2}$ assuming independence of $P_{1}$ and $P_{2}$. Here
this becomes very explicit:

(60) $(P_{1}\otimes P_{2})((a,b)|x)=P_{1}(a|x)P_{2}(b|x).$

However, since $\mathbb{B}^{2}$ is small we can easily write down the general
Markov predicate lifting $P_{1}$ and $P_{2}$:

(61) $\displaystyle P_{12}((\mathbf{F},\mathbf{F})|x)$
$\displaystyle=p_{FF}(x)$ $\displaystyle P_{12}((\mathbf{F},\mathbf{T})|x)$
$\displaystyle=p_{FT}(x)$ (62) $\displaystyle
P_{12}((\mathbf{T},\mathbf{F})|x)$ $\displaystyle=p_{TF}(x)$ $\displaystyle
P_{12}((\mathbf{T},\mathbf{T})|x)$ $\displaystyle=p_{TT}(x)$

where the $p_{ab}(x)\geq 0$ for all $x$ and

(63) $\displaystyle p_{FF}(x)+p_{FT}(x)$ $\displaystyle=1-p_{1}(x)$
$\displaystyle p_{FF}(x)+p_{TF}(x)$ $\displaystyle=1-p_{2}(x)$ (64)
$\displaystyle p_{TF}(x)+p_{TT}(x)$ $\displaystyle=p_{1}(x)$ $\displaystyle
p_{FT}(x)+p_{TT}(x)$ $\displaystyle=p_{2}(x).$

These equations are equivalent to

(65) $\displaystyle p_{FF}(x)+p_{FT}(x)+p_{TF}(x)+p_{TT}(x)$ $\displaystyle=1$
(66) $\displaystyle p_{TF}(x)+p_{TT}(x)$ $\displaystyle=p_{1}(x)$ (67)
$\displaystyle p_{FT}(x)+p_{TT}(x)$ $\displaystyle=p_{2}(x)$

which has solutions

(68) $\displaystyle p_{FF}(x)$ $\displaystyle=q(x)$ $\displaystyle p_{TF}$
$\displaystyle=(1-p_{2}(x))-q(x)$ (69) $\displaystyle p_{FT}(x)$
$\displaystyle=(1-p_{1}(x))-q(x)$ $\displaystyle p_{TT}$
$\displaystyle=p_{1}(x)+p_{2}(x)-1+q(x).$

Thus in this parametrisation (there can be others) the joint Markov predicate
is determined by a “belief function” $q(x)$ for _both_ predicates to be wrong.
The inequalities $p_{ab}(x)\geq 0$ translate inequalities

(70) $0\leq q_{\min}(x)\leq q(x)\leq q_{\max}(x)\leq 1$

for $q(x)$ where

(71) $\displaystyle q_{\min}(x)$
$\displaystyle:=\max(0,1-(p_{1}(x)+p_{2}(x)))$ (72) $\displaystyle
q_{\max}(x)$ $\displaystyle:=1-\max(p_{1}(x),p_{2}(x)).$

We also define

(73) $q_{\mathrm{indep}}(x):=(1-p_{1}(x))(1-p_{2}(x)).$

Given the joint Markov predicate we can now _by abuse of notation_ define a
Markov predicate $P_{1}\wedge_{12}P_{2}$ for the “and” connective by
functoriality

(74) $\displaystyle
P_{1}\wedge_{12}P_{2}:=P_{1}\wedge_{P_{12}}P_{2}:=\mathbf{and}_{*}P_{12}.$

As for the sum kernel in 17, the notation is a bit silly because $P_{1}$ and
$P_{2}$ are already determined by $P_{12}$ while $P_{1}$ and $P_{2}$ do not
determine the joint Markov predicate $P_{12}$. It is completely determined by
its belief function

(75) $\displaystyle(P_{1}\wedge_{12}P_{2})(\mathbf{T}|x)$
$\displaystyle:=(\mathbf{and}_{*}P_{12})(\mathbf{T}|x)$ (76)
$\displaystyle=P_{12}(\mathbf{and}^{-1}(\mathbf{T})|x)$ (77)
$\displaystyle=P_{12}(\\{(\mathbf{T},\mathbf{T})\\}|x)$ (78)
$\displaystyle=p_{TT}(x)$ (79) $\displaystyle=p_{1}(x)+p_{2}(x)+q(x)-1$

i.e. the belief function for the fuzzy “and” defined by the joint kernel is

(80)
$p_{1}(x)\wedge_{q(x)}p_{2}(x):=(P_{1}\wedge_{12}P_{2})(\mathbf{T}|x)=p_{1}(x)+p_{2}(x)+q(x)-1.$

By the inequalities (70) for $q(x$ we get

(81) $\max(0,p_{1}(x)+p_{2}(x)-1)\leq p_{TT}(x)\leq\min(p_{1}(x),p_{2}(x))$

and these inequalities are sharp if $q(x)=q_{\min}(x)$ (for the lower bound),
or $q(x)=q_{\max}(x)$ (for the upperbound). By definitions (37), (39)

(82) $\displaystyle p_{1}(x)\underline{\wedge}p_{2}(x)$
$\displaystyle=p_{1}(x)\wedge_{q_{\min}}p_{2}(x)$ (83) $\displaystyle
p_{1}(x)\overline{\wedge}p_{2}(x)$
$\displaystyle=p_{2}(x)\wedge_{q_{\max}}p_{2}(x)$

and so

(84) $p_{1}(x)\underline{\wedge}p_{2}(x)\leq p_{1}(x)\wedge_{q(x)}p_{2}(x)\leq
p_{1}\overline{\wedge}p_{2}(x).$

In case $P_{1}$ and $P_{2}$ are conditionally independent and
$P_{12}=P_{1}\otimes P_{2}$ we have

(85) $\displaystyle p_{FF}(x)$
$\displaystyle=(1-p_{1}(x))(1-p_{2}(x))=q_{\mathrm{indep}}$ (86)
$\displaystyle p_{TT}(x)$
$\displaystyle=p_{1}(x)p_{2}(x)=p_{1}(x)+p_{2}(x)+q_{\mathrm{indep}}-1$

hence comparing with (38) we find

(87) $p_{1}(x)\curlywedge
p_{2}(x)=p_{1}(x)\wedge_{q_{\mathrm{indep}}}p_{2}(x)$

###  4.3. logical connective “or”

We can proceed similarly for the “or” connective. Using the same abuse of
notation as for “and” we define a Markov predicate

(88) $P_{1}\vee_{12}P_{2}:=P_{1}\vee_{P_{12}}P_{2}:=\mathbf{or}_{*}P_{12}$

with belief function
$p_{1}(x)\vee_{q(x)}p_{2}(x):=P_{1}\vee_{P_{12}}P_{2}(\mathbf{T}|x)$. We then
find

(89)
$p_{1}(x)\vee_{q(x)}p_{2}(x):=(\mathbf{or}_{*}P_{12})(\mathbf{T}|x)=1-q(x).$

By the inequalities (70) for $q(x)$ we get inequalities

(90) $\max(p_{1},p_{2})\leq 1-q(x)\leq\min(1,p_{1}(x)+p_{2}(x))$

which are sharp if $q(x)=q_{\max}(x)$ (for the lowerbound) and $q=q_{\min}$
(for the upperbound). By (41) and (43) the inequalities (90) then translate to

(91) $p_{1}(x)\underline{\vee}p_{2}(x)\leq p_{1}(x)\vee_{q(x)}p_{2}(x)\leq
p_{1}(x)\overline{\vee}p_{2}(x).$

Finally for the conditionally independent case $P_{12}=P_{1}\otimes P_{2}$ we
have

(92) $(\mathbf{or}_{*}(P_{1}\otimes
P_{2}))(\mathbf{T}|x)=1-(1-p_{1})(1-p_{2})=p_{1}(x)+p_{2}(x)-p_{1}(x)p_{2}(x),$

and by (42) and (73) we find

(93) $p_{1}(x)\vee_{q_{\mathrm{indep}}}p_{2}(x)=p_{1}(x)\curlyvee p_{2}(x)$

###  4.4. logical implication

We can do exactly the same with other logic connectives in particular
implication. For $a,b\in\mathbb{B}$ we have

(94) $\mathbf{impl}(a,b)=a\mathbf{\to}b=\neg a\vee b=\neg(a\wedge\neg b).$

We therefore define a Markov predicate

(95)
$P_{1}\mathbf{\to}_{12}P_{2}:=P_{1}\mathbf{\to}_{P_{12}}P_{2}:=\mathbf{impl}_{*}P_{12}$

with a belief function

(96)
$p_{1}(x)\mathbf{\to}_{q(x)}p_{2}(x)=\mathbf{impl}_{*}P_{12}(\mathbf{T}|x)\\\
=1-P_{TF}(x)=p_{2}(x)+q(x).$

Now defining

(97) $\displaystyle p_{1}(x)\underline{\mathbf{\to}}p_{2}(x)$
$\displaystyle=p_{2}(x)+q_{\min}(x)=\max(p_{2}(x),1-p_{1}(x))$ (98)
$\displaystyle p_{1}(x)\rightsquigarrow p_{2}(x)$
$\displaystyle=p_{2}(x)+q_{\mathrm{indep}}(x)=1-p_{1}(x)+p_{1}(x)p_{2}(x)$
(99) $\displaystyle p_{1}(x)\overline{\mathbf{\to}}p_{2}(x)$
$\displaystyle=p_{2}(x)+q_{\max}(x)=\min(1,1+p_{2}(x)-p_{1}(x))$

we have

(100) $p_{1}(x)\underline{\mathbf{\to}}p_{2}(x)\leq
p_{1}(x)\mathbf{\to}_{q(x)}p_{2}(x)\leq
p_{1}(x)\overline{\mathbf{\to}}p_{2}(x)$

###  4.5. de Morgan’s law

It is easy to prove a general fuzzy de Morgan’s law using direct computation.
We will also give a more conceptual proof showing it is a consequence of the
classical de Morgan’s law (45) and functoriality. We have the identity

(101) $q=(1-p_{1})+(1-p_{2})+(p_{1}+p_{2}+q-1)-1=(\neg p_{1})+(\neg
p_{2})+(p_{1}\wedge_{q}p_{2})-1.$

In other words, the general fuzzy analogue of the de Morgan’s law (45) is

(102) $\neg\left(p_{1}(x)\vee_{q(x)}p_{2}(x)\right)=(\neg
p_{1}(x))\wedge_{(p_{1}(x)\wedge_{q(x)}p_{2}(x))}(\neg p_{2}(x))$

or

(103) $p_{1}(x)\vee_{q(x)}p_{2}(x)=\neg\left((\neg
p_{1}(x))\wedge_{(p_{1}(x)\wedge_{q(x)}p_{2}(x))}(\neg p_{2}(x))\right).$

For a more conceptual derivation of (102) rewrite the classic Boolean the
Morgan’s law (45) as

(104)
$\mathbf{not}\circ\mathbf{or}=\mathbf{and}\circ(\mathbf{not}\times\mathbf{not}).$

Then by functoriality

(105) $\displaystyle\mathbf{not}_{*}\mathbf{or}_{*}$
$\displaystyle=(\mathbf{not}\circ\mathbf{or})_{*}$ (106)
$\displaystyle=(\mathbf{and}\circ(\mathbf{not}\times\mathbf{not}))_{*}$ (107)
$\displaystyle=\mathbf{and}_{*}(\mathbf{not}\times\mathbf{not})_{*}.$

Thus we get

(108)
$\mathbf{not}_{*}(\mathbf{or}_{*}P_{12})=\mathbf{and}_{*}\left((\mathbf{not}\times\mathbf{not})_{*}P_{12}\right),$

and so

(109)
$(\mathbf{not}_{*}(\mathbf{or}_{*}P_{12}))(\mathbf{T}|x)=\mathbf{and}_{*}\left((\mathbf{not}\times\mathbf{not})_{*}P_{12}\right)(\mathbf{T}|x)$

But the “q” of the joint kernel $(\mathbf{not}\times\mathbf{not})_{*}P_{12}$
is

(110)
$((\mathbf{not}\times\mathbf{not})_{*}P_{12})(\mathbf{F},\mathbf{F})|x)=P_{12}((\mathbf{T},\mathbf{T})|x)=p_{1}(x)\wedge_{q(x)}p_{2}(x)$

whereas the marginals “$p_{1}$” and “$p_{2}$” of
$(\mathbf{not}\times\mathbf{not})_{*}P_{12}$ are

(111)
$\displaystyle\left(\pi_{i*}(\mathbf{not}\times\mathbf{not})_{*}P_{12})\right(\mathbf{T}|x)$
$\displaystyle=\left(\mathbf{not}_{*}\pi_{i*}P_{12}\right)(\mathbf{T}|x)$
(112) $\displaystyle=(\mathbf{not}_{*}P_{i})(\mathbf{T}|x)$ (113)
$\displaystyle=\neg p_{i}(x)$

Thus writing the left and right hand side of (109) in terms of belief
functions we get the Morgan’s law (102).

##  5\. Systems of logic expressions

It should now be clear that in order to use the Markov predicate
interpretation to define fuzzy logic predicates in multiple variables, we
require, in general, a joint distribution probability distributions for the
logic variables. A set of general logic formulas can be considered as a map
$L:\mathbb{B}^{n}\to\mathbb{B}^{m}$, $0\leq n,m<\infty$. In fact a single map
$L_{0}:\mathbb{B}^{n}\to\mathbb{B}$ can always be considered as evaluating an
element $\tilde{L}_{0}$ in $\mathbb{B}[t_{1},...,t_{n}]$, the free boolean
algebra on $n$ letters. For notational simplicity consider the $n=2$ case, the
general case being similar. We can define

(114) $\tilde{L}_{0}(t_{1},t_{2})=L_{0}(\mathbf{T},\mathbf{T})t_{1}t_{2}\vee
L_{0}(\mathbf{F},\mathbf{T})\bar{t}_{1}t_{2}\vee
L_{0}(\mathbf{T},\mathbf{F})t_{1}\bar{t}_{2}\vee
L_{0}(\mathbf{F},\mathbf{F})\bar{t}_{1}\bar{t}_{2}$

where juxtaposition is $\wedge$ and $\bar{t}=\neg t$. Since every logic
formula in $\mathbb{B}[t_{1},...,t_{n}]$ can be written uniquely in normal
form as monomials of $t$’s and $\bar{t}$’s “or”ed together, we see that the
map $L_{0}$ defines $\tilde{L}_{0}$ and vice versa. We will therefore no
longer make a notational difference.

Suppose then that we are given _joint_ Markov predicates
$P_{\\{1,\ldots,n\\}}:X\to\mathbb{B}^{n}$ or equivalently a full set of
$2^{n}$ _joint_ belief (confidence) functions

(115) $p_{a_{1}\ldots a_{n}}(x)=P_{\\{1,\ldots,n\\}}((a_{1}\ldots a_{n})|x)$

one for each “bitstring” $(a_{1},\ldots a_{m})\in\mathbb{B}^{n}$. They are
subject to the usual conditions for a probability distribution on a discrete
set

(116) $\sum_{(a_{1},\ldots a_{n})}p_{a_{1},\ldots a_{n}}(x)=1\text{ and
}p_{a_{1},\ldots a_{n}}(x)\geq 0\
\forall(a_{1},\ldots,a_{n})\in\mathbb{B}^{n}.$

Thus, we have $2^{n}-1$ degrees of freedom subject to $n$ conditions, one for
each of the believe (confidence) functions $p_{1}(x)\ldots p_{n}(x)$ from the
the marginals $P_{i}=\pi_{i*}P_{\\{1,...n\\}}$. Of course one can also
consider intermediate marginals, e.g. for every pair. By the the above $n=2$
case, these are defined by the belief functions $p_{i}(x)$ and for every pair,
the belief

(117) $q_{ij}=\pi_{ij*}P_{\\{1,...n\\}}((\mathbf{F},\mathbf{F})|x)=q_{ji}$

that neither is true, giving $n+n(n-1)/2$ conditions. 444Further assumptions
that may be natural depending on the application, may reduce the degrees of
freedom further. E.g. if we can assume conditional independence for
$P_{1},\ldots,P_{n}$ we have $p_{a_{1}\ldots
a_{n}}(x)=\prod_{i=1}^{n}p_{a_{i}}(x)$. Invariance of the joined Markov
predicate under permutations ensures that $p_{a_{1}\ldots a_{n}}$ only depends
on the number of $\mathbf{T}$’s in the “bitstring” $a_{1},\ldots,a_{n}$,
giving $(n+1)-1=n$ degrees of freedom, with 1 condition for the marginal
$p_{1}(x)=\cdots=p_{n}(x)$.

In any case, by functoriality, given a a system of joint Markov predicates
$P_{\\{1,\ldots,n\\}}:X\to\mathbb{B}^{n}$ and a system of logic formulas
$L:\mathbb{B}^{n}\to B^{m}$ we can define fuzzy version of $L$ applied to
$P_{\\{1,\ldots,n\\}}$ as a joint Markov predicate $X\to\mathbb{B}^{m}$

(118) $L_{1\ldots n}(P_{1},\ldots P_{n}):=L_{P_{1\ldots n}}(P_{1},\ldots
P_{n}):=L_{*}P_{1\ldots n}$

Again it is a silly _abuse of notation_ to pretend it is a fuzzy logic
predicate of the Markov predicates $P_{1},\ldots P_{n}$ separately. The Markov
predicate is completely determined by the joint belief (confidence) functions

(119) $p_{L,a_{1}\ldots a_{m}}(x)=(L_{*}P_{\\{1,\ldots,m\\}})((a_{1}\ldots
a_{m})|x)$

one for each “bitstring” $(a_{1},\ldots a_{m})\in\mathbb{B}^{m}$, which are
likewise subject to the conditions for a probability distribution
$\sum_{(a_{1},\ldots a_{m})}p_{L,a_{1},\ldots a_{m}}(x)=1$ and
$p_{L,a_{1},\ldots a_{m}}(x)\geq 0$.

###  5.1. Fuzzy quantification

The general setup assuming $n,m$ finite, does still not capture all natural
examples. In particular we want to use the logical quantifiers $\exists_{i\in
I}P_{I}=\vee_{i\in I}P_{i}$and $\forall_{i\in I}P_{i}=\wedge_{i\in I}P_{i}$.
In the case of finite sets this is covered by the above, but already in the
case of countable sets we have an extension. However in the countable case
quantifiers still define a map $B^{\mathbb{N}}\to\mathbb{B}$. The space
$\mathbb{B}^{\mathbb{N}}$ is a (non trivial) measurable space 555the
measurable sets are the $\sigma$-algebra generated by the inverse images of
subsets of $\mathbb{B}^{M}$, $M<\infty$ under the projection
$\mathbb{B}^{\mathbb{N}}\to B^{M}$ and we study measurable “logical
connectives”: $\mathbb{B}^{\mathbb{N}}\to\mathbb{B}$. Unsurprisingly,
$\exists$ and $\forall=\neg\circ\exists\circ(\neg^{\mathbb{N}})$ are
measurable. We will only discuss the $\exists$ quantifier to which the
$\forall$ quantifier can be reduced using the Morgan’s law or treated
similarly, _mutatis mutandis_. See also [LK] Our goal will be to derive
sensible lower and upper bounds for its belief function that they only require
finite marginals, and are therefore easier to handle.

Consider the countable infinite case first

(120)
$\mathbf{or}_{\mathbb{N}}(t_{1},t_{2},\ldots)=\vee_{i\in\mathbb{N}}.t_{i}$

with for each $I\subset\mathbb{N}$ a counterparts
$\mathbf{or}_{I}((t_{i})_{i\in I}=\vee_{i\in I}t_{i}$. A joint Markov
predicate $P_{\mathbb{N}}:X\to\mathbb{B}^{\mathbb{N}}$ gives marginals
$P_{I}:X\to\mathbb{B}^{I}$ and for $I_{1}\subseteq I_{2}\subseteq\mathbb{N}$

(121) $\displaystyle(\mathbf{or}_{I_{1}*}P_{I_{1}})(\mathbf{T}|x)$
$\displaystyle=1-P_{I_{1}}((\mathbf{F}|x)_{i\in I_{1}})$ (122)
$\displaystyle\leq 1-P_{I_{2}}((\mathbf{F}|x)_{i\in I_{2}})$ (123)
$\displaystyle=(\mathbf{or}_{I_{2}*}P_{I_{2}})(\mathbf{T}|x).$

In particular from the subsets $I=\\{i\\}$ we get the lower bound.

(124)
$\sup_{i}\mathbb{P}_{i}(\mathbf{T}|x)=\sup_{i}p_{i}\leq(\mathbf{or}_{\mathbb{N}*}\mathbb{P}_{\mathbb{N}})(\mathbf{T}|x)$

and for $I=\\{i,j\\}$ the lowerbound

(125)
$\sup_{ij}(\mathbf{or}_{ij*}P_{ij})(\mathbf{T}|x)=\sup_{ij}p_{i}(x)\vee_{q_{ij}(x)}p_{j}(x)\leq(\mathbf{or}_{\mathbb{N}*}\mathbb{P}_{\mathbb{N}})(\mathbf{T}|x)$

A sensible upper bound is a little harder to get by. Of course we have the
trivial bound

(126) $(\mathbf{or}_{\mathbb{N}*}\mathbb{P}_{\mathbb{N}})(\mathbf{T})\leq 1$

We get potentially better bounds as follows: Define

(127) $C_{I}=\\{t_{i}=\mathbf{T}\text{ for some }i\in
I\\}\subset\mathbb{B}^{\mathbb{N}}.$

Starting from a disjoint partition $\mathbb{N}=\coprod_{n}I_{n}$ we then have

(128)
$\mathbb{B}^{\mathbb{N}}=\\{(\mathbf{F},\mathbf{F},\ldots)\\}\coprod\cup_{n}C_{I_{n}}$

and therefore

(129) $\displaystyle 1=P_{\mathbb{N}}(\mathbb{B}^{\mathbb{N}}|x)$
$\displaystyle\leq\mathbb{P}_{\mathbb{N}}((\mathbf{F},\mathbf{F},\ldots)|x)+\sum_{n}P_{\mathbb{N}}(C_{I_{n}}|x)$
(130)
$\displaystyle=\mathbb{P}_{\mathbb{N}}((\mathbf{F},\mathbf{F},\ldots)|x)+\sum_{n}P_{I_{n}}(C_{I_{n}}|x)$

where we used $C_{I_{n}}$ for both $\\{t_{i}=\mathbf{T}\text{ for some }i\in
I_{n}\\}\subset\mathbb{B}^{I_{n}}\\}$ and $\\{t_{i}=\mathbf{T}\text{ for some
}i\in I_{n}\\}\subset\mathbb{B}^{\mathbb{N}}\\}$. We then have

(131)
$(\mathbf{or}_{\mathbb{N}*}P_{\mathbb{N}})(\mathbf{T}|x)\leq\sum_{n}(\mathbf{or}_{I_{n}*}P_{I_{n}})(\mathbf{T}|x)$

in particular for $I_{n}=\\{n\\}$

(132)
$(\mathbf{or}_{\mathbb{N}*}P_{\mathbb{N}})(\mathbf{T}|x)\leq\sum_{n}\mathbb{P}_{\\{n\\}}(\mathbf{T}|x)=\sum_{n}p_{n}(x)$

and for $I_{n}=\\{i_{n},j_{n}\\}$

(133)
$(\mathbf{or}_{\mathbb{N}*}P)(\mathbf{T}|x)\leq\inf_{\mathbb{N}=\cup_{n}\\{i_{n},j_{n}\\}}\sum_{n}p_{i_{n}}(x)\vee_{q_{i_{n}j_{n}}(x)}p_{j_{n}}(x)$

Of course these bounds can be rather poor. Better bounds can be obtained from
decomposing $\mathbb{B}^{\mathbb{N}}$ in disjoint pieces. E.g. we can
decompose as

(134) $\mathbb{B}^{\mathbb{N}}=\\{(\mathbf{F},\mathbf{F},\ldots)\\}\coprod
C_{1}\coprod\coprod_{n\geq 2}C_{n}\setminus C_{n-1}.$

However we can then no longer reduce to marginal joint Markov predicates with
a finite number of predicates although additional assumptions that give Markov
predicate for predicate $n$ conditional on the value of predicates $1,\ldots
n-1$ determine the $P_{\mathbb{N}}(C_{n}\setminus C_{n+1}|x)$, but we will not
pursue this further. Finally the finite marginals do determine the infinite
“or”: defining $C_{M}=C_{\\{1,\ldots,M\\}}$, we have

(135) $C_{\mathbb{N}}=\cup_{M}C_{M}$

and therefore

(136)
$(\mathbf{or}_{\mathbb{N}*}P_{\mathbb{N}})(\mathbf{T}|x)=P_{\mathbb{N}}(C_{\mathbb{N}}|x)=\lim_{M\to\infty}P_{\mathbb{N}}(C_{M}|x)=\lim_{M\to\infty}(\mathbf{or}_{M*}P_{\\{1\ldots,M\\}})(\mathbf{T}|x)$

by the monotone convergence theorem.

Where this countably infinite case shows up is in trying to quantify over the
(not necessarily countable) set $X$, i.e. making fuzzy sense of a classical
predicate expression like

(137) $\exists x\in X:P(x)=\vee_{x\in X}P(x)$

From the above, for a sensible fuzzy definition we should start with a lift to
joint Markov predicate that lifts the (possibly uncountably many)
$P(\mathbf{T}|x)_{x\in X}$. Since $P(\mathbf{T}|x)$ is a measurable function
of $x$, it is a the supremum of a countable sequence of simple functions
taking on a finite number of values. Thus we can taking on a finite number of
values, and countable limit of a function taking on on finite number of
values. This suggests that we should consider a lift

(138) $P_{\mathbb{N}}:X^{\mathbb{N}}\to\mathbb{B}^{\mathbb{N}}$

such that

(139)
$(\pi_{i*}P_{\mathbb{N}})(\mathbf{T}|(x_{1},x_{2},\ldots,))=P_{\mathbb{N}}(\\{t_{i}=\mathbf{T}\\}|(x_{1},\ldots
x_{n}))=P(\mathbf{T}|x_{i})$

Again such a lift can be constructed using a conditional independence
construction, but is not necessarily unique. We can then follow the above and
define the most general fuzzy version of the $\exists$ quantification as

(140) $p_{\exists x:P(x)}=\sup_{(x_{1},x_{2},\ldots)\in
X^{\mathbb{N}}}(\mathbf{or}_{\mathbb{N}*}P_{\mathbb{N}})(\mathbf{T}|(x_{1},x_{2},\ldots))$

We can now play the game of getting useful bounds by going to marginals. The
case $n=1$ gives

(141) $\displaystyle\sup_{x,\in
X}p(x)=\sup_{(x,x_{2},\ldots)}(\pi_{1*}P_{\mathbb{N}})(\mathbf{T}|(x,x_{2},\ldots))$
$\displaystyle\leq p_{\exists x:P(x)}$ (142) $\displaystyle\sum_{x\in
X}p(x)=\sup_{(x_{1},x_{2},\ldots)}\sum_{i}(\pi_{i*}P_{\mathbb{N}})(\mathbf{T}|(x_{1},x_{2},\ldots))$
$\displaystyle\geq p_{\exists x:P(x)}$

The $n=2$ case is more interesting. Spelled out, It means that in addition to
the belief function $p(x)$ we have a measurable function
$q(x_{1},x_{2}):X^{2}\to[0,1]$ such that $q_{\min}(p(x_{1}),p(x_{2}))\leq
q(x_{1},x_{2})\leq q_{\max}(p(x_{1}),p(x_{2}))$ which together with the belief
function $p(x)$ defines a lift $P_{1,2}:X^{2}\to\mathbb{B}^{2}$ such that
(compare (68) (69))

(143) $\displaystyle p_{FF}(x_{1},x_{2})$ $\displaystyle=q(x_{1},x_{2})$
$\displaystyle p_{FT}(x_{1},x_{2})$ $\displaystyle=1-p(x_{2})-q(x_{1},x_{2})$
(144) $\displaystyle P_{TF}(x_{1},x_{2})$
$\displaystyle=1-p(x_{1})-q(x_{1},x_{2})$ $\displaystyle p_{TT}(x_{1},x_{2})$
$\displaystyle=p(x_{1})+p(x_{2})-1-q(x_{1},x_{2}).$

We then have upper and lower bounds

(145) $\displaystyle\sup_{(x_{1},x_{2})\in
X^{2}}p(x_{1})\vee_{q(x_{1},x_{2})}p(x_{2})$ $\displaystyle\leq p_{\exists
x:P(x)}$ (146) $\displaystyle\sup_{(x_{1},x_{2},\ldots)\in
X^{\mathbb{N}}}\inf_{\mathbb{N}=\cup_{n}\\{i_{n},j_{n}\\}}\sum_{n}p(x_{i_{n}})\vee_{q(x_{i_{n}},x_{j_{n}})}p(x_{j_{n}})$
$\displaystyle\geq p_{\exists x:P(x)}$

A final note: the way we defined a fuzzy exists fails to take into account any
consideration of the way we sample the space $X$ or more precisely
$X^{\mathbb{N}}$. Fortunately this is easy to incorporate in the Markov kernel
framework and in fact makes things simpler. Choose a probability distribution
$\mathbb{P}:\star\to X^{\mathbb{N}}$, i.e. a sampling strategy. Then the
composition

(147)
$\star\xrightarrow{\mathbb{P}}X^{\mathbb{N}}\xrightarrow{P_{\mathbb{N}}}\mathbb{B}^{\mathbb{N}}\xrightarrow{\mathbf{or}_{\mathbb{N}}}\mathbb{B}$

is a probability distribution on $\mathbb{B}$, which gives a single number
$p_{\mathbb{P}}$ for the probability of $\mathbf{T}$

(148) $\displaystyle p_{\mathbb{P}}$
$\displaystyle=\int_{X^{\mathbb{N}}}(\mathbf{or}_{\mathbb{N}*}P_{\mathbb{N}})(\mathbf{T}|(x_{1},x_{2},\ldots)\mathbb{P}(d(x_{1},x_{2},\ldots))$
(149)
$\displaystyle=\sum_{(a_{1},a_{2}\ldots),\vee_{i}a_{i}=\mathbf{T}}\int_{X^{\mathbb{N}}}P_{\mathbb{N}}((a_{1},a_{2},\ldots)|(x_{1},x_{2},\ldots)\mathbb{P}(d(x_{1},x_{2},\ldots)$

This number has a different interpretation than existence: it is the expected
confidence to find a tuple $(x_{1},x_{2},\ldots)$ such that the predicate $P$
is true for at least one $x_{i}$ _given our sampling strategy_ $\mathbb{P}$.
It does gives a lower bound

(150)
$p_{\mathbb{P}}\leq\operatorname{\mathrm{ess\,sup}}\limits_{\mathbb{P}}(\mathbf{or}_{\mathbb{N}*}P_{\mathbb{N}})(\mathbf{T}|(x_{1},\ldots))\leq\sup(\mathbf{or}_{\mathbb{N}*}P_{\mathbb{N}})(\mathbf{T}|(x_{1},\ldots)=p_{\exists
x:P(x)}$

and may well be more useful than $p_{\exists x:P(x)}$ for many purposes.

##  6\. Conclusion

We have described fuzzy logic as a functorial“shadow” of ordinary boolean
logic. All the subtlety comes from the applying boolean logic not to to
boolean predicates but to boolean valued Markov kernels (Markov predicates)
that express the confidence in the truth of a predicate. However this means
that logic formulas become sensitive to the joint confidence in the
predicates. We have described how this works out in the simple cases of a
binary logic relation and arbitrary finite relations and simple infinte ones
like the existential quantifier.

##  7\. acknowledgements

The author is grateful for useful discussions with prof. Uzay Kaymak, and dr.
Hans Onvlee.

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