# First JWST Observations of JAGB Stars in the SN Ia Host Galaxies:
NGC 7250, NGC 4536, NGC 3972

Abigail J. Lee  Department of Astronomy & Astrophysics, University of Chicago,
5640 South Ellis Avenue, Chicago, IL 60637  Kavli Institute for Cosmological
Physics, University of Chicago, 5640 South Ellis Avenue, Chicago, IL 60637
Wendy L. Freedman  Department of Astronomy & Astrophysics, University of
Chicago, 5640 South Ellis Avenue, Chicago, IL 60637  Kavli Institute for
Cosmological Physics, University of Chicago, 5640 South Ellis Avenue, Chicago,
IL 60637  In Sung Jang  Department of Astronomy & Astrophysics, University of
Chicago, 5640 South Ellis Avenue, Chicago, IL 60637  Kavli Institute for
Cosmological Physics, University of Chicago, 5640 South Ellis Avenue, Chicago,
IL 60637  Barry F. Madore  Observatories of the Carnegie Institution for
Science 813 Santa Barbara St., Pasadena, CA 91101  Department of Astronomy &
Astrophysics, University of Chicago, 5640 South Ellis Avenue, Chicago, IL
60637  Kayla A. Owens  Department of Astronomy & Astrophysics, University of
Chicago, 5640 South Ellis Avenue, Chicago, IL 60637  Kavli Institute for
Cosmological Physics, University of Chicago, 5640 South Ellis Avenue, Chicago,
IL 60637 Abigail J. Lee<EMAIL_ADDRESS>
###### Abstract

The J-region Asymptotic Giant Branch (JAGB) method is a standard candle that
leverages the constant luminosities of color-selected, carbon-rich AGB stars,
measured in the near infrared at 1.2 microns. The Chicago-Carnegie Hubble
Program (CCHP) has obtained JWST imaging of the SN Ia host galaxies NGC 7250,
NGC 4536, and NGC 3972. With these observations, the JAGB method can be
studied for the first time using JWST. Lee et al. (2022) demonstrated the JAGB
magnitude is optimally measured in the outer disks of galaxies, because in the
inner regions the JAGB magnitude can vary significantly due to a confluence of
reddening, blending, and crowding effects. However, determining where the
‘outer disk’ lies can be subjective. Therefore, we introduce a novel method
for systematically selecting the outer disk. In a given galaxy, the JAGB
magnitude is first separately measured in concentric regions, and the ‘outer
disk’ is then defined as the first radial bin where the JAGB magnitude
stabilizes to a few hundredths of a magnitude. After successfully employing
this method in our JWST galaxy sample, we find the JAGB stars are well-
segregated from other stellar populations in color-magnitude space, and have
observed dispersions about their individual $F115W$ modes of
$\sigma_{N7250}=0.32$ mag, $\sigma_{N4536}=0.34$ mag, and
$\sigma_{N3972}=0.35$ mag. These measured dispersions are similar to the
scatter measured for the JAGB stars in the LMC using 2MASS data ($\sigma=0.33$
mag, Weinberg & Nikolaev 2001). In conclusion, the JAGB stars as observed with
JWST clearly demonstrate their considerable power both as high-precision
extragalactic distance indicators and as SN Ia supernova calibrators.

Observational cosmology (1146), Distance indicators (394), Asymptotic Giant
Branch stars (2100), Carbon stars (199), Galaxy distances (590), JWST (2291)

††software: NumPy (Harris et al., 2020), Matplotlib (Hunter, 2007), Pandas
(McKinney, 2010)††facilities: JWST (NIRCam)

##  1 Introduction

Determining a consistent and accurate value for the expansion rate of the
universe, as parameterized by the Hubble constant ($H_{0}$), has proven to be
an extraordinarily difficult endeavor. Measurements of $H_{0}$ from local
distance ladders compared with indirectly inferred measurements from the
Cosmic Microwave Background (Planck Collaboration et al., 2020; Aiola et al.,
2020) currently disagree at the $5\sigma$ level, potentially indicating new
physics and theories (Di Valentino et al., 2021). However, significant
evidence still points to the possibility that systematic uncertainties in the
local distance scale may be the reason for this tension. For example, the
current two most precise local distance ladder measurements of $H_{0}$ from
the Tip of the Red Giant Branch (TRGB) by the CCHP (Freedman, 2021) and from
the Cepheid Leavitt Law by the SH0ES group (Riess et al., 2022) themselves
differ by about $2\sigma$. It would seem prudent to resolve this local problem
(involving stars) before invoking new physics (involving the foundations of
contemporary cosmology).

One means of uncovering systematic errors in the local (stellar) distance
ladder is by comparing the individual distances measured by the Cepheids and
TRGB stars with a third equally precise and accurate distance indicator.
Pertinently, two independent studies in the recent literature have presented a
novel means of measuring distances to galaxies using carbon-rich, asymptotic
giant branch stars (Madore & Freedman, 2020; Freedman & Madore, 2020; Ripoche
et al., 2020), denominated the J-region Asymptotic Giant Branch (JAGB) method.
JAGB stars are a well-defined class of carbon-rich, thermally-pulsating AGB
stars whose narrowly-defined, constant luminosity in the near infrared (NIR)
has been shown to be constant from galaxy to galaxy. Easily identified on the
basis of their NIR colors, the mean luminosity of a galaxy’s JAGB stars
provide us with an excellent and simple, empirical standard candle (Nikolaev &
Weinberg, 2000; Weinberg & Nikolaev, 2001).

Carbon stars’ narrow range of luminosities have been predicted by stellar
modeling for more than half a century (Iben, 1973; Iben & Renzini, 1983). JAGB
stars are also so photometrically distinctive due to their extremely red
color, a product of their carbon-enhanced atmospheres. This phenomenon results
from the AGB stars’ third dredge-up event, which transports carbon from the
helium-burning shells up to the stellar surface through convective pulses
(Habing & Olofsson, 2003). The third dredge-up event is only effective for a
small range of AGB star masses ($2-5M_{\odot}$) and thus a small range of
luminosities (e.g., Marigo et al. 2008; Madore & Freedman 2020), therefore
explaining the JAGB stars’ low intrinsic scatter in their NIR luminosities
($\pm 0.3$ mag).

The JAGB method has been shown to have comparable accuracy and precision in
measuring distances to nearby galaxies with the Leavitt Law and TRGB (Freedman
& Madore, 2020; Lee et al., 2021; Parada et al., 2021; Zgirski et al., 2021;
Lee et al., 2022; Parada et al., 2023), and therefore will be a useful cross-
check for both of these distance indicators. The JAGB method also has several
clear advantages as a distance indicator: first, JAGB stars are brighter in
the near infrared than both the TRGB and 10-day Cepheids. Second, JAGB stars
are mostly ubiquitous in all galaxy morphologies111JAGB stars are found in all
galaxies with stellar populations between 200 Myr to 1 Gyr. Therefore, they
will not be found in old ellipticals and some dwarf galaxies. and
inclinations, unlike the Cepheid Leavitt Law which can only be applied to
late-type spiral galaxies with low to moderate inclinations. Third, only one
epoch of observing is needed to measure a JAGB distance, unlike Cepheids and
Miras for which more than a dozen observations are required to measure their
periods, amplitudes, mean magnitudes, and colors.

In this paper we show results for the first JWST NIRCam (Rigby et al., 2023;
Rieke et al., 2023; Gardner et al., 2023) observations of JAGB stars in the
first three galaxies of our JWST Cycle 1 program 1995 (PI: W. Freedman): NGC
7250, NGC 4536, and NGC 3972. The purpose of this program was to measure
distances to 10 SN Ia host galaxies with three independent distance
indicators, using the same imaging. Each of these methods, the Cepheids, the
TRGB method, and the JAGB method, have been shown to be high-precision
extragalactic distance indicators. Our research program, the CCHP, will
provide a path to $H_{0}$ based on these three independent methods, aiming to
decrease the systematic uncertainties below the 2% level for each method. This
paper in particular marks the beginning of a future measurement of $H_{0}$ via
the JAGB stars.

##  2 Data

Data presented here were taken with the JWST’s NIRCam. Imaging were
simultaneously obtained in the short-wavelength channel (pixel scale of
$0.031\arcsec~{}\rm{pixel^{-1}}$) and the long-wavelength channel
($0.063\arcsec~{}\rm{pixel^{-1}}$). The $F115W$ filter (J band equivalent) was
chosen for this program as the short-wavelength filter following Madore &
Freedman (2020), who found that the JAGB stars have a constant mean luminosity
in the J band. $F444W$ was originally chosen as the long-wavelength filter to
have a large $(F115W-F444W)$ color baseline to easily separate O-rich AGB
stars and C-rich AGB stars via their colors. The first two galaxies in our
program to be imaged, NGC 7250 and NGC 4536, therefore have $F115W$ and
$F444W$ data. However, it became clear that the pixel resolution of $F444W$
was poorer than even the HST WFC3/IR camera. We elected to therefore change
our LW filter to $F356W$ for the remaining nine galaxies to take advantage of
its increased resolution over $F444W$. We found that the $(F115W-F356W)$ color
was still excellent at separating the O-rich and C-rich AGB stars in NGC 3972.

Details on the calibration and the reduction of these data are extensively
described in a companion paper, Owens et al. (2023, submitted). In short, PSF
photometry was extracted from the images using the software package DOLPHOT
(Dolphin, 2000, 2016) via the the beta testing JWST/NIRCAM module (Weisz et
al., 2023).

The photometry used in this analysis remains preliminary for two reasons:
photometry was extracted via the beta-testing DOLPHOT JWST/NIRCAM module
(which is still being actively updated; D. Weisz, private communication), and
the absolute photometric zeropoints of NIRCam still remain relatively
uncertain (Boyer et al., 2022). A JWST cycle 1 proposal aims to more
accurately calibrate the NIRCam photometric zeropoints (Gordon et al., 2022),
after which we expect to be able to publicly provide final photometry closer
to the requisite $<2\%$ precision needed for extragalactic distance scale
measurements of $H_{0}$.

Nevertheless, even while using this preliminary photometry, the JAGB stars
still exhibit low scatter and are clearly delineated from other stellar
populations. However, we still cautiously emphasize that the magnitude axes
should be taken as arbitrary. We have added the same additional random
magnitude offset between $-0.1$ and $0.1$ mag to both our F115W and
F444W/F356W photometry. This offset will be removed once our photometric
zeropoints are finalized in the next paper in this series. Below, we now
describe the data for the three galaxies analyzed in this paper.

Figure 1: Left: a color mosaic of the $F115W$ and $F444W$ images for the
observed NGC 7250 field. Our analysis excluded the stars in NGC 7250’s inner
disk, delineated by the dotted white ellipse. Right: zoom-in views of the
inner (top) and outer (bottom) rectangles of the left image. The JAGB stars
are highlighted within the green circles. North is up and east is to the left
in all the panels.

###  2.1 NGC 7250

The irregular galaxy NGC 7250 (also known as PGC 68535 or UGC 11980) was the
first SN Ia host galaxy in this program to be observed by JWST in November
2022. Total exposures of 3769 seconds were simultaneously obtained in the
$F115W$ short-wavelength channel and the $F444W$ long-wavelength channel. A
color mosaic image of the observations is shown in Figure 1. In 2013, NGC 7250
was host to the type 1a SN 2013dy explosion, therefore uniquely positioning it
as a SN Ia calibrator for measuring the Hubble constant. Indeed, Riess et al.
(2016, 2022) used NGC 7250 as a SN Ia calibrator using 21 Cepheids observed
with the Hubble Space Telescope (HST) as part of the SH0ES program, measuring
a Leavitt law distance modulus of $31.628\pm 0.126$ mag (or $21.2\pm 1.3$
Mpc). Photometry for $\sim 514,000$ sources was extracted from the images;
after performing cleaning cuts designed to remove non-stellar sources (e.g.,
artifacts, cosmic rays, and extended sources) (DOLPHOT type = 1,
$\rm{SNR}_{F115W}>3$, $\rm{SNR}_{F444W}>1$, $\rm{sharp}^{2}<0.04$,
$\rm{crowd}_{F115W}<0.2$, $\sigma_{F115W}<0.02\times
0.0025\rm{e}^{F115W-24.05}$), $\sim 112,000$ sources remained in the final
catalog. In particular, our chosen quality cuts do a reasonable job of
eliminating background galaxies, but we plan to more thoroughly investigate
the optimal quality cut parameters in our full sample of SN Ia host galaxies.
We note, however, that the mode (used to measure the JAGB magnitude) is
particularly robust to contamination from background galaxies, which are
mostly $\sim 1$ mag fainter than the JAGB stars (Madore et al., 2022).

###  2.2 NGC 4536

NGC 4536 (also known as PGC 41823 or UGC 07732) is a spiral galaxy that was
observed with JWST in January 2023, and was host to SN 1981B. Its distance was
measured by the CCHP via the TRGB (Hatt et al. 2018,
$M_{F814W}^{TRGB}=-4.05\pm 0.04$ mag, $A_{F814W}=0.03$ mag) to be
$\mu_{0}=31.04\pm 0.06$ mag (or $16.1\pm 0.5$ Mpc). Its distance was also
measured using the Leavitt law via 40 Cepheids (Riess et al., 2016, 2022) to
be $\mu_{0}=30.838\pm 0.051$ mag (or $14.7\pm 0.3$ Mpc). Total exposures of
2802 seconds were simultaneously obtained in the $F115W$ short-wavelength
channel and the $F444W$ long-wavelength channel. Photometry for $\sim 566,000$
sources was extracted from the images; after performing cleaning cuts (DOLPHOT
type = 1, $\rm{SNR}_{F115W}>3$, $\rm{SNR}_{F444W}>1$, $\rm{sharp}^{2}<0.04$,
$\rm{crowd}_{F115W}<0.2$), $\sim 210,000$ sources remained in the final
catalog.

###  2.3 NGC 3972

NGC 3972 (also known as UGC 6904) is a spiral galaxy that was observed with
JWST in April 2023, and was host to SN 2011by. Riess et al. (2016, 2022) used
52 Cepheids to measure a distance modulus of $\mu_{0}=31.644\pm 0.090$ mag (or
$21.3\pm 0.9$ Mpc). Total exposures of 3768 seconds were simultaneously
obtained in the $F115W$ short-wavelength channel and the $F356W$ long-
wavelength channel. Photometry for $\sim 688,000$ sources was extracted from
the images; after performing cleaning cuts (DOLPHOT type = 1,
$\rm{SNR}_{F115W}>3$, $\rm{SNR}_{F444W}>1$, $\rm{sharp}^{2}<0.04$,
$\rm{crowd}_{F115W}<0.2$, $\sigma_{F115W}<0.02\times
0.0025\rm{e}^{F115W-24.05}$), $\sim 78,000$ sources remained in the final
catalog.

##  3 Measuring the JAGB magnitude

Figure 2:  Color-magnitude diagrams for the outer regions of NGC 7250, NGC
4536, and NGC 3972. The JAGB stars were measured to be within the light-blue
shaded regions. In the right-hand panel, the GLOESS-smoothed luminosity
functions for the JAGB stars is shown in light blue, over-plotted the 0.01 mag
binned luminosity functions in grey. The range of the F115W y-axis is 4.5 mag
for all three color magnitude diagrams.

First, the JAGB stars were identified by their near-infrared color. For NGC
7250 and NGC 4536, JAGB stars were selected as having colors of
$2.6<(F115W-F444W)<3.2$ mag. For NGC 3972, JAGB stars were selected as having
colors of $2.4<(F115W-F356W)<3.0$ mag. Tests of these color limits are
described later in this section.

Then, the $F115W$ magnitudes of the JAGB stars were finely binned using bin
sizes of 0.01 mag. To control for Poisson noise, the binned luminosity
function was smoothed using the GLOESS (Gaussian-windowed, LOcally weighted
Scatterplot Smoothing) algorithm (introduced by Persson et al. 2004), a data-
smoothing interpolating technique effective at suppressing false (noise-
induced) edges and peaks in luminosity functions. We also describe our tests
of varying the smoothing parameters used in the GLOESS algorithm later in this
section. The mode of the smoothed luminosity function then marks the JAGB
magnitude.

###  3.1 Defining the ‘outer disk’

Lee et al. (2022) partitioned the photometry of M33 into four concentric
regions, showing that the measured mode of the JAGB star luminosity function
stabilized to within 0.01 mag in the two outer regions, which covered the
outer disk and halo of M33, respectively. The two distance moduli measured in
the outer regions agreed with independent TRGB and Leavitt law distance moduli
by 2% (Lee et al., 2022). On the other hand, the mode measured in the inner
regions of M33 differed from the mode measured in the outer regions up to
$\sim 0.7$ mag. This test first demonstrated that the JAGB magnitude is less
accurate when measured in the inner disks of galaxies due to a confluence of
crowding, blending, and reddening errors. However, where the ‘outer disk’ lies
can be subjective.

We introduce a novel method for identifying the outer disk of galaxies for the
purpose of measuring JAGB method distances, where reddening, crowding, and
blending effects taper off. We separated the photometry of a given galaxy into
eight concentric regions split by semi-major axis (SMA), and measured the
change in JAGB magnitude from region to region. To compute the SMA, we
obtained the galaxy’s center coordinates from NED, and its inclination angle
and position angle from HyperLeda. The boundaries of the regions were chosen
so that each region had an equal number of JAGB stars. These boundaries are
shown in the left panels of Figures 3, 4, and 5, for NGC 7250, NGC 4536, and
NGC 3972, respectively, overlaid onto color mosaics of our NIRCam imaging.

To measure relative changes in mode throughout the different regions, we
computed $\Delta m_{JAGB}$, which was defined as the change in mode from the
fiducial mode. In NGC 7250, the fiducial mode was measured using JAGB stars
with colors between $2.6<(F115W-F444W)<3.2$ mag and an LF smoothed with a
smoothing parameter of $\sigma_{s}=0.25$ mag. In NGC 4536, the fiducial mode
was measured using JAGB stars with colors between $2.6<(F115W-F444W)<3.2$ mag
and an LF smoothed with a smoothing parameter of $\sigma_{s}=0.30$ mag. For
NGC 3972, the fiducial mode was measured using JAGB stars with colors between
$2.4<(F115W-F444W)<3.0$ mag and an LF smoothed with a smoothing parameter of
$\sigma_{s}=0.25$ mag. Regarding the spatial selection, the fiducial JAGB
modal magnitude was computed from the final selected ‘outer disk’ region
(i.e., the merged outer regions to the left of the black line in Figure 7.).
We now describe our process for selecting this outer disk region below. The
color magnitude diagrams with these chosen parameters are shown in Figure 2.

We then compared $\Delta m_{JAGB}$ to the relative average surface brightness
in each radial region.222Using a galaxy’s surface brightness profile to
determine the optimal halo location has also been used by the CCHP in
measuring the TRGB (e.g., Jang et al. 2021). In the right panels of Figures 3,
4, and 5, we show the change in $\Delta m_{JAGB}$ as a function of the average
sky_F115W333Labeled the total sky value in the DOLPHOT catalog. value in each
radial bin, divided by the maximum sky_F115W value for the whole galaxy (all
concentric regions). The sky_F115W parameter was derived from DOLPHOT for
every star, and is a relative measure of the sky background at the star’s
position in a given image (Dolphin, 2016). In Appendix A, we show plots of
this sky_F115W parameter as a function of the semi-major axis, with
corresponding power-law fits. We emphasize the sky_F115W parameter is not an
exact measure of surface brightness but is a simple parameter returned from
DOLPHOT that can be used to track changes in $\Delta m_{JAGB}$. We also
highlight that the choice of region separation (we used SMA here) is
irrelevant to the final result, as long as $\Delta m_{JAGB}$ converges in the
outermost regions.

The red points in Figures 3, 4, and 5 show the change in $\Delta m_{JAGB}$ as
a function of the sky_F115W parameter in each galaxy. The three panels show
the effects of using different color cuts and smoothing parameters. In all
cases, the JAGB magnitude leveled out in the outermost regions of the galaxy;
therefore, the choice of color cuts and smoothing parameter did not
significantly affect the final result. For example in NGC 7250 and NGC 4536,
where we used a color cut of $2.6<(F115W-F444W)<3.2$ mag to measure the
fiducial JAGB magnitude, we varied the blue color cut through {2.0, 2.1, 2.2,
2.3, 2.4, 2.5, 2.6, 2.7} and the red color through {2.9, 3.0, 3.1, 3.2, 3.3,
3.4}. For NGC 3972 where we used a color cut of $2.4<(F115W-F356W)<3.0$ mag to
measure the fiducial JAGB magnitude, we varied the blue color cut through
{2.2, 2.3, 2.4, 2.5, 2.6} and the red color through {2.8, 2.9, 3.0, 3.1, 3.2}.
We also tested how different smoothing parameters $\sigma_{s}$ in the GLOESS
smoothing algorithm affected the mode. We varied $\sigma_{s}$ from 0.15 to
0.40 mag in steps of 0.05 mag and quantified the change in the mode.

In all three galaxies, the mode $m_{JAGB}$ stabilized in the outer regions of
the galaxy. In NGC 7250, each region had 335 JAGB stars. The mode $m_{JAGB}$
stabilized to within 0.01 in the three outermost regions. In NGC 4536, each
region had 461 JAGB stars. The mode $m_{JAGB}$ stabilized to within 0.03 in
the five outermost regions. In NGC 3972, each region had 491 JAGB stars. The
mode $m_{JAGB}$ stabilized to within 0.04 in the four outermost regions.
Therefore, we selected the outer bin of the innermost stable region to be the
radial cutoff for that galaxy. In NGC 7250, this was $9.6$ kpc; for NGC 4536,
this was $SMA=10.3$ kpc; for NGC 4972, this was $SMA=7.6$ kpc. We recommend
this procedure be used for measuring the JAGB magnitude for all future studies
utilizing the JAGB method; the inner disk regions of galaxies can therefore be
cut out in a procedural way, instead of visually.

Figure 3: (Left) 8 regions in NGC 7250, split by semi-major axis. (Right)
Change in $\Delta m_{JAGB}$ as a function of the normalized average sky value
in that region for the 8 regions. $\Delta m_{JAGB}$ is defined as the change
in mode from the fiducial $m_{JAGB}$, which is measured from the outer disk
stars between $2.6<(F115W-F444W)<3.2$ mag, using a smoothing parameter for the
JAGB star luminosity function of $\sigma_{s}=0.25$ mag. The top panel shows
the effect of this test for different red color cuts while keeping the blue
color cut constant, the middle panel shows the effect of this test for
different blue color cuts while keeping the red color cut constant, and the
bottom panel shows the effect of this test for different smoothing parameters.
The black line shows the fiducial surface brightness cutoff used for each
galaxy. The grey region in the right plots covers $-0.05$ to 0.05 mag,
highlighting the convergence of the JAGB magnitude to within 0.01 mag in
Regions 6, 7, and 8.  Figure 4: (Left) 8 regions in NGC 4536, split by semi-
major axis. (Right) Change in $\Delta m_{JAGB}$ as a function of the
normalized average sky value in that region for the 8 regions. $\Delta
m_{JAGB}$ is defined as the change in mode from the fiducial $m_{JAGB}$, which
is measured for the outer disk stars between $2.6<(F115W-F444W)<3.2$ mag,
using a fixed smoothing parameter for the luminosity function of
$\sigma_{s}=0.30$ mag. The top panel shows the effect of this test for
different red color cuts while keeping the blue color cut constant, the middle
panel shows the effect of this test for different blue color cuts while
keeping the red color cut constant, and the bottom panel shows the effect of
this test for different smoothing parameters. The black line shows the
fiducial surface brightness cutoff used for each galaxy. The grey region in
the right plots covers $-0.05$ to 0.05 mag, highlighting the convergence of
the JAGB magnitude to within 0.03 mag in regions 5, 6, 7, and 8.  Figure 5:
(Left) 8 regions in NGC 3972, split by semi-major axis. (Right) Change in
$\Delta m_{JAGB}$ as a function of the normalized average sky value in that
region for the 8 regions. $\Delta m_{JAGB}$ is defined as the change in mode
from the fiducial $m_{JAGB}$, which is measured for the outer disk stars
between $2.4<(F115W-F444W)<3.0$ mag, using a smoothing parameter for the
luminosity function of $\sigma_{s}=0.25$ mag. The top panel shows the effect
of this test for different red color cuts while keeping the blue color cut
constant, the middle panel shows the effect of this test for different blue
color cuts while keeping the red color cut constant, and the bottom panel
shows the effect of this test for different smoothing parameters. The black
line shows the fiducial surface brightness cutoff used for each galaxy. The
grey region in the right plots covers $-0.05$ to 0.05 mag, highlighting the
convergence of the JAGB magnitude to within 0.04 mag in Regions 5, 6, 7, and
8.

###  3.2 Color-magnitude diagrams

The color-magnitude diagrams and smoothed luminosity functions for all three
galaxies after performing the inner disk spatial cut are shown in Figure 2.
The scatter on the mode $m_{JAGB}$ of the JAGB stars between $m_{JAGB}\pm
0.75$ mag was measured to be $\sigma=0.32$ mag, $\sigma=0.34$ mag, and
$\sigma=0.35$ mag for NGC 7250, NGC 4536, and NGC 3972, respectively. To
measure the scatter on the mode, we used the following formula:

$\sqrt{\frac{\Sigma{(m_{i}-m_{JAGB})^{2}}}{N_{JAGB}}},$ (1)

where $m_{i}$ are the F115W magnitudes of the JAGB stars and $N_{JAGB}$ is the
total number of JAGB stars.

These measured scatters from JWST are similar to the scatter measured from
ground-based telescopes for the JAGB stars in the LMC, a galaxy 400 times
closer than NGC 7250. Weinberg & Nikolaev (2001), the first study to use JAGB
stars as standard candles in the near infrared, measured a scatter of
$\sigma=0.33$ mag in their LMC sample using 2MASS data.

###  3.3 Artificial star tests

Artificial stars were computed with DOLPHOT for the first galaxy in our
program, NGC 7250, to assess the robustness of the JAGB star photometry. In
Figure 6, we show results for the $\sim 4200$ injected stars in the JAGB color
range and having a $SMA>9.6$ kpc. The range of the x-axis in Figure 6
corresponds to the same range of the y-axis in Figure 2 for NGC 7250. At the
average magnitude of the JAGB stars, the median offset was measured to be
$F115W_{in}-F115W_{out}=-0.01$ mag, demonstrating that crowding is a
negligible source of systematic error for the JAGB stars in the outer regions
of NGC 7250.

Figure 6: Difference between the input and output magnitudes from the DOLPHOT
artificial star tests for the outer region of NGC 7250 ($d>9.6$ kpc). The red
dots represent the median offset for 1 mag bins. The same random offset
between $-0.1$ and $0.1$ mag discussed in Section 2 has been applied to both
magnitude axes.

##  4 Conclusion

This paper demonstrates JWST’s impressive capabilities for utilizing JAGB
stars as standard candles. Our results exemplify we now have the potential to
study JAGB stars in galaxies at distances $\gtrsim 20$ Mpc away with JWST,
with comparable or greater resolution than galaxies $50$ kpc away with ground-
based telescopes.

The high sensitivity and resolution of JWST allows us to resolve JAGB and TRGB
stars in the outer regions of galaxies, at much farther distances than what
was previously possible with HST. For example, all of the Cepheid variable
stars in NGC 7250 have been discovered in the inner region (Riess et al. 2016,
2022, Owens et al., submitted), i.e., within the white ellipse in Figure 1,
where the JAGB measurement was excluded in this paper. Thus, a significant
advantage of the JAGB and TRGB methods to the Cepheid P-L relation is that
carbon stars can be found in the significantly less crowded halos of galaxies,
whereas Cepheids are mainly found in the more crowded star-forming disks.

A future calibration of the JAGB stars absolute magnitude in the $F115W$
filter will soon be possible via imaging of the JAGB stars in the water mega-
maser host galaxy NGC 4258. These images are now scheduled for early 2024 and
will also provide a zero point for the TRGB and Leavitt law. This calibration
will allow us to determine the distance to NGC 7250, NGC 4536, and NGC 3972
via the JAGB stars, as well as the rest of the SN Ia host galaxies in our
observing program, allowing us to independently measure $H_{o}$ via the JAGB
method.

Suggestions and comments from Saurabh Jha, Adam Riess, and Dan Scolnic at the
2023 MIAPbP extragalactic distance scale workshop, regarding the selection of
the location for the JAGB measurement, are gratefully acknowledged. We thank
Dan Weisz and Andy Dolphin for their insight and advice on the use of the
latest version of DOLPHOT tailored to JWST imaging data, and Jane Rigby for
updates on the calibration of the NIRCam filters. Finally, we thank the
anonymous referee for their constructive and helpful suggestions that improved
this work. AJL was supported by the Future Investigators in NASA Earth and
Space Science and Technology (FINESST) award number 80NSSC22K1602 during the
completion of this work. AJL thanks the LSSTC Data Science Fellowship Program,
which is funded by LSSTC, NSF Cybertraining Grant #1829740, the Brinson
Foundation, and the Moore Foundation; her participation in the program has
benefited this work. AJL was supported by the Munich Institute for Astro-,
Particle and BioPhysics (MIAPbP) which is funded by the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) under Germany´s
Excellence Strategy – EXC-2094 – 390783311. We also thank the University of
Chicago and the Observatories of the Carnegie Institution for their support of
our long-term research into the calibration and determination of the expansion
rate of the universe. This work is based on observations made with the
NASA/ESA/CSA JWST. The data were obtained from the Mikulski Archive for Space
Telescopes at the Space Telescope Science Institute, which is operated by the
Association of Universities for Research in Astronomy, Inc., under NASA
contract NAS 5-03127 for JWST. These observations are associated with program
1995. This research has made use of NASA’s Astrophysics Data System
Bibliographic Services. All the JWST data used in this paper can be found in
MAST: http://dx.doi.org/10.17909/ffv8-5279 (catalog 10.17909/ffv8-5279).

##  Appendix A Sky count profiles

In this section, we show plots of the average sky_F115W parameter in each of
the 8 regions as a function of semi-major axis for the three galaxies. Power-
law functions were fit to the data points, and are shown as the black line in
Figure 7.

Figure 7: Average normalized sky_F115W parameter returned from DOLPHOT as a
function of SMA for NGC 7250, NGC 4536, and NGC 3972, for the 8 defined
regions split by SMA. The average sky_F115W and SMA for each region is shown
by the black points. Each region has the same number of JAGB stars. Power law
functions were fit to the points and shown here as black lines. The analytical
form of the power law function is shown on each plot.

The dotted black line is the fiducial radial cut that was used for each
galaxy. The corresponding sky_F115W returned from the power-law fit for the
chosen radial cutoff is shown in Figures 3, 4, and 5.

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# Characterization of AGM Belief Contraction
in Terms of Conditionals

Giacomo Bonanno I am grateful to three anonymous reviewers for their
comments.Department of Economics
University of California
Davis, California, USA<EMAIL_ADDRESS>
###### Abstract

We provide a semantic characterization of AGM belief contraction based on
frames consisting of a Kripke belief relation and a Stalnaker-Lewis selection
function. The central idea is as follows. Let $K$ be the initial belief set
and $K\div\phi$ be the contraction of $K$ by the formula $\phi$; then $\psi\in
K\div\phi$ if and only if, at the actual state, the agent believes $\psi$ and
believes that if $\lnot\phi$ is (were) the case then $\psi$ is (would be) the
case.

##  1 Introduction

Belief contraction is the operation of removing from the set $K$ of initial
beliefs a particular belief $\phi$. One reason for doing so is, for example,
the discovery that some previously trusted evidence supporting $\phi$ was
faulty. For instance, a prosecutor might form the belief that the defendant is
guilty on the basis of his confession; if the prosecutor later discovers that
the confession was extorted, she might abandon the belief of guilt, that is,
become open minded about whether the defendant is guilty or not. In their
seminal contribution to belief change, Alchourrón, Gärdenfors and Makinson
([2]) defined the notion of ”rational and minimal” contraction by means of a
set of eight properties, known as the AGM axioms or postulates. They did so
within a syntactic approach where the initial belief set $K$ is a consistent
and deductively closed set of propositional formulas and the result of
removing $\phi$ from $K$ is a new set of propositional formulas, denoted by
$K\div\phi$.

We provide a new characterization of AGM belief contraction based on a so-far-
unnoticed connection between the notion of belief contraction and the
Stalnaker-Lewis theory of conditionals ([35, 22]). Stalnaker introduced the
notion of a selection function $f$ taking as input a possible world $w$ and a
set of worlds $E$ (representing a proposition) and giving as output a world
$w^{\prime}=f(w,E)\in E$, interpreted as the closest $E$-world to $w$ (an
$E$-world is a world that belongs to $E$). Lewis generalized this by allowing
$f(w,E)$ to be a set of worlds. In the Stalnaker-Lewis theory the (indicative
or subjunctive) conditional ”if $\phi$ is (were) the case then $\psi$ is
(would be) the case”, denoted by $\phi>\psi$, is declared to be true at a
world $w$ if and only if $\psi$ is true at all the worlds in $f(w,\|\phi\|)$
($\|\phi\|$ denotes the set of worlds at which $\phi$ is true).

We consider semantic frames consisting of a Kripke belief relation on a set of
states $S$, representing the agent’s initial beliefs, and a Stalnaker-Lewis
selection function on $S\times 2^{S}$ representing conditionals. Adding a
valuation to such a frame yields a model. Given a model, we define the initial
belief set $K$ as the set of formulas that the agent believes at the actual
state and $K\div\phi$ (the contraction of $K$ by $\phi$) as the set of
formulas that the agent believes initially and also on the supposition that
$\lnot\phi$: $\psi\in K\div\phi$ if and only if, at the actual state, the
agent (1) believes $\psi$ and (2) believes the conditional $\lnot\phi>\psi$.
We show that, when the selection function satisfies some natural properties,
the contraction operation so defined captures precisely the set of AGM belief
contraction functions.

##  2 AGM contraction functions

Let At be a countable set of atomic formulas. We denote by $\Phi_{0}$ the set
of Boolean formulas constructed from At as follows:
$\texttt{At}\subset\Phi_{0}$ and if $\phi,\psi\in\Phi_{0}$ then $\neg\phi$ and
$\phi\vee\psi$ belong to $\Phi_{0}$. Define $\phi\rightarrow\psi$,
$\phi\wedge\psi$, and $\phi\leftrightarrow\psi$ in terms of $\lnot$ and $\vee$
in the usual way.

Given a subset $K$ of $\Phi_{0}$, its deductive closure
$Cn(K)\subseteq\Phi_{0}$ is defined as follows: $\psi\in Cn(K)$ if and only if
there exist $\phi_{1},...,\phi_{n}\in K$ (with $n\geq 0$) such that
$(\phi_{1}\wedge...\wedge\phi_{n})\rightarrow\psi$ is a tautology. A set
$K\subseteq\Phi_{0}$ is consistent if $Cn(K)\neq\Phi_{0}$; it is deductively
closed if $K=Cn(K)$. Given a set $K\subseteq\Phi_{0}$ and a formula
$\phi\in\Phi_{0}$, the _expansion_ of $K$ by $\phi$, denoted by $K+\phi$, is
defined as follows: $K+\phi=Cn\left(K\cup\\{\phi\\}\right)$.

Let $K\subseteq\Phi_{0}$ be a consistent and deductively closed set
representing the agent’s initial beliefs and let $\Psi\subseteq\Phi_{0}$ be a
set of formulas representing possible candidates for withdrawal. A _belief
contraction function_ (based on $K$ and $\Psi$) is a function
$\div_{\Psi}:\Psi\rightarrow 2^{\Phi_{0}}$ (where $2^{\Phi_{0}}$ denotes the
set of subsets of $\Phi_{0}$) that associates with every formula $\phi\in\Psi$
a set $K\div_{\Psi}\,\phi\subseteq\Phi_{0}$ (interpreted as the result of
removing $\phi$ from $K$). If $\Psi\neq\Phi_{0}$ then $\div_{\Psi}$ is called
a _partial_ contraction function, while if $\Psi=\Phi_{0}$ then
$\div_{\Phi_{0}}$ is called a _full-domain_ contraction function; in this case
we simplify the notation and omit the subscript $\Phi_{0}$.

######  Definition 1.

Let $\div_{\Psi}:\Psi\rightarrow 2^{\Phi_{0}}$ be a partial contraction
function and $\div^{\prime}:\Phi_{0}\rightarrow 2^{\Phi_{0}}$ a full-domain
contraction function (both of them based on $K$). We say that $\div^{\prime}$
is an _extension_ of $\div_{\Psi}$ if, for every $\phi\in\Psi$,
$K\div^{\prime}\phi=K\div_{\Psi}\phi$.

A full-domain contraction function is called an AGM contraction function if it
satisfies the following properties, known as the AGM postulates:

($K-$1) | [Closure] $K\div\phi=Cn(K\div\phi)$.
---|---
($K-$2) | [Inclusion] $K\div\phi\subseteq K$.
($K-$3) | [Vacuity] If $\phi\notin K$ then $K\subseteq K\div\phi$.
($K-$4) | [Success] If $\phi$ is not a tautology, then $\phi\notin K\div\phi$.
($K-$5) | [Recovery] If $\phi\in K$ then $K\subseteq(K\div\phi)+\phi$.
($K-$6) | [Extensionality] If $\phi\leftrightarrow\psi$ is a tautology, then $K\div\phi=K\div\psi$.
($K-$7) | [Conjunctive overlap] $(K\div\phi)\cap(K\div\psi)\subseteq K\div(\phi\wedge\psi)$.
($K-$8) | [Conjunctive inclusion] If $\phi\notin K\div(\phi\wedge\psi)$, then $K\div(\phi\wedge\psi)\subseteq K\div\phi.$

($K-$1) requires the result of contracting $K$ by $\phi$ to be a deductively
closed set.
($K-$2) requires the contraction of $K$ by $\phi$ not to contain any beliefs
that were not in $K$.
($K-$3) requires that if $\phi$ is not in the initial belief set, then every
belief in $K$ should also be present in $K\div\phi$ (thus, by ($K-$2) and
($K-$3), if $\phi\notin K$ then the contraction of $K$ by $\phi$ coincides
with $K$).
($K-$4) requires that $\phi$ not be contained in $K\div\phi$, unless $\phi$ is
a tautology (in which case, by ($K-$1), it must be in $K\div\phi$).
($K-$5) is a conservativity requirement: when $\phi\in K$, contracting by
$\phi$ and then expanding the resulting set $K\div\phi$ by $\phi$ should
involve no loss of beliefs relative to $K$ (the converse inclusion
$(K\div\phi)+\phi\subseteq K$ follows from ($K-$2) and the hypothesis that
$K=Cn(K)$).
($K-$6) says that logically equivalent formulas should lead to the same result
in terms of contraction.
By ($K-$7), if a formula $\chi\in K$ is neither removed in the contraction of
$K$ by $\phi$ nor in the contraction of $K$ by $\psi$, then $\chi$ should not
be removed in the contraction of $K$ by the conjunction $\phi\wedge\psi$.
($K-$8), on the other hand, requires that if $\phi$ is removed when we
contract by $\phi\wedge\psi$, then every formula that survives the contraction
of $K$ by $\phi\wedge\psi$ survives also when $K$ is contracted by $\phi$
alone.

For an extensive discussion of the above postulates see [12, 18, 7].

The notion of AGM belief contraction has been given alternative
characterizations. One characterization is in terms of a binary relation
$\leqslant$ of ”epistemic entrenchment” on $K$, with the interpretation of
$\phi\leqslant\psi$ as ”$\phi$ is either less entrenched than, or as
entrenched as, $\psi$”. Gärdenfors ([12, Theorem 4.30, p. 96]) shows that if
the relation $\leqslant$ satisfies five properties and a contraction function
is defined by ‘$\psi\in K\div\phi$ if and only if $\psi\in K$ and either
$\phi$ is a tautology or $\phi<(\phi\vee\psi)$’, then such contraction
function is an AGM contraction function and, conversely, if an AGM contraction
function is used to define the relation $\leqslant$ by ‘$\phi\leqslant\psi$ if
and only if either $\phi\notin K\div(\phi\wedge\psi)$ or $\phi\wedge\psi$ is a
tautology’ then such relation satisfies those five properties. Another
characterization makes use of the set $W$ of possible worlds, where a possible
world is defined as a maximally consistent set of formulas in $\Phi_{0}$;
within this approach, contraction has been characterized either in terms of
systems of spheres ([14, 22]) or in terms of a plausibility relation on $W$ or
in terms of propositional selection functions (see [7, Chapter 4]).

In this paper we provide an alternative characterization in terms of
Stalnaker-Lewis conditionals.

##  3 An alternative semantic characterization of AGM contraction

Given a binary relation $R\subseteq S\times S$ on a set $S$, for every $s\in
S$ we define $R(s)=\\{x\in S:(s,x)\in R\\}$.

######  Definition 2.

A _pointed frame_ is a quadruple
$\left\langle{S,s_{@},\mathcal{B},f}\right\rangle$ where

  1. 1.

$S$ is a set of _states_ ; subsets of $S$ are called _events_.

  2. 2.

$s_{@}\in S$ is a distinguished element of $S$ interpreted as the _actual
state_.

  3. 3.

$\mathcal{B}\subseteq S\times S$ is a binary _belief relation_ on $S$ which is
serial: $\forall s\in S$, $\mathcal{B}(s)\neq\varnothing$.

  4. 4.

$f:\mathcal{B}(s_{@})\times 2^{S}\setminus\varnothing\rightarrow 2^{S}$ is a
_Stalnaker-Lewis selection function_ 111Note that, for the purpose of this
paper, the domain of $f$ can be taken to be $\mathcal{B}(s_{@})\times
2^{S}\setminus\varnothing$ rather than $S\times 2^{S}\setminus\varnothing$.
However, it can easily be extended to $S\times 2^{S}\setminus\varnothing$ as
follows: first, fix an arbitrary function
$g:S\setminus\mathcal{B}(s_{@})\rightarrow\mathcal{B}(s_{@})$ and then define,
for every $s\in S\setminus\mathcal{B}(s_{@})$ and every $\varnothing\neq
E\subseteq S$, $f(s,E)=f\left(g(s),E\right)$.  that associates with every
state-event pair $(s,E)$ (with $s\in\mathcal{B}(s_{@})$ and $\varnothing\neq
E\subseteq S$) a set of states $f(s,E)\subseteq S$ such that,

    1. (a)

(a.1) $f(s,E)\neq\varnothing$ and (a.2) (Success) $f(s,E)\subseteq E$,

    2. (b)

(Weak Centering) if $s\in E$ then $s\in f(s,E)$,

    3. (c)

(Doxastic Priority 1) if $\mathcal{B}(s_{@})\cap E\neq\varnothing$ then
$f(s,E)\subseteq\mathcal{B}(s_{@})\cap E$,

    4. (d)

(Intersection) $f(s,E)\cap F\subseteq f(s,E\cap F)$,

    5. (e)

(Doxastic Priority 2) Let $B_{EF}=\\{s\in\mathcal{B}(s_{@}):f(s,E)\cap
F\neq\varnothing\\}$. If $B_{EF}\neq\varnothing$ then

      1. (e.1)

if $s\in B_{EF}$ then $f(s,E\cap F)\subseteq f(s,E)\cap F$,

      2. (e.2)

if $s\notin B_{EF}$ then $f(s,E\cap F)\subseteq f(\hat{s},E\cap F)$ for some
$\hat{s}\in B_{EF}$.

The set $\mathcal{B}(s)$ is the set of states that the agent considers
possible at state $s$, so that $\mathcal{B}(s_{@})$ is the set of doxastic
possibilities at the actual state $s_{@}$ and represents the agent’s initial
beliefs. $f(s,E)$ is the set of states that the agent considers closest, or
most similar, to state $s$ conditional on event $E$.
(4.a) of Definition 2 requires $f(s,E)$ to be non-empty and, furthermore, that
every state in $f(s,E)$ be an $E$-state.
(4.b) postulates that if $s$ is an $E$-state then it belongs to $f(s,E)$, that
is, $s$ itself is one of the $E$-states that are closest to $s$.
By (4.c) if there exists an $E$-state among those initially considered
possible ($\mathcal{B}(s_{@})\cap E\neq\varnothing$), then, for every
$s\in\mathcal{B}(s_{@})$, the closest $E$-states to $s$ must belong to
$\mathcal{B}(s_{@})\cap E$.
By (4.d), the closest $E$-states to $s$ that are also $F$-states must belong
to the set of closest $(E\cap F)$-states to $s$.
(4.e) can be viewed as an extension of (4.c): it says that if, among the
states initially considered possible, there is at least one state, call it
$s$, that satisfies the property that among its closest $E$-states there is at
least one that is also an $F$-state, then (1) the closest $(E\cap F)$-states
to $s$ must belong to the intersection $f(s,E)\cap F$ and (2) for any other
state that does not satisfy the property, the closest $(E\cap F)$-states to it
are contained in the set of closest $(E\cap F)$-states to some state that does
satisfy the property.

Adding a valuation to a pointed frame yields a model. Thus a _model_ is a
tuple $\left\langle{S,s_{@},\mathcal{B},f,V}\right\rangle$ where
$\left\langle{S,s_{@},\mathcal{B},f}\right\rangle$ is a pointed frame and
$V:\texttt{At}\rightarrow 2^{S}$ is a valuation that assigns to every atomic
formula $p\in\texttt{At}$ the set of states where $p$ is true. Given a model
$\left\langle{S,s_{@},\mathcal{B},f,V}\right\rangle$ define truth of a Boolean
formula $\phi\in\Phi_{0}$ at a state $s\in S$, denoted by $s\models\phi$, in
the usual way:

######  Definition 3.

Truth of a formula at a state is defined as follows:

  1. 1.

if $p\in\texttt{At}$ then $s\models p$ if and only if $s\in V(p)$,

  2. 2.

$s\models\neg\phi$ if and only if $s\not\models\phi$,

  3. 3.

$s\models\phi\vee\psi$ if and only if $s\models\phi$ or $s\models\psi$ (or
both),

We denote by $\|\phi\|$ the truth set of $\phi$: $\|\phi\|=\\{s\in
S:s\models\phi\\}$.

Fix a model $M=\left\langle{S,s_{@},\mathcal{B},f,V}\right\rangle$ and let
$K=\\{\phi\in\Phi_{0}:\mathcal{B}(s_{@})\subseteq\|\phi\|\\}$ (to simplify the
notation, we omit the subscript denoting the model and thus write $K$ rather
than $K_{M}$); thus a Boolean formula $\phi$ belongs to $K$ if and only if at
the actual state $s_{@}$ the agent believes $\phi$. It is shown in the
Appendix (Lemma 1) that the set $K\subseteq\Phi_{0}$ so defined is deductively
closed and consistent. Next, for every $\phi\in\Phi_{0}$ such that
$\|\lnot\phi\|\neq\varnothing$, define $K\div\phi\subseteq\Phi_{0}$ as
follows:

$\begin{array}[]{*{20}{l}}\psi\in K\div\phi\text{ if and only if
}&(1)\,\mathcal{B}(s_{@})\subseteq\|\psi\|,\text{ and }\\\ {}&(2)\,\forall
s\in\mathcal{B}(s_{@}),f\left(s,\|\lnot\phi\|\right)\subseteq\|\psi\|.\end{array}$
(1)

In (2) below we rewrite (1) in an extended language containing a belief
operator and a conditional operator, thus making the interpretation more
transparent: $\psi\in K\div\phi$ if and only if, at the actual state $s_{@}$,
the agent believes $\psi$ initially as well as on the supposition that
$\lnot\phi$.222We take “believing $\psi$ on the supposition that $\lnot\phi$”
to mean “believing that if $\lnot\phi$ is (were) the case then $\psi$ is
(would be) the case”. .

Since, in general, not every $\phi\in\Phi_{0}$ is such that
$\|\lnot\phi\|\neq\varnothing$, this definition gives rise to a _partial_
belief contraction function. The next proposition says that this partial
contraction function can be extended to a full-domain AGM contraction
function; conversely, given a full-domain AGM contraction function based on a
consistent and deductively closed set $K$, there exists a model
$M=\left\langle{S,s_{@},\mathcal{B},f,V}\right\rangle$ such that
$K=\\{\phi\in\Phi_{0}:\mathcal{B}(s_{@})\subseteq\|\phi\|\\}$ and, for every
$\phi\in\Phi_{0}$ such that $\|\lnot\phi\|\neq\varnothing$, $K\div\phi$
satisfies (1). Thus the proposed semantics provides an alternative
characterization of AGM belief contraction. The proof of the following
proposition is given in the Appendix.

######  Proposition 1.

.

  1. (A)

Given a model $\left\langle{S,s_{@},\mathcal{B},f,V}\right\rangle$ let
$K=\\{\phi\in\Phi_{0}:\mathcal{B}(s_{@})\subseteq\|\phi\|\\}$ and, for every
$\phi\in\Phi_{0}$ such that $\|\lnot\phi\|\neq\varnothing$, let $K\div\phi$ be
defined by (1). Then $K$ is consistent and deductively closed and the
(partial) belief contraction function so defined can be extended to a full-
domain AGM belief contraction function.

  2. (B)

Let $K\subset\Phi_{0}$ be consistent and deductively closed and let
$\div:\Phi_{0}\rightarrow 2^{\Phi_{0}}$ be an AGM belief contraction function.
Then there exists a model $\left\langle{S,s_{@},\mathcal{B},f,V}\right\rangle$
such that $K=\\{\phi\in\Phi_{0}:\mathcal{B}(s_{@})\subseteq\|\phi\|\\}$ and,
for every $\phi\in\Phi_{0}$ such that $\|\lnot\phi\|\neq\varnothing$,
$K\div\phi$ satisfies (1).

The proposed semantics becomes more transparent if we extend the language by
introducing two modal operators: a unimodal belief operator $\mathbb{B}$,
corresponding to the belief relation $\mathcal{B}$, and a bimodal conditional
operator $>$, corresponding to the selection function $f$. Recall that
$\Phi_{0}$ is the set of Boolean (or factual) formulas. Let $\Phi_{1}$ be the
modal language constructed as follows.

  * •

$\Phi_{0}\subset\Phi_{1}$,

  * •

if $\phi,\psi\in\Phi_{0}$ then $\phi>\psi\in\Phi_{1}$,

  * •

all the Boolean combinations of formulas in $\Phi_{1}$.

Thus, for the purpose of this paper, the conditional $\phi>\psi$ (interpreted
as the indicative or subjunctive conditional ”if $\phi$ is (were) the case
then $\psi$ is (would be) the case”) is defined only for Boolean formulas.
Finally, let $\Phi$ be the modal language constructed as follows:

  * •

$\Phi_{1}\subset\Phi$,

  * •

if $\phi\in\Phi_{1}$ then $\mathbb{B}\phi\in\Phi$,

  * •

all the Boolean combinations of formulas in $\Phi$.

Thus formulas in $\Phi$ are either Boolean or formulas of the form
$\phi>\psi$, with $\phi$ and $\psi$ Boolean, or of the form $\mathbb{B}\phi$
where $\phi$ is either Boolean or of the form $\psi>\chi$ with $\psi$ and
$\chi$ Boolean, or a Boolean combination of such formulas. We can now extend
the definition of truth of a formula at a state (Definition 3) to the set
$\Phi$ as follows:

######  Definition 4.

If $\phi\in\Phi_{0}$ then $s\models\phi$ according to the rules of Definition
3. Furthermore,

  * •

$s\models(\phi>\psi)$ (with $\phi,\psi\in\Phi_{0}$) if and only if either
$\|\phi\|=\varnothing$, or $\|\phi\|\neq\varnothing$ and
$f(s,\|\phi\|)\subseteq\|\psi\|$,

  * •

$s\models\mathbb{B}\phi$ if and only if $\mathcal{B}(s)\subseteq\|\phi\|$.

Then we can re-write the definition of $K\div\phi$ given in (1) in terms of
the modal operators $\mathbb{B}$ and $>$ as follows:

$\psi\in K\div\phi\text{ if and only if }\phi,\psi\in\Phi_{0}\text{ and
}s_{@}\models\mathbb{B}\psi\wedge\mathbb{B}\left(\lnot\phi>\psi\right).$ (2)

Thus, in the statement of Proposition 1,
$K=\\{\phi\in\Phi_{0}:\mathcal{B}(s_{@})\subseteq\|\phi\|\\}$ can be replaced
by $K=\\{\phi\in\Phi_{0}:s_{@}\models\mathbb{B}\phi\\}$ and reference to (1)
can be replaced by reference to (2). Note that only a fragment of the extended
language is used in the characterization result of Proposition 1. In
particular, nesting of conditionals and beliefs is disallowed. The study of
whether the extended language can be used to obtain generalizations of AGM-
style belief change that go beyond merely Boolean expressions is a topic left
for future research.

##  4 Related literature

There is a vast literature that deals with AGM belief contraction (for a
survey see, for example, [6, 7]). Because of space limitations we will only
focus on a few issues.

The recovery postulate (AGM axiom ($K-5$)) appears to be a natural way of
capturing a “minimal” way of suspending belief in $\phi$, but has been subject
to extensive scrutiny (see [26, 10, 15, 20, 23, 29, 16, 17]). In Makinson’s
terminology ([26]), contraction operations that do not satisfy the recovery
postulates are called _withdrawals_. Alternative types of withdrawal operators
have been studied in the literature: contraction without recovery ([5]), semi-
contraction ([8]), severe withdrawal ([33]), systematic withdrawal ([27]),
mild contraction ([21]). If one interprets belief contraction as a form of
_actual belief change_ (in response to some input), then perhaps the recovery
postulate is open to scrutiny. However, in the interpretation of belief
contraction proposed in this paper, the recovery postulate is entirely
natural. Indeed, if $\psi$ belongs to the contraction of $K$ by $\phi$ then
$\psi$ is believed both initially and on the supposition that $\lnot\phi$; if
this supposition is removed then one naturally falls back to the initial
beliefs $K$.

There have been attempts in the literature to establish a link between notions
of AGM belief change and Stalnaker-Lewis conditionals. Within the context of
AGM belief revision this was done by [11], who considered the language that we
called $\Phi_{1}$, which includes conditionals of the form $\phi>\psi$.
Gäerdenfors introduced the following postulate (where $K*\phi$ denotes the
revised belief set in response to information $\phi$): $(\phi>\psi)\in K\text{
if and only if }\psi\in K{\Large*}\phi$. This postulate was taken to be an
expression of the so-called Ransey test.333The expression ”Ramsey Test” refers
to the following passage from [30, p. 247]: ”If two people are arguing ”If $p$
will $q$?” and are both in doubt as to $p$, they are adding $p$ hypothetically
to their stock of knowledge and arguing on that basis about $q$”.  Gäerdenfors
showed that this postulate can be satisfied only in cases where the revision
operation is trivial; in other words, there cannot be interesting revision
theories based on conditionals if one requires that the conditionals
themselves be incorporated in the initial belief set. Several attempts have
been made to circumvent Gäerdenfors’ “triviality result”. Different routes
have been taken: weakening or re-interpretating the theorem ([23, 25, 24, 31,
32], generalizing from belief revision functions to belief change systems
(consisting of a set of epistemic states, an assignment of a belief set to
each epistemic state and a transition function function that determines how
the epistemic state changes as a result of learning new information: [9]),
considering an alternative semantics, namely Moss and Parikh’s epistemic logic
of subsets logic ([28]), and augmenting it with conditionals ([13]), and, in
the context of iterated belief contraction, defining the notion of
”contractional” in the context of belief states ( [34]: if $\Psi$ denotes a
belief state and $[\beta|\alpha]$ is interpreted as “belief in $\beta$ even in
the absence of $\alpha$”, then the contractional is defined as
$\Psi\models[\beta|\alpha]$ if and only if $\Psi\div\alpha\models\beta$). None
of the approaches described above coincides with the framework considered in
this paper.

##  5 Conclusion

We proposed a semantic characterization of AGM belief contraction in terms of
a semantics consisting of a Kripke belief relation $\mathcal{B}$ (with
associated modal operator $\mathbb{B}$) and a Stalnaker-Lewis selection
function $f$ (with associated conditional bimodal operator $>$). The proposed
semantics can also be used to characterize AGM belief revision (see [3]).
Indeed all three operations: belief expansion, belief contraction and belief
revision, can be captured within this framework. Letting $s_{@}$ denote the
actual state, we have:

  1. 1.

Expansion: $\psi\in K+\phi$ if and only if
$s_{@}\models\lnot\mathbb{B}\lnot\phi\wedge\mathbb{B}(\phi\rightarrow\psi)$,

  2. 2.

Contraction: $\psi\in K\div\phi$ if and only if
$s_{@}\models\mathbb{B}\psi\wedge\mathbb{B}(\lnot\phi>\psi)$,

  3. 3.

Revision: $\psi\in K*\phi$ if and only if $s_{@}\models\mathbb{B}(\phi>\psi)$.

There are several issues that can be studied within this framework and are
left for future work, for example, whether the extended modal language can
provide a way to generalize AGM-style belief change and whether the proposed
framework can accommodate iterated belief contraction/revision.

##  Appendix A Appendix

In this Appendix we prove Proposition 1. In order to make the proof entirely
self-contained we include the proofs of known auxiliary results (e.g. the
lemmas).444Which can be found, for example, in [12, 18].

######  Lemma 1.

Fix a model $M=\left\langle{S,s_{@},\mathcal{B},f,V}\right\rangle$ and let
$K=\\{\phi\in\Phi_{0}:\mathcal{B}(s_{@})\subseteq\|\phi\|\\}$. Then $K$ is
deductively closed and consistent.

###### Proof.

First we show that $K$ is deductively closed, that is, $K=Cn(K)$. If $\psi\in
K$ then $\psi\in Cn(K)$, because $\psi\rightarrow\psi$ is a tautology; thus
$K\subseteq Cn(K)$. To show that $Cn(K)\subseteq K$, let $\psi\in Cn(K)$, that
is, there exist $\phi_{1},...,\phi_{n}\in K$ ($n\geq 0$) such that
$(\phi_{1}\wedge...\wedge\phi_{n})\rightarrow\psi$ is a tautology. Since
$\|\phi_{1}\wedge...\wedge\phi_{n}\|=\|\phi_{1}\|\cap...\cap\|\phi_{n}\|$ and,
for all $i=1,...,n$, $\phi_{i}\in K$ (that is,
$\mathcal{B}(s_{@})\subseteq\|\phi_{i}\|$), it follows that
$\mathcal{B}(s_{@})\subseteq\|\phi_{1}\wedge...\wedge\phi_{n}\|$. Since
$(\phi_{1}\wedge...\wedge\phi_{n})\rightarrow\psi$ is a tautology,
$\|(\phi_{1}\wedge...\wedge\phi_{n})\rightarrow\psi\|=S$, that is,
$\|\phi_{1}\wedge...\wedge\phi_{n}\|\subseteq\|\psi\|.$ Thus
$\mathcal{B}(s_{@})\subseteq\|\psi\|$, that is, $\psi\in K.$ Next we show that
$Cn(K)\neq\Phi_{0}$, that is, $K$ is consistent. Let $p\in\texttt{At}$ be an
atomic formula. Then $\|p\wedge\lnot p\|=\varnothing$. By seriality of
$\mathcal{B}$, $\mathcal{B}(s_{@})\neq\varnothing$ so that
$\mathcal{B}(s_{@})\nsubseteq\|p\wedge\lnot p\|$, that is, $(p\wedge\lnot
p)\notin K$ and hence, since $K=Cn(K)$, $(p\wedge\lnot p)\notin Cn(K).$ ∎

Proof of Part (A) of Proposition 1.
Fix a model $\left\langle{S,s_{@},\mathcal{B},f,V}\right\rangle$ and let
$K=\\{\phi\in\Phi_{0}:\mathcal{B}(s_{@})\subseteq\|\phi\|\\}$ and, for every
$\phi\in\Phi_{0}$ such that $\|\lnot\phi\|\neq\varnothing$, let $K\div\phi$ be
defined as follows ((A1) below reproduces (1) above):

$\begin{array}[]{*{20}{l}}\psi\in K\div\phi\text{ if and only if
}&(1)\,\mathcal{B}(s_{@})\subseteq\|\psi\|,\text{ (that is, }\psi\in K)\text{
and }\\\ {}&(2)\,\forall
s\in\mathcal{B}(s_{@}),f\left(s,\|\lnot\phi\|\right)\subseteq\|\psi\|.\end{array}$
(A1)

Let ‘$\div^{\prime}$’ be following extension to $\Phi_{0}$ of the operator
‘$\div$’ defined in (A1):

$K\div^{\prime}\phi=\left\\{{\begin{array}[]{*{20}{l}}K\div\phi&\text{if
}\|\lnot\phi\|\neq\varnothing\\\ K\cap Cn(\lnot\phi)&\text{if
}\|\lnot\phi\|=\varnothing.\end{array}}\right.$ (A2)

We want to show that the contraction operator defined in (A2) satisfies the
AGM axioms.

### ($K-$1)

We need to show that, for every $\phi\in\Phi_{0}$,
$K\div^{\prime}\phi=Cn\left(K\div^{\prime}\phi\right)$. If
$\|\lnot\phi\|=\varnothing$ then this is true by construction, since $K$ is
deductively closed and the intersection of deductively closed sets is
deductively closed. Assume, therefore, that $\|\lnot\phi\|\neq\varnothing$, so
that $K\div^{\prime}\phi=K\div\phi$. Note first that, by (A1), letting

$\Psi_{\lnot\phi}=\left\\{\psi\in\Phi_{0}:f(s,\|\lnot\phi\|)\subseteq\|\psi\|,\forall
s\in\mathcal{B}(s_{@})\right\\},$ (A3)

$K\div\phi=K\cap\Psi_{\lnot\phi}$. Since the intersection of two deductively
closed sets is deductively closed and $K$ is deductively closed, it suffices
to show that $\Psi_{\lnot\phi}$ is deductively closed, that is,
$\Psi_{\lnot\phi}=Cn(\Psi_{\lnot\phi})$. The inclusion
$\Psi_{\lnot\phi}\subseteq Cn(\Psi_{\lnot\phi})$ follows from the fact that,
for every $\chi\in\Psi_{\lnot\phi}$, $\chi\rightarrow\chi$ is a tautology.
Next we show that $Cn(\Psi_{\lnot\phi})\subseteq\Psi_{\lnot\phi}$. Since
$\|\lnot\phi\|\neq\varnothing$, $f(s,\|\lnot\phi\|)$ is defined for every
$s\in\mathcal{B}(s_{@})$. Fix an arbitrary $\psi\in Cn(\Psi_{\lnot\phi})$;
then there exist $\phi_{1},...,\phi_{n}\in\Psi_{\lnot\phi}$ ($n\geq 0$) such
that $(\phi_{1}\wedge...\wedge\phi_{n})\rightarrow\psi$ is a tautology, so
that $\|(\phi_{1}\wedge...\wedge\phi_{n})\rightarrow\psi\|=S$, that is,
$\|\phi_{1}\wedge...\wedge\phi_{n}\|\subseteq\|\psi\|$. Fix an arbitrary
$s\in\mathcal{B}(s_{@})$ and an arbitrary $i=1,...,n$. Then, since
$\phi_{i}\in\Psi_{\lnot\phi}$, $f(s,\|\lnot\phi\|)\subseteq\|\phi_{i}\|$.
Hence $f(s,\|\lnot\phi\|)\subseteq\|\phi_{1}\wedge...\wedge\phi_{n}\|$. Since
$\|(\phi_{1}\wedge...\wedge\phi_{n})\|\subseteq\|\psi\|$ it follows that
$f(s,\|\lnot\phi\|)\subseteq\|\psi\|$, that is, $\psi\in\Psi_{\lnot\phi}$.

### ($K-$2)

We need to show that $K\div^{\prime}\phi\subseteq K$. If
$\|\lnot\phi\|=\varnothing$ then $K\div^{\prime}\phi=K\cap
Cn(\lnot\phi)\subseteq K$. If $\|\lnot\phi\|\neq\varnothing$ then
$K\div^{\prime}\phi=K\div\phi=K\cap\Psi_{\lnot\phi}\subseteq K$.

### ($K-$3)

We need to show that if $\phi\notin K$ then $K\subseteq K\div^{\prime}\phi$.
Assume that $\phi\notin K$, that is,
$\mathcal{B}(s_{@})\cap\|\lnot\phi\|\neq\varnothing$. Then
$\|\lnot\phi\|\neq\varnothing$ and thus $K\div^{\prime}\phi=K\div\phi$. Fix an
arbitrary $\psi\in K$, that is, $\mathcal{B}(s_{@})\subseteq\|\psi\|$. We need
to show that, $\forall s\in\mathcal{B}(s_{@})$,
$f(s,\|\lnot\phi\|)\subseteq\|\psi\|$. Since
$\mathcal{B}(s_{@})\cap\|\lnot\phi\|\neq\varnothing$, by 4(c) of Definition 2,
for every $s\in\mathcal{B}(s_{@})$,
$f(s,\|\lnot\phi\|)\subseteq\mathcal{B}(s_{@})\cap\|\lnot\phi\|$ and thus,
since $\mathcal{B}(s_{@})\subseteq\|\psi\|$,
$f(s,\|\lnot\phi\|)\subseteq\|\psi\|$.

### ($K-$4)

We need to show that if $\phi$ is not a tautology then $\phi\notin
K\div^{\prime}\phi$. Suppose that $\phi$ is not a tautology, so that
$\phi\notin Cn(\lnot\phi)$. If $\|\lnot\phi\|=\varnothing$ then
$K\div^{\prime}\phi=K\cap Cn(\lnot\phi)$ and thus $\phi\notin
K\div^{\prime}\phi$. Next, suppose that $\|\lnot\phi\|\neq\varnothing$ so that
$K\div^{\prime}\phi=K\div\phi$. Since $K\div\phi=K\cap\Psi_{\lnot\phi}$ (where
$\Psi_{\lnot\phi}$ is given by (A3)) it is sufficient to show that
$\phi\notin\Psi_{\lnot\phi}$, that is,
$f(s,\|\lnot\phi\|)\not\subseteq\|\phi\|$, for some $s\in\mathcal{B}(s_{@})$.
This follows from the fact that, by 4(a) of Definition 2, for every
$s\in\mathcal{B}(s_{@})$, $f(s,\|\lnot\phi\|)\subseteq\|\lnot\phi\|$.

### ($K-$5)

We need to show that if $\phi\in K$ then
$K\subseteq(K\div^{\prime}\phi)+\phi=Cn(K\div^{\prime}\phi\cup\\{\phi\\})$.
Assume that $\phi\in K$ and fix an arbitrary $\psi\in K$. Then
$(\phi\rightarrow\psi)\in K$. If $\|\lnot\phi\|=\varnothing$ then
$K\div^{\prime}\phi=K\cap Cn(\lnot\phi)$. Since $\lnot\phi\in Cn(\lnot\phi)$,
$\phi\rightarrow\psi\in Cn(\lnot\phi)$ and thus $\phi\rightarrow\psi\in
K\div^{\prime}\phi$, from which it follows (since, by ($K-1$),
$K\div^{\prime}\phi$ is deductively closed) that $\psi\in
Cn(K\div^{\prime}\phi\cup\\{\phi\\})$. Suppose now that
$\|\lnot\phi\|\neq\varnothing$ so that
$K\div^{\prime}\phi=K\div\phi=K\cap\Psi_{\lnot\phi}$ (where $\Psi_{\lnot\phi}$
is given by (A3)). By 4(a) of Definition 2, for every
$s\in\mathcal{B}(s_{@})$, $f(s,\|\lnot\phi\|)\subseteq\|\lnot\phi\|$ and thus
$f(s,\|\lnot\phi\|)\subseteq\|\phi\rightarrow\psi\|=\|\lnot\phi\|\cup\|\psi\|$.
Hence (recall that $(\phi\rightarrow\psi)\in K$) $(\phi\rightarrow\psi)\in
K\div\phi$ so that $\psi\in Cn(K\div\phi\cup\\{\phi\\})$.

### ($K-$6)

We need to show that if $\phi\leftrightarrow\psi$ is a tautology then
$K\div^{\prime}\phi=K\div^{\prime}\psi$. Assume that $\phi\leftrightarrow\psi$
is a tautology. Then $Cn(\lnot\phi)=Cn(\lnot\psi)$ and
$\|\lnot\phi\|=\|\lnot\psi\|$. Thus $\|\lnot\phi\|=\varnothing$ if and only if
$\|\lnot\psi\|=\varnothing$, in which case $K\div^{\prime}\phi=K\cap
Cn(\lnot\phi)=K\cap Cn(\lnot\psi)=K\div^{\prime}\psi$. Furthermore,
$\|\lnot\phi\|\neq\varnothing$ if and only if $\|\lnot\psi\|\neq\varnothing$,
in which case $\\{\chi\in\Phi_{0}:f(s,\|\lnot\phi\|)\subseteq\|\chi\|,\forall
s\in\mathcal{B}(s_{@})\\}=\\{\chi\in\Phi_{0}:f(s,\|\lnot\psi\|)\subseteq\|\chi\|,\forall
s\in\mathcal{B}(s_{@})\\}$, from which it follows that $K\div\phi=K\div\psi$.

### ($K-$7)

We have to show that $(K\div^{\prime}\phi)\cap(K\div^{\prime}\psi)\subseteq
K\div^{\prime}(\phi\wedge\psi)$. We need to consider several cases.
Case 1: $\|\lnot\phi\|=\|\lnot\psi\|=\varnothing$ so that
$\|\lnot\phi\|\cup\|\lnot\psi\|=\|\lnot\phi\vee\lnot\psi\|=\|\lnot(\phi\wedge\psi)\|=\varnothing$.
In this case $K\div^{\prime}\phi=K\cap Cn(\lnot\phi)$,
$K\div^{\prime}\psi=K\cap Cn(\lnot\psi)$ and
$K\div^{\prime}(\phi\wedge\psi)=K\cap Cn(\lnot(\phi\wedge\psi))$. Since
$Cn(\lnot\phi)\cap Cn(\lnot\psi)\subseteq
Cn(\lnot\phi\vee\lnot\psi)=Cn(\lnot(\phi\wedge\psi))$ it follows that
$(K\div^{\prime}\phi)\cap(K\div^{\prime}\psi)\subseteq
K\div^{\prime}(\phi\wedge\psi)$.
Case 2: $\|\lnot\phi\|=\varnothing$ and $\|\lnot\psi\|\neq\varnothing$, so
that
$\|\lnot(\phi\wedge\psi)\|=\|\lnot\phi\vee\lnot\psi\|=\|\lnot\phi\|\cup\|\lnot\psi\|=\|\lnot\psi\|\neq\varnothing$.
In this case $K\div^{\prime}\phi=K\cap Cn(\lnot\phi)$,
$K\div^{\prime}\psi=K\div\psi=K\cap\\{\chi\in\Phi_{0}:f(s,\|\lnot\psi\|)\subseteq\|\chi\|,\forall
s\in\mathcal{B}(s_{@})\\}$ and
$K\div^{\prime}(\phi\wedge\psi)=K\div(\phi\wedge\psi)=K\cap\\{\chi\in\Phi_{0}:f(s,\|\lnot(\phi\wedge\psi)\|)\subseteq\|\chi\|,\forall
s\in\mathcal{B}(s_{@})\\}$. Since $\|\lnot(\phi\wedge\psi)\|=\|\lnot\psi\|$,
$f(s,\|\lnot(\phi\wedge\psi)\|)=f(s,\|\lnot\psi\|)$ and thus
$K\div(\phi\wedge\psi)=K\div\psi$. Hence the inclusion
$(K\div^{\prime}\phi)\cap(K\div\psi)\subseteq K\div(\phi\wedge\psi)$ reduces
to $(K\div^{\prime}\phi)\cap(K\div\psi)\subseteq K\div\psi$, which is
trivially true.
Case 3: $\|\lnot\phi\|\neq\varnothing$ and $\|\lnot\psi\|=\varnothing$, so
that
$\|\lnot\phi\vee\lnot\psi\|=\|\lnot\phi\|\cup\|\lnot\psi\|=\|\lnot\phi\|\neq\varnothing$.
In this case, by an argument similar to the one used in Case 2,
$K\div^{\prime}(\phi\wedge\psi)=K\div(\phi\wedge\psi)=K\div\phi=K\div^{\prime}\phi$,
so that the inclusion $(K\div^{\prime}\phi)\cap(K\div^{\prime}\psi)\subseteq
K\div^{\prime}(\phi\wedge\psi)$ reduces to
$(K\div\phi)\cap(K\div^{\prime}\psi)\subseteq K\div\phi$, which is trivially
true.
Case 4: $\|\lnot\phi\|\neq\varnothing$ and $\|\lnot\psi\|\neq\varnothing$, so
that
$\|\lnot(\phi\wedge\psi)\|=\|\lnot\phi\vee\lnot\psi\|=\|\lnot\phi\|\cup\|\lnot\psi\|\neq\varnothing$.
In this case
$K\div^{\prime}\phi=K\div\phi=K\cap\\{\chi\in\Phi_{0}:f(s,\|\lnot\phi\|)\subseteq\|\chi\|,\forall
s\in\mathcal{B}(s_{@})\\}$,
$K\div^{\prime}\psi=K\div\psi=K\cap\\{\chi\in\Phi_{0}:f(s,\|\lnot\psi\|)\subseteq\|\chi\|,\forall
s\in\mathcal{B}(s_{@})\\}$ and
$K\div^{\prime}(\phi\wedge\psi)=K\div(\phi\wedge\psi)=K\cap\\{\chi\in\Phi_{0}:f(s,\|\lnot(\phi\wedge\psi)\|)\subseteq\|\chi\|,\forall
s\in\mathcal{B}(s_{@})\\}$. Fix an arbitrary
$\chi\in(K\div\phi)\cap(K\div\psi)$ (thus, in particular, $\chi\in K$). We
need to show that $\chi\in K\div(\phi\wedge\psi)$, that is, that, $\forall
s\in\mathcal{B}(s_{@})$, $f(s,\|\lnot(\phi\wedge\psi)\|)\subseteq\|\chi\|$.
Since $\chi\in(K\div\phi)\cap(K\div\psi)$,

$f(s,\|\lnot\phi\|)\subseteq\|\chi\|\text{ and
}f(s,\|\lnot\psi\|)\subseteq\|\chi\|.$ (A4)

By Property 4(a) of Definition 2,
$f(s,\|\lnot(\phi\wedge\psi)\|)\subseteq\|\lnot(\phi\wedge\psi)\|=\|\lnot\phi\|\cup\|\lnot\psi\|$.
It follows from this that

$f(s,\|\lnot(\phi\wedge\psi)\|)\,=\,\left(f(s,\|\lnot(\phi\wedge\psi)\|)\cap\|\lnot\phi\|\right)\,{{{\cup}}}\,\left(f(s,\|\lnot(\phi\wedge\psi)\|)\cap\|\lnot\psi\|\right).$
(A5)

By Property 4(d) of Definition 2 (with
$E=\|\lnot\phi\|\cup\|\lnot\psi\|=\|\lnot(\phi\wedge\psi)\|$ and
$F=\|\lnot\phi\|$)

$f(s,\|\lnot(\phi\wedge\psi)\|)\cap\|\lnot\phi\|\,\,{{\subseteq}}\,\,f(s,\|\lnot\phi\|).$
(A6)

A second application of Property 4(d) of Definition 2 (with
$E=\|\lnot\phi\|\cup\|\lnot\psi\|=\|\lnot(\phi\wedge\psi)\|$ and, this time,
with $F=\|\lnot\psi\|$) gives

$f(s,\|\lnot(\phi\wedge\psi)\|)\cap\|\lnot\psi\|\,\,{{\subseteq}}\,\,f(s,\|\lnot\psi\|).$
(A7)

It follows from (A5), (A6), (A7) that
$f(s,\|\lnot(\phi\wedge\psi)\|\subseteq(f(s,\|\lnot\phi\|)\cup
f(s,\|\lnot\psi\|))$ and thus, by (A4),
$f(s,\|\lnot(\phi\wedge\psi)\|)\subseteq\|\chi\|$.

### ($K-$8)

We need to show that if $\phi\notin K\div^{\prime}(\phi\wedge\psi)$ then
$K\div^{\prime}(\phi\wedge\psi)\subseteq K\div^{\prime}\phi$. Assume that
$\phi\notin K\div^{\prime}(\phi\wedge\psi)$.
Suppose first that $\|\lnot\phi\|=\varnothing$, that is, $\|\phi\|=S$. Then
$\mathcal{B}(s_{@})\subseteq\|\phi\|$ and thus $\phi\in K$. If
$\|\lnot(\phi\wedge\psi)\|=\|\lnot\phi\|\cup\|\lnot\psi\|\neq\varnothing$ then
$K\div^{\prime}(\phi\wedge\psi)=K\div(\phi\wedge\psi)=K\cap\\{\chi\in\Phi_{0}:f(s,\|\lnot\phi\|\cup\|\lnot\psi\|)\subseteq\|\chi\|,\forall
s\in\mathcal{B}(s_{@})\\}$ and, since $\|\phi\|=S$, for all
$s\in\mathcal{B}(s_{@})$ we have that
$f(s,\|\lnot\phi\|\cup\|\lnot\psi\|)\subseteq\|\phi\|$, implying that $\phi\in
K\div(\phi\wedge\psi)$, contradicting our assumption. Thus the case where
$\|\lnot\phi\|=\varnothing$ and
$\|\lnot\phi\|\cup\|\lnot\psi\|\neq\varnothing$ is ruled out and we are left
with only two cases to consider.
Case 1: $\|\lnot\phi\|\cup\|\lnot\psi\|=\varnothing$ so that
$\|\lnot\phi\|=\varnothing$. In this case
$K\div^{\prime}(\phi\wedge\psi)=K\cap Cn(\lnot(\phi\wedge\psi))$ and
$K\div^{\prime}\phi=K\cap Cn(\lnot\phi)$. Fix an arbitrary $\chi\in
K\div^{\prime}(\phi\wedge\psi)$. Then $\chi\in K$ and $\chi\in
Cn(\lnot(\phi\wedge\psi))$. We need to show that $\chi\in K\div^{\prime}\phi$,
that is, that $\chi\in Cn(\lnot\phi)$. Since $\chi\in
Cn(\lnot(\phi\wedge\psi))$, $\lnot(\phi\wedge\psi)\rightarrow\chi$ is a
tautology. Thus, since $\lnot\phi\rightarrow\lnot(\phi\wedge\psi)$ is also a
tautology, $\lnot\phi\rightarrow\chi$ is a tautology and thus $\chi\in
Cn(\lnot\phi)$.
Case 2: $\|\lnot\phi\|\neq\varnothing$ and thus
$\|\lnot(\phi\wedge\psi)\|=\|\lnot\phi\|\cup\|\lnot\psi\|\neq\varnothing$.
Then
$K\div^{\prime}(\phi\wedge\psi)=K\div(\phi\wedge\psi)=K\cap\\{\chi\in\Phi_{0}:f(s,\|\lnot\phi\|\cup\|\lnot\psi\|)\subseteq\|\chi\|,\forall
s\in\mathcal{B}(s_{@})\\}$ and
$K\div^{\prime}\phi=K\div\phi=K\cap\\{\chi\in\Phi_{0}:f(s,\|\lnot\phi\|)\subseteq\|\chi\|,\forall
s\in\mathcal{B}(s_{@})\\}$. Recall the assumption that $\phi\notin
K\div(\phi\wedge\psi)$. Then two sub-cases are possible.
Case 2.1: $\phi\notin K$, that is,
$\mathcal{B}(s_{@})\cap\|\lnot\phi\|\neq\varnothing$. Then, by 4(c) of
Definition 2,

$\forall
s\in\mathcal{B}(s_{@}),\,f(s,\|\lnot\phi\|)\subseteq\mathcal{B}(s_{@})\cap\|\lnot\phi\|\subseteq\mathcal{B}(s_{@}).$
(A8)

Fix an arbitrary $\chi\in K\div(\phi\wedge\psi)$. Then $\chi\in K$, that is,
$\mathcal{B}(s_{@})\subseteq\|\chi\|$ and thus, by (A8), $\forall
s\in\mathcal{B}(s_{@})$, $f(s,\|\lnot\phi\|)\subseteq\|\chi\|$ so that
$\chi\in K\div\phi$.
Case 2.2: $\phi\in K$ and $B_{\lnot\phi\lnot\psi}\neq\varnothing$, where
$B_{\lnot\phi\lnot\psi}=\\{s\in\mathcal{B}(s_{@}):f(s,\|\lnot\phi\|\cup\|\lnot\psi\|)\cap\|\lnot\phi\|\neq\varnothing\\}$.555Note
that the case where $\phi\in K$ and $B_{\lnot\phi\lnot\psi}=\varnothing$ is
ruled out by our initial assumption that $\phi\notin K\div(\phi\wedge\psi)$.
In fact, $B_{\lnot\phi\lnot\psi}=\varnothing$ means that, $\forall
s\in\mathcal{B}(s_{@}),f(s,\|\lnot\phi\|\cup\|\lnot\psi\|)\cap\|\lnot\phi\|=\varnothing$,
that is, $f(s,\|\lnot\phi\|\cup\|\lnot\psi\|)\subseteq\|\phi\|$, which, in
conjunction with the hypothesis that $\phi\in K$, yields $\phi\in
K\div(\phi\wedge\psi)$.  Then, by 4(e.1) of Definition 2 (with
$E=\|\lnot\phi\|\cup\|\lnot\psi\|$ and $F=\|\lnot\phi\|$)

$\forall s\in\mathcal{B}_{\lnot\phi\lnot\psi},f(s,\|\lnot\phi\|)\subseteq
f(s,\|\lnot\phi\|\cup\|\lnot\psi\|)\,\cap\,\|\lnot\phi\|$ (A9)

and, by 4(e.2) of Definition 2 (again, with $E=\|\lnot\phi\|\cup\|\lnot\psi\|$
and $F=\|\lnot\phi\|$),

$\forall s\in\mathcal{B}(s_{@}),f(s,\|\lnot\phi\|)\subseteq
f(s^{\prime},\|\lnot\phi\|)\text{ for some }s^{\prime}\in
B_{\lnot\phi\lnot\psi}.$ (A10)

Fix an arbitrary $\chi\in K\div(\phi\wedge\psi)$. Then, $\chi\in K$ and
(recall that $\|\lnot(\phi\wedge\psi)\|=\|\lnot\phi\|\cup\|\lnot\psi\|$)
$f(s,\|\lnot\phi\|\cup\|\lnot\psi\|)\subseteq\|\chi\|$, $\forall
s\in\mathcal{B}(s_{@})$; it follows from this, (A9) and (A10) that, $\forall
s\in\mathcal{B}(s_{@})$, $f(s,\|\lnot\phi\|)\subseteq\|\chi\|$. Thus $\chi\in
K\div\phi$.

Before we proceed to the proof of Part (B) of Proposition 1, we establish the
following lemma.

######  Lemma 2.

Let $A\subseteq\Phi_{0}$ be such that $A=Cn(A)$. Then,
$\forall\alpha\in\Phi_{0}$,
$\|Cn\left(A\cup\\{\alpha\\}\right)\|=\|A\|\cap\|\alpha\|.$

###### Proof.

Since $A$ is deductively closed, $\forall\beta\in\Phi_{0}$,

$\beta\in Cn\left(A\cup\\{\alpha\\}\right)\text{ if and only if
}(\alpha\rightarrow\beta)\in A.$ (A12)

First we show that
$\|A\|\cap\|\alpha\|\subseteq\|Cn\left(A\cup\\{\alpha\\}\right)\|.$ Fix an
arbitrary $s\in\|A\|\cap\|\alpha\|$; we need to show that
$s\in\|Cn\left(A\cup\\{\alpha\\}\right)\|$, that is, that $\forall\beta\in
Cn\left(A\cup\\{\alpha\\}\right),\,\beta\in s$. Since $s\in\|\alpha\|$,
$\alpha\in s$. Fix an arbitrary $\beta\in Cn\left(A\cup\\{\alpha\\}\right)$;
then, by (A12), $(\alpha\rightarrow\beta)\in A$; thus, since $s\in\|A\|$,
$(\alpha\rightarrow\beta)\in s$. Hence, since both $\alpha$ and
$\alpha\rightarrow\beta$ belong to $s$ and $s$ is deductively closed,
$\beta\in s$.
Next we show that
$\|Cn\left(A\cup\\{\alpha\\}\right)\|\subseteq\|A\|\cap\|\alpha\|.$ Let
$s\in\|Cn\left(A\cup\\{\alpha\\}\right)\|$. Then, since $\alpha\in
Cn\left(A\cup\\{\alpha\\}\right)$, $\alpha\in s$, that is, $s\in\|\alpha\|$.
It remains to show that $s\in\|A\|$, that is, that, for every $\beta\in A$,
$\beta\in s$. Fix an arbitrary $\beta\in A$; then, since $A$ is deductively
closed, $(\alpha\rightarrow\beta)\in A$. Thus, by (A12), $\beta\in
Cn\left(A\cup\\{\alpha\\}\right)$ and thus, since
$s\in\|Cn\left(A\cup\\{\alpha\\}\right)\|$, $\beta\in s$. ∎

Proof of Part (B) of Proposition 1.
We need to show that if $K\subset\Phi_{0}$ is consistent and deductively
closed and $\div:\Phi_{0}\rightarrow 2^{\Phi_{0}}$ is an AGM belief
contraction function based on $K$, then there exists a model
$\left\langle{S,s_{@},\mathcal{B},f,V}\right\rangle$ such that
$K=\\{\phi\in\Phi_{0}:\mathcal{B}(s_{@})\subseteq\|\phi\|\\}$ and, for all
$\phi,\psi\in\Phi_{0}$, $\psi\in K\div\phi$ if and only if (A1) is satisfied.
Define the following model
$\left\langle{S,s_{@},\mathcal{B},f,V}\right\rangle$:

  1. 1.

$S$ is the set of maximally consistent sets of formulas in $\Phi_{0}$.

  2. 2.

The valuation $V:\textit{At}\rightarrow S$ is defined by $V(p)=\\{s\in S:p\in
s\\}$, so that, for every $\phi\in\Phi_{0}$, $\|\phi\|=\\{s\in S:\phi\in
s\\}$. If $\Psi\subseteq\Phi_{0}$, define $\|\Psi\|=\\{s\in
S:\forall\phi\in\Psi,\phi\in s\\}$.

  3. 3.

Choose an arbitrary $s_{@}\in S$ and define $\mathcal{B}(s_{@})=\|K\|$.

  4. 4.

Let $\mathcal{E}=\\{E\subseteq S:\varnothing\neq E=\|\phi\|\text{ for some
}\phi\in\Phi_{0}\\}$. Define $f:\mathcal{B}(s_{@})\times\mathcal{E}\rightarrow
2^{S}$ as follows:

$\forall
s\in\mathcal{B}(s_{@}),\,\,f(s,\|\phi\|)=\|K\div\lnot\phi\|\,\cap\,\|\phi\|.$
(A11)

######  Remark 1.

If $\phi$ is a tautology then $\lnot\phi$ is a contradiction and thus (since,
by hypothesis, $K$ is consistent) $\lnot\phi\notin K$. It follows from ($K-2$)
and ($K-3$) that $K\div\lnot\phi=K$. Furthermore, since $\phi$ is a tautology
and $K$ is deductively closed, $\phi\in K$, that is $\|K\|\subseteq\|\phi\|$
so that $\|K\|\cap\|\phi\|=\|K\|$. Hence, by (A11), $\forall
s\in\mathcal{B}(s_{@}),\,f(s,\|\phi\|)=\|K\|$. On the other hand, if
$\lnot\phi$ is a tautology then $\|\phi\|=\varnothing$ and thus
$\|\phi\|\notin\mathcal{E}$, that is, $\|\phi\|$ is not in the domain of $f$.

First we show that the selection function defined in (A11) satisfies
Properties 4(a)-4(e) of Definition 2. In view of Remark 1, we can restrict
attention to contingent formulas, that is, to formulas $\phi$ such that
neither $\phi$ nor $\lnot\phi$ is a tautology. Denote by
$\Phi_{cont}\subseteq\Phi_{0}$ the set of contingent formulas.
Recall that $S$ is the set of maximally consistent sets of formulas in
$\Phi_{0}$ and, for $A\subseteq\Phi_{0}$, $\|A\|=\\{s\in S:\chi\in
s,\,\forall\chi\in A\\}$.

### Property 4(a)

We need to show that if $\phi\in\Phi_{cont}$ then
$\|K\div\lnot\phi\|\,\cap\,\|\phi\|\subseteq\|\phi\|$, which is obviously
true, and $\|K\div\lnot\phi\|\,\cap\,\|\phi\|\neq\varnothing$. Since
$\phi\in\Phi_{cont}$, $\|\phi\|\neq\varnothing$ and, by ($K-4$),
$\lnot\phi\notin K\div\lnot\phi$. By ($K-1$)
$K\div\lnot\phi=Cn(K\div\lnot\phi)$ and thus $\lnot\phi\notin
Cn(K\div\lnot\phi)$, that is, $K\div\lnot\phi$ is consistent and hence
$\|K\div\lnot\phi\|\neq\varnothing$.

### Property 4(b)

Fix an arbitrary $s\in\mathcal{B}(s_{@})$ and an arbitrary
$\phi\in\Phi_{cont}$. We need to show that if $s\in\|\phi\|$ then $s\in
f(s,\|\phi\|)=\|K\div\lnot\phi\|\cap\|\phi\|$. By construction,
$\mathcal{B}(s_{@})=\|K\|$; thus, $s\in\|K\|$. By ($K-2$),
$K\div\lnot\phi\subseteq K$ so that $\|K\|\subseteq\|K\div\lnot\phi\|$. Hence
$s\in\|K\div\lnot\phi\|$. Thus if $s\in\|\phi\|$ then
$s\in\|K\div\lnot\phi\|\cap\|\phi\|$.

### Property 4(c)

We need to show that if $\mathcal{B}(s_{@})\cap\|\phi\|\neq\varnothing$ then
(since $\mathcal{B}(s_{@})=\|K\|$ and, $\forall s\in\mathcal{B}(s_{@})$,
$f(s,\|\phi\|)=\|K\div\lnot\phi\|\,\cap\,\|\phi\|$)
$\|K\div\lnot\phi\|\,\cap\,\|\phi\|\subseteq\|K\|\cap\|\phi\|$. If
$\|K\|\cap\|\phi\|\neq\varnothing$ then $\lnot\phi\notin K$ and thus, by
($K-3$), $K\subseteq K\div\lnot\phi$, so that
$\|K\div\lnot\phi\|\subseteq\|K\|$ and thus
$\|K\div\lnot\phi\|\,\cap\,\|\phi\|\subseteq\|K\|\cap\|\phi\|$.

### Property 4(d)

We need to show that if $\phi\in\Phi_{cont}$ and $\psi\in\Phi_{0}$, then
$\forall s\in\mathcal{B}(s_{@})$, $f(s,\|\phi\|)\,\cap\,\|\psi\|\subseteq
f(s,\|\phi\|\cap\|\psi\|)$, that is, using (A11) and the fact that
$\|\phi\|\cap\|\psi\|=\|\phi\wedge\psi\|$,

$\|K\div\lnot\phi\|\,\cap\,\|\phi\|\,\cap\,\|\psi\|\,\subseteq\,\|K\div\lnot(\phi\wedge\psi)\|\,\cap\,\|\phi\wedge\psi\|$
(A13)

By ($K-7$),
$\forall\alpha,\beta\in\Phi_{0},\,(K\div\alpha)\cap(K\div\beta)\subseteq
K\div(\alpha\wedge\beta)$. Thus applying ($K-7$) to
$\alpha=\lnot(\phi\wedge\psi)$ and $\beta=\phi\rightarrow\psi$ we get

$K\div\lnot(\phi\wedge\psi)\,\cap\,K\div(\phi\rightarrow\psi)\,\subseteq\,K\div\left(\lnot(\phi\wedge\psi)\wedge(\phi\rightarrow\psi)\right)$
(A14)

Since $\lnot(\phi\wedge\psi)\wedge(\phi\rightarrow\psi)$ is logically
equivalent to $\lnot\phi$, by ($K-6$)
$K\div\left(\lnot(\phi\wedge\psi)\wedge(\phi\rightarrow\psi)\right)=K\div\lnot\phi$.
Thus, by (A14)

$K\div\lnot(\phi\wedge\psi)\,\cap\,K\div(\phi\rightarrow\psi)\,\subseteq\,K\div\lnot\phi.$
(A15)

Next we show that

$Cn\left(K\div\lnot(\phi\wedge\psi)\,\cup\,\\{\phi\wedge\psi\\}\right)\,\subseteq\,Cn\left(K\div\lnot\phi\,\cup\,\\{\phi\wedge\psi\\}\right).$
(A16)

Fix an arbitrary $\chi\in
Cn\left(K\div\lnot(\phi\wedge\psi)\,\cup\,\\{\phi\wedge\psi\\}\right)$. Then,
since, by ($K-1$), $K\div\lnot(\phi\wedge\psi)$ is deductively closed,

$\left((\phi\wedge\psi)\rightarrow\chi\right)\in K\div\lnot(\phi\wedge\psi).$
(A17)

By ($K-2$), $K\div\lnot(\phi\wedge\psi)\subseteq K$ and thus, by (A17),

$\left((\phi\wedge\psi)\rightarrow\chi\right)\in K.$ (A18)

Next we show that

$\left((\phi\wedge\psi)\rightarrow\chi\right)\in K\div(\phi\rightarrow\psi).$
(A19)

If $(\phi\rightarrow\psi)\notin K$ then, by ($K-3)$, $K\subseteq
K\div(\phi\rightarrow\psi)$ and thus (A19) follows from (A18). If
$(\phi\rightarrow\psi)\in K$ then, by ($K-5)$, $K\subseteq
Cn(K\div(\phi\rightarrow\psi)\cup\\{\phi\rightarrow\psi\\})$ so that, by
(A18), $\left((\phi\wedge\psi)\rightarrow\chi\right)\in
Cn\left(K\div(\phi\rightarrow\psi)\cup\\{\phi\rightarrow\psi\\}\right)$, that
is (since, by ($K-1$), $K\div(\phi\rightarrow\psi)$ s deductively closed)
$(\phi\rightarrow\psi)\rightarrow\left((\phi\wedge\psi)\rightarrow\chi\right)\in
K\div(\phi\rightarrow\psi)$. Since
$(\phi\rightarrow\psi)\rightarrow\left((\phi\wedge\psi)\rightarrow\chi\right)$
is logically equivalent to
$\left((\phi\rightarrow\psi)\wedge(\phi\wedge\psi)\right)\rightarrow\chi$,
which, in turn is logically equivalent to $(\phi\wedge\psi))\rightarrow\chi$,
(A19) is satisfied. It follows from (A18), (A19) and (A15) that
$\big{(}(\phi\wedge\psi)\rightarrow\chi\big{)}\in K\div\lnot\phi$, that is,
that $\chi\in Cn\big{(}K\div\lnot\phi\,\cup\,\\{\phi\wedge\psi\\}\big{)}$,
thus establishing (A16). From (A16) we get that

$\|Cn\left(K\div\lnot\phi\,\cup\,\\{\phi\wedge\psi\\}\right)\|\,\subseteq\,\|Cn\left(K\div\lnot(\phi\wedge\psi)\,\cup\,\\{\phi\wedge\psi\\}\right)\|$
(A20)

By Lemma 2 (with $A=K\div\lnot\phi$ and $\alpha=\phi\wedge\psi$),
$\|Cn\left(K\div\lnot\phi\,\cup\,\\{\phi\wedge\psi\\}\right)\|$
$=\|K\div\lnot\phi\|\,\cap\,\|\phi\wedge\psi\|$ which in turn (since
=$\|\phi\wedge\psi\|=\|\phi\|\cap\|\psi\|$) is equal to
$\|K\div\lnot\phi\|\cap\|\phi\|\cap\|\psi\|$. By Lemma 2 again (with
$A=K\div\lnot(\phi\wedge\psi)$ and $\alpha=\phi\wedge\psi$),
$\|Cn(K\div\lnot(\phi\wedge\psi)\,\cup\,\\{\phi\wedge\psi\\})\|=\|K\div\lnot(\phi\wedge\psi)\|\,\cap\,\|\psi\wedge\psi\|$.
Hence (A13) follows from (A20).

### Property 4(e)

Since, by (A11), $\forall s,s^{\prime}\in\mathcal{B}(s_{@})$,
$f(s,\|\phi\|)=f(s^{\prime},\|\phi\|)=\|K\div\lnot\phi\|\,\cap\,\|\phi\|$, it
is sufficient to show that if
$\|K\div\lnot\phi\|\,\cap\,\|\phi\|\,\cap\,\|\psi\|\neq\varnothing$ then
$\|K\div\lnot(\phi\wedge\psi)\|\,\cap\,\|\phi\wedge\psi\|\,\subseteq\,\|K\div\lnot\phi\|\,\cap\,\|\phi\|\,\cap\,\|\psi\|$.
Assume that
$\|K\div\lnot\phi\|\,\cap\,\|\phi\|\,\cap\,\|\psi\|=\|K\div\lnot\phi\|\,\cap\,\|\phi\wedge\psi\|\neq\varnothing$.
Then

$\lnot(\phi\wedge\psi)\notin\,K\div\lnot\phi.$ (A21)

Since $\lnot\phi$ is logically equivalent to
$\lnot(\phi\wedge\psi)\wedge\lnot\phi$, by ($K-6$)

$K\div\lnot\phi=K\div\left(\lnot(\phi\wedge\psi)\wedge\lnot\phi\right).$ (A22)

Thus, by (A21) and (A22),

$\lnot(\phi\wedge\psi)\notin\,K\div\left(\lnot(\phi\wedge\psi)\wedge\lnot\phi\right).$
(A23)

By ($K-8$), $\forall\alpha,\beta\in\Phi_{0}$, if $\alpha\notin
K\div(\alpha\wedge\beta)$ then $K\div(\alpha\wedge\beta)\subseteq
K\div\alpha$. Thus, by (A23) and ($K-8$) (with $\alpha=\lnot(\phi\wedge\psi)$
and $\beta=\lnot\phi$),
$K\div\left(\lnot\phi\wedge\lnot(\phi\wedge\psi)\right)\subseteq
K\div\lnot(\phi\wedge\psi)$. It follows from this and (A22) that
$K\div\lnot\phi\subseteq K\div\lnot(\phi\wedge\psi)$ and thus

$\|K\div\lnot(\phi\wedge\psi)\|\subseteq\|K\div\lnot\phi\|.$ (A24)

Intersecting both sides of (A24) with
$\|\phi\wedge\psi\|=\|\phi\|\cap\|\psi\|$ we get
$\|K\div\lnot(\phi\wedge\psi)\|\cap\|\phi\wedge\psi\|\,\subseteq\,\|K\div\lnot\phi\|\cap\|\phi\|\cap\|\psi\|$,
as desired.

To complete the proof of Part (B) of Proposition 1 we need to show that

$\begin{array}[]{*{20}{l}}\psi\in K\div\phi\text{ if and only if
}&(1)\,\mathcal{B}(s_{@})\subseteq\|\psi\|,\text{ and }\\\ {}&(2)\,\forall
s\in\mathcal{B}(s_{@}),f\left(s,\|\lnot\phi\|\right)\subseteq\|\psi\|.\end{array}$

By (A11), $\forall s\in\mathcal{B}(s_{@})=\|K\|$,
$f\left(s,\|\lnot\phi\|\right)=\|K\div\phi\|\cap\|\lnot\phi\|$. Thus we have
to show that

$\psi\in K\div\phi\text{ if and only if }\|K\|\subseteq\|\psi\|\text{ and
}\|K\div\phi\|\cap\|\lnot\phi\|\subseteq\|\psi\|.$ (A25)

First we establish a lemma.

######  Lemma 3.

$\forall\phi\in\Phi_{0}$,

  1. ($i$)

if $A\subseteq\Phi_{0}$ is such that $A=Cn(A)$, then
$A=Cn\left(A\cup\\{\phi\\}\right)\,\cap\,Cn(A\cup\\{\lnot\phi\\})$

  2. ($ii$)

$K\div\phi=K\cap Cn(K\div\phi\cup\\{\lnot\phi\\})$

###### Proof.

($i$) Let $A\subseteq\Phi_{0}$ be such that $A=Cn(A)$. Since $A\subseteq
Cn\left(A\cup\\{\phi\\}\right)$ and $A\subseteq
Cn\left(A\cup\\{\lnot\phi\\}\right)$, $A\subseteq
Cn\left(A\cup\\{\phi\\}\right)\,\cap\,Cn\left(A\cup\\{\lnot\phi\\}\right)$.
Conversely, suppose that $\chi\in
Cn\left(A\cup\\{\phi\\}\right)\,\cap\,Cn\left(A\cup\\{\lnot\phi\\}\right)$.
Then both $\phi\rightarrow\chi$ and $\lnot\phi\rightarrow\chi$ belong to $A$
and thus so does their conjunction. Since
$(\phi\rightarrow\chi)\wedge(\lnot\phi\rightarrow\chi)$ is logically
equivalent to $\chi$ it follows that $\chi\in A$.
($ii$) We need to consider two cases.
Case 1: $\phi\in K$. Then, by ($K-5$), $K\subseteq
Cn\left(K\div\phi\cup\\{\phi\\}\right)$. By ($K-2$), $K\div\phi\subseteq K$,
so that $Cn\big{(}K\div\phi\cup\\{\phi\\}\big{)}\subseteq
Cn\big{(}K\cup\\{\phi\\}\big{)}=Cn(K)=K$ (by hypothesis, $K$ is deductively
closed). Thus

$K=Cn\left(K\div\phi\cup\\{\phi\\}\right)$ (A26)

By Part ($i$) (with $A=K\div\phi$, which, by ($K-1$), is deductively closed),

$K\div\phi=Cn\left(K\div\phi\cup\\{\phi\\}\right)\,\cap\,Cn\left(K\div\phi\cup\\{\lnot\phi\\}\right)$
(A27)

Thus, by (A26) and (A27), $K\div\phi=K\cap Cn(K\div\phi\cup\\{\lnot\phi\\})$.
Case 2: $\phi\notin K$. Then, by ($K-2$) and ($K-3$),

$K\div\phi=K$ (A28)

By Part ($i$) (with $A=K$)

$K=Cn\left(K\cup\\{\phi\\}\right)\cap Cn\left(K\cup\\{\lnot\phi\\}\right)$
(A29)

From (A29) we get that $K\cap
Cn\left(K\cup\\{\lnot\phi\\}\right)=Cn\left(K\cup\\{\phi\\}\right)\cap
Cn\left(K\cup\\{\lnot\phi\\}\right)=K$. Thus, by (A28), $K\div\phi=K\cap
Cn\left(K\cup\\{\lnot\phi\\}\right)$, from which, by using (A28) again to
replace the second instance of $K$ with $K\div\phi$, we get $K\div\phi=K\cap
Cn\left(K\div\phi\cup\\{\lnot\phi\\}\right)$ ∎

Now we are ready to prove (A25), namely that

$\psi\in K\div\phi\text{ if and only if }\|K\|\subseteq\|\psi\|,\text{ and
}\|Cn\left(K\div\phi\cup\\{\lnot\phi\\}\right)\|\subseteq\|\psi\|.$

Let $\psi\in K\div\phi$. By ($ii$) of Lemma 3, $K\div\phi=K\cap
Cn\left(K\div\phi\cup\\{\lnot\phi\\}\right)$; thus $\psi\in K$, that is,
$\|K\|\subseteq\|\psi\|$, and $\psi\in
Cn\left(K\div\phi\cup\\{\lnot\phi\\}\right)$, that is,
$\|Cn(K\cup\\{\lnot\phi\\})\|\subseteq\|\psi\|$. Conversely, suppose that
$\|K\|\subseteq\|\psi\|$ and
$\|Cn\left(K\div\phi\cup\\{\lnot\phi\\}\right)\|\subseteq\|\psi\|$, that is,
$\psi\in K\,\cap\,Cn\left(K\div\phi\cup\\{\lnot\phi\\}\right)$. Then, by
($ii$) of Lemma 3, $\psi\in K\div\phi$. $\square$

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