1 Non-Transferability in Communication Channels and Tarskis Truth Theorem Farhad Naderian

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Non-Transferability in Communication Channels and Tarski’s Truth Theorem

Farhad Naderian

System Integration Specialist, Freelance; Tehran, Iran

naderian51@gmail.com

ORCID:0000-0002-1771-2325

Abstract

This article explores the concept of transferability within communication channels, with a

particular focus on the inability to transmit certain situations through these channels. The Channel

Non-Transferability Theorem establishes that no encoding-decoding mechanism can fully transmit

all propositions, along with their truth values, from a transmitter to a receiver. The theorem

underscores that when a communication channel attempts to transmit its own error state, it

inevitably enters a non-transferable condition. I argue that Tarski's Truth Undefinability Theorem

parallels the concept of non-transferability in communication channels. As demonstrated in this

article, the existence of non-transferable codes in communication theory is mathematically

equivalent to the undefinability of truth as articulated in Tarski's theorem. This equivalence is

analogous to the relationship between the existence of non-computable functions in computer

science and Gödel’s First Incompleteness Theorem in mathematical logic. This new perspective

sheds light on additional aspects of Tarski's theorem, enabling a clearer expression and

understanding of its implications.

Keywords: Non-Transferability, Channel Theory, Tarski's Truth Theorem, Semantic.

1. Introduction

In modern telecommunications, data transmission relies on the standard model of communication

proposed by Shannon and Weaver [1]. As shown in Figure 1 , this traditional model comprises

three main components.

Fig.1 The traditional scheme of a communication channel

1. Source/Channel Coding: It is responsible for getting data D and converting it to the

suitable string .

2. Channel: It takes String  and sends it toward the receiver.

3. Source/Channel Decoding: It constructs the final data  based on the received signal 

from the channel.

In an ideal communication channel, =, meaning the final received data is identical or as close

as possible to the transmitted data [2].

This model primarily addresses syntax, not the origin or meaning of data D. Consequently, it

cannot address semantic issues unless the source of D is defined. To account for meaning, the

model must be expanded appropriately.

This article aims to address semantic issues by introducing a new concept: “non-transferability of

meaning”. This concept, previously unaddressed, is introduced here for the first time1.

Additionally, the article explores a fundamental question about the “existence of non-transferable

situations in communication channels.”

Investigating the non-transferability of meaning in communication channels depends on

foundational research into the semantics and logic of meaning. Fortunately, the concept of meaning

has been extensively examined from both logical and semantic perspectives [3]. The article builds

on these studies to examine non-transferability issues in communication channels and establishes

its relationship with a significant result in mathematical logic.

One of the key restrictive results in mathematical logic is Tarski’s undefinability theorem, which

states that there is no formal method to define “Truth” within the language of arithmetic [4].

Initially formulated to address the semantic concept of truth in mathematical logic, this theorem

has not been explored in other fields until now. This article draws a parallel between Tarski’s

theorem and non-transferability in communication channels, finding a similarity to the relationship

between Gödel’s incompleteness theorem and halting problem in Turing machines. Gödel’s

1 The term “non-transferability” is used in this article in a new meaning for the first time. A formal definition of it is at the end of

Section 2.

incompleteness theorem reveals that some true propositions in arithmetic cannot be proven within

arithmetic itself, analogous to some codes that, when given to a universal Turing machine, cause

it not to halt. This relationship connects mathematical logic and computer science and clarified by

Scott Aaronson in his 2011 blog at “Rosser's theorem via Turing machines. Shtetl-Optimized”.

Likewise, Tarski’s truth theorem can be seen as connected to non-transferability issues in

communication channels, suggesting that the undefinability of truth in logic aligns with limitations

in communication theory.

In the following, Section 2 will provide foundational definitions to establish the groundwork for

the article, including a generalization of the communication model to address the semantic issues.

A definition of transferability in communication channels appear at the end of Section 2. The

occurrence of non-transferable circumstances in communication channels is demonstrated in

Section 3. The article's central claim—that non-transferability in communication channels is

equivalent to the undefinability of truth in arithmetic—is presented in Section 4. The translation

of the Liar Paradox into communication theory, an immediate corollary of this theorem, is also

covered in this section. The findings and conclusions of this study are summed up in Section 5.

2. Communication channel model

In practical communication channels, data originates from real-world scenarios or sets of facts.

For instance, it may represent the switching status of devices in the utility system, alarms within a

telecommunication network, or measurements collected by sensor arrays. Additionally, it could

involve signals derived from real-time sources such as voice, images or video. While some data

sources are inherently analog, they are converted into digital streams to facilitate transmission over

digital channels. In Figure 2 , a communication channel scheme is represented to model semantic

issues. In this figure, set or situation S refers to some state of affairs in the world. The

communication channel must be able to send those states of affairs to the receiver.

Fig.2 A communication channel scheme to model semantic issues

There are various methods to represent these states of affairs; however, for the purposes of this

paper, they will be considered as sets of simple first-order sentences expressed through unary (1-

ary) relations or properties assigned to different objects denoted as #:

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1Non-TransferabilityinCommunicationChannelsandTarski’sTruthTheoremFarhadNaderianSystemIntegrationSpecialist,Freelance;Tehran,Irannaderian51@gmail.comORCID:0000-0002-1771-2325AbstractThisarticleexplorestheconceptoftransferabilitywithincommunicationchannels,withaparticularfocusontheinabilitytotransmit...

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