∎

11institutetext: A. Türkan 22institutetext: Özyeğin University, Department of
Natural and Mathematical Sciences, Çekmeköy,Istanbul,Turkey
Tel.:+90(216) 564 90 00
Fax:+90(216) 564 99 99
22email<EMAIL_ADDRESS>33institutetext: J.Y. Süngü, E. Güngör,
H. Reisoğlu, E. Veli Veliev 44institutetext: Department of Physics, Kocaeli
University, 41001 Izmit, Turkey

# Dissociation of Pseudotensor Mesons through Critical Temperature

A. Türkan  J.Y. Süngü  E. Güngör  H. Reisoğlu  E. Veli Veliev

(Received: date / Accepted: date)

###### Abstract

We have computed the masses and decay constants of isotriplet $\pi_{2}(1670)$,
the isoscalar $\eta_{2}(1645)$ and $\eta_{2}(1870)$ states as the ground-state
nonet of $1^{1}D_{2}$ case within the Thermal QCD sum rules framework. This
method is applied to the spectral changes of pseudotensor mesons at high
temperatures including quark, gluon, and mixed condensates up to the five
dimensions. We found that their masses and decay constants are insensitive to
temperature in low-temperature region. At the near critical temperature, we
observed an exponential decrease of the masses of $\pi_{2}$, $\eta_{2}(1645)$,
and $\eta_{2}(1870)$, but while the decay constant of $\pi_{2}$ is fallen,
that of $\eta_{2}(1645)$ and $\eta_{2}(1870)$ is increased as a function of
temperature. The modification of condensates at $T\neq 0$ is crucial to the
change of considered mesons properties in hot medium. We discuss and interpret
the implications of the physical meaning of our results.

###### Keywords:

Thermal QCD sum Rules Pseudotensor mesons Mass and decay constant at finite
temperature

##  1 Introduction

Conventional quark-antiquark meson states are classified as scalar
$(1^{3}P_{0},~{}J^{PC}=0^{++})$, pseudoscalar $(1^{1}S_{0},~{}J^{PC}=0^{-+})$,
vector $(1^{3}S_{1},~{}J^{PC}=1^{--})$, axial-vector
$(1^{3}P_{1},~{}J^{PC}=1^{++},1^{1}P_{1},~{}J^{PC}=1^{+-})$, pseudotensor
$(1^{1}D_{2},~{}J^{PC}=2^{-+},1^{3}D_{2},~{}J^{PC}=2^{--})$ and tensor
$(1^{3}P_{2},~{}J^{PC}=2^{++})$. Among them, the tensor meson nonet is not
well established Wang:2014sea  and there aren’t many studies in the
literature, but it has been one of the issues that attracted attention in
recent years.

$\pi_{2}(1670)$ was first detected at Aachen-Berlin-CERN collaboration in the
reaction $\pi^{-}p\rightarrow p\pi^{+}\pi^{-}\pi^{-}$. Its mass and width are
found to be $(1660\pm 16)$ MeV and $(270\pm 60)$ MeV Ascoli:1973 ,
respectively. Another family member of the pseudotensor mesons, the
$\eta_{2}(1870)$ has been observed in $\gamma\gamma$ reactions Karch:1991sm ,
$p\bar{p}$ annihilation Anisovich:2000mv , and radiative $J/\psi$ decays in
BES Collaboration Bai:1999tg . This state has been detected by the Crystal
Ball collaboration Karch:1991sm  in the final state $\eta\pi^{0}\pi^{0}$ of a
$\gamma\gamma$ reaction as a resonant structure finding mass $(1881\pm 32\pm
40)$ MeV and width $(221\pm 92\pm 44)$ MeV, and in the reaction
$p\bar{p}\longrightarrow\eta\pi^{0}\pi^{0}\pi^{0}$ by the Crystal Barrel
collaboration, with mass $(1875\pm 20\pm 25)$ MeV, and width $(200\pm 25\pm
45)$ MeV and for the $\eta_{2}(1645)$ state, $(1645\pm 14\pm 15)$ MeV,
$(180^{+40}_{-21}\pm 25)$ MeV, respectively Adomeit:1996nr .

The study of the Regge Trajectory model indicates that both $\eta_{2}{(1645)}$
and $\eta_{2}{(1870)}$ are the eigenvectors of a mass-squared matrix, the
physical isoscalar states can be related to the bare states as:

$\left(\begin{array}[]{c}|\eta_{2}{(1645)}\rangle\\\
|\eta_{2}{(1870)}\rangle\end{array}\right)=\left(\begin{array}[]{cc}\cos\theta&\sin\theta\\\
-\sin\theta&\cos\theta\end{array}\right)\left(\begin{array}[]{c}\Psi_{1}\\\
\Psi_{8}\end{array}\right),$ (1)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}\Psi_{1}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{3}}(u\bar{u}+d\bar{d}+s\bar{s}),$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}\Psi_{8}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{6}}(u\bar{u}+d\bar{d}-2s\bar{s}),$ (2)

where the mixing angle $\theta$ is an important input parameter determined by
the experimental data. From now on, we will use $\eta_{2}^{(1)}$ and
$\eta_{2}^{(2)}$ symbols for the $\eta_{2}{(1645)}$ and $\eta_{2}{(1870)}$
mesons, respectively in text for brevity. Although we have no clear
information on the mixing angle of these particles Zyla:2020zbs , this angle
is estimated to be small. So, we assume that there is no mixing between them.

In this study, we use the interpolating current associated with quantum
numbers $J^{PC}=2^{-+}$ to evaluate the two-point correlation function at non-
zero temperatures. We employ the Thermal QCD sum rules (TQCDSR) technique for
analyses and extract the masses and decay constants sum rules of the
considered pseudotensor states depending on temperature. This method has been
applied successfully for the tensor mesons in the literature Turkan:2019anj ;
Sungu:2020azn . A comparison of our results with predictions at vacuum
obtained from other approaches is also made. These kinds of works are helpful
for both fulfilling the hadron spectrum and allowing us to achieve some hints
about quark-gluon plasma (QGP) which is predicted to cover the universe at the
early stage of the universe. Quarks and gluons condense to create a gas of
nucleons and light mesons, the latter decay subsequently. Jet quenching for
light mesons at RHIC is well explained by radiative energy loss calculations
Vitev:2005he ; Lajoie:2006ha . As a result, probing the variations of hadronic
parameters depending on temperature can provide us valuable information on the
new phase of matter, i.e. QGP.

The outline of the article is arranged as follows. In Section 2, we shortly
describe the TQCDSR method used in the computation. The masses and decay
constants of the light-unflavored $\pi_{2}$, $\eta_{2}^{(1)}$,and
$\eta_{2}^{(2)}$ mesons are extracted from the TQCDSR. In the following,
numerical analysis is introduced in Section 3. Next, the numerical results
will be discussed in Section 4 and compared the results with other models and
lastly, the explicit forms of thermal spectral functions are given in Appendix
A.

##  2 The Methodology

The TQCDSR approach is a beautiful tool to describe the variations of the
hadronic parameters (mass, coupling constant, etc.) according to thermal
expectation values of quark and gluon condensates (or light-cone distribution
amplitudes) at low energy Bochkarev:1985ex . TQCDSR use dispersion
representations to compute QCD Green functions, the time-ordered
$(\mathcal{T})$ products of local hadron interpolating currents built of quark
and gluon fields, in two different schemes: These are the QCD part (or named
as theoretical or OPE side) which is defined in terms of quark and gluon
degrees of freedom, and the hadronic part (called the physical or
phenomenological side) related to hadronic degrees of freedom. The QCD side is
computed by converting the time-ordered product into a sum of local operators,
i.e thermal quark and gluon condensates via Wilson expansion (OPE) separating
systematically short and long-distance effects. Thermal quark and gluon
condensates are nonperturbative in nature and seem to power corrections to the
leading logarithmic, namely perturbative behaviour. They have greater
importance than higher-order $\alpha_{s}$ corrections and leading logarithmic
contributions Shifman:1978bx ; Shifman:1978by .

Thermal condensates parametrize the nonperturbative properties of the QCD in
hot medium. However, at finite temperatures, additional operators arise in the
rest frame of the heat bath. They can be expressed as Lorentz scalars in a
general frame by employing the four-velocity vector $u_{\mu}$. They come out
in 4-dimension, which are $u^{\mu}\Theta^{f}_{\mu\nu}u^{\nu}$ and
$u^{\mu}\Theta^{g}_{\mu\nu}u^{\nu}$, where $\Theta^{f,g}_{\mu\nu}$ are the
fermionic and gluonic parts of energy-momentum tensors of QCD, respectively
Mallik:1997kj . We will mention this later in detail in the numerical
illustration section.

We first begin to consider the usual current correlator to the computations
Bochkarev:1985ex -Shifman:1978by :

$\Pi_{\mu\nu,\alpha\beta}(q,T)=i\int
d^{4}x~{}e^{iq\cdot(x-y)}\langle\varrho\mathcal{T}\big{(}J_{\mu\nu}^{PT}(x)J_{\alpha\beta}^{\dagger
PT}(y)\big{)}\rangle,$ (3)

here $PT$ symbolizes the pseudotensor states $\pi_{2}$, $\eta_{2}^{(1)}$ and
$\eta_{2}^{(2)}$, $T$ is temperature, $\mathcal{T}$ represents the time-
ordered product and

$\varrho=\frac{e^{-H/T}}{Tr(e^{-H/T})},$ (4)

is the thermal density matrix of QCD.

In this paper, we restrict ourselves to the currents with quantum numbers
$J^{PC}=2^{-+}$ for the considered pseudotensor states
$\pi_{2},\eta_{2}^{(1)}$ and $\eta_{2}^{(2)}$, respectively Reinders:1984sr :

$\displaystyle J^{\pi_{2}}_{\mu\nu}$ $\displaystyle=$
$\displaystyle\frac{i}{2\sqrt{2}}\bigg{\\{}\bar{u}_{a}\left({\gamma_{\mu}}{\gamma_{5}}\partial_{\nu}+{\gamma_{\nu}}{\gamma_{5}}\partial_{\mu}+{\frac{2}{3}}{\eta_{\mu\nu}}{\gamma_{5}}\not{\partial}\right){u_{a}}$
$\displaystyle-$
$\displaystyle\bar{d}_{a}\left({\gamma_{\mu}}{\gamma_{5}}\partial_{\nu}+{\gamma_{\nu}}{\gamma_{5}}\partial_{\mu}+{\frac{2}{3}}{\eta_{\mu\nu}}{\gamma_{5}}\not{\partial}\right){d_{a}}\bigg{\\}},$
$\displaystyle J^{\eta_{2}^{(1)}}_{\mu\nu}$ $\displaystyle=$
$\displaystyle\frac{i}{2\sqrt{3}}\bigg{\\{}\bar{u}_{a}\left({\gamma_{\mu}}{\gamma_{5}}\partial_{\nu}+{\gamma_{\nu}}{\gamma_{5}}\partial_{\mu}+{\frac{2}{3}}{\eta_{\mu\nu}}{\gamma_{5}}\not{\partial}\right){u_{a}}$
$\displaystyle+$
$\displaystyle\bar{d}_{a}\left({\gamma_{\mu}}{\gamma_{5}}\partial_{\nu}+{\gamma_{\nu}}{\gamma_{5}}\partial_{\mu}+{\frac{2}{3}}{\eta_{\mu\nu}}{\gamma_{5}}\not{\partial}\right){d_{a}}$
$\displaystyle+$
$\displaystyle\bar{s}_{a}\left({\gamma_{\mu}}{\gamma_{5}}\partial_{\nu}+{\gamma_{\nu}}{\gamma_{5}}\partial_{\mu}+{\frac{2}{3}}{\eta_{\mu\nu}}{\gamma_{5}}\not{\partial}\right){s_{a}}\bigg{\\}},$
$\displaystyle J^{\eta_{2}^{(2)}}_{\mu\nu}$ $\displaystyle=$
$\displaystyle\frac{i}{2\sqrt{6}}\bigg{\\{}\bar{u}_{a}\left({\gamma_{\mu}}{\gamma_{5}}\partial_{\nu}+{\gamma_{\nu}}{\gamma_{5}}\partial_{\mu}+{\frac{2}{3}}{\eta_{\mu\nu}}{\gamma_{5}}\not{\partial}\right){u_{a}}$
(5) $\displaystyle+$
$\displaystyle\bar{d}_{a}\left({\gamma_{\mu}}{\gamma_{5}}\partial_{\nu}+{\gamma_{\nu}}{\gamma_{5}}\partial_{\mu}+{\frac{2}{3}}{\eta_{\mu\nu}}{\gamma_{5}}\not{\partial}\right){d_{a}}$
$\displaystyle-$ $\displaystyle
2\bar{s}_{a}\left({\gamma_{\mu}}{\gamma_{5}}\partial_{\nu}+{\gamma_{\nu}}{\gamma_{5}}\partial_{\mu}+{\frac{2}{3}}{\eta_{\mu\nu}}{\gamma_{5}}\not{\partial}\right){s_{a}}\bigg{\\}},$

here, $a$ is the color index, and $\eta_{\mu\nu}$ is:

$\eta_{\mu\nu}=\frac{q_{\mu}q_{\nu}}{q^{2}}-g_{\mu\nu}.$ (6)

At the hadron level, i.e. in the physical side, a complete set of intermediate
hadronic states, which has the same quantum numbers as the current operator is
embedded into the correlator to get the hadronic representation. Then,
subtracting the ground state contribution from other states and taking the
integration by $x$ and setting $y=0$, we can obtain the physical side of the
correlator as follows and from now on we shortly call it the left-hand side
(LHS) in formulae:

$\displaystyle\Pi_{\mu\nu,\alpha\beta}^{\mathrm{LHS}}(q,T)$ $\displaystyle=$
$\displaystyle\frac{\langle\Omega|J_{\mu\nu}^{PT}|PT\rangle\langle
PT|J_{\alpha\beta}^{\dagger PT}|\Omega\rangle}{m_{PT}^{2}(T)-q^{2}}+\ldots,$
(7)

where $m_{PT}(T)$ shows the temperature-dependent masses of pseudotensor
states handled. $\Omega$ indicates the heat bath and dots symbolize the higher
resonances and continuum states.

To continue the computation of the physical side’s thermal sum rules, we
present the matrix elements through masses and decay constants of pseudotensor
mesons. The decay constants and matrix element of the pseudotensor current
between the one meson state and the thermal medium are related to each other
as follows:

$\displaystyle\langle\Omega|J_{\mu\nu}^{PT}|PT\rangle=f_{PT}(T)m_{PT}^{3}(T)\varepsilon_{\mu\nu}.$
(8)

In Eq. (8), $\varepsilon_{\mu\nu}$ is the polarization tensor of the
considered mesons and obey the following relation:

$\varepsilon_{\mu\nu}\varepsilon^{*}_{\alpha\beta}=\frac{1}{2}{T_{\mu\alpha}}{T_{\nu\beta}}+\frac{1}{2}{T_{\mu\beta}}{T_{\nu\alpha}}-{\frac{1}{3}}{T_{\mu\nu}}{T_{\alpha\beta}},$
(9) $T_{\mu\nu}=\frac{q_{\mu}q_{\nu}}{m^{2}_{PT}(T)}-g_{\mu\nu}.$ (10)

In our calculations, we select following structure in
$\Pi_{\mu\nu,\alpha\beta}^{\mathrm{LHS}}(q,T)$:

$\displaystyle\Pi_{\mu\nu,\alpha\beta}^{\mathrm{LHS}}(q,T)=\Pi^{\mathrm{LHS}}(q^{2},T)\Bigg{[}\frac{1}{2}(g_{\mu\alpha}g_{\nu\beta}+g_{\mu\beta}g_{\nu\alpha})\Bigg{]}+other~{}structures,~{}$
(11)

here
$\Pi^{\mathrm{LHS}}(q^{2},T)=\frac{m_{PT}^{6}(T)f_{PT}^{2}(T)}{m^{2}_{PT}(T)-q^{2}}$
represents the scalar function belonging to the chosen structure. In the next
step, the Borel transformation is applied to Eq. (11) and only the coefficient
of the selected structure is written as:

$\displaystyle B(q^{2})\Pi^{\mathrm{LHS}}(q^{2},T)$ $\displaystyle=$
$\displaystyle m_{PT}^{6}(T)f_{PT}^{2}(T)e^{-m_{PT}^{2}(T)/{M}^{2}},$ (12)

where $M^{2}$ is the arbitrary Borel mass parameter in the theory.

The QCD side of the thermal sum rule should be expressed in terms of the quark
propagators. Note that it will be called as right-hand side (RHS) with a
superscript on the correlator hereafter. The currents given in Eq. (2) are
substituted in Eq. (3) and the quark fields are contracted using Wick theorem.
After some straightforward calculations, we obtain the below expressions for
the considered states:

$\displaystyle\Pi_{\mu\nu,\alpha\beta}^{\mathrm{RHS(\pi_{2})}}(q,T)$
$\displaystyle=$ $\displaystyle-\frac{3i}{8}\int
d^{4}x~{}e^{iq\cdot(x-y)}~{}\bigg{\\{}\Gamma_{u}+\Gamma_{u\rightarrow
d}\bigg{\\}},$
$\displaystyle\Pi_{\mu\nu,\alpha\beta}^{\mathrm{RHS(\eta_{2}^{(1)})}}(q,T)$
$\displaystyle=$ $\displaystyle-\frac{i}{4}\int
d^{4}x~{}e^{iq\cdot(x-y)}~{}\bigg{\\{}\Gamma_{u}+\Gamma_{u\rightarrow
d}+\Gamma_{u\rightarrow s}\bigg{\\}},$
$\displaystyle\Pi_{\mu\nu,\alpha\beta}^{\mathrm{RHS(\eta_{2}^{(2)})}}(q,T)$
$\displaystyle=$ $\displaystyle-\frac{i}{8}\int
d^{4}x~{}e^{iq\cdot(x-y)}~{}\bigg{\\{}\Gamma_{u}+\Gamma_{u\rightarrow
d}+4\Gamma_{u\rightarrow s}\bigg{\\}},$ (13) $\displaystyle\Gamma_{u}$
$\displaystyle=$
$\displaystyle\bigg{(}\mathrm{Tr}~{}\big{[}\partial_{\beta}(y)S_{u}(y-x)\gamma_{\mu}\gamma_{5}\partial_{\nu}(x)S_{u}(x-y)\gamma_{5}\gamma_{\alpha}\big{]}$
(14) $\displaystyle+$
$\displaystyle(\beta\rightarrow\alpha)+(\mu\rightarrow\nu)+(\beta\rightarrow\alpha,\mu\rightarrow\nu)\bigg{)}$
$\displaystyle+$
$\displaystyle\frac{2}{3}\eta_{\alpha\beta}\bigg{(}\mathrm{Tr}~{}\big{[}\partial_{\phi}(y)S_{u}(y-x)\gamma_{\mu}\gamma_{5}\partial_{\nu}(x)S_{u}(x-y)\gamma_{\phi}\gamma_{5}\big{]}+(\mu\rightarrow\nu)\bigg{)}$
$\displaystyle+$
$\displaystyle\frac{2}{3}\eta_{\mu\nu}\bigg{(}\mathrm{Tr}~{}\big{[}\partial_{\beta}(y)S_{u}(y-x)\gamma_{5}\gamma_{\theta}\partial_{\theta}(x)S_{u}(x-y)\gamma_{5}\gamma_{\alpha}\big{]}+(\beta\rightarrow\alpha)\bigg{)}$
$\displaystyle+$
$\displaystyle\frac{4}{9}\eta_{\mu\nu}\eta_{\alpha\beta}\bigg{(}\mathrm{Tr}~{}\big{[}\partial_{\phi}(y)S_{u}(y-x)\gamma_{5}\gamma_{\theta}\partial_{\theta}(x)S_{u}(x-y)\gamma_{\phi}\gamma_{5}\big{]}\bigg{)}.$

Then we will set $y=0$ after applying derivative with respect to $y$. In Eq.
(2) $S_{q}^{ij}$ is the thermal light quark propagator Mallik:1997pq :

$\displaystyle S_{q}^{ij}(x-y)$ $\displaystyle=$ $\displaystyle
i\frac{\not{x}-\not{y}}{2\pi^{2}(x-y)^{4}}\delta_{ij}-\frac{m_{q}}{4\pi^{2}(x-y)^{2}}\delta_{ij}-\frac{\langle\bar{q}q\rangle_{T}}{12}\delta_{ij}-\frac{(x-y)^{2}}{192}m_{0}^{2}\langle\bar{q}q\rangle_{T}$
(15) $\displaystyle\times$
$\displaystyle\Big{[}1-i\frac{m_{q}}{6}(\not{x}-\not{y})\Big{]}\delta_{ij}+\frac{i}{3}\Big{[}(\not{x}-\not{y})\Big{(}\frac{m_{q}}{16}\langle\bar{q}q\rangle_{T}-\frac{1}{12}\langle
u^{\mu}\Theta_{\mu\nu}^{f}u^{\nu}\rangle\Big{)}$ $\displaystyle+$
$\displaystyle\frac{1}{3}\Big{(}u\cdot(x-y)\Big{)}\not{u}\langle
u^{\mu}\Theta_{\mu\nu}^{f}u^{\nu}\rangle\Big{]}\delta_{ij}-\frac{ig_{s}G_{ij}^{\alpha\beta}}{32\pi^{2}(x-y)^{2}}$
$\displaystyle\times$
$\displaystyle\Big{(}(\not{x}-\not{y})\sigma_{\alpha\beta}+\sigma_{\alpha\beta}(\not{x}-\not{y})\Big{)}-i\delta_{ij}\frac{(x-y)^{2}(\not{x}-\not{y})\langle\bar{q}q\rangle^{2}_{T}}{7776}g_{s}^{2},$

here we use the notation $G_{ij}^{\alpha\beta}=G_{A}^{\alpha\beta}t_{ij}^{A}$,
$A=1,\,2\,\ldots 8$ are gluon color indices, $t^{A}=\lambda^{A}/2$ with Gell-
Mann matrices $\lambda^{A}$, and the gluon field strength tensor
$G_{\alpha\beta}^{A}\equiv G_{\alpha\beta}^{A}(0)$ is fixed at $x=0$.

The TQCDSR of the related pseudotensor states can be extracted equalizing the
same structures in both $\Pi_{\mu\nu,\alpha\beta}^{\mathrm{LHS}}(q,T)$ and
$\Pi_{\mu\nu,\alpha\beta}^{\mathrm{RHS}}(q,T)$. The expression of the light
quark propagator is inserted in Eq. (2), the derivatives are taken and $y$ is
set to zero, after calculating the integrals, we obtain the explicit tensor
structure of $\Pi_{\mu\nu,\alpha\beta}^{\mathrm{RHS}}(q,T)$. To continue our
calculations, the above structures in Eq. (11) are chosen from the
$\Pi_{\mu\nu,\alpha\beta}^{\mathrm{RHS}}(q,T)$ and then we get the related
scalar function $\Pi^{\mathrm{RHS}}(q^{2},T)$ which can be separated as a sum
of perturbative and non-perturbative terms:

$\Pi^{\mathrm{RHS}}(q^{2},T)=\Pi^{Pert.}(q^{2},T)+\Pi^{Non-Pert.}(q^{2},T).$
(16)

The expression of spectral density $\rho$ for the perturbative part is
obtained, which is given by the dispersion relation as follows:

$\Pi^{\mathrm{RHS}}(q^{2},T)=\int_{s_{min}}^{\infty}ds~{}\frac{\rho^{Pert.}(s,T)}{s-q^{2}}+\mathrm{subtracted~{}terms},$
(17) $\rho^{Pert.}(s,T)=\frac{1}{\pi}Im[\Pi^{RHS}(s,T)].$ (18)

In Eq. (17), $s_{min}$ is determined with the masses of quarks of the related
meson. The TQCDSR approach connects the perturbative and nonperturbative
territory with the assumption of quark-hadron duality. To eliminate the
contribution coming from the higher resonances and continuum states, the Borel
transformation is performed. After calculating the non-perturbative terms with
the direct Borel method, the effective mass and decay constant of the related
states are obtained as:

$m_{PT}(T)=\sqrt{\frac{\int_{s_{min}}^{s_{0}(T)}ds~{}\rho^{Pert.}(s)~{}s~{}e^{-s/M^{2}}+\frac{d}{d(-1/M^{2})}B\Pi^{Non-
Pert.}(q,T)}{\int_{s_{min}}^{s_{0}(T)}ds~{}\rho^{Pert.}(s)~{}e^{-s/M^{2}}+B\Pi^{Non-
Pert.}(q,T)}},$ (19) $\displaystyle
f_{PT}(T)=\sqrt{\frac{\int_{s_{min}}^{s_{0}(T)}ds~{}\rho^{Pert.}(s)~{}e^{-s/M^{2}}+B\Pi^{Non-
Pert.}(q,T)}{m_{PT}^{6}(T)~{}e^{-m^{2}_{PT}(T)/M^{2}}}}.$ (20)

The explicit expressions of perturbative spectral density term and Borel
transformed non-perturbative contributions are presented in the Appendix A. In
Eq. (19) and Eq. (20) there are two free parameters in the models, which
should be chosen to obtain the stable thermal sum rules: the Borel parameter
$M^{2}$ which establishes convergence of the OPE and the cut-off parameter
$s_{0}$ isolate the contribution of ground state from the higher resonances
and continuum.

##  3 Numerical Illustration

In this part, finite temperature effects on the pseudotensor meson properties
will be discussed and analyzed numerically in TQCDSR approach. By handling
analytic results in Eq. (19) and Eq. (20), we try to define the temperature-
dependent variations of the hadronic parameters belonging to the considered
states. We used the following input parameters for the quark masses,
$m_{u}=2.16^{+0.49}_{-0.26}~{}\mathrm{MeV}$,
$m_{d}=4.67^{+0.48}_{-0.17}~{}\mathrm{MeV}$,
$m_{s}=93^{+11}_{-5}~{}\mathrm{MeV}$ Zyla:2020zbs ; Narison:2010cg  and we
assume the critical temperature value $T_{c}=155~{}\mathrm{MeV}$ for the phase
transition to QGP as predicted in Refs. Aoki:2006br -Bazavov:2017dus . For the
$u$ and $d$ quarks vacuum condensate, we employ $\langle
0|\bar{q}q|0\rangle=(-275(5))^{3}~{}\mathrm{MeV}^{3}$ and for the $s$ quark
vacuum condensate $\langle
0|\bar{s}s|0\rangle=(-296(11))^{3}~{}\mathrm{MeV}^{3}$ Gubler:2018ctz .

Additionally, during our calculations, we will take into account the
temperature dependent quark condensate. The below fit function for the thermal
version of quark condensate is obtained from Ref. Gubler:2018ctz ;
Dominguez:2016roi  by fitting lattice data and is defined via light quark
vacuum condensate $\langle 0|\bar{q}q|0\rangle$:

$\displaystyle\langle\bar{q}q\rangle_{T}=(\kappa_{1}e^{c_{1}T}+\kappa_{2})\langle
0|\bar{q}q|0\rangle,$ (21)

here $q$ symbolizes the $u$ and $d$ quarks. Whereas for the $s$ quark
condensate:

$\displaystyle\langle\bar{s}s\rangle_{T}=(\kappa_{3}e^{c_{2}T}+\kappa_{4})\langle
0|\bar{s}s|0\rangle,$ (22)

with $c_{1}=\mathrm{0.040~{}MeV^{-1}}$, $c_{2}=\mathrm{0.516~{}MeV^{-1}}$,
$\kappa_{1}$$=-6.534\times 10^{-4}$, $\kappa_{2}=1.015$,
$\kappa_{3}=-2.169\times 10^{-5}$, $\kappa_{4}$$=1.002$ Azizi:2019cmj . To
carry on the calculation, the temperature-dependent continuum threshold for
the handled pseudotensor states also needs to be specified. The continuum
threshold expression is generated by Dominguez:2016roi ; Borsanyi:2010bp ;
Bhattacharya:2014ara :

$\displaystyle\frac{s_{0}(T)}{s_{0}(0)}=\bigg{[}\frac{\langle\bar{q}q\rangle_{T}}{\langle
0|\bar{q}q|0\rangle}\bigg{]}^{2/3}.$ (23)

The hadronic quantities rely on the Borel mass $M^{2}$ and cut-off parameter
$s_{0}$ in TQCDSR. However, the selection of these free parameters has to
satisfy standard restrictions. Also, we can not say the $s_{0}$ is totally
arbitrary because it is connected with the energy of the first excited state
of the considered particle. $s_{0}$ range is determined from the condition
that assures the sum rules to have the best stability in the chosen $M^{2}$
range. Maximum and minimum bounds of $M^{2}$ are settled analysing the pole
contribution. To make a reliable thermal sum rule analysis, we have to follow
two criteria to determine suitable working ranges for the two auxiliary
parameters $M^{2}$ and $s_{0}$:

$\bullet$ Pole dominance of the OPE; to determine the upper boundary of
$M^{2}$, the pole contribution has to form more than half of the full part.
This condition makes us certain that the contribution of continuum to the
correlator should not be large from the pole one, i.e. pole contribution (PC)
can be written as follows:

$\mathrm{PC}(s_{0},M^{2})=\frac{\Pi(M_{\mathrm{max}}^{2},\
s_{0})}{\Pi(M_{\mathrm{max}}^{2},\infty)}>\frac{1}{2}.$ (24)

$\bullet$ Convergence of the OPE; to get the lowest boundary of $M^{2}$, we
can use the usual convergence criterion as in the traditional QCD sum rules so
that the contribution of highest-order operators of any dimension is less than
the $\sim 20\%$ of the sum of all OPE terms. We found contribution of five-
dimensional mixed condensate $\Pi^{(\mathrm{Dim5})}$ is:

$\frac{\Pi^{(\mathrm{Dim5})}(M_{\mathrm{min}}^{2},\
s_{0})}{\Pi^{Total}(M_{\mathrm{min}}^{2},\ s_{0})}\cong 0.002.$ (25)

In addition to this criteria, derived quantities should have the slightest
dependence on $M^{2}$ while the chosen of $s_{0}$ interval is another
constraint that has to be imposed. As a result of the analysis made by the
above-mentioned conditions, our analysis yields to the below-working regions
as in Table 1 for the $M^{2}$ and $s_{0}$ belonging to the considered
pseudotensor states.

Table 1: The working ranges of Borel mass $M^{2}$ and cut-off parameter $s_{0}$. Parameter $[\mathrm{GeV}^{2}]$ | $\pi_{2}(1670)$ | $\eta_{2}(1645)$ | $\eta_{2}(1870)$
---|---|---|---
$M^{2}$ | 1.6-1.8 | 1.4-1.8 | 1.8 -2.0
$s_{0}$ | 4-4.7 | 4.0-4.6 | 5.0- 5.6

In the next step, using the defined free parameters, we arrive at the
numerical results of our calculations at $T=0$ and present them in Table 2 and
3. It is seen from these Tables that all masses is approximately consistent
with the experimental data. Theoretical errors in for $m_{PT}$ and $f_{PT}$
arising from uncertainties of the auxiliary parameters as well as errors in
input parameters stay inside the acceptable domain for theoretical errors in
the model.

Table 2: The masses of the pseudotensor mesons with $J^{PC}=2^{-+}$ at $T=0$. Mass [$\mathrm{GeV}$] | $m_{\pi_{2}(1670)}$ | $m_{\eta_{2}(1645)}$ | $m_{\eta_{2}(1870)}$
---|---|---|---
Experiment Zyla:2020zbs  | $1.671^{+2.9}_{-1.2}$ | $1.617\pm 0.05$ | $1.773\pm 0.08$
This Work | $1.659^{+0.063}_{-0.062}$ | $1.629^{+0.07}_{-0.007}$ | $1.809^{+0.053}_{-0.052}$
Table 3: The decay constants of the pseudotensor mesons with $J^{PC}=2^{-+}$ at $T=0$. Decay constant | $f_{\pi_{2}(1670)}$ | $f_{\eta_{2}(1645)}$ | $f_{\eta_{2}(1870)}$
---|---|---|---
This Work | $6.50^{+0.01}_{-0.02}\times 10^{-2}$ | $6.54^{+0.08}_{-0.07}\times 10^{-2}$ | $6.49^{+0.01}_{-0.01}\times 10^{-2}$
Other Theories | - | - | -

Also, we tested the masses and decay constants of pseudotensor mesons are
whether stable against variations at $M^{2}$ and $s_{0}$ and give the plot
only belonging to $\pi_{2}(1670)$ as an example in Figure 1.

Figure 1: The 3D plot of the mass of $\pi_{2}$ state against Borel parameter
$M^{2}$ and continuum threshold parameter $s_{0}$ at $T=0$.

Drawing the below Figure 2, we clearly see the deviation of hadronic
parameters of pseudotensor mesons $\pi_{2}(1670)$, $\eta_{2}(1645)$ and
$\eta_{2}(1870)$ on critical temperature at fixed $M^{2}$ and $s_{0}$.

Figure 2: The masses and decay constants of the $\pi_{2}(1670),\eta_{2}(1645)$
and $\eta_{2}(1870)$ as a function of temperature at some fixed values of
continuum threshold parameter $s_{0}$.

Our numerical results depict that the mass values of the $\pi_{2}(1670)$,
$\eta_{2}(1645)$ and $\eta_{2}(1870)$ states are stable at low temperatures,
but they decrease approximately by $49\%$, $38\%$ and $39\%$ at above nearly
critical temperature as shown in Figure 2. Meanwhile decay constant value of
$\pi_{2}(1670)$ drops by $10\%$, but $f_{\eta_{2}(1645)}$ and
$f_{\eta_{2}(1870)}$ increases by $13\%$, $15\%$, respectively when analysed
up to $T=180~{}\mathrm{MeV}$. These changes may be signal of deconfinement
phase transition in hot medium.

##  4 Summary

Analysing hadrons at finite temperatures is the popular current research area
to explore the QGP which is thought to be available in the early stages of the
universe and possibly in the inner core of neutron stars. In this context,
surveying the hadrons in hot medium can provide precious hints on the dynamics
of QCD. In a hot bath, we can probe the changes of QCD vacuum, by examining
the thermal effects on the condensates and observing the melting of hadrons at
estimated deconfined temperatures. Investigations lead to the results
consistent with the deconfinement temperature $T_{c}=155~{}\mathrm{MeV}$
Aoki:2006br -Bazavov:2017dus .

In this paper, we explored the pseudotensor mesons $\pi_{2}(1670)$,
$\eta_{2}(1645)$ and $\eta_{2}(1870)$ in hot bath. To describe the effects of
hot medium on the hadronic parameters of these resonances, the Thermal QCD sum
rule model is employed taking into account the contributions of condensates up
to dimension five. In calculations, we assume that the OPE and quark-hadron
duality stay valid at finite temperatures, however, we substitute the vacuum
condensates by their thermal expectation values. In addition, our data give
approximately the same values as our previous work, which we made using a
different fit function Sungu:2019imas ; Turkan:2019imas  and pseudotensor
current, our results at $T=0$ limit are also in reasonable agreement with the
available experimental data.

A remarkable changes in the values of mass and decay constant in hot medium
can be interpreted as the signal of QGP, called a new state of matter. Also,
it might indicate the deconfinement phase transition in QGP which may occur in
the primordial universe. So the behaviour of pseudotensor mesons in terms of
temperature can be a useful tool to analyse the heavy-ion collision
experiments. We hope that our predictions for the hadronic features of the
considered mesons in hot medium can be tested in near future experiments.

##  Appendix A Mesonic Spectral Functions at Finite Temperature

The explicit form of the perturbative part of the spectral density in Eq. (19)
and Eq. (20) is found as for the $\pi_{2}$ meson:

$\displaystyle\rho^{{Pert.}}(s)=\frac{[10(m_{d}^{2}+m_{u}^{2})-9s]s}{240\pi^{2}}.$
(26)

The nonperturbative contributions in Eq. (19) and Eq. (20) is evaluated via
the Borel transformation and has the following form, respectively:

$\displaystyle B\Pi^{\langle q\bar{q}\rangle}(q^{2},T)$ $\displaystyle=$
$\displaystyle-\frac{1}{3}M^{2}\Big{(}\langle\bar{d}d\rangle
m_{d}+m_{u}\langle\bar{u}u\rangle\Big{)},$ (27) $\displaystyle
B\Pi^{\mathrm{\langle G^{2}\rangle+\langle\Theta_{00}\rangle}}(q^{2},T)$
$\displaystyle=$ $\displaystyle 0,$ (28) $\displaystyle
B\Pi^{\langle\bar{q}Gq\rangle}(q^{2},T)$ $\displaystyle=$
$\displaystyle\frac{1}{4}m_{0}^{2}\Big{(}\langle\bar{d}d\rangle
m_{d}+m_{u}\langle\bar{u}u\rangle\Big{)},$ (29)

and for the $\eta_{2}(1645)$ as follows:

$\displaystyle\rho^{{Pert.}}(s)$ $\displaystyle=$
$\displaystyle\frac{[20(m_{d}^{2}+m_{s}^{2}+m_{u}^{2})-27s]s}{720\pi^{2}},$
(30) $\displaystyle B\Pi^{\langle q\bar{q}\rangle}(q^{2},T)$ $\displaystyle=$
$\displaystyle-\frac{2}{9}M^{2}\Big{(}\langle\bar{d}d\rangle
m_{d}+m_{s}\langle\bar{s}s\rangle+m_{u}\langle\bar{u}u\rangle\Big{)},$ (31)
$\displaystyle B\Pi^{\mathrm{\langle
G^{2}\rangle+\langle\Theta_{00}\rangle}}(q^{2},T)$ $\displaystyle=$
$\displaystyle 0,$ (32) $\displaystyle
B\Pi^{\langle\bar{q}Gq\rangle}(q^{2},T)$ $\displaystyle=$
$\displaystyle\frac{5}{18}m_{0}^{2}\Big{(}\langle\bar{d}d\rangle
m_{d}+m_{s}\langle\bar{s}s\rangle+m_{u}\langle\bar{u}u\rangle\Big{)},$ (33)

and also for the $\eta_{2}(1870)$ we get the below expressions:

$\displaystyle\rho^{{Pert.}}(s)$ $\displaystyle=$
$\displaystyle\frac{(10m_{d}^{2}+40m_{s}^{2}+10m_{u}^{2}-27s)s}{720\pi^{2}},$
(34) $\displaystyle B\Pi^{\langle q\bar{q}\rangle}(q^{2},T)$ $\displaystyle=$
$\displaystyle-\frac{1}{9}M^{2}\Big{(}\langle\bar{d}d\rangle
m_{d}+4m_{s}\langle\bar{s}s\rangle+m_{u}\langle\bar{u}u\rangle\Big{)},$ (35)
$\displaystyle B\Pi^{\mathrm{\langle
G^{2}\rangle+\langle\Theta_{00}\rangle}}(q^{2},T)$ $\displaystyle=$
$\displaystyle 0,$ (36) $\displaystyle
B\Pi^{\langle\bar{q}Gq\rangle}(q^{2},T)$ $\displaystyle=$
$\displaystyle\frac{5}{36}m_{0}^{2}\Big{(}\langle\bar{d}d\rangle
m_{d}+4m_{s}\langle\bar{s}s\rangle+m_{u}\langle\bar{u}u\rangle\Big{)}$ (37)

with $m^{2}_{0}=(0.8\pm 0.2)~{}\mathrm{GeV}^{2}$

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11institutetext: Friedrich-Alexander-Universität Erlangen-Nürnberg

# Trichotomic Argumentation Representation††thanks: Work forms part of the DFG
Project “Reconstructing Arguments from Noisy Text” (RANT) within SPP 1999
RATIO.

Merlin Göttlinger  0000-0002-2251-8519 Lutz Schröder  0000-0002-3146-5906

###### Abstract

The Aristotelian trichotomy distinguishes three aspects of argumentation:
Logos, Ethos, and Pathos. Even rich argumentation representations like the
Argument Interchange Format (AIF) are only concerned with capturing the Logos
aspect. Inference Anchoring Theory (IAT) adds the possibility to represent
ethical requirements on the illocutionary force edges linking locutions to
illocutions, thereby allowing to capture some aspects of ethos. With the
recent extensions AIF+ and Social  (S-AIF), which embed dialogue and speakers
into the AIF argumentation representation, the basis for representing all
three aspects identified by Aristotle was formed. In the present work, we
develop the Trichotomic  (T-AIF), building on the idea from S-AIF of adding
the speakers to the argumentation graph. We capture Logos in the usual known
from AIF+, Ethos in form of weighted edges between actors representing trust,
and Pathos via weighted edges from actors to illocutions representing their
level of commitment to the propositions. This extended structured
argumentation representation opens up new possibilities of defining semantic
properties on this rich graph in order to characterize and profile the
reasoning patterns of the participating actors.

###### Keywords:

Logos Ethos Pathos Structured Argumentation Trust Dialogue Structure Argument
Interchange Format Inference Anchoring

##  1 Introduction

Argumentation plays a central role in society for forming rational opinions
about controversial topics by providing a way to resolve conflicting views.
For understanding complex argumentative discussions it is helpful to draw
diagrams called _argument maps_ that visualize how the propositions and
arguments interact with one another. Having such a formal representation for
argumentation also allows for having computational semantics for them to aid
humans in evaluating argumentation.

Over the years there have been many developments of such argumentation
frameworks with varying information content and complexity. One of the
simplest forms of argumentation frameworks are the so called Abstract
Argumentation Frameworks that treat arguments as abstract objects without
having attached internal structure.

#### Abstract Argumentation Frameworks

consist of a graph having arguments as nodes without any additional structure
or information apart from the relations connecting them to other arguments.
Dung [19] popularized the simplest form of these frameworks (see 1) with only
a single relation representing attack between arguments.

######  Definition 1 (Dung AAF [20])

An _abstract argumentation framework_ $AF=(AR,\to)$ is composed of a set $AR$
of _arguments_ and an _attack_ relation $\to\;\subseteq AR\times AR$. An
argument $A\in AR$ is called an _attacker_ of $B\in AR$ if $A\to B$. A set of
arguments $S\subseteq AR$ _attacks_ $B$ if there exists $A\in S$ s.t. $A\to
B$. $S$ _defends_ $A$ if $S$ attacks each attacker of $A$. $S$ is _conflict-
free_ if it does not attack its own arguments. $S$ is _admissible_ if it is
conflict-free and defends each of its arguments. The _characteristic function_
of $AF$ is defined by $F_{AF}(S)\coloneqq\\{A\in AR\mid S\text{ defends
}A\\}$. $S\subseteq AR$ is:

  * •

a _stable extension_ if it is conflict-free and attacks each argument
$A\not\in S$;

  * •

a _preferred extension_ if it is a maximal (w.r.t. set inclusion) admissible
set of arguments;

  * •

a _complete extension_ if it is admissible and contains each argument it
defends (or equivalently a conflict-free fixed point of $F_{AF}$);

  * •

a _grounded extension_ if it is the least complete extension (or equivalently
the least fixed point of $F_{AF}$).

While being algorithmically and conceptionally simple, having only an attack
relation limits the expressivity of the framework. In argumentation, it is
often the case that one argument supports the truth of another argument, and
it is not always easy to express this as an attack. Bipolar Argumentation
Frameworks [15] add a support relation between arguments to capture those
situations. The semantics of BAFs is a priori somewhat less clear-cut than
that of AAF, however, as one needs to determine how support and attach
interact.

One approach, extensively discussed by Cohen et al. [17], is to derive
_complex attacks_ from joining support and attack relations and thus reduce to
Dung semantics. Another way is to move from crisp notions of _accept_ and
_reject_ towards a weighted acceptance where supporters and attackers are
accumulated into the acceptability of an argument [1, 29, 2]. The weighted
approach however comes at the cost of requiring acyclicity of the graph as
evident in the overview of current bipolar semantics by Amgoud and Ben-Naim
[1].

#### Structured Argumentation Frameworks

go beyond this abstract view of argumentation and attach structure to the
argument map. There are multiple points where one can add information to
obtain a richer argument representation. One way is to add structure to the
nodes, e.g. in the form of logical formulas representing the propositions and
reasoning patterns used in an argument [5, 13, 3, 28]. This allows using
logical reasoning to find conflicts and inferences between the arguments and
generating new arguments from existing facts and inference rules. Structure
can also be added to the edges in the form of identifying different kinds of
inferences, conflicts, or exceptions. Argument structure and the
classification of different types of arguments have been studied widely [39,
33, 42, 45, 41].

Toulmin [39] proposed that arguments generally obey a basic structure,
commonly referred to as the _Toulmin Scheme_ , consisting of identifiable
parts that justify why a listener should believe the qualified conclusion.
Perelman and Olbrechts-Tyteca [33] informally categorized different types of
arguments and analysed what makes them convincing. Walton [42] combined these
efforts and created a compendium of different _Argumentation Schemes_ with a
common structure of identifying multiple premises necessary to reasonably
derive the truth of an argument’s claim. Additionally, he identified so called
_critical questions_ specific to each scheme, which a critically thinking
audience can put forth to undermine arguments employing that scheme, as in the
following example.

######  Example 1 (Scheme: Argument from Position to Know [45])

Major Premise:

Source $a$ is in a position to know about things in a certain subject domain
$S$ containing proposition $A$

Minor Premise:

$a$ asserts that $A$ is true (false)

Conclusion:

$A$ is true (false)

Critical Questions:

  1. 1.

Is $a$ in a position to know whether $A$ is true (false)?

  2. 2.

Is $a$ an honest (trustworthy, reliable) source?

  3. 3.

Did $a$ assert that $A$ is true (false)?

For an in-depth review of this style of argumentation schemes as well as a
historical overview of classifying different patterns of argumentation see the
book by Walton et al. [45]. Due to their nature of clearly identifying
premises and claim of an argument, Walton-style schemes are a good candidate
for complex edge labels in an argumentation map.

#### The AIF

is a formal representation of argumentation general enough to include many
existing representations in order to establish a standard for storing and
exchanging argumentation [16]. Inferences in AIF can be labelled with
instances of _rule schemes_ which can be Walton–style argumentation schemes
but are not limited to those. Some of these schemes are implemented in the
_AIF ontology_, which specifies the overall format and accompanying concepts
intended as labels for the participating propositions.

To represent argumentative discourse between multiple actors, the AIF was
later extended to AIF+ by Modgil and McGinnis [27] and by Reed et al. [36],
who introduce new types of nodes representing the dialogue structure through
dialogue moves relating locutions.

IAT introduced the idea to incorporate illocutionary force into argument maps
for relating locutions and illocutions as well as transitions and
inferences/conflicts [12, 10, 11]. Having such links between statements and
their content also allows specifying requirements on force edges, e.g. that
the speaker is trustworthy. This can be seen as capturing ethical requirements
similar to premises of Walton-style argumentation schemes, expressing that a
witness is trustworthy in an _Argument from Witness Testimony_. In the
extension by Reed et al. [35], AIF+ already has the capabilities necessary to
represent IAT annotated argumentation maps.

Recently, Lawrence et al. [22] formalized an adjunct ontology to AIF+ called
the _S-AIF_ , which incorporates the authors of locutions into the model. They
use this to calculate various statistics about the discussion, such as
participation and agreement-based author clustering. We build on this idea in
Section 2 where we propose the _Trichotomic_ T-AIF, which formally captures
even more aspects of argumentation. S-AIF distinguishes between the roles of
Argument Web participants and is tailored towards augmented argument
construction and statistics, while our aim is to represent existing arguments
in more detail.

Argumentation in AIF+ is represented through different types of connected
nodes with corresponding semantic meaning. The nodes in the actual
argumentation graph are instances of the concepts from the _upper ontology_
with related instances of concepts from the _forms ontology_ specifying the
schemes for the S-nodes as well as concepts to classify the I-nodes into the
various forms (see the appendix).

_I-nodes_ represent Illocutions and hence contain the propositions relevant
for the arguments as premises, exceptions, and conclusions. The AIF defines
concepts necessary to classify these nodes to participate in Walton-style
argumentation schemes.

_L-nodes_ represent Locutions and contain the raw utterances that make up the
represented dialogue. They can be seen as illocutions in the sense that they
carry the propositional content that their text was uttered in the
conversation. In S-AIF, L-nodes are connected with nodes representing the
authors of the locution.

_S-nodes_ represent Schemes for interconnecting nodes. These schemes are
associated with scheme application nodes for the various kinds of
interconnections between nodes:

RA

Instances of rule schemes, such as Walton-style schemes.

TA

Instances of transition schemes representing speech acts in dialogue.

CA

Instances of conflict schemes capture that truth of propositions is in
conflict.

PA

Instances of preference schemes, which resolve conflicts via preference rules.

YA

Instances of illocutionary schemes modelling illocutionary force as a link
between locutions, illocutions, inferences, and transitions.

The semantics of AIF graphs is defined by translating into ASPIC [6] and then
evaluating using, e.g., the TOAST algorithm [38].

#### ASPIC

is a framework for building structured argumentation systems with a strong
logical background following the ideas of Besnard and Hunter [5] where the
nodes are sets of formulas representing a derivation tree [34, 6]. They
distinguish between strict and defeasible inferences and require the syntax of
the logic language of the nodes to contain each instance of a defesaible
inference to allow attacks on inferences. Apart from this form of attack they
also have attacks on premises and conclusions via a unidirectional relation
specifying that one formula is contrary to another. ASPIC builds on an
extended notion of _abstract argumentation system_ [40] and was generalised to
ASPIC+ [34] by partitioning inferences and facts into an undeniable and a
defeasible part [26].

##  2 The Trichotomic  T-AIF

Aristotle [4] noted that argument content is not the only relevant aspect for
human argument evaluation. He distinguished between three means of persuasion
forming the so called _Aristotelian Trichotomy_ :

Logos:

Appeal to logic. The structure of the argumentation both of individual
reasoning steps and the overall interaction of multiple arguments, e.g.
argumentation schemes used.

Ethos:

Appeal to authority. The properties of the speaker relevant for the evaluation
of argument, e.g. credibility or moral values.

Pathos:

Appeal to emotion. The emotional aspects of an argument, e.g. enthusiasm of
the speaker or intended emotional reaction from the audience.

Surprisingly, to our knowledge all formal representations for argumentation
are mostly focused on the logos aspect of argumentation. The T-AIF aims to
incorporate all three aspects into the formal representation to allow
representing arguments in a richer way. It consists of three interconnected
parts representing different aspects of the argumentation.

###  2.1 Trust Network

Building on ideas of Lawrence et al. [22], we include the speakers into the
argumentation representation. Other entities such as politicians, news
organisations, or other groups that might participate in the discussion
through citation or quotation are also included. We call these _E-nodes_ given
that they represent Entities as unifying concept over active and passive
participants of the conversation.

While Lawrence et al. use their speaker nodes mostly for statistics and
grouping, we also use them to represent ethos and pathos through weighted
edges in the graph. The ethos aspect is incorporated by weighted edges between
E-nodes capturing the notion of trust of active participants towards other
entities forming a trust network [32]. One could have multiple different trust
relations between the entities to capture the different means of trust
described in the literature [14, 18, 21, 31, 37] but in the following we are
mostly concerned with trust as the confidence an actor has in another actor’s
utterances.

The grade of commitment of the speakers towards their illocutions is a pathos
aspect represented in the graph. Distinguishing between different levels of
commitment or tracking the commitment of actors is mostly studied in the
context of dialogue games [44, 24, 25, 46, 11] but we believe it has relevant
interconnections with a speaker’s ethos. Strongly committing to propositions
obligates an actor to defend them when challenged and should have a negative
effect on her trustworthiness when this fails [30]. In contrast, an actor
voicing arguments for the opposing view besides her own should receive
increased trust due to seeming well-informed and not being afraid of attacks
on her own stance.

###  2.2 Dialogue Structure

This part of the argumentation representation captures the raw locutions the
actors have put forth. The Locutions are represented as textual labels in _L-
nodes_. Connecting the L-nodes via the reply relation forms a lattice
structure. Intuitively one would assume this to form a tree structure but an
utterance may reference multiple other locutions resulting in multiple
parents.

Apart from the reply relation, locutions are connected to moves from a
dialogue game giving meaning to their interconnections. Moreover, they are
connected to illocution nodes (described in Section 2.3) via illocutionary
force edges as seen widely in previous work [35, 12, 10, 11]. Note that in the
interest of clarity, our representation uses typed labelled edges to represent
illocutionary force and dialog relations in favour of having YA- and TA-nodes.

It is beneficial to keep the locutions of the dialogue in the representation
of the argumentation to enable a broader applicability of the format as a
source structure containing both argumentation and dialogue. This structure
serves as witness for the temporal development of the argumentation, and thus
e.g. allows penalizing unfavourable moves or decisions in the dialogue game as
proposed by Walton [43]. Penalizing or rewarding based on logical content as
briefly mentioned at the end of Section 2.1 also becomes expressible as a
result of having the locutions as nodes in T-AIF graphs.

###  2.3 Argument Map

Due to unknown background knowledge of the discussion participants and
enthymemes dominating real argumentative discussions, we cannot assume the
represented argument maps to be complete without unknown parts. From the
dialogue relations we can infer argumentative relations even if the precise
reasoning patterns in use are not clear. We assume that the participants of
the discussion voice complaints if the connections between propositions are
unclear to a human participant with relevant background knowledge. Hence, we
assume that enthymemes and skipped reasoning steps are not fallacious but
covered by knowledge available at the time.

Our argument map is similar to the one defined in the AIF specification [16]
but our _I-nodes_ are labelled with logical formulas (without committing to a
particular logic) representing the propositional content of the Illocution.
Attacks and supports have schemes similar to the ones proposed by the AIF but
restrict the shape of participating propositions according to their nature
(e.g. as specified by Walton [42] or Parsons et al. [31]). The Application of
these schemes is represented in the graph by _SA-nodes_ and _AA-nodes_ for
Support and Attacks, respectively. Exceptions to these schemes are represented
as propositions of specified shape connecting to the application nodes
directly. We allow E-nodes to be connected via attack and support nodes to
express ethical requirements on entities or infer ethical properties of these
respectively.

We leave out the preference nodes from AIF in our formalism as preferences are
inherently a per entity concept and our argument map represents all actors’
contributions. Preferences of the actors are not inexpressible this way but
are rather expressed through trust and commitment in the sense that an actor
likely prefers her own arguments or those made by highly truted entities over
those made by untrusted entities.

Due to elided reasoning steps the uttered propositions in real argumentative
conversations might not have the required shapes for the reasoning patterns
they are involved in (see 2). Filling these reasoning steps from common sense
and common knowledge is an easy task for humans (albeit not uniquely) but a
very hard task for machines as analysed by Boltužić and Šnajder [8].

######  Example 2 (Elided Steps in Typical Reasoning)

‘Experts say that Brexit would hurt the economy, so we should vote against
Brexit.’ Here the main argumentation scheme involved is _Argument from Expert
Opinion_ based on the fact that experts are quoted. The drawn conclusion by
the definition of that scheme [45] would be that ‘Brexit would hurt the
economy’ and not that ‘UK citizens should vote against Brexit’.

We resort to accepting this lack of information and capturing the unknown
reasoning patterns by having default inferences connecting the perceived to
the required proposition. So in the case of 2 we would have the conclusion of
the argument from expert opinion be that ‘Brexit would hurt the economy’ and
add a default inference from there to ‘UK citizens should vote against
Brexit’. Similarly, we add required premises and exceptions to the scheme
applications to complete enthymemes without committing entities to these.

BrexitEU harms UKEU has bad intentionsEU harms UK fishermenEU must harm EU
fishermen$\neg$ EU harms Norway fishermenNorway is in the EUEU protects
employeesEU protects natureB0.80.60.90.80.9A0.60.70.6 Figure 1: Simplified
extract from a Twitter discussion between A and B on Brexit. Red arrows
represent attack while green arrows represent support. The weighted black
edges represent the commitment of the actors to the propositions.

##  3 Semantics

For reasoning about T-AIF we define a reduced representing structure and
define properties on it as fuzzy formulas to give an example of what is
possible within this formalism. We use the standard Łukasiewicz semantics of
fuzzy logic (see Lukasiewicz and Straccia [23] for an overview):

######  Definition 2 (Łukasiewicz Semantics [23])

$\displaystyle\mathcal{I}(\phi\land\psi)$
$\displaystyle=\max(\mathcal{I}(\phi)+\mathcal{I}(\psi)-1,0)$
$\displaystyle\mathcal{I}(\phi\lor\psi)$
$\displaystyle=\min(\mathcal{I}(\phi)+\mathcal{I}(\psi),1)$
$\displaystyle\mathcal{I}(\phi\rightarrow\psi)$
$\displaystyle=\min(1-\mathcal{I}(\phi)+\mathcal{I}(\psi),1)$
$\displaystyle\mathcal{I}(\neg\phi)$ $\displaystyle=1-\mathcal{I}(\phi)$

In particular, $\phi\leftrightarrow\psi$ is interpreted into
$1-|\mathcal{I}(\phi)-\mathcal{I}(\psi)|$ in Łukasiewicz Logic, which matches
the intuition we intend in our formulas. All sets mentioned in the following
definitions are assumed to be finite unless mentioned otherwise.

######  Definition 3 ( Trichotomic Argumentation Framework (T-AF))

A _Trichotomic Argumentation Framework_
$(P_{i},P_{e},S,\operatorname{as},\operatorname{es},(\operatorname{si}_{s},I_{s})_{s\in
S},(O_{x})_{x\in P_{e}})$ is a structure over sets

  * •

$P=P_{i}\uplus P_{e}$ of _propositions_ formed from _illocutions_ $P_{i}$ and
_entities_ $P_{e}$,

  * •

$S=S_{att}\uplus S_{sup}$ of _atomic argumentation schemes_ formed from
_attack schemes_ $S_{att}$ and _support schemes_ $S_{sup}$,

with additional data

  * •

$\operatorname{as}:S\to\mathbb{N}$ representing the _arity_ of schemes,

  * •

$\operatorname{es}:S\to\mathbb{N}$ representing the _exception arity_ of
schemes,

  * •

$\operatorname{si}_{s}:[0,1]^{\operatorname{as}(s)}\to[0,1]^{\operatorname{es}(s)}\to[0,1]\to[0,1]$
representing the _interpretation_ of atomic schemes,

  * •

$I_{s}\subseteq P^{\operatorname{as}(s)}\times P^{\operatorname{es}(s)}\times
P$ representing _atomic argumentation actions_ , relating the _givens_ or
_premises_ and the _exceptions_ to the _claim_ or _conclusion_ , and

  * •

$O_{x}:P\to([0,1]\to[0,1])$ representing _belief_.

######  Remark 1 (Entities as Propositions)

Argumentation actions can include entities as premises and conclusions. An
entity seen as a proposition is to be read as the trustworthiness of the
entity. Therefore, an _Ad Hominem_ argument against entity $x$ would be an
argumentation action with $e$ as conclusion prescribing low trustworthiness.
Trust establishment or propagation schemes as described by Parsons et al. [31]
can be modelled as inferences perscribing high trustworthiness to an entity.
Additionally, treating entities as propositions enables the trust relations
decribed in Section 2.1 to be incorporated into the belief predicate $O$.

######  Example 3 (Kinds of inferences in T-AFs)

Note that the signature of the scheme interpretation function
$\operatorname{si}$ has as additional argument the acceptability of the
conclusion. It can therefore be seen as a fuzzy predicate over the combination
of givens, exceptions, and claim where _givens_ and _exceptions_ are the
propositions related to a _claim_ by an argumentation action. This allows
expressing different concepts, and combinations thereof, via this single
notion:

  * •

_Exceptions_ can produce a predicate that accepts all truth values of the
claim as the exception defeats the argumentation action;

  * •

_Weighted premises_ as seen in the specification of Walton-style schemes can
be incorporated through linear combination of the givens;

  * •

_Necessary conditions_ can produce a predicate specifying that the claim can
not be accepted unless the givens are accepted: $\neg p\rightarrow\neg c$
(read, e.g., as a fuzzy formula). This is what Boudhar et al. [9] call
_necessary support_ ;

  * •

_Sufficient conditions_ can produce a predicate specifying that the claim has
to be accepted when the givens are accepted: $p\rightarrow c$. This is what
Boella et al. [7] call _deductive support_ ;

  * •

_Inhibiting conditions_ can produce a predicate specifying that the claim can
not be accepted unless the givens are rejected: $p\rightarrow\neg c$;

  * •

_Disjunctive conditions_ can produce a predicate specifying that the givens or
the claim need to be accepted: $\neg p\rightarrow c$;

Similarly, belief is interpreted as a predicate $O$ rather than a single truth
value, thus capturing belief and disbelief as well as uncertainty.

######  Definition 4 (Scheme Interpretation)

The scheme interpretation is required to adhere to the given type of
argumentation scheme: An attack (support) is interpreted as a fuzzy predicate
that allows increasing (decreasing) acceptance of the claim if the acceptance
of the premises increases. This is made explicit in the following constraints
for scheme interpretation:

$\displaystyle\bigwedge_{s\in
S_{sup};\bar{p},\bar{q}\in[0,1]^{\operatorname{as}(s)};c\in[0,1]}\bar{p}\leq\bar{q}\rightarrow\bigvee_{c^{\prime}\in[0,1]}c\leq
c^{\prime}\land\operatorname{si}_{s}(\bar{p},\bot^{\operatorname{es}(s)},c)\leq\operatorname{si}_{s}(\bar{q},\bot^{\operatorname{es}(s)},c^{\prime})$
$\displaystyle\bigwedge_{s\in
S_{att};\bar{p},\bar{q}\in[0,1]^{\operatorname{as}(s)};c\in[0,1]}\bar{p}\leq\bar{q}\rightarrow\bigvee_{c^{\prime}\in[0,1]}c^{\prime}\leq
c\land\operatorname{si}_{s}(\bar{p},\bot^{\operatorname{es}(s)},c)\leq\operatorname{si}_{s}(\bar{q},\bot^{\operatorname{es}(s)},c^{\prime})$

######  Example 4 (Logically: Argument from Position to Know)

Consider a logical representation of _Argument from Position to Know_ (1)
using an epistemic modality $B_{a}$, read “$a$ believes that”, and a modality
$L_{a}$ for locution, read “$a$ said that”:

$\prftree[r]{(PositiontoKnow)}{B_{a}A\leftrightarrow
A}{L_{a}A}{L_{a}A\rightarrow B_{a}A}{A}$

Critical questions 1 and 3 are modelled as attacks against truth of the
premises while question 2 would be an attack against an added honesty premise
$L_{a}A\rightarrow B_{a}A$. As E-nodes are treated as propositions for their
honesty the third premise would be represented simply by $a$. Note that the
inference is deductively valid and therefore can have no exceptions but only
attacks on premises. This also leads to a uniform weighting of premises; hence
the interpretation for the scheme would correspond to the formula
$\Big{(}(B_{a}A\leftrightarrow A)\land L_{a}A\land a\Big{)}\rightarrow A$
involving illocutions $B_{a}A\leftrightarrow A$, $L_{a}A$, and $A$ as well as
the entity $a$.

Due to having diverse and joint notions of inference, forming complex attacks
to derive sensible semantics becomes more challenging [1, 17].

######  Definition 5 (Composite Schemes)

Define the extended sets of _complex support (attack) schemes_
$S_{sup}^{\prime}$ ($S_{att}^{\prime}$) and their union $S^{\prime}$ of
_complex schemes_ inductively:

  * •

$S_{sup}\subseteq S_{sup}^{\prime}$ and $S_{att}\subseteq S_{att}^{\prime}$

  * •

Given schemes $s\in S_{sup}^{\prime},t\in S^{\prime}$ and
$i\in\operatorname{as}(t)$ define the _composite scheme_ $s;_{i}t$. Depending
on wether $t$ was an attack or a support, $s;_{i}t$ belongs to
$S_{att}^{\prime}$ or $S_{sup}^{\prime}$, respectively.

The composite scheme $s;_{i}t$ is interpreted as follows:

$\displaystyle\operatorname{as}(s;_{i}t)=\operatorname{as}(s)+\operatorname{as}(t)-1$
$\displaystyle\operatorname{es}(s;_{i}t)=\operatorname{es}(s)+\operatorname{es}(t)$
$\displaystyle
I_{s;_{i}t}=\\{(u_{1},\dots,u_{i-1},\bar{v},u_{i+1},\dots,u_{\operatorname{as}(t)},\bar{e},\bar{f},p)\mid(\bar{v},\bar{e},\pi_{i}\bar{u})\in
I_{s},(\bar{u},\bar{f},p)\in I_{t}\\}$
$\displaystyle\operatorname{si}_{s;_{i}t}(u_{1},\dots,u_{i-1},\bar{v},u_{i+1},\dots,u_{\operatorname{as}(t)},\bar{e},\bar{f},p)=$
$\displaystyle\qquad\textstyle\bigvee_{b\in[0,1]}\operatorname{si}_{s}(\bar{v},\bar{e},b)\land\operatorname{si}_{t}(u_{1},\dots,b,\dots,u_{\operatorname{as}(t)},\bar{f},p)$

Note that $S_{att}^{\prime}$ or $S_{sup}^{\prime}$ may be infinite if there
are cycles in the graph that allow infinite composition.

For classifying extensions in our T-AFs, we follow the approach of graded
bipolar frameworks, which define a fuzzy labelling specifying the
acceptability of an argument. Using graded acceptance, uncertainty and
preferences can be incorporated.

###### Definition 6

A _labelling_ for a T-AF is a function $l:P\to[0,1]$ assigning to each
proposition a degree of acceptance. Let $L$ denote the set of all labellings.

Properties of labellings and actors can then be expressed as fuzzy formulas
over T-AFs. Entities can be grouped based on the similarity of their beliefs,
e.g. in order to reason about group dynamics.

######  Definition 7 (Similarity)

Two actors $x,y$ in a T-AF are considered _similar_ if they have voiced
similar belief.

$\mathit{Sim}(x,y)\coloneqq\textstyle\bigwedge_{l\in L;p\in
P}O_{x}(p,l(p))\leftrightarrow O_{y}(p,l(p))$

######  Definition 8 (Attack & Support)

A proposition $p$ is _attacked_ under a given labelling $l$ if there is an
attack (complex or atomic) with accepted givens, rejected exceptions, and
claim $p$:

$\textstyle\mathit{Att}_{l}(p)\coloneqq\bigvee_{s\in
S_{att}^{\prime};(\bar{q},p)\in
I_{s}}\bigwedge_{i\in\operatorname{as}(s)}l(\pi_{i}\bar{q})\land\bigwedge_{i\in\operatorname{es}(s)}\neg
l(\pi_{\operatorname{as}(s)+i}\bar{q})$

The definition of _support_ $\mathit{Sup}_{l}(p)$ is similar but with $s\in
S_{sup}^{\prime}$.

In Dung-style AAFs (1), the semantics is mostly defined in terms of a notion
of _defense_ , which needs to be suitable adapted in the presence of supports
and exceptions:

######  Definition 9 (Defense)

A proposition $p$ is _defended_ under a labelling $l$ if every attack against
$p$ is _defeated_. That is, either a given of the attack is attacked or an
exception is supported by accepted propositions:

$\textstyle\mathit{Def}_{l}(p)\coloneqq\bigwedge_{s\in
S_{att}^{\prime};(\bar{q},p)\in
I_{s}}\bigvee_{i\in\operatorname{as}(s)}\mathit{Att}_{l}(\pi_{i}\bar{q})\lor\bigvee_{i\in\operatorname{es}(s)}\mathit{Sup}_{l}(\pi_{\operatorname{as}(s)+i}\bar{q})$

In AAFs, the notion of attack induces a notion of _conflict-freeness_. For
T-AFs, we generalize this concept to a notion of _consistency_ effectively
incorporating both attack and support.

######  Definition 10 (Consistency)

A labelling $l$ for a T-AF is _consistent_ if the assigned labels are in
accordance with the argumentation actions of the system.

$\textstyle\mathit{Co}(l)\coloneqq\bigwedge_{s\in S;(\bar{v},p)\in
I_{s}}\operatorname{si}_{s}(l[\bar{v}],l(p))$

(To see that this does indeed generalize the standard notion of conflict-
freeness, introduce a unary attack scheme and equip it with mutual exclusion
between the given and the claim as the schema interpretation.)

###### Remark 2

Note that $\mathit{Co}(l)$ can be seen as the evaluation of the formula
$\bigwedge_{s\in S;(\bar{v},p)\in I_{s}}s(\bar{v},p)$ under the labelling $l$
but we prefer to write it as defined because we later need to talk about
different labellings in the same formula.

The _extensions_ in the style of Dung [19] become fuzzy predicates defined in
terms of our definition of defense and consistency — that is, a given
labelling will not be absolutely stable, preferred etc., but rather have these
properties to a certain degree, in accordance with our general consideration
of weights. We emphasize that this applies in particular also to preferred and
grounded labellings, which are now ‘maximal’ or ‘least’ not in an absolute but
in a fuzzy sense.

######  Definition 11 (Admissible Labelling)

A labelling $l$ is _admissible_ if it is consistent and each accepted
proposition is defended in $l$:

$\textstyle\mathit{Adm}(l)\coloneqq\mathit{Co}(l)\land\bigwedge_{p\in
P}l(p)\rightarrow Def_{l}(p)$

######  Definition 12 (Stable Labelling)

A labelling $l$ is _stable_ if it is consistent and each rejected proposition
is attacked under $l$:

$\textstyle\mathit{Sl}(l)\coloneqq\mathit{Co}(l)\land\bigwedge_{p\in P}\neg
l(p)\rightarrow\mathit{Att}_{l}(p)$

######  Definition 13 (Preferred Labelling)

A labelling is _preferred_ if it is a maximal admissible labelling:

$\mathit{Pref}(l)\coloneqq\mathit{Adm}(l)\land\bigwedge_{l^{\prime}\in
L}\bigg{(}\mathit{Adm}(l^{\prime})\rightarrow\Big{(}\bigwedge_{p\in
P}l(p)\rightarrow l^{\prime}(p)\Big{)}\rightarrow\bigwedge_{p\in
P}l^{\prime}(p)\rightarrow l(p)\bigg{)}$

######  Definition 14 (Complete Labelling)

A labelling is _complete_ if it is admissible and accepts each defended
proposition:

$\textstyle\mathit{Cl}(l)\coloneqq\mathit{Adm}(l)\land\bigwedge_{p\in
P}\mathit{Def}_{l}(p)\rightarrow l(p)$

######  Definition 15 (Grounded Labelling)

A labelling is _grounded_ if it is a least complete labelling of the T-AF:

$\textstyle\mathit{Gl}(l)\coloneqq\mathit{Cl}(l)\land\bigwedge_{l^{\prime}\in
L}\mathit{Cl}(l^{\prime})\rightarrow\bigwedge_{p\in P}l(p)\rightarrow
l^{\prime}(p)$

Apart from translating the definitions of Dung AAFs [19] we can define
exemplary properties of the actors in the dialogue specific to our extended
setting.

######  Definition 16 (Agreement)

A labelling $l$ _agrees_ with an actor $x$ if $x$’s beliefs are retained in
$l$:

$\textstyle\mathit{Ag}(l,x)\coloneqq\bigwedge_{p\in P}O_{x}(p,l(p))$

Having defined agreement opens possibilities to express interesting properties
like rationality of an actor.

######  Definition 17 (Rationality)

An actor $x$ is _rational_ if her belief can be extended to as a consistent
position:

$\textstyle\mathit{Ra}(x)\coloneqq\bigvee_{l\in
L}\mathit{Co}(l)\land\mathit{Ag}(l,x)$

######  Definition 18 (Justified Trust)

An actor $x$ has _justified trust_ if the belief of actors $x$ trusts can be
consistent in the same labelling:

$\textstyle\mathit{Jt}(x)\coloneqq\bigvee_{l\in
L}\mathit{Co}(l)\land\bigwedge_{y\in P_{e}}\Big{(}(O_{x}(y,l(y))\land
l(y))\rightarrow\mathit{Ag}(l,y)\Big{)}$

An actor $x$ having justified trust not only expresses that the actors $x$
trusts have rational beliefs but also that these beliefs are consistent with
eachother.

######  Definition 19 (Trust Compliance)

A labelling $l$ is _trust compliant_ for an actor $x$ if belief of actors
trusted by $x$ is reflected in $l$:

$\textstyle\mathit{Tc}(l,x)\coloneqq\bigwedge_{p\in P_{i}}\Big{(}\bigvee_{y\in
P_{e}}O_{x}(y,l(y))\land l(y)\land O_{y}(p,l(p))\Big{)}$

Trust compliance expresses that $x$ trusts her trusted actors for validity in
the sense of Demolombe [18] and reflects that in her beliefs (the labelling
$l$). Combining all trusted actors via disjunction captures a similar notion
to parallel path composition as defined by Parsons et al. [32].

##  4 ASPIC+ Translation

Given that the argument map in T-AIF is very similar to AIF and hence can be
easily converted, we obtain ASPIC+ semantics using an existing translation
[6]. That translation can be improved with the additional information
available in our extended formalism to obtain per-actor semantics from the
translation.

For constructing the argumentation theory we modify the translation [6] by
forming a separate knowledge base for each actor. An ordering on the formulas
in the knowledge base can be derived from the trust network via the
accumulated trust and commitment of each author. This semantically results in
different systems for each participant that reflect her own beliefs in the
propositions as well as her trust in the other entities participating in the
discussion.

##  5 Conclusion and future research

We have proposed the _Trichotomic_ , aimed at representing argumentation
resulting from dialogue between multiple actors, capturing aspects from all
three areas of the aristotelian trichotomy. Our format is inspired by the AIF
and especially its extensions AIF+ and S-AIF. The ethos aspect is captured by
relating entities via trust and allowing their ethical properties to be
involved in arguments. Relating entities to their illocutions incorporates the
strength of their commitment as an aspect from the domain of pathos. Finally,
the argument map represents the logos aspect of argumentation, capturing the
logical connections between the propositions. Given that the AIF and the
existing extending formalisms are specified using the Web Ontology Language
(OWL), we plan to formalize our format in an ontology as well.

Our main contribution in this paper is to provide a formalism for representing
and reasoning about more than just the logos aspect of argumentation. This
enables a very natural treatment, e.g., of Ad Hominem arguments, which affect
the acceptance of all arguments made by the attacked actor. Having the
dialogue game [27, 36] in the representation can interact nicely with ethical
aspects of argumentation by e.g. penalizing trustworthiness when illogical
moves are made or undefendable stances are taken. The representation is also a
good basis for deriving a per-participant semantics incorporating voiced
beliefs and trust relations of actors to approximate what they are likely to
believe.

We have proposed T-AFs as an exemplary simplified version of T-AIF, aimed at
defining properties profiling the participating actors. T-AFs do not include
the dialogue part of T-AIF, and there are meta-arguments — that is, arguments
talking about the discussion itself, which in fact occur rather frequently —
that become expressible when the dialogue is included in the structure. An
example of this kind are arguments against an actor’s credibility on the
premise that she has retracted previous statements she had high commitment to
when counterarguments were given.

Since the tooling around the AIF ecosystem only supports some features of
T-AIF, we plan to provide tool support for creating, storing, and visualizing
argument structures in our format. The envisioned tool support will also
include reasoning capabilities beyond the described ASPIC+ translation for
working on a given T-AIF graph directly.

As maybe apparent from our choice of examples, we plan to evaluate our
formalism on a dataset of argumentative discourse on Twitter, specifically
concerning Brexit (pre-referendum). Ultimately, we intend to construct T-AIF
representations automatically from written discussions using a structured
argumentation mining algorithm.

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##  Appendix 0.A Appendix

NodeS-NodeI-NodeL-NodeCA-NodeTA-NodeRA-NodePA-NodeYA-NodeUpper
OntologyFormPremiseConclusionPresumptionSchemeConflictPreferenceRulePresumptiveInductiveDeductiveTransitionIllocutionaryForms
Ontology Figure 2: The AIF+ Ontology. The boxes are concepts and the arrows
represent the inclusion relation. L-nodes, TA-nodes, and Transition Schemes
[36] as well as YA-nodes and Illocutionary Schemes [35] were added to AIF.

  *[AIF]: Argument Interchange Format
  *[IAT]: Inference Anchoring Theory
  *[S-AIF]: Social
  *[T-AIF]: Trichotomic
  *[AAF]: Abstract Argumentation Framework
  *[BAFs]: Bipolar Argumentation Framework
  *[T-AF]: Trichotomic Argumentation Framework
  *[T-AFs]: Trichotomic Argumentation Framework
  *[AAFs]: Abstract Argumentation Framework
  *[OWL]: Web Ontology Language