# Nonpositively curved surfaces are Loewner

Mikhail G. Katz  Department of Mathematics, Bar Ilan University, Ramat Gan
5290002 Israel<EMAIL_ADDRESS>and  Stéphane Sabourau  Univ Paris Est
Creteil, CNRS, LAMA, F-94010 Creteil, France Univ Gustave Eiffel, LAMA,
F-77447 Marne-la-Vallée, France<EMAIL_ADDRESS>
###### Abstract.

We show that every closed nonpositively curved surface satisfies Loewner’s
systolic inequality. The proof relies on a combination of the Gauss–Bonnet
formula with an averaging argument using the invariance of the Liouville
measure under the geodesic flow. This enables us to find a disk with large
total curvature around its center yielding a large area.

###### Key words and phrases:

systole, area, Loewner’s inequality, nonpositive curvature, Gauss–Bonnet
formula, comparison theorem

###### 2020 Mathematics Subject Classification:

Primary 53C20; Secondary 53C23

Partially supported by the ANR project Min-Max (ANR-19-CE40-0014).

_Dedicated to Eugenio Calabi’s memory._

##  1\. Introduction

The systole of a closed nonsimply connected surface $M$ endowed with a
Riemannian metric, denoted by $\operatorname{{\rm sys}}(M)$, is defined as the
length of the shortest noncontractible loop of $M$. It is attained by the
length of a noncontractible closed geodesic. The _systolic area_ of $M$ is
defined as

$\sigma(M)=\frac{\operatorname{{\rm area}}(M)}{\operatorname{{\rm
sys}}(M)^{2}}.$

We will say that $M$ is _Loewner_ if its systolic area satisfies

$\sigma(M)\geq\frac{\sqrt{3}}{2}\approx 0.866.$

The first systolic inequality, due to Loewner, asserts that every metric on
the torus ${\mathbb{T}}^{2}$ is Loewner; see [10]. By [6], all surfaces with a
Riemannian metric in a hyperelliptic conformal class are Loewner. In
particular, every genus two surface is Loewner since every conformal class in
genus two is hyperelliptic. By [5], surfaces of genus at least $20$ are
Loewner. Pushing this technique further, Li and Su announce that this still
holds true for surfaces of genus $18$ and $19$, and even for surfaces of genus
$\geq 11$ when one restricts to nonpositively curved metrics; see [9]. For the
other cases the problem is still open.

In this article, we resolve this problem for nonpositively curved surfaces of
any genus (as well as for nonorientable surfaces) relying on a new approach.

######  Theorem 1.1.

Every closed nonpositively curved surface is Loewner.

Actually, we prove a stronger statement (see Corollary 4.2): every closed
nonpositively curved surface other than a torus has a systolic area at least

$\frac{\pi}{8}\left(1+\sqrt{\tfrac{7}{3}}\right)\approx 0.992.$

Our strategy differs from previous works. Here is the rough idea. Suppose that
$M$ is a genus $g$ surface which is not Loewner. Normalize the metric so that
$\operatorname{{\rm sys}}(M)=1$. Since the metric is nonpositively curved,
every disk of radius $\frac{1}{4}$ has area at least $\frac{\pi}{16}$, which
represents at least $\frac{\pi}{16}/\frac{\sqrt{3}}{2}\approx 22.6\%$ of the
total area. Now, by an averaging argument using the invariance of the
Liouville measure under the geodesic flow, one should be able to find a disk
$D$ of radius $\frac{1}{4}$ with at least $22.6\%$ of the total curvature. The
disk $D_{+}$ of radius $\frac{1}{2}$ centered at the same point has a lot of
(negative) curvature around its center, namely in $D$. This should force the
disk $D_{+}$ to have a lot of area contradicting the
$\frac{\sqrt{3}}{2}$-bound on the surface area. At implementation level, the
existence of a curvature-rich disk relies on an integral-geometric formula
relating the weighted average of the curvature $K$ on tangent disks in $TM$
with the Euler characteristic and a comparison result between this weighted
average and another weighted average of $K$ on metric disks in $M$. The area
lower bound on this curvature-rich disk follows from an expression relating
the area of this disk with the previous weighted average of $K$ on the same
disk.

##  2\. Disks and curvature

Let $M$ be a closed surface of nonpositive Euler characteristic $\chi(M)$
endowed with a Riemannian metric. Let $\pi:UM\to M$ be the canonical
projection defined on the unit tangent bundle $UM$ of $M$. We will sometimes
denote a unit tangent vector $u\in UM$ by $u_{x}\in U_{x}M$ when we want to
emphasize its basepoint $x=\pi(u)$ in $M$.

The Liouville measure on $UM$ decomposes as

$du=dx\,du_{x}$ (2.1)

where $dx$ is the area measure of $M$ and $du_{x}$ is the canonical length
measure of $U_{x}M$; see [2, §1.M]. Note that the Liouville measure is
invariant under the geodesic flow $\varphi_{t}:UM\to UM$ of $M$ and that

$\operatorname{{\rm vol}}(UM)=2\pi\operatorname{{\rm area}}(M).$

Let $K$ be the Gaussian curvature of $M$. The Gauss–Bonnet formula for a
domain $D$ of $M$ with piecewise smooth boundary $\partial D$ can be written

$\int_{D}K(x)\,dx+\int_{\partial D}\kappa(s)\,ds+\tau_{\partial
D}=2\pi\chi(D)$ (2.2)

where $\chi(D)$ is the Euler characteristic of $D$, $\kappa$ is the geodesic
curvature of $\partial D$ and $\tau_{\partial D}$ is the sum of the angular
differences of the tangent vectors at the corner points of $\partial D$. When
$D=M$, the Gauss–Bonnet formula for $M$ takes the form

$\int_{M}K(x)\,dx=2\pi\chi(M).$ (2.3)

It will be convenient to introduce the function

$\overline{K}:UM\to{\mathbb{R}}$

defined as $\overline{K}=K\circ\pi$.

The results of this section can be summarized as follows. First, there is an
integral-geometric formula relating a weighted average of $\overline{K}$ of
certain horizontal disks of radii at most $r$ in $UM$ in terms of the Euler
characteristic and $r$. Furthermore, the curvature condition yields an
inequality between this weighted average and the average of $K$ of certain
disks in $M$. The inequality between the two averages entails the existence of
a curvature-rich disk.

The following function will play a key role in our approach.

######  Definition 2.1.

Let $r>0$. Define $F_{r}:M\to{\mathbb{R}}$ as

$F_{r}(x)=\int_{0}^{r}(r-\rho)\int_{U_{x}M}\int_{0}^{\rho}\overline{K}(\varphi_{t}(u_{x})))\,t\,dt\,du_{x}\,d\rho.$

Let us compute the integral of $F_{r}$.

######  Lemma 2.2.

We have

$\int_{M}F_{r}(x)\,dx=2\pi\chi(M)\cdot\frac{\pi r^{4}}{12}.$

###### Proof.

By the Liouville measure decomposition (2.1), we derive

$\displaystyle\int_{M}F_{r}(x)\,dx$
$\displaystyle=\int_{M}\int_{0}^{r}(r-\rho)\int_{U_{x}M}\int_{0}^{\rho}\overline{K}(\varphi_{t}(u_{x})))\,t\,dt\,du_{x}\,d\rho\,dx$
$\displaystyle=\int_{0}^{r}(r-\rho)\int_{0}^{\rho}\int_{UM}\overline{K}(\varphi_{t}(u)))\,du\,t\,dt\,d\rho.$

Now, by the invariance of the Liouville measure under the geodesic flow, we
obtain

$\displaystyle\int_{M}F_{r}(x)\,dx$
$\displaystyle=\int_{0}^{r}(r-\rho)\int_{0}^{\rho}\int_{UM}\overline{K}(u))\,du\,t\,dt\,d\rho$
$\displaystyle=2\pi\,\int_{M}K(x)\,dx\,\int_{0}^{r}(r-\rho)\int_{0}^{\rho}t\,dt\,d\rho$
$\displaystyle=2\pi\chi(M)\cdot\frac{\pi r^{4}}{12}$

where the last equality follows from the Gauss–Bonnet formula (2.3). ∎

Let us introduce the following function in connection with the area lower
bound in Proposition 3.1.

######  Definition 2.3.

Let $r\in(0,\frac{1}{2}\operatorname{{\rm sys}}(M))$. Define
$G_{r}:M\to{\mathbb{R}}$ as

$G_{r}(x)=\int_{0}^{r}(r-\rho)\int_{B_{x}(\rho)}K(y)\,dy\,d\rho$

where $B_{x}(\rho)$ is the ball of radius $\rho$ centered at $x$.

The functions $F_{r}$ and $G_{r}$ are related through the following comparison
result.

######  Lemma 2.4.

Suppose $M$ is nonpositively curved. Let $r\in(0,\frac{1}{2}\operatorname{{\rm
sys}}(M))$. Then, for every $x\in M$,

$F_{r}(x)\geq G_{r}(x).$

###### Proof.

By Gauss’ Lemma and since $r<\frac{1}{2}\operatorname{{\rm sys}}(M)$, the
exponential map at $x$ in polar coordinates induces a diffeomorphism

$\displaystyle U_{x}M\times(0,r]$ $\displaystyle\longrightarrow
B_{x}(r)\setminus\\{x\\}$ $\displaystyle(u_{x},t)$ $\displaystyle\longmapsto
y=\pi(\varphi_{t}(u_{x}))$

Since $M$ is nonpositively curved, this map is distance nondecreasing by
Rauch’s comparison theorem, see [4, §1.11]. It follows that

$t\,dt\,du_{x}\leq dy.$

After integration and since $K\leq 0$, this implies

$\int_{0}^{r}(r-\rho)\int_{U_{x}M}\int_{0}^{\rho}\overline{K}(\varphi_{t}(u_{x})))\,t\,dt\,du_{x}\,d\rho\geq\int_{0}^{r}(r-\rho)\int_{B_{x}(\rho)}K(y)\,dy\,d\rho.$

That is, $F_{r}(x)\geq G_{r}(x)$. ∎

We can now derive our key estimate.

######  Proposition 2.5.

Suppose that $M$ is nonpositively curved. Let
$r\in(0,\frac{1}{2}\operatorname{{\rm sys}}(M))$. Then there exists $x_{0}\in
M$ such that

$G_{r}(x_{0})\leq\frac{2\pi\chi(M)}{\operatorname{{\rm
area}}(M)}\cdot\frac{\pi r^{4}}{12}.$

###### Proof.

Taking the average integral $\fint_{M}$ over $M$ in Lemma 2.4 leads to

$\fint_{M}G_{r}(x)\,dx\leq\fint_{M}F_{r}(x)\,dx=\frac{1}{\operatorname{{\rm
area}}(M)}\,\int_{M}F_{r}(u)\,du.$

By Lemma 2.2, this yields

$\fint_{M}G_{r}(x)\,dx\leq\frac{2\pi\chi(M)}{\operatorname{{\rm
area}}(M)}\cdot\frac{\pi r^{4}}{12}$

and the result immediately follows. ∎

##  3\. Disks and area

In this section, we show that disks with large (negative) curvature have a
large area, and proceed to the proof of the main theorem.

We will need the following area lower bound.

######  Proposition 3.1.

Let $r\in(0,\frac{1}{2}\operatorname{{\rm sys}}(M))$. Then

$\operatorname{{\rm area}}B_{x}(r)\geq\pi r^{2}-G_{r}(x).$ (3.1)

###### Proof.

First, approximate the metric on $M$ by a real analytic metric. For this new
metric, the component $\mathcal{C}_{x}(s)$ of the circle of radius $s\leq r$
centered at $x$ surrounding its center is a piecewise smooth curve. Moreover,
the length function $s\mapsto L(\mathcal{C}_{x}(s))$ is differentiable except
for a finite number of values of $s$, and its derivative is given by the first
variation formula. Specifically, as long as $B_{x}(s)$ is nonempty, we have

$L^{\prime}(\mathcal{C}_{x}(s))=\int_{\mathcal{C}_{x}(s)}\kappa(t)\,dt+\tau_{s}$

for almost every $s$, where $\kappa$ is the geodesic curvature of the curve
$\mathcal{C}_{x}(s)$ and $\tau_{s}$ is the sum of the angular difference of
the tangent vectors at the corner points of $\mathcal{C}_{x}(s)$. By the
Gauss–Bonnet formula (2.2) for domains with boundary and since
$\mathcal{C}_{x}(s)$ bounds a topological disk $\mathcal{D}_{x}(s)$, we derive

$L^{\prime}(\mathcal{C}_{x}(s))=2\pi-\int_{\mathcal{D}_{x}(s)}K(y)\,dy.$

Integrating this relation twice and using the coarea formula lead to

$\displaystyle\operatorname{{\rm area}}\mathcal{D}_{x}(r)$
$\displaystyle\geq\pi
r^{2}-\int_{0}^{r}\int_{0}^{\rho}\int_{\mathcal{D}_{x}(s)}K(y)\,dy\,ds\,d\rho$
(3.2) $\displaystyle\geq\pi
r^{2}-\int_{0}^{r}(r-\rho)\,\int_{\mathcal{D}_{x}(\rho)}K(y)\,dy\,d\rho.$

To conclude, simply observe that, when the real analytic metric approaches the
initial metric on $M$, the domains $\mathcal{D}_{x}(\rho)$ Hausdorff converge
to the balls $B_{x}(\rho)$ (uniformly in $\rho$). In particular, the area of
$\mathcal{D}_{x}(r)$ converges to the one of $B_{x}(r)$, and
$\int_{\mathcal{D}_{x}(\rho)}K(y)\,dy$ uniformly converges to
$\int_{B_{x}(\rho)}K(y)\,dy$. Hence,

$\operatorname{{\rm area}}B_{x}(r)\geq\pi
r^{2}-\int_{0}^{r}(r-\rho)\,\int_{B_{x}(\rho)}K(y)\,dy\,d\rho.$

∎

######  Remark 3.2.

When the surface $M$ is nonpositively curved, there is an equality in (3.1).
Indeed, in this case, we can approximate the metric by a real analytic metric
of negative curvature. In this case, the curve $\mathcal{C}_{x}(s)$ represents
the circle of radius $s$ around $x$ and the domain $\mathcal{D}_{x}(s)$ it
surrounds coincides with the metric ball of radius $s$ around $x$. Because of
the curvature condition, this ball is a topological disk. In this case, the
coarea formula leads to an equality in (3.2) and in the following
inequalities.

Putting everything together, we obtain the following area lower bound.

######  Proposition 3.3.

Suppose that $M$ is nonpositively curved. Let
$r\in(0,\frac{1}{2}\operatorname{{\rm sys}}(M))$. Then there exists a ball
$B(r)$ of radius $r$ with

$\operatorname{{\rm area}}B(r)\geq\pi
r^{2}-\frac{2\pi\chi(M)}{\operatorname{{\rm
area}}(M)}\cdot\frac{\pi}{12}r^{4}.$

###### Proof.

By Proposition 2.5, there exists $x_{0}\in M$ such that

$G_{r}(x_{0})\leq\frac{2\pi\chi(M)}{\operatorname{{\rm
area}}(M)}\cdot\frac{\pi r^{4}}{12}.$

By Proposition 3.1, it follows that

$\operatorname{{\rm area}}B_{x_{0}}(r)\geq\pi
r^{2}-\frac{2\pi\chi(M)}{\operatorname{{\rm
area}}(M)}\cdot\frac{\pi}{12}r^{4}.$

∎

We can conclude the proof of Theorem 1.1 as follows.

###### Proof of Theorem 1.1.

Apply Proposition 3.3 with $r=\frac{1}{2}\operatorname{{\rm sys}}(M)$. We
obtain

$\operatorname{{\rm area}}(M)\geq\operatorname{{\rm
area}}B(\tfrac{1}{2}\operatorname{{\rm
sys}}(M))\geq\left(\frac{\pi}{4}-\frac{\pi^{2}\chi(M)}{96}\cdot\frac{\operatorname{{\rm
sys}}(M)^{2}}{\operatorname{{\rm area}}(M)}\right)\operatorname{{\rm
sys}}(M)^{2}.$

Expressed in terms of the systolic area, this inequality takes the following
form

$\sigma(M)\geq\lambda+\frac{\mu}{\sigma(M)},$

where $\lambda=\frac{\pi}{4}$ and $\mu=-\frac{\pi^{2}\chi(M)}{96}\geq 0$. That
is, $\sigma(M)^{2}-\lambda\,\sigma(M)-\mu\geq 0$. Hence,

$\sigma(M)\geq\frac{\lambda+\sqrt{\lambda^{2}+4\mu}}{2}.$

Thus,

$\sigma(M)\geq\frac{\pi}{8}\left(1+\sqrt{1-\tfrac{2}{3}\chi(M)}\right)\geq\frac{\pi}{8}\left(1+\sqrt{\tfrac{5}{3}}\right)\approx
0.899$ (3.3)

for $\chi(M)\leq-1$. Therefore, the surface $M$ is Loewner in this case.

For the Klein bottle (where $\chi=0$), the minimal value of the systolic area
over nonpositively curved metrics is attained by a square flat metric and is
equal to $1$. (Alternatively, the minimal value of the systolic area of a
Riemannian Klein bottle is equal to $\frac{2\sqrt{2}}{\pi}\approx 0.9$; see
[1].)

Hence, every closed nonpositively curved surface is Loewner. ∎

##  4\. Corollaries

In [6] and [7], we computed the minimal value of the systolic area over all
nonpositively curved metrics on the genus $2$ surface $\Sigma_{2}$ and Dyck’s
surface $3{\mathbb{R}\mathbb{P}}^{2}$. For the genus $3$ surface, Calabi [3]
presented a CAT$(0)$ piecewise flat metric with the lowest systolic area we
know of. See Remark 4.4 below. Combined with the systolic inequality (3.3),
this leads to the following result.

######  Corollary 4.1.

We obtain the following data, where the third column represents the minimal
value of the systolic area over nonpositively curved metrics.

$\begin{array}[]{|c|c|c|c|}\hline\cr M&\chi(M)&\sigma_{\leq 0}(M)&source\\\
\hline\cr\hline\cr{\mathbb{T}}^{2}&0&\frac{\sqrt{3}}{2}\approx
0.866&\text{\cite[cite]{[\@@bibref{}{pu}{}{}]}}\\\
\hline\cr{\mathbb{K}}^{2}=2{\mathbb{R}\mathbb{P}}^{2}&0&1&\textrm{obvious}\\\
\hline\cr
3{\mathbb{R}\mathbb{P}}^{2}&-1&1+\frac{(169-38\sqrt{19})^{\frac{1}{2}}}{12}\approx
1.152&\text{\cite[cite]{[\@@bibref{}{KS15}{}{}]}}\\\
\hline\cr\Sigma_{2}&-2&3(\sqrt{2}-1)\approx
1.242&\text{\cite[cite]{[\@@bibref{}{KS06pams}{}{}]}}\\\
\hline\cr\Sigma_{3}&-4&<\frac{7\sqrt{3}}{8}\approx
1.515&\text{\cite[cite]{[\@@bibref{}{cal}{}{}]}}\\\ \hline\cr\Sigma_{g\geq
3}&2-2g&\geq\frac{\pi}{8}\left(1+\sqrt{\tfrac{4g-1}{3}}\right)\geq
1.144&\eqref{eq:sqrt}\\\ \hline\cr\Sigma_{g\geq
4}&2-2g&\geq\frac{\pi}{8}\left(1+\sqrt{\tfrac{4g-1}{3}}\right)\geq
1.270&\eqref{eq:sqrt}\\\ \hline\cr n{\mathbb{R}\mathbb{P}}^{2}\text{(with
$n\geq 4$)}&2-n&\geq\frac{\pi}{8}\left(1+\sqrt{\tfrac{2n-1}{3}}\right)\geq
0.992&\eqref{eq:sqrt}\\\ \hline\cr n{\mathbb{R}\mathbb{P}}^{2}\text{(with
$n\geq 7$)}&2-n&\geq\frac{\pi}{8}\left(1+\sqrt{\tfrac{2n-1}{3}}\right)\geq
1.210&\eqref{eq:sqrt}\\\ \hline\cr\end{array}$

We obtain the following consequences.

######  Corollary 4.2.

Every closed nonpositively curved surface $M$ other than a torus has a
systolic area at least

$\frac{\pi}{8}\left(1+\sqrt{\tfrac{7}{3}}\right)\approx 0.992.$

######  Corollary 4.3.

Every orientable nonpositively curved surface $\Sigma_{g}$ of genus $g\geq 2$
has a systolic area at least

$\frac{\pi}{8}\left(1+\sqrt{\tfrac{11}{3}}\right)\approx 1.144.$

######  Remark 4.4.

Observe that nonpositively curved surfaces of genus $g\geq 4$ have a systolic
area greater than the minimal systolic area of a nonpositively curved surface
of genus $2$, which is equal to $3(\sqrt{2}-1)\approx 1.242$; see [6]. It
would be interesting to know if this still holds true in genus $3$, showing a
monotonicity of the minimal systolic area in terms of the genus for low
genera. Observe that the best value of the systolic area we know of in genus
$3$ is given by a CAT$(0)$ piecewise flat surface in the conformal class of
the Klein quartic described by Calabi [3] and is equal to

$\frac{7\sqrt{3}}{8}\approx 1.515.$

This metric is a critical with respect to some metric variations (see [11])
and might be extremal among all metrics without any curvature assumption. It
is not surprising that such a metric is piecewise flat since optimal CAT$(0)$
metrics are flat with finitely many singularities in every genus by [8]. Note
that Calabi’s surface has only a slightly better systolic area than the one
given by the triangle hyperbolic surface $(2,3,12)$ of the same genus
described by Schmutz [12] (and conjectured extremal among hyperbolic metrics),
which is equal to

$\frac{2\pi}{\operatorname{arcsinh}(2+\sqrt{3})^{2}}\approx 1.528.$

It would be interesting to find the optimal CAT$(0)$ metric in the conformal
class of the surface described by Schmutz (or at least find a good
approximation by piecewise flat metrics ) to see if it has a lower systolic
area than the one of Calabi’s surface.

Similarly, nonpositively curved surfaces $n{\mathbb{R}\mathbb{P}}^{2}$ with
$n\geq 7$ have a systolic area greater than the minimal systolic area of a
nonpositively curved Dyck’s surface $3{\mathbb{R}\mathbb{P}}^{2}$, which is
equal to

$1+\frac{(169-38\sqrt{19})^{\frac{1}{2}}}{12}\approx 1.152$

(see [7]) and the question is open for $n=4,5,6$.

## References

  * [1] Bavard, C.: Inégalité isosystolique pour la bouteille de Klein. _Math. Ann._ 274 (1986), 439–441.
  * [2] Besse, A. L.: Manifolds all of whose geodesics are closed. _Ergeb. Math. Grenzgeb._ 93, Springer-Verlag, 1978.
  * [3] Calabi, E.: Extremal isosystolic metrics for compact surfaces. Actes de la Table Ronde de Géométrie Différentielle, _Sémin. Congr._ 1 (1996), Soc. Math. France, 146–166.
  * [4] Cheeger, J.; Ebin, D.: Comparison theorems in Riemannian geometry. _North-Holland Math. Library_ , Vol. 9, North-Holland Publishing Co., 1975
  * [5] Katz, M.; Sabourau, S.: Entropy of systolically extremal surfaces and asymptotic bounds. _Ergodic Theory Dynam. Systems_ 25 (2005), no. 4, 1209–1220.
  * [6] Katz, M.; Sabourau, S.: Hyperelliptic surfaces are Loewner. _Proc. Amer. Math. Soc._ 134 (2006), no. 4, 1189–1195.
  * [7] Katz, M.; Sabourau, S.: Dyck’s surfaces, systoles, and capacities. _Trans. Amer. Math. Soc._ 367 (2015), no. 6, 4483–4504.
  * [8] Katz, M.; Sabourau, S.: Systolically extremal nonpositively curved surfaces are flat with finitely many singularities. _J. Topol. Anal._ 13 (2021), no. 2, 319–347.
  * [9] Li, Q.; Su, W.: Every closed surface of genus at least $18$ is Loewner. See arXiv:2401.00720v2.
  * [10] Pu, P.M.: Some inequalities in certain nonorientable Riemannian manifolds. _Pacific J. Math._ 2 (1952), 55–71.
  * [11] Sabourau, S.: Isosystolic genus three surfaces critical for slow metric variations. _Geom. Topol._ 15 (2011), no. 3, 1477–1508.
  * [12] Schmutz, P.: Riemann surfaces with shortest geodesic of maximal length. _Geom. Funct. Anal._ 3 (1993), no. 6, 564–631 .


* An (1998) M. Y. An.  Logconcavity versus logconvexity: a complete characterization.  _Journal of Economic Theory_ , 80(2):350–369, 1998.
  * Attar et al. (2019) A. Attar, T. Mariotti, and F. Salanié.  On competitive nonlinear pricing.  _Theoretical Economics_ , 14(1):297–343, 2019\.
  * Back and Baruch (2013) K. Back and S. Baruch.  Strategic liquidity provision in limit order markets.  _Econometrica_ , 81(1):363–392, 2013.
  * Bertsekas (1999) D. Bertsekas.  _Nonlinear programming_.  Athena Scientific, Belmont, MA, second edition, 1999.
  * Biais et al. (2000) B. Biais, D. Martimort, and J.-C. Rochet.  Competing mechanisms in a common value environment.  _Econometrica_ , 68(4):799–837, 2000.
  * Biais et al. (2013) B. Biais, D. Martimort, and J.-C. Rochet.  Corrigendum to “Competing mechanisms in a common value environment”.  _Econometrica_ , 81(1):393–406, 2013.
  * Bielagk et al. (2019) J. Bielagk, U. Horst, and S. Moreno-Bromberg.  Trading under market impact: Crossing networks interacting with dealer markets.  _Journal of Economic Dynamics and Control_ , 100:131–151, 2019.
  * Cetin and Waelbroeck (2021) U. Cetin and H. Waelbroeck.  An equilibrium analysis of price impact and order flow.  Preprint, 2021.
  * Efron (1965) B. Efron.  Increasing properties of Polya frequency functions.  _Annals of Mathematical Statistics_ , 36(1):272–279, 1965.
  * Glosten (1989) L. R. Glosten.  Insider trading, liquidity, and the role of the monopolist specialist.  _Journal of Business_ , 62(2):211–235, 1989.
  * Ho and Stoll (1981) T. Ho and H. R. Stoll.  Optimal dealer pricing under transactions and return uncertainty.  _Journal of Financial Economics_ , 9(1):47–73, 1981.
  * Miravete (2002) E. J. Miravete.  Preserving log-concavity under convolution: Comment.  _Econometrica_ , 70(3):1253–1254, 2002.
  * Saumard and Wellner (2014) A. Saumard and J. A. Wellner.  Log-concavity and strong log-concavity: A review.  _Statistics Surveys_ , 8:45, 2014.
  * Treynor (1971) J. Treynor.  The only game in town.  _Financial Analysts Journal_ , 22:12–14, 1971.
  * Walter (1998) W. Walter.  _Ordinary differential equations_.  Springer, New York, 1998.


# Routley Star in Information-Based Semantics††thanks: This paper is an
outcome of the project Logical Structure of Information Channels, no.
21-23610M, supported by the Czech Science Foundation and realized at the
Institute of Philosophy of the Czech Academy of Sciences.

Vít Punčochář Institute of Philosophy,
Czech Academy of Sciences,
The Czech Republic<EMAIL_ADDRESS>Institute of Philosophy,
Czech Academy of Sciences,
The Czech Republic Igor Sedlár Institute of Philosophy,
Czech Academy of Sciences,
The Czech Republic<EMAIL_ADDRESS>
###### Abstract

It is common in various non-classical logics, especially in relevant logics,
to characterize negation semantically via the operation known as Routley star.
This operation works well within relational semantic frameworks based on prime
theories. We study this operation in the context of “information-based”
semantics for which it is characteristic that sets of formulas supported by
individual information states are theories that do not have to be prime. We
will show that, somewhat surprisingly, the incorporation of Routley star into
the information-based semantics does not lead to a collapse or a
trivialization of the whole semantic system. On the contrary, it leads to a
technically elegant though quite restricted semantic framework that determines
a particular logic. We study some basic properties of this semantics. For
example, we show that within this framework double negation law is valid only
in involutive linear frames. We characterize axiomatically the logic of all
linear frames and show that the logic of involutive linear frames coincides
with a system that Mike Dunn coined Kalman logic. This logic is the
$\\{\and,\vee,\neg\\}$-fragment of the “semi-relevant” logic known as
$\mathsf{R}$-mingle. Finally, we characterize by a deductive system the logic
of all information frames equipped with Routley star.

##  1 Introduction

This paper is a contribution to the general study of the operation known as
Routley star, standardly denoted by the symbol $*$ (see [4, 5, 7, 11]).
Routley star is used in non-classical, especially relevant logics for the
following semantic characterization of negation: $w\vDash\neg\alpha$ iff
$w^{*}\nvDash\alpha$. This operation works so well because under some basic
assumptions there is a corresponding operation on prime theories, mapping a
prime theory $\Omega$ to the prime theory
$\Omega^{*}=\\{\alpha\mid\neg\alpha\notin\Omega\\}$ so that one obtains the
analogue of the semantic clause $\neg\alpha\in\Omega$ iff
$\alpha\notin\Omega^{*}$. This fact is used in the canonical model
constructions and completeness proofs for logics equipped with semantics using
Routley star. Since primeness of the theories used in canonical models is
crucial for this strategy it seems that Routley star can work well only within
relational semantic frameworks in which sets of formulas that are true at
individual points of evaluation are always prime theories. In this paper we
study Routley star in the context of a semantics that is based on theories
rather than prime theories. We show that the incorporation of Routley star in
this sematnics significantly restricts its flexibility but it does not lead to
its collapse or a trivialization. On the contrary, it leads to a non-trivial
semantics that determines a particular interesting logic.

We call our semantic framework information-based sematnics. It has been used
in the context of inquisitive semantics (see [2, 8, 9]). Similar semantic
frameworks were employed for example in [3, 6, 12]. The main idea behind this
semantics is that there is a crucial conceptual difference between the notion
of truth and the notion of informational support. Both can be viewed as
relations between some specific entities on one side and formulas on the other
side. In the case of truth the specific relatum is some ontic entity, a
possible world or a situation. We will use the expression $w\vDash\alpha$ for
the claim that the sentence $\alpha$ is true in a world (or situation) $w$. In
the case of informational support the specific relatum is a body of
information or an information state and we will write $s\Vdash\alpha$
expressing that $\alpha$ is supported by the information state $s$. Truth and
informational support seem to be often confused but arguably they are
different notions.

The clearest way to show this is to compare the natural truth and support
conditions for disjunction. It makes a perfect sense to claim that a
disjunction $\alpha\vee\beta$ is true in a world (or situation) $w$ if and
only if $\alpha$ or $\beta$ is true in $w$. However, informational support of
disjunction cannot be characterized by an analogous condition. It is quite
common that a body of information supports a disjunction without supporting
any of its disjuncts. Technically speaking, this means that the sets of
formulas supported by individual information states are theories that do not
have to be prime. Our aim here is to study Routley star negation in the
context of such information-based semantics (a different treatment of negation
in information-based semantics, characterized in terms of an incompatibility
relation, was studied in [10]). Our main result is a syntactic
characterization of the logic of all information frames equipped with Routley
star.

The article is structured as follows. In Section 2 we recall the Routley star
in the context of the standard framework where points of evaluation correspond
to prime theories. Section 3 introduces information-based semantics with the
Routley star and proves a series of characterization results; in particular,
it is shown that the double negation law forces linearity of frames. Section 4
studies linear frames. In particular, the axiomatizations of the logic of all
linear frames and all involutive linear frames, respectively, are provided.
The latter logic coincides with what Dunn called Kalman logic. Section 5
establishes a completeness result for the logic of all information frames with
the Routley star.

##  2 Routley star in the standard framework

Let $L$ be the language built up from a set of atomic formulas $At$ using
conjunction $\and$, disjunction $\vee$ and negation $\neg$. We will now
reconstruct the standard semantic framework used for this language in non-
classical logics. For more details, see [4, 5].

A standard frame (SF, for short) is a tuple $\mathcal{F}=\langle
W,\leq,*\rangle$ where $\langle W,\leq\rangle$ is a partially ordered set and
$*$ is a unary operation on $W$ such that $v\leq w$ implies $w^{*}\leq v^{*}$,
for all $v,w\in W$. A standard model (SM) is a pair
$\mathcal{M}=\langle\mathcal{F},V\rangle$, where $\mathcal{F}$ is an SF and
$V$ is a valuation defined as a function assigning to each atomic formula an
upward closed set in $\mathcal{F}$. Given such a model $\mathcal{M}$ the
following semantic clauses specify recursively a relation of truth $\vDash$
between the elements of $\mathcal{F}$ and formulas of $L$:

  * $w\vDash p$ iff $w\in V(p)$, for every $p\in At$,

  * $w\vDash\alpha\and\beta$ iff $w\vDash\alpha$ and $w\vDash\beta$,

  * $w\vDash\alpha\vee\beta$ iff $w\vDash\alpha$ or $w\vDash\beta$,

  * $w\vDash\neg\alpha$ iff $w^{*}\nvDash\alpha$.

A crucial feature of the clauses is that they yield persistence, the property
required for atomic formulas in the definition of valuation: for any
$L$-formula $\alpha$, if $w\vDash\alpha$ and $w\leq v$ then $v\vDash\alpha$.
In other words, the propositions expressed by formulas are always upward
closed sets and the algebra of propositions is the algebra of such sets.

A pair of $L$-formulas $\left\langle\alpha,\beta\right\rangle$ will be called
a (consequence) $L$-pair. We will write $\alpha\vdash\beta$ instead of
$\left\langle\alpha,\beta\right\rangle$. We say that $\alpha\vdash\beta$ is
valid in $\mathcal{M}$ if for every $w$ in $\mathcal{M}$ if $w\vDash\alpha$
then $w\vDash\beta$. We say that $\alpha\vdash\beta$ is valid in a class of
standard models if it is valid in every model of that class.

It is known that the set of $L$-pairs valid in all standard models is
generated by the deductive system consisting of the axioms and rules of the
distributive lattice logic $\mathsf{DLL}$:

(ID) $\alpha\vdash\alpha$ | (e$\and_{\\_}1$) $\alpha\and\beta\vdash\alpha$ | (e$\and_{\\_}2$) $\alpha\and\beta\vdash\beta$ | (i$\vee_{\\_}1$) $\alpha\vdash\alpha\vee\beta$ | (i$\vee_{\\_}2$) $\beta\vdash\alpha\vee\beta$
---|---|---|---|---

(D) $\alpha\and(\beta\vee\gamma)\vdash(\alpha\and\beta)\vee(\alpha\and\gamma)$
---

(T) $\alpha\vdash\beta,\beta\vdash\gamma/\alpha\vdash\gamma$ | (i$\and$) $\alpha\vdash\beta,\alpha\vdash\gamma/\alpha\vdash\beta\and\gamma$ | (e$\vee$) $\alpha\vdash\gamma,\beta\vdash\gamma/\alpha\vee\beta\vdash\gamma$
---|---|---

plus the following axioms and rule characterizing the behaviour of negation:

(DM1) $\neg(\alpha\and\beta)\vdash\neg\alpha\vee\neg\beta$ | (DM2) $\neg\alpha\and\neg\beta\vdash\neg(\alpha\vee\beta)$
---|---

(N) $\alpha\vdash\beta/\neg\beta\vdash\neg\alpha$
---

We can denote this system, or more precisely, the set of $L$-pairs generated
by this system, as Distributive Lattice Logic with Routley Star Negation, or
DLLR, for short.

The elegance of the treatment of negation via the Routley star is clear form
the completeness proof which is based on the standard canonical model
construction. The canonical model is build out of prime theories. A set of
$L$-formulas $\Omega$ is a prime DLLR-theory if it satisfies the following
three conditions: (a) if $\alpha\in\Omega$ and
$\alpha\vdash\beta\in\textsf{DLLR}$ then $\beta\in\Omega$; (b) if
$\alpha\in\Omega$ and $\beta\in\Omega$ then $\alpha\and\beta\in\Omega$; (c) if
$\alpha\vee\beta\in\Omega$ then $\alpha\in\Omega$ or $\beta\in\Omega$. Note
that the set of all formulas that are true at a given $w$ is always a prime
DLLR-theory.

Now it is crucial that there is an operation on prime DLLR-theories that
corresponds to the Routley star operation. In particular, for any prime DLLR-
theory $\Omega$ let us define the set of formulas
$\Omega^{*}=\\{\beta~{}|~{}\neg\beta\notin\Omega\\}$. One can easily observe
that $\Omega^{*}$ is again a prime DLLR-theory (to show this the axioms (DM1)
and (DM2) and the rule (N) are needed). Moreover, it follows directly from the
definition of $*$ that if $\Omega$, $\Gamma$ are prime DLLR-theories such that
$\Omega\subseteq\Gamma$, then $\Gamma^{*}\subseteq\Omega^{*}$. So, one can
take the set of prime DLLR-theories ordered by inclusion and equipped with $*$
and the resulting structure is indeed a standard frame. Moreover, if valuation
is defined in the expected way ($\Omega\in V(p)$ iff $p\in\Omega$) one can
prove by induction the truth lemma: $\Omega\vDash\alpha$ iff
$\alpha\in\Omega$, from which completeness follows by a Lindenbaum-style
extension lemma.

If the standard frames are required to satisfy generally the property
$w=w^{**}$ then the resulting logic is axiomatized by the system above
enriched with the double negation laws (DN1) $\alpha\vdash\neg\neg\alpha$ and
(DN2) $\neg\neg\alpha\vdash\alpha$. Classical logic is then obtained
semantically by requiring $w=w^{*}$ (from which it follows that $\leq$ is
reduced to the identity relation) and syntactically by enriching the logic
with the law of excluded middle (EM) $\beta\vdash\alpha\vee\neg\alpha$ and the
principle of explosion (EFQ) $\alpha\and\neg\alpha\vdash\beta$.

##  3 Information-based semantics

The standard semantics presented in the previous section was based on truth
conditions as is clear from the semantic clause for disjunction. We now want
to move to a semantics in which the central notion is informational support
rather than truth. So, we must lift the assumption that a disjunction holds at
a point of evaluation only if some of its disjuncts hold at this point. In
other words, we must lift the assumption that sets of formulas that hold at
particular points of evaluation are always prime theories. This undermines the
use of Routley star. The goal of this paper is to explore whether an
information-based semantics with the Routley star treatment of negation
completely breaks down or whether it leads to a non-trivial framework.
Somewhat surprisingly, it will turn out that the resulting framework leads to
an interesting though rather restricted semantics. Let us introduce the frames
of information-based semantics.

######  Definition 1.

A Routley information proto-frame (or just Routley proto-IF) is a structure
$\mathcal{I}=\linebreak\left\langle S,\circ,*,i,e\right\rangle$, where
$\left\langle S,\circ\right\rangle$ is a semilattice, i.e. $S$ is a non-empty
set and $\circ$ is an associative, commutative and idempotent binary operation
on $S$; moreover, $*:S\rightarrow S$ and $i,e$ are two distinct elements of
$S$ for which the following conditions are required: (a) $s\circ i=s$, for any
$s\in S$, (b) $s\circ e=e$, for any $s\in S$, and (c) for any $t,u\in S$, if
$e=t\circ u$ then $e=t$ or $e=u$.

We define an order on $\mathcal{I}$ in the following way: $s\leq t$ iff
$s\circ t=s$. It follows that $e$ is the least element, and $i$ is the top
element of the structure.

The set $S$ represents a space of information states. The operation $\circ$
assigns to any two states $t,u$ the common (informational) content of $t$ and
$u$, i.e. a state $t\circ u$ consisting of information that the two states
have in common. The element $i$ represents the state of absolute
inconsistency, and the state $e$ the state of absolute ignorance. That is, $i$
is intended to satisfy every formula, and $e$ no formula. The informal claim
that $t\circ u$ is the common informational content of $t$ and $u$ translates
into the formal requirement that for any formula $\alpha$, $t\circ u$
satisfies $\alpha$ iff both $t$ and $u$ satisfy $\alpha$. In other words, the
set of states satisfying a formula should form a (proper) filter in
$\mathcal{I}$. This determines a restriction on valuations.

######  Definition 2.

Let $\mathcal{I}=\left\langle S,\circ,*,i,e\right\rangle$ be a Routley proto-
IF. A proper filter in $\mathcal{I}$ is any subset $F$ of $S$ such that $i\in
F$, $e\notin F$, and for any $t,u\in S$ it holds that $t\circ u\in F$ iff
$t\in F$ and $u\in F$. A valuation in $\mathcal{I}$ is a function that assigns
to each atomic formula from $At$ a proper filter in $\mathcal{I}$. A Routley
information proto-model (or just Routley proto-IM) is a Routley proto-IF
equipped with a valuation.

Note that one can equivalently define filters as non-empty proper subsets of
$S$ that are upward closed (w.r.t. $\leq$) and closed under $\circ$. Relative
to a given Routley proto-IM, we can introduce the following semantic clauses
for $L$-formulas (the non-standard semantic clause for disjunction captures
formally the idea that disjunction is what two states supporting the disjuncts
have in common):

  * $s\Vdash p$ iff $s\in V(p)$, for any atomic formula $p$,

  * $s\Vdash\alpha\and\beta$ iff $s\Vdash\alpha$ and $s\Vdash\beta$,

  * $s\Vdash\alpha\vee\beta$ iff $t\circ u\leq s$ for some states $t,u$ such that $t\Vdash\alpha$ and $u\Vdash\beta$,

  * $s\Vdash\neg\alpha$ iff $s^{*}\nVdash\alpha$.

If $s\Vdash\alpha$, we say that the state $s$ supports the formula $\alpha$.
Let $\mathcal{N}$ be a proto-IM, and $\alpha$ an $L$-formula. Then
$||\alpha||_{\\_}{\mathcal{N}}$ is the set of states in $\mathcal{N}$ that
support $\alpha$ (the subscript will usually be omitted). We will call it the
proposition expressed by $\alpha$ in $\mathcal{N}$.

We want to have that propositions expressed by $L$-formulas are always proper
filters, which corresponds to the claim that the set of formulas supported by
a state $s\circ t$, i.e. by the common content of $s$ and $t$, is the
intersection of the set of formulas supported by $s$ and the set of formulas
supported by $t$. For atomic formulas this property is guaranteed directly by
the definition of a valuation. The semantic clauses for conjunction and
disjunction form proper filters from proper filters. However, the semantic
clause for negation does not generally preserve this property. We will
restrict ourself to the largest class of Routley proto-IFs where this property
is preserved. This class is described in the following proposition.

######  Proposition 1.

Let $\mathcal{I}$ be a Routley proto-IF. Then $||\alpha||$ is a proper filter
in $\left\langle\mathcal{I},V\right\rangle$, for every valuation $V$ and every
$L$-formula $\alpha$, iff the following conditions are satisfied:

  * (a)

$i^{*}=e$ and $e^{*}=i$,

  * (b)

for any $t,u\in S$, if $t\leq u$ then $u^{*}\leq t^{*}$,

  * (c)

for any $t,u\in S$, $(t\circ u)^{*}\leq t^{*}$ or $(t\circ u)^{*}\leq u^{*}$.

###### Proof.

Assume that the conditions (a)-(c) are satisfied. We need to show that for any
$L$-formula $\alpha$, if $||\alpha||$ is a proper filter, then
$||\neg\alpha||$ is a proper filter. Assume that $||\alpha||$ is a proper
filter. Since $i\vDash\alpha$ and $e^{*}=i$, we obtain
$e\notin||\neg\alpha||$. Since $e\nvDash\alpha$ and $i^{*}=e$, we obtain
$i\in||\neg\alpha||$. Now, we show that $||\neg\alpha||$ is closed under
$\leq$ and $\circ$. First, assume that $t\vDash\neg\alpha$ and $t\leq u$. Then
$u^{*}\leq t^{*}$, and $t^{*}\nvDash\alpha$. So, $u^{*}\nvDash\alpha$, i.e.
$u\vDash\neg\alpha$. Second, assume that $t\vDash\neg\alpha$ and
$u\vDash\neg\alpha$, i.e. $t^{*}\nvDash\alpha$ and $u^{*}\nvDash\alpha$. Then,
due to (c), we obtain $(t\circ u)^{*}\nvDash\alpha$, i.e. $t\circ
u\vDash\neg\alpha$.

Now, we will assume that some of the conditions (a)-(c) is not satisfied and
we will show that we can define a valuation such that the formula $\neg p$
will not semantically correspond to a filter.

Assume that $i^{*}\neq e$ and take a valuation $V$ such that
$V(p)=\\{s~{}|~{}i^{*}\leq s\\}$. The assumption $i^{*}\neq e$ implies that
$\\{s~{}|~{}i^{*}\leq s\\}$ is a proper filter and thus such a valuation
exists. Then $i^{*}\vDash p$ and so $i\nVdash\neg p$. Now assume that
$e^{*}\neq i$ and take a valuation $V$ such that $V(p)=\\{i\\}$. Then
$e^{*}\nvDash p$ and thus $e\vDash\neg p$.

Next assume that (b) is not satisfied. So, there are $t,u\in S$ such that
$t\leq u$ but not $u^{*}\leq t^{*}$. Take a valuation $V$ such that
$V(p)=\\{s~{}|~{}u^{*}\leq s\\}$. Then $u^{*}\vDash p$, but $t^{*}\nvDash p$,
i.e. $u\nvDash\neg p$, but $t\vDash\neg p$.

Finally, assume that (c) is not satisfied. We can also assume that (b) is
satisfied (if not, we can proceed as in the previous paragraph). Since (c) is
not satisfied, there are $t,u\in S$ such that neither $(t\circ u)^{*}\leq
t^{*}$ nor $(t\circ u)^{*}\leq u^{*}$. Take a valuation $V$ such that
$V(p)=\\{s~{}|~{}(t\circ u)^{*}\leq s\\}$. Then $(t\circ u)^{*}\vDash p$ but
$t^{*}\nvDash p$ and $u^{*}\nvDash p$. So, $t\vDash\neg p$, $u\vDash\neg p$
but $t\circ u\nvDash\neg p$. ∎

In the light of the previous proposition it is desirable to introduce the
following definition.

######  Definition 3.

Any Routley proto-IF (proto-IM) that satisfies the conditions (a)-(c) from
Proposition 1 will be called a Routley IF (IM).

Note that due to Proposition 1, we obtain that the state $i$ in any Routley IM
supports every formula, and the state $e$ supports no formula. This also
explains the role of these two elements in the semantics. The presence of $i$
guarantees that if a state $s$ supports $\alpha$ then $s$ supports also
$\alpha\vee\beta$ because $i$ supports $\beta$ and $s=s\circ i$. The empty
state $e$ is needed as the $*$-image of the state $i$.

In general disjunction can be supported by a state even if none of the
disjuncts is. But the properties of Routley IFs lead to an asymmetry between
positive formulas and their negations. It follows from the definition of
Routley IFs that disjunctions of negations can be characterized by the truth-
conditional semantic clause.

######  Proposition 2.

For any Routley IM, any state $s$ in that model, and all $L$-formulas
$\alpha,\beta$:

  * $s\Vdash\neg\alpha\vee\neg\beta$ iff $s\Vdash\neg\alpha$ or $s\Vdash\neg\beta$.

###### Proof.

We need to show only the left-to-right implication. Assume
$s\Vdash\neg\alpha\vee\neg\beta$. Then there are states $t,u$ such that
$t^{*}\nVdash\alpha$, $u^{*}\nVdash\beta$ and $t\circ u\leq s$. It follows
that $s^{*}\leq(t\circ u)^{*}$, and thus $s^{*}\leq t^{*}$ or $s^{*}\leq
u^{*}$. Hence, $s^{*}\nVdash\alpha$ or $s^{*}\nVdash\beta$, that is
$s\Vdash\neg\alpha$ or $s\Vdash\neg\beta$. ∎

The condition (c) from the definition of Routley IFs is quite strong and
restricting, as shown by the following proposition.

######  Proposition 3.

Let $\mathcal{I}$ be a Routley IF. Then it holds:

  * (a)

for every $t,u\in S$, $(t\circ u)^{*}=t^{*}$ or $(t\circ u)^{*}=u^{*}$,

  * (b)

for every $t,u\in S$, $t^{*}\leq u^{*}$ or $u^{*}\leq t^{*}$,

  * (c)

for every $t\in S$, $t^{*}\leq t^{**}$ or $t^{**}\leq t^{*}$.

###### Proof.

(a) Since $t\circ u\leq t$ and $t\circ u\leq u$ we have $t^{*}\leq(t\circ
u)^{*}$ and $u^{*}\leq(t\circ u)^{*}$. Moreover, $(t\circ u)^{*}\leq t^{*}$ or
$(t\circ u)^{*}\leq u^{*}$, and thus $(t\circ u)^{*}=t^{*}$ or $(t\circ
u)^{*}=u^{*}$.

(b) Since $t^{*}\leq(t\circ u)^{*}$ and $u^{*}\leq(t\circ u)^{*}$, and
$(t\circ u)^{*}=t^{*}$ or $(t\circ u)^{*}=u^{*}$, we have $t^{*}\leq u^{*}$ or
$u^{*}\leq t^{*}$.

(c) This claim is obtained just by applying (b) to $t$ and $t^{*}$. ∎

We say that an $L$-pair $\alpha\vdash\beta$ is valid in a given Routley IM
$\mathcal{N}$ if $||\alpha||\subseteq||\beta||$. An $L$-pair is valid in a
Routely IF $\mathcal{I}$ if it is valid in every Routely IM based on
$\mathcal{I}$. An $L$-pair is valid in a class of Routely IFs (IMs) if it is
valid in every member of that class.

######  Proposition 4.

Let $\mathcal{N}$ be a Routley IM. Then it holds for all $L$-formulas
$\alpha$, $\beta$:

  * (a)

The schematic rules (T), (i$\and$), (e$\vee$), (N) preserve validity in
$\mathcal{N}$, i.e. if the premises are valid in $\mathcal{N}$ then the
conclusion is valid in $\mathcal{N}$ as well.

  * (b)

All instances of the axiom schemata (ID), (e$\and_{\\_}1$), (e$\and_{\\_}2$),
(i$\vee_{\\_}1$), (i$\vee_{\\_}2$), (DM1) are valid in $\mathcal{N}$.

  * (c)

All instances of the axiom schemata (DM2∗)
$\neg\neg\alpha\and\neg\neg\beta\vdash\neg(\neg\alpha\vee\neg\beta)$ and (D∗)
$\alpha\and(\neg\beta\vee\neg\gamma)\vdash(\alpha\and\neg\beta)\vee(\alpha\and\neg\gamma)$
are valid in $\mathcal{N}$.

###### Proof.

(a) We will show that (N) preserves validity: Assume that $\alpha\vdash\beta$
is valid in $\mathcal{N}$. Assume for some state $s$, that $s\Vdash\neg\beta$.
Then $s^{*}\nVdash\beta$, and thus $s^{*}\nVdash\alpha$. Hence,
$s\Vdash\neg\alpha$.

(b) We will show the validity of (DM1): Assume for some state $s$, that
$s\Vdash\neg(\alpha\and\beta)$, i.e. $s^{*}\nVdash\alpha\and\beta$. Then
$s^{*}\nVdash\alpha$ or $s^{*}\nVdash\beta$, i.e. $s\Vdash\neg\alpha$ or
$s\Vdash\neg\beta$. Hence $s\Vdash\neg\alpha\vee\neg\beta$.

(c) First, assume for some state $s$, that
$s\Vdash\neg\neg\alpha\and\neg\neg\beta$, i.e. $s^{*}\nVdash\neg\alpha$ and
$s^{*}\nVdash\neg\beta$. It follows from Proposition 2 that
$s^{*}\nVdash\neg\alpha\vee\neg\beta$, i.e.
$s\Vdash\neg(\neg\alpha\vee\neg\beta)$. Second, assume
$s\Vdash\alpha\and(\neg\beta\vee\neg\gamma)$, i.e. $s\Vdash\alpha$ and
$s\Vdash\neg\beta\vee\neg\gamma$. So, $s\Vdash\neg\beta$ or
$s\Vdash\neg\gamma$, by Proposition 2. If $s\Vdash\neg\beta$, then
$s\Vdash\alpha\and\neg\beta$, and hence
$s\Vdash(\alpha\and\neg\beta)\vee(\alpha\and\neg\gamma)$. If
$s\Vdash\neg\gamma$, we can reason in an analogous way. ∎

The claim (b) in Proposition 3 shows that in any Routley IF the image of the
operation $*$ forms a linearly ordered set. However, the whole frame does not
have to be linearly ordered. Consider the two non-linear examples of RIFs from
Fig. 1.

$s$$t$$u$$v$$e$$i$ | $s$$t$$u$$v$$e$$i$
---|---
Figure 1: Examples of non-linear Roultey IFs

The vertical position determines the order so that, for example, in both cases
$s\leq t$ and $u\leq v$. The states $t,u$ are the only incomparable states. It
holds that $t\circ u=s$. The arrows determine the behaviour of $*$ so that,
for example, on the left $t^{*}=v$, and on the right $t^{*}=s$. These frames
can also be used to construct counterexamples to various consequence
$L$-pairs. For example, we can observe the following:

  * •

$\neg\neg p\vdash p$ is not valid in the frame on the left (assume that
$V(p)=\\{v,i\\}$; then $t\vDash\neg\neg p$ but $t\nvDash p$),

  * •

$p\vdash\neg\neg p$ is not valid in the frame on the right (assume that
$V(p)=\\{v,i\\}$; then $v\vDash p$ but $v\nvDash\neg\neg p$),

  * •

$\neg p\and\neg q\vdash\neg(p\vee q)$ is not valid in the frame on the right
(assume that $V(p)=\\{t,v,i\\}$ and $V(q)=\\{u,v,i\\}$; then $s\vDash\neg
p\and\neg q$ but $s\nvDash\neg(p\vee q)$).

From the axioms of the distributive lattice logic $\mathsf{DLL}$ we lose the
distributive axiom (D), which is not generally valid in the class of all
Roultey IFs. This can be illustrated with the example in Fig. 2. The states
$s,t,u,v,w$ are ordered as in the non-distributive lattice $N_{\\_}5$, and the
Routley star, for instance, maps $i$ to $e$, $e$ to $i$, and every other
element to $u$ (Routley star does not play any role in the example, it is
introduced just to make the structure a Routley IF).

$s$$t$$v$$w$$u$$e$$i$
---
Figure 2: A counterexample to distributivity

For a counterexample to $p\and(q\vee r)\vdash(p\and q)\vee(p\and r)$ it
suffices to take the valuation $V$ such that $V(p)$ is the principle filter
generated by $w$, $V(q)$ is the principle filter generated by $t$, and $V(r)$
is the principal filter generated by $v$. In the resulting model $w$ supports
$p\and(q\vee r)$ but not $(p\and q)\vee(p\and r)$.

######  Definition 4.

We say that a consequence $L$-pair $\alpha\vdash\beta$ characterizes a class
of Roultey IFs $\mathcal{C}$ if it holds for any Routley IF $\mathcal{I}$ that
$\alpha\vdash\beta$ is valid in $\mathcal{I}$ iff $\mathcal{I}\in\mathcal{C}$.

######  Proposition 5.

The following holds:

  * (a)

$p\vdash\neg\neg p$ characterizes the class of Routley IFs satisfying $s\leq
s^{**}$, for each state $s$,

  * (b)

$\neg\neg p\vdash p$ characterizes the class of Routley IFs satisfying
$s^{**}\leq s$, for each state $s$,

  * (c)

$\neg p\and\neg q\vdash\neg(p\vee q)$ characterizes the class of Routley IFs
satisfying the following property: for all states $s,t,u$, if $t\circ u\leq
s^{*}$ then $t\leq s^{*}$ or $u\leq s^{*}$.

###### Proof.

(a) Assume that $s\leq s^{**}$, for each state $s$. Assume $s\vDash\alpha$.
Then $s^{**}\vDash\alpha$, i.e. $s\vDash\neg\neg\alpha$. Now assume that there
is a state $s$ such that $s\nleq s^{**}$. Take a valuation $V$ such that
$V(p)=\\{t~{}|~{}s\leq t\\}$. Then $s^{**}\nvDash p$, and so $s\nvDash\neg\neg
p$. But $s\vDash p$.

(b) Assume that $s^{**}\leq s$, for each state $s$. Assume
$s\vDash\neg\neg\alpha$. Then $s^{**}\vDash\alpha$, and hence $s\vDash\alpha$.
Now assume that there is a state $s$ such that $s^{**}\nleq s$. Take a
valuation $V$ such that $V(p)=\\{t~{}|~{}s^{**}\leq t\\}$. Then $s\nvDash p$.
But $s\vDash\neg\neg p$.

(c) First, assume that for all states $s,t,u$, if $t\circ u\leq s^{*}$ then
$t\leq s^{*}$ or $u\leq s^{*}$. Assume $s\nvDash\neg(\alpha\vee\beta)$, i.e.
$s^{*}\vDash\alpha\vee\beta$. So, there are $t,u$ such that $t\vDash\alpha$,
$u\vDash\beta$, and $t\circ u\leq s^{*}$. Then $t\leq s^{*}$ or $u\leq s^{*}$
and thus $s^{*}\vDash\alpha$ or $s^{*}\vDash\beta$. Hence
$s\nvDash\neg\alpha\and\neg\beta$. Second, assume that there are states
$s,t,u$ such that $t\circ u\leq s^{*}$ but neither $t\leq s^{*}$ nor $u\leq
s^{*}$. Take a valuation $V$ such that $V(p)=\\{v~{}|~{}t\leq v\\}$ and
$V(q)=\\{v~{}|~{}u\leq v\\}$. So, $s^{*}\nvDash p$ and $s^{*}\nvDash q$, i.e.
$s\vDash\neg p\and\neg q$. But since $t\circ u\vDash p\vee q$, we obtain
$s^{*}\vDash p\vee q$, and so $s\nvDash\neg(p\vee q)$. ∎

Note that the frame pictured in Fig. 1 on the left satisfies the conditions
(a) and (c) but not (b), from Proposition 5, and the frame on the right
satisfies (b) but neither (a) nor (c). As a consequence of Proposition 5 we
obtain the following rather surprising fact.

######  Proposition 6.

Let $\mathcal{I}$ be a Routley IF. If $p\vdash\neg\neg p$ and $\neg\neg
p\vdash p$ are both valid in $\mathcal{I}$ then $\mathcal{I}$ is linearly
ordered by $\leq$.

###### Proof.

If $p\vdash\neg\neg p$ and $\neg\neg p\vdash p$ are both valid in
$\mathcal{I}$, then according to (a) and (b) of Proposition 5, $s=s^{**}$, for
every state $s$. So, the image of $*$ is the whole set of states. Then,
according to (b) of Proposition 3, $\mathcal{I}$ must be linearly ordered. ∎

The converse of the previous proposition is not true since there are linearly
ordered IFs in which $s=s^{**}$ does not hold generally. These frames are
discussed in the following section.

##  4 Linearly ordered Routley IFs

Linearly ordered Routley IFs (i.e. those Routley IFs in which $\leq$ is a
linear order) are special in that they are in the common area of the
information-based semantics and the standard semantics. The reason is that in
the linearly ordered Routley IFs our semantic condition for disjunction
reduces to the standard one.

######  Proposition 7.

For any linear Routley IM, any state $s$ in that model, and every $L$-formulas
$\alpha,\beta$:

  * $s\vDash\alpha\vee\beta$ iff $s\vDash\alpha$ or $s\vDash\beta$.

###### Proof.

In any linearly ordered Routley IF, $t\circ u=t$ or $t\circ u=u$. ∎

Let us characterize the logic of all linearly ordered Routley IFs by a
deductive system. For any natural number $n$ let $\neg^{n}\alpha$ denote the
formula $\neg\ldots\neg\alpha$ with $n$ occurrences of $\neg$ (in particular,
$\neg^{0}\alpha=\alpha$). We will also need the following notation. If $n$ is
a natural number and $s$ is a state in a Routley IM, then $s^{n(*)}$ is the
state $s^{*\ldots*}$ with $n$ occurrences of $*$ (in particular,
$s^{0(*)}=s$). Note that for any natural number $k$, $s^{2k(*)}\Vdash\alpha$
iff $s\Vdash\neg^{2k(*)}\alpha$, and $s^{2k+1(*)}\Vdash\alpha$ iff
$s\nVdash\neg^{2k+1(*)}\alpha$.

Consider the deductive system consisting of the axioms and rules of
$\mathsf{DLLR}$ enriched with the following two axioms for each natural number
$k$:

(L1) $\alpha\and\neg^{2k}\beta\vdash\neg^{2k}\alpha\vee\beta$ | (L2) $\alpha\and\neg^{2k+1}\alpha\vdash\beta\vee\neg^{2k+1}\beta$
---|---

Let us denote this logic, or more precisely the set of $L$-pairs derivable in
the resulting system, as $\mathsf{LinDLLR}$.

######  Theorem 1.

$\alpha\vdash\beta$ is valid in all linearly ordered Routley IFs iff
$\alpha\vdash\beta\in\mathsf{LinDLLR}$.

###### Proof.

The right-to-left implication amounts to soundness of the system with respect
to linear Routley IFs. Let us check the soundness of the two axioms (L1) and
(L2).

(L1) Let $s\Vdash\alpha\and\neg^{2k}\beta$ in a linear Routley IM, that is
$s\Vdash\alpha$ and $s^{2k(*)}\Vdash\beta$. We will consider two possible
cases. First, assume that $s\leq s^{2k(*)}$. Then $s^{2k(*)}\Vdash\alpha$, and
hence $s\Vdash\neg^{2k}\alpha$. It follows that
$s\Vdash\neg^{2k}\alpha\vee\beta$. Second, assume that $s^{2k(*)}\leq s$. Then
$s\Vdash\beta$ and thus $s\Vdash\neg^{2k}\alpha\vee\beta$.

(L2) Let $s\Vdash\alpha\and\neg^{2k+1}\alpha$ in a linear Routley IM, that is
$s\Vdash\alpha$ and $s^{2k+1(*)}\nVdash\alpha$. It follows that $s\nleq
s^{2k+1(*)}$ and hence $s^{2k+1(*)}\leq s$. If $s\Vdash\beta$ then also
$s\Vdash\beta\vee\neg^{2k+1}\beta$. If $s\nVdash\beta$ then
$s^{2k+1(*)}\nVdash\beta$ and so $s\Vdash\neg^{2k+1}\beta$. Hence, we obtain
also in this case $s\Vdash\beta\vee\neg^{2k+1}\beta$.

Now we will prove the left-to-right implication, i.e. completeness of the
system. For any set of formulas $\Omega$ we define $\Omega^{*}=\\{\alpha\in
L\mid\neg\alpha\notin\Omega\\}$. Note that if $\Omega$ is a prime
$\mathsf{LinDLLR}$-theory then $\Omega^{*}$ is also a prime
$\mathsf{LinDLLR}$-theory. As before, $\Omega^{n(*)}$ denotes
$\Omega^{*\ldots*}$ with $n$ occurrences of $*$. Now we will prove the
following auxiliary claim:

######  Lemma 1.

Let $\Omega$ be a prime $\mathsf{LinDLLR}$-theory and $n$ a natural number.
Then $\Omega\subseteq\Omega^{n(*)}$ or $\Omega^{n(*)}\subseteq\Omega$.

For the sake of contradiction, assume that $\Omega\nsubseteq\Omega^{n(*)}$ and
$\Omega^{n(*)}\nsubseteq\Omega$. So, there is $\alpha\in\Omega$ such that
$\alpha\notin\Omega^{n(*)}$, and there is $\beta\in\Omega^{n(*)}$ such that
$\beta\notin\Omega$. We distinguish two possible cases. First, assume that
$n=2k$. Then $\neg^{n}\alpha\notin\Omega$ and $\neg^{n}\beta\in\Omega$. Hence,
we have $\alpha\and\neg^{n}\beta\in\Omega$ and
$\neg^{n}\alpha\vee\beta\notin\Omega$. This is a contradiction with the
assumption that (L1) is in $\mathsf{LinDLLR}$. Second, assume that $n=2k+1$.
Then $\neg^{n}\alpha\in\Omega$ and $\neg^{n}\beta\notin\Omega$. Then
$\alpha\and\neg^{n}\alpha\in\Omega$ and $\beta\vee\neg^{n}\beta\notin\Omega$
which is in contradiction with the assumption that (L2) is in
$\mathsf{LinDLLR}$. This finishes the proof of Lemma 1.

Now assume that $\alpha\vdash\beta\notin\mathsf{LinDLLR}$. By the standard
procedure we can construct a prime $\mathsf{LinDLLR}$-theory $\Delta$ such
that $\alpha\in\Delta$ and $\beta\notin\Delta$. Relative to
$\alpha\vdash\beta$ we construct a Routley IM $\mathcal{M}=\left\langle
S,\circ,*,i,e,V\right\rangle$ where $i$ is the set of all $L$-formulas; $e$ is
the empty set; $S=\\{i,e\\}\cup\\{\Delta^{n(*)}\mid n\in\mathbb{N}\\}$; the
star operation is extended to $S$ by fixing $i^{*}=e$ and $e^{*}=i$; $\circ$
is defined as intersection; and $V(p)=\\{\Omega\in S\mid p\in\Omega\\}$.

Note that Lemma 1 guarantees that $S$ is closed under intersection. Since it
is also closed under $*$ and satisfies all the desired properties we obtain
that $\mathcal{M}$ is ineed a Routley IM. Lemma 1 further guarantees that
$\mathcal{M}$ is linearly ordered. Now it can be shown by a straightforward
induction that support corresponds to membership in this particular model.

######  Lemma 2.

For any $\Omega\in S$ and any $L$-formula $\alpha$, $\Omega\Vdash\alpha$ in
$\mathcal{M}$ iff $\alpha\in\Omega$.

It follows from this lemma that $\Delta\Vdash\alpha$ but $\Delta\nVdash\beta$,
so $\alpha\vdash\beta$ is not valid in all linearly ordered IFs. This finishes
the proof of Theorem 1. ∎

Now we will focus on those linearly ordered Routley IFs that validate double
negation laws. As is shown in Proposition 5 these frames are characterized by
general validity of $s=s^{**}$. If this condition is satisfied we say that the
frame is involutive. Fig. 3 shows schematically the structure of involutive
linearly ordered Routley IFs.

$\vdots$$\bullet$$\bullet$$\vdots$$\vdots$$\bullet$$\bullet$$\vdots$$\vdots$$i$$e$
---
Figure 3: Involutive linearly ordered Roultey IFs

We will prove that the logic of all involutive linearly ordered Routley IFs
coincides with what Dunn in [5] called Kalman logic ($\mathsf{KL}$). This
logic is axiomatized by $\mathsf{DLLR}$ enriched with double negation laws
(DN1) and (DN2), and the following axiom:

(KA) $\alpha\and\neg\alpha\vdash\beta\vee\neg\beta$
---

Interestingly, this logic is the $\\{\and,\vee,\neg\\}$-fragment of the
relatively well-known and studied logic $\mathsf{R}$-mingle (see [5]). We now
prove completeness of this logic with respect to all involutive linearly
ordered Routley IFs, and also with respect to the particular Routley IF
$\mathcal{I}(\mathsf{KL})$ depicted in Fig. 4.

######  Theorem 2.

$\alpha\vdash\beta$ is valid in all involutive linearly ordered Routley IFs
iff $\alpha\vdash\beta\in\mathsf{KL}$. Moreover, $\mathsf{KL}$ is also the set
of all $L$-pairs that are valid in the Routley IF $\mathcal{I}(\mathsf{KL})$.

###### Proof.

Notice that the axioms (L1) and (L2) collapse into (KA) in the context of the
double negation laws. It follows that (KA) is valid in all involutive linearly
ordered Routley IFs which gives us soundness, i.e. the right-to-left
implication. In order to prove completeness, i.e. the left-to-right
implication, one can show that $\mathsf{KL}$ is sound and complete with
respect to the particular involutive linearly ordered Routley IF
$\mathcal{I}(\mathsf{KL})$.

$s$$t$$i$$e$
---
Figure 4: The Routley IF $\mathcal{I}(\mathsf{KL})$

To prove this claim one can proceed as in Theorem 1 and construct for a given
$\alpha\vdash\beta\notin\mathsf{KL}$ a prime $\mathsf{KL}$-theory $\Delta$
such that $\alpha\in\Delta$ and $\beta\notin\Delta$. But now, due to the
double negation laws, we obtain that $\Delta^{**}=\Delta$. Since $i$ is the
set of all $L$-formulas and $e$ is the empty set, it follows that $\Delta\neq
i$ and $\Delta\neq e$. Moreover, by the double negation laws we obtain
$\neg\alpha\notin\Delta^{*}$ and $\neg\beta\in\Delta^{*}$, so we have also
$\Delta^{*}\neq i$ and $\Delta^{*}\neq e$. If $\Delta\neq\Delta^{*}$ then the
Routley IM described in the proof of Theorem 1 has now the structure of
$\mathcal{I}(\mathsf{KL})$. It can happen that $\Delta=\Delta^{*}$ but in that
special case we can just take two copies of $\Delta$ and define the Routley
star operation accordingly to obtain again the structure of
$\mathcal{I}(\mathsf{KL})$. ∎

Note that since $\mathsf{KL}$ is the logic of one finite frame, it also
follows from our proof that this logic is decidable. This is however not a new
result because $\mathsf{KL}$ has been characterized, e.g. in [5], as a three-
valued logic.

##  5 The logic of all Routley IFs

The logic of all Routley IFs can be axiomatized in the following way. It
contains the axioms and rules of the system $\mathsf{DLLR}$ with a modified
distributivity axiom and one of the DeMorgan laws:

(ID) $\alpha\vdash\alpha$ | (e$\and_{\\_}1$) $\alpha\and\beta\vdash\alpha$ | (e$\and_{\\_}2$) $\alpha\and\beta\vdash\beta$ | (i$\vee_{\\_}1$) $\alpha\vdash\alpha\vee\beta$ | (i$\vee_{\\_}2$) $\beta\vdash\alpha\vee\beta$
---|---|---|---|---

(DM1) $\neg(\alpha\and\beta)\vdash\neg\alpha\vee\neg\beta$ | (DM2∗): $\neg\neg\alpha\and\neg\neg\beta\vdash\neg(\neg\alpha\vee\neg\beta)$
---|---

(D∗)
$\alpha\and(\neg\beta\vee\neg\gamma)\vdash(\alpha\and\neg\beta)\vee(\alpha\and\neg\gamma)$
---

(T) $\alpha\vdash\beta,\beta\vdash\gamma/\alpha\vdash\gamma$ | (i$\and$) $\alpha\vdash\beta,\alpha\vdash\gamma/\alpha\vdash\beta\and\gamma$ | (e$\vee$) $\alpha\vdash\gamma,\beta\vdash\gamma/\alpha\vee\beta\vdash\gamma$
---|---|---

(N) $\alpha\vdash\beta/\neg\beta\vdash\neg\alpha$
---

Moreover, for each positive number $k$ the system contains the following two
axioms, which modify the characteristic axioms of $\mathsf{LinDLLR}$:

(L1∗) $\neg\alpha\and\neg^{2k+1}\beta\vdash\neg^{2k+1}\alpha\vee\neg\beta$ | (L2∗) $\neg\alpha\and\neg^{2k}\alpha\vdash\neg\beta\vee\neg^{2k}\beta$
---|---

Let us call this logic $\mathsf{LRIF}$ (the logic of Routley information
frames). Note that even though Routley star is often used in the context of
relevant logics, the logic $\mathsf{LRIF}$ (similarly to the logics
$\mathsf{LinDLLR}$ and $\mathsf{KL}$ discussed in the previous sections) is
not “relevant” in a strict sense of the word because the axiom (L2∗) violates
one of the essential features of relevant logics, namely the variable sharing
property.

The rest of this paper is devoted to proving completeness of $\mathsf{LRIF}$
with respect to Routley IFs. The proof is similar to the proof of Theorem 1
with some additional complications. First, we have to replace the notion of
prime theory with the following one.

######  Definition 5.

A set of $L$-formulas $\Omega$ is a negative prime $\mathsf{LRIF}$-theory of
the just described system if it satisfies the following three conditions: (a)
if $\alpha\in\Omega$ and $\alpha\vdash\beta\in\mathsf{LRIF}$ then
$\beta\in\Omega$; (b) if $\alpha\in\Omega$ and $\beta\in\Omega$ then
$\alpha\and\beta\in\Omega$; (c) if $\neg\alpha\vee\neg\beta\in\Omega$ then
$\neg\alpha\in\Omega$ or $\neg\beta\in\Omega$.

In order to prove our main result, we will need several lemmas.

######  Lemma 3.

If $\alpha\vdash\beta\notin\mathsf{LRIF}$, then there is a negative prime
$\mathsf{LRIF}$-theory $\Omega$ such that $\alpha\in\Omega$ and
$\beta\notin\Omega$.

###### Proof.

This result is obtained by standard Lindenbaum-style construction using the
modified distributivity axiom (D∗). ∎

######  Lemma 4.

If $\Omega$ is a negative prime $\mathsf{LRIF}$-theory, then $\Omega^{*}$ is
also a negative prime $\mathsf{LRIF}$-theory.

###### Proof.

Assume that $\neg\alpha\vee\neg\beta\in\Omega^{*}$. So,
$\neg(\neg\alpha\vee\neg\beta)\notin\Omega$. Using the axiom (DM2∗) we obtain
$\neg\neg\alpha\and\neg\neg\beta\notin\Omega$. So,
$\neg\neg\alpha\notin\Omega$ or $\neg\neg\beta\notin\Omega$ and thus
$\neg\alpha\in\Omega^{*}$ or $\neg\beta\in\Omega^{*}$. ∎

######  Lemma 5.

If $\Omega$ is a negative prime $\mathsf{LRIF}$-theory then
$\Omega^{m(*)}\subseteq\Omega^{n(*)}$ or
$\Omega^{n(*)}\subseteq\Omega^{m(*)}$, for all positive natural numbers $m,n$.

###### Proof.

We can reason as in the proof of Lemma 1 using the axioms (L1∗) and (L2∗). ∎

######  Lemma 6.

If $\Omega$ is a negative prime $\mathsf{LRIF}$-theory then
$\Omega\cap\Omega^{n(*)}$ is a negative prime $\mathsf{LRIF}$-theory, for
every positive natural number $n$.

###### Proof.

Assume that we have $\neg\alpha\notin\Omega\cap\Omega^{n(*)}$, and
$\neg\beta\notin\Omega\cap\Omega^{n(*)}$. Without loss of generality, assume
that $\neg\alpha\notin\Omega$ and $\neg\beta\notin\Omega^{n(*)}$. Hence,
$\alpha\in\Omega^{*}$ and $\beta\in\Omega^{n+1(*)}$. By Lemma 5, we obtain
$\alpha,\beta\in\Omega^{*}$ or $\alpha,\beta\in\Omega^{n+1(*)}$, and thus
$\neg\alpha,\neg\beta\notin\Omega$ or
$\neg\alpha,\neg\beta\notin\Omega^{n(*)}$. Since, by Lemma 4, $\Omega$ and
$\Omega^{n(*)}$ are both negative prime $\mathsf{LRIF}$-theories, we obtain
$\neg\alpha\vee\neg\beta\notin\Omega$ or
$\neg\alpha\vee\neg\beta\notin\Omega^{n(*)}$. It follows that
$\neg\alpha\vee\neg\beta\notin\Omega\cap\Omega^{n(*)}$. ∎

######  Lemma 7.

If $\Omega$ is a negative prime $\mathsf{LRIF}$-theory then
$(\Omega\cap\Omega^{n(*)})^{*}=\Omega^{*}$ or
$(\Omega\cap\Omega^{n(*)})^{*}=\Omega^{n+1(*)}$, for every positive natural
number $n$.

###### Proof.

Since $\Omega\cap\Omega^{n(*)}\subseteq\Omega$, we obtain
$\Omega^{*}\subseteq(\Omega\cap\Omega^{n(*)})^{*}$. By the same reasoning,
$\Omega^{n+1(*)}\subseteq(\Omega\cap\Omega^{n(*)})^{*}$. We have to prove that
$(\Omega\cap\Omega^{n(*)})^{*}\subseteq\Omega^{*}$ or
$(\Omega\cap\Omega^{n(*)})^{*}\subseteq\Omega^{n+1(*)}$. For the sake of
contradiction assume that there are $\alpha,\beta$ such that
$\alpha,\beta\in(\Omega\cap\Omega^{n(*)})^{*}$, $\alpha\notin\Omega^{n+1(*)}$
and $\beta\notin\Omega^{*}$. On one hand, we obtain
$\neg\alpha,\neg\beta\notin\Omega\cap\Omega^{n(*)}$. By Lemma 6,
$\Omega\cap\Omega^{n(*)}$ is a negative prime $\mathsf{LRIF}$-theory and hence
$\neg\alpha\vee\neg\beta\notin\Omega\cap\Omega^{n(*)}$. On the other hand, we
obtain $\neg\alpha\in\Omega^{n(*)}$ and $\neg\beta\in\Omega$. It follows that
$\neg\alpha\vee\neg\beta\in\Omega\cap\Omega^{n(*)}$, which is a contradiction.
∎

######  Theorem 3.

$\alpha\vdash\beta$ is valid in all Routley IFs iff
$\alpha\vdash\beta\in\mathsf{LRIF}$.

###### Proof.

We will prove completeness, i.e. the left-to-right implication. Assume that
$\alpha\vdash\beta\notin\mathsf{LRIF}$. By Lemma 3 there is a negative prime
$\mathsf{LRIF}$-theory $\Delta$ such that $\alpha\in\Delta$ and
$\beta\notin\Delta$. Relative to $\alpha\vdash\beta$ we construct a Routley IM
$\mathcal{M}=\left\langle S,\circ,*,i,e,V\right\rangle$ where $i$ is the set
of all $L$-formulas; $e$ is the empty set;
$S=\\{i,e\\}\cup\\{\Delta^{m(*)}\cap\Delta^{n(*)}\mid m,n\in\mathbb{N}\\}$;
the star operation is defined as usual; $\circ$ is defined as intersection;
and $V(p)=\\{\Omega\in S\mid p\in\Omega\\}$. The structure of the model is
illustrated with Fig. 5 (for the case where $\Delta^{*}\subseteq\Delta^{**}$).

Note that $S$ is closed under intersection and $*$. In particular, it is
closed under intersection because of Lemmas 5 and 6 and it is closed under $*$
because of Lemmas 4 and 7. Moreover, Lemma 7 guarantees that $\mathcal{M}$ is
a Routley IM. Again, it can be shown by straightforward induction that
$\Omega\Vdash\alpha$ in $\mathcal{M}$ iff $\alpha\in\Omega$. It follows that
$\Delta\Vdash\alpha$ but $\Delta\nVdash\beta$, so $\alpha\vdash\beta$ is not
valid in all IFs. ∎

$\Delta$$\Delta^{1(*)}$$\Delta^{2(*)}$$\Delta^{3(*)}$$\Delta^{4(*)}$$\Delta^{5(*)}$$\Delta\cap\Delta^{1(*)}$$\Delta\cap\Delta^{2(*)}$$\Delta\cap\Delta^{3(*)}$$\Delta\cap\Delta^{4(*)}$$\Delta\cap\Delta^{5(*)}$$\vdots$$\vdots$$\vdots$$\vdots$$e=i^{*}$$i=e^{*}$
---
Figure 5: The structure of a Routley IM that provides a counterexample to an
$\mathsf{LRIF}$-unprovable $\alpha\vdash\beta$

##  6 Conclusion

The goal of this paper was to explore the behaviour of Routley star in the
context of “information-based semantics” in which states are not in general
prime. We succeeded in characterizing syntactically the logic of all
information frames equipped with Routley star. We also showed that if double
negation laws are added, the resulting logic coincides with Kalman logic that
is known to be the $\\{\and,\vee,\neg\\}$-fragment of $\mathsf{R}$-mingle. We
observed that it can be characterized as the logic of involutive linearly
ordered information frames. We also characterized the logic of all linear
frames without the requirement of involution.

## References

  * [1]
  * [2] Ivano Ciardelli & Floris Roelofsen (2011): _Inquisitive Logic_.  Journal of Philosophical Logic 40, pp. 55–94, 10.1007/s10992-010-9142-6.
  * [3] Kosta Došen (1989): _Sequent Systems and Groupoid Models. II_.  Studia Logica 48, pp. 41–65, 10.1007/BF00370633.
  * [4] Jon Michael Dunn (1993): _Star and Perp: Two Treatments of Negation_.  Philosophical Perspectives 7, pp. 331–357, 10.2307/2214128.
  * [5] Jon Michael Dunn (1999): _A Comparative Study of Various Model-theoretic Treatments of Negation: A History of Formal Negation_.  In Dov Gabbay & Heinrich Wansing, editors: What is Negation?, Applied Logic Series 13, Kluwer, pp. 23–53, 10.1007/978-94-015-9309-0_2.
  * [6] Lloyd Humberstone (1988): _Operational Semantics for Positive R_.  Notre Dame Journal of Formal Logic 29, pp. 61–80, 10.1305/ndjfl/1093637771.
  * [7] Hannes Leitgeb (2019): _HYPE: A System of Hyperintensional Logic (with an Application to Semantic Paradoxes)_.  Journal of Philosophical Logic 48, pp. 305–405, 10.1007/s10992-018-9467-0.
  * [8] Vít Punčochář (2019): _Substructural Inquisitive Logics_.  Review of Symbolic Logic 12, pp. 296–330, 10.1017/S1755020319000017.
  * [9] Vít Punčochář & Igor Sedlár (2021): _Epistemic Extensions of Substructural Inquisitive Logics_.  Journal of Logic and Computation 31, pp. 1820–1844, 10.1093/logcom/exab008.
  * [10] Vít Punčochář & Andrew Tedder (2021): _Disjunction and Negation in Information Based Semantics_.  In Alexandra Silva, Renata Wassermann & Ruy de Queiroz, editors: Logic, Language, Information, and Computation, WoLLIC 2021, Lecture Notes in Computer Science 13038, Springer, pp. 355–371, 10.1007/978-3-030-88853-4_22.
  * [11] Greg Restall (1999): _Negation in Relevant Logics: How I Stopped Worrying and Learned to Love the Routley Star_.  In Dov Gabbay & Heinrich Wansing, editors: What is Negation?, Applied Logic Series 13, Kluwer, pp. 53–76, 10.1007/978-94-015-9309-0_3.
  * [12] Heinrich Wansing (1993): _Informational Interpretation of Substructural Logics_.  Journal of Logic, Language, and Information 2, pp. 285–308, 10.1007/BF01181683.