# Unknown Truths and Unknowable Truths

Jie Fan
Institute of Philosophy, Chinese Academy of Sciences;
School of Humanities, University of Chinese Academy of Sciences
<EMAIL_ADDRESS>
###### Abstract

Notions of unknown truths and unknowable truths are important in formal
epistemology, which are related to each other in e.g. Fitch’s paradox of
knowability. Although there have been some logical research on the notion of
unknown truths and some philosophical discussion on the two notions, there
seems to be no logical research on unknowable truths. In this paper, we
propose a logic of unknowable truths, investigate the logical properties of
unknown truths and unknowable truths, which includes the similarities of the
two notions and the relationship between the two notions, and axiomatize this
logic.

Keywords: unkown truths, unknowable truths, axiomatization, Fitch’s paradox of
knowability

##  1 Introduction

This article investigates notions of unknown truths and unknowable truths. A
proposition is an unknown truth, if it is true but unknown; a proposition is
an unknowable truth, if it is true but unknowable. The two notions are related
to each other in e.g. Fitch’s paradox of knowability, which states that if
there is an unknown truth, then there is an unknowable truth.

The notion of unknown truths is important in formal epistemology. For example,
it is related to verification thesis, which says that all truths can be known.
By the thesis and two uncontroversial principles of knowledge, it follows from
the notion that all truths are actually known. In this way, the notion gives
rise to a well-known counterexample to the verification thesis. This is the
so-called ‘Fitch’s paradox of knowability’. To take another example: the
notion gives us an important type of Moore sentences, which is in turn crucial
to Moore’s paradox, saying that one cannot claim the paradoxical sentence “$p$
but I do not know it”. A well-known result is that such a Moore sentence is
unsuccessful and self-refuting [17, 10].

The notion of unknowable truths is crucial in the dispute of realists and
anti-realists. Anti-realists holds the aforementioned verificationist thesis,
which states that all truths can be known, thereby denying the very
possibility of unknowable truths. In contrast, realists believe that there are
parts of reality, representable in some conceptually accessible language, that
it is impossible for any agent ever to know. As guessed in [19, p. 119],
“Perhaps, in truths that cannot be known lies ‘the mystery at the heart of
things’.”

Note that the notion of unknowable truths also gives an important type of
Moore sentences, since one cannot claim the paradoxical sentence “$p$ is true
but I cannot know it”. We can also show that such a Moore sentence is
unsuccessful and self-refuting.

Some researchers in the literature discuss about whether unknowable truths
exist or not. For example, Edgington [8] admits that there are unknowable
truths in the sense of Fitch’s argument. Horsten [18] gives an interesting
example of unknowable truths, by using a revision of the Gödel sentence. Cook
[5] argues that even if it is impossible for it to be known of any particular
sentence that it is both true and unknowable, we may still know that there are
unknowable truths.111However, it is shown in [7] to be unsuccessful since the
argument depends on a paradoxical reasoning. Besides, it is argued in [19,
Sec. 2] that there are unknowable truths, that is, necessary limits to
knowledge. As we will show below, under our interpretation of knowability,
there are indeed unknowable truths. Thus our interpretation is in line with
the aforementioned philosophical discussion.

In the literature, there have been several interpretations for the notion of
knowability. For instance, in [20, 3], knowability means ‘known after an
announcement’; in [6], knowability means the existence of proofs; in [12],
knowability means capabilities to know; in [19], it is interpreted as a
capacity-knowledge modality ‘it can be known that’; in [23], knowability means
‘known after an information update’; whereas in [16], knowability means ‘
dynamically possible knowledge of what was true before the update of the
epistemic state of the agent’. In this paper, we follow [20, 3] to interpret
‘$\varphi$ is knowable’ as ‘$\varphi$ is known after some truthful public
announcement’, so that ‘$\varphi$ is an unknowable truth’ is interpreted as
‘$\varphi$ is true and after any truthful public announcement, $\varphi$ is
unknown’.

Unknowable truths (and unknown truths) is a subjective concept, since it is
possible that a proposition is an unknowable truth (an unknown truth, resp.)
for an agent but not for another. For instance, consider the true proposition
“it is raining but Ann does not know it”. This proposition is an unknowable
truth (an unknown truth, resp.) for Ann, but is not an unknowable truth (an
unknown truth, resp.) for another agent Bob, who may be aware of Ann’s
ignorance. Therefore we move to the multiagent cases.

Although there have been some logical research on the notion of unknown truths
(see e.g. [14, 11]), and also some philosophical discussion on the two notions
[13, 5, 7, 19], there seems to be no logical research on unknowable truths.
For instance, there have been no axiomatization on the notion of unknowable
truths. For another example, it is unnoticed that there are many similarities
and relationship between the notions of unknown truths and unknowable truths.
In this paper, we propose a logic of unknowable truths, investigate the
logical properties of unknown truths and unknowable truths, which include the
similarities and relationship between the two notions, and axiomatize the
logic of unknowable truths.

The remainder of the article is organized as follows. Section 2 introduces the
language and semantics of the logic of unknowable truths. Section 3
investigates logical properties of unknown truths and unknowable truths.
Section 4 axiomatizes the logic of unknowable truths and shows its soundness.
Section 5 demonstrates the completeness. We conclude with some discussion in
Section 6.

##  2 Syntax and Semantics

Throughout the paper, we let P be the denumerable set of propositional
variables, and let I be a finite set of agents.

######  Definition 1.

Where $p\in\textbf{P}$, $i\in\textbf{I}$, the language of the logic of
unknowable truths, denoted LUT, is defined recursively as follows:

$\begin{array}[]{lll}\textbf{LUT}&\varphi::=&p\mid\neg\varphi\mid(\varphi\land\varphi)\mid\textit{K}_{i}\varphi\mid[\varphi]\varphi\mid
U_{i}\varphi\\\ \end{array}$

$\textit{K}_{i}\varphi$ is read “agent $i$ knows that $\varphi$”,
$[\psi]\varphi$ is read “after every truthfully public announcement of $\psi$,
it is the case that $\varphi$”, and $U_{i}\varphi$ is read “$\varphi$ is an
unknowable truth for $i$”. Other connectives are defined as usual; in
particular, $\bullet_{i}\varphi$, read “$\varphi$ is an unknown truth”,
abbreviates $\varphi\land\neg\textit{K}_{i}\varphi$, and
$\langle\psi\rangle\varphi$ abbreviates $\neg[\psi]\neg\varphi$. Without the
construct $U_{i}\varphi$, we obtain public announcement logic PAL; without the
construct $[\varphi]\varphi$ further, we obtain epistemic logic EL. The
language LUT is interpreted on models.

######  Definition 2.

A model $\mathcal{M}$ is a triple $\langle S,\\{R_{i}\mid
i\in\textbf{I}\\},V\rangle$, where $S$ is a nonempty set of states, each
$R_{i}$ is the accessibility relation for $i$, and $V$ is a valuation. We
assume that every $R_{i}$ is reflexive.222Note that here for simplicity, we
only consider the minimal restriction of knowledge. We can also consider extra
properties of knowledge, for instance, transitivity, Euclidicity, etc., in
which cases we add the corresponding axioms for knowledge in the
axiomatization. A frame is a model without valuations. A pointed model is a
pair of a model and a state in the model.

######  Definition 3.

Given a model $\mathcal{M}=\langle S,\\{R_{i}\mid i\in\textbf{I}\\},V\rangle$,
the semantics of LUT is defined recursively as follows.

$\begin{array}[]{|lll|}\hline\cr\mathcal{M},s\vDash p&\iff&s\in V(p)\\\
\mathcal{M},s\vDash\neg\varphi&\iff&\mathcal{M},s\nvDash\varphi\\\
\mathcal{M},s\vDash\varphi\land\psi&\iff&\mathcal{M},s\vDash\varphi\text{ and
}\mathcal{M},s\vDash\psi\\\
\mathcal{M},s\vDash\textit{K}_{i}\varphi&\iff&\text{for all }t,\text{ if
}sR_{i}t,\text{ then }\mathcal{M},t\vDash\varphi\\\
\mathcal{M},s\vDash[\psi]\varphi&\iff&\text{if }\mathcal{M},s\vDash\psi\text{
then }\mathcal{M}|_{\psi},s\vDash\varphi\\\ \mathcal{M},s\vDash
U_{i}\varphi&\iff&\mathcal{M},s\vDash\varphi\text{ and for all
}\psi\in\textbf{EL},\mathcal{M},s\vDash{[\psi]\neg\textit{K}_{i}}\varphi\\\
\hline\cr\end{array}$

where $\mathcal{M}|_{\psi}$ is the model restriction of $\mathcal{M}$ to
$\psi$-states, and the restriction of announced formula in the semantics of
$U_{i}$ is to avoid circularity. We say that $\varphi$ is valid over a frame
$F$, if for all models $\mathcal{M}$ based on $F$ and all states $s$, we have
$\mathcal{M},s\vDash\varphi$. We say that $\varphi$ is valid, denoted
$\vDash\varphi$, if it is valid over all (reflexive) frames. We say that
$\varphi$ is satisfiable, if there is a pointed model $(\mathcal{M},s)$ such
that $\mathcal{M},s\vDash\varphi$.

Intuitively, $U_{i}\varphi$ means that $\varphi$ is true and unknowable for
$i$; in more details, $\varphi$ is true and after each truthfully public
announcement, $\varphi$ is unknown to $i$.

One may compute the semantics of $\bullet_{i}$ and $\langle\psi\rangle\varphi$
as follows:

$\begin{array}[]{lll}\mathcal{M},s\vDash\bullet_{i}\varphi&\iff&\mathcal{M},s\vDash\varphi\text{
and for some }t,sR_{i}t\text{ and }\mathcal{M},t\nvDash\varphi\\\
\mathcal{M},s\vDash\langle\psi\rangle\varphi&\iff&\mathcal{M},s\vDash\psi\text{
and }\mathcal{M}|_{\psi},s\vDash\varphi.\\\ \end{array}$

##  3 Logical properties

In this section, we investigate logical properties of unknown truths and
unknowable truths.

###  3.1 Similarities

First, we can summarize some similarities between the logical properties of
unknown truths and the logical properties of unknowable truths in the
following diagram.

$\begin{array}[]{|c||c|c|}\hline\cr&\text{Unknown Truths}&\text{Unknowable
Truths}\\\ \hline\cr&\bullet_{i}\varphi\to\varphi&U_{i}\varphi\to\varphi\\\
&\bullet_{i}(\varphi\to\psi)\to(\bullet_{i}\varphi\to\bullet_{i}\psi)&U_{i}(\varphi\to\psi)\to(U_{i}\varphi\to
U_{i}\psi)\\\
&\bullet_{i}\varphi\land\bullet_{i}\psi\to\bullet_{i}(\varphi\land\psi)&U_{i}\varphi\land
U_{i}\psi\to U_{i}(\varphi\land\psi)\\\
&\bullet_{i}\varphi\to\bullet_{i}{\bullet_{i}\varphi}&U_{i}\varphi\to
U_{i}U_{i}\varphi\\\
\text{validities}&\bullet_{i}\varphi\leftrightarrow\bullet_{i}{\bullet_{i}\varphi}&U_{i}\varphi\leftrightarrow
U_{i}U_{i}\varphi\\\
&\neg\textit{K}_{i}{\bullet_{i}\varphi}&\neg\textit{K}_{i}U_{i}\varphi\\\
&\neg{\bullet_{i}\textit{K}_{i}\varphi}~{}(\text{transitive})&\neg
U_{i}\textit{K}_{i}\varphi~{}(\text{transitive})\\\
&\neg{\bullet_{i}\neg\textit{K}_{i}\varphi}~{}(\text{Euclidean})&\neg
U_{i}\neg\textit{K}_{i}\varphi~{}(\text{Euclidean})\\\
&[\bullet_{i}p]\neg{\bullet_{i}p}&[U_{i}p]\neg U_{i}p\\\
\hline\cr\hline\cr\text{invalidities}&\neg{\bullet_{i}\varphi}\to\bullet_{i}\neg{\bullet_{i}\varphi}&\neg
U_{i}\varphi\to U_{i}\neg U_{i}\varphi\\\ \hline\cr\end{array}$

In what follows, we take some of the validities and invalidities as examples,
and leave others to the reader.

######  Proposition 4.

$\vDash U_{i}(\varphi\to\psi)\to(U_{i}\varphi\to U_{i}\psi)$.

###### Proof.

Let $\mathcal{M}=\langle S,\\{R_{i}\mid i\in\textbf{I}\\},V\rangle$ be a model
and $s\in S$. Suppose that $\mathcal{M},s\vDash U_{i}(\varphi\to\psi)$ and
$\mathcal{M},s\vDash U_{i}\varphi$, to show that $\mathcal{M},s\vDash
U_{i}\psi$. By supposition, we have that $\mathcal{M},s\vDash\varphi$ and for
all $\chi\in\textbf{EL}$,
$\mathcal{M},s\vDash[\chi]\neg\textit{K}_{i}\varphi$, and that
$\mathcal{M},s\vDash\varphi\to\psi$ and for all $\chi\in\textbf{EL}$,
$\mathcal{M},s\vDash[\chi]\neg\textit{K}_{i}(\varphi\to\psi)$. From
$\mathcal{M},s\vDash\varphi$ and $\mathcal{M},s\vDash\varphi\to\psi$, it
follows that $\mathcal{M},s\vDash\psi$. From for all $\chi\in\textbf{EL}$,
$\mathcal{M},s\vDash[\chi]\neg\textit{K}_{i}(\varphi\to\psi)$, we can derive
that for all $\chi\in\textbf{EL}$,
$\mathcal{M},s\vDash[\chi]\neg\textit{K}_{i}\psi$. Therefore,
$\mathcal{M},s\vDash U_{i}\psi$. ∎

######  Proposition 5.

$\vDash U_{i}\varphi\land U_{i}\psi\to U_{i}(\varphi\land\psi)$.

###### Proof.

Let $\mathcal{M}=\langle S,\\{R_{i}\mid i\in\textbf{I}\\},V\rangle$ be a model
and $s\in S$. Suppose that $\mathcal{M},s\vDash U_{i}\varphi$ and
$\mathcal{M},s\vDash U_{i}\psi$, to show that $\mathcal{M},s\vDash
U_{i}(\varphi\land\psi)$. By supposition, $\mathcal{M},s\vDash\varphi$ and for
all $\chi\in\textbf{EL}$,
$\mathcal{M},s\vDash[\chi]\neg\textit{K}_{i}\varphi$, and
$\mathcal{M},s\vDash\psi$ and for all $\chi\in\textbf{EL}$,
$\mathcal{M},s\vDash[\chi]\neg\textit{K}_{i}\psi$. It follows that
$\mathcal{M},s\vDash\varphi\land\psi$ and for all $\chi\in\textbf{EL}$,
$\mathcal{M},s\vDash[\chi]\neg\textit{K}_{i}(\varphi\land\psi)$, and therefore
$\mathcal{M},s\vDash U_{i}(\varphi\land\psi)$. ∎

Intuitively, one cannot know the unknowable truths. This is because unknowable
truths are themselves unknowable truths, as shown below. One can show that it
is equivalent to the statement that it is impossible for it to be known of any
particular sentence that it is both true and unknowable, where the latter is
argued by Cook [5].

######  Proposition 6.

$\vDash U_{i}\varphi\to U_{i}U_{i}\varphi$, and thus $\vDash
U_{i}\varphi\leftrightarrow U_{i}U_{i}\varphi$.

###### Proof.

Let $\mathcal{M}=\langle S,\\{R_{i}\mid i\in\textbf{I}\\},V\rangle$ be a model
and $s\in S$. Suppose that $\mathcal{M},s\vDash U_{i}\varphi$, to show that
$\mathcal{M},s\vDash U_{i}U_{i}\varphi$. By supposition, it suffices to show
that for all $\psi\in\textbf{EL}$,
$\mathcal{M},s\vDash[\psi]\neg\textit{K}_{i}U_{i}\varphi$.

If not, then for some $\psi\in\textbf{EL}$ we have
$\mathcal{M},s\vDash\langle\psi\rangle\textit{K}_{i}U_{i}\varphi$, thus
$\mathcal{M}|_{\psi},s\vDash\textit{K}_{i}U_{i}\varphi$. From this it follows
that for all $t$ such that $sR_{i}t$, $\mathcal{M}|_{\psi},t\vDash
U_{i}\varphi$, and then $\mathcal{M}|_{\psi},t\vDash\varphi$, and for all
$\chi\in\textbf{EL}$,
$\mathcal{M}|_{\psi},t\vDash[\chi]\neg\textit{K}_{i}\varphi$, which implies
that $\mathcal{M}|_{\psi},t\vDash\neg\textit{K}_{i}\varphi$. Since $sR_{i}s$,
we obtain $\mathcal{M}|_{\psi},s\vDash\neg\textit{K}_{i}\varphi$. On the other
hand, since for all $t$ such that $sR_{i}t$ we have
$\mathcal{M}|_{\psi},t\vDash\varphi$, we infer that
$\mathcal{M}|_{\psi},s\vDash\textit{K}_{i}\varphi$: a contradiction.

Based on the above analysis, we conclude that $\mathcal{M},s\vDash
U_{i}U_{i}\varphi$, as desired. ∎

Intuitively, one cannot know the unknowable truths, since otherwise one would
know the truths that cannot be known, which is impossible. In other words,
unknowable truths are necessary limits to knowledge. This follows immediately
from the result below.

######  Proposition 7.

$\vDash\neg\textit{K}_{i}U_{i}\varphi$.

###### Proof.

Suppose, for a contradiction, that there is a model $\mathcal{M}=\langle
S,\\{R_{i}\mid i\in\textbf{I}\\},V\rangle$ and $s\in S$ such that
$\mathcal{M},s\vDash\textit{K}_{i}U_{i}\varphi$. Then for all $t$, if
$sR_{i}t$ then $\mathcal{M},t\vDash U_{i}\varphi$, thus
$\mathcal{M},t\vDash\varphi$ and for all $\psi\in\textbf{EL}$,
$\mathcal{M},t\vDash[\psi]\neg\textit{K}_{i}\varphi$. On one hand, we can
obtain that $\mathcal{M},s\vDash\textit{K}_{i}\varphi$; on the other hand, by
letting $t$ be $s$ and $\psi$ be $\top$, we infer that
$\mathcal{M},s\vDash\neg\textit{K}_{i}\varphi$: a contradiction. ∎

######  Remark 8.

If we assume the property of transitivity, then $\neg
U_{i}\textit{K}_{i}\varphi$ is valid, which says that all knowledge is not an
unknowable truth. The proof goes as follows. Assume, for a contradiction, that
there is a model $\mathcal{M}=\langle S,\\{R_{i}\mid
i\in\textbf{I}\\},V\rangle$ and a state $s\in S$ such that
$\mathcal{M},s\vDash U_{i}\textit{K}_{i}\varphi$. Then
$\mathcal{M},s\vDash\textit{K}_{i}\varphi$, which implies that
$\mathcal{M},s\vDash\textit{K}_{i}\textit{K}_{i}\varphi$ due to transitivity.
We have also that for all $\psi\in\textbf{EL}$,
$\mathcal{M},s\vDash[\psi]\neg\textit{K}_{i}\textit{K}_{i}\varphi$. By letting
$\psi$ be $\top$, we infer that
$\mathcal{M},s\vDash\neg\textit{K}_{i}\textit{K}_{i}\varphi$. Contradiction.
Similarly, if we assume the property of Euclidicity, we can show that $\neg
U_{i}\neg\textit{K}_{i}\varphi$ is valid, which says that all non-knowledge is
not an unknowable truth. Similar results apply to the case for the notion of
unknown truths: $\neg{\bullet_{i}\textit{K}_{i}\varphi}$ is valid over
transitive frames, and $\neg{\bullet_{i}\neg\textit{K}_{i}\varphi}$ is valid
over Euclidean frames.

The following states that not all non-unknowable-truths are themselves
unknowable truths.

######  Proposition 9.

$\nvDash\neg U_{i}\varphi\to U_{i}\neg U_{i}\varphi$.

###### Proof.

We show that $\nvDash\neg U_{i}\top\to U_{i}\neg U_{i}\top$. First, it should
be clear that $\vDash\neg U_{i}\top$, since there is an announcement (say
$\top$) such that after that the agent knows $\top$. However, $\nvDash
U_{i}\neg U_{i}\top$, since validities are always knowable truths, thus not
unknowable truths (we will defer the statement to Prop. 17). ∎

As mentioned in the introduction, Moore sentences such as $\bullet_{i}p$ and
$U_{i}p$ are unsuccessful and self-refuting. It is shown in [10] that
$\bullet_{i}p$ is unsuccessful and self-refuting. Here we show that $U_{i}p$
is also unsuccessful and self-refuting.

######  Proposition 10.

$\vDash[U_{i}p]\neg U_{i}p$, and thus $\nvDash[U_{i}p]U_{i}p$.

###### Proof.

Suppose that $\mathcal{M},s\vDash U_{i}p$, to show that
$\mathcal{M}|_{U_{i}p},s\nvDash U_{i}p$. By supposition, $\mathcal{M},s\vDash
p$. Thus $\mathcal{M}|_{U_{i}p}\subseteq\mathcal{M}|_{p}$, and then
$\mathcal{M}|_{U_{i}p}|_{U_{i}p}\subseteq\mathcal{M}|_{p}|_{U_{i}p}$. Since
$\mathcal{M}|_{p},s\vDash\textit{K}_{i}p$, we obtain that
$\mathcal{M}|_{p},s\nvDash U_{i}p$, that is,
$s\notin\mathcal{M}|_{p}|_{U_{i}p}$. It then follows that
$s\notin\mathcal{M}|_{U_{i}p}|_{U_{i}p}$, and therefore
$\mathcal{M}|_{U_{i}p},s\nvDash U_{i}p$. ∎

One may ask if there are any differences between the logical properties of
unknown truths and the logical properties of unknowable truths. The answer is
positive. We have seen from [10] that
$\bullet_{i}(\varphi\land\psi)\to\bullet_{i}\varphi\lor\bullet_{i}\psi$ is
valid. In contrast, $U_{i}(\varphi\land\psi)\to U_{i}\varphi\vee U_{i}\psi$ is
invalid.

######  Proposition 11.

$\nvDash U_{i}(\varphi\land\psi)\to U_{i}\varphi\lor U_{i}\psi$.

###### Proof.

Consider the following model $\mathcal{M}$:

$\textstyle{t:p,\neg
q\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i}$$\textstyle{s:p,q\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i}$$\scriptstyle{i}$$\scriptstyle{i}$$\textstyle{u:\neg
p,q\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i}$

We show that $\mathcal{M},s\vDash
U_{i}(\neg\textit{K}_{i}p\land\neg\textit{K}_{i}q)$ but $\mathcal{M},s\nvDash
U_{i}\neg\textit{K}_{i}p$ and $\mathcal{M},s\nvDash U_{i}\neg\textit{K}_{i}q$.

  * •

$\mathcal{M},s\vDash U_{i}(\neg\textit{K}_{i}p\land\neg\textit{K}_{i}q)$.
Clearly, $\mathcal{M},s\vDash\neg\textit{K}_{i}p\land\neg\textit{K}_{i}q$.
Suppose, for reductio, that there exists $\psi\in\textbf{EL}$ such that
$\mathcal{M},s\vDash\langle\psi\rangle\textit{K}_{i}(\neg\textit{K}_{i}p\land\neg\textit{K}_{i}q)$,
and then $\mathcal{M},s\vDash\psi$ and
$\mathcal{M}|_{\psi},s\vDash\textit{K}_{i}(\neg\textit{K}_{i}p\land\neg\textit{K}_{i}q)$.
Because the announcement is interpreted via world-elimination, we consider the
following three cases.

    * –

$t$ is retained in the updated model $\mathcal{M}|_{\psi}$. On the one hand,
since
$\mathcal{M}|_{\psi},s\vDash\textit{K}_{i}(\neg\textit{K}_{i}p\land\neg\textit{K}_{i}q)$
and $sR_{i}t$, we obtain $\mathcal{M}|_{\psi},t\vDash\neg\textit{K}_{i}p$; on
the other hand, as $t$ can only see itself and $t\vDash p$, we have
$\mathcal{M}|_{\psi},t\vDash\textit{K}_{i}p$. A contradiction.

    * –

$u$ is retained in the updated model $\mathcal{M}|_{\psi}$. Similar to the
first case, we can show that $\mathcal{M}|_{\psi},u\vDash\textit{K}_{i}q$ and
$\mathcal{M}|_{\psi},u\vDash\neg\textit{K}_{i}q$, a contradiction.

    * –

Neither $t$ nor $u$ is retained in $\mathcal{M}|_{\psi}$. Then in
$\mathcal{M}|_{\psi}$, there is only a state $s$ which is reflexive. Similarly
to the first case, we can also show that
$\mathcal{M}|_{\psi},s\vDash\textit{K}_{i}p$ and
$\mathcal{M}|_{\psi},s\vDash\neg\textit{K}_{i}p$, a contradiction.

  * •

$\mathcal{M},s\nvDash U_{i}\neg\textit{K}_{i}p$. We have seen that
$\mathcal{M},s\vDash\neg\textit{K}_{i}p$. It suffices to show that for some
$\psi\in\textbf{EL}$ we have
$\mathcal{M},s\vDash\langle\psi\rangle\textit{K}_{i}\neg\textit{K}_{i}p$. This
is indeed the case, because announcing $q$ makes $t$ be deleted, and in the
remaining model, $\neg\textit{K}_{i}p$ is true at $s$ and $u$.

  * •

$\mathcal{M},s\nvDash U_{i}\neg\textit{K}_{i}q$. The proof is analogous to
that of $\mathcal{M},s\nvDash U_{i}\neg\textit{K}_{i}p$, by announcing $p$
instead.

∎

###  3.2 Interactions

In what follows, we study the interactions between unknown truths and
unknowable truths. First, every unknowable truth is an unknown truth, but not
vise versa. This indicates that the notion of unknowable truths is stronger
than that of unknown truths.

######  Proposition 12.

$\vDash U_{i}\varphi\to\bullet_{i}\varphi$, and $\nvDash\bullet_{i}\varphi\to
U_{i}\varphi$.

###### Proof.

Let $\mathcal{M}=\langle S,R,V\rangle$ be a model and $s\in S$. Assume that
$\mathcal{M},s\vDash U_{i}\varphi$, then $\mathcal{M},s\vDash\varphi$ and for
all $\psi\in\textbf{EL}$,
$\mathcal{M},s\vDash[\psi]\neg\textit{K}_{i}\varphi$. Letting $\psi=\top$, we
obtain that $\mathcal{M},s\vDash\neg\textit{K}_{i}\varphi$, and therefore
$\mathcal{M},s\vDash\bullet_{i}\varphi$.

For the invalidity, consider an instance where $\varphi=p$. It is possible
that $p$ is true but possibly false, but $p$ must be knowable. For the strict
proof, consider the following model:

$\textstyle{\mathcal{M}}$$\textstyle{s:p\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i}$$\scriptstyle{i}$$\textstyle{t:\neg
p\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i}$

We will show that $\mathcal{M},s\nvDash{\bullet_{i}p}\to U_{i}p$.

First, since $\mathcal{M},s\vDash p$ and $sR_{i}t$ and $\mathcal{M},t\nvDash
p$, we have $\mathcal{M},s\vDash\bullet_{i}p$. Moreover, because
$\mathcal{M},s\vDash\langle p\rangle\textit{K}_{i}p$ we have that
$\mathcal{M},s\nvDash U_{i}p$. ∎

Also, intuitively, one cannot know the unknown truths. This is because unknown
truths are themselves unknowable truths. In other words, if it is an unknown
truth that $p$, it is an unknowable truth that it is an unknown truth that
$p$, which is argued in [13, Thm. 2] (see also [24, p. 154]).

######  Proposition 13.

$\vDash\bullet_{i}\varphi\to{U_{i}\bullet_{i}}\varphi$, thus
$\vDash\bullet_{i}\varphi\leftrightarrow{U_{i}\bullet_{i}}\varphi$.

###### Proof.

Let $\mathcal{M}=\langle S,\\{R_{i}\mid i\in\textbf{I}\\},V\rangle$ be a model
and $s\in S$. Suppose that $\mathcal{M},s\vDash\bullet_{i}\varphi$, to show
that $\mathcal{M},s\vDash U_{i}{\bullet_{i}\varphi}$. For this, it suffices to
prove that for all $\psi\in\textbf{EL}$,
$\mathcal{M},s\vDash[\psi]\neg\textit{K}_{i}{\bullet_{i}\varphi}$.

If not, then for some $\psi\in\textbf{EL}$, we have
$\mathcal{M},s\vDash\langle\psi\rangle\textit{K}_{i}{\bullet_{i}\varphi}$,
then $\mathcal{M}|_{\psi},s\vDash\textit{K}_{i}{\bullet_{i}\varphi}$. Then for
all $t$ such that $sR_{i}t$, we can obtain that
$\mathcal{M}|_{\psi},t\vDash\varphi$ and
$\mathcal{M}|_{\psi},t\nvDash\textit{K}_{i}\varphi$. Then
$\mathcal{M}|_{\psi},s\vDash\textit{K}_{i}\varphi$ and
$\mathcal{M}|_{\psi},s\nvDash\textit{K}_{i}\varphi$ (since $sR_{i}s$), a
contradiction. ∎

From Prop. 12 and Prop. 13, we can see that $\vDash U_{i}\varphi\to
U_{i}{\bullet_{i}\varphi}$. Despite this, the converse formula is not valid.
This follows directly from Prop. 12 and Prop. 13.

######  Proposition 14.

$\nvDash U_{i}{\bullet_{i}\varphi}\to U_{i}\varphi$.

The following states that unknowable truths are themselves unknown truths. In
other words, if it is an unknowable that $\varphi$, then it is an unknown
truth that it is an unknowable truth that $\varphi$.

######  Proposition 15.

$\vDash U_{i}\varphi\to{\bullet_{i}U_{i}}\varphi$, thus $\vDash
U_{i}\varphi\leftrightarrow{\bullet_{i}U_{i}}\varphi$.

###### Proof.

Let $\mathcal{M}=\langle S,\\{R_{i}\mid i\in\textbf{I}\\},V\rangle$ be a model
and $s\in S$. Suppose that $\mathcal{M},s\vDash U_{i}\varphi$, to demonstrate
that $\mathcal{M},s\vDash\bullet_{i}{U_{i}\varphi}$. For this, it suffices to
show that for some $t$, $sR_{i}t$ and $\mathcal{M},t\nvDash U_{i}\varphi$.

If not, then for all $t$ such that $sR_{i}t$, we have $\mathcal{M},t\vDash
U_{i}\varphi$, thus $\mathcal{M},t\vDash\varphi$ and for all
$\psi\in\textbf{EL}$, $\mathcal{M},t\vDash[\psi]\neg\textit{K}_{i}\varphi$,
which gives us $\mathcal{M},t\vDash\neg\textit{K}_{i}\varphi$. Since
$sR_{i}s$, we derive that $\mathcal{M},s\vDash\neg\textit{K}_{i}\varphi$;
however, from $\mathcal{M},t\vDash\varphi$ for all $t$ with $sR_{i}t$, it
follows that $\mathcal{M},s\vDash\textit{K}_{i}\varphi$: a contradiction. ∎

Fitch’s paradox of knowability states that if all truths are knowable, then
all truths are actually known. This can be shown as follows.

######  Proposition 16.

$\vDash\neg U_{i}\varphi$ for all $\varphi$, then
$\vDash\neg{\bullet_{i}\varphi}$ for all $\varphi$.

###### Proof.

It suffices to show that $\nvDash\neg U_{i}\varphi$ for some $\varphi$, which
intuitively says that there are unknowable truths. This follows from Prop. 13,
since $\bullet_{i}\varphi$ is satisfiable. ∎

Although not every truth (e.g. unknown truths) is knowable, every logical
truth (namely, validity) is knowable.

######  Proposition 17.

If $\vDash\varphi$, then $\vDash\neg U_{i}\varphi$.

##  4 Axiomatization and Soundness

In this section, we will give a proof system for LUT and show its soundness.
For this, we need a notion of admissible form. This notion is originated from
[15, pp. 55–56], which is called ‘necessity forms’ in [2, 3, 1, 4].

######  Definition 18 (Admissible forms).

Let $\varphi\in\textbf{LUT}$ and $i\in\textbf{I}$. Admissible forms
$\eta(\sharp)$ are defined recursively in the following.

$\begin{array}[]{ll}\eta(\sharp)::=&\sharp\mid\varphi\to\eta(\sharp)\mid{\textit{K}_{i}}\eta(\sharp)\mid[\varphi]\eta(\sharp)\\\
\end{array}$

Note that $\sharp$ is a placeholder, not a formula. By replacing $\sharp$ in
an admissible form $\eta(\sharp)$ with a formula $\chi$, we obtain a formula
$\eta(\psi)$, defined as follows.

$\begin{array}[]{lll}\sharp(\chi)&=&\chi\\\
(\varphi\to\eta(\sharp))(\chi)&=&\varphi\to\eta(\chi)\\\
{\textit{K}_{i}}\eta(\sharp)(\chi)&=&\textit{K}_{i}\eta(\chi)\\\
([\varphi]\eta(\sharp))(\chi)&=&[\varphi]\eta(\chi)\\\ \end{array}$

With the previous preparation, we now axiomatize the logic of unknowable
truths LUT.

######  Definition 19.

The proof system $\mathbb{LUT}$ consists of the following axioms and inference
rules.

$\begin{array}[]{ll}\text{PL}&\text{All instances of tautologies}\\\
\text{K}&\textit{K}_{i}(\varphi\to\psi)\to(\textit{K}_{i}\varphi\to\textit{K}_{i}\psi)\\\
\text{KA}&[\chi](\varphi\to\psi)\to([\chi]\varphi\to[\chi]\psi)\\\
\text{T}&\textit{K}_{i}\varphi\to\varphi\\\
\text{AP}&[\psi]p\leftrightarrow(\psi\to p)\\\
\text{AN}&[\psi]\neg\varphi\leftrightarrow(\psi\to\neg[\psi]\varphi)\\\
\text{AC}&[\psi](\varphi\land\chi)\leftrightarrow([\psi]\varphi\land[\psi]\chi)\\\
\text{AK}&[\psi]\textit{K}_{i}\varphi\leftrightarrow(\psi\to\textit{K}_{i}[\psi]\varphi)\\\
\text{AA}&[\psi][\chi]\varphi\leftrightarrow[\psi\land[\psi]\chi]\varphi\\\
\text{AU}&U_{i}\varphi\to\varphi\land[\psi]\neg\textit{K}_{i}\varphi,\text{
where }\psi\in\textbf{EL}\\\
\text{MP}&\dfrac{\varphi~{}~{}~{}\varphi\to\psi}{\psi}\\\
\text{GEN}&\dfrac{\varphi}{\textit{K}_{i}\varphi}\\\
\text{GENA}&\dfrac{\varphi}{[\chi]\varphi}\\\
\text{RU}&\dfrac{\eta(\varphi\land[\psi]\neg\textit{K}_{i}\varphi)\text{ for
all }\psi\in\textbf{EL}}{\eta(U_{i}\varphi)}\\\ \end{array}$

The novel thing is the axiom AU and inference rule RU, which together
characterize the semantics of $U_{i}$ in a certain way: AU says that if
$\varphi$ is an unknowable truth for $i$, then $\varphi$ is true and after
every truthfully public announcement, $\varphi$ is unknown to $i$; RU roughly
says that if $\varphi$ is true but unknown to $i$ after every truthfully
public announcement, then $\varphi$ is an unknowable truth for $i$. Other
axioms and inference rules are familiar from public announcement logic, cf.
e.g. [22].

We say a formula $\varphi$ is a theorem of $\mathbb{LUT}$, or $\varphi$ is
provable in $\mathbb{LUT}$, notation: $\vdash\varphi$, if $\varphi$ is either
an instantiation of some axiom, or obtained from axioms with an application of
some inference rule. Let Thm be the set of all theorems of $\mathbb{LUT}$.

######  Proposition 20.

$\mathbb{LUT}$ is sound with respect to the class of all frames.

###### Proof.

We only consider the axiom AU and the inference rule RU. The validity of axiom
AC is straightforward by semantics. For the soundness of RU, we need to show
that if $\vDash\eta(\varphi\land[\psi]\neg\textit{K}_{i}\varphi)\text{ for all
}\psi\in\textbf{EL}$, then $\vDash\eta(U_{i}\varphi)$. To do this, we show the
following stronger result:

$(\star)$ for all $(\mathcal{M},s)$, if
$\mathcal{M},s\vDash\eta(\varphi\land[\psi]\neg\textit{K}_{i}\varphi)\text{
for all }\psi\in\textbf{EL}$, then $\mathcal{M},s\vDash\eta(U_{i}\varphi)$.

The proof proceeds by induction on the structure of admissible forms. The base
case and the inductive cases $\varphi\to\eta(\sharp)$ and
$\textit{K}_{i}\eta(\sharp)$ and $[\varphi]\eta(\sharp)$ follow from the
semantics and the induction hypothesis. ∎

In the next section, we will show the completeness of $\mathbb{LUT}$, where
the canonical model will be based on a notion of maximal consistent theories,
rather than the more familiar notion of maximal consistent sets. This is
because we need to use the condition of closure under RU, which is
indispensible in the completeness proof.

######  Definition 21 (theory, consistent theory, maximal theory, maximal
consistent theory).

Given a set of formulas $s$, we say it is a theory, if the following holds:

(a) $\textbf{Thm}\subseteq s$, and

(b) $s$ is closed under MP and RU.
Moreover, if $\bot\notin s$, then $s$ is a consistent; if for all $\varphi$,
either $\varphi\in s$ or $\neg\varphi\in s$, then $s$ is a maximal; if $s$ is
consistent and maximal, then $s$ is maximally consistent.

One may check that Thm is the smallest theory.

Define $s+\varphi=\\{\psi\mid\varphi\to\psi\in s\\}$.

######  Proposition 22.

Let $s$ be a theory and $\varphi$ be a formula. Then

(1) $s+\varphi$ is a theory that contains $s\cup\\{\varphi\\}$, and

(2) $s+\varphi$ is consistent iff $\neg\varphi\notin s$.

###### Proof.

(1) Suppose that $s$ is a theory. First, we show that $s+\varphi$ is a theory.

  * •

Contains Thm: given any $\psi\in\textbf{Thm}$, we have
$\varphi\to\psi\in\textbf{Thm}$. Since $\textbf{Thm}\subseteq s$ (as $s$ is a
theory), $\varphi\to\psi\in s$, and thus $\psi\in s+\varphi$.

  * •

Closed under MP: assume that $\psi,\psi\to\chi\in s+\varphi$, then
$\varphi\to\psi\in s$ and $\varphi\to(\psi\to\chi)\in s$. Since $s$ contains
Thm and is closed under MP, we infer that $\varphi\to\chi\in s$, and therefore
$\chi\in s+\varphi$.

  * •

Closed under RU: suppose that $\eta(\chi\land[\psi]\neg\textit{K}_{i}\chi)\in
s+\varphi$ for all $\psi\in\textbf{EL}$, then
$\varphi\to\eta(\chi\land[\psi]\neg\textit{K}_{i}\chi)\in s$. Because $s$ is
closed under RU for the admissible form $\varphi\to\eta(\sharp)$, we derive
that $\varphi\to\eta(U_{i}\chi)\in s$, so $\eta(U_{i}\chi)\in s+\varphi$.

Next, we show that $s\cup\\{\varphi\\}\subseteq s+\varphi$. Given any $\psi\in
s\cup\\{\varphi\\}$, either $\psi\in s$ or $\psi=\varphi$. If $\psi\in s$,
then as $\psi\to(\varphi\to\psi)\in\textbf{Thm}$ and $\textbf{Thm}\subseteq
s$, $\psi\to(\varphi\to\psi)\in s$; since $s$ is closed under MP,
$\varphi\to\psi\in s$, that is, $\psi\in s+\varphi$. If $\psi=\varphi$, then
since $\varphi\to\varphi\in s$, we immediately have $\varphi\in s+\varphi$,
that is, $\psi\in s+\varphi$. Therefore, $s\cup\\{\varphi\\}\subseteq
s+\varphi$.

(2) Suppose that $\neg\varphi\in s$, then by (1), $\neg\varphi\in s+\varphi$.
Moreover, by (1) again, $\varphi\in s+\varphi$. We have also
$\neg\varphi\to(\varphi\to\bot)\in s+\varphi$. Since $s+\varphi$ is closed
under MP (by (1) again), we conclude that $\bot\in s+\varphi$, and therefore
$s+\varphi$ is inconsistent.

Conversely, assume that $s+\varphi$ is inconsistent, then $\bot\in s+\varphi$,
and thus $\varphi\to\bot\in s$. Since $(\varphi\to\bot)\to\neg\varphi\in s$
and $s$ is closed under MP, it follows that $\neg\varphi\in s$. ∎

Lindenbaum’s Lemma can be proved as [3, Lem.4.12].

######  Lemma 23 (Lindenbaum’s Lemma).

Every consistent theory can be extended to a maximal consistent theory.

##  5 Completeness

To demonstrate the completeness of $\mathbb{LUT}$, we adapt the method in [4]
for APAL. The point is as follows. In the proof of the truth lemma, the
hardest part is the cases involving $U_{i}\varphi$. To handle this, we need to
‘reduce’ the case $U_{i}\varphi$ to the easy-to-fix case
$\varphi\land[\psi]\neg\textit{K}_{i}\varphi$ (where $\psi\in\textbf{EL}$),
via a formula-based complexity measure $<^{S}_{d}$ (defined below), and then
use the $<^{S}_{d}$-induction hypothesis to deal with the cases involving
$U_{i}\varphi$ in the truth lemma.

######  Definition 24 (Canonical Model).

The canonical model for $\mathbb{LUT}$ is a tuple $\mathcal{M}^{c}=\langle
S^{c},\\{R^{c}_{i}\mid i\in\textbf{I}\\},V^{c}\rangle$ where

  * •

$S^{c}=\\{s\mid s\text{ is a maximal consistent theory for }\mathbb{LUT}\\}$.

  * •

$sR_{i}^{c}t$ iff $\lambda_{i}(s)\subseteq t$, where
$\lambda_{i}(s)=\\{\varphi\mid\textit{K}_{i}\varphi\in s\\}$.

  * •

$V^{c}(p)=\\{s\in S^{c}\mid p\in s\\}$.

An equivalent definition of $R^{c}_{i}$ is as follows.

$sR^{c}_{i}t$ iff for all $\varphi$, if $\textit{K}_{i}\varphi\in s$, then
$\varphi\in t$.

######  Proposition 25.

Where $s\in S^{c}$, $\lambda_{i}(s)$ is a consistent theory.

###### Proof.

Suppose that $s\in S^{c}$. We show that $\lambda_{i}(s)$ is a consistent
theory.

  * •

Contains Thm: given any $\varphi\in\textbf{Thm}$, using GEN we have
$\textit{K}_{i}\varphi\in\textbf{Thm}$. As $s\in S^{c}$,
$\textit{K}_{i}\varphi\in s$, hence $\varphi\in\lambda_{i}(s)$.

  * •

Closed under MP: suppose that $\varphi,\varphi\to\psi\in\lambda_{i}(s)$, then
$\textit{K}_{i}\varphi\in s$ and $\textit{K}_{i}(\varphi\to\psi)\in s$. By
axiom K and $s\in S^{c}$, we can obtain that $\textit{K}_{i}\psi\in s$, and
therefore $\psi\in\lambda_{i}(s)$.

  * •

Closed under RU: assume that
$\eta(\varphi\land[\psi]\neg\textit{K}_{i}\varphi)\in\lambda_{i}(s)$ for all
$\psi\in\textbf{EL}$, to show that $\eta(U_{i}\varphi)\in\lambda_{i}(s)$. By
assumption,
$\textit{K}_{i}\eta(\varphi\land[\psi]\neg\textit{K}_{i}\varphi)\in s$. Since
$s$ is closed under RU for the admissible form $\textit{K}_{i}\eta(\sharp)$,
it follows that $\textit{K}_{i}\eta(U_{i}\varphi)\in s$. Therefore,
$\eta(U_{i}\varphi)\in\lambda_{i}(s)$.

  * •

Contains no $\bot$: since $s\in S^{c}$, $\bot\notin s$, which plus the axiom T
implies that $\textit{K}_{i}\bot\notin s$, and thus
$\bot\notin\lambda_{i}(s)$.

∎

######  Proposition 26.

Let $s\in S^{c}$. The following conditions are equivalent.

(1) $\textit{K}_{i}\psi\in s$,

(2) for all $t\in S^{c}$, if $sR^{c}_{i}t$, then $\psi\in t$.

###### Proof.

The direction from (1) to (2) is straightforward by definition of $R^{c}_{i}$.
For the converse, suppose that (1) fails (that is, $\textit{K}_{i}\psi\notin
s$), to show that (2) fails, that is, for some $t\in S^{c}$, $sR^{c}_{i}t$ and
$\psi\notin t$.

By Prop. 25, $\lambda_{i}(s)$ is a theory. Then by the item (1) of Prop. 22,
$\lambda_{i}(s)+\neg\psi$ is a theory and
$\lambda_{i}(s)\cup\\{\neg\psi\\}\subseteq\lambda_{i}(s)+\neg\psi$. Moreover,
by supposition, we infer that $\psi\notin\lambda_{i}(s)$. By Prop. 25, we can
obtain that $\neg\neg\psi\notin\lambda_{i}(s)$. Now using the item (2) of
Prop. 22, we derive that $\lambda_{i}(s)+\neg\psi$ is consistent. We have now
shown that $\lambda_{i}(s)+\neg\psi$ is a consistent theory. Then by
Lindenbaum’s Lemma (Lemma 23), there exists $t\in S^{c}$ such that
$\lambda_{i}(s)+\neg\psi\subseteq t$. This implies that
$\lambda_{i}(s)\cup\\{\neg\psi\\}\subseteq t$. From $\lambda_{i}(s)\subseteq
t$, it follows that $sR^{c}_{i}t$; from $\neg\psi\in t$ and $t\in S^{c}$, it
follows that $\psi\notin t$, as desired. ∎

As mentioned before, in the proof of the truth lemma below, we need to use a
formula-based complexity measure to ‘reduce’ the hard case $U_{i}\varphi$ to
the easy-to-fix case $\varphi\land[\psi]\neg\textit{K}_{i}\varphi$ (where
$\psi\in\textbf{EL}$). Now we introduce the notion of complexity, which
combines the notions of size and $U$-depth.

######  Definition 27 (Size, $U$-depth, Complexity).

The complexity of a formula consists of two aspects: size and $U$-depth, which
are defined as follows.

The size of a formula $\varphi$, denoted $S(\varphi)$, is a positive natural
number, defined recursively as follows.

$\begin{array}[]{lll}S(p)&=&1\\\ S(\neg\varphi)&=&S(\varphi)+1\\\
S(\varphi\land\psi)&=&S(\varphi)+S(\psi)+1\\\
S(\textit{K}_{i}\varphi)&=&S(\varphi)+1\\\
S([\varphi]\psi)&=&(5+S(\varphi))\cdot S(\psi)\\\
S(U_{i}\varphi)&=&S(\varphi)+1\\\ \end{array}$

The $U$-depth of a formula $\varphi$, denoted $d_{U}(\varphi)$, is a natural
number, defined recursively as follows.

$\begin{array}[]{lll}d_{U}(p)&=&0\\\ d_{U}(\neg\varphi)&=&d_{U}(\varphi)\\\
d_{U}(\varphi\land\psi)&=&max\\{d_{U}(\varphi),d_{U}(\psi)\\}\\\
d_{U}(\textit{K}_{i}\varphi)&=&d_{U}(\varphi)\\\
d_{U}([\varphi]\psi)&=&d_{U}(\varphi)+d_{U}(\psi)\\\
d_{U}(U_{i}\varphi)&=&d_{U}(\varphi)+1\\\ \end{array}$

We say that $\varphi$ is less complex than $\psi$, denoted
$c(\varphi)<c(\psi)$, if either $d_{U}(\varphi)<d_{U}(\psi)$, or
$d_{U}(\varphi)=d_{U}(\psi)$ and $S(\varphi)<S(\psi)$.

One may easily check that $S(\varphi\to\psi)=S(\varphi)+S(\psi)+3$ and
$d_{U}(\varphi\to\psi)=max\\{d_{U}(\varphi),d_{U}(\psi)\\}$. Moreover,
$d_{U}(\varphi)=0$ for all $\varphi\in\textbf{EL}$.

Note that our definition of $S(\varphi\land\psi)$ is different from that in
[4, 21, 9]. The reason of so defining is that, for us, it is more suitable for
the intuition of the size of a conjunction. Also note that the number 5 in the
definition of $S([\varphi]\psi)$ is the least natural number that gives us the
following properties (more precisely, (4) and (5) in Prop. 28), in contrast to
the least natural number 4 in the complexity measure for public announcement
logic in [22, Chap. 7].

######  Proposition 28.

  1. (1)

$c(\psi)<c(\neg\psi)$

  2. (2)

$c(\psi)<c(\psi\land\chi)$, $c(\chi)<c(\psi\land\chi)$.

  3. (3)

$c(\psi)<c(\textit{K}_{i}\psi)$

  4. (4)

$c(\psi\to p)<c([\psi]p)$

  5. (5)

$c(\psi\to\neg[\psi]\chi)<c([\psi]\neg\chi)$

  6. (6)

$c([\psi]\chi_{1})<c([\psi](\chi_{1}\land\chi_{2}))$,
$c([\psi]\chi_{2})<c([\psi](\chi_{1}\land\chi_{2}))$

  7. (7)

$c(\psi\to{\textit{K}_{i}[\psi]\chi})<c({[\psi]\textit{K}_{i}}\chi)$

  8. (8)

$c([\psi\land[\psi]\chi]\delta)<c([\psi][\chi]\delta)$

  9. (9)

$c([\psi]\chi)<c([\psi]U_{i}\chi)$,$c([\psi]{[\delta]\neg\textit{K}_{i}}\chi)<c([\psi]U_{i}\chi)$,
where $\delta\in\textbf{EL}$.

  10. (10)

$c(\psi)<c(U_{i}\psi)$, $c({[\chi]\neg\textit{K}_{i}}\psi)<c(U_{i}\psi)$,
where $\chi\in\textbf{EL}$.

###### Proof.

We show some of them as examples.

(4) One may easily verify that $d_{U}(\psi\to p)=d_{U}(\psi)=d_{U}([\psi]p)$.
Moreover, $S(\psi\to p)=S(\psi)+S(p)+3=S(\psi)+4<5+S(\psi)=S([\psi]p)$. Thus
$c(\psi\to p)<c([\psi]p)$.

(5) One may easily verify that
$d_{U}(\psi\to\neg[\psi]\chi)=d_{U}(\psi)+d_{U}(\chi)=d_{U}([\psi]\neg\chi)$.
Moreover, $S(\psi\to\neg[\psi]\chi)=S(\psi)+5\cdot S(\chi)+S(\psi)\cdot
S(\chi)+4<S(\psi)+5\cdot S(\chi)+S(\psi)\cdot S(\chi)+5=S([\psi]\neg\chi)$.
Therefore, $c(\psi\to\neg[\psi]\chi)<c([\psi]\neg\chi)$.

(7) One may easily verify that
$d_{U}(\psi\to{\textit{K}_{i}[\psi]}\chi)=d_{U}(\psi)+d_{U}(\chi)=d_{U}({[\psi]\textit{K}_{i}}\chi)$.
Moreover, $S(\psi\to{\textit{K}_{i}[\psi]}\chi)=S(\psi)+5\cdot
S(\chi)+S(\psi)\cdot S(\chi)+4<S(\psi)+5\cdot S(\chi)+S(\psi)\cdot
S(\chi)+5=S({[\psi]\textit{K}_{i}}\chi)$. Therefore,
$c(\psi\to{\textit{K}_{i}[\psi]}\chi)<c({[\psi]\textit{K}_{i}}\chi)$.

(8) One may easily verify that
$d_{U}([\psi\land[\psi]\chi]\delta)=d_{U}(\psi)+d_{U}(\chi)+d_{U}(\delta)=d_{U}([\psi][\chi]\delta)$.
Moreover, $S([\psi\land[\psi]\chi]\delta)=6\cdot S(\delta)+S(\psi)\cdot
S(\delta)+5\cdot S(\chi)\cdot S(\delta)+S(\psi)\cdot S(\chi)\cdot S(\delta)$,
and $S([\psi][\chi]\delta)=25\cdot S(\delta)+5\cdot S(\psi)\cdot
S(\delta)+5\cdot S(\chi)\cdot S(\delta)+S(\psi)\cdot S(\chi)\cdot S(\delta)$,
thus $S([\psi\land[\psi]\chi]\delta)<S([\psi][\chi]\delta)$. Therefore,
$c([\psi\land[\psi]\chi]\delta)<c([\psi][\chi]\delta)$.

(10) Since $d_{U}(\psi)<d_{U}(\psi)+1=d_{U}(U_{i}\psi)$, we obtain
$c(\psi)<c(U_{i}\psi)$. Also, one may verify that
$d_{U}({[\chi]\neg\textit{K}_{i}}\psi)=d_{U}(\psi)<d_{U}(\psi)+1=d_{U}(U_{i}\psi)$,
thus $c({[\chi]\neg\textit{K}_{i}}\psi)<c(U_{i}\psi)$. ∎

Now we are ready to show the truth lemma.

######  Lemma 29.

For all $s\in S^{c}$, for all $\varphi\in\textbf{LUT}$, we have

$\mathcal{M}^{c},s\vDash\varphi\text{ iff }\varphi\in s.$

###### Proof.

By induction on the complexity of $\varphi$. Recall that the notion of
complexity is given in Def. 27.

  * •

$\varphi=p\in\textbf{P}$. Straightforward by the definition of $V^{c}$.

  * •

$\varphi=\neg\psi$. By the item (1) of Prop. 28, $c(\psi)<c(\varphi)$. Then by
induction hypothesis, we have that $\mathcal{M}^{c},s\vDash\psi$ iff $\psi\in
s$. Thus $\mathcal{M}^{c},s\vDash\varphi$ iff $\mathcal{M}^{c},s\nvDash\psi$
iff $\psi\notin s$ iff $\varphi\in s$.

  * •

$\varphi=\psi\land\chi$. By the item (2) of Prop. 28,
$c(\psi),c(\chi)<c(\varphi)$. Then by induction hypothesis,
$\mathcal{M}^{c},s\vDash\psi$ iff $\psi\in s$, and
$\mathcal{M}^{c},s\vDash\chi$ iff $\chi\in s$. Therefore,
$\mathcal{M}^{c},s\vDash\varphi$ iff ($\mathcal{M}^{c},s\vDash\psi$ and
$\mathcal{M}^{c},s\vDash\chi$) iff ($\psi\in s$ and $\chi\in s$) iff
$\varphi\in s$.

  * •

$\varphi=\textit{K}_{i}\psi$. By the item (3) of Prop. 28,
$c(\psi)<c(\varphi)$. Then we have the following equivalences.

$\begin{array}[]{lll}\mathcal{M}^{c},s\vDash\textit{K}_{i}\psi&\iff&\text{for
all }t\in S^{c},\text{ if }sR^{c}_{i}t,\text{ then
}\mathcal{M}^{c},t\vDash\psi\\\ &\iff&\text{for all }t\in S^{c},\text{ if
}sR^{c}_{i}t,\text{ then }\psi\in t\\\ &\iff&\textit{K}_{i}\psi\in s,\\\
\end{array}$

where the penultimate equivalence holds by induction hypothesis, whereas the
last one has been shown in Prop. 26.

  * •

$\varphi=[\psi]p$. By the item (4) of Prop. 28, $c(\psi\to p)<c(\varphi)$.
Then we have the following equivalences.

$\begin{array}[]{lll}\mathcal{M}^{c},s\vDash[\psi]p&\iff&\mathcal{M}^{c},s\vDash\psi\to
p\\\ &\iff&\psi\to p\in s\\\ &\iff&[\psi]p\in s,\\\ \end{array}$

where the penultimate equivalence holds by induction hypothesis, whereas other
two equivalences holds by axiom AP.

  * •

$\varphi=[\psi]\neg\chi$. By the item (5) of Prop. 28,
$c(\psi\to\neg[\psi]\chi)<c(\varphi)$. Then we have

$\begin{array}[]{lll}\mathcal{M}^{c},s\vDash[\psi]\neg\chi&\iff&\mathcal{M}^{c},s\vDash\psi\to\neg[\psi]\chi\\\
&\iff&\psi\to\neg[\psi]\chi\in s\\\ &\iff&[\psi]\neg\chi\in s,\\\ \end{array}$

where the penultimate equivalence holds by induction hypothesis, whereas other
two equivalences holds by axiom AN.

  * •

$\varphi=[\psi](\chi_{1}\land\chi_{2})$. By the item (6) of Prop. 28,
$c([\psi]\chi_{1}),c([\psi]\chi_{2})<c(\varphi)$. Then we have

$\begin{array}[]{lll}\mathcal{M}^{c},s\vDash[\psi](\chi_{1}\land\chi_{2})&\iff&\mathcal{M}^{c},s\vDash[\psi]\chi_{1}\text{
and }\mathcal{M}^{c},s\vDash[\psi]\chi_{2}\\\ &\iff&[\psi]\chi_{1}\in s\text{
and }[\psi]\chi_{2}\in s\\\ &\iff&[\psi](\chi_{1}\land\chi_{2})\in s,\\\
\end{array}$

where the penultimate equivalence holds by induction hypothesis, whereas other
two equivalences holds by axiom AC.

  * •

$\varphi={[\psi]\textit{K}_{i}}\chi$. By the item (7) of Prop. 28,
$c(\psi\to{\textit{K}_{i}[\psi]}\chi)<c(\varphi)$. Then we have

$\begin{array}[]{lll}\mathcal{M}^{c},s\vDash{[\psi]\textit{K}_{i}}\chi&\iff&\mathcal{M}^{c},s\vDash(\psi\to{\textit{K}_{i}[\psi]}\chi)\\\
&\iff&\psi\to{\textit{K}_{i}[\psi]}\chi\in s\\\
&\iff&{[\psi]\textit{K}_{i}}\chi\in s,\\\ \end{array}$

where the penultimate equivalence holds by induction hypothesis, whereas other
two equivalences holds by axiom AK.

  * •

$\varphi=[\psi][\chi]\delta$. By the item (8) of Prop. 28,
$c([\psi\land[\psi]\chi]\delta)<c(\varphi)$. Then we have

$\begin{array}[]{lll}\mathcal{M}^{c},s\vDash[\psi][\chi]\delta&\iff&\mathcal{M}^{c},s\vDash[\psi\land[\psi]\chi]\delta\\\
&\iff&[\psi\land[\psi]\chi]\delta\in s\\\ &\iff&[\psi][\chi]\delta\in s,\\\
\end{array}$

where the penultimate equivalence holds by induction hypothesis, whereas other
two equivalences holds by axiom AA.

  * •

$\varphi=[\psi]U_{i}\chi$. By the item (9) of Prop. 28,
$c([\psi]\chi)<c([\psi]U_{i}\chi)$,$c([\psi]{[\delta]\neg\textit{K}_{i}}\chi)<c([\psi]U_{i}\chi)$,
where $\delta\in\textbf{EL}$. Then we have

$\begin{array}[]{lll}\mathcal{M}^{c},s\vDash[\psi]U_{i}\chi&\iff&\mathcal{M}^{c},s\vDash\psi\text{
implies }\mathcal{M}^{c}|_{\psi},s\vDash U_{i}\chi\\\
&\iff&\mathcal{M}^{c},s\vDash\psi\text{ implies
}(\mathcal{M}^{c}|_{\psi},s\vDash\chi\text{ and for all
}\delta\in\textbf{EL},\mathcal{M}^{c}|_{\psi},s\vDash{[\delta]\neg\textit{K}_{i}}\chi)\\\
&\iff&(\mathcal{M},s\vDash\psi\text{ implies
}\mathcal{M}^{c}|_{\psi},s\vDash\chi)\text{ and }\\\
&&(\mathcal{M},s\vDash\psi\text{ implies for all
}\delta\in\textbf{EL},\mathcal{M}^{c}|_{\psi},s\vDash{[\delta]\neg\textit{K}_{i}}\chi)\\\
&\iff&\mathcal{M},s\vDash[\psi]\chi\text{ and for all
}\delta\in\textbf{EL},\mathcal{M}^{c},s\vDash[\psi]{[\delta]\neg\textit{K}_{i}}\chi\\\
&\stackrel{{\scriptstyle(\ast)}}{{\iff}}&[\psi]\chi\in s\text{ and for all
}\delta\in\textbf{EL},[\psi]{[\delta]\neg\textit{K}_{i}}\chi\in s\\\
&\stackrel{{\scriptstyle(\ast\ast)}}{{\iff}}&[\psi](\chi\land{[\delta]\neg\textit{K}_{i}}\chi)\in
s\text{ for all }\delta\in\textbf{EL}\\\
&\stackrel{{\scriptstyle(\ast\ast\ast)}}{{\iff}}&[\psi]U_{i}\chi\in s,\\\
\end{array}$

where $(\ast)$ holds by induction hypothesis, $(\ast\ast)$ holds due to axiom
AC and $s\in S^{c}$, and $(\ast\ast\ast)$ uses the fact that $s$ is closed
under RU for the admissible form $[\psi]\sharp$, axiom AU, and the rule
$\text{RM}[\cdot]$, which is in turn derivable from the axiom KA and the rule
GENA.

  * •

$\varphi=U_{i}\psi$. By the item (10) of Prop. 28, $c(\psi)<c(U_{i}\psi)$,
$c({[\chi]\neg\textit{K}_{i}}\psi)<c(U_{i}\psi)$, where $\chi\in\textbf{EL}$.
Then we have

$\begin{array}[]{lll}\mathcal{M}^{c},s\vDash
U_{i}\psi&\iff&\mathcal{M}^{c},s\vDash\psi\text{ and for all
}\chi\in\textbf{EL},\mathcal{M}^{c},s\vDash{[\chi]\neg\textit{K}_{i}}\psi\\\
&\iff&\psi\in s\text{ and }{[\chi]\neg\textit{K}_{i}}\psi\in s\text{ for all
}\chi\in\textbf{EL}\\\ &\iff&\psi\land{[\chi]\neg\textit{K}_{i}}\psi\in
s\text{ for all }\chi\in\textbf{EL}\\\ &\iff&U_{i}\psi\in s,\\\ \end{array}$

where the second equivalence holds by induction hypothesis, the third one uses
the closure of $s$ under MP, and the last one uses the closure of $s$ under RU
for the admissible form $\sharp$ and the axiom AU.

∎

It is now a standard exercise to show the following result.

######  Theorem 30.

$\mathbb{LUT}$ is sound and complete with respect to the class of all frames.
That is, if $\vDash\varphi$, then $\vdash\varphi$.

With the completeness in hand, we can give a syntactic proof for Fitch’s
paradox of knowability, which is much simpler than those in the literature,
e.g. in [19].

######  Proposition 31.

If $\vdash\neg U_{i}\varphi$ for all $\varphi$, then
$\vdash\neg{\bullet_{i}\varphi}$ for all $\varphi$.

###### Proof.

Suppose that $\vdash\neg U_{i}\varphi$ for all $\varphi$, then $\vdash\neg
U_{i}{\bullet_{i}\varphi}$. Then by Prop. 13 and Thm. 30,
$\vdash\bullet_{i}\varphi\to U_{i}{\bullet_{i}\varphi}$, and therefore
$\vdash\neg{\bullet_{i}\varphi}$ for all $\varphi$. ∎

##  6 Conclusion and Discussion

In this paper, we proposed a logic of unknowable truths, investigated the
logical properties of unknown truths and the logical properties of unknowable
truths, which includes the similarities and relationship between the two
notions, and finally axiomatized the logic of unknowable truths.

So far we have seen that the semantics of unknowable truths depends on that of
propositional knowledge. However, we have mainly focused on the relationship
between unknowable truths and unknown truths, instead of between unknowable
truths and propositional knowledge, and the notion of unknown truths is weaker
than that of propositional knowledge. So there seems to be some asymmetry
between the semantics of unknowable truths and our concerns. Then a natural
question is: is there any other semantics for unknowable truths which relates
the notion to that of unknown truths more directly/properly? Our answer is
positive. We will introduce this new semantics in the future work.

For another future work, a natural extension would be to add propositional
quantifiers. This can increase the expressive power of the current logic. For
instance, we can express Fitch’s paradox of knowability in the new language as
follows: $\forall p\neg U_{i}p\to\forall p\neg{\bullet_{i}p}$, or
equivalently, $\exists p{\bullet_{i}p}\to\exists pU_{i}p$.

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Non-bi-orderable one-relator groups without generalized torsion

Azer Akhmedov and James Thorne

ABSTRACT: We construct examples of non-bi-orderable one-relator groups without
generalized torsion. This answer a question asked in [2].

##  1\. Introduction

A left-order on a group is a linear order which is invariant under the left
translations, and a bi-order is a linear order which is invariant under both
left and right translations. Bi-orderable groups turn out to be, perhaps not
too narrow, but still a rather special sub-class in the class of all left-
orderable groups.

Existence of a torsion element is clearly an obstruction to left-ordera-
bility of a group. Similarly, existence of a generalized torsion element is an
obstruction to bi-orderability. There exist examples of groups without a
torsion which are not left-orderable, and examples of non-bi-orderable groups
without a generalized torsion also exist ([9], [3]). On the other hand, for
many interesting classes of groups, the absence of a torsion element turns out
to be equivalent to left-orderability. One of the non-trivial examples is the
class of one-relator groups ([1], [5], [6], [3]). Interestingly, torsion-
freeness of a one-relator group can be immediately detected from the relator,
namely, a one-relator group on generators $a,b,\dots,c$ contains a torsion
element iff the relation of a group is in the form $r^{n}$ for some $n\geq 2$
and $r=r(a,b,\dots,c)\in F$ where $F$ is the free group on generators
$a,b,\dots,c$ (see [8]).

In [2], the authors study bi-orderability of one-relator groups and prove that
1) if a one-relator group with the so-called tidy relation is bi-orderable,
then its Alexander polynomial has at least one positive real root; 2) if all
the roots of Alexander polynomial of a one-relator group with monic relation
are real and positive, then the group is bi-orderble. They also ask (See
Question 3) if every one-relator group without a generalized torsion is bi-
orderable.

In this note we produce non-bi-orderable examples of one-relator groups which
do not possess a generalized torsion. Let us recall that for a group $G$, an
element $g\in G\backslash\\{1\\}$ is called a generalized torsion if there
exist $x_{1},\dots x_{n}\in G$ for some $n\geq 1$ such that
$(x_{1}gx_{1}^{-1})\dots(x_{n}gx_{n}^{-1})=1$.

The following theorem produces an explicit example of a one-relator non-bi-
orderable group without a generalized torsion.

######  Theorem 1.1.

A one-relator group

$\langle t,a\ |\
tat^{-1}at^{2}at^{-1}a^{-1}ta^{-1}t^{-2}a^{-1}t^{2}a^{-1}t^{-2}a^{-1}\rangle$

is not bi-orderable and does not contain a generalized torsion.

Let us note that we can write $\Gamma$ as

$\Gamma=\langle t,a\ |\
tat^{-1}(at^{2}at^{-2})ta^{-1}t^{-1}=(at^{2}at^{-2})^{2}\rangle;$

we will be using this presentation throughout this note.

From the proof of Theorem 1.1 we immediately obtain the following more general
family of examples.

######  Theorem 1.2.

Let $s\geq 1,m\geq 2$, $W(x_{1},\dots,x_{s})$ be a nontrivial positive word in
the free group $\mathbb{F}_{s}$ generated by $x_{1},\dots,x_{s}$, and in the
free group formally generated by letters $t,x$, let $z_{i}=t^{i}xt^{-i},0\leq
i\leq s+1$. Then the one-relator group

$\Gamma_{W}=\langle t,x\ |\
z_{s}(W(z_{0},\dots,z_{s-1})z_{s+1})z_{s}^{-1}=(W(z_{0},\dots,z_{s-1})z_{s+1})^{m}\rangle$

is bi-orderable and does not contain a generalized torsion.

Let us notice that the Alexander polynomial ([2]) of $\Gamma$ is $t^{2}+1$ (so
it does not have a positive real root), and the Alexander polynomial of
$\Gamma_{W}$ is in the form $1+t^{2}p(t)$ where $p(t)$ is a polynomial with
non-negative integer coefficients (so the Alexander polynomial again does not
have a positive real root).

In the class of 3-manifold groups, it is conjectured in [10] that the
existence of a generalized torsion is equivalent to non-bi-orderability.
Although this conjecture is still open, the authors of [10] verify it for all
non-hyperbolic, geometric 3-manifolds (in particular for Seifert fibered
manifolds and for Sol manifolds) and in [7], further evidence for the
conjecture are provided. Interestingly, our groups $\Gamma_{W}$ are not
3-manifold groups which follows from the main result of [11].

For any class $\mathcal{C}$ of groups, we say $\mathcal{C}$ satisfies property
$T_{O}$ if for any group $G$ in this class, $G$ is left-orderable iff $G$ is
torsion-free. Similarly, we say $\mathcal{C}$ satisfies property $T_{BO}$ if
for any group $G$ in $\mathcal{C}$, $G$ is bi-orderable iff $G$ has no
generalized torsion. For the class of 3-manifold groups, property $T_{O}$
fails (see [4]) whereas property $T_{BO}$ is an open conjecture; and for the
class of one-relator groups, property $T_{O}$ holds, whereas property $T_{BO}$
fails.

##  2\. Proof of Theorem 1.1

A one-relator group $G$ in $n\geq 3$ generators can always be embedded into a
group $H$ with two generators and one relator. For the latter, by a sequence
of Shreier moves we can obtain a one-relator presentation $\langle t,h\
|r\rangle$ of it such that sum of exponents of one of the generators (say, of
the generator $t$) is zero. This allows to view $H$ as a semidirect product
$\mathbb{Z}\ltimes H_{0}$ where the acting group $\mathbb{Z}$ is generated by
$t$, the normal subgroup $H_{0}$ is generated by elements
$h_{n}=t^{n}ht^{-n},n\in Z$ and the $\mathbb{Z}$ action is given by the
conjugation $th_{n}t^{-1}=h_{n+1},n\in\mathbb{Z}$.

The relation $r$ can be written as a reduced word $R(h_{0},\dots,h_{k})$ in
generators $h_{0},\dots,h_{k}$ for some $k\geq 0$ and this allows to view $H$
as an HNN extension $(H,A,B,t)$ where

$A=\langle h_{0},\dots,h_{k}\ |\ R(h_{0},\dots,h_{k})\rangle,\ \mathrm{and}\
B=\langle h_{1},\dots,h_{k+1}\ |\ R(h_{1},\dots,h_{k+1})\rangle$

where one can even claim that the word $R$ is shorter than the word $r$ in
length. This inductive phenomenon together with the HNN extension structure
become very helpful in understanding one-relator groups. One of the strong
tools in utilising the HNN extension structure is given by Britton’s Lemma. In
our approach we will be using the amalgamated free product structure of one-
relator groups (rather than HNN extension) since the group $H_{0}$ above is a
direct limit of groups $H_{m,n}=\langle h_{m},\dots,h_{n}\rangle,m\leq n$ and
for all $m<n$ with $n-m\geq k$, we can write $H_{m,n}$ as the amalgamated free
product $H_{m,n-1}\ast_{H_{m+1,n-1}}H_{m+1,n}$. (In our proof, we will be
using a slightly different but related amalgamated free product decomposition
of the groups $H_{m,n}$.)

The following result is well known (see [8]); it can be viewed as an analog of
Britton’s Lemma and plays an important role in understanding amalgamated free
products similar to the role that Britton’s Lemma plays in understanding HNN
extensions.

######  Proposition 2.1.

Let $A,B,C$ be groups with monomorphisms $\phi:C\to A$ and $\psi:C\to B$ and
$G=A*_{C}B$ be the amalgamated free product with respect to these
monomorphisms. Let also $T_{A}=A\backslash C,T_{B}=B\backslash C$ and
$w=g_{0}g_{1}\dots g_{n},n\geq 1$ such that for all $1\leq i\leq n$ if

(i) if $g_{i-1}\in A$, then $g_{i}\in T_{B}$

(ii) if $g_{i-1}\in B$, then $g_{i}\in T_{A}$

Then $w\neq 1\in A*_{C}B$.

We are now ready to start the proof of Theorem 1.1. First, we prove that
$\Gamma$ is not bi-orderable.

Let $b=tat^{-1},c=t^{2}at^{-2}$. Then we have $bacb^{-1}=(ac)^{2}$.

Now, assume that $<$ is a bi-order on $\Gamma$. Without loss of generality, we
may assume that $a>1$. Then $b>1$ and $c>1$. If $b>a$, since the positive cone
is invariant under a conjugation, we obtain that $c>b$; and if $b<a$, then
$c<b$. Thus either $1<a<b<c$ or $1<c<b<a$. Then, in the former case we have
$bacb^{-1}<bac<cac<(ac)^{2}$ which is a contradiction. In the latter case, we
have $bacb^{-1}<ba<a^{2}<(ac)^{2}$ which is again a contradiction. Hence
$\Gamma$ is not bi-orderable.

Now, assume that $\Gamma$ contains a generalized torsion $\tau$.

Let $a_{n}=t^{n}at^{-n},n\in\mathbb{Z}$ and $H$ be the subgroup of $\Gamma$
generated by all $a_{n},n\in\mathbb{Z}$. Then $H$ is a normal subgroup and
$\Gamma\cong\mathbb{Z}\ltimes H$ where $\mathbb{Z}$ acts on $H$ by shift, i.e.
by the $t$-conjugation $ta_{n}t^{-1}=a_{n+1},n\in\mathbb{Z}$. Then necessarily
$\tau\in H$.

Elements of $H$ can be written (not uniquely) as reduced words in the alphabet
$\mathcal{A}=\\{a_{n},n\in\mathbb{Z}\\}$. Then $H$ can be given by the
presentation

$H=\langle a_{n},n\in\mathbb{Z}\ |\
a_{n+1}(a_{n}a_{n+2})a_{n+1}^{-1}=(a_{n}a_{n+2})^{2},n\in\mathbb{Z}\rangle$

Let $\mathcal{F}$ be the free group formally generated by the alphabet
$\mathcal{A}$ and let $\tau=U$ where $U$ be a cyclically reduced word in the
alphabet $\mathcal{A}$ of minimal length. $\mathbb{Z}$ acts on the alphabet
$\mathcal{A}$ by shift $a_{n}\to a_{n+1},n\in\mathbb{Z}$ hence also on the
group $\mathcal{F}$. By abusing the notation we will write $twt^{-1}$ for the
shift of a word $w\in\mathcal{F}$. Also, we will write $|w|$ to denote the
length of $w\in\mathcal{F}$ in the Cayley graph of $\mathcal{F}$ with respect
to the generating set $\mathcal{A}$.

Since $\tau$ is a generalized torsion, we have

$(t^{n_{1}}g_{1}\tau g_{1}^{-1}t^{-n_{1}})(t^{n_{2}}g_{2}\tau
g_{2}^{-1}t^{-n_{2}})\dots(t^{n_{r}}g_{r}\tau g_{r}^{-1}t^{-n_{r}})=1$

for some $r\geq 1$ and $g_{j}\in H,n_{j}\in\mathbb{Z},1\leq j\leq r$ where
$g_{i},n_{i},1\leq i\leq r$ are chosen such that if
$t^{n_{i}}g_{i}t^{-n_{i}},t^{n_{i}}\tau t^{-n_{i}}$ are written as reduced
words $V_{i},\tau_{i}$ respectively in the alphabet $\mathcal{A}$, then the
word

$W=V_{1}\tau_{1}V_{1}^{-1}\dots V_{r}\tau_{r}V_{r}^{-1}$

viewed as a reduced word in the alphabet $\mathcal{A}$ has the shortest
possible word length in $\mathcal{F}$. Let us emphasize that

$|W|\leq\displaystyle\mathop{\sum}_{i=1}^{r}|V_{i}|+|\tau_{i}|+|V_{i}^{-1}|=\displaystyle\mathop{\sum}_{i=1}^{r}(2|V_{i}|+|\tau|).$

In addition, we can write $W$ as

$W^{\prime}=V_{1}^{\prime}\tau_{1}^{\prime}(V_{1}^{\prime})^{-1}\dots
V_{r}^{\prime}\tau_{r}^{\prime}(V_{r}^{\prime})^{-1}$

where the following conditions hold:

i) for all $1\leq i\leq r$, the reduced word $\tau_{i}^{\prime}$ is just a
cyclic permutation of letters of $\tau_{i}=t^{n_{i}}\tau t^{-n_{i}}$;

ii) for all $1\leq i\leq r$,
$|V_{i}^{\prime}\tau_{i}^{\prime}|=|V_{i}^{\prime}|+|\tau_{i}^{\prime}|=|\tau_{i}(V_{i}^{\prime})^{-1}|$.

Let

$(t^{n_{1}}g_{1}\tau g_{1}^{-1}t^{-n_{1}})(t^{n_{2}}g_{2}\tau
g_{2}^{-1}t^{-n_{2}}))\dots(t^{n_{r}}g_{r}\tau
g_{r}^{-1}t^{-n_{r}})=W(a_{l},a_{l+1},\dots,a_{k})$

where $l\leq k$ and for any $a_{i}$ occurring in the reduced word $W$, we have
$l\leq i\leq k$ and both $a_{0}$ and $a_{k}$ occur in $W$. Without loss of
generality, we may and will assume that $l=0$.

Now, for all $n\geq 0$, let $G_{n}$ be the subgroup of $H$ generated by
$a_{0},\dots,a_{n}$ and for all $m\leq n$, let $H_{m,n}$ be the subgroup of
$H$ generated by $a_{m},\dots,a_{n}$. Then $H_{m,n}\cong G_{m-n}$ by the
obvious isomorphism sending $a_{m+j}$ to $a_{j},0\leq j\leq n-m$.

Furthermore, $G_{0}\cong\mathbb{Z},G_{1}\cong\mathbb{F}_{2}$ and
$G_{3}\cong\langle a,b,c\ |\ b(ac)b^{-1}=(ac)^{2}\rangle$. Additionally, for
all $n\geq 2$ we make the following key observations about $G_{n}$:

a) The group $G_{n}$ is isomorphic to an amalgamated free product
$G_{n-1}\ast_{\mathbb{Z}}BS(1,2)$, more precisely, we have

$G_{n}\cong G_{n-1}\ast_{H_{n-1,n-1}}B\cong G_{n-1}\ast_{\mathbb{Z}}BS(1,2)$

where $BS(1,2)=\langle\alpha,\beta\ |\
\alpha\beta\alpha^{-1}=\beta^{2}\rangle\cong\mathbb{Z}\ltimes\mathbb{Z}[\frac{1}{2}]$
is the Baumslag-Solitar group, $B$ is the subgroup of $G_{n}$ generated by
$c:=a_{n-1}$ and $x:=a_{n-2}a_{n}$, the monomorphism $\phi_{1}:H_{n-1,n-1}\to
G_{n-1}$ is just the inclusion map given by $\phi_{1}(a_{n-1})=a_{n-1}$, the
monomorphism $\phi_{2}:H_{n-1,n-1}\to B$ is given by
$\phi_{2}(a_{n-1})=a_{n-1}$, and finally the isomorphism $\psi:B\to BS(1,2)$
is given by $\psi(a_{n-1})=\alpha,\psi(x)=\beta$.

b) Any element of $G_{n}$ can be written as $U_{1}c^{m_{1}}X_{1}\dots
U_{s}c^{m_{s}}X_{s}U_{s+1}$ where $U_{1},\dots,U_{s+1}$ are reduced (possibly
trivial) words in the alphabet $\\{a_{0},a_{1},\dots,a_{n-1}\\}$ not ending
with $c^{\pm 1}$ for $1\leq i\leq s$ and not starting with $c^{\pm 1}$ for
$2\leq i\leq s+1$, $X_{i}=x^{l_{i}},1\leq l\leq s$ with
$l_{i}=\frac{q_{i}}{2^{p_{i}}},p_{i},q_{i}\in\mathbb{Z},1\leq i\leq s$.

c) For all $Y\in G_{n-1}\backslash\\{c^{n}\ |\ n\in\mathbb{Z}\\}$ and
$p,q\in\mathbb{Z}$, $c^{p}Yc^{q}\neq Y^{-1}$. 111Indeed, by replacing $n$ with
$n+1$, it suffices to prove that for all $Y\in G_{n}\backslash\\{d^{n}\ |\
n\in\mathbb{Z}\\}$ and $p,q\in\mathbb{Z}$, $d^{p}Yd^{q}\neq Y^{-1}$, where
$d:=a_{n}$. But $d=b^{-1}x$ with $b:=a_{n-2}$, so we need to show that
$(b^{-1}x)^{p}Y(b^{-1}x)^{q}\neq Y^{-1}$ but the latter follows immediately
from b) and Proposition 2.1

Let us recall that any element of $BS(1,2)$ can be written as
$\alpha^{i}\beta^{l}$ where $l=\frac{p}{2^{q}}$ is a dyadic rational number
(i.e. $l\in\mathbb{Z}[\frac{1}{2}]$; we will write $\mathbb{Z}[\frac{1}{2}]$
to denote the ring of dyadic rationals as well as normal subgroup of $BS(1,2)$
upon which the cyclic group $\mathbb{Z}$ acts). Then, by Proposition 2.1, for
some $s\geq 0$, the generalized torsion $\tau$ can be written as
$\tau=\Omega_{1}c^{m_{1}}X_{1}\dots\Omega_{s}c^{m_{s}}X_{s}\Omega_{s+1}$ where
$\Omega_{1},\dots,\Omega_{s+1}$ are reduced words in the alphabet
$\\{a_{0},a_{1},\dots,a_{n-1}\\}$, $\Omega_{j}$ not ending with $c^{\pm 1}$
for $1\leq j\leq s$ and not starting with $c^{\pm 1}$ for $2\leq j\leq s+1$,
$\Omega_{j}\notin C,2\leq j\leq s$, $X_{j}=x^{l_{i}},1\leq j\leq s$ where $C$
denotes the infinite cyclic subgroup generated by $c$, and $l_{j},1\leq j\leq
s$ are dyadic rational numbers.

Now, let $n=k$. We distinguish three cases for $\tau$:

1) No power of $x$ occurs in $\tau$ (this happens when $s=0$);

2) The powers of $x$ occurs only once in $\tau$ (i.e. $s=1$);

3) The powers of $x$ occurs at least twice in $\tau$ (i.e. $s\geq 2)$.

Now, we can rule out Case 1) and 2). Indeed, by adding the relations
$[a_{i},x]=1,0\leq i\leq n-2$ to $G_{n}$ we obtain a quotient
$\overline{G_{n}}$ of it such that $\overline{G_{n}}/\langle x=1\rangle\cong
G_{n-1}$. In Case 2, $\tau$ (the image of $\tau$; by abuse of notation, we
denote the images in the quotient by the same letters here) can be written as
$\Omega c^{m}x^{l}$ where $\Omega$ is a reduced word in the alphabet
$\\{a_{0},\dots,a_{n-1}\\}$ not ending with $c^{\pm 1}$ and
$l\in\mathbb{Z}[\frac{1}{2}]\backslash\\{0\\}$. Then the product
$W^{\prime}=V_{1}^{\prime}\tau_{1}^{\prime}(V_{1}^{\prime})^{-1}\dots
V_{r}^{\prime}\tau_{r}^{\prime}(V_{r}^{\prime})^{-1}$ will also be in the form
$\Omega^{\prime}c^{m^{\prime}}x^{l^{\prime}}$ again with $l^{\prime}\neq 0$
(because $l^{\prime}$ is positive if $l$ is positive, and negative if $l$ is
negative). Hence $W^{\prime}\neq 1$ in the quotient. Then $W^{\prime}\neq 1$
in $G_{n}$ hence also in $\Gamma$. In Case 1, the image of $\tau$ can be
written as $\Omega c^{m}$ where $\Omega$ is a reduced word in the alphabet
$\\{a_{0},\dots,a_{n-1}\\}$ not ending with $c^{\pm 1}$. Then
$W^{\prime}=\Omega^{\prime}c^{m^{\prime}}x^{l^{\prime}}$ and we must have
$l^{\prime}=0$ to have $\tau$ a generalized torsion, but then $\tau$ is also a
generalized torsion in $G_{n-1}$ and we may proceed by induction.

We are now left with Case 3. First, let us notice that in the identity
$W^{\prime}=V_{1}^{\prime}\tau_{1}^{\prime}(V_{1}^{\prime})^{-1}\dots
V_{r}^{\prime}\tau_{r}^{\prime}(V_{r}^{\prime})^{-1}=1$, clearly we have
$r\geq 2$. Let

$\tau_{i}^{\prime}=\Omega_{1}^{(i)}c^{m_{1}^{(i)}}X_{1}^{(i)}\dots\Omega_{s}^{(i)}c^{m_{s}^{(i)}}X_{s}^{(i)}\Omega_{s+1}^{(i)}$

for all $1\leq i\leq r$ where again
$\Omega_{1}^{(i)},\dots,\Omega_{s+1}^{(i)}$ are reduced words in the alphabet
$\\{a_{0},a_{1},\dots,a_{n-1}\\}$, $\Omega_{j}^{(i)}$ not ending with $c^{\pm
1}$ for $1\leq j\leq s$ and not starting with $c^{\pm 1}$ for $2\leq j\leq
s+1$, $\Omega_{j}^{(i)}\notin C,2\leq j\leq s$,
$X_{j}^{(i)}=x^{l_{j}^{(i)}},1\leq j\leq s$ where $l_{j}^{(i)},1\leq j\leq s$
are dyadic rational numbers.

In the simplest case of $r=2$, we have

$\tau_{1}^{\prime}=\Omega_{1}^{(1)}c^{m_{1}^{(1)}}X_{1}^{(1)}\dots\Omega_{s}^{(1)}c^{m_{s}^{(1)}}X_{s}^{(1)}\Omega_{s+1}^{(1)}=A_{1}c^{k_{1}}X^{p_{1}}\dots
A_{s}c^{k_{s}}X^{p_{s}}A_{s+1}$

and

$\tau_{2}^{\prime}=\Omega_{1}^{(2)}c^{m_{1}^{(2)}}X_{1}^{(2)}\dots\Omega_{s}^{(2)}c^{m_{s}^{(2)}}X_{s}^{(2)}\Omega_{s+1}^{(2)}=B_{1}c^{l_{1}}X^{q_{1}}\dots
B_{s}c^{l_{s}}X^{q_{s}}B_{s+1}$

where (for simplicity of notations) we have denoted
$A_{i}=\Omega_{i}^{(1)},B_{i}=\Omega_{i}^{(2)},1\leq i\leq s+1$ and
$x^{p_{i}}=X_{i}^{(1)},x^{q_{i}}=X_{i}^{(2)},k_{i}=m_{i}^{(1)},l_{i}=m_{i}^{(2)},1\leq
i\leq s$.

Then by conditions (i)-(ii) and by Proposition 2.1, we have

$\Omega_{s+1}^{(1)}(V_{1}^{\prime})^{-1}V_{2}^{\prime}\Omega_{1}^{(2)}=A_{s+1}(V_{1}^{\prime})^{-1}V_{2}^{\prime}B_{1}\in
C\ \mathrm{and}\ sgn(p_{s})=-sgn(q_{1}).$

Then, again by Proposition 2.1 we have that $s$ is even,
$\Omega_{s}^{(1)}c^{k}\Omega_{2}^{(2)}=A_{s}c^{k}B_{2}\in C$ for some
$k\in\mathbb{Z}[\frac{1}{2}]$ and inductively, we obtain that for all $1\leq
i\leq s$, $sgn(p_{s-i+1})=-sgn(q_{i})$, and for all $1\leq i\leq s-1$ and
$A_{s+1-i}cB_{1+i}\in C$ for some $c\in C$. (Indeed, we will use these
conditions only for $1\leq i\leq s/2$.)

But $\tau_{2}^{\prime}$ is a cyclic permutation of $\tau_{1}^{\prime}$ hence
there exists $k\in\\{1,\dots,s-1\\}$ such that $q_{i}=p_{i+k},1\leq i\leq s$
(we identify $p_{i}$ with $p_{j}$ if $i\equiv j(\mod s)$) and $B_{i}=A_{i+k}$,
for all $2\leq i\leq s-k$; $B_{i}=A_{i+k-1}$, for all $s-k+2\leq i\leq s$ (we
identify $A_{i}$ with $A_{j}$ if $1\leq i\leq s,j>s+1$ and $i\equiv j(\mod
s)$).

Notice that if $k$ is odd, then there exists $1\leq i<s/2$ such hat
$i+k=s-i+1$. Then $p_{s-i+1}=p_{i+k}=q_{i}$ but $sgn(p_{s-i+1})=-sgn(q_{i})$;
contradiction. Thus $k$ is even. Since both $n$ and $k$ are even and
$s-k+1\neq\frac{s-k}{2}-1$, we obtain that there exists $1\leq j\leq s$ such
hat $A_{j}cA_{j}\in C$ for some $c\in C$. This contradicts condition c).

In the general case of an arbitrary $r\geq 2$, again by Proposition 2.1, $s$
is even. Also, there will be $1\leq i\leq r-1$ such that if

$\tau_{i}^{\prime}=\Omega_{1}^{(i)}c^{m_{1}^{(i)}}X_{1}^{(i)}\dots\Omega_{s}^{(i)}c^{m_{s}^{(i)}}X_{s}^{(i)}\Omega_{s+1}^{(i)}=A_{1}c^{k_{1}}X^{p_{1}}\dots
A_{s}c^{k_{s}}X^{p_{s}}A_{s+1}$

and

$\tau_{i+1}^{\prime}=\Omega_{1}^{(i+1)}c^{m_{1}^{(i+1)}}X_{1}^{(i+1)}\dots\Omega_{s}^{(i+1)}c^{m_{s}^{(i+1)}}X_{s}^{(i+1)}\Omega_{s+1}^{(i+1)}=B_{1}c^{l_{1}}X^{q_{1}}\dots
B_{s}c^{l_{s}}X^{q_{s}}B_{s+1}$

then for all $1\leq i\leq s/2$, $sgn(p_{n-i+1})=-sgn(q_{i})$. Moreover, since
$\tau_{i+1}^{\prime}$ is a cyclic permutation of $\tau_{i}^{\prime}$ we again
have that there exists $k\in\\{1,\dots,s-1\\}$ such that (with the same
modular identifications) $q_{i}=p_{i+k},1\leq i\leq s$ and $B_{i}=A_{i+k}$,
for all $2\leq i\leq s-k$; $B_{i}=A_{i+k-1}$, for all $s-k+2\leq i\leq s$.
Again, we obtain that $k$ is even and there exists $1\leq j\leq s$ and $c\in
C$ such hat $A_{j}cA_{j}\in C$; this contradicts condition c).

##  3\. Proof of Theorem 1.2

To see that $\Gamma_{W}$ is non-bi-orderable, let $z_{i},0\leq i\leq s-1$ be
any letter which occur in $W$. We let
$a=z_{i},b=z_{s},c=z_{s+1},D=W(z_{0},\dots,z_{s-1})$.

Again, without loss of generality we may assume that $a>1$. Then either
$1<a<b<c$ or $1<c<b<a$. In the former case, we have
$bDcb^{-1}<bDc<cDc<(Dc)^{2}$ which is a contradiction. In the latter case, we
have $1<c<b<a\leq D$ hence $bDcb^{-1}<bD<aD\leq D^{2}<(Dc)^{2}$ which is again
a contradiction. Hence $\Gamma$ is not bi-orderable.

The rest of the proof goes without any change, except, for the groups
$G_{n},n\geq 0$ we will have $G_{n}\cong\mathbb{F}_{n+1}$ for all $0\leq n\leq
s$ if the ositive word $W$ involves both $z_{0}$ and $z_{s-1}$, and the
representation of the group $H$ is naturally given as

$H=\langle a_{n},n\in\mathbb{Z}\ |\
a_{n+1}(W(a_{n-s+1},\dots,a_{n})a_{n+2})a_{n+1}^{-1}=(W(a_{n-s+1},\dots,a_{n})a_{n+2})^{2},n\in\mathbb{Z}\rangle$

In additionr, we define $x=W(a_{0},\dots,a_{s-1})a_{s+1}\in G_{i}$ only for
indices $i\geq s+1$, moreover, instead of $BS(1,2)$, we consider the group

$BS(1,m)=\langle\alpha,\beta\ |\
\alpha\beta\alpha^{-1}=\beta^{m}\rangle\cong\mathbb{Z}\ltimes\mathbb{Z}[\frac{1}{m}]$

where the $\mathbb{Z}$-action is by multiplication by $m$, and instead of
dyadic rationals we consider the numbers from the ring
$\mathbb{Z}[\frac{1}{m}]=\\{\frac{p}{m^{l}}\ |\ p,l\in\mathbb{Z}\\}$.

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