# Dynamic Many-Valued Logic Systems in Theoretical Economics

Daniel Lu

(March, 2024)

###### Abstract

This paper is an original attempt to understand the foundations of economic
reasoning. It endeavors to rigorously define the relationship between
subjective interpretations and objective valuations of such interpretations in
the context of theoretical economics. This analysis is substantially expanded
through a dynamic approach, where the truth of a valuation results in an
updated interpretation or changes in the agent’s subjective belief regarding
the effectiveness of the selected action as well as the objective reality of
the effectiveness of all other possible actions (i.e. consequence
realization). Complications arise when the economic agent is presented with a
set of actions that render ambiguous preference, or when the effectiveness of
an action cannot be perceived upon its selection, thereby necessitating a
different theory of choice and consequence realization.

##  1 Introduction

Suppose you are interested in selecting a portfolio that maximizes future
returns. We may express each portfolio as an element, $p_{i}$, in the set
$\mathbb{P}$. As opposed to traditional statistical approaches, we shall view
each $p_{i}$ as a assertion. For example, $p_{0}$ can be defined as saying
“Investing 10% of one’s wealth in asset A, 20% in asset B… would result in the
maximization of future returns.” By defining each member of $\mathbb{P}$ in
this manner, we may assign truth values to each sentence. According to the law
of the excluded middle, either a statement is true or false; either it is the
case that a portfolio results in the maximization of future returns, or it
does not. In this naive model, the investor is confronted with a simplistic
computational task where a true $p_{i}$ is selected. To express the selection
of a portfolio, we may use the sentence $s$, which is syntactically equivalent
to saying “This portfolio shall be selected.” To formally denote a valid
selection, we shall write $p_{i}\rightarrow s$. Under classical logic, the
meaning of the statement $p_{i}\rightarrow s$ has no bearing on its logical
form. In this case, the assertion in question is an instantiation of an
“If…then” relationship. We could equally replace $p_{i}$ and $s$ with
arbitrary variables such as $x$ and $y$ without necessarily dispensing with
the fundamental logical abstraction. In fact, classical logic posits that the
truthfulness of the statements $p_{i}$ and $s$ mirror that of the cohesive
statement $p_{i}\rightarrow s$ by following the law of the excluded middle.
This invites us to examine two epistemological question: can a portfolio’s
ability to maximize future returns be $true$ in the same way that its
selection is $true$? (Or, in the same way that the statement $p_{i}\rightarrow
s$ is interpreted as being a $true$ assertion?) Secondly, can the logical form
of a statement be entirely separate its meaning?

##  2 Motivation

This paper is predicated on the position that the answers to the preceding
questions are no. Classical logic offers an inadequate explanation of how
objects can possess truth values and be grouped through logical connectives.
This is because reasoning within an object language is intrinsically different
from reasoning about its semantic properties, and the semantic properties
underpinning the ascription of such properties. The primary motivation behind
the creation of many-valued logical systems, where one abandons the
absoluteness of binary notions such as $True$ or $False$, is to elucidate this
difference that underpins the essence of a logical system.

An analogy for this is to imagine a box in which a person is asked to examine
the semantic properties of a list of statements on a piece of paper, whatever
they may be. Another person outside the box, who observes this, can be asked
about the semantic properties of the reasoning used by the person within the
box. The nature of these semantic properties, whether true, false, unknown,
etc. and their interrelations can be flexibly designed in a logical manner
that fits desired purposes.

Returning to the introductory example, let us consider two portfolios, $p_{k}$
and $p_{j}$. By interpreting both assertions individually, we may examine
their truthfulness in terms of whether they are $conducive$ to the goal of
maximizing future returns. Whereas the formation of a complex sentence, such
as $p_{k}$ $\lor$ $p_{j}$, should be deemed as an entirely new statement that
can be interpreted as asking whether either portfolio would be $selected$. In
other words, an interpretation on such a sentence occupies an entirely
different epistemological realm, and their respective $truths$ are not
structurally identical. Any truth value attributable to the statement $p_{k}$
$\lor$ $p_{j}$ is an assertion regarding the selection of either portfolio,
whereas any truth value attributable to each individual portfolio is an
assertion pertaining to the validity of its maximal quality.

Additionally, instances where the logical form of a sentence cannot adequately
capture its inherent meaning are prevalent. Most notably, propositional logic
(PL, a formal language that does not feature quantifiers, such as $\forall$ or
$\exists$) cannot be used to express the validity of syllogisms such as:

$A$:= All investment options with a Sharpe ratio greater than 1 are considered
to have a good risk-adjusted return.

$B$:= Mutual Fund $\mathfrak{K}$ has a Sharpe ratio of 1.5.

$C$:= Therefore, Mutual Fund $\mathfrak{K}$ is considered to have a good risk-
adjusted return.
This is because PL does not have the expressive power of quantifiers and
therefore does not allow the derivation of any statement $C$ from the validity
of statements $A$ and $B$.

Another example would be the underlying epistemology of the conditional
“If…then” relation. In classical logic, a conditional relation is a false
assertion if and only if a false consequent arises from a true antecedent.
However, there exist conditionals where the truth of the antecedent is
necessary though insufficient to guarantee the truth of the consequent. For
example, consider the statement “If you invest in stock $X$ then you will reap
its returns.” Under the usual interpretation of the “If…then” relation, if an
investment is made in stock $X$ and future losses are permanently incurred,
then the conditional relation is false. However, the relation only suggests
that an investment is necessary though insufficient for one to reap its
returns. In other words, the conditional relation is false if and only if the
antecedent is false and the consequent is true, since the consequent is
necessarily contingent upon the truth of the antecedent.

First order logic lacks the expressive power of modal logic, which features
nuanced notions such as necessarily true or possibly true. This is symptomatic
of an inherent problem in the pathological insistence on the notion that
meaning is distinct from logical form when, in fact, they are inextricably
linked. As opposed to making further abstractions regarding the logical form
of sentences through the introduction of additional logical operators, (e.g.,
going beyond quantifiers such as $\forall$ or $\exists$ or modal operators
like $\Box p$ a.k.a it is necessarily the case that $p$), this paper assumes
that each sentence, whether complex or atomic, could assume any truth value as
they may address fundamentally different epistemic questions.

The objective of this essay is to approach the application of dynamic many-
valued logic in theoretical economics in an abstract, rigorous, and accessible
manner. In addition, this essay endeavors to offer a fundamental understanding
of economic reasoning. We shall begin by introducing the basic definitions
that make up a logical system.

Definition 2.1 An atomic sentence, $\varphi$, is a letter for which a semantic
truth value can be assigned. We denote the set of all atomic sentences with
$\mathcal{A}$.

Recall from the introductory example the set $\mathbb{P}$ which stands for the
collection of all assertions regarding the optimal portfolio.

Definition 2.2 A semantic truth value, $T_{k}$, can be construed as an
arbitrary denotation that says something about the properties of an object.
Let there be a set such that $T_{k}$ belongs to the collection
$\mathbb{T}_{d}$, or equivalently, $\\{T_{i}\\}_{i=0}^{j}$ for some $j$ or
$d\in\mathbb{N}$, where each $T_{i}$ is a truth value. Formally, we define
$T_{k}$ as follows:

$T_{k}\in\\{T_{i}\ |\ i\in\\{0,\ldots,j\\},j\in\mathbb{N}\\}$

Recall from the introductory example how a truth value can be used to denote
whether an individual portfolio is capable of rendering the maximum level of
future returns, or indicate whether it has been selected.

To interpret a sentence is to acknowledge an unambiguous many-to-one mapping
from sentence letters to a unique truth value. Though multiple sentences may
be understood as having the same truth value, the same sentence cannot be
interpreted in a singular way that results in the emergence of multiple truth
values.

Definition 2.3 An interpretation is a function that assigns an atomic sentence
a unique truth value in $\mathbb{T}_{d}$ for some $d$ in $\mathbb{N}$:

$I:\mathcal{A}\rightarrow\mathbb{T}_{d}$

Example:  $I$$(p_{k})=T$ i.e. Portfolio k is maximal.

Definition 2.4 A complex sentence consists of atomic sentences that are
grouped by logical connectives. We denote the set of all complex sentences
with $\mathcal{S}$.

Example: ($p_{k}$ $\lor$ $p_{j}$) $\in$ $\mathcal{S}$

Definition 2.5 (Logical Connective) A logical connective ($\oplus_{j}$) is a
function that specifies the logical relationship between a sequence of atomic
sentence letters of length $k$ in $\mathbb{N}$ and their respective roles in a
complex sentence. For some $j$ in $\mathbb{N}$, there exists a $\oplus_{j}$
such that:

$\oplus_{j}:\mathcal{A}^{k}\rightarrow\mathcal{S}$

Example: $\lor$($p_{k}$, $p_{j}$) := ($p_{k}$ $\lor$ $p_{j}$)

Definition 2.6 A valuation function, $V_{I}$, is a function that assigns a
semantic truth value in $\mathbb{T}_{d}$ for some $d$ in $\mathbb{N}$ to a
complex sentence on the basis of the truth values of its individual parts,
regardless of whether or not the range of the complex sentence’s truth value
coincides with that of its atomic counterparts.

$V_{I}:\mathcal{S}\rightarrow\mathbb{T}_{d}$

Example: $V_{I}$($p_{k}$ $\lor$ $p_{j})=T$ i.e. The selection of portfolios
$k$ or $j$ is true.

Although classical logic conflates the codomain of the valuation function with
the codomain of the interpretational function, it is necessary to specify an
important metalogical distinction between the two. This distinction is
necessary because logical connectives may be used to express arbitrary
semantic consequences given an arbitrary relationship between the truth values
of multiple atomic sentences. Let $\psi$ and $\varphi$ be sentence letters and
$\oplus_{0}$ an arbitrary binary connective; let the codomain of an
interpretational function be $\mathfrak{\\{M,F\\}}$; let that of the
connective be $\\{A,B,C,D\\}$. This allows us to define $\oplus_{0}$ in the
following arbitrary manner:

$\varphi\oplus_{0}\psi$ $\varphi\smallsetminus\psi$ | $I(\psi)=\mathfrak{M}$ | $I(\psi)=\mathfrak{F}$
---|---|---
$I(\varphi)=\mathfrak{M}$ | $V_{I}\left(\varphi\oplus_{0}\psi\right)=A$ | $V_{I}\left(\varphi\oplus_{0}\psi\right)=C$
$I(\varphi)=\mathfrak{F}$ | $V_{I}\left(\varphi\oplus_{0}\psi\right)=B$ | $V_{I}\left(\varphi\oplus_{0}\psi\right)=D$
$\psi\oplus_{0}\varphi$ $\psi\smallsetminus\varphi$ | $I(\varphi)=\mathfrak{M}$ | $I(\varphi)=\mathfrak{F}$
---|---|---
$I(\psi)=\mathfrak{M}$ | $V_{I}\left(\psi\oplus_{0}\varphi\right)=A$ | $V_{I}\left(\psi\oplus_{0}\varphi\right)=C$
$I(\psi)=\mathfrak{F}$ | $V_{I}\left(\psi\oplus_{0}\varphi\right)=B$ | $V_{I}\left(\psi\oplus_{0}\varphi\right)=D$
$\varphi\oplus_{0}\varphi$ $\varphi$ | $\varphi\oplus_{0}\varphi$
---|---
$I(\varphi)=\mathfrak{M}$ | $V_{I}\left(\varphi\oplus_{0}\varphi\right)=A$
$I(\varphi)=\mathfrak{F}$ | $V_{I}\left(\varphi\oplus_{0}\varphi\right)=D$
$\psi\oplus_{0}\psi$ $\psi$ | $\psi\oplus_{0}\psi$
---|---
$I(\psi)=\mathfrak{M}$ | $V_{I}\left(\psi\oplus_{0}\psi\right)=A$
$I(\psi)=\mathfrak{F}$ | $V_{I}\left(\psi\oplus_{0}\psi\right)=D$

The preceding tables are known as truth-table. A truth table consists of
different interpretations of two atomic sentences on the rows and columns.
Each square in the middle of the table shows the outcome of a valuation on a
binary connective when the interpretation of each sentence is taken into
consideration (e.g. $V_{I}\left(\psi\oplus_{0}\psi\right)=A$). The exact
meaning of each truth value in $\mathfrak{\\{M,F\\}}$ or $\\{A,B,C,D\\}$ is
irrelevant here, as they may be used to address different epistemic questions.

Note that the limit to the number of semantic truth values of the valuation of
a $j$-nary connective, where each object can have $n$ semantic
interpretations, is $n^{j}$. When this is the case, it is possible to derive
the exact interpretation of every sentence letter solely on the basis of the
truth value of the valuation. For example, the valuation
$V_{I}(\psi\oplus_{0}\varphi)=C$ allows us to derive that
$I(\psi)=\mathfrak{M}$ whereas $I(\varphi)=\mathfrak{F}$.

##  3 Note to Reader

The reader should bear the following in mind. Firstly, a formal language
cannot be understood intuitively (Falkenstein et al.). When applied to
theoretical economics, a formal language can be thought of as a counter-
intuitive rendition of the nuances of economic reasoning. When an $informal$
justification is provided, the reader should bear in mind that the reasoning
or idiosyncratic notations being used should be interpreted as aids to one’s
comprehension, but not an accurate representation of what is being conveyed.
The reader should consider this work as an exercise in philosophy. It is an
attempt to understand the interplay between economic reasoning and the results
of its application. Due to the density of the notations and concepts featured,
the reader may be frequently asked to recall concepts in preceding sections to
aid one’s understanding.

##  4 Six-Valued Logic System

Consider the following set of truth values which specifies the codomain the
interpretational function:

$\mathbb{T}_{0}=\\{T,F,U\\}$

Where $T$ stands for “Interpretationally True”, $F$ stands for
“Interpretationally False”, and $U$ stands for “Interpretationally Unknown”.
These are truth values that can only ascribed to atomic setences. In contrast,
consider the codomain for the valuation function:

$\mathbb{T}_{1}=\\{t,f,u\\}$

Where $t$ stands for “Cohesively True”, $f$ stands for “Cohesively False”, and
$u$ stands for “Cohesively Unknown”. The exact meaning of each truth value is
irrelevant here as we are trying to incorporate the epistemology of each
statement into a particular logical interpretation. It is only necessary for
us to understand that $\mathbb{T}_{1}$ stands for the collection of all truth
values that can be assumed by complex sentences.

Consider the following logical connectives in propositional calculus: namely,
not ($\neg$), and ($\land$), or ($\lor$), and if…then ($\rightarrow$). To
specify what the logical connectives say about the relationship between
sentence letters and the truth value/valuation of the whole sentence, the
following tables can be created.

$\neg\varphi$ $\begin{array}[]{|c|c|}\hline\cr\varphi&\neg\varphi\\\ \hline\cr
I(\varphi)=T&V_{I}(\neg\varphi)=f\\\ \hline\cr
I(\varphi)=F&V_{I}(\neg\varphi)=t\\\ \hline\cr
I(\varphi)=U&V_{I}(\neg\varphi)=u\\\ \hline\cr\end{array}$ $\varphi\land\psi$
$\begin{array}[]{|c|c|c|c|}\hline\cr\varphi\smallsetminus\psi&I(\psi)=T&I(\psi)=F&I(\psi)=U\\\
\hline\cr
I(\varphi)=T&V_{I}(\varphi\land\psi)=t&V_{I}(\varphi\land\psi)=f&V_{I}(\varphi\land\psi)=u\\\
\hline\cr
I(\varphi)=F&V_{I}(\varphi\land\psi)=f&V_{I}(\varphi\land\psi)=f&V_{I}(\varphi\land\psi)=f\\\
\hline\cr
I(\varphi)=U&V_{I}(\varphi\land\psi)=u&V_{I}(\varphi\land\psi)=f&V_{I}(\varphi\land\psi)=u\\\
\hline\cr\end{array}$

$\varphi\lor\psi$
$\begin{array}[]{|c|c|c|c|}\hline\cr\varphi\smallsetminus\psi&I(\psi)=T&I(\psi)=F&I(\psi)=U\\\
\hline\cr
I(\varphi)=T&V_{I}(\varphi\lor\psi)=t&V_{I}(\varphi\lor\psi)=t&V_{I}(\varphi\lor\psi)=t\\\
\hline\cr
I(\varphi)=F&V_{I}(\varphi\lor\psi)=t&V_{I}(\varphi\lor\psi)=f&V_{I}(\varphi\lor\psi)=u\\\
\hline\cr
I(\varphi)=U&V_{I}(\varphi\lor\psi)=t&V_{I}(\varphi\lor\psi)=u&V_{I}(\varphi\lor\psi)=u\\\
\hline\cr\end{array}$

$\varphi\rightarrow\psi$
$\begin{array}[]{|c|c|c|c|}\hline\cr\varphi\smallsetminus\psi&I(\psi)=T&I(\psi)=F&I(\psi)=U\\\
\hline\cr
I(\varphi)=T&V_{I}(\varphi\rightarrow\psi)=t&V_{I}(\varphi\rightarrow\psi)=f&V_{I}(\varphi\rightarrow\psi)=f\\\
\hline\cr
I(\varphi)=F&V_{I}(\varphi\rightarrow\psi)=u&V_{I}(\varphi\rightarrow\psi)=u&V_{I}(\varphi\rightarrow\psi)=t\\\
\hline\cr
I(\varphi)=U&V_{I}(\varphi\rightarrow\psi)=u&V_{I}(\varphi\rightarrow\psi)=u&V_{I}(\varphi\rightarrow\psi)=t\\\
\hline\cr\end{array}$

An $informal$ justification for each arbitrary table can be offered:

  * •

The Negation Table specifies that if a sentence is known to be true, then its
negation must be false, and vice-versa. However, if the sentence’s truth value
cannot be determined, then neither can its negation. More specifically, when
the interpretation of $\varphi$ is $U$, the valuation of
$(\varphi\lor\neg\varphi)$ is $u$. This may seem counter-intuitive at first;
one may argue that although $\varphi$ may be unknown, either $\varphi$ or
$\neg\varphi$ is true. However, that is the exact subject in question; we
cannot determine whether $\varphi$ is true or not true. Hence, the valuation
must be $u$.

  * •

The And Table states that if one of the interpretations of the sentences is
false, then the valuation of the whole sentence must also be cohesively false.
However, if one is true and the other is unknown, then the valuation is also
unknowable since the interpretation of a sentence with an unknowable valuation
can either be true or false. To note the consistency of this multi-valued
system, one may argue that $(\varphi\land\neg\varphi)$ must be false even if
the interpretation of $\varphi$ is unknown because $\varphi$ cannot have two
values at once. However, if $(\varphi\land\neg\varphi)$ were false, then
either $\varphi$ is false, or its negation is false. But since the valuation
of $\varphi$ is unknown, then the valuation of its negation must be unknown as
well, hence the valuation of $(\varphi\land\neg\varphi)$ is not false, but
rather, unknown. $U/u$ must be interpreted as truth values whose definitions
are independent of the meaning of other truth values; absolute ambiguity is a
valid logical state.

  * •

The Or Table states that if one of the sentence values is interpreted as being
true, then the whole sentence is true, regardless of whether the valuation of
the other sentence is true, false, or unknowable. However, if both are false
then the whole sentence is false since neither sentence is true. But if one
sentence is false/unknowable, and the interpretation of the other is
unknowable, then the valuation of the whole sentence is also unknown.

  * •

The If…Then Table states that if $\varphi$ is interpreted to be true, then
$\psi$ must also be true: in which case $\psi$ cannot be false or unknown,
otherwise that would result in the whole sentence being valued as false.
However, if $\varphi$ were false or unknown, then what can be reasonably
inferred is that the valuation for the whole construction is also unknown.
This is because the If…Then statement only specifies the resulting value of
$\psi$ when and only when $\varphi$ is true. Otherwise, it may either be true
or false, and only its unknowability can be affirmed.

We may now proceed with this analysis by developing a set of inference rules
through a reversal of the direction in which we interpret the previous tables.
This allows us to examine what can be said of the truth values of an
individual sentence(s) given the valuation of the complex sentence, and in
some instances, given, also, the interpretation of a different sentence. Let
$\Gamma$ represent a set of premises from which we infer a conclusion, $\phi$.
An inference is valid insofar as all the premises in $\Gamma$ are true and the
resulting conclusion is not false. A premise must be explicitly specified to
have meaning. For example, suppose we are privy to the fact that the valuation
of a particular interpretation of the sentence “$\neg\varphi$” is false. From
this premise, we derive that that exact interpretation must have assessed
“$\varphi$” to be true. This practice of explicitly enunciating the truth
values of premise sentences allows for greater flexibility and coherence in
the analysis of many-valued logic.

We shall define the premises in a way that ascertains the individual
interpretations of the atomic sentences and infers what the valuation on the
interpretation says about a particular logical connective. This requires us to
define an Equivalence Valuation Function, $\mathfrak{v}$, which ascertains
whether the individual interpretations or valuations are semantically
equivalent to their purported truth values. The codomain of this function is
$\mathfrak{\\{T,F\\}}$. For example:

From $\Gamma:=\\{\mathfrak{v}((I(\psi)=T)\land(I(\varphi)=T))=\mathfrak{T}\\}$
infer $V_{I}(\varphi\land\psi)=t$

From
$\Gamma:=\\{\mathfrak{v}(I(\varphi)=T)=\mathfrak{T},\mathfrak{v}(I(\psi)=T)=\mathfrak{T}\\}$
infer $V_{I}(\varphi\land\psi)=t$

$\vdots$

Definition 4.1 (Equivalence Valuation Function): Consider the logical system
defined by the atomic sentences ($\mathcal{A}$), complex sentences
($\mathcal{S}$), the interpretation function $I$, the valuation function
$V_{I}$, and the set of semantic truth values ($\mathbb{T}_{d}$) as outlined
in Definitions 2.1 to 2.6. The equivalence valuation function, $\mathfrak{v}$,
is a mechanism for assessing the fidelity of the semantic assignments made by
$I$ and $V_{I}$ against a predetermined equivalence criterion with respect to
the truth value $T_{k}$ in $\mathbb{T}_{d}$. Formally, $\mathfrak{v}$ operates
as follows:

  * •

For any atomic sentence $\varphi\in\mathcal{A}$, and any truth value $T_{k}$,
$\mathfrak{v}$ evaluates the interpretation $I(\varphi)$ against $T_{k}$ based
on a predefined equivalence relation. Specifically,
$\mathfrak{v}(I(\varphi),T_{k})$ yields $\mathfrak{T}$ if the truth value
assigned by $I$ to $\varphi$ is deemed equivalent to $T_{k}$, and
$\mathfrak{F}$ otherwise.

  * •

For any complex sentence $\psi\in\mathcal{S}$, and any truth value $T_{k}$,
$\mathfrak{v}$ assesses the valuation $V_{I}(\psi)$ against $T_{k}$ according
to the same equivalence relation. In this case,
$\mathfrak{v}(V_{I}(\psi),T_{k})$ results in $\mathfrak{T}$ if the truth value
determined by $V_{I}$ for $\psi$ is considered equivalent to $T_{k}$, and
$\mathfrak{F}$ otherwise.

Thus, the function $\mathfrak{v}$ can be expressed in terms of its domain and
range as follows:

$\mathfrak{v}:(\mathcal{A}\cup\mathcal{S})\times\mathbb{T}_{d}\rightarrow\\{\mathfrak{T,F}\\}$

where the domain consists of pairs of either atomic or complex sentences and a
particular semantic truth value $T_{k}$, and the range is the set
$\\{\mathfrak{T,F}\\}$, representing the truthfulness of the equivalence
relation between the sentence’s assigned truth value and $T_{k}$.

Notice that when the Equivalence Valuation Function is used, a fraktur capital
$T$, $\mathfrak{T}$, is used to denote the validity of the statement
“$I(\psi)=T$” in lieu of $T$ or $t$. This reflects a subtle though important
distinction between statements within and about a metalanguage (i.e. the meta-
metalanguage). Even if a many-valued logical system features more than two
truth values, the concept of an equivalence relation is fundamentally binary.
For example, it is either the case that the interpretation of $\psi$ assumes a
particular truth value, whether it may be $T,F$ or $U$, or it is not. The
previous 3-valued logic system offers a variety of inference relations. An
incomplete list of which is presented here.

Negation Inferences:

From $\Gamma:=\\{\mathfrak{v}(V_{I}(\neg\varphi)=f)=\mathfrak{T}\\}$ infer
$\phi:=I(\varphi)=T$

From $\Gamma:=\\{\mathfrak{v}(V_{I}(\neg\varphi)=t)=\mathfrak{T}\\}$ infer
$\phi:=I(\varphi)=F$

From $\Gamma:=\\{\mathfrak{v}(V_{I}(\neg\varphi)=u)=\mathfrak{T}\\}$ infer
$\phi:=I(\varphi)=U$

And Inferences:

From
$\Gamma:=\\{\mathfrak{v}(V_{I}(\varphi\land\psi)=t)=\mathfrak{T},\mathfrak{v}(I(\psi)=T)=\mathfrak{T}\\}$
infer $\phi:=I(\varphi)=T$

From
$\Gamma:=\\{\mathfrak{v}(V_{I}(\varphi\land\psi)=t)=\mathfrak{T},\mathfrak{v}(I(\psi)=T)=\mathfrak{T}\\}$
infer $\phi:=I(\varphi)=T$

From
$\Gamma:=\\{\mathfrak{v}(V_{I}(\varphi\land\psi)=f)=\mathfrak{T},\mathfrak{v}(I(\psi)=T)=\mathfrak{T}\\}$
infer $\phi:=I(\varphi)=F$

From
$\Gamma:=\\{\mathfrak{v}(V_{I}(\varphi\land\psi)=f)=\mathfrak{T},\mathfrak{v}(I(\varphi)=T)=\mathfrak{T}\\}$
infer $\phi:=I(\psi)=F$

$\vdots$

Definition 4.2 (Entailment): Entailment is a relationship between a set of
statements or propositions (premises) and an individual statement
(conclusion), where the truth of the premises logically guarantees the truth
of the conclusion. This concept is crucial for understanding logical inference
and reasoning processes.

To denote a valid inference, where a set of premises $\Gamma$ logically leads
to a conclusion $\phi$, we write $\Gamma\models\phi$. This notation signifies
that if all the statements in $\Gamma$ are true, then $\phi$ must also be
true. The symbol $\models$ represents the entailment or logical consequence,
indicating that the conclusion $\phi$ is an entailment of the premises in
$\Gamma$.

###  4.1 Note on $\mathfrak{v}$ and the Law of the Excluded Middle

The law of the excluded middle states that for all statements, either it is
the case that the statement is true, or it is not the case that the statement
is true, or it is the case that a statement is false, or it is not the case
that a statement is false. To say that it is not the case that a statement is
true is equivalent to saying that it is the case that a statement is false,
and to say that it is not the case that a statement is false is equivalent to
saying that it is the case that a statement is true.

To express this definition formally, let $\phi$ be an arbitrary sentence
letter. Let $I(\phi)$ be an interpretational function that assigns a
particular truth value to the sentence letter. Let the domain of $I$ be any
sentence letter and let its codomain be $\\{T,F\\}$. Recall the definition of
an equivalence valuation function whose domain is any equivalence relation
between an interpretation and a semantic property and whose codomain is
strictly limited to $\\{\mathfrak{T},\mathfrak{F}\\}$. We may therefore
formalize the law of the excluded middle in the following manner.

$\forall\phi\left(\mathfrak{v}\left(I(\phi)=T\right)=\mathfrak{T}\right)\lor\left(\mathfrak{v}\left(I(\phi)=F\right)=\mathfrak{T}\right)\lor\left(\mathfrak{v}\left(I(\phi)=T\right)=\mathfrak{F}\right)\lor\left(\mathfrak{v}\left(I(\phi)=F\right)=\mathfrak{F}\right)$
(1)
$\left(\mathfrak{v}\left(I(\phi)=T\right)=\mathfrak{F}\right)\Leftrightarrow\left(\mathfrak{v}\left(I(\phi)=F\right)=\mathfrak{T}\right)$
(2)
$\left(\mathfrak{v}\left(I(\phi)=F\right)=\mathfrak{F}\right)\Leftrightarrow\left(\mathfrak{v}\left(I(\phi)=T\right)=\mathfrak{T}\right)$
(3)

Consider the following sentence:

$\psi:=\left(\mathfrak{v}\left(I(\psi)=F\right)=\mathfrak{T}\right)$ (4)

Since $\psi$ is a statement about an equivalence relation, we must also
account for a surreptitiously embedded assumption in classical logic, namely,
the universality of truth/falsity in characterizing the correctness of any
statement, regardless of whether or not it features an equivalence relation.
To show this, we shall also assume the following about the statement $\psi$:

$\left(\mathfrak{v}\left(I(\psi)=T\right)=\mathfrak{T}\right)\Leftrightarrow\left(\mathfrak{v}\left(I(\psi)=F\right)=\mathfrak{T}\right)$
(5)
$\left(\mathfrak{v}\left(I(\psi)=F\right)=\mathfrak{T}\right)\Leftrightarrow\left(\mathfrak{v}\left(I(\psi)=F\right)=\mathfrak{F}\right)$
(6)

More specifically, we are assuming that if it is the case that the
interpretation of $\psi$ is false, then it satisfies the definition of $\psi$,
which means that it is the case that the interpretation of $\psi$ is true, and
vice versa. And, if it is not the case that $\psi$ is false, then the
definition of $\psi$ is not satisfied (the criterion being the falsity of
$\psi$), which means that it is the case that the interpretation of $\psi$
being false is true. We can immediately see that this contradicts the criteria
set by the law of the exclusive middle. But let us continue by demonstrating
the contradiction explicitly.

###### Proof.

$\displaystyle\psi$ $\displaystyle:=\mathfrak{v}(I(\psi)=F)=\mathfrak{T}$
(Assumption) $\displaystyle\mathfrak{v}(I(\psi)=F)=\mathfrak{T}$
$\displaystyle\Rightarrow\mathfrak{v}(I(\psi)=T)=\mathfrak{T}$ (see (5))
$\displaystyle\mathfrak{v}(I(\psi)=F)=\mathfrak{T}$
$\displaystyle\Rightarrow\mathfrak{v}(I(\psi)=T)=\mathfrak{F}$ (see (2))
$\displaystyle\bot$ (Contradiction)

∎

We have now arrived at a contradiction in the equivalence relation between the
interpretation on a sentence letter and its corresponding semantic property.
Similarly, let us assume that $\psi$ is interpreted to be false:

###### Proof.

$\displaystyle\psi$ $\displaystyle:=\mathfrak{v}(I(\psi)=F)=\mathfrak{F}$
(Assumption) $\displaystyle\mathfrak{v}(I(\psi)=F)=\mathfrak{F}$
$\displaystyle\Rightarrow\mathfrak{v}(I(\psi)=F)=\mathfrak{T}$ (see (6))
$\displaystyle\bot$ (Contradiction)

∎

Since $\psi$ can only be interpreted as being either true or false, we have
exhausted all possible values and have shown that both lead to contradictions.
The problem with the law of the excluded middle is that it cannot adequately
account for statements that feature self-referential equivalence relations,
though we can reconcile with these apparent paradoxes by featuring more
nuanced values that reflect other semantic properties which cannot be
adequately characterized as being universally true or false but occupy an
entirely different epistemological realm.

An ostensible paradox appears to be present in this reasoning. To prove that
the law of the excluded middle does not work, we appear to have assumed that
it does work for the equivalence valuation function, where an equivalence
relation between an interpretation of a sentence and its semantic property
either holds or does not hold. However, we have effectively prevented such a
paradox by not falling into the fallacy of conflating the codomain of the
interpretational function with the codomain of the equivalence valuation
function. In other words, $equivalence$ and $truth$ are fundamentally
distinct.

##  5 Formalization of the Action Axiom

The introduction of a new truth value in the metalanguage has practical
implications. Consider an economic agent who endeavors to raise their utility
by selecting an action from the set of all possible actions that can be taken
to improve welfare: $\\{A_{j}\\}_{\forall j|j\in N}$. The interpretational
utility function, $I(A_{i})$, interprets the action and determines whether it
will genuinely improve (i.e. $I(A_{i})=T$), degenerate (i.e. $I(A_{i})=F$), or
have an indeterminate effect on economic circumstance (i.e. $I(A_{i})=U$). For
the purposes of this demonstration let us assume that the cardinality of the
action set is 2: in other words, the agent can choose one of the following:
$\mathbb{A}=\\{A_{0},A_{1}\\}$.

A set of practical restrictions on behavior is expected; when one action is
perceived to be conducive to the outcome of higher utility as opposed to the
other action, which is interpreted to be false, then such an action shall be
selected. When both actions result in equivalent outcomes, then either action
shall be selected, since either consequence is unavoidable, resulting in an
ambiguous state. If one action will lead to an unknown outcome whereas the
other one will make the agent worse off, then the first action shall be
selected. If one action leads to an unknown outcome but the other one will
result in a positive outcome, then the agent may or may not select the action
that results in the positive outcome depending on one’s preference for risk.
The action matrix for a risk-averse agent is shown in the following table:

Informal Action Selection Matrix

$A_{1}\backslash A_{0}$ | $I(A_{0})=T$ | $I(A_{0})=F$ | $I(A_{0})=U$
---|---|---|---
$I(A_{1})=T$ | $A_{1}\lor A_{0}$ | $A_{1}$ | $A_{1}$
$I(A_{1})=F$ | $A_{0}$ | $A_{1}\lor A_{0}$ | $A_{0}$
$I(A_{1})=U$ | $A_{0}$ | $A_{1}$ | $A_{1}\lor A_{0}$

Consider two binary connectives that take the truth values of the actions as
inputs and output another truth value to show whether a particular action has
been chosen. For example, $A_{1}\oplus_{A_{1}}A_{0}$ (i.e.
$\oplus_{A_{1}}(A_{1},A_{0})$) outputs a truth value to note whether action 1
will $necessarily$ be selected by the agent given the underlying values of
$A_{1}$ and $A_{0}$; $\oplus_{A_{0}}(A_{1},A_{0})$ performs the exact same
function for action 0. The formal definitions of the connectives are presented
in the following tables:

$\text{A}_{1}\,\oplus_{\text{A}_{1}}\,\text{A}_{0}$
$\begin{array}[]{|c|c|c|c|}\hline\cr A_{1}\backslash
A_{0}&I(A_{0})=T&I(A_{0})=F&I(A_{0})=U\\\ \hline\cr
I(A_{1})=T&V_{I}(A_{1}\oplus_{A_{1}}A_{0})=u&V_{I}(A_{1}\oplus_{A_{1}}A_{0})=t&V_{I}(A_{1}\oplus_{A_{1}}A_{0})=t\\\
\hline\cr
I(A_{1})=F&V_{I}(A_{1}\oplus_{A_{1}}A_{0})=f&V_{I}(A_{1}\oplus_{A_{1}}A_{0})=u&V_{I}(A_{1}\oplus_{A_{1}}A_{0})=f\\\
\hline\cr
I(A_{1})=U&V_{I}(A_{1}\oplus_{A_{1}}A_{0})=f&V_{I}(A_{1}\oplus_{A_{1}}A_{0})=t&V_{I}(A_{1}\oplus_{A_{1}}A_{0})=u\\\
\hline\cr\end{array}$

$\text{A}_{1}\oplus_{\text{A}_{0}}\text{A}_{0}$
$\begin{array}[]{|c|c|c|c|}\hline\cr A_{1}\backslash
A_{0}&I(A_{0})=T&I(A_{0})=F&I(A_{0})=U\\\ \hline\cr
I(A_{1})=T&V_{I}(A_{1}\oplus_{A_{0}}A_{0})=u&V_{I}(A_{1}\oplus_{A_{0}}A_{0})=f&V_{I}(A_{1}\oplus_{A_{0}}A_{0})=f\\\
\hline\cr
I(A_{1})=F&V_{I}(A_{1}\oplus_{A_{0}}A_{0})=t&V_{I}(A_{1}\oplus_{A_{0}}A_{0})=u&V_{I}(A_{1}\oplus_{A_{0}}A_{0})=t\\\
\hline\cr
I(A_{1})=U&V_{I}(A_{1}\oplus_{A_{0}}A_{0})=t&V_{I}(A_{1}\oplus_{A_{0}}A_{0})=f&V_{I}(A_{1}\oplus_{A_{0}}A_{0})=u\\\
\hline\cr\end{array}$

To formalize the contents of this matrix, let us consider two sets that
specify the circumstances under which an action is selected. The sets are
named $\Gamma_{A_{1}}$ and $\Gamma_{A_{0}}$ and each logically entails the
selection of their respective actions.

$\displaystyle\Gamma_{A_{1}}:=\\{$
$\displaystyle\mathfrak{v}\bigl{(}((I(A_{1})=T)\land((I(A_{0})=F)\lor(I(A_{0})=U)))$
$\displaystyle\lor((I(A_{1})=U)\land(I(A_{0})=F)))=\mathfrak{T}\\}$
$\displaystyle\Gamma_{A_{0}}:=\\{$
$\displaystyle\mathfrak{v}\bigl{(}((I(A_{0})=T)\land((I(A_{1})=F)\lor(I(A_{1})=U)))$
$\displaystyle\lor((I(A_{0})=U)\land(I(A_{1})=F)))=\mathfrak{T}\\}$

Annotated Proof of $\Gamma_{A_{1}}\models V_{I}(A_{1}\oplus_{A_{1}}A_{0})=t$:

###### Proof.

$\displaystyle\overbrace{P_{1}}^{\text{Premise 1}}$
$\displaystyle:=\underbrace{\mathfrak{v}(((I(A_{1})=T)\land((I(A_{0})=F)\lor(I(A_{0})=U)))}_{\text{Condition
A}}$
$\displaystyle\quad\lor\underbrace{((I(A_{1})=U)\land(I(A_{0})=F)))=\mathfrak{T},}_{\text{Condition
B}}$ $\displaystyle\overbrace{P_{2}}^{\text{Premise 2}}$
$\displaystyle:=\underbrace{V_{I}(A_{1}\oplus_{A_{1}}A_{0})\neq
t,}_{\text{Assertion}}$ $\displaystyle P_{1}$
$\displaystyle\Rightarrow\overbrace{(\mathfrak{v}((I(A_{1})=T)\land((I(A_{0})=F)\lor(I(A_{0})=U)))=\mathfrak{T})}^{\text{Consequence
1A}}$
$\displaystyle\quad\lor\underbrace{(\mathfrak{v}((I(A_{1})=U)\land(I(A_{0})=F))=\mathfrak{T}),}_{\text{Consequence
1B}}$ $\displaystyle P_{1}$
$\displaystyle\Rightarrow\overbrace{((\mathfrak{v}((I(A_{1})=T)\land(I(A_{0})=F))=\mathfrak{T})}^{\text{Consequence
1.1A}}$
$\displaystyle\quad\lor\underbrace{(\mathfrak{v}((I(A_{1})=T)\land(I(A_{0})=U))=\mathfrak{T}))}_{\text{Consequence
1.2A}}$
$\displaystyle\quad\lor\underbrace{(\mathfrak{v}((I(A_{1})=U)\land(I(A_{0})=F))=\mathfrak{T}),}_{\text{Consequence
1B}}$ $\displaystyle P_{1}$
$\displaystyle\Rightarrow\overbrace{((V_{I}(A_{1}\oplus_{A_{1}}A_{0})=t)\lor(V_{I}(A_{1}\oplus_{A_{1}}A_{0})=t))}^{\text{Final
Conclusion via Discharge of A}}$
$\displaystyle\quad\lor\underbrace{(V_{I}(A_{1}\oplus_{A_{1}}A_{0})=t),}_{\text{Final
Conclusion via Discharge of B}}$ $\displaystyle P_{1}$
$\displaystyle\Rightarrow\overbrace{V_{I}(A_{1}\oplus_{A_{1}}A_{0})=t,}^{\text{Final
Conclusion}}$ $\displaystyle P_{1}$ $\displaystyle\Rightarrow\underbrace{\lnot
P_{2}.}_{\text{Contradiction}}$

Therefore, $\Gamma_{A_{1}}\models V_{I}(A_{1}\oplus_{A_{1}}A_{0})=t$. ∎

Proof of $\Gamma_{A_{0}}\models V_{I}(A_{1}\oplus_{A_{0}}A_{0})=t$:

###### Proof.

$\displaystyle P_{3}$
$\displaystyle:=\mathfrak{v}(((I(A_{0})=T)\land((I(A_{1})=F)\lor(I(A_{1})=U)))$
$\displaystyle\lor((I(A_{0})=U)\land(I(A_{1})=F)))=\mathfrak{T},$
$\displaystyle P_{4}$ $\displaystyle:=V_{I}(A_{1}\oplus_{A_{0}}A_{0})\neq t,$
$\displaystyle P_{3}$
$\displaystyle\Rightarrow(\mathfrak{v}((I(A_{0})=T)\land((I(A_{1})=F)\lor(I(A_{1})=U)))=\mathfrak{T})$
$\displaystyle\lor(\mathfrak{v}((I(A_{0})=U)\land(I(A_{1})=F))=\mathfrak{T}),$
$\displaystyle P_{3}$
$\displaystyle\Rightarrow((\mathfrak{v}((I(A_{0})=T)\land(I(A_{1})=F))=\mathfrak{T})$
$\displaystyle\lor(\mathfrak{v}((I(A_{0})=T)\land(I(A_{1})=U))=\mathfrak{T}))$
$\displaystyle\lor(\mathfrak{v}((I(A_{0})=U)\land(I(A_{1})=F))=\mathfrak{T}),$
$\displaystyle P_{3}$
$\displaystyle\Rightarrow((V_{I}(A_{1}\oplus_{A_{0}}A_{0})=t)\lor(V_{I}(A_{1}\oplus_{A_{0}}A_{0})=t))$
$\displaystyle\lor(V_{I}(A_{1}\oplus_{A_{0}}A_{0})=t),$ $\displaystyle P_{3}$
$\displaystyle\Rightarrow V_{I}(A_{1}\oplus_{A_{0}}A_{0})=t,$ $\displaystyle
P_{3}$ $\displaystyle\Rightarrow\lnot P_{4}.$

Therefore, $\Gamma_{A_{0}}\models V_{I}(A_{1}\oplus_{A_{0}}A_{0})=t$. ∎

Hence, the selection criterion is as follows: if there exists a $\Gamma$ such
that for all $\mathfrak{v}\in\Gamma$, $\mathfrak{v}=\mathfrak{T}$, then
$V_{I}(A_{1}\oplus_{a}A_{0})=t$. This is an important assumption. It states
that the selection of human actions is based on whether they are conducive to
the realization of desired outcomes. We may now expand upon this analysis by
showing that an agent may respond to an objective reality based on one’s
subjective perception; this, in turn, is shaped by the objective consequences
of the actions selected.

###  5.1 Intuition behind Proofs

As the proofs show, the equivalence valuation function plays a substantial
role in making deductive inferences. Essentially, the preceding proofs start
from assertions regarding what the truth values of the atomic sentences are,
and proceeds to show that the featured combinations of the atomic truth
values, when considered alongside the valuation on a particular connective,
would result in the selection of a particular action.

##  6 Dynamic Six-Valued Logic System

###  6.1 Motivation and Basic Ideas

Having established a system in which an agent may weigh the relative merits of
each action before making a unique selection, we may now turn to a system in
which we distinguish between subjective perceptions and objective realities.
Subjective perceptions play a significant role in economic theory. The
subjective theory of value, for instance, posits that the value of any good is
determined by the eye of the beholder, or the perceived utility that can be
derived from it (this is a tautology; utility is perceived as it is a
subjective experience). As opposed to saying that an agent makes a decision on
the basis of objective circumstances, it is more accurate for us to depict the
agent’s decisions as a consequence of his or her perception of these
circumstances. Once an action has been selected, both the agent’s belief
regarding that action, and the objective reality of whether an action is
conducive to a particular objective, shall be updated, and the selection
process iterates (hence giving rise to a $dynamic$ system).

###  6.2 Introduction

Following from our previous assumptions regarding the selection of a
particular course of action: consider a set, $\mathbb{T}$, of six truth
values, namely, subjectively true ($T_{s}$), subjectively false ($F_{s}$),
subjectively unknown ($U_{s}$), objectively true ($T_{o}$), objectively false
($F_{o}$), and objectively unknowable ($U_{o}$):

$\mathbb{T}=\\{T_{s},F_{s},U_{s},T_{o},F_{o},U_{o}\\}$

It is assumed that the agent now selects one action from the action set, which
contains at least two possible actions. In other words:

$\mathbb{A}=\\{A_{j}|j\in\mathbb{N},j\geq 2\\}$

An interpretation of an action is a function that maps an element of the set
$A$ to two truth values, each taken from a partition of $\mathbb{T}$.

$S=\\{T_{s},F_{s},U_{s}\\}$

$O=\\{T_{o},F_{o},U_{o}\\}$

$\forall i,\forall t,\exists!s\in S,\exists!o\in O:I_{t}(A_{i})=[s/o]$

A set of j-nary connectives, $\oplus$, is defined. Each connective determines
whether an action is selected on the basis of the interpretational value of
all possible actions.

$\oplus=\left\\{\begin{array}[]{@{}l@{}}\oplus_{A_{0}}(A_{0},A_{1},\dots,A_{j})\\\
\quad\quad\quad\quad\vdots\\\
\oplus_{A_{j}}(A_{0},A_{1},\dots,A_{j})\end{array}\right\\}$

The codomain of the valuation of each connective is $O$. This is because the
selection of an action can only be interpreted as an objective realization of
individual behavior.

$\forall
I_{t},\forall\oplus_{(A\in\mathbb{A})},V_{I_{t}}(\oplus_{(A\in\mathbb{A})}(A_{0},A_{1},\ldots,A_{j}))\in
O$

For each individual connective,

$\displaystyle
V_{I_{t}}(\oplus_{(A_{m}\in\mathbb{A})}(A_{0},A_{1},\ldots,A_{j}))=T_{o}\Leftrightarrow\exists!A_{m}\in\mathbb{A}:\forall
s\in\\{F_{s},U_{s}\\},\forall o_{0},o_{1}\in O,$ $\displaystyle\forall
A_{n}\in\mathbb{A}:A_{n}\neq
A_{m},((\mathfrak{v}(I_{t}(A_{m})=[T_{s}/o_{0}])=\mathfrak{T})\land(\mathfrak{v}(I_{t}(A_{n})=[s/o_{1}])=\mathfrak{T}))$
$\displaystyle\quad\lor((\mathfrak{v}(I_{t}(A_{m})=[U_{s}/o_{0}])=\mathfrak{T})\land(\mathfrak{v}(I_{t}(A_{n})=[F_{s}/o_{1}])=\mathfrak{T}))$

More succinctly, an action is selected if its ability to contribute to welfare
is subjectively true whereas that of all other alternatives is subjectively
false or unknown, or if its ability is subjectively unknown whereas that of
all other alternatives is subjectively false.

###  6.3 Belief Update Mechanism

We have established that the selection of an action is solely predicated on
its subjective interpretation. However, a subjective truth value can be
morphed by an objective outcome, which is inhered in the objective truth
value. Additionally, the objective truth values of all actions can be morphed
through the selection of an action. This gives rise to a dynamic logical
system where beliefs can be updated by the feedback of an environment.

$\displaystyle\forall t,\forall a,b,c,d,e,f\in\\{T,F,U\\},\forall
A_{n},A_{m}\in\mathbb{A}:A_{n}\neq A_{m},$
$\displaystyle(V_{I_{t}}(\oplus_{A_{m}}(A_{0},A_{1},\ldots,A_{j}))=T_{o})\Rightarrow$
$\displaystyle((\mathfrak{v}(I_{t}(A_{n})=[a_{s}/b_{o}])=\mathfrak{T}\Rightarrow
I_{(t+1)}(A_{n})=[a_{s}/c_{o}])\land$
$\displaystyle\quad(\mathfrak{v}(I_{t}(A_{m})=[d_{s}/e_{o}])=\mathfrak{T}\Rightarrow
I_{(t+1)}(A_{m})=[e_{s}/f_{o}]))$

Note that this definition relies on a strict axiom of preservation. In other
words, it relies on the fact that beliefs do not spontaneously change and are
preserved unless altered through the manifestation of an action. In contrast,
an omniscient agent is privy to the objective truth values of all actions at
any point in time and does not need to allow the consequence of the action to
manifest to adjust their beliefs. In other words:

$\displaystyle\forall t,$ $\displaystyle\forall a,b,c,d\in\\{T,F,U\\},\forall
A_{n},A_{m}\in\mathbb{A}:A_{n}\neq A_{m},$
$\displaystyle(V_{I_{t}}(\oplus_{A_{m}}(A_{0},A_{1},\ldots,A_{j}))=T_{o})\Rightarrow$
$\displaystyle((\mathfrak{v}(I_{t}(A_{n})=[a_{s}/a_{o}])=\mathfrak{T}\Rightarrow
I_{(t+1)}(A_{n})=[b_{s}/b_{o}])\land$
$\displaystyle(\mathfrak{v}(I_{t}(A_{m})=[c_{s}/c_{o}])=\mathfrak{T}\Rightarrow
I_{(t+1)}(A_{m})=[d_{s}/d_{o}]))$

However, this is an eminently unrealistic assertion in practice. A realistic
model should incorporate latency and discrepancies to arrive at a more
accurate depiction of human action. Returning to the preservative model, the
previous selection process continues unless the following occurs:

$\displaystyle\forall
k\in\mathbb{N},V_{I_{(t+k)}}(\oplus_{(A_{m}\in\mathbb{A})}(A_{0},A_{1},\ldots,A_{j}))=T_{o}$
$\displaystyle\qquad\Leftrightarrow\exists!A_{m}\in\mathbb{A}:\forall
k\in\mathbb{N},\forall s\in\\{F_{s},U_{s}\\},\forall o_{0},o_{1}\in O,\forall
A_{n}\in\mathbb{A}:A_{n}\neq A_{m},$
$\displaystyle\qquad((\mathfrak{v}(I_{(t+k)}(A_{m})=[T_{s}/o_{0}])=\mathfrak{T})\land(\mathfrak{v}(I_{(t+k)}(A_{n})=[s/o_{1}])=\mathfrak{T}))$
$\displaystyle\qquad\lor((\mathfrak{v}(I_{(t+k)}(A_{m})=[U_{s}/o_{0}])=\mathfrak{T})\land(\mathfrak{v}(I_{(t+k)}(A_{n})=[F_{s}/o_{1}])=\mathfrak{T}))$

This can be characterized as an equilibrium state in which the further
selection of $A_{m}$ will not result in a different objective truth value:
resulting in it being selected indefinitely and irrespective of the objective
truth values of all other actions.

###  6.4 Discussion

So far, we have neglected cases where the agent is confronted with at least
two actions that are both perceived to be conducive to a particular objective
or when the subjective truth values for all actions are false or unknown. This
is because in such cases there does not exist a rational basis for selection.
Contemporary economic methods purport to address this by assigning weightings
to actions, thereby conveniently denoting the degree to which each action is
conducive to particular outcomes. But even these methods suggest that the
agent should be indifferent between virtually identical alternatives that
render similar levels of utility. To address this, an indifference curve
analysis can be employed to discriminate between such outcomes. If each action
incurs a cost, the outcome that yields the maximal level of utility given a
particular budget is chosen. It is possible for us to formalize this notion in
the metalanguage by assigning truth values on a nuanced criterion. For
example, a sentence letter, $A_{i}$, can be interpreted as asserting: “Given a
budget, $B$, allocating $X$ dollars to purchase good A, where $X\leq B$, and
$(B-X)$ dollars to purchase good B would allow us to reach the maximal amount
of utility derivable from B.” Whereas $A_{i+1}$ can be interpreted as saying:
“Given a budget, $B$, allocating $(X+1)$ dollars to purchase good A, where
$(X+1)\leq B$, and $(B-(X+1))$ dollars to purchase good B would allow us to
reach the maximal amount of utility derivable from B.” In some instances, this
may mathematically preclude the truth value $U$, since the indifference curve
approach is inherently deterministic and offers no room for absolute
ambiguity. Yet again, this is not necessarily true. For example, if the
marginal rate of substitution were increasing and the indifference curve
function which renders the maximal level of possible utility derivable from
the budget is concave, it would be possible for a linear budget constraint
function to reach the concave curve at more than one point, resulting in an
ambiguous state for which no unique allocation of one budget is present.

####  6.4.1 Example

As a demonstration, consider an example of an agent who is interested in
maximizing their utility through the selection of two perfectly complementary
goods. There are two actions in our model: either the agent can select good A
($A_{0}$), or it can select good B ($A_{1}$). Let us suppose that the
objective truth values for conducting either action are initially false. This
is because selecting either good on its own would not allow the agent to
realize the utility of their purchase.

The agent is initially skeptical of the need to purchase a complementary good.
In other words:

$\mathbb{A}=\\{A_{0},A_{1}\\}$ $I_{0}(A_{0})=[U_{s}/F_{o}]$
$I_{0}(A_{1})=[T_{s}/F_{o}]$

We now consider two connectives that specify whether or not a particular
action is selected, namely, $\oplus_{A_{0}}$ and $\oplus_{A_{1}}$. We have
previously specified that the valuation on such a connective is objectively
true under the following circumstances:

Recall from 6.2:

$V_{I_{t}}(\oplus_{(A_{m}\in\mathbb{A})}(A_{0},A_{1}))=T_{o}\Leftrightarrow\exists!A_{m}\in\mathbb{A}:\\\
\forall s\in\\{F_{s},U_{s}\\},\forall o_{0},o_{1}\in O,\forall
A_{n}\in\mathbb{A}:A_{n}\neq A_{m},\\\
((\mathfrak{v}(I_{t}(A_{m})=[T_{s}/o_{0}])=\mathfrak{T})\land(\mathfrak{v}(I_{t}(A_{n})=[s/o_{1}])=\mathfrak{T}))\\\
\lor\big{(}(\mathfrak{v}(I_{t}(A_{m})=[U_{s}/o_{0}])=\mathfrak{T})\land(\mathfrak{v}(I_{t}(A_{n})=[F_{s}/o_{1}])=\mathfrak{T})\big{)}$

Since the interpretation of $A_{0}$ when $t=0$ is $[U_{s}/F_{o}]$ and that of
$A_{1}$ is $[T_{s}/F_{o}]$, $V_{I}(\oplus_{A_{1}}(A_{0},A_{1}))=T_{o}$, hence,
$A_{1}$ is selected. Upon selecting this action, the agent’s subjective belief
collapses with the objective truth value of the action:

Recall from 6.3:

$\forall t,\forall a,b,c,d,e,f\in\\{T,F,U\\},\forall
A_{n},A_{m}\in\mathbb{A}:A_{n}\neq A_{m},\\\
(V_{I_{t}}(\oplus_{A_{m}}(A_{0},A_{1}))=T_{o})\Rightarrow\\\
((\mathfrak{v}(I_{t}(A_{n})=[a_{s}/b_{o}])=\mathfrak{T}\Rightarrow
I_{t+1}(A_{n})=[a_{s}/c_{o}])\land\\\
(\mathfrak{v}(I_{t}(A_{m})=[d_{s}/e_{o}])=\mathfrak{T}\Rightarrow
I_{t+1}(A_{m})=[e_{s}/f_{o}]))$

Since $A_{1}$ is selected, let $I_{1}(A_{1})=[F_{s}/F_{o}]$. In other words,
upon selecting the action, its consequences manifest, and the agent realizes
that the purchase of good B is insufficient. On the other hand, let
$I_{1}(A_{0})=[U_{s}/T_{o}]$. This implies that the selection of $A_{1}$ has
affected the objective truth value of $A_{0}$. Since the subjective truth
value of $A_{0}$ is unknown, whereas that of $A_{1}$ is false, $A_{0}$ will be
selected. This results in $I_{2}(A_{0})=[T_{s}/T_{o}]$. The process of
selection now comes to a meaningful halt because the subjective truth value of
the selected action now reflects its objective truth value, which is true
(i.e. conducive to the objective of realizing utility).

##  7 Theories of Choice and Alternative Forms of Consequence Realization

We may now consider the diverse ways in which we could define the behavior of
$I_{t}$, and correspondingly, $V_{I_{t}}$, across $t$. We have previously
assumed that upon selecting an arbitrary action, $A_{m}$, the subjective truth
values of all other actions, $a_{s}$, are preserved at $I_{t+1}$.
Additionally, we have also assumed that the only way in which an action can
manifest its objective truth value is by selecting that action. However, it is
also possible for the subjective truth values of other un-selected actions to
change on the basis of the realization of an objective truth value of any
particular action, whether it has been selected or not. In other words, it
could be useful to consider a lenient form of consequence realization:

$\forall t,\forall a,b,c,d\in\\{T,F,U\\},\forall
A_{n}\in\mathbb{A},\mathfrak{v}(I_{t}(A_{n})=[a_{s}/b_{o}])=\mathfrak{T}\Rightarrow
I_{t+1}(A_{n})=[c_{s}/d_{o}]$

On its own, this definition does not offer any meaningful information on how
we can make predictions regarding dynamic beliefs or changing realities.
Hence, for us to continue our analysis it becomes necessary for us to
formalize the circumstances in which an action is selected and the
corresponding architecture of the individual’s belief.

###  7.1 Monty Hall Problem

An example that illustrates the need to formalize the “theories” under which
actions are selected is the Monty Hall Problem. Consider being in a game show
where three doors are present before you; a goat lies behind two doors whereas
a car lies behind the third. The objective of the utility-maximizing agent is
to select a door that results in the obtainment of the car. Upon selecting a
door, Monty Hall reveals one of the doors that did not have a car behind it.
You are then given the choice of either selecting your current door or
selecting the other door which has not yet been revealed or selected.

Let us define each action in the following manner:
$\mathbb{A}=\\{D_{1},D_{2},D_{3}\\}$

Where $D_{i}$ is syntactically equivalent to saying that the act of selecting
the $i^{th}$ door would result in the obtainment of the car. Let us suppose
that the interpretational truth value of each action is informally defined:

$\displaystyle I_{0}(D_{1})=[U_{s}\textfractionsolidus F_{s}/T_{o}]$
$\displaystyle I_{0}(D_{2})=[U_{s}\textfractionsolidus F_{s}/F_{o}]$
$\displaystyle I_{0}(D_{3})=[T_{s}/F_{o}]$

It is assumed that the action we have selected, namely, $D_{3}$, is
subjectively true (whereas the subjective truth values of $D_{1}$ and $D_{2}$
are unknown, or false) precisely because the game requires us to select an
action over others, thereby revealing our preference. However, upon selecting
the action, the objective truth value of selecting the 3rd door does not
manifest itself directly. On the contrary, we become privy to the objective
truth value of the second door, which is revealed to be false. The updated
truth values are as follows.

$\displaystyle I_{1}(D_{1})=[U_{s}\textfractionsolidus F_{s}/T_{o}]$
$\displaystyle I_{1}(D_{2})=[F_{s}/F_{o}]$ $\displaystyle
I_{1}(D_{3})=[T_{s}/F_{o}]$

According to our previous set of axioms, we have demonstrated that when an
agent believes that an action leads to a particular outcome, then the agent
should have no reason to spontaneously alter their beliefs unless the
consequences of that action materialize and are revealed to be contrary to
what is expected. In other words, if the agent’s conviction in their initial
selection (3rd door) is genuinely true, the agent should have no reason to
then select the first door: which they initially did not select because its
subjective truth value was false/unknown as opposed to being true. This runs
athwart the conventional result from probabilistic reasoning, namely: if the
likelihood of initially selecting a goat is comparatively high, then the
likelihood of being able to obtain a car upon the reversal of one’s initial
selection would be comparatively high as well. Thus, from an epistemological
perspective, the selection of an action is not necessarily a reflection of
one’s faith in its ability to be conducive to a particular outcome. Moreover,
upon selecting an action, it may not manifest its objective truth value
directly but result in the actualization of the objective truth values of
other courses of action, which in turn affect the subjective truth values of
the remaining set of actions (despite the fact that they have not been
selected).

###  7.2 Overview of Preceding Theory

We have previously stated that when an agent is confronted with at least two
actions that are both perceived to be conducive to a particular objective or
when the subjective truth values of all actions are false or unknown, there
does not exist a rational basis for selection. The definition of rationality
espoused is limited to cases where we can denote an agent’s clear preference
for one course of action over another. To gain a comprehensive view of the
theory we have developed thus far, consider a set, $\Gamma_{R}$, which denotes
the theory for strictly rational behavior under belief preservation:

$\displaystyle\Gamma_{R}:=\Bigg{\\{}$ $\displaystyle\begin{aligned}
&\overbrace{{\mathbb{A}=\\{{A_{j}|j\in\mathbb{N},j\geq 2\\}}}}^{\text{Action
Set}},\overbrace{{S=\\{{T_{s},F_{s},U_{s}\\}}}}^{\text{Subjective
Faith}},\overbrace{{O=\\{{T_{o},F_{o},U_{o}\\}}}}^{\text{Objective
Realities}},\\\ &\underbrace{{(\forall i,\forall t,\exists!s\in S,\exists!o\in
O,I_{t}(A_{i})=[s/o])}}_{\text{Interpretation of an action}},\\\
&\overbrace{{\oplus=\left\\{\begin{array}[]{@{}l@{}}\oplus_{A_{0}}(A_{0},A_{1},\ldots,A_{j})\\\
\quad\quad\quad\quad\vdots\\\
\oplus_{A_{j}}(A_{0},A_{1},\ldots,A_{j})\end{array}\right\\}}}^{\text{Action
Selector}}\end{aligned}$ $\displaystyle\begin{aligned} &,\underbrace{{(\forall
I_{t},\forall\oplus_{(A\in\mathbb{A})},V_{I_{t}}(\oplus_{(A\in\mathbb{A})}(A_{0},A_{1},\ldots,A_{j}))\in
O)}}_{\text{Selections have an Objective Presence}},\\\
&\overbrace{({V_{I_{t}}(\oplus_{(A_{m}\in\mathbb{A})}(A_{0},A_{1},\ldots,A_{j}))=T_{o}})}^{\text{Outcome:
Selection of $A_{m}$}}\Leftrightarrow(\exists!A_{m}\in\mathbb{A}:\forall
s\in\\{F_{s},U_{s}\\},\forall o_{0},o_{1}\in O,\\\ &\underbrace{\forall
A_{n}\in\mathbb{A}:A_{n}\neq A_{m}}_{\text{Specifies non-selected
action}},\underbrace{(({\mathfrak{v}(I_{t}(A_{m})=[T_{s}/o_{0}])=\mathfrak{T}})\land(\mathfrak{v}(I_{t}(A_{n})=[s/o_{1}])=\mathfrak{T}))}_{\text{First
condition for selection}}\lor\\\
&\overbrace{((\mathfrak{v}(I_{t}(A_{m})=[U_{s}/o_{0}])=\mathfrak{T})\land(\mathfrak{v}(I_{t}(A_{n})=[F_{s}/o_{1}])=\mathfrak{T}))}^{\text{Alternative
condition for selection}})\end{aligned}$ $\displaystyle\begin{aligned}
&(\forall t,\forall a,b,c,d,e,f\in\\{T,F,U\\},\forall
A_{n},A_{m}\in\mathbb{A}:A_{n}\neq A_{m},\\\
&(\overbrace{{V_{I_{t}}(\oplus_{A_{m}}(A_{0},A_{1},\ldots,A_{j}))=T_{o}}}^{\text{Condition:
If an action is
selected...}})\Rightarrow((\overbrace{{\mathfrak{v}(I_{t}(A_{n})=[a_{s}/b_{o}])=\mathfrak{T}}}^{\text{Specifies
initial value}}\Rightarrow\\\
&\underbrace{I_{(t+1)}(A_{n})=[a_{s}/c_{o}])}_{\text{Specifies updated
value}}\land\underbrace{(\mathfrak{v}(I_{t}(A_{m})=[d_{s}/e_{o}])=\mathfrak{T}\Rightarrow
I_{(t+1)}(A_{m})=[e_{s}/f_{o}])}_{\text{Transmission of $e_{o}$ to $e_{s}$ via
selection}}))\Bigg{\\}}\end{aligned}$

In concise terms, we have defined the set of actions, the set of possible
truth values, the relationship between an interpretation of an action and its
corresponding truth value, the semantic properties of a $j$-nary connective,
and the process by which the selection of an action (i.e. a valuation on an
action that results in an objective truth, $T_{o}$) generates updated
interpretations.

The theory for strict rational choice, $\Gamma_{R}$, does not address the
selection of an action in the following cases, or addresses them trivially by
assigning an unknown valuation:

$\displaystyle\exists t:\forall o_{1}\in O,\forall
A_{w}\in\mathbb{A},\mathfrak{v}(I_{t}(A_{w})=[T_{s}/o_{1}])=\mathfrak{T}$
$\displaystyle\exists t:\forall o_{1}\in O,\forall
A_{w}\in\mathbb{A},\mathfrak{v}(I_{t}(A_{w})=[F_{s}/o_{1}])=\mathfrak{T}$
$\displaystyle\exists t:\forall o_{1}\in O,\forall
A_{w}\in\mathbb{A},\mathfrak{v}(I_{t}(A_{w})=[U_{s}/o_{1}])=\mathfrak{T}$
$\displaystyle\exists t:\forall j>2,\forall a\in\\{F_{s},U_{s}\\},\forall
o_{1},o_{2}\in O,\forall A_{w},A_{n}\in\mathbb{A},$ $\displaystyle\quad\exists
d\in\mathbb{N},j>d\geq 2:\\{A_{n}\\}_{n=1}^{d}\subset\mathbb{A}:\forall
n,A_{n}\neq A_{w},$
$\displaystyle\quad(\mathfrak{v}(I_{t}(A_{n})=[T_{s}/o_{1}])=\mathfrak{T})\land(\mathfrak{v}(I_{t}(A_{w})=[a/o_{2}])=\mathfrak{T})$
$\displaystyle\exists t:\forall j>2,\forall o_{1},o_{2}\in O,\forall
A_{w},A_{n}\in\mathbb{A},$ $\displaystyle\quad\exists d\in\mathbb{N},j>d\geq
2:\\{A_{n}\\}_{n=1}^{d}\subset\mathbb{A}:\forall n,A_{n}\neq A_{w},$
$\displaystyle\quad(\mathfrak{v}(I_{t}(A_{n})=[U_{s}/o_{1}])=\mathfrak{T})\land(\mathfrak{v}(I_{t}(A_{w})=[F_{s}/o_{2}])=\mathfrak{T})$

####  7.2.1 Translations from formal language

The following cases are not addressed due to the implicit notion of revealed
preference:
1\. Cases where every action is interpreted as being conducive to a particular
objective.
2\. Cases where every action is not interpreted as being conducive to a
particular objective.
3\. Cases where the interpretation of every action is unknown.
4\. Cases where the interpretation of at least two actions are true, whereas
that of all other actions are false or unknown.
5\. Cases where the interpretation of at least two actions are unknown, and
the rest are interpreted as being false.

Recall from section 5 where the definition of a $j$-nary connective is
presented, we claimed that such a connective can only show whether it is
$necessarily$ the case that a unique action has been selected. If more than
one action is conducive to the realization of a particular objective, the
agent should have no reason to make a selection on the basis of one’s
preference, since the $explicit$ $selection$ of an action necessarily implies
that the agent $explicitly$ $prefers$ one action over the other. As we have
stated in section 6.4, these are cases where there does not exist a strictly
rational basis of selection, thereby necessitating alternative theories.

###  7.3 Alternative Theory

The primary justification for the selection of any action is to check whether
its objective truth value would prove to be ephemerally true, or robustly and
consistently true. However, different philosophical beliefs underpin such
methodologies. To demonstrate this, consider how the selection of an action
occurs under the belief of philosophical skepticism, that is, the belief that
objective truths are unknowable (Klein). Insofar as we are only capable of
perceiving the extent to which an action is conducive to a particular outcome
through subjective truth values, any method of selection on the basis of this
perception must be unreliable. Although this position results in the
invalidation of all selection methods, we may consider a similar though less
stringent position of empirical skepticism ($\Gamma_{ES}$). Under
$\Gamma_{ES}$, an action is selected when its probability of realizing a
particular objective exceeds that of all other actions.

$\displaystyle\Gamma_{ES}:=\Bigg{\\{}$ $\displaystyle\begin{aligned}
&\overbrace{{\mathbb{A}=\\{{A_{j}|j\in\mathbb{N},j\geq 2\\}}}}^{\text{Action
Set}},\overbrace{{S=\\{{x\in\mathbb{R}|0\leq x\leq 1\\}}}}^{\text{Subjective
Faith}},\overbrace{{O=\\{{y\in\mathbb{R}|0\leq y\leq 1\\}}}}^{\text{Objective
Realities}},\\\ \\\ &\underbrace{{(\forall i,\forall t,\exists!s\in
S,\exists!o\in O,I_{t}(A_{i})=[s/o])}}_{\text{Interpretation of an
action}},\\\
&\overbrace{{\oplus=\left\\{\begin{array}[]{@{}l@{}}\oplus_{A_{0}}(A_{0},A_{1},\ldots,A_{j})\\\
\quad\quad\quad\quad\vdots\\\
\oplus_{A_{j}}(A_{0},A_{1},\ldots,A_{j})\end{array}\right\\}}}^{\text{Action
Selector}}\end{aligned}$ $\displaystyle\begin{aligned} &,\underbrace{{(\forall
I_{t},\forall\oplus_{(A\in\mathbb{A})},V_{I_{t}}(\oplus_{(A\in\mathbb{A})}(A_{0},A_{1},\ldots,A_{j}))\in\\{T_{o},F_{o},U_{o}\\})}}_{\text{Selections
have an Objective Presence}},\\\
&\overbrace{{(V_{I_{t}}(\oplus_{(A_{m}\in\mathbb{A})}(A_{0},A_{1},\ldots,A_{j}))=T_{o})}}^{\text{Outcome:
Selection of $A_{m}$}}\Leftrightarrow(\exists!A_{m}\in\mathbb{A}:\forall
s_{0},s_{1}\in S,\forall o_{0},o_{1}\in O,\\\ &\underbrace{{\forall
A_{n}\in\mathbb{A}:A_{n}\neq A_{m}}}_{\text{Specifies non-selected
action}},\underbrace{{((\mathfrak{v}(I_{t}(A_{m})=[s_{0}/o_{0}])=\mathfrak{T})\land(\mathfrak{v}(I_{t}(A_{n})=[s_{1}/o_{1}])=\mathfrak{T}))}}_{\text{Specifies
the initial values of a selected and a non-selected action}}\\\
&\land\overbrace{{(s_{1}>s_{0})}}^{\text{Condition}}),(\forall t,\forall
a,d,e\in S,\forall b,c,e,f\in O,\forall A_{n},A_{m}\in\mathbb{A}:A_{n}\neq
A_{m},\\\
&(\overbrace{{V_{I_{t}}(\oplus_{A_{m}}(A_{0},A_{1},\ldots,A_{j}))=T_{o}}}^{\text{Condition:
If an action is
selected...}})\Rightarrow((\overbrace{{\mathfrak{v}(I_{t}(A_{n})=[a/b])=\mathfrak{T}}}^{\text{Specifies
initial value}}\Rightarrow\\\
&\underbrace{{I_{(t+1)}(A_{n})=[a/c]}}_{\text{Specifies updated
value}})\land\underbrace{{(\mathfrak{v}(I_{t}(A_{m})=[d/e])=\mathfrak{T}\Rightarrow
I_{(t+1)}(A_{m})=[e/f])}}_{\text{Transmission of $e$ via
selection}}))\Bigg{\\}}\end{aligned}$

##  8 Areas of Further Discussion

One area of further discussion regarding the preceding model is whether
government intervention could lead to permanent distortions in a rational
agent’s subjective perception of his or her objective circumstances, thereby
producing inefficient social outcomes. This can be modelled by limiting the
type of actions that are featured in the agent’s action set, or by inflating
the agent’s subjective perception of certain actions to artificially high
values (to indicate their perceived effectiveness, or in discrete cases,
$True$). Admittedly, such a system would go beyond what this paper could
possibly cover, and the notations used to express $\Gamma_{R}$ and
$\Gamma_{ES}$ would probably pale in complexity.

##  9 Conclusion

The current analysis of dynamic many-valued logic is far from complete and
deserves further analyses. A variety of applications are predicated on this
foundational approach to economic reasoning. Additionally, further analyses
could be made by examining an efficient method of defining an updated
interpretation on the basis of a selected action, or a series of selected
actions; this, however, ultimately rests on the structure of the dynamic
problem being investigated. It is also worth considering a set of sets of
actions that exist for each moment of interpretation, $t$, thereby allowing us
to model certain decisions that expand or constrict the domain of possible
options. The explicit enunciation of the underlying philosophy behind the
selection of an action in cases where no action results in a strict preference
would allow for the rigorous identification of underlying assumptions,
resulting in a robust method of selection.

## References

1\. Falkenstein, Lorne, et al. Logic Works : A Rigorous Introduction to Formal
Logic. New York, Routledge, Taylor and Francis Group, 2022.
2\. Klein, Peter. “Skepticism (Stanford Encyclopedia of Philosophy).”
Stanford.edu, 2015, plato.stanford.edu/entries/skepticism/.