# Two Modes of Nonmonotonic Consequence†

Alexei Muravitsky  Louisiana Scholars’ College,
Northwestern State University
Natchitoches, LA 71497, USA<EMAIL_ADDRESS>
###### Abstract.

We discuss two ways to implement a semantic approach to nonmonotonic
consequence relations in an arbitrary propositional language. For one
particular language, we also discuss the proof-theoretic framework that we
connect with this semantic approach. This article is an addition to
[Muravitsky, 2021].

†This is the text of my speech at the Logica Universalis webinar, which took
place on May 11, 2022.

##  1\. What do we know and how do we know about nonmonotonic consequences?

We discuss non-monotony expressed in formalized languages. All main objective
languages that will be involved are propositional.111First-order language is
used only as an auxiliary tool.

Aristotle is credited with distinguishing between _what we know_ and _how we
know_. G. Kreisel who made this remark in [Kreisel, 2019], p. 42., adds,

> The point I wish to stress, again as a matter of accumulated experience, is
> how much _easier_ the former question is to answer (to our satisfaction)
> than the latter. Of the _what_ , whose aspects strike the (mind’s) eye, we
> have easily accessible practical knowledge (which constitutes the threshold
> for additions by further theory). We also would like to know about the _how_
> , but the price is high. (Ibid)

We note that it is not only the possible easiness of answering the _what_
question that prompts us to start with it but also the view that “knowing how
we know is one small department of knowing what we know” (Russell).222Here is
a full quote:  I reverse the process which has been common in philosophy since
Kant. It has been common among philosophers to begin with how we know and
proceed afterwards to what we know. I think this is a mistake, because knowing
how we know is one small department of knowing what we know. ([Russell, 1959],
chapter II)

Speaking of “easily accessible practical knowledge”, Kreisel probably had in
mind, first of all, modern mathematics and formal logic. As is known, both are
based on monotonic reasoning. On the contrary, with respect to nonmonotonic
reasoning, it seems that answering both _what_ and _how_ questions is not an
easy task. We do not think that the following statement by D. Makinson can be
taken seriously.

> Of course, human have been reasoning nonmonotonically for as long as they
> have been reasoning at all. ([Makinson, 1994], section 1.1)

Despite Makinson’s remark, it is precisely because of the lack of rich
_contextual practices_ involving nonmonotonic reasoning that it is so
difficult to answer _what we know_ and _how we know_ about it.

However, non-monotonicity does occur in colloquial speech. R. Stalnaker
[Stalnaker, 1994] gives an example of nonmonotonic argument based on Grice’s
concept of (conversational) ‘implicature’.333See [Grice, 1989], part I. This
example is remarkable only in one respect — it indicates the need to
distinguish between the concepts of argumentation and logic. Logic deals
exclusively with forms, while content can be involved in argumentation. An
example of the latter is that of Stalnaker — the nonmonotonic effect of his
example is based on the interaction of contents, more precisely, between
implicit and explicit contents. Understanding logic as a science dealing with
forms of judgment, we will pay attention to sentential variables and, as a
consequence, to the role of substitution, understood both as an operation and
as a rule of inference. But first, we outline the _monotonicity-non-
monotonicity_ dichotomy as such.

Before proceeding to a formal presentation, we note that we are using the
concept of logic in its dialectical, and not demonstrative, manner. That is,
we will consider the relationship between a set of premises and the
conclusions that can be drawn from these premises according to some standards.

For this, we fix a propositional language, $\mathcal{L}$, (later we will be
more specific), whose formulas are denoted by letters $\alpha$, $\beta$, etc.;
sets of formulas by $X$, $Y$, etc.; and the set of all formulas by
$\mathbf{Fm}_{\mathcal{L}}$. Given sets $X$ and $Y$ of formulas, we write
$X\Subset Y$ if $X$ is a finite subset of $Y$.

The operation of (uniform or simultaneous) substitution is understood in the
usual way. We denote substitutions in $\mathcal{L}$ by $\sigma$, $\delta$,
$\ldots$ (with or without subscripts or other marks). The result of
substitution $\sigma$ in a formula $\alpha$ is denoted by $\sigma(\alpha)$;
accordingly,

$\sigma(X):=\\{\sigma(\alpha)~{}:~{}\alpha\in X\\}.$

Let $\mathbf{C}$ be a mapping
$\mathbf{C}:\mathcal{P}(\mathbf{Fm}_{\mathcal{L}})\longrightarrow\mathcal{P}(\mathbf{Fm}_{\mathcal{L}})$.
Given a set $X\subseteq\mathbf{Fm}_{\mathcal{L}}$, $\mathbf{C}(X)$ is
understood as the set of all conclusions obtained from $X$.

The definition

$X\vdash_{\mathbf{C}}\alpha~{}\stackrel{{\scriptstyle\text{df}}}{{\Longleftrightarrow}}~{}\alpha\in\mathbf{C}(X)$
(1)

will be very useful subsequently.

The last definition implies conversion; namely, given a relation
$\vdash\subseteq\mathcal{P}(\mathbf{Fm}_{\mathcal{L}})\times\mathbf{Fm}_{\mathcal{L}}$,
we define:

$\alpha\in\mathbf{C}_{\vdash}(X)~{}\stackrel{{\scriptstyle\text{df}}}{{\Longleftrightarrow}}~{}X\vdash\alpha.$
(2)

The definitions (1) and (2) allow to use relations of the above type and the
corresponding operators interchangeably.

The following properties of the $\mathbf{C}$ operator have long been in the
spotlight.444See e.g. [Makinson, 2005a].

  * (con-1)

$X\subseteq\mathbf{C}(X)$; (_reflexivity_)

  * (con-2)

$X\subseteq Y$ implies $\mathbf{C}(X)\subseteq\mathbf{C}(Y)$; (_monotonicity_)

  * (con-3)

$\mathbf{C}(\mathbf{C}(X))\subseteq\mathbf{C}(X)$; (_closedness_)

  * (con-4)

$\mathbf{C}(X)\subseteq\bigcup\\{\mathbf{C}(Y)~{}:~{}Y\Subset X\\}$;
(_finitariness_ or _compactness_)

  * (con-5)

$X\subseteq\mathbf{C}(Y)$ implies $\mathbf{C}(X)\subseteq\mathbf{C}(Y)$;
(_cumulative transitivity_)

  * (con-6)

$X\subseteq Y\subseteq\mathbf{C}(X)$ implies
$\mathbf{C}(X)\subseteq\mathbf{C}(Y)$; (_weak cumulative transitivity_ 555This
property is also known as _cautious monotony_.)

  * (con-7)

$X\subseteq\mathbf{C}(Y)$ implies $\mathbf{C}(X\cup Y)\subseteq\mathbf{C}(Y)$;
(_strong cumulative transitivity_)

  * (con-8)

$\bigcup\\{\mathbf{C}(Y)~{}:~{}Y\Subset X\\}\subseteq\mathbf{C}(X)$;
(_finitary inclusion_)

  * (con-9)

if $X\neq\varnothing$, then
$\mathbf{C}(X)\subseteq\bigcup\\{\mathbf{C}(Y)~{}:~{}Y\neq\varnothing~{}\text{and}~{}Y\Subset
X\\}$.666We eliminate curly brackets for singleton premises; thus
$\mathbf{C}(\alpha)$ stands for $\mathbf{C}(\\{\alpha\\}$).
(_strong finitariness_)

A. Tarski, who introduced the operator $\mathbf{C}$ into the discourse in his
address to the Polish Mathematical Society in 1928,777He used the notation
‘$\mathbf{Cn}$’ instead of ‘$\mathbf{C}$’. began with the following remark.

> Our object in this communication is to define the meaning, and to establish
> the elementary properties, of some important concepts belonging to the
> _methodology of deductive sciences_ , which, following Hilbert, it is
> customary to call _metamathematics_. ([Tarski, 1930])

In the published version of this address, Tarski formulated (con-1)–(con-3).
Given a set $X\subseteq\mathbf{Fm}_{\mathcal{L}}$, he called $\mathbf{C}(X)$
“the _consequences of the set_ $X$,” which suggests to call any operator
$\mathbf{C}$ satisfying these properties a _consequence operator_. It was the
rich variety of contextual practices in mathematics and the deductive sciences
that prompted him to introduce this notion. In his own worlds,

> The concept of _logical consequence_ is one of those whose introduction into
> the field of strict formal investigation was not a matter of arbitrary
> decision on the part of this or that investigator; $\ldots$ ([Tarski, 1936])

Tarski also considered (con-4) but not other properties. One interesting
property was introduced over a quarter of a century later in [Łoś and Suszko,
1958]. Namely,

  * (con-10)

$\alpha\in\mathbf{C}(X)$ implies $\sigma(\alpha)\in\mathbf{C}(\sigma(X))$.
(_structurality_)

As for the _what / how_ questions about monotonic reasoning, at least with
regard to its forms as presented in mathematics and the deductive sciences, it
is generally agreed that the properties (con-1)–(con-3) seem to answer the
first question, or at least can be regarded as “the threshold for additions by
further theory” (Kreisel).

Frege and Tarski can be credited for pioneering to answer the second question.
In particular, Tarski pointed out the following two ways of implementing
monotonic reasoning in metamathematics. According to him, the first way is as
follows.

> From the sentences of any set $X$ certain other sentences can be obtained by
> means of certain operations called _rules of inference_. ([Tarski, 1930])

The second way is through semantics.888Cf. [Tarski, 1936]. In the most general
terms, one can imagine a non-linguistic machinery $\mathfrak{M}$ such that it
is natural to say that the sentence $\alpha$ follows from the set of premises
$X$, when $\mathfrak{M}$ accepts $\alpha$ whenever $\mathfrak{M}$ accepts all
premises of $X$. For propositional languages, this leads to matrix
consequences.999See e.g. [Wójcicki, 1988], chapter 3, or [Citkin and
Muravitsky, 2022], chapter 4. And although from this level of abstraction the
implementation of this idea may vary, we will not discuss here all the
alternatives.101010See [Citkin and Muravitsky, 2022], section 4.3.3.

Unfortunately, at this point, little can be said about answers to the _what_
question in relation to nonmonotonic reasoning. To reiterate, we argue that it
is the lack of contextual practices of nonmonotonic reasoning that has led to
speculating about “the minimum conditions that a consequence relation should
satisfy to represent a bona fide nonmonotonic _logic_ $\ldots$”111111Cf.
[Kraus et al., 1990], section 1.2; comp. [Gabbay, 1985]. In contrast to this
optimistic hope, one can see a growing confusion in the endeavor of trying to
learn something about the objective world of reasoning through pure thinking.
Another confusion, the _what-how_ confusion, occurs when someone, in an
attempt to give a general definition of nonmonotonic reasoning, gives examples
of answers to the _how_ question.

If contextual practices of nonmonotonic reasoning are lacking, what can the
theory of nonmonotonic reasoning be based on? One possible answer is that it
can be based on conceptualized practices of monotonic reasoning. Recall that
algebraic logic began with George Boole’s algebraization of Aristotle’s
syllogistic. And to give credit to Kreisel’s insight, to answer the _what_
question, we should simply propose a $\mathbf{C}$ operator that would not
satisfy (con-2). However, there are many candidates for answering the _how_
question, not all of them are indisputable. What seems certain is that any
mode of nonmonotonic reasoning worth exploring should be considered together
with its monotonic counterpart.121212This idea was systematically implemented
in [Makinson, 2005a], although only as an explanatory tool, and not as a
methodological guide. Following this observation, we introduce the notation
‘$\mathbf{Cn}$’ to indicate that the last operator satisfies the properties
(con-1)–(con-3). The seeming contradiction with the last paragraph is resolved
by establishing relations between the operators $\mathbf{Cn}$ and
$\mathbf{C}$.

##  2\. Logicality of (logical) consequences

First of all, we must explain the title of this section.

Even in the case of monotonic reasoning, if the answer to the _what_ question
can be explicated in the form of conditions (con-1)–(con-3), any possible
answer to the _how_ question is fraught with danger. Here are two views on the
issue. The first comes from the corner of constructive philosophy.

> A logical inference is a movement from certain propositions (the premises)
> to a further proposition (the conclusion). We still need to know, however,
> which of these movements are to be called logical. ([Lorenzen, 1987],
> section 4)

The second point of view sounds from the corner of proof theory.

> If we simply axiomatize or define the notion of logical consequence with the
> understanding that a logical consequence holds when this follows logically
> (!) from the axioms or the definitions, then one may rightly say whether
> anything really is achieved. ([Prawitz, 1974], section 2)

Such sentiments sometimes, though rarely, appeared in the camp of researchers
of nonmonotonic reasoning; in particular, in the second sentence of the
following quote (if you forgive the _what-how_ confusion in the first).

> $[\ldots]$ what is wanted is not a specific technical definition for a
> specific non-monotonic logic, but a general account of what consequence is
> supposed to mean—of what concept some specific technical definition is
> trying to capture. And one should expect such an account to explain why it
> is reasonable to call the concept by the name “consequence”. ([Stalnaker,
> 1993])

Apparently, D. Makinson [Makinson, 1994], section 2.2, was the first to
propose a clear criterion for the logicality of a nonmonotonic consequence;
namely, he formulated that for two operators $\mathbf{Cn}$ (monotonic) and
$\mathbf{C}$ (presumably nonmonotonic), the operator $\mathbf{C}$ is logical
(with respect to $\mathbf{Cn}$) if

$\mathbf{Cn}\mathbf{C}=\mathbf{C}=\mathbf{C}\mathbf{Cn}.$ (3)

It should be clarified that any standard on the logicality of nonmonotonic
reasoning is an answer to the _what_ question, not to the _how_ question. In
formulating such a standard, we want to prescribe the conditions of what we
expect it to be.

To facilitate the analysis of (3), we introduce two relations similar to (1).
So we define:

$X\vdash\alpha~{}\stackrel{{\scriptstyle\text{df}}}{{\Longleftrightarrow}}~{}\alpha\in\mathbf{Cn}(X),$
(4)

and

$X\mid\\!\sim\alpha~{}\stackrel{{\scriptstyle\text{df}}}{{\Longleftrightarrow}}~{}\alpha\in\mathbf{C}(X).$
(5)

We find that (3) is too strong. Let us consider first the inequality

$\mathbf{Cn}\mathbf{C}\leq\mathbf{C},$ (6)

and express it as follows:

$\\{\beta~{}:~{}X\mid\\!\sim\beta\\}\vdash\alpha~{}\Longrightarrow~{}X\mid\\!\sim\alpha,$

or in a shorter form

$\mathbf{C}(X)\vdash\alpha~{}\Longrightarrow~{}X\mid\\!\sim\alpha.$

The last conditional does not look convincing; it is rather unjustifiably
specific.131313Although (6), and even (3), is fulfilled in any epsilon
inference probabilistic operator (cf. [Makinson, 1994], observation 3.5.2),
Makinson, however, makes the following remark.  It must be said, however, that
there is a rather unsatisfying gap between the initial intuitions behind the
epsilon approach and their technical formulation [$\dots$].

Turning to the inequality

$\mathbf{C}\leq\mathbf{C}\mathbf{Cn},$

the following generalization seems to be more flexible:

$(X\mid\\!\sim\alpha~{}\text{and}~{}Y\subseteq\mathbf{Cn}(X))~{}\Longrightarrow~{}X\cup
Y\mid\\!\sim\alpha.$ (7)

Besides, (7) can be seen as an attempt to make the relation $|\\!\\!\\!\sim$
somewhat monotonic. As we will show in Section LABEL:section:M-r-consequence,
in some interesting models, (7) does not hold. Nonetheless, as will be seen in
the sequel, limited monotonicity can be achieved, although on a different
path.

We call a nonmonotonic relation $|$$\sim$ logical relative to a given
monotonic consequence relation $\vdash$ if it satisfies the conditions:

$X\vdash\alpha~{}\Longrightarrow~{}X\mid\\!\sim\alpha,$ (8)

and

$(X\mid\\!\sim\alpha~{}\text{and}~{}\alpha\vdash\beta)~{}\Longrightarrow~{}X\mid\\!\sim\beta.$
(9)

The conditions (8) and (9) can be formulated in terms of operators
$\mathbf{Cn}$ and $\mathbf{C}$ that are defined by (4) and (5), respectively,
as follows.

We say that an operator $\mathbf{C}$ is logical relative to $\mathbf{Cn}$ if,
and only if, the following conditions are satisfied:

  * (log-1)

$\mathbf{Cn}\leq\mathbf{C}$;

  * (log-2)

$\alpha\in\mathbf{C}(X)$ and $\beta\in\mathbf{Cn}(\alpha)$ imply
$\beta\in\mathbf{C}(X)$.

Continuing the discussion of the _what_ question in relation to nonmonotonic
reasoning, let us turn to the map of properties (con-1)–(con-9) presented in
the next proposition. However, first, we will divide these properties into
five (colored) groups: the _reflexivity property_ (con-1), the _closedness
property_ (con-3), the _monotonicity property_ (con-2), the _cumulativity
properties_ (con-5)–(con-7), and the _finitariness properties_ (con-4),
(con-8) and (con-9).

######  Proposition 2.1 (cf. [Citkin and Muravitsky, 2022], proposition
4.2.5).

The following implications hold:

  * i)

(con-1) and (con-6) imply (con-3);

  * ii)

(con-2) and (con-3) imply (con-5);

  * iii)

(con-5) implies (con-6);

  * iv)

(con-7) implies (con-6);

  * v)

(con-1) and (con-6) imply (con-7);

  * vi)

(con-1) and (con-7) imply (con-3);

  * vii)

(con-1) and (con-2) and (con-3) imply (con-7);

  * viii)

(con-9) implies (con-4);

  * ix)

(con-2) and (con-4) imply (con-9);

  * x)

(con-2) implies (con-8);

  * xi)

(con-4) and (con-8) imply (con-2).151515The implication (xi) was noted by
Makinson who wrote:  The compactness biconditional must [$\dots$] fail in any
nonmonotonic logic, and although the right-to-left half of it [$\ldots$] if
$A_{0}\mid\\!\sim x$ for some $A_{0}$ included in $A$, then $A\mid\\!\sim x$]
seems the more clearly inappropriate, even its converse is rather
dubious$\dots$ ([Makinson, 1994], p. 41)

As is clear from the last proposition, if we intend to challenge the
monotonicity property (con-2), and we do so, we must be aware that other
properties will be affected.

Note that (con-1) is always satisfied if the consequence operator in question
is logical. We also observe that the cumulative properties and the
finitariness properties do not interact. Further, in the presence of (con-1),
if (con-7) does not hold, it is problematic that (con-3) holds.

In the mode of nonmonotonic reasoning that we are about to propose (thereby
answering the _how_ question), the closedness and cumulative properties that
have been the focus of nonmonotonic research161616See e.g. [Makinson, 1989].
will fail. Therefore, we will be interested in the finitariness properties.
But even in this case, by Proposition 2.1-xi, we cannot have both (con-4) and
(con-8). As is expected, (con-4) will take precedence.

##  3\. Matrix (monotonic) consequence and two restricted matrix consequences

In this section, we show some ways of passing from monotonic consequences to
nonmonotonic consequences. We note that, although our starting point will be
language-independent, in rejection of (con-2) we are forced to deal with
specifics of the language involved. However, further, in showing which
properties of the list (con-1)-(con-9) are preserved, we again can deal with a
language-independent framework.

Fixing a language $\mathcal{L}$, we also fix the following notations:

  * •

$\mathcal{V}_{\mathcal{L}}$ is an infinite set of propositional variables;

  * •

$\mathcal{F}_{\mathcal{L}}$ is a set of logical constants, also known as
logical connectives, including $0$-ary logical constants, also known as
constants;

  * •

$\mathcal{A}_{\mathcal{L}}$ is the set of all atomic formulas, that is, the
set of all variables and constants;

  * •

the universal algebra
$\mathfrak{F}_{\mathcal{L}}:=\langle\mathbf{Fm}_{\mathcal{L}},\mathcal{F}_{\mathcal{L}}\rangle$
is called the $\mathcal{L}$-formula algebra or simply formula algebra when
$\mathcal{L}$ is unambiguous;

  * •

given a set $X\subseteq\mathbf{Fm}_{\mathcal{L}}$, $\mathcal{V}(X)$ is the set
of propositional variables occurring in all formulas from $X$.

Interpreting elements of $\mathcal{F}_{\mathcal{L}}$ as operations on a
nonempty set A, we obtain an algebra
$\mathbf{A}:=\langle\textsf{A},\mathcal{F}_{\mathcal{L}}\rangle$.
Subsequently, we also use the notation $|\mathbf{A}|:=\textsf{A}$.

A system $\mathbf{M}:=\langle\mathbf{A},D\rangle$, where
$D\subseteq|\mathbf{A}|$ is called a (logical) matrix.

Any homomorphism $v:\mathfrak{F}_{\mathcal{L}}\longrightarrow\mathbf{A}$ is
called a valuation in $\mathbf{A}$; if $\mathbf{A}$ is part of a matrix
$\mathbf{M}$, $v$ is also a valuation in $\mathbf{M}$. The result of the
application of $v$ to $\alpha$ is denoted by $v[\alpha]$; and for
$X\subseteq\mathbf{Fm}_{\mathcal{L}}$,

$v[X]:=\\{v[\alpha]~{}:~{}\alpha\in X\\}.$

We note that any mapping
$v:\mathcal{V}_{\mathcal{L}}\longrightarrow\mathbf{A}$ can be uniquely
extended to a homomorphism
$\hat{v}:\mathfrak{F}_{\mathcal{L}}\longrightarrow\mathbf{A}$. Because of
this, we often regard $v$ as a valuation in $\mathbf{A}$

###  3.1. Matrix consequence

In defining a monotonic consequence, in order to keep a language-independent
framework, we use one of Tarski’s modes of semantic consequence (Section 1).

Given a matrix $\mathbf{M}=\langle\mathbf{A},D\rangle$, a relation
$\models_{\mathbf{M}}\subseteq\mathcal{P}(\mathbf{Fm}_{\mathcal{L}})\times\mathbf{Fm}_{\mathcal{L}}$
is defined as follows:

$X\models_{\mathbf{M}}\alpha~{}\stackrel{{\scriptstyle\text{df}}}{{\Longleftrightarrow}}~{}\text{for
any valuation $v$, $v[X]\subseteq D$ implies $v[\alpha]\in D$}.$ (10)

The last relation can be generalized.

Let $\mathcal{M}$ be a nonempty set of matrices. Then we define:

$X\models_{\mathcal{M}}\alpha\stackrel{{\scriptstyle\text{df}}}{{\Longleftrightarrow}}~{}X\models_{\mathbf{M}}\alpha~{}\text{for
every $\mathbf{M}\in\mathcal{M}$}.$ (11)

We will use the notations $\mathbf{Cn}_{\mathbf{M}}$ and
$\mathbf{Cn}_{\mathcal{M}}$, thereby indicating that the former is the
operator corresponding to $\models_{\mathbf{M}}$ and the latter is the
operator corresponding to $\models_{\mathcal{M}}$, both in the light of
definition (2).171717It must be clear that if $\mathcal{M}=\\{\mathbf{M}\\}$,
$\models_{\mathbf{M}}$ and $\models_{\mathcal{M}}$ can be used
interchangeably. These notations are justified by the following proposition.

######  Proposition 3.1.

Any operator $\mathbf{Cn}_{\mathcal{M}}$ satisfies the conditions
(con-1)–(con-3) and (con-10).181818This statement goes back to [Łukasiewicz
and Tarski, 1930], theorem 3; For modern expositions, the reader is offered to
consult [Wójcicki, 1988], theorem 3.1.3, or [Dunn and Hardegree, 2001],
corollary 6.13.3, or [Citkin and Muravitsky, 2022], proposition 4.3.11.

We note that the next equality is an immediate consequence of (11).

$\mathbf{Cn}_{\mathcal{M}}(X)=\bigcap_{\mathbf{M}\in\mathcal{M}}\mathbf{Cn}_{\mathbf{M}}(X).$
(12)

###  3.2. M-r-consequence and $\mathcal{M}$-r-consequence

Let $\mathcal{V}\subseteq\mathcal{V}_{\mathcal{L}}$ and $v$ be a valuation in
an algebra $\mathbf{A}$, regarded as a mapping on $\mathcal{V}_{\mathcal{L}}$.
We call $v\upharpoonright\mathcal{V}$ a restriction (of $v$) to $\mathcal{V}$.
Given $\mathcal{V}\subseteq\mathcal{V}_{\mathcal{L}}$, we denote by
$v_{\mathcal{V}}$ a _restricted valuation_ with $\mathbf{dom}(v)=\mathcal{V}$,
where $\mathbf{dom}(v)$ is the domain of $v$.

Given two restrictions $v$ and $w$ in the same algebra, we write $v\leq w$ (or
$w\geq v$) and say that $w$ is an extension of $v$ if
$\mathbf{dom}(v)\subseteq\mathbf{dom}(w)$.

In light of these notions, the relation $\models_{\mathbf{M}}$ (see (10)) can
be reformulated as follows:

$\begin{array}[]{rcl}X\models_{\mathbf{M}}\alpha&\Longleftrightarrow&\text{for
any restriction $v$ with $\mathbf{dom}(v)=\mathcal{V}(X)$},\\\ &&\text{if
$v[X]\subseteq D$, then for any extension $w\geq v$}\\\ &&\text{with
$\mathcal{V}(X\cup\\{\alpha\\})\subseteq\mathbf{dom}(w)$, $w[\alpha]\in
D$}.\end{array}$

Changing the right-hand side of the last equivalence, we obtain the following
definition.191919The idea of this definition is due to D. Makinson, who used
it with a two-element Boolean matrix; cf. [Makinson, 2005b, Makinson, 2007].

$\begin{array}[]{rcl}X\models^{r}_{\mathbf{M}}\alpha&\stackrel{{\scriptstyle\text{df}}}{{\Longleftrightarrow}}&\text{for
any restriction $v$ with $\mathbf{dom}(v)=\mathcal{V}(X)$},\\\ &&\text{if
$v[X]\subseteq D$, then there is an extension $w\geq v$}\\\ &&\text{with
$\mathcal{V}(X\cup\\{\alpha\\})\subseteq\mathbf{dom}(w)$, such that
$w[\alpha]\in D$}.\end{array}$ (13)

Generalizing the last definition to any nonempty set $\mathcal{M}$ of
matrices, we obtain the following.

$X\models^{r}_{\mathcal{M}}\alpha~{}\Longleftrightarrow~{}X\models^{r}_{\mathbf{M}}\alpha~{}\text{for
every $\mathbf{M}\in\mathcal{M}$}.$ (14)

The last definitions define M-r-consequence and $\mathcal{M}$-r-consequence,
respectively; we call them collectively restricted matrix consequences.

We denote by $\mathbf{C}_{\mathbf{M}}^{r}$ and $\mathbf{C}_{\mathcal{M}}^{r}$
the operators corresponding to $\models^{r}_{\mathbf{M}}$ and
$\models^{r}_{\mathcal{M}}$, respectively, and then we observe:

$\mathbf{C}_{\mathcal{M}}^{r}(X)=\bigcap_{\mathbf{M}\in\mathcal{M}}\mathbf{C}_{\mathbf{M}}^{r}(X),$
(15)

which immediately follows from (13)

To prove the next proposition, we borrow a definition from [Muravitsky, 2021]
and then state a lemma.

######  Definition 3.1 (comp. [Muravitsky, 2021], definition 3.5).

Let $\mathbf{M}$ be a matrix. Given a nonempty set $X$ of formulas and a
valuation $v$ in $\mathbf{M}$, we say that $v$ adopts $X$ in $\mathbf{M}$, in
symbols $v\rightsquigarrow_{\mathbf{M}}X$, if there is an extension $w$ of $v$
with $\mathbf{dom}(w)=\mathcal{V}(X)\cup\mathbf{dom}(v)$ which validates all
formulas of $X$. If $X=\\{\alpha\\}$, we write
$v\rightsquigarrow_{\mathbf{M}}\alpha$.

######  Lemma 3.1 (cf. [Muravitsky, 2021], proposition 3.4).

Given $X\cup\\{\alpha\\}\subseteq\mathbf{Fm}_{\mathcal{L}}$, let us denote
$\mathcal{V}_{0}:=\mathcal{V}(X)\cap\mathcal{V}(\alpha)$. Then for any matrix
$\mathbf{M}$,

$X\models^{r}_{\mathbf{M}}\alpha~{}\Longleftrightarrow~{}(\text{for any
$v_{\mathcal{V}_{0}}$ in
$\mathbf{M}$},~{}v_{\mathcal{V}_{0}}\rightsquigarrow_{\mathbf{M}}X~{}\Longrightarrow~{}v_{\mathcal{V}_{0}}\rightsquigarrow_{\mathbf{M}}\alpha).$
(16)

######  Proposition 3.2.

Any operator $\mathbf{C}_{\mathcal{M}}^{r}$ is logical relative to
$\mathbf{Cn}_{\mathcal{M}}$.

###### Proof.

To prove the statement, we have to verify the conditions (log-1) and (log-2)
for $\mathbf{Cn}_{\mathcal{M}}$ and $\mathbf{C}_{\mathcal{M}}^{r}$. It must be
clear that for this, it suffices to verify (log-1) and (log-2) for
$\mathbf{Cn}_{\mathbf{M}}$ and $\mathbf{C}_{\mathbf{M}}^{r}$, where
$\mathbf{M}\in\mathcal{M}$.

Let us fix such $\mathbf{M}=\langle\mathbf{A},D\rangle$.

The condition (log-1), which is
$\mathbf{Cn}_{\mathbf{M}}\leq\mathbf{C}_{\mathbf{M}}^{r}$, obviously is true.

To prove (log-2), we use its form (9), that is,

$X\models^{r}_{\mathbf{M}}\alpha~{}\text{and}~{}\alpha\models_{\mathbf{M}}\beta~{}\text{imply}~{}X\models^{r}_{\mathbf{M}}\beta.$

Intending to apply (16), we denote
$\mathcal{V}_{0}=\mathcal{V}(X)\cap\mathcal{V}(\beta)$ and assume that
$v_{\mathcal{V}_{0}}\rightsquigarrow_{\mathbf{M}}X$.

This implies that there is a restricted valuation $v\geq v_{\mathcal{V}_{0}}$
such that $\mathbf{dom}(v)=\mathcal{V}(X)$ and $v[X]\subseteq D$. In virtue of
the first assumption, there is a valuation $v^{\prime}\geq v$ such that
$\mathbf{dom}(v^{\prime})=\mathcal{V}(X)\cup\mathcal{V}(\alpha)$ and
$v^{\prime}[X\cup\\{\alpha\\}]\subseteq D$. Now, given an arbitrary
$a\in|\mathbf{A}|$, we define a valuation $w$ as follows:

$w[q]:=\begin{cases}\begin{array}[]{cl}v^{\prime}[p]&\text{if
$p\in\mathcal{V}(X)\cup\mathcal{V}(\alpha)$}\\\
a&\text{otherwise}.\end{array}\end{cases}$

It is obvious that $w$ validates $\alpha$ and hence, in virtue of the second
assumption, $w$ also validates $\beta$. Since $v_{\mathcal{V}_{0}}\leq w$, we
conclude that $v_{\mathcal{V}_{0}}\rightsquigarrow_{\mathbf{M}}\beta$. It
remains to apply Lemma 3.1. ∎

The next two definitions were introduced in [Muravitsky, 2021].

######  Definition 3.2 (cf. [Muravitsky, 2021], definition 3.6).

A relation
$\vdash\subseteq\mathcal{P}(\mathbf{Fm}_{\mathcal{L}})\times\mathbf{Fm}_{\mathcal{L}}$,
or the operator $\mathbf{C}$ corresponding to this relation, is weakly
monotonic if for any set $X\cup
Y\cup\\{\alpha\\}\subseteq\mathbf{Fm}_{\mathcal{L}}$ with $X\subseteq Y$ and
$\mathcal{V}(X)\cap\mathcal{V}(\alpha)=\mathcal{V}(Y)\cap\mathcal{V}(\alpha)$,
$X\vdash\alpha$ implies $Y\vdash\alpha$.

######  Definition 3.3 (cf. [Muravitsky, 2021], definition 5.1).

A relation
$\vdash\subseteq\mathcal{P}(\mathbf{Fm}_{\mathcal{L}})\times\mathbf{Fm}_{\mathcal{L}}$,
or the operator $\mathbf{C}$ corresponding to this relation, is called very
strongly finitary if for any nonempty set $X$, if $X\vdash\alpha$, then there
is a nonempty $Y\Subset X$ with
$\mathcal{V}(Y)\cap\mathcal{V}(\alpha)=\mathcal{V}(X)\cap\mathcal{V}(\alpha)$
such that $Y\vdash\alpha$.

The following properties of $\mathcal{M}$-r-consequence were established
earlier.

######  Proposition 3.3 ([Muravitsky, 2021], corollary 3.1).

Any $\mathcal{M}$-r-consequence is weakly monotonic.

######  Proposition 3.4 ([Muravitsky, 2021], proposition 5.1).

If $\mathcal{M}$ is a finite set of finite matrices, then
$\mathcal{M}$-r-consequence is very strongly finitary.

In general, an operator $\mathbf{C}_{\mathbf{M}}^{r}$ can fail (con-2), and
the corresponding relation $\models^{r}_{\mathbf{M}}$ fails the transitivity
property:

$X\models^{r}_{\mathbf{M}}\beta,~{}\text{for every $\beta\in Y$, and $Y\cup
Z\vdash_{\mathbf{C}}\alpha$ imply $X\cup Z\models^{r}_{\mathbf{M}}\alpha$},$

which is equivalent to the property:

$Y\subseteq\mathbf{C}_{\mathbf{M}}^{r}(X)~{}\text{implies}~{}\mathbf{C}_{\mathbf{M}}^{r}(Y\cup
Z)\subseteq\mathbf{C}_{\mathbf{M}}^{r}(X\cup Z).$ (16)

######  Example 3.1.

Let $\mathbf{A}=\langle\textsf{A},\land,\lor,\neg,\bm{1}\rangle$ be a
nontrivial Boolean algebra with a unit $\bm{1}$, and let
$\mathbf{M}=\langle\mathbf{A},\\{\bm{1}\\}\rangle$. Then, although
$p\models^{r}_{\mathbf{M}}\neg q$, we observe that
$p,q\not\models^{r}_{\mathbf{M}}\neg q$; and although
$p\models^{r}_{\mathbf{M}}q$ and $q\models^{r}_{\mathbf{M}}\neg p$, we observe
that $p\not\models^{r}_{\mathbf{M}}\neg p$.

We note that (16) is a generalized form of (con-5). The property (con-6) also
fails. Indeed, consider inclusion $\\{p\\}\subseteq\\{p,q\\}$. Although
$\\{p,q\\}\subseteq\mathbf{C}_{\mathbf{M}}^{r}(p)$, $\neg
q\in\mathbf{C}_{\mathbf{M}}^{r}(p)\setminus\mathbf{C}_{\mathbf{M}}^{r}(\\{p,q\\})$.

In virtue of Proposition 2.1-iv, (con-7) also fails.

Thus all cumulative properties can fail for some operators
$\mathbf{C}_{\mathbf{M}}^{r}$, even when the matrix $\mathbf{M}$ is finite,
while for all $\mathbf{M}$, $\mathbf{C}_{\mathbf{M}}^{r}$ is weakly monotonic,
and for all finite $\mathbf{M}$, $\mathbf{C}_{\mathbf{M}}^{r}$ is very
strongly finitary.

###  3.3. M-r$\ast$-consequence and $\mathcal{M}$-r$\ast$-consequence

Although restricted matrix consequence exhibits interesting properties, there
is a problem with it.

######  Example 3.2.

Let $\mathbf{A}=\langle\textsf{A},\rightarrow,\bm{1}\rangle$ be a nontrivial
implicative algebra; cf. [Rasiowa, 1974], chapter II. And let
$\mathbf{M}=\langle\mathbf{A},\\{\bm{1}\\}\rangle$. Then
$p\models^{r}_{\mathbf{M}}q$ but $p,q\rightarrow
q\not\models^{r}_{\mathbf{M}}q$, although $\models_{\mathbf{M}}q\rightarrow q$
and $p,\alpha\models^{r}_{\mathbf{M}}q$ for any formula $\alpha$ which does
not contain $q$.

This observation leads to the following consideration.

######  Definition 3.4.

Let $\mathbf{M}$ be a matrix and $\alpha$ be a formula. Given
$p\in\mathcal{V}(\alpha)$, $p$ is $\mathbf{M}$-essential in $($or of$)$
$\alpha$ if there are valuations $v$ and $w$ in $\mathbf{M}$ such that
$v[\alpha]\neq w[\alpha]$, although for any
$q\in\mathcal{V}(\alpha)\setminus\\{p\\}$, $v[q]=w[q]$; otherwise $p$ is
$\mathbf{M}$-inessential. Given a family $\mathcal{M}$ of matrices, $p$ is
$\mathcal{M}$-essential (in $\alpha$) if it is $\mathbf{M}$-essential for some
$\mathbf{M}\in\mathcal{M}$; otherwise $p$ is $\mathcal{M}$-inessential.

Given a matrix $\mathbf{M}$ and formula $\alpha$, we denote by
$\mathcal{V}^{\ast}_{\mathbf{\mathbf{M}}}(\alpha)$ the set of all
$\mathbf{M}$-essential variables of $\alpha$; and denote:

$\mathcal{V}^{\ast}_{\mathbf{\mathbf{M}}}(X):=\bigcup_{\alpha\in
X}\mathcal{V}^{\ast}_{\mathbf{\mathbf{M}}}(\alpha).$

We state the following two proposition that can be easily verified.

######  Proposition 3.5.

Let $\mathbf{M}$ be a matrix. Given $p\in\mathcal{V}(\alpha)$, if $p$ is
$\mathbf{M}$-inessential, then $v[\alpha]=v[\sigma(\alpha)]$ for any valuation
$v$ in $\mathbf{M}$ and any substitution $\sigma$ satisfying the condition:
$\sigma(q)=q$ whenever $q\neq p$.

_Proof_ is obvious.

######  Proposition 3.6.

Let $\alpha$ be a formula and $v$ and $v^{\prime}$ be valuations in a matrix
$\mathbf{M}$ such that
$v\upharpoonright\mathcal{V}^{\ast}_{\mathbf{\mathbf{M}}}(\alpha)=v^{\prime}\upharpoonright\mathcal{V}^{\ast}_{\mathbf{\mathbf{M}}}(\alpha)$.
Then $v[\alpha]=v^{\prime}[\alpha]$.

_Proof_ can be carried out without much efforts by induction on the
construction of $\alpha$.

These two last propositions will be used without reference.

The concept of an inessential variable induces an additional concept.

######  Definition 3.5.

Let $\mathbf{M}$ be a matrix and $a\in|\mathbf{M}|$. We add a new constant
$\mathbf{c}_{a}$ to the language $\mathcal{L}$ along with the agreement that
each valuation $v$ in $\mathbf{M}$, being extended to include $\mathbf{c}_{a}$
in its domain, satisfies the condition: $v[\mathbf{c}_{a}]=a$. Now, Given an
$\mathcal{L}$-formula $\alpha$, any formula that is obtained from the former
by replacing all $\mathbf{M}$-inessential variables of $\alpha$ by
$\mathbf{c}_{a}$ is called a $\mathbf{c}_{a}$-instance of $\alpha$. If
$\alpha$ does not contain $\mathbf{M}$-inessential variables, a
$\mathbf{c}_{a}$-instance of $\alpha$ is coincident with $\alpha$.

######  Proposition 3.7.

Let $\mathbf{M}$ be a matrix, $a,b\in|\mathbf{M}|$ and $v$ be an arbitrary
valuation in $\mathbf{M}$ extended by the conditions $v[\mathbf{c}_{a}]=a$ and
$v[\mathbf{c}_{b}]=b$. If $\alpha^{\ast}$ and $\alpha^{\ast\ast}$ are
$\mathbf{c}_{a}$-instance and $\mathbf{c}_{b}$-instance of $\alpha$,
respectively, then $v[\alpha]=v[\alpha^{\ast}]=v[\alpha^{\ast\ast}]$.

_Proof_ can be carried out by induction on the complexity of $\alpha$.

The last proposition induces the following definition.

######  Definition 3.6 ($\mathbf{c}$-instance).

Let $\mathbf{M}$ be a matrix. An arbitrary fixed $\mathbf{c}_{a}$-instance of
a formula $\alpha$ is called simply a $\mathbf{c}_{\mathbf{M}}$-instance of
$\alpha$ is denoted by $\alpha_{\mathbf{c}_{\mathbf{M}}}$. If
$X\subseteq\mathbf{Fm}_{\mathcal{L}}$, we denote:

$X_{\mathbf{c}_{\mathbf{M}}}:=\\{\alpha_{\mathbf{c}_{\mathbf{M}}}~{}:~{}\alpha\in
X\\}.$

The next equality and the following conditional will be used throughout
without reference.

$\mathcal{V}(X_{\mathbf{c}_{\mathbf{M}}})=\mathcal{V}_{\mathbf{M}}^{\ast}(X)~{}\text{for
every matrix $\mathbf{M}$}.$ $Y\subseteq
X~{}\Longrightarrow~{}Y_{\mathbf{c}_{\mathbf{M}}}\subseteq
X_{\mathbf{c}_{\mathbf{M}}}~{}\text{for every matrix $\mathbf{M}$}.$

######  Definition 3.7.

Let $\mathbf{M}=\langle\mathbf{A},D\rangle$ be a matrix. A formula $\alpha$ is
called an $\mathbf{M}$-constant if there is an element $a\in|\mathbf{A}|$ such
that for any valuation $v$ in $\mathbf{M}$, $v[\alpha]=a$. Given a family
$\mathcal{M}$ of matrices, $\alpha$ is an $\mathcal{M}$-constant if it is an
$\mathbf{M}$-constant for each $\mathbf{M}\in\mathcal{M}$.

######  Proposition 3.8.

Let $\mathcal{M}$ be a family of matrices and $\alpha$ be a formula. If all
variables of $\alpha$ are $\mathcal{M}$-inessential, then $\alpha$ is an
$\mathcal{M}$-constant.

###### Proof.

Suppose $\mathbf{M}$ is an arbitrary matrix from $\mathcal{M}$ and $v$ is a
valuation in $\mathbf{M}$. We denote: $a:=v[\alpha]$. Since each variable of
$\alpha$ is $\mathbf{M}$-inessential, it is not difficult to conclude that for
any valuation $v^{\prime}$ in $\mathbf{M}$, $v^{\prime}[\alpha]=a$. Therefore,
$\alpha$ is $\mathbf{M}$-constant. ∎

Refining definition (13), we arrive at the main definition of this subsection.

Let $\mathbf{M}=\langle\mathbf{A},D\rangle$ and
$X\cup\\{\alpha\\}\subseteq\mathbf{Fm}_{\mathcal{L}}$.

$\begin{array}[]{rl}X\models^{r\ast}_{\mathbf{M}}\alpha~{}\stackrel{{\scriptstyle\text{df}}}{{\Longleftrightarrow}}&\text{for
any restricted $v$ in $\mathbf{M}$ with
$\mathbf{dom}(v)=\mathcal{V}_{\mathbf{M}}^{\ast}(X)$},\\\ &\text{there is an
extension $w\geq v$ with
$\mathcal{V}(X\cup\\{\alpha\\})\subseteq\mathbf{dom}(w)$}\\\ &\text{such that
if $v[X_{\mathbf{c}_{\mathbf{M}}}]\subseteq D$, then $w[\alpha]\in
D$}.\end{array}$ (17)

We note:

$\varnothing\models^{r\ast}_{\mathbf{M}}\alpha~{}\Longleftrightarrow~{}\varnothing\models^{r}_{\mathbf{M}}\alpha.$
(18)

Similarly to (14), we define the relation of $\mathcal{M}$-r∗-consequence.

$X\models^{r\ast}_{\mathcal{M}}\alpha~{}\stackrel{{\scriptstyle\text{df}}}{{\Longleftrightarrow}}~{}X\models^{r\ast}_{\mathbf{M}}\alpha~{}\text{for
every $\mathbf{M}\in\mathcal{M}$}.$ (19)

The corresponding operators are denoted by $\mathbf{C}_{\mathbf{M}}^{r\ast}$
and $\mathbf{C}_{\mathcal{M}}^{r\ast}$. Similarly to (15), we have:

$\mathbf{C}_{\mathcal{M}}^{r\ast}(X)=\bigcap_{\mathbf{M}\in\mathcal{M}}\mathbf{C}_{\mathbf{M}}^{r\ast}(X).$
(20)

First, we focus on M-r∗-consequence. From definition (17), there follows the
following equivalence.

$X\models^{r\ast}_{\mathbf{M}}\alpha~{}\Longleftrightarrow~{}X_{\mathbf{c}_{\mathbf{M}}}\models^{r}_{\mathbf{M}}\alpha.$
(21)

_Proof_ is obvious.

We note that (21) could be taken as a definition of relation
$\models^{r\ast}_{\mathbf{M}}$.

######  Proposition 3.9.

For any matrix $\mathbf{M}=\langle\mathbf{A},D\rangle$ and a set
$X\cup\\{\alpha\\}\subseteq\mathbf{Fm}_{\mathcal{L}}$,

$X\models^{r}_{\mathbf{M}}\alpha~{}\Longrightarrow~{}X\models^{r\ast}_{\mathbf{M}}\alpha.$

The converse, in general, does not hold.

###### Proof.

Assume that a valuation $v$ in $\mathbf{M}$ with
$\mathbf{dom}(v)=\mathcal{V}^{\ast}_{\mathbf{\mathbf{M}}}(X)$ satisfies the
condition that $v[X_{\mathbf{c}_{\mathbf{M}}}]\subseteq D$. We recall that
$v[\mathbf{c}_{\mathbf{M}}]=a$, where $a\in|\mathbf{A}|$. Then, we extend $v$
to a restriction $v^{\prime}$ with $\mathbf{dom}(v^{\prime})=\mathcal{V}(X)$
in such a way that $v^{\prime}[p]=a$ for any inessential variable $p$ of $X$.
It is clear that $v^{\prime}[X]\subseteq D$. This, by premise, implies that
there is an extension $v^{\prime\prime}$ with
$\mathbf{dom}(v^{\prime\prime})=X\cup\\{\alpha\\}$ such that
$v^{\prime\prime}[\alpha]\in D$. Thus, we obtain that
$X_{\mathbf{c}_{\mathbf{M}}}\models^{r}_{\mathbf{M}}\alpha$. It remains to
apply (21).

Example 3.2 shows that the converse can fail. ∎

######  Proposition 3.10.

Relation $\models^{r\ast}_{\mathbf{M}}$ is logical relative to
$\models_{\mathbf{M}}$.

###### Proof.

Indeed, property (8) follows from Proposition 3.2 and Proposition 3.9.

To prove property (9), we assume that $X\models^{r\ast}_{\mathbf{M}}\alpha$
and $\alpha\models_{\mathbf{M}}\beta$. In virtue of (21),
$X_{\mathbf{c}_{\mathbf{M}}}\models^{r}_{\mathbf{M}}\alpha$. According to
Proposition 3.2, $X_{\mathbf{c}_{\mathbf{M}}}\models^{r}_{\mathbf{M}}\beta$.
Then we apply (21). ∎

We should not expect that relation $\models^{r\ast}_{\mathbf{M}}$ will enjoy
the weak monotonicity property (Definition 3.2). However, we obtain the
following.

######  Proposition 3.11.

Given a matrix $\mathbf{M}$, for any set $X\cup
Y\cup\\{\alpha\\}\subseteq\mathbf{Fm}_{\mathcal{L}}$ with $X\subseteq Y$ and
$\mathcal{V}^{\ast}_{\mathbf{M}}(X)\cap\mathcal{V}(\alpha)=\mathcal{V}^{\ast}_{\mathbf{M}}(Y)\cap\mathcal{V}(\alpha)$,
$X\models^{r\ast}_{\mathbf{M}}\alpha$ implies
$Y\models^{r\ast}_{\mathbf{M}}\alpha$.

###### Proof.

Let $X\models^{r\ast}_{\mathbf{M}}\alpha$ and all other premises be fulfilled.
Then, in virtue of (21),
$X_{\mathbf{c}_{\mathbf{M}}}\models^{r}_{\mathbf{M}}\alpha$. Since
$\mathcal{V}(X_{\mathbf{c}_{\mathbf{M}}})\cap\mathcal{V}(\alpha)=\mathcal{V}(Y_{\mathbf{c}_{\mathbf{M}}})\cap\mathcal{V}(\alpha)$
and $X_{\mathbf{c}_{\mathbf{M}}}\subseteq Y_{\mathbf{c}_{\mathbf{M}}}$, we can
apply the property that $\models^{r}_{\mathbf{M}}$ is weakly monotonic
(Proposition 3.3) in order to conclude that
$Y_{\mathbf{c}_{\mathbf{M}}}\models^{r}_{\mathbf{M}}\alpha$, that is,
$Y\models^{r\ast}_{\mathbf{M}}\alpha$. ∎

As to finitariness, we obtain the following proposition.

######  Proposition 3.12.

Given a matrix $\mathbf{M}$, for any nonempty
$X\subseteq\mathbf{Fm}_{\mathcal{L}}$, if
$X\models^{r\ast}_{\mathbf{M}}\alpha$, then there exists a nonempty $Y\Subset
X$ such that
$\mathcal{V}^{\ast}_{\mathbf{M}}(Y)\cap\mathcal{V}(\alpha)=\mathcal{V}^{\ast}_{\mathbf{M}}(X)\cap\mathcal{V}(\alpha)$
and $Y\models^{r\ast}_{\mathbf{M}}\alpha$. Moreover, for any set $Z$ with
$Y\subseteq Z$ and
$\mathcal{V}^{\ast}_{\mathbf{M}}(Z)\cap\mathcal{V}(\alpha)=\mathcal{V}^{\ast}_{\mathbf{M}}(Y)\cap\mathcal{V}(\alpha)$,
$Z\models^{r\ast}_{\mathbf{M}}\alpha$.

###### Proof.

Suppose $X\models^{r\ast}_{\mathbf{M}}\alpha$, where $X\neq\varnothing$. By
(21), $X_{\mathbf{c}_{\mathbf{M}}}\models^{r}_{\mathbf{M}}\alpha$. In virtue
of Proposition 3.4, there is a nonempty $X_{0}\Subset
X_{\mathbf{c}_{\mathbf{M}}}$ such that
$\mathcal{V}(X_{0})\cap\mathcal{V}(\alpha)=\mathcal{V}(X_{\mathbf{c}_{\mathbf{M}}})\cap\mathcal{V}(\alpha)$
and $X_{0}\models^{r}_{\mathbf{M}}\alpha$.

It must be clear that for some nonempty $Y\Subset X$,
$Y_{\mathbf{c}_{\mathbf{M}}}=X_{0}$. This implies that
$\mathcal{V}(Y_{\mathbf{c}_{\mathbf{M}}})\cap\mathcal{V}(\alpha)=\mathcal{V}(X_{\mathbf{c}_{\mathbf{M}}})\cap\mathcal{V}(\alpha)$
and $Y_{\mathbf{c}_{\mathbf{M}}}\models^{r}_{\mathbf{M}}\alpha$. Applying
(21), we obtain that
$\mathcal{V}^{\ast}_{\mathbf{M}}(Y)\cap\mathcal{V}(\alpha)=\mathcal{V}^{\ast}_{\mathbf{M}}(X)\cap\mathcal{V}(\alpha)$.

The last part of the statement follows by Proposition 3.11. ∎

Now we will turn to $\mathcal{M}$-r$\ast$-consequence. First of all, we
observe the following.

######  Proposition 3.13.

Any $\mathcal{M}$-r$\ast$-consequence is logical relative to the corresponding
$\mathcal{M}$-consequence.

###### Proof.

To verify (log-1), we notice that, in virtue of Proposition 3.10, for every
$\mathbf{M}\in\mathcal{M}$,
$\mathbf{Cn}_{\mathbf{M}}\leq\mathbf{C}_{\mathbf{M}}^{r\ast}$. Then, in virtue
of (15) and (20),
$\mathbf{Cn}_{\mathcal{M}}\leq\mathbf{C}_{\mathcal{M}}^{r\ast}$.

To verify (log-2), assume that $\alpha\in\mathbf{C}_{\mathcal{M}}^{r\ast}(X)$
and $\beta\in\mathbf{Cn}_{\mathcal{M}}(\alpha)$, that is,
$X\models^{r\ast}_{\mathcal{M}}\alpha$ and $\alpha\models_{\mathcal{M}}\beta$.
In virtue of Proposition 3.10, for every $\mathbf{M}\in\mathcal{M}$,
$X\models^{r\ast}_{\mathbf{M}}\beta$, that is,
$\beta\in\mathbf{C}_{\mathcal{M}}^{r\ast}(X)$. ∎

######  Proposition 3.14.

If $X\models^{r\ast}_{\mathcal{M}}\alpha$, then for any set $Y$ with
$X\subseteq Y$ and
$\mathcal{V}^{\ast}_{\mathbf{\mathbf{M}}}(Y)\cap\mathcal{V}(\alpha)=\mathcal{V}^{\ast}_{\mathbf{\mathbf{M}}}(X)\cap\mathcal{V}(\alpha)$
for every $\mathbf{M}\in\mathcal{M}$, $Y\models^{r\ast}_{\mathcal{M}}\alpha$.

Proof immediately follows from Proposition 3.11.

######  Proposition 3.15.

Let $\mathcal{M}$ be a finite family of finite matrices. For any nonempty
$X\subseteq\mathbf{Fm}_{\mathcal{L}}$, if
$X\models^{r\ast}_{\mathcal{M}}\alpha$, then there exists a nonempty $Y\Subset
X$ such that $Y\models^{r\ast}_{\mathcal{M}}\alpha$ and
$\mathcal{V}^{\ast}_{\mathbf{\mathbf{M}}}(Y)\cap\mathcal{V}(\alpha)=\mathcal{V}^{\ast}_{\mathbf{\mathbf{M}}}(X)\cap\mathcal{V}(\alpha)$
for every $\mathbf{M}\in\mathcal{M}$. Moreover, for any set $Z$ with
$Y\subseteq Z$ and
$\mathcal{V}^{\ast}_{\mathbf{\mathbf{M}}}(Z)\cap\mathcal{V}(\alpha)=\mathcal{V}^{\ast}_{\mathbf{\mathbf{M}}}(Y)\cap\mathcal{V}(\alpha)$
for every $\mathbf{M}\in\mathcal{M}$, $Z\models^{r\ast}_{\mathcal{M}}\alpha$.

###### Proof.

Suppose $\mathcal{M}=\\{\mathbf{M}_{1},\ldots,\mathbf{M}_{n}\\}$ and
$X\models^{r\ast}_{\mathcal{M}}\alpha$, that is,

$X\models^{r\ast}_{\mathbf{M}_{i}}\alpha~{}\text{for each
$i\in\\{1,\ldots,n\\}$},$

where $X\neq\varnothing$.

According to Proposition 3.12, for each $i$, there is a nonempty $Y_{i}\Subset
X$ such that
$\mathcal{V}^{\ast}_{\mathbf{M}_{i}}(Y_{i})\cap\\{\alpha\\}=\mathcal{V}^{\ast}_{\mathbf{M}_{i}}(X)\cap\mathcal{V}(\alpha)$.
We also note that
$\mathcal{V}^{\ast}_{\mathbf{M}_{i}}(Y_{j})\cap\\{\alpha\\}\subseteq\mathcal{V}^{\ast}_{\mathbf{M}_{i}}(X)\cap\mathcal{V}(\alpha)$
for any $i,j\in\\{1,\ldots,n\\}$.

Next we denote:

$Y:=Y_{1}\cup\ldots\cup Y_{n}.$

We observe that:

  * •

$Y\neq\varnothing$ and $Y\Subset X$;

  * •

$\mathcal{V}^{\ast}_{\mathbf{M}_{i}}(Y)\cap\\{\alpha\\}=\mathcal{V}^{\ast}_{\mathbf{M}_{i}}(X)\cap\mathcal{V}(\alpha)$
for each $i\in\\{1,\ldots,n\\}$.

Applying Proposition 3.11, we conclude that

$Y\models^{r\ast}_{\mathbf{M}_{i}}\alpha~{}\text{for each
$i\in\\{1,\ldots,n\\}$,}$

that is, $Y\models^{r\ast}_{\mathcal{M}}\alpha$.

Now if for each $i\in\\{1,\ldots,n\\}$, for some $Z$ with $Y\subseteq Z$ and
$\mathcal{V}^{\ast}_{\mathbf{\mathbf{M}_{i}}}(Z)\cap\mathcal{V}(\alpha)=\mathcal{V}^{\ast}_{\mathbf{\mathbf{M}_{i}}}(Y)\cap\mathcal{V}(\alpha)$,
then, according to Proposition 3.14, $Z\models^{r\ast}_{\mathbf{M}_{i}}\alpha$
for each $\mathbf{M}_{i}\in\mathcal{M}$, that is,
$Z\models^{r\ast}_{\mathcal{M}}\alpha$. ∎

######  Corollary 3.1.

Let $\mathcal{M}$ be a finite family of finite matrices. Then
$\mathcal{M}$-r$\ast$-consequence is strongly finitary.

##  4\. Substitution

According to Proposition 3.1, any $\mathbf{Cn}_{\mathbf{M}}$ operator is
structural. In terms of consequence relations, this means that, given a matrix
$\mathbf{M}$, for any substitution $\sigma$,

$X\models_{\mathbf{M}}\alpha~{}\Longrightarrow~{}\sigma(X)\models_{\mathbf{M}}\sigma(\alpha).$

The structurality property can hardly be expected for
$\models^{r}_{\mathbf{M}}$ or $\models^{r\ast}_{\mathbf{M}}$ relations.
However, the following rule can be considered for these relations, although
not without some restriction.

$X\models^{r}_{\mathbf{M}}\sigma(\alpha)~{}\Longrightarrow~{}X\models^{r}_{\mathbf{M}}\alpha,$
(22)

providing that $\mathcal{V}(X)\cap\mathcal{V}(\alpha)=\varnothing$. (In
Section 5.2 this property is formulated as the rule of reverse substitution.)

It is obvious that (22) is true. Indeed, the restriction
‘$\mathcal{V}(X)\cap\mathcal{V}(\alpha)=\varnothing$’ allows us to reduce (22)
to the implication: for any valuation $v$ in $\mathbf{M}$, if $v$ validates
$\sigma(\alpha)$, then $v\circ\sigma$ validates $\alpha$. On the other hand,
if we drop the restriction, we face with the problem that when a valuation $v$
validates $X$ and its extension $w$ validates $\sigma(\alpha)$, the valuation
$w\circ\sigma$, which validates $\alpha$, is not necessarily an extension of
$v$.

It is also true that (22) can be extended to the following:

$X\models^{r}_{\mathcal{M}}\sigma(\alpha)~{}\Longrightarrow~{}X\models^{r}_{\mathcal{M}}\alpha$

and

$X\models^{r\ast}_{\mathcal{M}}\sigma(\alpha)~{}\Longrightarrow~{}X\models^{r\ast}_{\mathcal{M}}\alpha,$

providing that $\mathcal{V}(X)\cap\mathcal{V}(\alpha)=\varnothing$.

The last implication can obviously be reduced to the case when $\mathcal{M}$
consists of a single matrix:

$X\models^{r\ast}_{\mathbf{M}}\sigma(\alpha)~{}\Longrightarrow~{}X\models^{r\ast}_{\mathbf{M}}\alpha,$

where $\mathcal{V}(X)\cap\mathcal{V}(\alpha)=\varnothing$.

Indeed, according to (21), the last implication is equivalent to the
following:

$X_{\mathbf{c}_{\mathbf{M}}}\models^{r}_{\mathbf{M}}\sigma(\alpha)~{}\Longleftrightarrow~{}X_{\mathbf{c}_{\mathbf{M}}}\models^{r}_{\mathbf{M}}\alpha.$

It remains to notice that
$\mathcal{V}(X_{\mathbf{c}_{\mathbf{M}}})\cap\mathcal{V}(\alpha)=\varnothing$
whenever $\mathcal{V}(X)\cap\mathcal{V}(\alpha)=\varnothing$.

##  5\. Logical friendliness relation and beyond

So far, the specifics of objective language have not been used in our
discussion to achieve positive results, except that the language was
sentential. But to get negative results, we had to turn to specific languages.
However, specifying an objective language can also open up more possibilities
for defining nonmonotonic consequence relations.

For the rest of this paper, we will use the language of Example 3.1, that is a
language with the logical connectives $\land$, $\lor$, $\rightarrow$, $\neg$
and the constant $\top$. We denote this formal language by
$\mathcal{L}^{\circ}$. Unspecified formulas of $\mathcal{L}^{\circ}$ we denote
by letters $A$, $B$, etc., and sets of such formulas by letters $\Gamma$,
$\Delta$, etc. We continue using the same notation for substitutions and
valuations. Since all our matrices will be Heyting algebras with a greatest
element $\bm{1}$, for each valuation $v$, we require the condition:
$v[\top]=\bm{1}$.

###  5.1. Logical friendliness

Our interest to nonmonotonic relation was inspired by [Makinson, 2005b] and
[Makinson, 2007], where the relation of _logical friendliness_ was
introduced.202020Makinson has proved that logical friendliness is strongly
finitary with some additional property for monotonicity, which is however
weaker than the one formulated in Proposition 5.1 of [Muravitsky, 2021]. To
define logical friendliness, we use the matrix $\mathbf{B}_{2}$ which is a
two-element Boolean algebra with the filter $\\{\bm{1}\\}$; namely

$\Gamma\mid\\!\sim_{F}A~{}\stackrel{{\scriptstyle\text{df}}}{{\Longleftrightarrow}}~{}\Gamma\models_{\mathbf{B}_{2}}^{r}A.$
(23)

Further, we define:

$\Gamma\mid\\!\sim_{F^{\ast}}A~{}\stackrel{{\scriptstyle\text{df}}}{{\Longleftrightarrow}}~{}\Gamma\models_{\mathbf{B}_{2}}^{r\ast}A.$
(24)

We denote the operators corresponding to $\mid\\!\sim_{F}$ and
$\mid\\!\sim_{F^{\ast}}$ by $\mathbf{C}_{F}$ and $\mathbf{C}_{F^{\ast}}$,
respectively.

In view of Example 3.1 and Example 3.2, we observe:

$\mathbf{Cn}_{\mathbf{B}_{2}}<\mathbf{C}_{F}<\mathbf{C}_{F^{\ast}}.$ (25)

The following proposition later will play the role of a navigator, but it is
convenient to discuss this property here. It reads that for defining logical
friendliness, we can use any nontrivial Boolean algebra.

######  Proposition 5.1.

Let $\mathbf{B}$ be a nontrivial Boolean algebra that is in combination with a
logical filter $\\{\bm{1}\\}$ forms a logical matrix $\mathbf{B}$. Then

$\Gamma\models_{\mathbf{B}}^{r}A~{}\Longleftrightarrow~{}\Gamma\models_{\mathbf{B}_{2}}^{r}A.$
(26)

###### Proof.

We recall that $\mathbf{B}$ (understood as an algebra, not matrix) is a
subdirect product $\mathbf{B}_{2}^{I}$ (the Cartesian product of copies of
$\mathbf{B}_{2}$), where $\mathbf{B}_{2}$ is also regarded here as an algebra.
This means that $\mathbf{B}$ is embedded to $\mathbf{B}_{2}^{I}$ so that each
projection, $g_{i}$, is an epimorphism. We also recall that each valuation in
an algebra is a homomorphism from the corresponding formula algebra to the
former.

Now assume that $\Gamma\models_{\mathbf{B}}^{r}A$ and consider a valuation $v$
in $\mathbf{B}_{2}$ such that $v[\Gamma]\subseteq\\{\bm{1}\\}$ and
$\mathbf{dom}(v)=\mathcal{V}(\Gamma)$.

We define a valuation $\hat{v}$ in $\mathbf{B}$ as follows.

$\hat{v}[p]:=\begin{cases}\begin{array}[]{cl}\bm{0}&\text{if
$v[p]=\bm{0}$},\\\ \bm{1}&\text{if $v[p]=\bm{1}$},~{}\text{for every
$p\in\mathcal{V}(\Gamma)$}.\end{array}\end{cases}$

It is clear that $\hat{v}[\Gamma]\subseteq\\{(\bm{1})_{I}\\}$. Therefore, by
premise, there is an extension $w\geq\hat{v}$ such that
$w[\Gamma\cup\\{\alpha\\}]\subseteq\\{(\bm{1})_{I}\\}$. Now,
$v^{\ast}:=g_{i}\circ w$ is a valuation in $\mathbf{B}_{2}$ such that
$v^{\ast}\geq v$ and $v^{\ast}[\Gamma\cup\\{A\\}]\subseteq\\{\bm{1}\\}$.

Next assume that $\Gamma\models_{\mathbf{B}_{2}}A$ and that a valuation $w$ in
$\mathbf{B}$ validates $\Gamma$, that is,
$w[\Gamma]\subseteq\\{(\bm{1})_{I}\\}$. Then, obviously, valuation
$v:=g_{i}\circ w$ validates $\Gamma$ in $\mathbf{B}_{2}$, that is,
$v[\Gamma]\subseteq\\{\bm{1}\\}$. Then there is an extension $v^{\prime}\geq
v$ such that $v^{\prime}[\Gamma\cup\\{A\\}]\subseteq\\{\bm{1}\\}$. Now we
define a valuation $w^{\prime}$ in $\mathbf{B}$ (regarded as a subalgebra of
$\mathbf{B}_{2}^{I}$) according to the rule:
$(w^{\prime}[p])_{i}:=v^{\prime}[p]$. It is clear that $w^{\prime}\geq w$ and
$w^{\prime}[\Gamma\cup\\{A\\}]\subseteq\\{(\bm{1})_{I}\\}$. ∎

###  5.2. Beyond logical friendliness

We aim to make logical friendliness the starting point for getting more
logical nonmonotonic consequences. This time we choose the method of proof
theory.

Let Int and Cl denote the classes of intuitionistic and classical
propositional tautologies, respectively. An _intermediate logic_ is a set $L$
of formulas, satisfying the following conditions: $\textsf{Int}\subseteq
L\subseteq\textsf{Cl}$, $L$ is closed under uniform substitution and under
modus ponens. The class of all intermediate logics is denoted by
$\mathbf{ExtInt}$.

Let $L\in\mathbf{ExtInt}$. A finite sequence

$A_{1},A_{2},\ldots,A_{n}$

is called an $L$-_derivation from a set_ $\Gamma$ if for each $A_{i}$, either
$A_{i}\in L$ or $A_{i}\in\Gamma$ or $A_{i}$ can be obtained by modus ponens
from two preceding formulas of the sequence. In addition, we say that the
sequence is an $L$-derivation of the last formula of the sequence, that is, of
$A_{n}$.

Next, we define:

$\Gamma\vdash_{L}A~{}\stackrel{{\scriptstyle\text{df}}}{{\Longleftrightarrow}}~{}\text{there
is an $L$-derivation of $A$ from $\Gamma$}.$

Each relation $\vdash_{L}$ obviously is a monotonic consequence relations. We
denote the corresponding consequence operator by $\mathbf{Cn}_{L}$.

We call a set $\Gamma$ a $\mathbf{Cn}_{L}$-_theory_ if
$\mathbf{Cn}_{L}(\Gamma)=\Gamma$. (This concept will be employed later.)

We call any expression of the form $\Gamma\\!\Rightarrow\\!A$ a sequent.

Next, we provide the list of sequential $L$-axioms (or sL-_axioms_ for short)
and that of sequential $L$-rules (or sL-rules for short).

  * sL-Axioms:
  * axiom 1:

$\\!\Rightarrow\\!p$, for any $p\in\mathcal{V}_{\mathcal{L}^{\circ}}$;

  * axiom 2:

$\Gamma\\!\Rightarrow\\!\top$;

  * axiom 3:

$\Gamma\\!\Rightarrow\\!\bigwedge\Delta$ if $\Delta\Subset\Gamma$
($\bigwedge\Delta:=\top$ if $\Delta=\varnothing$);

  * axiom 4:

$\Gamma\\!\Rightarrow\\!A$ whenever $\Gamma\vdash_{L}A$.

  * sL-Rules:
  * rule 1:

$\dfrac{\\!\Rightarrow\\!A}{\Gamma\\!\Rightarrow\\!A}$, providing that
$\mathcal{V}(\Gamma)\cap\mathcal{V}(A)=\varnothing$;

  * rule 2:

$\dfrac{\Gamma,A\\!\Rightarrow\\!C~{}\text{and}~{}\Delta,B\\!\Rightarrow\\!C}{\Gamma\cup\Delta,A\lor
B\\!\Rightarrow\\!C}$, providing that
$\mathcal{V}(\Gamma\cup\\{A\\})=\mathcal{V}(\Delta\cup\\{B\\})$;

  * rule 3:

$\dfrac{\Gamma\\!\Rightarrow\\!\sigma(A)}{\Gamma\\!\Rightarrow\\!A},$
providing that $\mathcal{V}(\Gamma)\cap\mathcal{V}(A)=\varnothing$; (Reverse
substitution)

  * rule 4:

$\dfrac{\Gamma\\!\Rightarrow\\!A,~{}\text{and}~{}A\\!\Rightarrow\\!B}{\Gamma\\!\Rightarrow\\!B},$
providing that either $\mathcal{V}(\Gamma)\subseteq\mathcal{V}(A)$ or
$\mathcal{V}(\Gamma)\cap\mathcal{V}(B)\subseteq\mathcal{V}(A)\subseteq\mathcal{V}(\Gamma)$;
(Cut)

  * rule 5:

$\dfrac{\Gamma,A\\!\Rightarrow\\!B~{}\text{and}~{}C\\!\Rightarrow\\!A}{\Gamma,C\\!\Rightarrow\\!B},$
providing that $\mathcal{V}(C)\subseteq\mathcal{V}(\Gamma\cup\\{A\\})$;
(Deductive replacement in antecedent)

  * rule 6:

$\dfrac{\Gamma\\!\Rightarrow\\!A~{}\text{and}~{}A\\!\Rightarrow\\!B}{\Gamma\\!\Rightarrow\\!B}$.
(Deductive replacement in consequent)

A finite nonempty list

$\Gamma_{1}\\!\Rightarrow\\!A_{1},\ldots,\Gamma_{n}\\!\Rightarrow\\!A_{n}$

is called an sL-derivation (of the last sequent on the list, i.
$\Gamma_{n}\\!\Rightarrow\\!A_{n}$) if each $\Gamma_{k}\\!\Rightarrow\\!A_{k}$
is either a sL-axiom or can be obtained from preceding sequents by one of the
rules 1–6. We say that $\Gamma\\!\Rightarrow\\!A$ holds in s$L$-calculus if
there is an sL-derivation ending with $\Gamma\\!\Rightarrow\\!A$.

This leads to the following definition: for each $l\in\mathbf{ExtInt}$,

$\Gamma\Vdash_{L}A~{}\stackrel{{\scriptstyle\text{df}}}{{\Longleftrightarrow}}~{}\Gamma\\!\Rightarrow_{L}\\!A~{}\text{holds
in s$l$-calculus}.$ (27)

The motivation for formulating the s$L$-calculi is based on the following
observation.

$\Gamma\mid\\!\sim_{F}A~{}\Longleftrightarrow~{}\Gamma\Vdash_{\textsf{Cl}}A~{}(\text{cf.~{}\cite[cite]{[\@@bibref{}{muravitsky2009}{}{}]},
theorems 4.5 and 4.8}).$

Reformulating the last equivalence, we obtain completeness theorem for the
nonmonotonic consequence $\Vdash_{\textsf{Cl}}$:

$\Gamma\Vdash_{\textsf{Cl}}A~{}\Longleftrightarrow~{}\Gamma\models_{\mathbf{B}_{2}}^{r}A.$

This suggests that there may be similar completeness theorems for other
$\Vdash_{L}$ relations. We will come back to this later, but now we will
discuss what is interesting about the $\Vdash_{L}$ relations.

According to [Muravitsky, 2021], proposition 7.3, each relation $\Vdash_{L}$
is reflexive and nonmonotonic. Further, for any nonempty set $\Gamma$ and any
formula $A$ with $\mathcal{V}(\Gamma)\cap\mathcal{V}(A)=\varnothing$, if
$\Gamma\Vdash_{L}A$, then there is a nonempty set $\Delta\Subset\Gamma$ such
that $\Delta\Vdash_{L}A$. To this, we add the following.

######  Proposition 5.2.

Each $\Vdash_{L}$ is logical relative to $\vdash_{L}$.

###### Proof.

Indeed, suppose $\Gamma\vdash_{L}A$. Then, by axiom 4,
$\Gamma\\!\Rightarrow_{L}\\!A$ holds, that is, $\Gamma\Vdash_{L}A$.

Next, assume that $\Gamma\Vdash_{L}A$ and $A\vdash_{L}B$. The first assumption
implies that there is an s$L$-derivation of the s$L$-sequent
$\Gamma\\!\Rightarrow_{L}\\!A$. Then we apply rule 6. Thus there is an
s$L$-derivation of the s$L$-sequent $\Gamma\\!\Rightarrow_{L}\\!B$, that is,
$\Gamma\Vdash_{L}B$. ∎

Now let us return to the question of the possibility of the completeness
theorem for each relation $\Vdash_{L}$.

We recall that, in the terminology of [Citkin and Muravitsky, 2022], each
$\vdash_{L}$ is a _unital assertional abstract logic_.212121For definitions,
the reader is offered to consult [Citkin and Muravitsky, 2022], chapter 6.

For a fixed $L\in\mathbf{ExtInt}$, we denote by $\Sigma_{L}$ the set of all
$L$-theories. Then, given $D\in\Sigma_{L}$, we denote by $\mathbf{LT}_{L}[D]$
the _Lindenbaum-Tarski algebra relative to_ $D$. Further, each
$\mathbf{LT}_{L}[D]$ is a homomorphic image of
$\mathbf{LT}_{L}[\mathbf{Cn}_{L}(\varnothing)]$ and, hence, is a Heyting
algebra. Moreover, each such algebra can be regarded as a unital matrix when
the logical filter consists of one element. In each $\mathbf{LT}_{L}[D]$, the
designated element is the unit of the algebra, that is, its greatest element.

In the case $L=\textsf{Cl}$, each $\mathbf{LT}_{\textsf{Cl}}[D]$ is a Boolean
algebra. This circumstance will play an important role soon.

By proposition 6.3.5 of [Citkin and Muravitsky, 2022],

$\Gamma\vdash_{L}A~{}\Longleftrightarrow~{}\Gamma\models_{\mathbf{LT}_{L}[D]}A~{}\text{for
every $D\in\Sigma_{L}$}.$ (28)

This allows us to say that each $\vdash_{L}$ is the
$\mathcal{M}_{L}$-consequence, where

$\mathcal{M}_{L}:=\\{\mathbf{LT}_{L}[D]~{}:~{}D\in\Sigma_{L}\\}.$

In the case $L=\textsf{Cl}$, we have:

$\Gamma\vdash_{\textsf{Cl}}A~{}\Longleftrightarrow~{}\Gamma\models_{\mathbf{LT}_{\textsf{Cl}}[D]}A~{}\text{for
every $D\in\Sigma_{\textsf{Cl}}$}.$

Now, when we move from $\mathcal{M}_{\textsf{Cl}}$ consequence to
$\mathcal{M}_{\textsf{Cl}}$-r-consequence, in virtue of Proposition 5.1, we
obtain the conclusion: relation $\mid\\!\sim_{F}$ is the
$\mathcal{M}_{\textsf{Cl}}$-r-consequence. This suggests the following.

######  Conjecture 1.

Each $\Vdash_{L}$ is the $\mathcal{M}_{L}$-r-consequence.

Of course, from $\mathcal{M}_{L}$-r-consequence, we can move to
$\mathcal{M}_{L}$-r$\ast$-consequence. Then, we define:

$\Gamma\Vdash_{L^{\ast}}A~{}\stackrel{{\scriptstyle\text{df}}}{{\Longleftrightarrow}}~{}\Gamma_{\mathbf{c}}\Vdash_{L}A,$

where

$\Gamma_{\mathbf{c}}:=\\{B_{\mathbf{c}_{\mathbf{LT}_{L}[X]}}~{}:~{}B\in\Gamma\\}$

and each $B_{\mathbf{c}_{\mathbf{LT}_{L}[X]}}$ is obtained from $B$ by
replacement of each occurrence of a variable inessential in
$\mathbf{LT}_{L}[\Gamma]$ with $\top$. (We recall that each
$\mathbf{LT}_{L}[\Gamma]$ is a Heyting algebra.)

The following conjecture suggests itself.

######  Conjecture 2.

$\Gamma\Vdash_{L^{\ast}}A$ if, and only if,
$\Gamma\models^{r\ast}_{\mathbf{M}}A$ for every
$\mathbf{M}\in\mathcal{M}_{L}$.

##  6\. What has been achieved?

We started our discussion about the concept of nonmonotonic reasoning from a
philosophical point of view. Popular belief says that time can render the
final verdict. But what is meant by saying this? We can suggest two possible
answers.

The first is contained in the following quote from R. von Mises.

> I aim at the construction of a rational theory, based on the simplest
> possible concepts, one which, although admittedly inadequate to represent
> the complexity of the real processes, is able to reproduce satisfactorily
> some of their essential properties. ([von Mises, 1981], The inadequacy of
> theories)

The second opinion is more difficult to express.

The success of the proposed approach is often provided by unexplored space for
its further development. The larger this space, the more questions left
unanswered, the more tools available to answer these questions, the more
likely it is that this approach will become the main focus of research. Such
an approach could be termed _open_ , that is, open for further development.

This view of mine echoes another statement by von Mises.

> $[\ldots]$ the value of a concept is not gauged by its correspondence with
> some usual group of notions, but only by its usefulness for further
> scientific development $[\ldots]$ ([von Mises, 1981], Synthetic definitions)

To be open in the above sense means not to be sterile in the understanding of
G. Kreisel, who wrote,

> Aristotle gives much attention to the matter of _choice_ of abstractions.
> This indicates a shift of emphasis away from matters of principle (e.g. mere
> validity of analogies) to a focus (on, of course, valid cases) which
> provides safeguards against _sterility_ (not merely against straight error).
> ([Kreisel, 2019], Choice of abstractions)

Whether a (consistent) approach is open enough to avoid sterility can only be
judged by time.

In a broader context, concerning the what and how questions, the difficulty of
the task should not overshadow the more important level associated with the
act of knowing. This thought is not mine, but a paraphrase of the thought of
Thomas Aquinas, who wrote,

> When something is more difficult, it is not for that reason necessarily more
> worthwhile, but it must be more difficult in such a way, as also to be at a
> higher level of goodness. [Read: “the act of knowing” instead of “goodness”]
> (Quoted from [Pieper, 2009], chapter II.)

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