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text:
Marking King Lincoln's Birthday
Posted by thetimman in Uncategorized
Feckless, soy-boy tyrant Justin Trudeau may be Fidel Castro's biological bastard child, but he is also the political bastard grandchild of the destroyer of our Republic, Abraham Lincoln. Today is his birthday, so I thought a few words to honor his historical importance were in honor. From Ilana Mercer:
The "pseudo-intellectuals who [are] devoted to pulling the wool over the public's eyes" have a lot to answer for. Lincoln's violent, unconstitutional revolution took the lives of 620,000 individuals (including 50,000 Southern civilians, blacks included), maimed thousands, and brought about "the near destruction of 40 percent of the nation's economy." "The costs of an action cannot be dismissed as irrelevant to morality," wrote the Mises Institute's David Gordon in "Secession, State & Liberty." Almost every other country at the time chose the path of peaceful emancipation. Yet today's Americans look upon the terrible forces Lincoln unleashed as glorious events, the native appetite having habituated to carnage over time.
Lincoln lovers like to claim that the Constitution ratified in 1788 forbade peaceful secession, and authorized a federal government of so-called limited, delegated powers to invade and occupy any seceding state; declare martial law, subdue secessionists by force, burn and ransack entire cities, and then establish a military dictatorship over those states lasting a dozen years.
Suppose this indeed is the case, and that it was perfectly constitutional to intentionally wage war on civilians, to imprison without trial thousands of Northern citizens, jail—even execute—people who refused to take an oath of loyalty to Lord Lincoln, declare martial law, confiscate private property, censor telegraph lines, and shut down newspapers for opposing the war, incarcerating their editors and owners. Say, for the sake of argument, that it was indeed lawful to suspend the Bill of Rights, the writ of habeas corpus, and the international law.
If it endorsed, or even accommodated, what Lincoln did, including his disregard for the Ninth and 10th amendments, and his violation of the Second — then the Constitution is categorically evil and self-contradictory.
The more plausible explanation is that, in 1861, Lincoln kidnapped and killed the Constitution. The Jacobins who lionize Lincoln's actions (by referring to his billowing prose) have been covering up his crimes and ignoring the consequences of his coup ever since.
3 thoughts on "Marking King Lincoln's Birthday"
Frank @TxTradCatholic said:
TM, y'all need to move down here to the Republic of Texas. A lot of us still think secession is perfectly Constitutional, and more and more people are raising it as a serious option. Plus the weather's better than dear old St. Louis.
thetimman said:
Understood, but I like snow too much. Missouri is a pretty good and free state in its own right. But if necessary when it goes down, Texas will be quite fine. If they'll have me…
I was never able to like the snow after the first day. 😆

redpajama_set_name: RedPajamaCommonCrawl


text:
The Supreme Court Weighing in on Frontloading?
No, this isn't happening (...yet), but I was asked about the possibility of the Supreme Court allowing the parties' wishes to take precedence over the actions of state governments setting a presidential primary date not in accordance with party rules. [This question arose over at DemConWatch from a discussion over my 2012 Primary Calendar Projection.]
The parties do have the right to set the rules of their nominations. Last year when the Clinton camp was up in arms over the "at-large" casino caucuses in Nevada -- the ones they thought would give Obama a distinct and unfair advantage -- the courts refused to hear the case, falling back on the precedent set in other cases that the party dictates who its nominee will be.
And this has applications in other areas as well. The main conflict has been over the rules concerning how open a primary is. On the question of who can participate in a party's nomination process, the Supreme Court has said that the parties get to decide. In other words, if there is a conflict between state law and the party's wishes, the party gets its way.
This has worked in both directions: parties not wanting closed primaries and parties not wanting open primaries. The case that started all of this was the Tashjian case. Basically, a thirty year old Connecticut law requiring closed primaries came under fire when the state's Republican Party wanted to open their process up to independents as well. The result: The party's right to free association was upheld and independents were allowed to participate in the primary.
On the flip side, California Democratic Party v. Jones established that if a party didn't accept a state-mandated blanket primary as the primary modus operandi, the requirement had to be scrapped. And as the two major parties (and others) didn't accept the California blanket primary, it hit the road.
So what does all this have to do with primary timing? [Yeah, I go on a bit, don't I?] Well, you'd think that if a state law requiring a state to hold its primary on a date the parties didn't like, the parties would get their way. It seems a reasonable extension of the legal doctrine established above, right?
If the DNC says, "Massachusetts, you're going on the third week in May," and the Bay state government doesn't like it, then too bad. Well, what about the RNC? Let's say the GOP says that Massachusetts is fine where it is during the first week in February. Fine, the GOP can go in February and the Democrats can go in May.
Here's the rub, though, and this is where the timing conflict breaks with the opened/closed issue. The state picks up the tab for conducting the primary election. Splitting the two parties up like that doubles the cost (at least theoretically). That places an undue financial burden on the state all of a sudden.
There are a couple of questions that emerge here:
1) What about a state like Montana, where in 2008, the GOP went on February 5 and the Democrats held their delegate selection on June 3? Well, in that case the Montana GOP voluntarily opted out of the state funding to fund its own caucus. Nebraska did the same thing on the Democratic side. And Idaho Democrats have opted out of the state-funded primary at the end of May for years.
2) This one is more important. What if the GOP went along with the plan. Take the Massachusetts example. Let's say that both the DNC and RNC agree that the Massachusetts primary should be in May. Well, now there's a case; one the parties can win because the financial burden is now gone. Massachusetts would likely be required by the courts to shift its primary back a few months since that's what both parties wanted.
But for the Supreme Court to weigh in on the side of the parties on this issue, it would likely require a coordinated calendar assembled in a bipartisan fashion. There are certainly efforts being made on this front, but those competing interests -- that zero sum game where one tiny shift could fundamentally shift the balance in an election toward one party -- will stand in the way of that vision coming to fruition.
Like a Kid in a Candy Store: A 2012 GOP Presidential Preference Poll
2012 Primary Reform: Previous General Election Margin as a Means of Setting the Calendar
Labels: court challenges, frontloading
Like a Kid in a Candy Store: A 2012 GOP Presidenti...
2012 Primary Reform: Previous General Election Mar...
If You Were Indiana, What Would You Do in 2012? A ...
Indiana and 2012
Democratic Ulterior Motives for 2012 and Frontloading
Florida in 2012: Primary on the Move?
Beebe's Signature Makes It Official: Arkansas Back...
North Carolina Bill to Move 2012 Primary to February
More on the Potential August Arkansas Primary
Arkansas Senate Unanimously Passes Primary Bill
Arkansas Senate Slated to Vote on 2012 Presidentia...
Chairman Steele and the 2012 Presidential Primary ...


text:
Michel Alibo (* 14. April 1959 in Martinique) ist ein französischer Weltmusik- und Jazz-Bassist.

Leben und Wirken
Michel Alibo wanderte als Kind nach Paris aus und brachte sich mit 14 Jahren selbst das Bassspielen bei. Mit 16 Jahren brach er die Schule ab und ging mit dem afrikanischen Saxofonisten Manu Dibango nach Kamerun auf Tournee. Er ist Gründungsmitglied der Fusionband Sixun, mit welcher er in mehr als 25 Jahren zahllose Konzerte gab und zwölf Alben aufnahm. Er spielte unter anderem mit Salif Keita, Michel Jonasz, Angélique Kidjo, Kassav, Touré Kunda, Youssou N'Dour, Karim Ziad, Eddy Louiss, Bojan Zulfikarpašić, Nguyên Lê, Trilok Gurtu, Vince Mendoza, Meddy Gerville und Andy Narell.

Stil/Musikalische Einflüsse
Michel Alibo hat viele Musikrichtungen aus seiner karibischen Heimat und Afrika sowie Jazzrock und Fusion assimiliert. Unter anderem sind hier Beguine, Reggae, Zouk, Soukous und Groka zu nennen. Er verbindet die Slaptechnik mit afrikanischen Rhythmen und bringt nordafrikanische Einflüsse in Fusion ein.

Diskografie (Auswahl)
Michel Alibo spielte bei fast 500 Aufnahmen mit; der größte Teil davon ist afrikanische Musik. Im Folgenden ist eine Auswahl der bekannteren Werke aufgeführt:
 Antoine Hervé's Bob 13  Tutti(1985)
 Sixun Nuit Blanche (1985)
 Manu Dibango Afrijazzy (1986)
 Alibo (erstes Soloalbum) (1986)
 Sixun Pygmees (1987)
 Explore mit Sixun (1988)
 "Live" mit Sixun (1989)
 Sixun (Live) mit Sixun (1990)
 L'Eau de Là mit Sixun (1990)
 Kintetik (Soloalbum 1991)
 Nomads' Land mit Sixun (1993)
 Flashback mit Sixun (1995)
 Nguyên Lê Maghreb & Friends (1998)
 Princesses Nubiennes mit Les Nubians  (1998)
 Tata Nzambé mit Bisso Na Bisso (1998)
 Michel Alibo & New Sakiyo Corporation (1998)
 Andy Narell Fire in the Engine Room (2000)
 Takfarinas  Yal (2000)
 Nouvelle Vague mit Sixun (2000)
 MC Solaar Cinquieme As (2001)
 Sakesho mit Sakesho  (2002)
 Nguyên Lê Purple - Celebrating Jimi Hendrix (2002)
 Angélique Kidjo Black Ivory Soul (2002)
 Huong Thanh Mangustao (2003)
 Les Enfants De La Terre mit Les Enfants De La Terre (2003)
 Mylène Farmer Les Mots (2003)
 We Want You To Say... mit Sakesho (2005)
 Mokhtar Samba Dounia (2005)
 Fete Ses 20 Ans: la Cigale Live mit Sixun (2006)
 MC Solaar Chapitre 7 (2007)
 Sixun DVD - Sixun fête ses 20 ans - Live à La Cigale (2006)
 Sixun Palabre (2008)
 Franck Nicolas Jazz Ka Philosophy 5 - Trio Zalizé (2012)
 Karim Ziad Jdid (2013)
 Mario Canonge Zouk Out (2018)
 Ray Lema Hommage á Franco Luambo: On entre KO, on sort OK (2020)

Einzelnachweise

Weblinks
 Michel Alibo: The Bassic Personification of World Fusion (englisch)
 Michel Alibo in MySpace

Jazz-Bassist
Fusion-Musiker
Weltmusik-Künstler
Franzose
Geboren 1959
Mann

redpajama_set_name: RedPajamaWikipedia


text:
\section{Properties of Probability Quantification}
\label{sec:proofs:quantifier}
In this section we present the proofs for properties of probability quantification.

\AxiomsNotPIQ*
\begin{proof}
We show the first claim as follows.
By the definition of semantics,
$\M, w \models \neg \PR{I} \psi$ is logically equivalent to
$\Pr\!\left[ s \randassign w :~ s \models \psi \right] \not\in I$,
which is equivalent to
$\Pr\!\left[ s \randassign w :~ s \models \psi \right] \in~I^c$,
namely, $\M, w \models \PR{I^c} \psi$.

Next we show the second claim as follows.
By the definition of semantics,
$\M, w \models \PR{I} \neg \psi$ is logically equivalent to
$1 - \Pr\!\left[ s \randassign w :~ s \models \psi \right] \in I$, i.e.,
$\Pr\!\left[ s \randassign w :~ s \models \psi \right] \in \ov{I}$.
This is equivalent to $\M, w \models \PR{\ov{I}} \psi$.
\qed
\end{proof}

\section{Properties of the Epistemic Operators}
\label{sec:proofs:epistemic}
In this section we present properties of our epistemic operators and their proofs.

\AxiomsMinimal*
\begin{proof}
We first show ({\bf N}) necessitation rule as follows.
Assume that $\M \models \phi$.
Then for any world $w$ in $\M$, we have $\M, w \models \phi$.
Hence $\M, w \models \MKa \phi$.
Therefore the necessitation rule holds.

Next we show ({\bf K}) distribution axiom as follows.
Let $w$ be a possible world in $\M$.
Assume that $\M, w \models \MKa (\phi_0 \rightarrow \phi_1)$, and that $\M, w \models \MKa \phi_0$.
Let $w'$ be any world such that $(w, w')\in\Raeps$.
Then we have $\M, w' \models \phi_0 \rightarrow \phi_1$ and $\M, w' \models \phi_0$, hence $\M, w' \models \phi_1$.
Thus we have $\M, w \models \MKa \phi_1$.
Therefore we obtain $\M \models \MKa (\phi_0 \rightarrow \phi_1) \rightarrow  (\MKa \phi_0 \rightarrow \MKa \phi_1)$.
\qed
\end{proof}

\AxiomsDivergence*
\begin{proof}
Let $w$ be a possible world in $\M$.

We first show ({\bf T}) reflexivity as follows.
Assume that $\M, w \models \MKaz \phi$.
By $(w, w)\in\Raz$, we have $\M, w \models \phi$.
Therefore, we obtain $\M \models \MKae \phi \rightarrow \phi$.

Next we show ($\bf{\ge}$) comparison of observability as follows.
Assume that $\M, w \models \MKae \phi$.
Let $w'$ be any world such that $(w, w')\in\Raeps$.
Then $\M, w' \models \phi$.
By $\varepsilon' \le \varepsilon$ and the definition of $\Raeps$, we have $\calr_{a,\varepsilon'}\subseteq\Raeps$, hence $(w, w')\in\calr_{a,\varepsilon'}$.
Then $\M, w \models {\mathop{\mathsf{K}_{a,\varepsilon'}}} \phi$.
Therefore we obtain
$\M \models {\mathop{\mathsf{K}_{a,\varepsilon}}} \phi \rightarrow {\mathop{\mathsf{K}_{a,\varepsilon'}}} \phi$.

Finally, we show ({\bf B}) symmetry when $\Raeps$ is symmetric.
Assume that $\M, w \models \phi$.
Let $w'$ be any world such that $(w, w')\in\Raeps$.
Since $\Raeps$ is symmetric, we have $(w', w)\in\Raeps$.
By $\M, w \models \phi$, we obtain $\M, w' \models \MPae \phi$.
Hence $\M, w \models \MKae \MPae \phi$.
Therefore we obtain $\M \models \phi \rightarrow \MKae \MPae \phi$.
\qed
\end{proof}

\AxiomsMetric*
\begin{proof}
Since a metric satisfies the definition of a divergence (in Section~\ref{sec:preliminaries}), a metric-based accessibility relation is also a divergence-based accessibility relation.
Therefore we obtain {\rm ({\bf T})} reflexivity and {\rm ({\bf B})} symmetry from Proposition~\ref{prop:axioms:divergence}.

Next we show ({\bf 4q}) quantitative transitivity as follows.
Let $w$ be a possible world in $\M$.
Assume that $\M, w \models {\mathop{\mathsf{K}_{a,\varepsilon+\varepsilon'}}} \phi$.
Let $w'$ be any world such that $(w, w')\in\Raeps$, and
$w''$ be any world such that $(w', w'')\in\calr_{a,\varepsilon'}$.
By definition, we have
$\diverge{\sigma_{w}(x)}{\sigma_{w'}(x)} \le \varepsilon$ and
$\diverge{\sigma_{w'}(x)}{\sigma_{w''}(x)} \le \varepsilon'$.
By the subadditivity of the divergence $D$, we have
$\diverge{\sigma_{w}(x)}{\sigma_{w''}(x)} \le
\diverge{\sigma_{w}(x)}{\sigma_{w'}(x)} + \diverge{\sigma_{w'}(x)}{\sigma_{w''}(x)} \le
\varepsilon + \varepsilon'$,
hence $(w, w'')\in\calr_{a,\varepsilon+\varepsilon'}$.
Then it follows from $\M, w \models {\mathop{\mathsf{K}_{a,\varepsilon+\varepsilon'}}} \phi$ that $\M, w'' \models \phi$.
By the definition of $w''$, we obtain
$\M, w \models \MKae {\mathop{\mathsf{K}_{a,\varepsilon'}}} \phi$.
Therefore we have
$\M \models {\mathop{\mathsf{K}_{a,\varepsilon+\varepsilon'}}} \phi \rightarrow \MKae {\mathop{\mathsf{K}_{a,\varepsilon'}}} \phi$.

We next show ({\bf 5q}) relaxed Euclidean as follows.
Let $w$ be a possible world in $\M$.
Assume that $\M, w \models {\mathop{\mathsf{P}_{a,\varepsilon}}} \phi$.
Then there exists a world $w'$ such that $\M, w' \models \phi$ and $(w, w') \in \Raeps$.
Let $w''$ be any world such that $(w, w'')\in\calr_{a,\varepsilon'}$.
Since $\calr_{a,\varepsilon'}$ is a metric-based accessibility relation, it is symmetric, hence $(w'', w) \in\calr_{a,\varepsilon'}$.
Then by $(w, w') \in \Raeps$, we obtain $(w'', w')\in\calr_{a,\varepsilon+\varepsilon'}$.
Hence
$\M, w \models {\mathop{\mathsf{K}_{a,\varepsilon'}}} {\mathop{\mathsf{P}_{a,\varepsilon+\varepsilon'}}} \phi$.
Therefore we obtain
$\M \models {\mathop{\mathsf{P}_{a,\varepsilon}}} \phi \rightarrow {\mathop{\mathsf{K}_{a,\varepsilon'}}} {\mathop{\mathsf{P}_{a,\varepsilon+\varepsilon'}}} \phi$.

Finally, for $\varepsilon = \varepsilon' = 0$, ({\bf 4}) transitivity and ({\bf 5}) Euclidean respectively follow from ({\bf 4q}) quantitative transitivity and ({\bf 5q}) relaxed Euclidean.
\qed
\end{proof}

\subsection{Differential Privacy}
\label{sub:DP}

\emph{Differential privacy}~\cite{Dwork:06:ICALP,dwork2014algorithmic} is a popular measure of data privacy guaranteeing that
by observing a statistics about a database $d$,
we cannot learn whether an individual user's record is included in $d$ or not.

As a toy example, let us assume that the body weight of individuals is sensitive information, and we publish the average weight of all users recorded in a database~$d$.
Then
we denote by $d'$ the database obtained by adding to $d$ a single record of a new user $u$'s weight.
If we also disclose the average weight of all users in $d'$, then you learn $u$'s weight from the difference between these two averages.

To mitigate such privacy leaks, many studies have proposed \emph{obfuscation mechanisms}, i.e., randomized algorithms that add random noise to the statistics calculated from databases.
In the above example, an obfuscation mechanism receives a database $d$ and outputs a statistics of average weight to which some random noise is added.
Then you cannot learn much information on $u$'s weight from the perturbed statistics of average weight.

The privacy achieved by such obfuscation is often formalized as differential privacy.
Intuitively, an $\varepsilon$-differential privacy mechanism makes every two ``adjacent'' (i.e., close) database $d$ and $d'$ indistinguishable with a degree of $\varepsilon$.

\begin{definition}[Differential privacy] \label{def:DP} \rm
Let $e$ be the base of natural logarithm,
$\varepsilon\ge 0$,
$\cald$ be the set of all databases, and $\Psi\subseteq\cald\times\cald$ be an adjacency relation between two databases.
A randomized algorithm $\alg: \cald \rightarrow \Dists\calo$ provides
\emph{$\varepsilon$-differential privacy}
w.r.t. $\Psi$ if for any $(d, d')\in\Psi$ and any $R\subseteq\calo$,
\[
\Prob[ \alg(d)\in R ] \leq e^{\varepsilon} \,\Prob[ \alg(d')\in R ]
\]
where the probability is taken over the randomness in $\alg$.
\end{definition}

For a smaller $\varepsilon$,
the protection of differential privacy is stronger.
It is known that differential privacy can be defined using
the max-divergence $\Dinf$
(Definition~\ref{def:max-divergence}) as follows~\cite{dwork2014algorithmic}.

\begin{restatable}{prop}{DP}
\label{prop:DP}
An obfuscation mechanism $\alg: \cald \rightarrow \Dists\calo$ provides
$\varepsilon$-differential privacy
w.r.t. $\Psi\subseteq\cald\times\cald$ iff for any $(d, d')\in\Psi$,\,
$\maxdiverge{\alg(d)}{\alg(d')} \le \varepsilon$ and
$\maxdiverge{\alg(d')}{\alg(d)} \le~\varepsilon$.
\end{restatable}

\subsection{Formalization of Differential Privacy}
\label{sub:formalize-DP}

Next we define a distributional Kripke model
$\M =(\calw, \Reps, (V_s)_{s\in\cals})$
where there is a possible world corresponding to each database in $\cald$.
We assume that each world is a probability distribution of states in each of which an obfuscation mechanism $\alg$ uses a different value of random seed for providing a probabilistically perturbed output.
Let $x$ (resp. $y$) be a measurement variable denoting the input (resp. output) of the obfuscation mechanism $\alg$.
In each world $w$,\, $\sigma_{w}(x)$ is the database that $\alg$ receives as input, and $\sigma_{w}(y)$ is the distribution of statistics that $\alg$ outputs.
Then the set of all databases is denoted by $\cald = \{ \sigma_{w}(x) \mid w\in\calw \}$.

Now we define the accessibility relation $\Reps$
in $\M$ by using
the max divergence $\Dinf$
as follows\footnote{Since the relation $\Reps$
is symmetric, the symmetry axiom {\rm ({\bf B})} also holds. }:
\[
\Reps
\eqdef
\left\{ (w, w') \in \calw\times\calw \mid
\maxdiverge{\sigma_{w}(y)\!}{\!\sigma_{w'}(y)} \le \varepsilon
\mbox{, }
\maxdiverge{\sigma_{w'}(y)\!}{\!\sigma_{w}(y)} \le \varepsilon
\right\}
.
\]
Intuitively,
$(w, w') \in \Reps$
represents that, when we observe an output $y$ of the obfuscation mechanism $\alg$, we do not know which of the two worlds $w$ and $w'$ we are located at.
Hence we do not see which of the two databases $\sigma_{w}(x)$ and $\sigma_{w'}(x)$ was the input to $\alg$.

For each $d\in\cald$, let $\phi_d$ be a formula representing that we have a database $d$.
Then the
$\varepsilon$-differential privacy
of $\alg$ w.r.t. an adjacency relation $\Psi$ is expressed~as:
\[
\M \models
\bigwedge_{d\in \cald}
\Bigl( \phi_d \rightarrow
\bigwedge_{d'\in \Psi(d)}
\MPe \phi_{d'}
\Bigr)
{.}
\]

Note that the privacy of user attributes defined as distribution privacy~\cite{Kawamoto:19:ESORICS} can also be expressed using \StatEL{}, since it is defined as the differential privacy w.r.t. a relation between the probability distributions that represent user attributes.
We will elaborate on this in future work.

\subsection{Statistical Hypothesis Testing}
\label{sub:HT}
A \emph{statistical hypothesis testing} is a method of statistical inference to check whether given datasets provide sufficient evidence to support some hypothesis.
Typically, given two datasets, a \emph{null hypothesis} $H_0$ is defined to claim that there is no statistical relationship between the two datasets (e.g., no difference between the result of a medical treatment and the placebo effect), while an \emph{alternative hypothesis} $H_1$ represents that there is some relationship between them (e.g., the result of a medical treatment is better than the placebo effect).

Before performing a hypothesis test, we specify a \emph{significance level} \journal{(\emph{type I error rate})} $\alpha$, i.e., the probability that the test might reject the null hypothesis $H_0$, given that $H_0$ is true.
Typically, $\alpha$ is $0.05$ or $0.01$.
$1 - \alpha$ is called a \emph{confidence level}.
\journal{Symmetrically, a \emph{type II error rate} $\beta$ is the probability that the test fails to reject $H_0$, given that $H_0$ is false.
$1 - \beta$ is called a \emph{power} of the test.}

\subsection{Formalization of Statistical Hypothesis Testing}
\label{sub:reasoning:HT}

Now we define a distributional Kripke model $\M$ with a universe $\calw$ that includes at least two worlds $\wre$ and $\wid$ corresponding to the two datasets we compare:
\begin{itemize}
\item the real world $\wre$ where we have a dataset sampled from actual experiments (e.g., from a medical treatment whose effectiveness we want to know);
\item the ideal world $\wid$ where we have a dataset that is synthesized from the null hypothesis setting (e.g., the dataset obtained from the placebo effect).
\end{itemize}
Note that $\calw$ may include other worlds corresponding to different possible datasets.

Let $\samp$ be the size of the dataset, and
$x$ be a measurement variable denoting a single data value chosen from the dataset we have.
We assume that each world $w$ has a state $s$ corresponding to each single data value $\sigma_{s}(x)$ in the dataset.
Then $\sigma_{\wre}(x)$ is the empirical distribution (histogram) calculated from the dataset observed in the actual experiments in $\wre$, while $\sigma_{\wid}(x)$ is the distribution calculated from the synthetic dataset in $\wid$.
Then the number of data having a value $v$ in the dataset in a world $w$ is given by $\samp \cdot \sigma_{w}(x)[v]$.

Assume that $\M$ has an accessibility relation $\calr_{c_{\alpha}/\samp}$ that is specific to the sample size $\samp$, the statistical hypothesis test, and the \emph{critical value} $c_{\alpha}$ for a significance level $\alpha$ we use.
For brevity let $\epsan = c_{\alpha} / \samp$.
Intuitively, $(\wre, \wid) \in \Rea$ represents that the hypothesis test cannot distinguish  the actual dataset from the synthetic one.
For instance, when we use Pearson's $\chi^2$-test as the hypothesis test, then $\Rea$ is defined by:
\[
\Rea \eqdef
\left\{ (w, w') \in \calw\times\calw \mid
D_{\chi^2}(\sigma_{w}(x) \parallel \sigma_{w'}(x)) \le \epsan
\right\}\!,
\]
where $D_{\chi^2}$ is Pearson's $\chi^2$ divergence (Definition~\ref{def:chi-divergence}).

Observe that
when the confidence level $1- \alpha$ increases, then $c_{\alpha}$ decreases, hence $\epsan = c_{\alpha} / \samp$ is smaller, i.e., the capability of distinguishing possible worlds is stronger.

Let $\phisyn$ be a formula representing that the dataset is synthesized from the null hypothesis setting (e.g., representing the placebo effect).
Then $\M, \wid \models \phisyn$.
Since each world in $\calw$ corresponds to a different dataset, it holds for any $w'\neq \wid$ that $\M, w' \models \neg \phisyn$.
For instance, $\M, \wre \models \neg \phisyn$, since the actual dataset is used in $\wre$ even when it looks indistinguishable from the synthetic dataset by the hypothesis test.

When the null hypothesis is rejected with a confidence level $1-\alpha$, then $(\wre, \allowbreak \wid) \not\in \Rea$.
Since $\M, w' \models \neg \phisyn$ holds for any $w'\neq \wid$,
this rejection of the null hypothesis implies:
\[
\M, \wre \models \MKea \neg \phisyn
{,}
\]
which is logically equivalent to $\M, \wre \models \neg \MPea \phisyn$.
This means that with the confidence level $1-\alpha$, we know we are not located in the world $\wid$, hence do not have a synthetic dataset.

On the other hand, when the null hypothesis is not rejected with a confidence level $1-\alpha$, then $(\wre, \wid) \in \Rea$.
Thus we obtain:
\begin{align}\label{eq:null}
\M, \wre \models \MPea \phisyn
{.}
\end{align}
This means that we cannot recognize whether we are located in the world $\wre$ or $\wid$, i.e.,
we are not sure which database we have.
To see this in details, let $\phi'$ be a formula representing that we have a third database (different from those in $\wre$ and $\wid$).
Suppose that another null hypothesis of satisfying $\phi'$ is not rejected with a confidence level $1-\alpha$.
Then we have $\M, \wre \models \MPea \phi'$.
Since each world in $\calw$ corresponds to a different database, we obtain
$\M, \wre \models \MPea \neg \phisyn$,
which implies
$\M, \wre \models \neg \MKea \phisyn$.
This represents that, when the null hypothesis is not rejected,
we are not sure whether the null hypothesis is true or false.

\subsection{Formalization of Statistical Secrecy}
\label{sub:secrecy}
Now let us formalize the coin flipping in Example~\ref{eg:coin} in Section~\ref{sec:running-example} by using \StatEL{} as follows.
Recall that $p(heads) = 0.5$ in $w_0$ and $p(heads) = 0.4$ in $w_1$.
Let $\psi$ be a static formula representing that the coin is a heads.
Then $\M, w_0 \models \PR{0.5} \psi$ and $\M, w_1 \models \PR{0.4} \psi$.
Assume that either $p(heads) = 0.5$ or $p(heads) = 0.4$ holds, i.e., $\M \models \PR{0.5} \psi \vee \PR{0.4} \psi$.

When we have a sufficient number $\samp$ of coin flips (e.g., $\samp=500$), we can distinguish $p(heads) = 0.5$ from $p(heads) = 0.4$ (i.e., $w_0$ from $w_1$) by a hypothesis test.
Hence we learn the probability $p(heads)$ with some confidence level $1 - \alpha$, i.e.,
$\M, w_0 \models \MKea \PR{0.5} \psi$
and
$\M, w_1 \models \MKea \PR{0.4} \psi$.
Therefore we obtain:
\[
\M \models \bigl( \PR{0.5} \psi \rightarrow \MKea \PR{0.5} \psi \bigr) \wedge
\bigl( \PR{0.4} \psi \rightarrow \MKea \PR{0.4} \psi \bigr)
{.}
\]
Note that for a larger sample size $\samp' > \samp$,\, we have $\epsanp = c_{\alpha} / \samp' < c_{\alpha} / \samp = \epsan$,
hence it follows from the axiom $({\bf{\ge}})$ in Proposition~\ref{prop:axioms:divergence} that:
\[
\M \models \bigl( \PR{0.5} \psi \rightarrow \MKeap \PR{0.5} \psi \bigr) \wedge
\bigl( \PR{0.4} \psi \rightarrow \MKeap \PR{0.4} \psi \bigr)
{.}
\]
This means that if our knowledge derived from a smaller sample is statistically significant, then we derive the same conclusion from a larger sample.

On the other hand, when we have a very small number $\samp''$ of coin flips, we cannot distinguish $w_0$ from $w_1$.
Then we are not sure about $p(heads)$ with a confidence level $1-\alpha$, i.e.,
$\M, w_0 \models \MPeapp \PR{0.5} \psi$ and
$\M, w_1 \models \MPeapp \PR{0.4} \psi$.
Hence:
\[
\M \models (\PR{0.5} \psi \vee \PR{0.4} \psi ) \rightarrow (\MPeapp \PR{0.5} \psi \wedge \MPeapp \PR{0.4} \psi)
{.}
\]
This expresses a secrecy of $p(heads)$.
We generalize this to introduce the following definition of secrecy.
\begin{definition}[$(\alpha,\samp)$-statistical secrecy]\label{def:secrecy}\rm
Let $\Phi$ be a finite set of formulas,
$\alpha \in [0, 1]$ be a significance level,
and $\samp$ be a sample size.
We say that $\Phi$ is \emph{$(\alpha,\samp)$-statistically secret} if we have:
\begin{align*}
\M \models \bigvee_{\phi\in\Phi} \phi \rightarrow \bigwedge_{\phi \in \Phi} \MPea \phi
{.}
\end{align*}
\end{definition}

In the above coin flipping example, $\{ \PR{0.5} \psi,\, \PR{0.4} \psi \}$ is $(\alpha,\samp)$-statistically secret for some significance level $\alpha$ and sample size $\samp$.
Syntactically, $(\alpha,\samp)$-statistical secrecy resembles the notion of \emph{total anonymity}~\cite{Halpern:03:CSFW}, whereas in our definition, the epistemic operator $\MPea$ deals with the statistical significance and $\phi$ is not limited to a formula representing an agent's action.

\subsection{Syntax}
\label{sub:syntax}

We first present the syntax of the statistical epistemic logic as follows.
To express both deterministic and probabilistic properties, we introduce two levels of formulas: \emph{static formulas} and \emph{epistemic formulas}.
Intuitively, a static formula represents a proposition that can be satisfied at a state with probability $1$, while an epistemic formula represents a proposition that can be satisfied at a probability distribution of states with some probability.

Formally, let $\Var$ be a set of symbols called \emph{measurement variables}, and
$\Gamma$ be a set of atomic formulas of the form $\gamma(x_1, x_2, \ldots, x_n)$ for a predicate symbol $\gamma$ and $x_1, x_2, \ldots, x_n\in\Var$ ($n \ge 0$).
Let $I \subseteq [0, 1]$ be a finite union of intervals, and $\cala$ be a finite set of indices (typically associated with the names of agents and/or statistical tests).
Then the static and epistemic formulas are defined by:
\begin{itemize}
\item[] Static formulas:~~
$\psi \mathbin{::=}

 \gamma(x_1, x_2, \ldots, x_n) \mid
 \neg \psi \mid \psi \wedge \psi$
\item[] Epistemic formulas:~~
$\phi \mathbin{::=}
 \PR{I} \psi \mid \neg \phi \mid \phi \wedge \phi \mid
 \psi \erightarrow \phi \mid \MKa \phi$
\end{itemize}
where
$a\in\cala$.
Let $\calf$ be the set of all epistemic formulas.
Note that we have no quantifiers over measurement variables. (See Section~\ref{sub:interpretation}.)

The \emph{probability quantification} $\PR{I} \psi$ represents that a static formula $\psi$ is satisfied with a probability belonging to a set $I$.
For instance, $\PR{(0.5, 1]} \psi$ represents that $\psi$ holds with a probability greater than $0.5$.
The \emph{non-classical implication} $\erightarrow$ is used to represent conditional probabilities.
For example, by $\psi_0 \erightarrow \PR{I} \psi_1$ we represent that the conditional probability of $\psi_1$ given $\psi_0$ is included in a set $I$.
The \emph{epistemic knowledge} $\MKa \phi$ expresses that an agent $a$ knows $\phi$.
The formal meaning of these operators will be shown in the definition of semantics.

As syntax sugar, we use \emph{disjunction} $\vee$, \emph{classical implication} $\rightarrow$, and \emph{epistemic possibility operator} $\MPa$, defined by:
$\phi_0 \vee \phi_1 \mathbin{::=} \neg (\neg \phi_0 \wedge \neg \phi_1)$,
$\phi_0 \rightarrow \phi_1 \mathbin{::=} \neg \phi_0 \vee \phi_1$,
and $\MPa{\phi} \mathbin{::=} \neg \MKa \neg \phi$.
When $I$ is a singleton $\{ i \}$, we abbreviate $\PR{[i, i]}$ as $\PR{i}$.

\subsection{Modeling of Systems}
\label{sub:system}

In this work we deal with a simple stochastic system with measurement variables.
Let $\calo$ be the finite set of all data that can be assigned to the measurement variables in $\Var$.
We assume that all possible worlds share the same domain $\calo$.
We define a \emph{stochastic system} as a pair $(S, \sigma)$ consisting of:
\begin{itemize}
\item a stochastic program $S$ that deals with input and output data through measurement variables in $\Var$, behaves deterministically or probabilistically (by using some randomly generated data), and terminates with probability $1$;
\item a \emph{stochastic assignment} $\sigma:\Var\rightarrow\Dists\calo$ representing that each measurement variable $x$ has an observed value $v$ with probability $\sigma(x)[v]$.
\end{itemize}
Here we present only a general model and do not specify the data type of those measurement variables,
which can be (sequences of) bit strings, floating point numbers, texts, or other types of data.
Thanks to the assumption on the program termination and on the finite range of data, the program $S$ can reach finitely many states.
For the sake of simplicity, our model does not take timing into account.
Extension to time and temporal modality is left for future work.

\subsection{Distributional Kripke Model}
\label{sub:Kripke}

To define a semantics for \StatEL{}, we recall the notion of a Kripke model~\cite{Kripke:63:MLQ}:
\begin{definition}[Kripke model] \label{def:Kripke-model} \rm
Given a set $\Gamma$ of atomic formulas, a \emph{Kripke model} is defined as a triple $(\calw, \calr, V)$ consisting of a non-empty set $\calw$, a binary relation $\calr$ on $\calw$, and a function
$V$ that maps each atomic formula $\gamma\in\Gamma$ to a subset $V(\gamma)$ of $\calw$.
The set $\calw$ is called a \emph{universe}, its elements are called \emph{possible worlds}, $\calr$ is called an \emph{accessibility relation}, and $V$ is called a \emph{valuation}.
\end{definition}

Now we introduce a Kripke model called a ``distributional'' Kripke model where each possible world is a probability distribution of states over $\cals$ and
each world $w$ is associated with a stochastic assignment $\sigma_w$ to measurement variables.

\begin{definition}[Distributional Kripke model] \label{def:dist-Kripke-model} \rm
Let $\cala$ be a finite set of indices (typically associated with the names of agents and/or statistical tests),
$\cals$ be a finite set of states\footnote{It is left for future work to investigate the case of infinite numbers of states.}, and $\calo$ be a finite set of data.
A \emph{distributional Kripke model}
is a tuple
$\M =(\calw, (\calr_a)_{a\in\cala}, (V_s)_{s\in\cals})$
consisting of:
\begin{itemize}
\item a non-empty set\footnote{Since $\calw$ is not a multiset, each world in $\calw$ is a different distribution of states. However, this is still expressive enough when we take $\cals$ to be sufficiently large.
} $\calw$ of probability distributions of states over $\cals$;
\item for each $a\in\cala$, an accessibility relation $\calr_a \subseteq \calw \times \calw$;
\item for each $s\in\cals$, a valuation $V_s$ that maps each $k$-ary predicate $\gamma$ to a set $V_s(\gamma) \subseteq \calo^k$.
\end{itemize}
We assume that each $w\in\calw$ is associated with a function $\rho_w: \Var\times\cals\rightarrow\calo$ that maps each measurement variable $x$ to its value $\rho_w(x, s)$ observed at a state~$s$.
We also assume that each state $s$ in a world $w$ is associated with the assignment $\sigma_s: \Var\rightarrow\calo$ defined by $\sigma_s(x) = \rho_w(x, s)$.
\end{definition}

Note that this model assumes a constant domain $\calo$; i.e., all measurement variables range over the same set $\calo$ in every world.
Since each world $w$ is a probability distribution of states, we denote by $w[s]$ the probability that a state $s$ is sampled from $w$.
Then the probability that a variable $x$ has a value $v$ in a world $w$ is given by:
\[
\sigma_w(x)[v] =
\sum_{\substack{s\in\supp(w),\, \sigma_s(x) = v}} w[s]
{.}
\]
This means that when a state $s$ is drawn from the distribution $w$, an input value $\sigma_s(x)$ is sampled from the distribution $\sigma_w(x)$.

\subsection{Divergence-based Accessibility Relation}
\label{sub:divergence-relation}

Next we introduce a family of accessibility relations used in typical statistical inferences.
Since many notions of statistical distance are not metrics but divergences, we introduce an accessibility relation based on a divergence as follows.

Suppose that an agent $a$ observes some data through a single measurement variable $x$.
Then the distribution of the observed data at a world $w$ is represented by $\sigma_{w}(x)$.
Assume that the agent $a$ distinguishes distributions in terms of a divergence $\diverge{\cdot\!}{\!\cdot}: \Dists\calo \times \Dists\calo \rightarrow \realsnng$.
Then given a threshold $\varepsilon \ge 0$, we define a \emph{divergence-based accessibility relation} $\Raeps$ by:
\[
\Raeps \eqdef
\left\{ (w, w') \in \calw\times\calw \mid D(\sigma_{w}(x) \parallel \sigma_{w'}(x)) \le \varepsilon
\right\}
{.}
\]
For a smaller value of $\varepsilon$, the capability of distinguishing worlds is stronger.

If $D$ is a metric instead,
we call $\Raeps$ a \emph{metric-based accessibility relation}.
We often omit $a$ to write $\Reps$ when we do not compare different agents' knowledge.

Intuitively, $(w, w')\in\Raeps$ represents that the distribution of the data observed in $w$ is indistinguishable from that in $w'$ in terms of $D$.
By the definition of a divergence/metric $D$,  $D(\sigma_{w}(x) \parallel \sigma_{w'}(x)) = 0$ implies $\sigma_{w}(x) = \sigma_{w'}(x)$.
Therefore, the relation $\calr_{a,0}$ expresses that the agent $a$ has an unlimited capability of observing the distributions $\sigma_{w}(x)$ and $\sigma_{w'}(x)$.
In Sections~\ref{sec:formal:HT} and~\ref{sec:formal:DP} we will show examples of divergence-based accessibility relations.

\subsection{Stochastic Semantics}
\label{sub:interpretation}

In this section we define the \emph{stochastic semantics} for the \StatEL{} formulas over a distributional Kripke model
$\M =(\calw, (\calr_a)_{a\in\cala}, (V_s)_{s\in\cals})$
with $\calw = \Dists\cals$.

The interpretation of static formulas $\psi$ at a state $s$ is given by:
\begin{align*}
s \models \gamma(x_1, x_2, \ldots, x_k)
& ~\mbox{ iff }~
(\sigma_s(x_1), \sigma_s(x_2), \ldots, \sigma_s(x_k)) \in V_s(\gamma)
\\
s \models \neg \psi
& ~\mbox{ iff }~
s \not\models \psi
\\
s \models \psi \wedge \psi'
& ~\mbox{ iff }~
s \models \psi
~\mbox{ and }~
s \models \psi'
{.}
\end{align*}
Note that the satisfaction of the static formulas does not involve probability.

To interpret the non-classical implication $\erightarrow$, we define the \emph{restriction} $w|_\psi$ of a world $w$ to a state formula $\psi$ as follows.
If there exists a state $s$ such that $w[s] > 0$ and $s \models \psi$, then
$w|_\psi$ can be defined as the distribution over the finite set $\cals$ of states such that:
\begin{align*}
w|_\psi[s] =
\begin{cases}
\frac{w[s]}{\sum_{s': s' \models \psi} w[s']}
& \mbox{ if $s \models \psi$}
\\
0 & \mbox{ otherwise.}
\end{cases}
\end{align*}
Then $\sum_{s} w|_\psi[s] = 1$.
Note that $w|_\psi$ is undefined if $w$ does not have a state $s$ that satisfies $\psi$ and has a non-zero probability in $w$.

Now we define the interpretation of epistemic formulas at a world $w$ in $\M$~by:
\begin{align*}
\M, w \models \PR{I} \psi
& ~\mbox{ iff }~
\Pr\!\left[ s \randassign w :~ s \models \psi \right] \in I
\\
\M, w \models \neg \phi
& ~\mbox{ iff }~
\M, w \not\models \phi
\\
\M, w \models \phi \wedge \phi'
& ~\mbox{ iff }~
\M, w \models \phi
~\mbox{ and }~
\M, w \models \phi'
\\
\M, w \models \psi \erightarrow \phi
& ~\mbox{ iff }~
\mbox{$w|_{\psi}$ is defined and }~
\M, w|_{\psi} \models \phi
\\
\M, w \models \MKa \phi
& ~\mbox{ iff }~
\mbox{for every $w'$ s.t. $(w, w') \in \calr_a$, }~
\M, w' \models \phi
{,}
\end{align*}
where $s \randassign w$ represents that a state $s$ is sampled from the distribution $w$.

Finally, the interpretation of an epistemic formula $\phi$ in $\M$ is given by:
\begin{align*}
\M \models \phi
& ~\mbox{ iff }~
\mbox{for every world $w$ in $\M$, }~
\M, w \models \phi
{.}
\end{align*}
We remark that in each world $w$, measurement variables can be interpreted using $\sigma_w$,  as shown in Section~\ref{sub:Kripke}.
This allows one to assign different values to distinct occurrences of a variable in a formula;
E.g., in $\phi(x) \rightarrow \MKa \phi'(x)$,\, the measurement variable $x$ occurring in $\phi(x)$ can be interpreted using $\sigma_{w}$ in a world $w$, while $x$ in $\phi'(x)$ can be interpreted using $\sigma_{w'}$ in another $w'$ s.t. $(w, w')\in \calr_a$.

Note that our semantics for probability quantification is different from that in the previous work.
Halpern~\cite{Halpern:90:AI} shows two approaches to defining semantics: giving probabilities (1) on the domain and (2) on possible worlds.
However, our semantics is different from both.
It defines probabilities on the states belonging to a possible world, while each world is not assigned a probability.
Hence, unlike Halpern's approaches, our model can deal with both probabilistic behaviours of systems and non-deterministic inputs from an external environment.

We also remark that
\StatEL{} can be used to formalize conditional probabilities.
If the conditional probability of satisfying a static formula $\psi_1$ given another static formula $\psi_0$ is included in a set $I$ at a world $w$, then we have
$\Pr\!\left[ s \randassign w|_{\psi_0}: s \models \psi_1 \right]\!\in~I$,
hence we obtain
$\M, w \models \psi_0 \erightarrow \PR{I} \psi_1$.

\section{Introduction}
\label{sec:intro}
\input{_intro}

\section{Preliminaries}
\label{sec:preliminaries}
\input{_preliminaries}

\section{Motivating Example}
\label{sec:running-example}
\input{_running}

\section{Statistical Epistemic Logic (\StatEL{})}
\label{sec:logic}
\input{_logic}

\section{Basic Properties of \StatEL{}}
\label{sec:properties}
\input{_properties}

\section{Modeling Statistical Hypothesis Testing Using \StatEL{}}
\label{sec:formal:HT}
\input{_HT}

\section{Modeling Statistical Data Privacy Using \StatEL{}}
\label{sec:formal:DP}
\input{_DP}

\section{Related Work}
\label{sec:related}
\input{_related}

\section{Conclusion}
\label{sec:conclude}
\input{_conclusion}

\section*{Acknowledgments}
I would like to thank the reviewers for their helpful and insightful comments.
I am also grateful to Ken Mano, Gergei Bana, and Ryuta Arisaka for their useful comments on preliminary manuscripts.

 \bibliographystyle{splncs04}

\subsection{Notations}
\label{subsec:notations-pd}

Let $\realsnng$ be the set of non-negative real numbers,
and $[0, 1] = \{ r\in\realsnng \mid r \le 1 \}$.
We denote by $\Dists\calo$ the set of all probability distributions over a set~$\calo$.
For a finite set $\calo$ and a distribution $\mu\in\Dists\calo$, the probability of sampling a value $y$ from $\mu$ is denoted by $\mu[y]$.
For a subset $R\subseteq\calo$, let $\mu[R] = \sum_{y\in R} \mu[y]$.
The \emph{support} of a distribution $\mu$ over a finite set $\calo$ is
$\supp(\mu) = \{ v \in \calo \colon \mu[v] > 0 \}$.
For a set $\cald$, a randomized algorithm $\alg:\cald\rightarrow\Dists\calo$ and a set $R\subseteq\calo$ we denote by $\alg(d)[R]$ the probability that given input $d\in\cald$, $\alg$ outputs one of the elements of $R$.

\subsection{Metric and Divergence}
\label{subsec:divergence}

A \emph{metric} over a non-empty set $\calo$ is a function $d: \calo\times\calo \rightarrow \realsnng$ such that for all $y, y', y'' \in \calo$,
(i)
$d(y, y') \ge 0$;
(ii)
$d(y, y') = 0$ iff $y = y'$;
(iii)
$d(y, y') = d(y', y)$;
(iv)
$d(y, y'') \le d(y, y') + d(y', y'')$.
Recall that (iii) and (iv) are respectively referred to as symmetry and subadditivity.

A \emph{divergence} over a non-empty set $\calo$ is a function $\diverge{\cdot}{\cdot}: \Dists\calo\times\Dists\calo\rightarrow\realsnng$ such that
for all $\mu,\mu'\in\Dists\calo$,
(i)
$\diverge{\mu}{\mu'}\ge0$ and
(ii)
$\diverge{\mu}{\mu'}=0$ iff $\mu=\mu'$.
Note that a divergence may not be symmetric or subadditive.

To describe a statistical hypothesis testing in Section~\ref{sec:formal:HT},
we recall the definition of $\chi^2$ divergence due to Pearson~\cite{Pearson:1900:LEDPMJS} as follows:

\begin{definition}[Pearson's $\chi^2$ divergence]\label{def:chi-divergence}\rm
Given two distributions $\mu, \mu'$ over a finite set $\calo$,\,
the \emph{$\chi^2$-divergence} $D_{\chi^2}(\mu \parallel  \mu')$ of $\mu$ from $\mu'$ is defined by:
\[
D_{\chi^2}(\mu \parallel  \mu') =
\!\sum_{y\in\supp(\mu)}\hspace{-1ex}
\frac{ (\mu'[y] - \mu[y])^2 }{ \mu[y] }
{.}
\]
\emph{$\chi^2$ statistics} is the multiplication of $\chi^2$-divergence with a sample size~$\samp$.
\end{definition}

To introduce a notion of statistical data privacy in Section~\ref{sec:formal:DP},
we recall the definition of
the max-divergence $\Dinf$
as follows.
\begin{definition}[Max divergence]\label{def:max-divergence}\rm
For two distributions $\mu, \mu'$ over a finite set~$\calo$,\,
the \emph{max divergence} $\maxdiverge{\mu}{\mu'}$
of $\mu$ from $\mu'$ is defined~by:
\[
\maxdiverge{\mu}{\mu'} =
\max_{\substack{R\subseteq\supp(\mu)}}\,
\ln \frac{ \mu[R] }{ \mu'[R] }
{.}
\]
\end{definition}

Note that neither $D_{\chi^2}$ nor
$\Dinf$
is symmetric.

\subsection{Properties of Probability Quantification}
\label{sub:property:PIQ}

We can define a dual operator of $\PR{I}$ as follows.
Given a finite union $I \subseteq [0, 1]$ of disjoint intervals,
let $I^c \eqdef [0,1] \setminus I$ and $\ov{I}\,\, \eqdef \{ 1 - p \mid p \in I \}$.
Then $\ov{I^c} = \ov{I}^c$.
Negation with $\PR{I}$ has the following properties.

\begin{restatable}[Negation with probability quantification]{prop}{AxiomsNotPIQ}
\label{prop:not-PIQ}
For any world $w$ in a model $\M$ and any static formula $\psi$, we have:
\begin{enumerate}
\item
$\M, w \models \neg \PR{I} \psi$ ~iff~
$\M, w \models \PR{I^c} \psi$
\item
$\M, w \models \PR{I} \neg \psi$ ~iff~
$\M, w \models \PR{\ov{I}} \psi$.
\end{enumerate}
\end{restatable}

By Proposition~\ref{prop:not-PIQ}, $\neg \PR{I} \neg \psi$ is logically equivalent to $\PR{\ov{I^c}} \psi$.
For instance, $\neg \PR{(0,1]} \neg \psi$ is equivalent to $\PR{1} \psi$, and
$\neg \PR{[0,1)} \neg \psi$ is equivalent to $\PR{0} \psi$.

\subsection{Properties of Epistemic Modality}
\label{sub:property:epistemic}
Next we show some properties of epistemic modality.
As with the standard modal logic, \StatEL{} satisfies the necessitation rule and distribution axiom.

\begin{restatable}[Minimal properties]{prop}{AxiomsMinimal}
\label{prop:axioms:basic}
For any distributional Kripke model $\M$, any $\phi, \phi_0, \phi_1\in\calf$, and any $a\in\cala$, we have:
\begin{itemize}
\item[]{\rm ({\bf N})} necessitation:~
$\M \models \phi$ implies $\M \models \MKa \phi$
\item[]{\rm ({\bf K})} distribution:~
$\M \models \MKa (\phi_0 \rightarrow \phi_1) \rightarrow  (\MKa \phi_0 \rightarrow \MKa \phi_1)$.
\end{itemize}
\end{restatable}

The satisfaction of other properties depends on the definition of the accessibility relation.
Since many notions of statistical distance are not metrics but divergences, we present some basic properties when $\M$ has a divergence-based accessibility relation:
$\Raeps =
\left\{ (w, w') \in \calw\times\calw \mid D(\sigma_{w}(x) \parallel \sigma_{w'}(x)) \le \varepsilon
\right\}$.

\begin{restatable}[Properties with divergence-based accessibility]{prop}{AxiomsDivergence}
\label{prop:axioms:divergence}
Let $a\in\cala$ and $\varepsilon \ge \varepsilon' \ge 0$.
For any distributional Kripke model $\M$ with a divergence-based accessibility relation $\Raeps$ and any $\phi\in\calf$, we have:
\begin{itemize}
\item[]{\rm ({\bf T})} reflexivity:~
$\M \models \MKae \phi \rightarrow \phi$
\item[]{\rm ($\bf{\ge}$)} comparison of observability:~
$\M \models {\mathop{\mathsf{K}_{a,\varepsilon}}} \phi \rightarrow {\mathop{\mathsf{K}_{a,\varepsilon'}}} \phi$.
\end{itemize}
If $\Raeps$ is symmetric (e.g., based on the Jensen-Shannon divergence~\cite{Lin:91:TIT}) then:
\begin{itemize}
\item[]{\rm ({\bf B})} symmetry:~
$\M \models \phi \rightarrow \MKae \MPae \phi$.
\end{itemize}
\end{restatable}
Here the axiom $({\bf{\ge}})$ represents that
an agent having a stronger capability of distinguishing worlds may have more beliefs.

Finally, we show some properties when $\Raeps$ is based on a metric (e.g. the $p$-Wasserstein metric~\cite{Vaserstein:69:PPI}, including the Earth mover's distance).

\begin{restatable}[Properties with metric-based accessibility]{prop}{AxiomsMetric}
\label{prop:axioms:metric}
Let $a\in\cala$ and $\varepsilon, \varepsilon' \ge 0$.
For any distributional Kripke model $\M$ with a metric-based accessibility relation $\Raeps$ and any $\phi\in\calf$,
we have {\rm ({\bf T})}reflexivity, {\rm ({\bf B})}symmetry,~and:
\begin{itemize}
\item[]{\rm ({\bf 4q})} quantitative transitivity:~
$\M \models {\mathop{\mathsf{K}_{a,\varepsilon+\varepsilon'}}} \phi \rightarrow \MKae {\mathop{\mathsf{K}_{a,\varepsilon'}}} \phi$
\item[]{\rm ({\bf 5q})} relaxed Euclidean:~
$\M \models {\mathop{\mathsf{P}_{a,\varepsilon}}} \phi \rightarrow {\mathop{\mathsf{K}_{a,\varepsilon'}}} {\mathop{\mathsf{P}_{a,\varepsilon+\varepsilon'}}} \phi$.
\end{itemize}
If the agent has an unlimited capability of observation (i.e., $\varepsilon = \varepsilon' = 0$), then:
\begin{itemize}
\item[]{\rm ({\bf 4})} transitivity:~
$\M \models \MKaz \phi \rightarrow \MKaz \MKaz \phi$
\item[]{\rm ({\bf 5})} Euclidean:~
$\M \models \MPaz \phi \rightarrow \MKaz \MPaz \phi$.
\end{itemize}
\end{restatable}

By this proposition, for $\varepsilon = 0$, \StatEL{} has the axioms of {\bf S5}, hence
the epistemic operator $\MKaz$ represents knowledge rather than beleif.

However, if the agent has a limited observability (i.e., $\varepsilon>0$), then neither transitivity nor Euclidean may hold.
This means that, even when he know whether $\phi$ holds or not with some confidence, he may not be perfectly confident that he knows it.

redpajama_set_name: RedPajamaArXiv


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From gym owners to big-name athletes, CrossFit acolytes are breaking ties over the brand's response to Black Lives Matter.
By Cheryl Wischhover@CherylAnneNY Jun 12, 2020, 12:20pm EDT
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CrossFit, which black acolytes say "lacks diversity and doesn't represent people of color well," is losing devotees over the former CEO's insensitive Black Lives Matter comments.
SOPA Images/LightRocket via Getty Images
CrossFit is known for its diehard fans. While many workout regimens have cult-like adherents, CrossFit still manages to stand out. Proponents have been known to vomit into large tires after workouts involving throwing those tires around, and then come back again the next day. The pain they willingly put themselves through with small groups of other devoted fans forges deep bonds within the community. Now, the company is losing acolytes right and left.
Since George Floyd's killing at the hands of police and the global protests that have followed, companies across industries have found themselves in the middle of a reckoning. Consumers have demanded that brands take real action on racial inequality rather than just posting platitudes on Instagram. CrossFit didn't even do that, remaining silent for over a week after Floyd's death, which some in the community found disconcerting and even damning.
Founder Greg Glassman then outraged people further when he wrote a controversial tweet about Floyd, comparing Floyd's killing and the subsequent uprising to the coronavirus. Then, he made offensive comments while on a call with CrossFit gym owners (the audio of which was leaked to BuzzFeed), including, "We're not mourning for George Floyd — I don't think me or any of my staff are." Glassman "retired" from his role as CEO on the evening of June 8 after backlash in the community grew, but at least 1,200 gyms of 15,000 so far have still threatened to pull their affiliations with the company.
High-level trainers, athletes, and owners have resigned, vowed to disaffiliate from the brand, or spoken out, including past CrossFit Games winners like Katrin Davidsdottir, who owes her career and lucrative brand partnerships to CrossFit. Reebok, Rogue Fitness, and other brands cut ties with CrossFit. And Glassman retains ownership in the brand, meaning that even if he is out as CEO, he still earns money from the enterprise.
"We're not mourning for George Floyd — I don't think me or any of my staff are"
CrossFit and its sometimes contentious founder have weathered other controversies but for the most part have seemed untouchable. The demand for systemic change amplified by the recent Black Lives Matter protests, however, have made this scandal seem more dire for the company, forcing actual action. And CrossFit has so far done everything wrong.
"Nothing has ever happened to these guys," says Maillard Howell, a co-founder of Dean Crossfit in Brooklyn, which he says is the largest black-owned CrossFit gym in New York City. "But it's been brewing for quite some time."
CrossFit's history is littered with controversies about LGBTQ rights, guns, and race
From its inception, CrossFit has been polarizing. Glassman, an outspoken libertarian, founded the workout in 2000 after spending the '90s getting kicked out of gyms where he worked as a personal trainer, according to a 2013 Inc. profile. He ended up teaching his quirky method of functional fitness techniques to the Santa Cruz sheriff's department, and the brand forged a strong connection to the law enforcement and military communities. Dave Castro, the new CEO who has been with Glassman since the early days, is a former Navy SEAL. Some of the workouts are dubbed "hero" workouts and honor military and first responders who died in the line of duty.
A CrossFit program stipulates a different routine each day, called WODs, or "workouts of the day." Gyms that feature the methodology are called "boxes," with none of the fancy bells and whistles you'd find at upscale gyms. It uses basic equipment like kettlebells, ropes, and sandbags, with the exercises reminiscent of calisthenics and PE class — burpees, pull-ups, push-ups, box jumps.
Gyms that use the CrossFit name are called affiliates and they pay $3,000 per year. It's not as strict as a franchise arrangement, meaning they can use their own logos and individualize their methods. Before this current controversy, there were more than 15,000.
CrossFit devotees are fanatical, and the brand has built a loyal community that views it as a lifestyle, not just a workout. Like SoulCycle and yoga, it's become almost a religion for its practitioners, as Tara Isabella Burton wrote at Vox in 2018. CrossFit is aggressive when it comes to protecting its IP and its reputation, and has sued researchers for faulty conclusions about the workout's injury risk. Glassman has been evangelical about touting CrossFit's benefits for the prevention of obesity and diabetes, and has clashed with traditional fitness organization gatekeepers and Big Soda, which has funded training organizations and health and medical nonprofits.
CrossFit also hosts the CrossFit Games, a televised annual event that has had Reebok as a sponsor for the past decade. It has worked closely with Rogue Fitness to provide equipment for CrossFit boxes. The Games, and CrossFit generally, have given rise to a whole mini-economy of brands and athletes that owe their livelihoods to the concept.
A CrossFit competition in Fort Lauderdale, Florida.
Michele Eve Sandberg/Corbis via Getty Images
Through the years, CrossFit has weathered some controversies. In 2018, it fired one of its original executives, Russell Berger, after he made anti-LGBTQ comments and called Pride festivals a "sin." That same year, it finally overturned its policy forbidding transgender people from competing in the division in which they identify.
In 2016, a month after the Pulse nightclub mass shooting in Orlando, CrossFit announced that the winners of its CrossFit Games would be receiving Glock handguns. The backlash was swift, with a petition circulated. Reebok put out the statement: "While we understand CrossFit's foundations are tied to military and first responders, we do not agree with this decision, particularly in light of current events in the United States." Castro was unapologetic, responding, "Unless the state and federal laws regarding gun ownership in California and the U.S. change in the next week, then no, nothing is changing."
The brand's devotees do skew white, as scrolling through CrossFit's social feeds makes clear. "I have always said that CrossFit lacks diversity and doesn't represent people of color well within HQ and in the community," says Christina Spencer, the black owner of Junction City CrossFit in Kansas, who has been an affiliate since 2012 after being introduced to the concept several years prior while on a deployment in the military.
CrossFit has even had problems with outright racist messaging in the past. In 2013, there was a backlash after CrossFit HQ shared a blog post on social media that was written by a group that believed in "race realism," a concept embraced by former Ku Klux Klan top leader David Duke. The brand apologized. This year, it reposted a meme (now deleted) of a member next to a sign reading: "Some guy eats a bat halfway around the world and now I can't go to CrossFit," referring to the origins of Covid-19.
Glassman has also been vocal about disagreeing with coronavirus lockdown policies.
Black Lives Matter helps bring CrossFit's issues with racism to the fore
On June 6, the Institute for Health Metrics and Evaluation tweeted a message that racism is a public health issue, along with the Black Lives Matter hashtag. Glassman replied in a tweet that is still posted: "It's FLOYD-19." The next day he tweeted: "Your failed model quarantined us and now you're going to model a solution to racism? George Floyd's brutal killing sparked riots nationally. Quarantine alone is accompanied in every age and under all political regimes by an undercurrent of suspicion, distrust, and riots. Thanks!" It seemed like he was blaming the Covid-19 quarantines for the "riots" (most of which were actually peaceful protests) that followed Floyd's killing.
In the meantime, a Glassman apology didn't come until a full day later, via CrossFit's Twitter account, saying in part, "I made a mistake by the words I chose yesterday. My heart is deeply saddened by the pain it has caused. It was a mistake, not racist but a mistake." He went on to apologize and explain he was trying to criticize the IHME's quarantine policies. He ended with: "Please hear me when I say, we stand by our community to fight for justice. I care about you, our community, and I am here for you."
"I made a mistake by the words I chose yesterday. My heart is deeply saddened by the pain it has caused."
But it was too late for some important partners. After these comments, Reebok, which has had a partnership with CrossFit since 2011, sponsoring the CrossFit Games and making branded apparel, canceled negotiations to renew its contract. (It will honor its contract through the 2020 CrossFit Games.) Rogue Fitness, a company that makes workout equipment, also strongly condemned the company and will be reconsidering partnerships. Reebok just signed on to sponsor Rogue's Invitational, a previously CrossFit-sanctioned competition, prompting fans to wonder if perhaps this was a new guard forming without CrossFit.
But what was unknown at the time of his first tweet was that Glassman had had a Zoom call with several CrossFit affiliate owners a few hours before his controversial tweet. BuzzFeed obtained a recording of it and published excerpts the day after the apology. On the call, Glassman doubles down on his assertion that he wasn't mourning for Floyd. "Can you tell me why I should mourn for him? Other than that it's the white thing to do — other than that, give me another reason," he said.
Later that day, CrossFit published a long note on its site, titled, "Why Didn't CrossFit Just Say Something?" The company apologized for waiting so long to vocally support the black community, but offered no concrete plans for how it would support its black members going forward. An excerpt:
"It is rare for us to speak on these types of issues. We struggled the past few weeks because we allowed our desire to get it right to paralyze us. Tackling social justice issues of this magnitude is not our strength as a company, because our varied points of view sometimes devolve into dysfunction. But we can agree on two things: We are a team dedicated to fitness and health. This team is anti-racist. Therefore, we are committed to scrutinizing ourselves internally, continuing to listen to the community, and taking actions in support of change."
The statement was widely seen as rambling and unfocused. "Successful crisis management requires both effective action and communication to happen simultaneously," says Melissa Agnes, a crisis management strategist and author. "[CrossFit's] statement doesn't do this. It has a whole lot of excuses."
Later that evening, CrossFit announced that Glassman would be retiring and longtime lieutenant and director of the CrossFit Games, Dave Castro, would step in as CEO. Shortly after, BuzzFeed published another story with a video of Castro at a 2019 press conference sitting at a podium with a row of athletes, all of whom were white. A reporter asked how he planned to add diversity to the roster. Castro ignored the question.
In a statement to BuzzFeed, Castro said, "The 2019 CrossFit Games panel in question was to discuss the day's events, and the question wasn't relevant to the topics of discussion at the panel." A CrossFit spokesperson also told BuzzFeed: "He's Brown and Mexican-American, so a question about lack of diversity might have initially caused some confusion." CrossFit did not respond to a request for comment from Vox for this story.
Early on Thursday morning, Castro tweeted: "Thanks to all those showing support to CF in these tough times and having the belief in us to earn your trust back." On his Instagram stories, he reposted several messages of support, stating, "We are in." CrossFit's account posted the admonition: "Work out today." One person responded: "Was this tweet scheduled for, um, last year?"
How CrossFit is managing the fallout with angry community members
Sunday, before the leaked Zoom call had been made public, black box owners joined a call with Glassman and others at CrossFit HQ. Howell, of Dean Crossfit, was on the call, as was another co-owner, Charmel Rodgers.
"When pressed on, 'Why didn't he say anything? Why had he been silent up to that point?' [Glassman] had no answer for that. He was repeatedly pressed on his stance on BLM and he didn't say anything about that. He was just silent, he never answered," says Rodgers. "You can be supportive of the jobs that police officers have to do and ... at the same time want them to do better. They're not mutually exclusive."
Castro was asked on Twitter and on CrossFit's site to address this as well, but the brand has not come out with a strong statement in support for Black Lives Matter or specifically how it will support the community going forward. Still, many commenters seemed supportive of the brand. "If someone inside the CF universe cannot deal and accept to live with multiple opinions on a subject like race, let them go! They do not belong here. It should not be accepted to call someone racist just because of a different vision on race that honors every person. It is an electoral year and the pressure to surrender will only rise. Keep your DNA CrossFit and you will be stronger in one year then you are today!" wrote one.
One high-profile box owner, Alyssa Royse of Rocket CrossFit (now Rocket Community Fitness) in Seattle, publicized her misgivings about the company and posted a disturbing email she received from Glassman before he stepped down, after she told him she was disaffiliating. She wrote: "CrossFit has stayed silent for too long as our country is at a time of reckoning for centuries of systemic racism. They have been called out by countless gyms and athletes and brands alike for their silence." She raised long-simmering issues about her concerns around CrossFit's brand identity, communication problems, and "moral ambiguity."
In response, Glassman wrote in part, "You're doing your best to brand us as racist and you know it's bullshit. That makes you a really shitty person. Do you understand that? You've let your politics warp you into something that strikes me as wrong to the point of evil. I am ashamed of you." Royse had no further comment for this story, preferring that black owners of black boxes be heard at this time.
CrossFit Games champion Katrin Tanja Davidsdottir is one of the high-profile athletes cutting ties with the brand.
Meg Oliphant/Getty Images
Howell says Dean Crossfit is considering disaffiliating from CrossFit, and they are not alone, according to a circulating spreadsheet, which indicates that hundreds of owners are also considering dropping their affiliation with the company. For Howell and his partners, remaining a part of the brand is dependent on the company meeting certain "deliverables," including creating a diverse board, establishing a black scholarship fund, and creating fitness programs in correctional facilities and schools in marginalized neighborhoods. These requests have not yet received a response from CrossFit HQ. They hope that CrossFit will engage black owners and members in the process.
Spencer, the owner of the Kansas box, will keep her affiliation until she sees how Castro and HQ respond. "If you want to see change, we have to be a part of it. I can't ask CrossFit to make changes if I'm not an affiliate," she says. "I would love to see an effort on HQ's part to ensure that they have a diverse HQ staff, that they show people of color on their Instagram and Facebook social media more regularly and not just savior stories — someone who was in prison or living in a core part of a city. I think this could be a learning opportunity and a chance for them to grow and become better for it."
CrossFit has lost crucial support of dozens of brands, hundreds of practitioners, and several prominent Games athletes. Since Castro's appointment, Davidsdottir has doubled down on her criticism, writing on Instagram: "I am disappointed, to say the least, with the solution that was provided and I do not see change. ... Doesn't Greg Glassman still own 100% of CrossFit?" It's clearly the biggest crisis in the company's history. One redditor on the CrossFit subreddit wrote: "Removed CrossFit from my tinder profile. In all seriousness, I'm happy the community is standing together."
But CrossFit obviously still has the support of thousands of people, and its brand equity is strong, according to Agnes, the crisis management pro. This split is evident on a post on Morning Chalk Up, a website devoted to CrossFit news. It characterized the community as "divided and angry."
Dean CrossFit's owners do not have a great deal of confidence that Castro's leadership will produce change in the organization.
"Castro is Glassman's right hand. This is all smoke and mirrors," says Howell. "I have no faith in Castro, and he's going to play the game and pretend way better than Glassman. We tried to make it as clear as possible that they need to spend some resources in the community."
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text:
Charlotte Whitton (8 mars 1896 - 25 janvier 1975) est une femme politique canadienne. Elle fut mairesse d'Ottawa à deux reprises.

Biographie
Charlotte Whitton est née le 8 mars 1896 à Renfrew en Ontario (Canada). Fille de John Edward Whitton et d'Elizabeth Langin, elle vécut la majeure partie de son enfance avec sa mère et sa grand-mère maternelle, son père étant souvent trop occupé par son travail.

Elle fit de brillantes études, tout d'abord au Refrew Collegiate Institute puis, malgré ses origines ouvrières, elle réussit à entrer à l'Université Queen's en 1914, alors que, à l'époque, très peu de jeunes femmes pouvaient y mettre les pieds. Elle étudia notamment l'anglais, l'histoire et la philosophie. Elle fut, de plus, la première femme à être éditée dans le Queen's Journal, le journal des étudiants, et parvint même à rejoindre l'équipe de hockey sur glace de l'université.

En 1918, après avoir quitté l'université, elle fut engagée au Social Service Council of Canada (SSCC) à Toronto. En 1922, elle quitta le SSCC pour devenir la secrétaire privée de Thomas Law, alors ministre du commerce. En 1926, elle devint le premier directeur féminin du Conseil canadien pour la sauvegarde de l'enfance jusqu'en 1941, pour ensuite se consacrer à un travail indépendant dans différents domaines. Elle lut, étudia et écrivit pendant plusieurs années.

En 1950, elle devint le City Controller d'Ottawa. Le 27 août 1951, le maire d'Ottawa, Grenville Goodwin, mourut soudainement de deux attaques cardiaques à quelques heures d'intervalle. Whitton devint immédiatement le nouveau maire, et fut ainsi la première femme à diriger une grande ville canadienne.

Bien que très conservatrice et francophobe notoire, elle était aussi très féministe, et avait un certain courage dans ses opinions, jusqu'à avoir de nombreuses altercations avec ses collègues masculins. Elle était connue pour sa pugnacité et ses citations frappantes. Elle fut réélue en 1952, 1954, 1960 et 1962, et eut l'occasion de passer au jeu télévisé What's my line ?. Elle accueillit chaleureusement la reine Elizabeth lors de sa visite à Ottawa en juin 1964.

Elle perdit les élections de 1964. Ses adversaires crurent enfin ne plus avoir à s'occuper d'elle, mais elle devint encore plus célèbre qu'auparavant. Entre 1965 et 1966, on lui dédia même un jeu télévisé, Dear Charlotte. En 1966, elle retourna à la vie politique, et acquit une place dans la cité jusqu'à sa retraite en 1972. Elle mourut le 25 janvier 1975 à l'âge de 78 ans.

Citations
Charlotte Whitton est aussi et surtout connue pour ses citations. En voici quelques-unes :
 « Quoi qu'elle fasse, une femme doit toujours travailler deux fois plus qu'un homme pour être aussi reconnue. Heureusement, ce n'est pas difficile ! »
 « Tourne ton visage vers le soleil et les ombres tomberont derrière toi. »
 « Nos réussites sont déterminées par notre manière de faire face à l'échec. »
 « Nous avons tous des capacités. Ce qui nous différencie est la manière dont nous les utilisons. »

Notes et références

Sources
 Patricia T. Rooke, Rodolph Leslie Schnell, No bleeding heart: Charlotte Whitton a feminist on the right, University of British Columbia Press, 1987

Article connexe
 Liste de dirigeantes politiques
 Andrée P. Boucher
 Francine Ruest-Jutras
 Valérie Plante
 Vicki-May Hamm

Maire d'Ottawa
Naissance en mars 1896
Naissance en Ontario
Décès en janvier 1975
Décès à 78 ans
Étudiant de l'Université Queen's
Commandeur de l'ordre de l'Empire britannique


text:
Siempre es el vigésimo cuarto álbum de estudio oficial de la banda chilena Quilapayún, lanzado en 2007.

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Créditos
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 Eduardo Carrasco
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 Hernán Gómez
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 Hugo Lagos
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 Ricardo Venegas
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 Sebastián Quezada

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Álbumes de 2007
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Brighton East carjacking victim left..
A distressed and shaken woman who was threatened with a pair of scissors in a terrifying carjacking in Brighton East says she is in shock.
Anna, 49, was in her car waiting at Landcox Park at Brighton East when she was approached by a man.
She said she's recovering but traumatised.
'Before I could think he opened the door really forcefully, he had a pair of scissors at my head, and said very loudly get out of the car,' she told Tony Jones on 3AW Mornings.
He then drove off in her car, a silver 2007 Toyota Camry sedan.
She said she felt violated and incredibly upset after the ordeal.
The man involved is described as being perceived to be African in appearance, about 185cm tall and spoke English well.
At the time he was wearing tracksuit pants, a blue top with black print, light coloured runners and a cap.
The registration of the stolen vehicle is UYZ 360.


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