# Presumptive Reasoning in a Paraconsistent Setting

Sabine Frittella Daniil Kozhemiachenko The research of Sabine Frittella and
Daniil Kozhemiachenko was funded by the grant ANR JCJC 2019, project PRELAP
(ANR-19-CE48-0006).INSA Centre Val de Loire, Univ. Orléans, LIFO EA 4022,
France<EMAIL_ADDRESS>Bernoulli
Institute, Rijksuniversiteit Groningen
Groningen, the Netherlands Bart Verheij Bart Verheij participates in the
Hybrid Intelligence Center, a 10-year programme funded by the Dutch Ministry
of Education, Culture and Science through the Netherlands Organisation for
Scientific Research, https://hybrid-intelligence-centre.nl.Bernoulli
Institute, Rijksuniversiteit Groningen
Groningen, the Netherlands<EMAIL_ADDRESS>
###### Abstract

We explore presumptive reasoning in the paraconsistent case. Specifically, we
provide semantics for non-trivial reasoning with presumptive arguments with
contradictory assumptions or conclusions. We adapt the case models proposed by
Verheij [26, 27] and define the paraconsistent analogues of the three types of
validity defined therein: coherent, presumptively valid, and conclusive ones.
To formalise the reasoning, we define case models that use
$\mathsf{BD}\triangle$, an expansion of the Belnap–Dunn logic with the Baaz
Delta operator. We also show how to recover presumptive reasoning in the
original, classical context from our paraconsistent version of case models.
Finally, we construct a two-layered logic over $\mathsf{BD}\triangle$ and
$\mathsf{biG}$ (an expansion of Gödel logic with a coimplication $\Yleft$ or
$\triangle$) and obtain a faithful translation of presumptive arguments into
formulas.

##  1 Introduction

When arguing for a given statement, it can happen that a person uses
contradictory assumptions. From the classical standpoint, every statement
trivially follows from a contradiction. This, however, is counter-intuitive as
an agent may not be willing to accept a completely arbitrary statement just
because their premises contain a contradiction.

In general, an argument from $\phi$ to $\chi$ (written formally as
$\langle\phi,\chi\rangle$) can be either _deductive_ (when $\phi$  _entails_
$\chi$) or _presumptive_ (otherwise). I.e., to verify the correctness of a
deductive argument, it suffices to utilise purely logical means while
establishing the correctness (acceptability) of a presumptive one requires an
extra-logical framework. Thus, from _the classical standpoint_ , every
argument from a contradictory premise is _deductive_. Hence, if one wants to
formalise _non-trivial_ presumptive reasoning from contradictory premises, one
has to use a _paraconsistent_ logic, i.e., a logic where the explosion
principle $p,\neg p\models q$ is not valid.

### Dung’s argumentative semantics vs case models

An influential approach to the formalisation of argumentation focuses on
argument attack [12]. The main idea is to represent the argumentative
framework as a directed graph where $\mathcal{A}\rightarrow\mathcal{B}$ is
interpreted as ‘argument $\mathcal{A}$ attacks $\mathcal{B}$’ (here, arguments
are treated as unified statements, and premises and conclusions are not
singled out). Then, $\mathcal{A}$ is _acceptable_ if it responds to every
attack (or, formally, if $\mathcal{A}\rightarrow\mathcal{B}$ for every
$\mathcal{B}$ s.t. $\mathcal{B}\rightarrow\mathcal{A}$). An argument’s
correctness depends on the argumentation semantics choice.

However, the connection of Dung’s approach to standard logical semantics may
not be straightforward. In addition, the support of arguments is abstracted.
Both issues have been addressed in several ways (cf., e.g. [10, 6, 15, 22]).
One such alternative to Dung’s approach was proposed in [26] and further
developed in [27]. In these works, the interpretation of presumptive arguments
was given by means of _case models_ : sets of classically incompatible
satisfiable propositional formulas called ‘cases’ (whence the name) with a
preference relation defined thereon. An argument in this framework has the
following form: $\mathcal{A}=\langle\phi,\chi\rangle$. Here, $\phi$ is the
premise, $\chi$ is the conclusion, and, in addition, $\mathcal{A}$ presents a
_case_ — $\phi\wedge\chi$. Three kinds of acceptable arguments were studied:
coherent (both the premise and the conclusion are supported by at least one
case), presumptively valid (both the premise and conclusion are supported by
the most preferred case), and conclusive (the conclusion is supported by all
cases that support the premise). Furthermore, a representation of case models
in terms of sample spaces with probability measures was devised and a
correspondence between different arguments and probabilities of the
corresponding events was provided. An important distinction between case
models and Dung’s semantics is that the validities in the former are defined
via the entailment in the classical logic. Thus, one can produce a non-
classical counterpart to case models by changing the underlying entailment
relation.

### Non-trivial contradictory arguments

In both approaches discussed above, it is assumed that the acceptable
arguments are not _self-contradictory_. Namely, if $\phi$ is an argument in
Dung’s framework, it should be classically satisfiable, and if
$\langle\chi,\psi\rangle$ is an argument over a case model, then both $\chi$
and $\psi$ must be classically satisfiable. This restriction is easy to
explain in Dung’s approach: indeed, we can claim that a contradictory argument
attacks itself. In the case model setting, however, it makes sense to consider
contradictory arguments and cases under the following interpretation.

Every ‘case’ in the model can be thought of as a source that gives some
information regarding a given set of statements. Accordingly, the preference
relation on cases shows which sources are trusted more or less. In this
interpretation, it is clear that even if a source is trusted, it can provide a
contradictory response to a question (e.g., a police officer testifying in
court can first claim that they were unarmed while on patrol and then say
‘when I saw the suspect, I immediately drew my pistol out of the holster’) or
fail to provide any information at all.

Let us now introduce the running example to illustrate the contexts that we
aim to formalise.

######  Running example, part 1 (Witnessing a robbery).

An investigator reads a report by a police officer who questioned several
witnesses on a bank robbery. The relevant information is whether the
perpetrator had a limp ($l$), whether they had a big bag for the robbed
valuables ($b$), and whether they used the lift or the staircase ($s$) to
leave the office. The report contains the following testimonies.

  * •

$\mathsf{c}_{1}$ tells that the perpetrator indeed had a limp but cannot say
anything about how they left the office; moreover, $\mathsf{c}_{1}$ tells that
the robber put all the loot in the pockets.

  * •

$\mathsf{c}_{2}$ tells nothing about whether the perpetrator was limping and
mentions that the perpetrator had a big shoulder bag; unfortunately,
$\mathsf{c}_{2}$ is confused: they claim that they saw the robber using the
lift but are also saying that ‘the lift has been out of order for half a
year’.

  * •

$\mathsf{c}_{3}$ testifies that the robber had a limp but walked down the
stairs; $\mathsf{c}_{3}$’s account is also contradictory: they describe the
bag as ‘huge’ but then say that the robber put it into the pocket.

All witnesses gave non-classical (incomplete or contradictory) responses,
whence we cannot straightforwardly represent them in the case models, nor in
Dung’s framework. An investigator, however, needs to draw conclusions from the
accounts at hand. E.g., they might want to know how the perpetrator in fact
left the building and for that, they need to know whether the perpetrator had
a limp.

### Plan of the paper

In this paper, we adapt the case models presented in [26, 27] to the
presumptive reasoning with possibly contradictory statements. To this end, we
will use $\mathsf{BD}\triangle$ — the expansion of the Belnap–Dunn logic [2,
13, 5] with a Baaz $\triangle$ operator (cf. [3] for the original presentation
of $\triangle$ in the context of fuzzy logics) originating from [25]. We
define the analogues of coherent, presumptively valid, and conclusive
arguments and show their relations to one another. Finally, we are going to
provide a logical representation of all these arguments.

The remainder of the paper is structured as follows. In Section 2, we present
the syntax and semantics of $\mathsf{BD}\triangle$. Then, in Section 3, we
develop the $\mathsf{BD}\triangle$ case models. In Section 4, we present a
logic that formalises reasoning in $\mathsf{BD}\triangle$ models and construct
a faithful translation of arguments to formulas. Finally, in Section 5, we
summarise the results and provide a plan for future research.

##  2 Logical preliminaries

The language of $\mathscr{L}_{\mathsf{BD}\triangle}$ and its $\triangle$-free
fragment $\mathscr{L}_{\mathsf{BD}}$ are defined via the following grammar
($\mathtt{Prop}$ is a fixed countable set of propositional variables).

$\displaystyle\mathscr{L}_{\mathsf{BD}\triangle}\ni\phi$
$\displaystyle\coloneqq
p\in\mathtt{Prop}\mid\neg\phi\mid(\phi\wedge\phi)\mid(\phi\vee\phi)\mid\triangle\phi$

There are several semantics for $\mathsf{BD}$ and its expansions (cf. [21] for
the examples). One of the simplest is a truth-table semantics from [5]. There,
a formula can have one of the following four values corresponding to the
available information regarding a statement $\phi$. $\mathbf{T}$ stands for
‘there is only information in support of $\phi$’; $\mathbf{F}$ for ‘there is
only information denying $\phi$’; $\mathbf{N}$ for ‘there is information
neither in support nor in denial of $\phi$’; $\mathbf{B}$ for ‘there is
information both in support and denial of $\phi$’. We also use frame semantics
(cf., e.g., [9, 17, 8]) as it is more convenient for the logical
representation of case models.

######  Definition 1 (Truth-table semantics of $\mathsf{BD}\triangle$).

A $\mathbf{4}$-valuation is a map
$v_{\mathbf{4}}:\mathtt{Prop}\rightarrow\\{\mathbf{T},\mathbf{B},\mathbf{N},\mathbf{F}\\}$
that is extended to complex formulas using the following definitions.

| $\neg$ |
---|---
$\mathbf{T}$ | $\mathbf{F}$
$\mathbf{B}$ | $\mathbf{B}$
$\mathbf{N}$ | $\mathbf{N}$
$\mathbf{F}$ | $\mathbf{T}$
| $\triangle$ |
---|---
$\mathbf{T}$ | $\mathbf{T}$
$\mathbf{B}$ | $\mathbf{T}$
$\mathbf{N}$ | $\mathbf{F}$
$\mathbf{F}$ | $\mathbf{F}$
| $\wedge$ | $\mathbf{T}$ | $\mathbf{B}$ | $\mathbf{N}$ | $\mathbf{F}$
---|---|---|---|---
$\mathbf{T}$ | $\mathbf{T}$ | $\mathbf{B}$ | $\mathbf{N}$ | $\mathbf{F}$
$\mathbf{B}$ | $\mathbf{B}$ | $\mathbf{B}$ | $\mathbf{F}$ | $\mathbf{F}$
$\mathbf{N}$ | $\mathbf{N}$ | $\mathbf{F}$ | $\mathbf{N}$ | $\mathbf{F}$
$\mathbf{F}$ | $\mathbf{F}$ | $\mathbf{F}$ | $\mathbf{F}$ | $\mathbf{F}$
| $\vee$ | $\mathbf{T}$ | $\mathbf{B}$ | $\mathbf{N}$ | $\mathbf{F}$
---|---|---|---|---
$\mathbf{T}$ | $\mathbf{T}$ | $\mathbf{T}$ | $\mathbf{T}$ | $\mathbf{T}$
$\mathbf{B}$ | $\mathbf{T}$ | $\mathbf{B}$ | $\mathbf{T}$ | $\mathbf{B}$
$\mathbf{N}$ | $\mathbf{T}$ | $\mathbf{T}$ | $\mathbf{N}$ | $\mathbf{N}$
$\mathbf{F}$ | $\mathbf{T}$ | $\mathbf{B}$ | $\mathbf{N}$ | $\mathbf{F}$

######  Definition 2 (Frame semantics for $\mathsf{BD}\triangle$).

Let $\phi,\phi^{\prime}\in\mathscr{L}_{\mathsf{BD}\triangle}$. For a model
$\mathfrak{M}=\langle W,v^{+},v^{-}\rangle$ with
$v^{+},v^{-}:\mathtt{Prop}\rightarrow 2^{W}$, we define notions of
$w\vDash^{+}\phi$ and $w\vDash^{-}\phi$ for $w\in W$ as follows.

$w\vDash^{+}p$ | iff | $w\in v^{+}(p)$ | $w\vDash^{-}p$ | iff | $w\in v^{-}(p)$
---|---|---|---|---|---
$w\vDash^{+}\neg\phi$ | iff | $w\vDash^{-}\phi$ | $w\vDash^{-}\neg\phi$ | iff | $w\vDash^{+}\phi$
$w\vDash^{+}\phi\wedge\phi^{\prime}$ | iff |  $w\vDash^{+}\phi$ and $w\vDash^{+}\phi^{\prime}$ | $w\vDash^{-}\phi\wedge\phi^{\prime}$ | iff | w$\vDash^{-}\phi$ or $w\vDash^{-}\phi^{\prime}$
$w\vDash^{+}\phi\vee\phi^{\prime}$ | iff |  $w\vDash^{+}\phi$ or $w\vDash^{+}\phi^{\prime}$ | $w\vDash^{-}\phi\vee\phi^{\prime}$ | iff |  $w\vDash^{-}\phi$ and $w\vDash^{-}\phi^{\prime}$
$w\vDash^{+}\triangle\phi$ | iff | $w\vDash^{+}\phi$ | $w\vDash^{-}\triangle\phi$ | iff | $w\nvDash^{+}\phi$

We define the _positive_ and _negative interpretations of $\phi$_ as follows:
$|\phi|^{+}=\\{w\in W\mid w\vDash^{+}\phi\\}$; $|\phi|^{-}=\\{w\in W\mid
w\vDash^{-}\phi\\}$.

We say that a sequent $\phi\vdash\chi$ is _satisfied on $\mathfrak{M}$_
(denoted, $\mathfrak{M}\models[\phi\vdash\chi]$) iff
$|\phi|^{+}\subseteq|\chi|^{+}$ and $|\chi|^{-}\subseteq|\phi|^{-}$.
$\phi\vdash\chi$ is _valid_ iff it is satisfied on every model. In this case,
we say that $\phi$ _entails_ $\chi$ and write
$\phi\models_{\mathsf{BD}\triangle}\chi$.

Let us make several quick observations regarding $\mathsf{BD}\triangle$.
First, the semantical conditions of $\neg$, $\wedge$, and $\vee$ coincide with
those from the classical logic. On the other hand, it is more intuitive to
interpret $w\vdash^{+}\phi$ as ‘$w$ gives evidence for (confirms) $\phi$’ and
$w\vdash^{-}\phi$ as ‘$w$ gives evidence against (denies) $\phi$’. Thus, $w$
confirms $\phi\wedge\phi^{\prime}$ when both conjuncts are confirmed by $w$
and $w$ denies $\phi\wedge\phi^{\prime}$ when at least one conjunct is denied.

The difference is that in $\mathsf{BD}\triangle$ the truth and falsity of a
formula _are independent_. Thus, in contrast to the classical logic, neither
$p\wedge\neg p\vdash q$ nor $p\vdash q\vee\neg q$ is valid. Second, the
addition of $\triangle$ (read ‘it is true that’) to $\mathsf{BD}$ makes it
weakly functionally complete (cf. [20] for further details). This allows us to
represent every testimony a source can give regarding $\phi$ (i.e., confirm
$\phi$, contradict itself regarding $\phi$, say nothing about $\phi$ or deny
$\phi$) as follows:

$\displaystyle\mathbf{t}(\phi)$
$\displaystyle\coloneqq\triangle\phi\wedge\neg\triangle\neg\phi$
$\displaystyle\mathbf{b}(\phi)$
$\displaystyle\coloneqq\triangle\phi\wedge\triangle\neg\phi$
$\displaystyle\mathbf{n}(\phi)$
$\displaystyle\coloneqq\neg\triangle\phi\wedge\neg\triangle\neg\phi$
$\displaystyle\mathbf{f}(\phi)$
$\displaystyle\coloneqq\neg\triangle\phi\wedge\triangle\neg\phi$

Note that $v_{\mathbf{4}}(\mathbf{x}(\phi))\\!=\\!\mathbf{T}$ if
$v_{\mathbf{4}}(\phi)\\!=\\!\mathbf{X}$; and
$v_{\mathbf{4}}(\mathbf{x}(\phi))\\!=\\!\mathbf{F}$ otherwise (with
$\mathbf{x}\\!\in\\!\\{\mathbf{t},\mathbf{b},\mathbf{n},\mathbf{f}\\}$ and
$\mathbf{X}\\!\in\\!\\{\mathbf{T},\mathbf{B},\mathbf{N},\mathbf{F}\\}$).
Furthermore, it is possible to define $\bot$ and $\top$ s.t. $|\top|^{+}=W$,
$|\top|^{-}=\varnothing$, $|\bot|^{+}=\varnothing$, $|\bot|^{-}=W$ as follows:
$\top\coloneqq\triangle p\vee\neg\triangle p$; $\bot\coloneqq\neg\top$.
$\triangle$ also allows for the internalisation of entailment: for
$\mathbf{x},\mathbf{x}^{\prime}\in\\{\mathbf{t},\mathbf{b},\mathbf{n},\mathbf{f}\\}$,
the formula below is valid iff $\phi\models_{\mathsf{BD}\triangle}\chi$.

$\displaystyle\phi\Rrightarrow\chi$
$\displaystyle\coloneqq\bigvee\limits_{\scriptsize{\begin{matrix}\mathbf{x}\leq_{\mathbf{4}}\mathbf{x}^{\prime}\end{matrix}}}\\!\\!\\!\\!(\mathbf{x}(\phi)\wedge\mathbf{x}^{\prime}(\chi))$
($\mathbf{f}\leq_{\mathbf{4}}\mathbf{b},\mathbf{n}\leq_{\mathbf{4}}\mathbf{t}$;
$\mathbf{b}\not\leq_{\mathbf{4}}\mathbf{n}$;
$\mathbf{n}\not\leq_{\mathbf{4}}\mathbf{b}$)

The following property will be useful in showing how classical case models can
be simulated in $\mathsf{BD}\triangle$.

######  Proposition 1.

Let $\phi\in\mathscr{L}_{\mathsf{BD}\triangle}$ be s.t. every occurrence of
every variable $p$ in $\phi$ is in the scope of $\triangle$. Then for every
$\mathsf{BD}\triangle$ model $\mathfrak{M}$ and $w\in\mathfrak{M}$, exactly
one of the following holds: $w\vDash^{+}\phi$ and $w\nvDash^{-}\phi$, or
$w\nvDash^{+}\phi$ and $w\vDash^{-}\phi$.

###### Proof.

Observe that $\phi$ is constructed from the formulas of the form
$\triangle\chi$ using $\neg$, $\wedge$, and $\vee$. We can now proceed by
induction on $\phi$. The basis case is simple. From Definition 2, we see that
$|\triangle\chi|^{+}=W\setminus|\triangle\chi|^{-}$, whence, indeed, either
$w\vDash^{+}\triangle\chi$ and $w\nvDash^{-}\triangle\chi$ or
$w\nvDash^{+}\triangle\chi$ and $w\vDash^{-}\triangle\chi$. The cases of
$\phi=\psi\vee\psi^{\prime}$, $\phi=\psi\wedge\psi^{\prime}$, and
$\phi=\neg\psi$ can be shown by straightforward application of the induction
hypothesis. ∎

##  3 $\mathsf{BD}\triangle$ case models

In this section, we introduce the $\mathsf{BD}\triangle$ case models. To make
the presentation clearer, let us first recall the case models from [26, 27]
and types of arguments over them that we will henceforth call _classical case
models_ and _classical arguments_ since they use the classical logic as
background.

######  Definition 3 (Classical case models).

A _classical case model_ is a tuple
$\mathfrak{C}_{\mathsf{CPL}}=\langle\mathtt{C},\preceq\rangle$ s.t.
$\mathsf{C}$ is a finite set of pairwise incompatible classically satisfiable
formulas and $\preceq$ is a total preorder on $\mathsf{C}$.

The strict preorder associated with $\preceq$ is interpreted as a preference
relation on the set of cases. I.e., $\phi\prec\phi^{\prime}$ means that the
agent prefers $\phi^{\prime}$ to $\phi$ (or trusts in $\phi^{\prime}$ more
than in $\phi$).

######  Definition 4 (Classical arguments and their types).

An _argument_ is a tuple $\langle\phi,\phi^{\prime}\rangle$ of classical
propositional formulas. The _case_ is the statement $\phi\wedge\phi^{\prime}$,
while a _premise_ (conclusion) is any formula $\chi$ s.t.
$\phi\models_{\mathsf{CPL}}\chi$ ($\phi^{\prime}\models_{\mathsf{CPL}}\chi$).
We say that the argument is _presumptive_ iff
$\phi\not\models_{\mathsf{CPL}}\phi^{\prime}$.

An argument $\langle\phi,\chi\rangle$ over a classical case model
$\mathfrak{C}_{\mathsf{CPL}}=\langle\mathtt{C},\preceq\rangle$ is

  * •

_classically coherent_ iff there is $\psi\in\mathsf{C}$ s.t.
$\psi\models_{\mathsf{CPL}}\phi\wedge\chi$;

  * •

_classically conclusive_ iff it is classically coherent and it holds
$\psi\models_{\mathsf{CPL}}\phi\wedge\chi$ for every $\psi\in\mathsf{C}$ s.t.
$\psi\models_{\mathsf{CPL}}\phi$;

  * •

_classically presumptively valid_ iff it is classically coherent and there is
$\psi\in\mathsf{C}$ s.t. $\psi\models_{\mathsf{CPL}}\phi\wedge\psi$ and
$\psi\succeq\psi^{\prime}$ for every $\psi^{\prime}$ s.t.
$\psi^{\prime}\models_{\mathsf{CPL}}\phi$.

Let us now present $\mathsf{BD}\triangle$ case models and the counterparts to
the coherent, conclusive, and presumptively valid arguments.

######  Definition 5 ($\mathsf{BD}\triangle$ case models).

A _$\mathsf{BD}\triangle$ case model_ is a tuple
$\mathfrak{C}_{\mathsf{BD}\triangle}=\langle\mathsf{C},\preceq\rangle$ with
$\mathsf{C}$ being a finite set of $\mathscr{L}_{\mathsf{BD}\triangle}$
formulas s.t. for any $\phi,\phi^{\prime}\in\mathsf{C}$, it holds that
$\phi\not\models_{\mathsf{BD}\triangle}\bot$ and
$\phi\wedge\phi^{\prime}\models_{\mathsf{BD}\triangle}\bot$, and $\preceq$ a
total preorder on $\mathsf{C}$.

######  Definition 6 (Arguments).

An _argument_ is a tuple $\langle\phi,\phi^{\prime}\rangle$ with
$\phi,\phi^{\prime}\in\mathscr{L}_{\mathsf{BD}\triangle}$. The _case_ is the
statement $\phi\wedge\phi^{\prime}$, while a _premise_ (conclusion) is any
formula $\chi$ s.t. $\phi\models_{\mathsf{BD}\triangle}\chi$
($\phi^{\prime}\models_{\mathsf{BD}\triangle}\chi$). We say that the argument
is _presumptive_ iff $\phi\not\models_{\mathsf{BD}\triangle}\phi^{\prime}$.

We can interpret $\psi\in\mathsf{C}$ as witnesses’ testimonies. A testimony
might be contradictory or omit information relevant to the case. Thus, given
an argument $\langle\phi,\chi\rangle$, it makes sense to differentiate between
three kinds of conclusions.

  1. 1.

Given $\phi$, $\chi$ is claimed to be _true_ but nothing is said whether it is
also non-false.

  2. 2.

Given $\phi$, $\chi$ is claimed to be _non-false_ but nothing is said about
whether it is true as well.

  3. 3.

Given $\phi$, $\chi$ is claimed to be _true and non-false_.

Let us now recall part 1 of the running example and build a case model.

######  Running example, part 2 (Witnessing a robbery, formalisation).

The investigator in part 1 can build the following case model $\mathfrak{C}$
(we omit the ordering for now):

$\displaystyle\mathsf{C}$
$\displaystyle=\\{\underbrace{\mathbf{t}(l)\wedge\mathbf{n}(s)\wedge\mathbf{f}(b)}_{\mathsf{c}_{1}},\underbrace{\mathbf{n}(l)\wedge\mathbf{b}(s)\wedge\mathbf{t}(b)}_{\mathsf{c}_{2}},\underbrace{\mathbf{t}(l)\wedge\mathbf{t}(s)\wedge\mathbf{b}(b)}_{\mathsf{c}_{3}}\\}$

Using part 2 of the running example, we define the counterparts to coherent
and conclusive arguments from [26, 27].

######  Definition 7 (Coherent arguments).

Let $\mathfrak{C}=\langle\mathsf{C},\preceq\rangle$. $\langle\phi,\chi\rangle$
is

  * •

_negatively coherent_ (denoted $\mathfrak{C}\models\phi\mapsto^{-}\chi$) over
$\mathfrak{C}$ iff there is $\psi\in\mathsf{C}$ s.t.
$\chi\models_{\mathsf{BD}\triangle}\phi\wedge\neg\triangle\neg\chi$;

  * •

_positively coherent_ (denoted $\mathfrak{C}\models\phi\mapsto^{+}\chi$) over
$\mathfrak{C}$ iff there is $\psi\in\mathsf{C}$ s.t.
$\chi\models_{\mathsf{BD}\triangle}\phi\wedge\triangle\chi$;

  * •

_strongly coherent_ (denoted $\mathfrak{C}\models\phi\mapsto^{\pm}\chi$) over
$\mathfrak{C}$ iff there is $\psi\in\mathsf{C}$ s.t.
$\chi\models_{\mathsf{BD}\triangle}\phi\wedge\mathbf{t}(\chi)$.

######  Definition 8 (Conclusive arguments).

Let $\mathfrak{C}=\langle\mathsf{C},\preceq\rangle$. $\langle\phi,\chi\rangle$
is

  * •

_negatively conclusive_ over $\mathfrak{C}$ (denoted
$\mathfrak{C}\\!\models\\!\phi\\!\Rightarrow^{-}\\!\chi$) iff it is negatively
coherent and it holds that if $\chi\\!\models_{\mathsf{BD}\triangle}\\!\phi$,
then
$\psi\\!\models_{\mathsf{BD}\triangle}\\!\phi\\!\wedge\\!\neg\triangle\neg\chi$
for any $\psi\in\mathsf{C}$;

  * •

_positively conclusive_ over $\mathfrak{C}$ (denoted
$\mathfrak{C}\\!\models\\!\phi\\!\Rightarrow^{+}\\!\chi$) iff it is positively
coherent and it holds that if $\psi\\!\models_{\mathsf{BD}\triangle}\\!\phi$,
then $\psi\\!\models_{\mathsf{BD}\triangle}\\!\phi\\!\wedge\triangle\chi$ for
any $\psi\in\mathsf{C}$;

  * •

_strongly conclusive_ over $\mathfrak{C}$ (denoted
$\mathfrak{C}\\!\models\\!\phi\\!\Rightarrow^{\pm}\\!\chi$) iff it is strongly
coherent, and it holds that if $\psi\models_{\mathsf{BD}\triangle}\phi$, then
$\psi\models_{\mathsf{BD}\triangle}\phi\wedge\mathbf{t}(\chi)$ for any
$\psi\in\mathsf{C}$.

######  Remark 1.

Let us provide an intuitive explanation of coherent and conclusive arguments.
We begin with coherent arguments:

  * •

for an argument to be _negatively coherent_ , there has to be a case that
supports the premise and _does not contradict the conclusion_ ;

  * •

for an argument to be _positively coherent_ , there has to be a case that
supports the premise and also _supports the conclusion_ ;

  * •

for an argument to be _strongly coherent_ , there has to be a case that
supports the premise, _does not contradict the conclusion_ , and _supports
it_.

Conclusive arguments can be construed as follows:

  * •

for an argument to be _negatively conclusive_ , no case satisfying the premise
should _contradict_ the conclusion of a argument;

  * •

for an argument to be _positively conclusive_ , all cases satisfying the
premises of an argument should support its conclusion.

Observe that the arguments that are both positively and negatively conclusive
are strongly conclusive as well. On the other hand, $\langle\phi,\chi\rangle$
can be both positively and negatively coherent but not strongly coherent if
there is no case $\mathsf{c}$ s.t.
$\mathsf{c}\models_{\mathsf{BD}\triangle}\mathbf{t}(\chi)$.

######  Remark 2 ($\mathsf{BD}\triangle$ and classical arguments).

Note that while there is no classical case model over which both
$\mathcal{A}=\langle\phi,\chi\rangle$ and
$\mathcal{B}=\langle\phi,\neg\chi\rangle$ are classically conclusive (albeit,
they can be presumptively valid), it is possible that they are both
_positively conclusive_ (_negatively conclusive_) if
$\mathsf{c}\models_{\mathsf{BD}\triangle}\mathbf{b}(\chi)$ (resp.,
$\mathsf{c}\models_{\mathsf{BD}\triangle}\mathbf{n}(\chi)$) for every
$\mathsf{c}\in\mathsf{C}$. Still, there is no $\mathsf{BD}\triangle$ case
model over which $\mathcal{A}$ and $\mathcal{B}$ are _strongly conclusive_.

In addition, it is clear that no argument of the form
$\langle\phi,\neg\phi\rangle$ is _classically coherent_ since
$\phi\wedge\neg\phi$ is classically unsatisfiable. On the other hand, $\langle
s,\neg s\rangle$ is _positively coherent_ (by $\mathsf{c}_{2}$) in the model
from the part 2 of the running example.

Finally, it is easy to see that every coherent deductive classical argument
$\langle\phi,\chi\rangle$ (i.e., the one where
$\phi\models_{\mathsf{CPL}}\chi$) is also classically conclusive. In the case
of $\mathsf{BD}\triangle$ arguments, only the weaker statement holds: _if
$\phi\models_{\mathsf{BD}\triangle}\chi$ and $\langle\phi,\chi\rangle$ is
positively coherent, then it is positively conclusive as well_. E.g.,
$p\wedge\neg p\wedge q\models_{\mathsf{BD}\triangle}p\wedge\neg p$ but
$\mathbf{t}(p\wedge\neg p)$ always has value $\mathbf{F}$, whence $\langle
p\wedge\neg p\wedge q,p\wedge\neg p\rangle$ can never be negatively or
strongly coherent (and thus, negatively or strongly conclusive).

Let us now define the $\mathsf{BD}\triangle$ counterparts of presumptively
valid arguments.

######  Definition 9 (Presumptively valid arguments).

An argument $\mathcal{A}=\langle\phi,\chi\rangle$ is:

  * •

_positively presumptively valid_ (denoted
$\mathfrak{C}\models\phi\\!\rightsquigarrow^{+}\\!\chi$) iff there is
$\psi\in\mathsf{C}$ s.t.
$\psi\models_{\mathsf{BD}\triangle}\phi\wedge\triangle\chi$ and
$\psi\succeq\psi^{\prime}$ for any $\psi^{\prime}$ s.t.
$\psi^{\prime}\models\phi$;

  * •

_negatively presumptively valid_ (denoted
$\mathfrak{C}\models\phi\\!\rightsquigarrow^{-}\\!\chi$) iff there is
$\psi\in\mathsf{C}$ s.t.
$\psi\models_{\mathsf{BD}\triangle}\phi\wedge\neg\triangle\neg\chi$ and
$\psi\succeq\psi^{\prime}$ for any $\psi^{\prime}$ s.t.
$\psi^{\prime}\models\phi$;

  * •

_strongly presumptively valid_ (denoted
$\mathfrak{C}\models\phi\\!\rightsquigarrow^{\pm}\\!\chi$) iff there is
$\psi\in\mathsf{C}$ s.t.
$\psi\models_{\mathsf{BD}\triangle}\phi\wedge\mathbf{t}(\chi)$ and
$\psi\succeq\psi^{\prime}$ for any $\psi^{\prime}$ s.t.
$\psi^{\prime}\models\phi$.

######  Convention 1.

We will further call $\psi$ the _witnessing case for $\mathcal{A}$_.

######  Remark 3.

We can now explain presumptively valid arguments similarly to how we
interpreted coherent and conclusive ones.

  * •

An argument is _negatively coherent_ when there is the most preferred case
that supports its premise and _does not contradict the conclusion_.

  * •

An argument is _positively coherent_ when there is the most preferred case
that supports _both its premise and conclusion_.

######  Running example, part 3 (Witnessing a robbery, preferences).

We return to the model in part 2. The investigator now wants to find out how
the robber escaped from the office. It is clear that neither
$\langle\top,s\rangle$ nor $\langle\top,\neg s\rangle$ is strongly conclusive.
On the other hand, nobody _explicitly denied_ that the robber was limping,
whence $\langle\top,l\rangle$ is _negatively conclusive_. The investigator
thinks that it is reasonable to take $l$ to be true. Unfortunately, even
assuming $l$, neither $\langle l,s\rangle$ nor $\langle l,\neg s\rangle$ is
conclusive.

The investigator rereads the accounts of $\mathsf{c}_{1}$, $\mathsf{c}_{2}$,
and $\mathsf{c}_{3}$ and notices that $\mathsf{c}_{3}$ was the only one to
follow the robber out of the office. On the other hand, $\mathsf{c}_{1}$ hid
under the table and was sitting there during the robbery. Thus, the preference
is given as $\mathsf{c}_{1}\prec\mathsf{c}_{2}\prec\mathsf{c}_{3}$. Hence,
$\langle l,s\rangle$ is _strongly presumptively valid_.

######  Remark 4.

It is important to note that _both following statements are false_ :

  * •

$\langle\phi,\chi\rangle$ is negatively coherent (resp., presumptively valid,
conclusive) iff $\langle\phi,\neg\chi\rangle$ is positively coherent (resp.,
presumptively valid, conclusive);

  * •

$\langle\phi,\chi\rangle$ is positively coherent (resp., presumptively valid,
conclusive) iff $\langle\phi,\neg\chi\rangle$ is negatively coherent (resp.,
presumptively valid, conclusive).

Indeed, recall part 2 of the running example. $\langle\top,l\rangle$ is
negatively coherent while $\langle\top,\neg l\rangle$ is not positively
coherent. $\langle b,\neg s\rangle$ is negatively presumptively valid but
$\langle s,\neg b\rangle$ is positively presumptively valid but $\langle
s,b\rangle$ is not negatively presumptively valid.

The following statement establishes the expected relations between coherent,
presumptively valid, and conclusive arguments and follows immediately from
Definitions 7–9.

######  Proposition 2.

The diagram in Fig. 1 depicts the inclusions between different types of
arguments.

$\textstyle{\phi\mapsto^{+}\chi\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\supseteq}$$\textstyle{\supseteq}$$\textstyle{\phi\rightsquigarrow^{+}\chi\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\supseteq}$$\textstyle{\supseteq}$$\textstyle{\phi\Rightarrow^{+}\chi\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\supseteq}$$\textstyle{\phi\mapsto^{\pm}\chi\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\supseteq}$$\textstyle{\phi\rightsquigarrow^{\pm}\chi\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\supseteq}$$\textstyle{\phi\Rightarrow^{\pm}\chi}$$\textstyle{\phi\mapsto^{-}\chi\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\supseteq}$$\textstyle{\supseteq}$$\textstyle{\phi\rightsquigarrow^{-}\chi\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\supseteq}$$\textstyle{\supseteq}$$\textstyle{\phi\Rightarrow^{-}\chi\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\supseteq}$
Figure 1: Conclusive ($\Rightarrow$), presumptively valid
($\rightsquigarrow$), and coherent ($\mapsto$) arguments with same statements.

It is instructive to see that $\mathsf{BD}\triangle$ models allow classical
presumptive reasoning if the values of formulas in the cases are classical. We
define a class of $\mathsf{BD}\triangle$ case models ‘simulating’ the
classical ones.

######  Definition 10 (Quasi-classical case models).

A $\mathsf{BD}\triangle$ case model
$\mathfrak{C}=\langle\mathsf{C},\preceq\rangle$ is called _quasi-classical_
iff every $\chi\in\mathsf{C}$ is constructed from $\mathbf{t}(p)$’s via
applications of $\neg$, $\wedge$, and $\vee$.

######  Proposition 3.

Let $\mathfrak{C}$ be a quasi-classical $\mathsf{BD}\triangle$ case model and
$\triangleright\in\\{\mapsto,\rightsquigarrow,\Rightarrow\\}$. Then
$\mathfrak{C}\models\phi\triangleright^{+}\chi$ iff
$\mathfrak{C}\models\phi\triangleright^{-}\chi$ iff
$\mathfrak{C}\models\phi\triangleright^{\pm}\chi$.

###### Proof.

We only consider the case of coherent arguments as conclusive and
presumptively valid ones can be tackled similarly. It suffices to prove that
positively coherent arguments and negatively coherent arguments are strongly
coherent. Let $\mathfrak{C}$ be quasi-classical and
$\mathfrak{C}\models\phi\mapsto^{+}\chi$. Then, there is $\psi\in\mathfrak{C}$
s.t. $\psi\models_{\mathsf{BD}\triangle}\phi\wedge\triangle\chi$. But then,
from Definition 2 and Proposition 1, it is clear that
$\psi\models_{\mathsf{BD}\triangle}\mathbf{t}(\chi)$. Likewise, let
$\mathfrak{C}\models\phi\mapsto^{-}\chi$, and, accordingly,
$\psi\models_{\mathsf{BD}\triangle}\phi\wedge\neg\triangle\neg\chi$. Again,
using Definition 2 and Proposition 1, we have
$\psi\models_{\mathsf{BD}\triangle}\mathbf{t}(\chi)$. The result now follows.
∎

We finish the section by showing how to build a $\mathsf{BD}\triangle$
counterpart of a classical case model that preserves all arguments.

######  Definition 11.

Let $\mathfrak{C}=\langle\mathsf{C},\preceq\rangle$ be a classical case model
s.t. all formulas in $\mathsf{C}$ are over $\\{\neg,\wedge,\vee\\}$. In
addition, for $\phi\in\mathscr{L}_{\mathsf{BD}}$, denote $\phi^{\mathbf{t}}$
the result of substitution of every variable $p$ occurring in $\phi$ with
$\mathbf{t}(p)$.

The _$\mathsf{BD}\triangle$ counterpart_ of $\mathfrak{C}$ is
$\mathfrak{C}_{\mathsf{BD}\triangle}=\langle\mathsf{C}_{\mathsf{BD}\triangle},\preceq_{\mathsf{BD}\triangle}\rangle$
with
$\mathsf{C}_{\mathsf{BD}\triangle}=\\{\chi^{\mathbf{t}}:\chi\in\mathsf{C}\\}$
and
$\chi^{\mathbf{t}}\preceq_{\mathsf{BD}\triangle}{\chi^{\prime}}^{\mathbf{t}}$
iff $\chi\preceq_{\mathsf{BD}}\chi^{\prime}$.

######  Theorem 1.

Let $\mathfrak{C}=\langle\mathsf{C},\preceq\rangle$ be a classical case model
s.t. all formulas in $\mathsf{C}$ are over $\\{\neg,\wedge,\vee\\}$.

  1. 1.

The $\mathsf{BD}\triangle$ counterpart $\mathfrak{C}_{\mathsf{BD}\triangle}$
of $\mathfrak{C}$ is quasi-classical.

  2. 2.

$\langle\phi,\chi\rangle$ is coherent (resp., presumptively valid, conclusive)
in $\mathfrak{C}$ iff $\langle\phi^{\mathbf{t}},\chi^{\mathbf{t}}\rangle$ is
strongly coherent (resp., strongly presumptively valid, strongly conclusive)
in $\mathfrak{C}_{\mathsf{BD}\triangle}$.

###### Proof.

1\. holds by Definitions 3 and 11. Let us now consider 2. We will only tackle
the case of presumptively valid arguments since coherent and conclusive ones
can be dealt with similarly.

Let $\mathfrak{C}=\langle\mathsf{C},\preceq\rangle$ be a classical case model
and $\langle\phi,\chi\rangle$ presumptively valid on $\mathfrak{C}$. Then,
there is $\psi\in\mathsf{C}$ s.t. $\psi\models_{\mathsf{CPL}}\phi\wedge\chi$
and $\psi\succeq\psi^{\prime}$ for every $\psi^{\prime}\in\mathsf{C}$ s.t.
$\psi^{\prime}\models_{\mathsf{CPL}}\phi$. Now observe from Definition 1 that
$\neg$, $\wedge$, and $\vee$ behave classically on $\mathbf{T}$ and
$\mathbf{F}$. Thus, it is clear that $\tau\models_{\mathsf{CPL}}\tau^{\prime}$
iff
$\tau^{\mathbf{t}}\models_{\mathsf{BD}\triangle}{\tau^{\prime}}^{\mathbf{t}}$
for every $\tau,\tau^{\prime}\in\mathscr{L}_{\mathsf{BD}}$. It now follows
that
$\psi^{\mathbf{t}}\models_{\mathsf{BD}\triangle}\phi^{\mathbf{t}}\wedge\mathbf{t}(\chi^{\mathbf{t}})$
and
$\psi^{\mathbf{t}}\succeq_{\mathsf{BD}\triangle}{\psi^{\prime}}^{\mathbf{t}}$
for every
${\psi^{\prime}}^{\mathbf{t}}\models_{\mathsf{BD}\triangle}\phi^{\mathbf{t}}$,
as required. Conversely, let $\langle\phi,\chi\rangle$ be not presumptively
valid on $\mathfrak{C}$. Then, there is no $\psi\in\mathsf{C}$ s.t.
$\psi\models_{\mathsf{CPL}}\phi\wedge\chi$ and $\psi\succeq\psi^{\prime}$ for
every $\psi^{\prime}\in\mathsf{C}$ s.t.
$\psi^{\prime}\models_{\mathsf{CPL}}\phi$. Again, from Definition 1, it
follows that there is no $\psi^{\mathbf{t}}$ s.t.
$\psi^{\mathbf{t}}\models_{\mathsf{BD}\triangle}\phi^{\mathbf{t}}\wedge\mathbf{t}(\chi^{\mathbf{t}})$
and
$\psi^{\mathbf{t}}\succeq_{\mathsf{BD}\triangle}{\psi^{\prime}}^{\mathbf{t}}$
for every
${\psi^{\prime}}^{\mathbf{t}}\models_{\mathsf{BD}\triangle}\phi^{\mathbf{t}}$.
∎

##  4 A two-layered logic for case models

Conclusive and presumptively valid arguments on _classical_ case models can be
represented in terms of conditional probabilities [26, 27]. In this section,
we provide a representation of $\mathsf{BD}\triangle$ models and arguments on
them in terms of a paraconsistent two-layered logic.

Two-layered logics form a class of formalisms designed to reason about
uncertainty: their languages consist of _inner-layer_ formulas that describe
events and _outer-layer_ formulas composed of _modal atoms_ of the form
$\mathsf{M}\phi$ ($\phi$ being an inner-layer formula and $\mathsf{M}$ the
modality interpreted as a measure on the set of events). There are two-layered
logics formalising reasoning with classical probabilities [4] and their
paraconsistent counterparts [8] presented in [17].111Note that [17] is not the
only paraconsistent interpretation of probabilities: alternative approaches
can be found. e.g., in [19, 14, 11, 24]. Furthermore, there are two-layered
logics formalising paraconsistent reasoning with belief and plausibility
functions [8].

These papers usually study _quantitative_ representations of uncertainty. On
the other hand, case models provide a _qualitative one_ via their preference
relations. This shows a degree of affinity between case models and
representations of different uncertainty measures by means of total preorders
as studied in [18] (for the case of probabilities) and [29, 28] (belief
functions). In [7], two-layered logics formalising reasoning with the
qualitative counterparts of belief functions and probabilities were presented.

In this section, we present a two-layered logic
$\mathsf{QG}_{\mathsf{BD}\triangle}$ which is a modification of $\mathsf{QG}$
from [7]. The inner layer of $\mathsf{MCB}_{\triangle}$ is
$\mathsf{BD}\triangle$, the outer one is $\mathsf{biG}$ — an expansion of
Gödel logic (cf., e.g., [16]) with a coimplication $\Yleft$ or the Baaz Delta
operator $\triangle$. To connect the layers, we use $\mathtt{B}$ (with
$\mathtt{B}\phi$ read as ‘the agent believes in $\phi$’). Since we do not
impose any restrictions on $\preceq$ in case models, we are interpreting
$\mathtt{B}$ as a _capacity_ on the set of events $W$, i.e., via a map
$\mu:2^{W}\rightarrow[0,1]$ which is monotone w.r.t. $\subseteq$ with
$\mu(W)=1$ and $\mu(\varnothing)=0$. The main goal of the paper is to
establish a correspondence between case models and
$\mathsf{QG}_{\mathsf{BD}\triangle}$ models as well as to show how given an
argument $\langle\phi,\phi^{\prime}\rangle$ to construct a
$\mathsf{QG}_{\mathsf{BD}\triangle}$ formula that is true in the corresponding
model iff the argument is (positively, negatively, or strongly) coherent,
conclusive, or presumptively valid.

Let us now recall $\mathsf{biG}$.

######  Definition 12.

The bi-Gödel algebra
$[0,1]_{\mathsf{G}}=\langle[0,1],0,1,\wedge_{\mathsf{G}},\vee_{\mathsf{G}},\rightarrow_{\mathsf{G}},\Yleft,\sim_{\mathsf{G}},\triangle_{\mathsf{G}}\rangle$
is defined as follows: for all $a,b\in[0,1]$, $\wedge_{\mathsf{G}}$ and
$\vee_{\mathsf{G}}$ are given by $a\wedge_{\mathsf{G}}b\coloneqq\min(a,b)$,
$a\vee_{\mathsf{G}}b\coloneqq\max(a,b)$. The remaining operations are defined
below:

$\displaystyle a\rightarrow_{\mathsf{G}}b$
$\displaystyle=\begin{cases}1,\text{ if }a\leq b\\\ b\text{ else}\end{cases}$
$\displaystyle a\Yleft_{\mathsf{G}}b$ $\displaystyle=\begin{cases}0,\text{ if
}a\leq b\\\ a\text{ else}\end{cases}$ $\displaystyle{\sim_{\mathsf{G}}}a$
$\displaystyle=\begin{cases}0,\text{ if }a>0\\\ 1\text{ else}\end{cases}$
$\displaystyle\triangle_{\mathsf{G}}a$ $\displaystyle=\begin{cases}0,\text{ if
}a<1\\\ 1\text{ else}\end{cases}$

######  Remark 5.

Note that constants $\top$ and $\bot$ are definable as, respectively,
$p\rightarrow p$ and $p\Yleft p$, and that $\triangle$ and $\Yleft$ are
interdefinable as follows: $\triangle\phi\coloneqq\top\Yleft(\top\Yleft\phi)$,
$\phi\Yleft\phi^{\prime}\coloneqq\phi\wedge{\sim}\triangle(\phi\rightarrow\phi^{\prime})$.

######  Definition 13 (Language and semantics of $\mathsf{biG}$).

We fix a countable set $\mathtt{Prop}$ of propositional variables and consider
the following language.

$\displaystyle\mathscr{L}_{\mathsf{biG}}\ni\phi$ $\displaystyle\coloneqq
p\in\mathtt{Prop}\mid{\sim}\phi\mid(\phi\wedge\phi)\mid(\phi\vee\phi)\mid(\phi\rightarrow\phi)\mid(\phi\Yleft\phi)\mid\triangle\phi$

Let $e:\mathsf{Prop}\rightarrow[0,1]$. For the complex formulas, we define
$e(\phi\circ\phi^{\prime})=e(\phi)\circ_{\mathsf{G}}e(\phi^{\prime})$.

Finally, let $\Gamma\cup\\{\phi\\}\subseteq\mathscr{L}_{\mathsf{biG}}$. We
define: $\Gamma\models_{\mathsf{biG}}\phi$ iff $\forall
e:\inf\\{e(\psi):\psi\in\Gamma\\}\leq e(\phi)$.

Using $\mathsf{biG}$, we can define $\mathsf{QG}_{\mathsf{BD}\triangle}$ as
follows.

######  Definition 14.

The language of $\mathsf{QG}_{\mathsf{BD}\triangle}$ is defined via the
following grammar:
$\mathscr{L}_{\mathsf{QG}_{\mathsf{BD}\triangle}}\\!\ni\\!\alpha\coloneqq\mathtt{B}\phi\mid\alpha\circ\alpha$
$(\circ\in\\{{\sim},\wedge,\vee,\rightarrow,\Yleft,\triangle\\},\phi\in\mathscr{L}_{\mathsf{BD}\triangle})$.
A $\mathsf{QG}_{\mathsf{BD}\triangle}$ model is a tuple $\mathscr{M}=\langle
W,v^{+},v^{-},\mu,e\rangle$ with $\langle W,v^{+},v^{-}\rangle$ being a
$\mathsf{BD}\triangle$ model (cf. Definition 2), $\mu:2^{W}\rightarrow[0,1]$
being a capacity. Semantic conditions of
$\mathscr{L}_{\mathsf{QG}_{\mathsf{BD}\triangle}}$ formulas are as follows:
$e(\mathtt{B}\phi)=\mu(|\phi|^{+})$ for modal atoms; the values of complex
formulas are computed according to Definition 13.

For a given model $\mathscr{M}$, we write $\mathscr{M}\models\alpha$ to
designate $e(\alpha)=1$. For a frame $\mathbb{F}=\langle W,\pi\rangle$ on a
$\mathsf{QG}_{\mathsf{BD}\triangle}$ model $\mathscr{M}$, we say that
$\alpha\in\mathscr{L}_{\mathsf{MCB}_{\triangle}}$ is valid on $\mathbb{F}$
($\mathbb{F}\models\alpha$) iff $e(\alpha)\\!=\\!1$ for every $e$ on
$\mathbb{F}$. Finally, for
$\Psi\cup\\{\alpha\\}\subseteq\mathscr{L}_{\mathsf{QG}_{\mathsf{BD}\triangle}}$,
we define the same entailment relation as in Definition 13.

Let us now establish the correspondence results for coherent, conclusive, and
presumptively valid arguments. To do this, we define a class of _$\mu$
-counterparts_ for every $\mathsf{BD}\triangle$ model.

######  Definition 15 ($\mu$-counterparts).

Let $\mathfrak{C}=\langle\mathsf{C},\preceq\rangle$ be a
$\mathsf{BD}\triangle$ case model and
$\mathsf{C}=\\{\mathsf{c}_{1},\ldots,\mathsf{c}_{n}\\}$. Its $\mu$-counterpart
is a $\mathsf{QG}_{\mathsf{BD}\triangle}$-model
$\mathscr{M}_{\mathfrak{C}}=\langle\\{w_{1},\ldots,w_{n}\\},v^{+},v^{-},\mu_{\preceq},e\rangle$
for which the following holds.

  1. 1.

For every $\mathsf{c}_{i}\in\mathsf{C}$ and every $\phi$, if
$\mathsf{c}_{i}\models_{\mathsf{BD}\triangle}\phi$
($\mathsf{c}_{i}\models_{\mathsf{BD}\triangle}\neg\phi$), then
$w_{i}\vDash^{+}\phi$ ($w_{i}\vDash^{-}\phi$).

  2. 2.

For every $\mathsf{c}_{i},\mathsf{c}_{j}\in\mathsf{C}$,
$\mathsf{c}_{i}\preceq\mathsf{c}_{j}$ iff
$\mu_{\preceq}(\\{w_{i}\\})\leq\mu_{\preceq}(\\{w_{j}\\})$.

  3. 3.

For every $\mathsf{c}_{i}\in\mathsf{C}$,
$\mu_{\preceq}(\\{\mathsf{c}_{i}\\})>0$.

######  Theorem 2.

Let $\mathfrak{C}=\langle\mathsf{C},\preceq\rangle$ be a
$\mathsf{BD}\triangle$ case model and $\mathscr{M}_{\mathfrak{C}}$ its
$\mu$-counterpart. Then the following holds.

  1. 1.

$\mathfrak{C}\models\phi\mapsto^{+}\phi^{\prime}$ iff
$\mathscr{M}_{\mathfrak{C}}\models{\sim\sim}\mathtt{B}(\phi\wedge\triangle\phi^{\prime})$.

  2. 2.

$\mathfrak{C}\models\phi\mapsto^{-}\phi^{\prime}$ iff
$\mathscr{M}_{\mathfrak{C}}\models{\sim\sim}\mathtt{B}(\phi\wedge\neg\triangle\neg\phi^{\prime})$.

  3. 3.

$\mathfrak{C}\models\phi\mapsto^{\pm}\phi^{\prime}$ iff
$\mathscr{M}_{\mathfrak{C}}\models{\sim\sim}\mathtt{B}(\phi\wedge\mathbf{t}(\phi^{\prime}))$.

###### Proof.

We consider 2. Other cases can be proved in the same way. Let
$\mathfrak{C}\models\phi\mapsto^{-}\phi^{\prime}$. Then, there is
$\mathsf{c}_{i}\in\mathfrak{C}$ s.t.
$\mathsf{c}_{i}\models_{\mathsf{BD}\triangle}\phi\wedge\neg\triangle\neg\phi^{\prime}$,
whence $w_{i}\vDash^{+}\phi\wedge\neg\triangle\neg\phi^{\prime}$ and
$\mu(|\phi\wedge\neg\triangle\neg\phi^{\prime}|^{+})>0$. Thus,
$e(\mathtt{B}(\phi\wedge\neg\triangle\neg\phi^{\prime}))>0$ and
$\mathscr{M}_{\mathfrak{C}}\models{\sim\sim}\mathtt{B}(\phi\wedge\neg\triangle\neg\phi^{\prime})$,
as required. Conversely, let
$\mathfrak{C}\not\models\phi\mapsto^{-}\phi^{\prime}$. Then, for every
$\mathsf{c}_{i}\in\mathfrak{C}$,
$\mathsf{c}_{i}\not\models_{\mathsf{BD}\triangle}\phi\wedge\neg\triangle\neg\phi^{\prime}$,
whence there is no $w_{i}$ s.t.
$w_{i}\vDash^{+}\phi\wedge\neg\triangle\neg\phi^{\prime}$. Hence,
$|\phi\wedge\neg\triangle\neg\phi^{\prime}|^{+}=\varnothing$ and
$\mu(|\phi\wedge\neg\triangle\neg\phi^{\prime}|^{+})=0$. Thus,
${\sim\sim}e(\mathtt{B}(|\phi\wedge\neg\triangle\neg\phi^{\prime}|^{+}))=0$,
as required. ∎

Observe from Definitions 8 and 9 that the classes of strongly conclusive
(presumptively valid) arguments on the one hand and both positively and
negatively conclusive (presumptively valid) arguments on the other hand
coincide. Thus, it suffices to provide representation for positively and
negatively conclusive (presumptively valid) arguments only.

######  Theorem 3.

Let $\mathfrak{C}=\langle\mathsf{C},\preceq\rangle$ be a
$\mathsf{BD}\triangle$ case model and $\mathscr{M}_{\mathfrak{C}}$ its
$\mu$-counterpart. Then the following holds.

  1. 1.

$\mathfrak{C}\models\phi\Rightarrow^{+}\phi^{\prime}$ iff
$\mathscr{M}_{\mathfrak{C}}\models{\sim}\mathtt{B}(\phi\wedge\neg\triangle\phi^{\prime})\wedge{\sim\sim}\mathtt{B}(\phi\wedge\triangle\phi^{\prime})$.

  2. 2.

$\mathfrak{C}\models\phi\Rightarrow^{-}\phi^{\prime}$ iff
$\mathscr{M}_{\mathfrak{C}}\models{\sim}\mathtt{B}(\phi\wedge\triangle\neg\phi^{\prime})\wedge{\sim\sim}\mathtt{B}(\phi\wedge\neg\triangle\neg\phi^{\prime})$.

###### Proof.

Again, for the sake of brevity, we consider only 1. We let
$\mathfrak{C}\models\phi\Rightarrow^{+}\phi^{\prime}$. Then,
$\langle\phi,\phi^{\prime}\rangle$ is positively coherent on $\mathfrak{C}$
and thus (by Theorem 2),
$\mathscr{M}_{\mathfrak{C}}\models{\sim\sim}\mathtt{B}(\phi\wedge\triangle\phi^{\prime})$.
Furthermore, since
$\psi\models_{\mathsf{BD}\triangle}\phi\wedge\triangle\phi^{\prime}$ for every
$\psi\in\mathfrak{C}$ s.t. $\psi\models_{\mathsf{BD}\triangle}\phi$, we have
that
$|\phi|^{+}\cap(\mathsf{C}\setminus|\triangle\phi^{\prime}|^{+})=\varnothing$,
whence
$\mathscr{M}_{\mathfrak{C}}\models{\sim}\mathtt{B}(\phi\wedge\neg\triangle\phi^{\prime})$,
as required. As the converse direction can be proved in the same manner, the
result follows. ∎

######  Theorem 4.

Let $\mathfrak{C}=\langle\mathsf{C},\preceq\rangle$ be a
$\mathsf{BD}\triangle$ case model and $\mathscr{M}_{\mathfrak{C}}$ its
$\mu$-counterpart. Then the following holds.

  1. 1.

$\mathfrak{C}\models\phi\rightsquigarrow^{+}\phi^{\prime}$ and $\chi$ is
$\langle\phi,\phi^{\prime}\rangle$’s witnessing case iff

$\mathscr{M}_{\mathfrak{C}}\models{\sim\sim}\mathtt{B}(\phi\wedge\triangle\phi^{\prime})\wedge\triangle\mathtt{B}(\chi\Rrightarrow(\phi\wedge\triangle\phi^{\prime}))\wedge\bigwedge_{\chi^{\prime}\in\mathsf{C}}\left(\triangle\mathtt{B}(\chi\Rrightarrow\phi)\rightarrow\triangle(\mathtt{B}\chi^{\prime}\rightarrow\mathtt{B}\chi)\right)$

  2. 2.

$\mathfrak{C}\models\phi\rightsquigarrow^{-}\phi^{\prime}$ and $\chi$ is
$\langle\phi,\phi^{\prime}\rangle$’s witnessing case iff

$\mathscr{M}_{\mathfrak{C}}\models{\sim\sim}\mathtt{B}(\phi\wedge\neg\triangle\neg\phi^{\prime})\wedge\triangle\mathtt{B}(\chi^{\prime}\Rrightarrow(\phi\wedge\neg\triangle\neg\phi^{\prime}))\wedge\bigwedge_{\chi^{\prime}\in\mathsf{C}}\left(\triangle\mathtt{B}(\chi^{\prime}\Rrightarrow\phi)\rightarrow\triangle(\mathtt{B}\chi^{\prime}\rightarrow\mathtt{B}\chi)\right)$

###### Proof.

We prove 1. as 2. can be proven in the same manner. Assume that
$\langle\phi,\phi^{\prime}\rangle$ is positively presumptively valid over
$\mathfrak{C}$ and that $\chi$ is its witnessing case. Then,
$\langle\phi,\phi^{\prime}\rangle$ is positively coherent (whence,
$\mathscr{M}\models{\sim\sim}\mathtt{B}(\phi\wedge\neg\triangle\neg\phi^{\prime})$)
and $\chi\models_{\mathsf{BD}\triangle}\phi\wedge\triangle\phi^{\prime}$.
Thus, $|\chi|^{+}\subseteq|\phi\wedge\triangle\phi^{\prime}|^{+}$ and
$|\chi|^{-}\supseteq|\phi\wedge\triangle\phi^{\prime}|^{-}$ _for every model
$\mathscr{M}$_, whence
$\mathscr{M}\models\triangle\mathtt{B}(\chi\Rrightarrow(\phi\wedge\neg\triangle\neg\phi^{\prime}))$.
Finally, we also have that $\chi^{\prime}\preceq\chi$ for every
$\chi^{\prime}\in\mathsf{C}$ s.t. $\chi\models_{\mathsf{BD}\triangle}\phi$.
But this means that for every such $\chi^{\prime}$,
$\mu(|\chi^{\prime}|^{+})\leq\mu(|\chi|^{+})$222Recall that since
$\mathsf{c}_{i}\wedge\mathsf{c}_{j}\models_{\mathsf{BD}\triangle}\bot$ for
every $\mathsf{c}_{i},\mathsf{c}_{j}\in\mathsf{C}$, we have that
$\mu(|\mathsf{c}_{k}|^{+})=\mu(\\{w_{k}\\})$ for all
$\mathsf{c}_{k}\in\mathsf{C}$. and thus,
$\mathscr{M}\models\triangle(\mathtt{B}\chi^{\prime}\rightarrow\mathtt{B}\chi)$.
Hence,
$\mathscr{M}\models\bigwedge_{\chi^{\prime}\in\mathsf{C}}\left(\triangle\mathtt{B}(\chi^{\prime}\Rrightarrow\phi)\rightarrow\triangle(\mathtt{B}\chi^{\prime}\rightarrow\mathtt{B}\chi)\right)$,
as required.

For the converse, let $\langle\phi,\phi^{\prime}\rangle$ be _not_ positively
presumptively valid argument with $\chi$ as the witnessing case. Then at least
one of the following holds: (1) $\langle\phi,\phi^{\prime}\rangle$ is not
positively coherent; (2)
$\chi\not\models_{\mathsf{BD}\triangle}\phi\wedge\triangle\phi^{\prime}$; (3)
there is some $\chi^{\prime}\in\mathsf{C}$ s.t.
$\chi^{\prime}\models_{\mathsf{BD}\triangle}\phi$ but
$\chi^{\prime}\succ\chi$. Now, for (1),
$\mathscr{M}\not\models{\sim\sim}\mathtt{B}(\phi\wedge\neg\triangle\neg\phi^{\prime})$;
for (2),
$\mathscr{M}\not\models\triangle\mathtt{B}(\chi\Rrightarrow(\phi\wedge\triangle\phi^{\prime}))$;
and finally, for (3),
$\mathscr{M}\not\models\bigwedge_{\chi^{\prime}\in\mathsf{C}}\left(\triangle\mathtt{B}(\chi^{\prime}\Rrightarrow\phi)\rightarrow\triangle(\mathtt{B}\chi^{\prime}\rightarrow\mathtt{B}\chi)\right)$.
∎

Recall that conclusive and coherent arguments over _classical_ case models
were represented by means of _conditional probabilities_ in [26, 27]. Here, we
did not need conditionalisations on capacities as we used a purely logical
representation and could employ $\rightarrow$ in order to ‘simulate’
conditionalised measures.

##  5 Conclusion

In this paper, we provided paraconsistent counterparts to the case models
discussed in [26, 27] that use $\mathsf{BD}\triangle$ as their underlying
logic. We showed how to recover classical presumptive reasoning from
$\mathsf{BD}\triangle$ case models (Theorem 1). Moreover, we constructed a
two-layered logic $\mathsf{QG}_{\mathsf{BD}\triangle}$ over
$\mathsf{BD}\triangle$ and $\mathsf{biG}$ and used it to establish a
representation of arguments with $\mathsf{QG}_{\mathsf{BD}\triangle}$ formulas
(Theorems 2–4).

The natural next steps would be as follows. First, it is instructive to
provide a complete axiomatisation of $\mathsf{QG}_{\mathsf{BD}\triangle}$.
Second, while in this paper we used a linear preference relation (as it is
traditionally done, cf., e.g., [23]), one could argue that if an agent cannot
choose between two cases $\mathsf{c}$ and $\mathsf{c}^{\prime}$, it does not
mean that they prefer them to the same degree. It is, hence, reasonable to
explore case models whose preference relation is a _partial_ preorder.
Finally, we managed to represent preference relations on case models as
capacities on their $\mathsf{BD}\triangle$ counterparts. An expected question
to ask is which properties we have to require from the case model so that its
preference relation be represented as a stronger measure: e.g., a belief
function, a plausibility function, or a probability measure.

## References

  * [1]
  * [2] A.R. Anderson & N.D. Belnap (1962): _Tautological entailments_.  Philosophical Studies 13(1), pp. 9–24, 10.1007/BF00818100.
  * [3] M. Baaz (1996): _Infinite-valued Gödel logics with $0$-$1$-projections and relativizations_.  In: Gödel’96: Logical foundations of mathematics, computer science and physics—Kurt Gödel’s legacy, Brno, Czech Republic, August 1996, proceedings, Association for Symbolic Logic, pp. 23–33.
  * [4] P. Baldi, P. Cintula & C. Noguera (2020): _Classical and Fuzzy Two-Layered Modal Logics for Uncertainty: Translations and Proof-Theory_.  International Journal of Computational Intelligence Systems 13, pp. 988–1001, 10.2991/ijcis.d.200703.001.
  * [5] N.D. Belnap (2019): _How a Computer Should Think_.  In H. Omori & H. Wansing, editors: New Essays on Belnap-Dunn Logic, Synthese Library (Studies in Epistemology, Logic, Methodology, and Philosophy of Science) 418, Springer, Cham, pp. 35–53, 10.1007/978-3-030-31136-0_4.
  * [6] P. Besnard & A. Hunter (2001): _A logic-based theory of deductive arguments_.  Artificial Intelligence 128(1–2), pp. 203–235, 10.1016/S0004-3702(01)00071-6.
  * [7] M. Bílková, S. Frittella, D. Kozhemiachenko & O. Majer (2023): _Qualitative reasoning in a two-layered framework_.  International Journal Approximate Reasoning 154, pp. 84–108, 10.1016/j.ijar.2022.12.011.
  * [8] M. Bílková, S. Frittella, D. Kozhemiachenko, O. Majer & S. Nazari (2022): _Reasoning with belief functions over Belnap–Dunn logic_.  https://arxiv.org/abs/2203.01060.
  * [9] M. Bílková, S. Frittella, O. Majer & S. Nazari (2020): _Belief Based on Inconsistent Information_.  In M.A. Martins & I. Sedlár, editors: Dynamic Logic. New Trends and Applications, Springer International Publishing, Cham, pp. 68–86, 10.1007/978-3-030-65840-3_5.
  * [10] A. Bondarenko, P.M. Dung, R.A. Kowalski & F. Toni (1997): _An abstract, argumentation-theoretic approach to default reasoning_.  Artificial Intelligence 93(1-2), pp. 63–101, 10.1016/S0004-3702(97)00015-5.
  * [11] J. Bueno-Soler & W. Carnielli (2016): _Paraconsistent probabilities: Consistency, contradictions and Bayes’ theorem_.  Entropy (Basel) 18(9), p. 325, 10.3390/e18090325.
  * [12] P.M. Dung (1995): _On the acceptability of arguments and its fundamental role in nonmonotonic reasoning, logic programming and $n$-person games_.  Artificial Intelligence 77(2), pp. 321–357, 10.1016/0004-3702(94)00041-X.
  * [13] J.M. Dunn (1976): _Intuitive semantics for first-degree entailments and ‘coupled trees’_.  Philosophical Studies 29(3), pp. 149–168, 10.1007/BF00373152.
  * [14] J.M. Dunn (2010): _Contradictory information: Too much of a good thing_.  Journal of Philosophical Logic 39, pp. 425–452, 10.1007/s10992-010-9134-6.
  * [15] A.J. García & G.R. Simari (2004): _Defeasible logic programming: an argumentative approach_.  Theory and Practice of Logic Programming 4(1+2), pp. 95–138, 10.1017/S1471068403001674.
  * [16] P. Hájek (1998): _Metamathematics of Fuzzy Logic_.  Trends in Logic 4, Springer, Dordrecht, 10.1007/978-94-011-5300-3_5.
  * [17] D. Klein, O. Majer & S. Rafiee Rad (2021): _Probabilities with gaps and gluts_.  Journal of Philosophical Logic 50(5), pp. 1107–1141, 10.1007/s10992-021-09592-x.
  * [18] C.H. Kraft, J.W. Pratt & A. Seidenberg (1959): _Intuitive probability on finite sets_.  The Annals of Mathematical Statistics 30(2), pp. 408–419, 10.1214/aoms/1177706260.
  * [19] E.D. Mares (1997): _Paraconsistent Probability Theory and Paraconsistent Bayesianism_.  Logique et Analyse 40(160), pp. 375–384.
  * [20] H. Omori & K. Sano (2015): _Generalizing Functional Completeness in Belnap-Dunn Logic_.  Studia Logica 103(5), pp. 883–917, 10.1007/s11225-014-9597-5.
  * [21] H. Omori & H. Wansing (2017): _40 years of FDE: An Introductory Overview_.  Studia Logica 105(6), pp. 1021–1049, 10.1007/s11225-017-9748-6.
  * [22] H. Prakken (2010): _An abstract framework for argumentation with structured arguments_.  Argument & Computation 1(2), pp. 93–124, 10.1080/19462160903564592.
  * [23] F.S. Roberts (1985): _Measurement Theory_.  Cambridge University Press.
  * [24] A. Rodrigues, J. Bueno-Soler & W. Carnielli (2021): _Measuring evidence: a probabilistic approach to an extension of Belnap–Dunn logic_.  Synthese 198(S22), pp. 5451–5480, 10.1007/s11229-020-02571-w.
  * [25] K. Sano & H. Omori (2014): _An expansion of first-order Belnap–Dunn logic_.  Logic Journal of IGPL 22(3), pp. 458–481, 10.1093/jigpal/jzt044.
  * [26] B. Verheij (2016): _Correct grounded reasoning with presumptive arguments_.  In: Logics in Artificial Intelligence, Lecture notes in computer science, Springer International Publishing, Cham, pp. 481–496, 10.1007/978-3-031-10769-6_26.
  * [27] B. Verheij (2017): _Proof with and without probabilities_.  Artificial Intelligence and Law 25(1), pp. 127–154, 10.1007/s10506-017-9199-4.
  * [28] S.K.M. Wong, Y. Yao & P. Bollmann (1992): _Characterization of comparative belief structures_.  International journal of man-machine studies 37(1), pp. 123–133, 10.1016/0020-7373(92)90094-2.
  * [29] S.K.M. Wong, Y.Y. Yao, P. Bollmann & H.C. Bürger (1991): _Axiomatization of qualitative belief structure_.  IEEE Transactions on Systems, Man, and Cybernetics 21(4), pp. 726–734, 10.1109/21.108290.


[1,2]Rachit Dhar

1]TCS Research, Tata Consultancy Services, Pune, Maharashtra, India

2]Department of Materials Science and Engineering, Indian Institute of
Technology Delhi, 110016, New Delhi, India

# Physics and data driven model for prediction of residual stresses in
machining

<EMAIL_ADDRESS>Ankur Krishna<EMAIL_ADDRESS>Bilal Muhammed
<EMAIL_ADDRESS>[  [

###### Abstract

Predicting residual stresses has always been a topic of significance due to
its implications in the development of enhanced materials and better
processing conditions. In this work, an analytical model for prediction of
residual stresses is developed for orthogonal machining. It consists of three
component models for force, temperature and stress computation. The Oxley
force model and Waldorf’s slip-line model are employed for obtaining cutting
force, thrust force, and temperatures at the shear zone and tool-chip
interface for the given parameters. The Komanduri-Hou two heat source model is
used for obtaining the temperature distribution in the workpiece. The effect
of coolant with differing mass flow rates has also been incorporated. The
residual stresses are obtained by combining the mechanical and thermal
components, followed by the loading and relaxation of the stresses. Optimal
values for unknown parameters are predicted by leveraging a cost function. The
residual stress distributions obtained give a tensile region near the surface
for Inconel 718, and a compressive region for Ti6Al4V, which are in line with
experimental results found in literature.

###### keywords:

Residual stress, Analytical modeling, Stress distribution

##  1 Introduction

Understanding and predicting residual stresses has been a significant focus of
research because controlling their formation can lead to more durable and
safer materials. Therefore, extensive research has been conducted on modeling
residual stress distributions in materials and studying the effects of
different materials and process parameters. Residual stresses in orthogonal
machining arise from two main factors: (a) local plastic deformation during
cutting, which persists after the cutting forces are removed, and (b) thermal
stresses due to heat generation at the tool-chip interface and in the shear
zone.

Initial work on residual stress modeling relied on empirical models, but their
applicability is limited to specific conditions and cannot be extended to
other situations [1]. Analytical models have been developed to address this
limitation by providing a general framework capable of handling various
parameters based on the underlying mechanisms of residual stress generation
[1]. Merwin and Johnson [2] proposed an analytical model based on classical
Hertz theory for calculating stress distribution, which was later refined by
Johnson [3]. McDowell [4] used cycloidal distributions to describe the
functions $p(s)$ and $q(s)$ in the Johnson stress model. Su and Liang [5]
utilized Oxley’s [6] cutting force model and Waldorf’s [7] plowing model to
predict residual stresses analytically. Wan et al. [8] modified the plowing
force model by incorporating the Johnson-Cook model [9] for flow shear stress
calculation, which became widely adopted. Initial experimental work on
temperature distribution was conducted by Matsumoto et al. [10], and a
temperature model was developed by Jacobus et al. [11]. Sekhon and Chenot [12]
derived an expression for $\gamma$, the friction heat partition coefficient.
Ulutan et al. [13] employed a finite-difference model for temperature
distribution. Komanduri and Hou [14, 15, 16] proposed a two heat source model.
Pan et al. [17] suggested a modified Johnson-Cook model for analytical
residual stress modeling. Drucker and Palgen [18] provided an expression for
the plastic modulus. Sehitoglu and Jiang [19] introduced a non-linear
kinematic hardening model for elasto-plastic loading, later modified by
McDowell [4]. Merwin and Johnson [2] provided boundary conditions to eliminate
certain components of the residual stress tensor based on the assumption of
plane deformation. Finite element methods (FEMs) represent a newer set of
techniques for simulating tool-workpiece interactions without requiring deep
mechanistic analysis. However, FEMs often suffer from low efficiency and high
computational demands [1]. Therefore, analytical models remain the most
suitable choice for meeting industrial needs.

This paper presents an analytical model for predicting residual stresses in
orthogonal machining, comprising three component analytical models for force,
temperature, and stress. To enhance accuracy, optimal values of unknown
parameters ($C_{o}$, $\delta$ in the force model, and the plastic modulus
function $h$ in stress loading and relaxation) are determined within a
specific range based on a cost function to minimize errors in force and stress
prediction. The Oxley cutting force model is used to predict cutting and
thrust forces, along with Waldorf’s model for plowing forces, as reported by
Su [20], Zhou and Ren [21]. Temperatures in the shear zone and tool-chip
interface are determined using Boothroyd’s [22] cutting temperature model. The
Johnson-Cook model is employed to model flow stresses. The temperature
distribution across the workpiece is obtained using the Komanduri-Hou model,
incorporating the additional effect of a coolant based on the work of Singh
and Sharma [23]. This temperature profile is then used to model mechanical [3]
and thermal stresses [24, 25], which are combined and incrementally relaxed
following the methods outlined by Shan et al. [26], Liang et al. [24, 25],
McDowell [4], and Merwin and Johnson [2], ultimately yielding the residual
stresses. The results of forces, temperature, and residual stress are plotted
and discussed for Inconel 718 and Ti6Al4V machining data from the literature.

The paper proceeds by discussing various models used, starting with the force
model in Section 2 for calculating cutting and thrust forces, followed by the
temperature distribution model in Section 3, and concluding with the residual
stress model in Section 4. The parameters used for validating the model are
presented in Section 5, while Section 6 discusses the results and their
implications when applying the model to machining data for two materials -
Inconel 718 and Ti6Al4V. Finally, the conclusions drawn from the study are
discussed in Section 7.

Figure 1: Overall workflow of the paper

##  2 Force modeling for orthogonal cutting

To determine the mechanical stresses and temperature changes within the
workpiece during orthogonal machining, it is necessary to ascertain the
cutting and thrust forces. These forces are computed using Oxley’s model,
which in turn relies on flow stresses derived from the Johnson-Cook model. The
following section provides a detailed description of these models.

###  2.1 Oxley cutting force model

Oxley [6] introduced a model to compute the forces encountered in orthogonal
machining. Su [20] presented an algorithm for utilizing Oxley’s model to
calculate forces. In accordance with their research, the resultant force $R$,
as well as the shear cutting and thrust forces ($F_{c}^{shear}$ and
$F_{t}^{shear}$), are expressed by the following equations:

$R=\dfrac{k_{AB}tw}{\sin\phi\cos\theta}$ (1)

$F_{c}^{shear}=R\cos(\lambda-\alpha)$ (2)

$F_{t}^{shear}=R\sin(\lambda-\alpha)$ (3)

where $k_{AB}$ is the flow stress in the shear zone, $t$ is the uncut chip
thickness (i.e., depth of cut), and $w$ is the width of cut. $\phi$, $\alpha,$
and $\lambda$ represent the shear angle, tool rake angle, and friction angle
respectively. The shear speed $v_{s}$ and chip speed $v_{c}$ are obtained
using the cutting speed $v$ as:

$v_{s}=\dfrac{v\cos\alpha}{\cos(\phi-\alpha)}$ (4)

$v_{c}=\dfrac{v\sin\phi}{\cos(\phi-\alpha)}$ (5)

The strain ($\varepsilon_{AB}$) and strain rate ($\dot{\varepsilon}_{AB}$) in
the shear zone are:

$\varepsilon_{AB}=\dfrac{\lambda^{{}^{\prime}}\cos\alpha}{\sqrt{3}\sin\phi\cos(\phi-\alpha)}$
(6)

$\dot{\varepsilon}_{AB}=\dfrac{C_{o}v_{s}}{\sqrt{3}(\dfrac{t}{\sin\phi})}$ (7)

where $\lambda^{{}^{\prime}}$ is given by
$\cos\phi\cos(\phi-\alpha)/\cos\alpha$. The angles $\theta$ and $\lambda$ are
obtained from:

$\theta=\arctan(1+2(\dfrac{\pi}{4}-\phi)-C_{o}n_{eq})$ (8)

$\lambda=\theta+\alpha-\phi$ (9)

Here, $n_{eq}$ is given by the following expression:

$n_{eq}=\dfrac{nB\varepsilon_{AB}^{n}}{A+B\varepsilon_{AB}^{n}}$ (10)

Also, the chip thickness $t_{c}$ is given by $t\cos(\phi-\alpha)/\sin\phi$.
The plowing forces $P_{c}$ and $P_{t}$ are modeled according to Waldorf’s
slip-line model [7], the equations for which are given as follows:

$\begin{split}P_{c}&=k_{AB}w\bigl{[}(1+2\theta+2\lambda+\sin(2\eta))\sin(\phi-\gamma+\eta)\\\
&+\cos(2\eta)\cos(\phi-\gamma+\eta)\bigr{]}CB\end{split}$ (11)

$\begin{split}P_{t}&=k_{AB}w\bigl{[}(1+2\theta+2\lambda+\sin(2\eta))\cos(\phi-\gamma+\eta)\\\
&-\cos(2\eta)\sin(\phi-\gamma+\eta)\bigr{]}CB\end{split}$ (12)

where $CB=R_{plow}/\sin(\eta_{plow})$. The angles $\eta_{plow},\gamma_{plow}$,
and $\theta_{plow}$ are calculated based on the work developed by Zhou and Ren
[21]. In this paper, the equation for calculating the sector radius $R_{plow}$
has been solved analytically, by taking the maximum of the two quadratic
roots, instead of using the iterative approach proposed by Zhou and Ren [21].
The total cutting force $F_{c}$ and total thrust force $F_{t}$ are obtained
by:

$\begin{split}F_{c}&=F_{c}^{shear}+P_{c}\\\
F_{t}&=F_{t}^{shear}+P_{t}\end{split}$ (13)

###  2.2 Boothroyd’s cutting temperature model

Boothroyd [22] described a model for calculating the average temperatures at
the shear zone ($T_{AB}$) and the tool-chip interface ($T_{int}$), which is
also proposed by Oxley [6]:

$T_{AB}=T_{0}+\eta\Delta T_{AB}$ (14)

$T_{int}=T_{0}+\Delta T_{AB}+\psi\Delta T_{M}$ (15)

$\Delta T_{AB}=\dfrac{1-\beta}{\rho
Stw}\dfrac{F_{s}\cos\alpha}{\cos(\phi-\alpha)}$ (16)

where $\Delta T_{M}$ and $\beta$ are calculated using the work done by Oxley
[6]. The value of the shear force $F_{s}$ has been computed using an iterative
algorithm given by Zhou and Ren [21].

###  2.3 Johnson-Cook model

The flow stresses in the shear zone $k_{AB}$, and in the tool-chip interface
$k_{int}$ are obtained using the Johnson-Cook model [9], the equations for
which have been shown.

$k_{AB}=\dfrac{1}{\sqrt{3}}(A+B\varepsilon_{AB}^{n})\left(1+C\ln{\dfrac{\dot{\varepsilon}_{AB}}{\dot{\varepsilon}_{0}}}\right)\left(1-\left(\dfrac{T_{AB}-T_{0}}{T_{m}-T_{0}}\right)^{m}\right)$

(17)

$k_{int}=\dfrac{1}{\sqrt{3}}(A+B\varepsilon_{int}^{n})\left(1+C\ln{\dfrac{\dot{\varepsilon}_{int}}{\dot{\varepsilon}_{0}}}\right)\left(1-\left(\dfrac{T_{int}-T_{0}}{T_{m}-T_{0}}\right)^{m}\right)$

(18)

The average strain $\varepsilon_{int}$ and strain rate
$\dot{\varepsilon}_{int}$ in the chip are approximated by the following
expressions:

$\varepsilon_{int}=2\varepsilon_{AB}+\dfrac{h}{\sqrt{3}\delta t_{c}}$ (19)

$\dot{\varepsilon}_{int}=\dfrac{v_{c}}{\sqrt{3}\delta t_{c}}$ (20)

Here, the thickness of the primary shear zone $h$ is given by:

$h=\dfrac{t\sin\theta}{\cos\lambda\sin\theta}\left(1+\dfrac{C_{o}n_{eq}}{3\tan\theta}\right)$
(21)

The shear stress along tool-chip interface is given by $\tau_{int}=F/(hw)$,
and the shear angle $\phi$ is chosen such that we obtain $\tau_{int}=k_{int}$.

##  3 Temperature modeling

Apart from the mechanical influences of deformation, stress formation in the
material is also influenced by temperature changes. This is determined by
integrating the heating effect within the workpiece from the shear zone and
tool-chip interface, followed by the cooling effect due to the presence of a
coolant.

###  3.1 The temperature model

Komanduri and Hou [14, 15, 16] developed a two heat source model. Based on
this model, the temperature change due to shearing and rubbing interactions
are modeled. The temperature change due to coolant is obtained using the work
done by Singh and Sharma [23]. The three equations for $\Delta T_{shear}$,
$\Delta T_{rubbing}$ and $\Delta T_{cool}$ are given.

$\begin{split}\Delta T_{shear}(X,Z)&=\dfrac{q_{shear}}{2\pi
k}\int_{0}^{L}e^{-\dfrac{(X-l_{i}sin\varphi)v}{2a}}\bigl{(}K_{0}\bigl{[}\dfrac{v}{2a}\sqrt{(X-l_{i}sin\varphi)^{2}+(Z-l_{i}cos\varphi)^{2}}\bigr{]}\\\
&+K_{0}\bigl{[}\dfrac{v}{2a}\sqrt{(X-l_{i}sin\varphi)^{2}+(Z+l_{i}cos\varphi)^{2}}\bigr{]}\bigr{)}dl_{i}\end{split}$

(22)

$\begin{split}\Delta T_{rubbing}(X,Z)=\dfrac{q_{rubbing}}{\pi
k}\int_{0}^{\text{VB}}\gamma
e^{-\dfrac{(X-x)v}{2a}}K_{0}\bigl{[}\dfrac{v}{2a}\sqrt{(X-x)^{2}+Z^{2}}\bigr{]}dx\end{split}$

(23)

$\begin{split}\Delta T_{cool}(X,Z)=-\dfrac{q_{cool}}{\pi
k}\int_{0}^{l}e^{-\dfrac{(X-x)v}{2a}}K_{0}\bigl{[}\dfrac{v}{2a}\sqrt{(X-x)^{2}+Z^{2}}\bigr{]}dx\end{split}$

(24)

Here, $\varphi=\phi-\pi/2$, $L=t/\sin\phi$, VB is the flank wear length, $a$
is the thermal diffusivity of the workpiece, $k$ is the thermal conductivity
of the workpiece, and $K_{0}$ is the modified Bessel function of the second
kind of order zero.

The values for heat sources $q_{shear}$ and $q_{rubbing}$ are given by:

$q_{shear}=\dfrac{(F_{c}\cos\phi-
F_{t}\sin\phi)(v\cos\alpha/\cos(\phi-\alpha))}{tw\csc\phi}$ (25)

$q_{rubbing}=\dfrac{P_{c}v}{w(\text{VB})}$ (26)

###  3.2 The effect of coolant

For a coolant with heat transfer coefficient $h$, the rate of heat loss
through convection per unit area is given by $q_{cool}=h(T-T_{0})$, as
proposed by Su [20]. For a coolant flowing with a mass flow rate per unit area
of $\rho v$, the heat transfer coefficient $h$ is obtained using the following
expressions given by Singh and Sharma [23]:

$1.86{Re}^{1/3}{Pr}^{1/3}=\dfrac{hd}{k}$ (27)

$0.023{Re}^{0.8}{Pr}^{0.4}=\dfrac{hd}{k}$ (28)

where, $Re=\rho vd/\mu$ is the Reynolds number, $Pr$ is the Prandtl number,
and the two equations above are for the cases of laminar and turbulent flow
respectively. The overall temperature change is then given by $\Delta
T(X,Z)=\Delta T_{shear}(X,Z)+\Delta T_{rubbing}(X,Z)+\Delta T_{cool}(X,Z)$.

##  4 Residual stress modeling

The mechanical and thermal stresses that develop within the material are now
calculated. These stresses are combined under either elastic or elasto-plastic
loading conditions to derive the components of total stress. Subsequently,
they are relaxed to ascertain the residual stress value. The previously
obtained temperature distribution is employed in computing thermal stresses.

###  4.1 Johnson model for mechanical Stress

Johnson [3] provided a model for obtaining the components of mechanical
stresses. These equations for $\sigma_{xx}^{m}$, $\sigma_{zz}^{m}$ and
$\tau_{xz}^{m}$ are:

$\sigma_{xx}^{m}=-\dfrac{2z}{\pi}\int_{-b}^{a}\dfrac{p(s)(x-s)^{2}}{[(x-s)^{2}+z^{2}]^{2}}ds-\dfrac{2}{\pi}\int_{-b}^{a}\dfrac{q(s)(x-s)^{3}}{[(x-s)^{2}+z^{2}]^{2}}ds$

(29)

$\sigma_{zz}^{m}=-\dfrac{2z^{3}}{\pi}\int_{-b}^{a}\dfrac{p(s)}{[(x-s)^{2}+z^{2}]^{2}}ds-\dfrac{2z^{2}}{\pi}\int_{-b}^{a}\dfrac{q(s)(x-s)}{[(x-s)^{2}+z^{2}]^{2}}ds$

(30)

$\tau_{xz}^{m}=-\dfrac{2z^{2}}{\pi}\int_{-b}^{a}\dfrac{p(s)(x-s)}{[(x-s)^{2}+z^{2}]^{2}}ds-\dfrac{2z}{\pi}\int_{-b}^{a}\dfrac{q(s)(x-s)^{2}}{[(x-s)^{2}+z^{2}]^{2}}ds$

(31)

where the normal and tangential stress distributions, $p(s)$ and $q(s)$, have
been assumed to be for a 2-D hertzian contact, with $q(s)$ proportional to
$p(s)$, as proposed by McDowell [4]:

$p(s)=p_{0}\sqrt{1-\left(s/a\right)^{2}}$ (32)

$q(s)=q_{0}\sqrt{1-\left(s/a\right)^{2}}$ (33)

where, $q_{0}=-\mu p_{0}$, and $\mu$ is the coefficient of friction. The
mechanical stress components obtained from these integrals are then
transformed from the $(x^{\prime},z^{\prime})$ coordinate system of the shear
zone, to the $(x,z)$ coordinate system of the workpiece, as proposed by [20].

###  4.2 Thermal stresses

The thermal stresses are derived using a model established by Liang et al.
[24, 25]. The equations for calculating the thermal stress components
$\sigma_{xx}^{t}$, $\sigma_{zz}^{t}$ and $\tau_{xz}^{t}$ are:

$\begin{split}\sigma_{xx}^{t}(x,z)&=-\dfrac{\alpha
E}{1-2\nu}\int_{0}^{\infty}\int_{-\infty}^{\infty}{\bigl{(}G_{xh}(x^{\prime},z^{\prime})\dfrac{\partial
T}{\partial x}(x,z)+G_{x\nu}(x^{\prime},z^{\prime})\dfrac{\partial T}{\partial
z}(x,z)\bigr{)}dx^{\prime}dz^{\prime}}\\\
&+\dfrac{2z}{\pi}\int_{-\infty}^{\infty}\dfrac{p(t)(t-x)^{2}}{((t-x)^{2}+z^{2})^{2}}dt-\dfrac{\alpha
ET(x,z)}{1-2\nu}\end{split}$

(34)

$\begin{split}\sigma_{zz}^{t}(x,z)&=-\dfrac{\alpha
E}{1-2\nu}\int_{0}^{\infty}\int_{-\infty}^{\infty}{\bigl{(}G_{zh}(x^{\prime},z^{\prime})\dfrac{\partial
T}{\partial x}(x,z)+G_{z\nu}(x^{\prime},z^{\prime})\dfrac{\partial T}{\partial
z}(x,z)\bigr{)}dx^{\prime}dz^{\prime}}\\\
&+\dfrac{2z^{3}}{\pi}\int_{-\infty}^{\infty}\dfrac{p(t)}{((t-x)^{2}+z^{2})^{2}}dt-\dfrac{\alpha
ET(x,z)}{1-2\nu}\end{split}$

(35)

$\begin{split}\tau_{xz}^{t}(x,z)&=-\dfrac{\alpha
E}{1-2\nu}\int_{0}^{\infty}\int_{-\infty}^{\infty}{\bigl{(}G_{xzh}(x^{\prime},z^{\prime})\dfrac{\partial
T}{\partial x}(x,z)+G_{xz\nu}(x^{\prime},z^{\prime})\dfrac{\partial
T}{\partial z}(x,z)\bigr{)}dx^{\prime}dz^{\prime}}\\\
&+\dfrac{2z^{2}}{\pi}\int_{-\infty}^{\infty}\dfrac{p(t)(t-x)}{((t-x)^{2}+z^{2})^{2}}dt\end{split}$

(36)

where $\alpha$ is the coefficient of thermal expansion of the workpiece, $E$
is the Young’s modulus, $\nu$ is the poisson ratio, and $G_{xh}$, $G_{x\nu}$,
$G_{zh}$, $G_{z\nu}$, $G_{xzh}$ and $G_{xz\nu}$ are Green’s functions,
calculated using their corresponding expressions developed by Saif et al.
[27]. The function $p(t)$ represents surface traction, and is given by Liang
et al. [24, 25] as:

$p(t)=\dfrac{\alpha ET(x=0,z=0)}{1-2\nu}$ (37)

Using the mechanical and thermal stresses obtained, the total stress
components are calculated for elastic loading condition as:

$\begin{split}\sigma_{xx}&=\sigma_{xx}^{m}+\sigma_{xx}^{t}\\\
\sigma_{zz}&=\sigma_{zz}^{m}+\sigma_{zz}^{t}\\\
\tau_{xz}&=\tau_{xz}^{m}+\tau_{xz}^{t}\\\
\sigma_{yy}&=\nu(\sigma_{xx}+\sigma_{zz})-\alpha E\Delta T\end{split}$ (38)

The model proposed by Shan et al. [26], based on the Jiang-Sehitoglu [19]
model, has been incorporated for calculating the stress and strain increments
for elasto-plastic loading. The stress increments are obtained through an
iterative process, which involves the use of deviatoric stresses, the shear
yield strength, and the plastic modulus function (denoted by $h$).

###  4.3 Stress relaxation to obtain residual stresses

After the cutting process is complete, the unloading (relaxation) of the
stresses must be performed in order to satisy certain boundary conditions for
the stresses and strains. The stress and strain increments for this process
are proposed by Merwin and Johnson [2], and Su [20]:

$\begin{split}\Delta\sigma_{zz}^{r}&=-\dfrac{\sigma_{zz}^{r}}{M},\Delta\tau_{xz}^{r}=-\dfrac{\tau_{xz}^{r}}{M}\\\
\Delta\varepsilon_{xx}^{r}&=-\dfrac{\varepsilon_{xx}^{r}}{M},\Delta
T^{r}=-\dfrac{T^{r}}{M}\end{split}$ (39)

Using these increments, either elastic or elastic-plastic unloading is
performed, based on the work of Shan et al. [26], which has also been proposed
by Su [20], McDowell [4], and Liang et al. [24].

##  5 Validation

Table 1: Process parameters taken for the model Parameters | Values
---|---
Tool rake angle, $\alpha$ | $12.62^{\circ}$
Width of cut, $w$ | 2 mm
Nose radius, $r_{e}$ | $30\,\mu m$
Flank wear length, VB | $10\,\mu m$
$\psi$ (in oxley temperature model) | 0.4
$\eta$ (in oxley temperature model) | 0.9
Friction factor, $m_{plow}$ | 0.99
$\rho_{plow}$ (in plowing force model) | $20^{\circ}$
|
Table 2: Properties of Inconel 718 and Ti6Al4V for material parameters Properties | Inconel 718 | Ti6Al4V
---|---|---
Melting temperature, $T_{m}$ | $1297^{\circ}C$ | $1604^{\circ}C$
Density, $\rho$ (in $kg/m^{3}$) | 8280 | 4430
Specific heat capacity, $S$ (in $J/kgK$) | 362 | 565
Thermal conductivity, $k$ (in $W/mK$) | 10.3 | 6.6
Thermal diffusivity, $a$ (in $m^{2}/s$) | $2.87\times 10^{-6}$ | $2.76\times 10^{-6}$
Young’s modulus, $E$ | 214.580 GPa | 113.8 GPa
Poisson ratio, $\nu$ | 0.305 | 0.342
Coefficient of friction, $\mu$ | 0.7 | 0.85
Shear yield stress |  $1100/\sqrt{3}$ MPa |  $880/\sqrt{3}$ MPa
Coefficient of thermal expansion, $\alpha$ (in $K^{-1}$) | $14.8\times 10^{-6}$ | $9.2\times 10^{-6}$
|  |

The authors have utilized machining data associated with two work materials -
Inconel 718 and Ti6Al4V - to validate the proposed residual stress model.
These materials have been selected due to their widespread application in the
aerospace industry, particularly in critical components such as turbine blades
and various structural parts, owing to their exceptional mechanical
properties, thermal stability, and corrosion resistance. The parameters for
the Johnson-Cook model adopted for Inconel 718 are as follows:
$A=1290\,\text{MPa}$, $B=895\,\text{MPa}$, $C=0.0252$, $n=0.526$, $m=1.55$,
and $\dot{\varepsilon}_{0}=10^{-5}$; for Ti6Al4V, they are:
$A=724.7\,\text{MPa}$, $B=683.1\,\text{MPa}$, $C=0.01$, $n=0.47$, $m=1.0$, and
$\dot{\varepsilon}_{0}=10^{-5}$. The ambient temperature $T_{0}$ is taken as
$20^{\circ}C$ (equivalent to 293 K). The parameters for the Johnson-Cook model
for Inconel 718 are derived from the work of Zhou and Ren [21], while those
for Ti6Al4V are obtained from Zhou et al. [28]. The process parameters
utilized for the model are displayed in Table 1, and the material properties
of Inconel 718 and Ti6Al4V are presented in Table 2.

The values for the tool rake angle, width of cut, nose radius, density,
specific heat, thermal conductivity, coefficient of friction, $m_{plow}$,
$\rho_{plow}$, melting temperature, $\psi$, and $\eta$ for both Inconel 718
and Ti6Al4V are sourced from the data provided by Zhou and Ren [21] and Zhou
et al. [28]. The values for the Young’s modulus, Poisson ratio, and
coefficient of thermal expansion for Inconel 718 are obtained from Shen et al.
[29], while those for Ti6Al4V are derived from Shan et al. [26]. The thermal
diffusivity of Ti6Al4V is sourced from Liang et al. [25].

The tool is modeled as an uncoated cemented carbide tool, with a density of
$14860\,kg/m^{3}$, thermal conductivity of $82\,W/mK$, and specific heat of
$249.8\,J/kgK$, based on data provided by Liu et al. [30]. The coolant is
modeled as water, with an absolute viscosity of $0.001\,kg/m.s$, Prandtl
number of 7.56, and a diameter of the circular duct of 2 mm, as per data from
Singh and Sharma [23]. For this model, a mass flow rate per unit area of
$2500\,kg/m^{2}s$ is considered to model the effect of the coolant on the
temperature distribution.

In the force model, the shear angle $\phi$ is iteratively computed within the
range of $10^{\circ}$ to $45^{\circ}$. The parameters $C_{0}$ and $\delta$ are
treated as hyperparameters, which are optimized by tuning them using the
training data. Specifically, $C_{o}$ is varied within the range of 0.1 to 4.0,
while $\delta$ is varied within the range of 0.005 to 0.20. The cost function
employed for determining the optimal values of these hyperparameters is
$(1+C_{mpe})(1+T_{mpe})(1+C_{sde})(1+T_{sde})$, where $C_{mpe}$ and $T_{mpe}$
denote the mean absolute percentage errors in the cutting and thrust forces,
respectively, and $C_{sde}$ and $T_{sde}$ represent the standard deviation of
errors in the cutting and thrust forces. The experimental data utilized for
training the model, and subsequently evaluating it to compute the errors in
$F_{c}$ and $F_{t}$, are obtained from Zhou and Ren [21] (for Inconel 718) and
Zhou et al. [28] (for Ti6Al4V). The plastic modulus function $h$ is tuned
within the range of 10 GPa to 60 GPa, employing a mean absolute percentage
error cost function to minimize the error in the predicted residual stress.
For stress relaxation, the number of iterations $M$ is set to 100.

The integrals employed in temperature and stress modeling are computed using
the quad and dblquad functions in the Python scipy-1.7.3 library. The model is
implemented in Python-3.9.12 and executed on a 64-bit system featuring an
Intel(R) Core(TM) i7-8565U processor, with a clock speed of 1.80 GHz and 16 GB
RAM.

##  6 Results and discussion

Figure 2: Force vs cutting speed plots at t = 0.05 mm and t = 0.10 mm for
Inconel 718. The black lines represent model predictions, and the red lines
are from experimental data.

Figure 3: Force vs cutting speed plots at t = 0.15 mm for Ti6Al4V. The black
lines represent model predictions, and the red lines are from experimental
data.

Figure 4: Cutting force vs depth of cut plots at v = 20 m/min and v = 30 m/min
for Inconel 718. The black lines represent model predictions, and the red
lines are from experimental data.

Figure 5: Force vs depth of cut plots at v = 20 m/min for Ti6Al4V. The black
lines represent model predictions, and the red lines are from experimental
data.

Figure 6: Temperature vs cutting speed, and temperature vs depth of cut plots
for Inconel 718. The blue lines represent predicted shear zone temperature
($T_{AB}$), and the red lines represent predicted tool-chip interface
temperature ($T_{int}$).

Figure 7: Temperature vs cutting speed, and temperature vs depth of cut plots
for Ti6Al4V. The blue lines represent predicted shear zone temperature
($T_{AB}$), and the red lines represent predicted tool-chip interface
temperature ($T_{int}$).

Figure 8: Shear angles predicted for each v-t pair for Inconel 718 [21] (top),
and Ti6Al4V [28] (bottom).

Figure 9: Errors in $F_{c}$ and $F_{t}$ calculated for each v-t pair for
Inconel 718 [21] (top), and Ti6Al4V [28] (bottom). The blue bars represent the
percentage error in $F_{c}$, and green bars represent percentage error in
$F_{t}$.

Figure 10: The predicted temperature distribution through the workpiece, for
Inconel 718 (top) and Ti6Al4V (bottom).

Figure 11: The predicted change in temperature through the workpiece due to
the coolant, for Inconel 718 (top) and Ti6Al4V (bottom).

Figure 12: The predicted residual stress vs depth into the workpiece, for
Inconel 718 (top) and Ti6Al4V (bottom). The red line represents experimental
data, and the blue and green lines represent $\sigma_{xx}$ and $\sigma_{yy}$
respectively. Figure 13: Errors in residual stress vs depth into the
workpiece, for Inconel 718 (blue bars) and Ti6Al4V (green bars).

The force model is employed for both Inconel 718 and Ti6Al4V, utilizing the
parameters detailed in Tables 1-2. For Inconel 718, the predicted cutting
forces ($F_{c}$) and thrust forces ($F_{t}$) yield mean percentage errors of
3.65% and 16.16%, respectively, with standard deviations of the errors
amounting to 24.46 N and 79.31 N, respectively. Correspondingly, for Ti6Al4V,
the errors stand at 4.37% for $F_{c}$ and 15.12% for $F_{t}$, with standard
deviations of 17.65 N and 20.05 N, respectively. Generally, the force model
demonstrates higher accuracy in predicting cutting force compared to thrust
force, a trend in line with observations by Zhou and Ren [21] regarding the
superior accuracy of cutting force prediction with various J-C models. As
proposed by Zhou and Ren [21], the predictive accuracy may be enhanced by
employing an improved plow force model.

Plots illustrating cutting force and thrust force against speed ($v$) and
depth of cut ($t$) are provided in Figures 2-5 for both Inconel 718 and
Ti6Al4V. Figures 6 and 7 depict plots for the average shear zone temperatures
($T_{AB}$) and average tool-chip interface temperatures ($T_{int}$) computed
from Boothroyd’s [22] model, in relation to cutting speed and depth of cut for
both materials. Furthermore, Figure 8 illustrates the predicted shear angles
obtained from the force model for Inconel 718 and Ti6Al4V, across all data
sets (numbered 1-15 in [21] for Inconel 718 and 1-24 in [28] for Ti6Al4V). The
percentage errors in $F_{c}$ and $F_{t}$ for each individual data set are
plotted in Figure 9. The values of $C_{o}$ and $\delta$ derived from the force
model are 1.8 and 0.185 for Inconel 718, and 0.1 and 0.145 for Ti6Al4V.

Temperature distribution plots within the workpiece and the impact of coolant
are presented in Figures 10 and 11 for both Inconel 718 and Ti6Al4V. The
maximum temperature drop due to coolant, as observed in Figure 11, falls
within the range of $50-70^{\circ}C$, notably smaller than the anticipated
drop (approximately 29% for laminar and 53% for turbulent flow, as suggested
by [23]). Possible explanations for this disparity include: (a) The convective
heat flow rate ($q_{cool}$) used in temperature drop calculations is directly
determined for the workpiece, without consideration of the flow rate for the
tool, a significant heat source. (b) The impact of coolant has not been
incorporated into the force model, resulting in higher forces and consequently
greater temperature values than expected in the presence of coolant.

Residual stresses ($\sigma_{xx}$ and $\sigma_{yy}$) are plotted alongside
experimental data in Figure 12. For Inconel 718, the results are plotted for a
cutting speed of 60 m/min and a depth of cut of 0.15 mm, along with
experimental data from Liu et al. [30]. Similarly, for Ti6Al4V, the results
are plotted for a cutting speed of 50 m/min and a depth of cut of 0.10 mm,
alongside experimental data from Liang et al. [24]. The values of the plastic
modulus function $h$ derived from the residual stress model are 40 GPa for
Inconel 718 and 25 GPa for Ti6Al4V. The model results indicate that residual
stresses are more pronounced near the surface, diminishing rapidly with
increasing depth, consistent with the expectation that mechanical and thermal
effects of machining primarily influence regions near the surface. Tensile
stress regions are observed near the surface for Inconel 718, attributed to
the dominance of tensile thermal effects over compressive mechanical effects
as explained by Liu et al. [30]. This observation aligns with the maximum
temperatures for Inconel 718 ($>600^{\circ}$C), higher than those for Ti6Al4V
($>500^{\circ}$C), partly due to larger $v$ and $t$ values used for modeling
residual stresses in Inconel 718. Conversely, compressive stress regions near
the surface are observed for Ti6Al4V, attributed to the dominance of
mechanical effects over thermal effects, as explained by Liang et al. [24],
given the lower heating due to friction for low cutting speeds (here, $v=50$
m/min) and smaller flank wear (here, $\text{VB}_{model}=10\,\mu m$,
$\text{VB}_{exp}=0\,mm$). The errors in residual stress (i.e., the difference
between model prediction and experimental data) are plotted against depth into
the workpiece for both Inconel 718 and Ti6Al4V in Figure 13. The residual
stress results exhibit similar trends as the experimental data but show larger
errors compared to the force model. This discrepancy can partly be attributed
to the fact that the parameters (aside from $v$ and $t$) for which the
experimental data are plotted differ from those used in modeling residual
stresses. Notably, variations in experimental values with parameters such as
flank wear (VB) are observed in results provided by Liu et al. [30] and Liang
et al. [24]. Additionally, the accumulation of errors from component force,
temperature, and stress analytical models contributes to larger errors.

##  7 Conclusions

This study introduces a physics-informed and data-driven model designed to
predict residual stresses arising from orthogonal machining. Employing
analytical modeling techniques, the authors model the forces and temperature
distributions, facilitating the subsequent computation of residual stress
variations with depth into the workpiece. Upon comparison with experimental
data, the force model demonstrates precise predictions for $F_{c}$ and $F_{t}$
values. Although the residual stress exhibits larger errors, it aligns with
anticipated trends from literature for both materials, considering their
respective input parameters - notably, a tensile region in Inconel 718 and a
compressive region in Ti6Al4V, closer to the surface. Key factors integrated
into the model encompass material properties, machining process parameters
(including tool rake angle, width of cut, nose radius, and flank wear), and
the influence of coolant on the workpiece owing to laminar and turbulent flow.

The proposed model exhibits adaptability for specific tool-work-process
combinations even with limited data availability, enhancing its practical
utility for industrial applications. This adaptability enables the prediction
of parameters for which data is scarce across various materials. Moreover, the
model holds promise for extension into a real-time framework for industrial
deployment. Such real-time integration would facilitate continual verification
of experimental measurements against their expected counterparts, facilitating
adjustments for any deviations attributable to improper processing conditions,
thereby enhancing machining efficiency and component quality. Future
enhancements to the model could involve the inclusion of additional factors
such as machining vibrations, the impact of coolant on both the force model
and temperature profiles in the shear zone and tool-chip interface,
utilization of refined literature data, and adoption of advanced optimization
techniques for hyperparameter tuning, with the aim of improving predictive
accuracy.

Acknowledgments

The authors would like to thank TCS Research for supporting this work.

## Declarations

Author contributions

Dhar, R.: methodology, investigation, model development, programming, formal
analysis, resources, writing - original draft and editing. Krishna, A. and
Muhammed, B.: study conception, methodology, resources, reviewing, supervision
and project administration. All authors have extensively revised and approved
the final manuscript.

Ethics approval

The authors declare that this manuscript was not submitted to more than one
journal for simultaneous consideration. Also, the submitted work was original
and has not been published elsewhere in any form or language.

Competing interests

The authors declare no competing interests.

## References

  *   * Wan et al. [2019] Wan, M., Ye, X.Y., Wen, D.Y., Zhang, W.H.: Modeling of machining-induced residual stresses. Journal of Materials Science 54, 1–35 (2019) https://doi.org/10.1007/s10853-018-2808-0
  * Merwin and Johnson [1963] Merwin, J.E., Johnson, K.L.: An analysis of plastic deformation in rolling contact. Archive Proc. Inst. Mech. Eng. 177(1), 676–690 (1963) https://doi.org/10.1243/PIME_PROC_1963_177_052_02
  * Johnson [1985] Johnson, K.: Contact Mechanics. Cambridge University Press, Cambridge (1985). https://doi.org/10.1002/zamm.19890690713
  * McDowell [1997] McDowell, D.L.: An approximate algorithm for elastic-plastic two-dimensional rolling/sliding contact. Wear 211(2), 237–246 (1997) https://doi.org/10.1016/s0043-1648(97)00117-8
  * Liang and Su [2007] Liang, S.Y., Su, J.C.: Residual stress modeling in orthogonal machining. CIRP Ann. Manuf. Technol. 56(1), 65–68 (2007) https://doi.org/10.1016/j.cirp.2007.05.018
  * Oxley [1989] Oxley, P.L.B.: The Mechanics of Machining: an Analytical Approach to Assessing Machinability. E. Horwood, Chichester [England] (1989). https://doi.org/%****␣sn-article.bbl␣Line␣125␣****10.1007/978-1-349-06281-2_35
  * Waldorf et al. [1998] Waldorf, D.J., Devor, R.E., Kapoor, S.G.: A slip-line field for ploughing during orthogonal cutting. J. Manuf. Sci. Eng. 120(4), 693–699 (1998) https://doi.org/10.1115/1.2830208
  * Wan et al. [2017] Wan, M., Ye, X.Y., Yang, Y., Zhang, W.H.: Theoretical prediction of machining-induced residual stresses in three-dimensional oblique milling processes. Int. J. Mech. Sci. 133, 426–437 (2017) https://doi.org/10.1016/j.ijmecsci.2017.09.005
  * Johnson and Cook [1983] Johnson, R., Cook, W.K.: A constitutive model and data for metals subjected to large strains high strain rates and high temperatures. The 7th International Symposium on Ballistics, The Hague, pp 541-547 (1983)
  * Matsumoto et al. [1991] Matsumoto, Y., Magda, D., Hoeppner, D.W., Kim, T.Y.: Effect of machining processes on the fatigue strength of hardened AISI 4340 steel. J. Eng. Ind. 113(2), 154–159 (1991) https://doi.org/10.1115/1.2899672
  * Jacobus et al. [2000] Jacobus, K., Devor, R.E., Kapoor, S.G.: Machining-induced residual stress: experimentation and modeling. J. Manuf. Sci. Eng. 122(1), 633–647 (2000) https://doi.org/10.1115/1.538906
  * Sekhon and Chenot [1993] Sekhon, G.S., Chenot, J.L.: Numerical simulation of continuous chip formation during non-steady orthogonal cutting. Eng. Comput. 10(1), 31–48 (1993) https://doi.org/10.1108/eb023893
  * Ulutan et al. [2007] Ulutan, D., Alaca, B.E., Lazoglu, I.: Analytical modelling of residual stresses in machining. J. Mater. Process. Tech. 183(1), 77–87 (2007) https://doi.org/10.1016/j.jmatprotec.2006.09.032
  * Komanduri and Hou [2000] Komanduri, R., Hou, Z.B.: Thermal modeling of the metal cutting process: part I—temperature rise distribution due to shear plane heat source. Int. J. Mech. Sci. 42(9), 1715–1752 (2000) https://doi.org/10.1016/s0020-7403(99)00070-3
  * Komanduri and Hou [2001a] Komanduri, R., Hou, Z.B.: Thermal modeling of the metal cutting process—part II: temperature rise distribution due to frictional heat source at the tool–chip interface. Int. J. Mech. Sci. 43(1), 57–88 (2001) https://doi.org/10.1016/s0020-7403(99)00104-6
  * Komanduri and Hou [2001b] Komanduri, R., Hou, Z.B.: Thermal modeling of the metal cutting process—part III: temperature rise distribution due to the combined effects of shear plane heat source and the tool–chip interface frictional heat source. Int. J. Mech. Sci. 43(1), 89–107 (2001) https://doi.org/10.1016/s0020-7403(99)00105-8
  * Pan et al. [2017] Pan, Z., Feng, Y., Ji, X., Liang, S.Y.: Turning induced residual stress prediction of AISI 4130 considering dynamic recrystallization. Mach. Sci. Technol. 9, 1–15 (2017) https://doi.org/10.1080/10910344.2017.1365900
  * Drucker and Palgen [1981] Drucker, D.C., Palgen, L.: On stress–strain relations suitable for cyclic and other loading. J. Appl. Mech. 48(3), 479–485 (1981) https://doi.org/10.1115/1.3157660
  * Jiang and Sehitoglu [1994] Jiang, Y., Sehitoglu, H.: An analytical approach to elastic–plastic stress analysis of rolling contact. J. Tribol. 116(3), 577–587 (1994) https://doi.org/10.1115/1.2928885
  * Su [2006] Su, J.C.: Residual stress modeling in machining processes. Ph.D. Thesis, Georgia Institute of Technology, Atlanta (2006)
  * Zhou and Ren [2020] Zhou, J., Ren, J.: Predicting cutting force with unequal division parallel-sided shear zone model for orthogonal cutting. International Journal of Advanced Manufacturing Technology 107, 4201–4211 (2020) https://doi.org/10.1007/s00170-020-05273-2
  * Boothroyd [1963] Boothroyd, G.: Temperatures in orthogonal metal cutting. Proc. Inst. Mech. Eng. 177(1), 789–810 (1963) https://doi.org/10.1243/pime_proc_1963_177_058_02
  * Singh and Sharma [2021] Singh, R., Sharma, V.: Numerical modelling of residual stresses during orthogonal cutting of Ti6Al4V using internally cooled cutting inserts. Journal of Manufacturing Processes 65, 502–511 (2021) https://doi.org/10.1016/j.jmapro.2021.03.042
  * Liang et al. [2021] Liang, X., Liu, Z., Wang, B., Song, Q., Cai, Y., Wan, Y.: Prediction of residual stress with multi-physics model for orthogonal cutting Ti-6Al-4V under various tool wear morphologies. Journal of Materials Processing Technology 288 (2021) https://doi.org/10.1016/j.jmatprotec.2020.116908
  * Liang et al. [2018] Liang, X., Liu, Z., Wang, B., Hou, X.: Modeling of plastic deformation induced by thermo-mechanical stresses considering tool flank wear in high-speed machining Ti-6Al-4V. International Journal of Mechanical Sciences 140, 1–12 (2018) https://doi.org/10.1016/j.ijmecsci.2018.02.031
  * Shan et al. [2020] Shan, C., Zhang, M., Zhang, S., Dang, J.: Prediction of machining-induced residual stress in orthogonal cutting of Ti6Al4V. The International Journal of Advanced Manufacturing Technology 107, 2375–2385 (2020) https://doi.org/10.1007/s00170-020-05181-5
  * Saif et al. [1993] Saif, M.T.A., Hui, C.Y., Zehnder, A.T.: Interface shear stresses induced by non-uniform heating of a film on a substrate. Thin Solid Films 224(2), 159–167 (1993) https://doi.org/10.1016/0040-6090(93)90427-q
  * Zhou et al. [2020] Zhou, J., Ren, J., Jiang, Y.: Inverse identification of modified Johnson–Cook model for cutting titanium alloy Ti6Al4V using firefly algorithm. Proceedings of the Institution of Mechanical Engineers, Part B: Journal of Engineering Manufacture 234(3), 584–599 (2020) https://doi.org/10.1177/0954405419864003
  * Shen et al. [2018] Shen, Q., Liu, Z., Hua, Y., Zhao, J., Lv, W., Mohsan, A.U.H.: Effects of cutting edge microgeometry on residual stress in orthogonal cutting of Inconel 718 by FEM. Materials 11(6), 1015 (2018) https://doi.org/10.3390/ma11061015
  * Liu et al. [2021] Liu, Y., Xu, D., Agmell, M., M’Saoubi, R., Ahadi, A., Stahl, J.E., Zhou, J.: Numerical and experimental investigation of tool geometry effect on residual stresses in orthogonal machining of Inconel 718. Simulation Modelling Practice and Theory 106 (2021) https://doi.org/%****␣sn-article.bbl␣Line␣500␣****10.1016/j.simpat.2020.102187