\documentclass[ocgsteps,ocgstepsdraft,larger]{../text/text}

\newcommand{\auth}[3]{%
	#1 \\[-1ex]
	{\small \href{https://twitter.com/#2}{@#2}} \\[-1ex]
	{\small \href{mailto:#3}{#3}}}

\title{\Large Space in the causal chain:\\The perspective from French and Biblical Hebrew}
\author{
	\auth{Camil Staps}{SemiCamil}{info@camilstaps.nl}
	\and
	\auth{Johan Rooryck}{JohanRooryck}{johan.rooryck@gmail.com}
}
\date{COCOA, 9\textsuperscript{th} November 2022}

\usepackage{amssymb}
\newcommand{\sv}[1]{\ensuremath{\llbracket{#1}\rrbracket}}

\tikzset{
	>=stealth,
	every picture/.style={
		baseline
	},
	every tree node/.style={
		align=center,
		anchor=north
	},
	level 1+/.style={
		level distance=4.5\baselineskip
	}
}

\usepackage{gb4e}
\let\eachwordone=\it
\setlist{nosep}

\usepackage{endnotes}

\let\footnote=\endnote

\newcommand{\imp}[1]{{\color{red}#1}}

\setlength\tabcolsep{.2222em} % \>

\DeclareMathOperator{\Agent}{Agent}
\DeclareMathOperator{\Origin}{Origin}
\DeclareMathOperator{\Net}{net}
\DeclareMathOperator{\FROM}{FROM}
\DeclareMathOperator{\THROUGH}{THROUGH}

\begin{document}

\maketitle

\begin{step}[section]
	\section{Outline}

	\begin{enumerate}
		\item
			Languages with multiple Agent prepositions are problematic for current accounts of Agent PPs

			\begin{enumerate}
				\item
					\sloppy
					These prepositions have interpretive differences,
						which we can explain with the Space $\Rightarrow$ Causation metaphor~\citep{Croft2012}
				\item
					We need a formalism that remains close to conceptual thinking about causation:
						force dynamics~\citep{CopleyEtAl2015,CopleyEtAl2020}
				\item
					This allows a polymorphic denotation of Agent prepositions with which we can predict the interpretive differences based on spatial meaning
			\end{enumerate}

		\item
			Space in the causal chain can be interpreted in different ways

			\begin{enumerate}
				\item
					In French, greater distance indicates a lack of influence \dots
				\item
					\dots\ but in Biblical Hebrew, greater distance indicates more control.
				\item
					This is acceptable if we understand that languages can have different {\sc perspectives on the causal chain}
			\end{enumerate}
	\end{enumerate}
\end{step}

\begin{step}[section]
	\section{Agent PPs}

	\subsection{\texorpdfstring{The Space $\Rightarrow$ Causation metaphor~\citep{Croft2012}}{The Space => Causation metaphor (Croft 2012)}}

	\citet{Croft2012} makes a distinction between roles that precede the Object in the causal chain and roles that follow it:

	\begin{exe}
		\ex
		\gls{Sue broke the coconut for Greg with a hammer.}
		\trailingcitation{\citep[224]{Croft2012}}
		\begin{tabular}{lllllll}
			Sue & $\rightarrow$ & hammer & $\rightarrow$ & coconut & $\dashrightarrow$ & Greg \\
			Subject &           & Antecedent oblique &   & Object  &                   & Subsequent oblique \\
		\end{tabular}
	\end{exe}
\end{step}

\begin{step}
	Antecedent obliques are commonly marked with ablative/perlative/proximative prepositions (\ref{antecedent}), and subsequent obliques with allative ones (\ref{subsequent}):

	\begin{exe}
		\ex \label{antecedent}
		\begin{xlist}
			\ex Cause: \gls{The rabbit died \imp{from}/\imp{of} thirst.}
				\trailingcitation{\citep[223]{Croft2012}}
			\ex Agent: \gls{The cat food was eaten \imp{by} raccoons.}
			\ex Means: \gls{I went downtown \imp{by} bus.}
			\ex Instrument: \gls{Sue broke the coconut \imp{with} a hammer.}
			\ex Comitative: \gls{I went to the park \imp{with} Carol.}
		\end{xlist}

		\ex \label{subsequent}
		\begin{xlist}
			\ex Result: \gls{They smashed the statue \imp{to} pieces.}
				\trailingcitation{\citep[223]{Croft2012}}
			\ex Result: \gls{The boy carved the stick \imp{into} a knife.}
			\ex Beneficiary: \gls{Sue broke the coconut \imp{for} Greg.}
		\end{xlist}
	\end{exe}
\end{step}

\begin{step}
	This suggests the {\sc Space $\Rightarrow$ Causation metaphor}:

	\begin{exe}
		\ex
		\begin{tabular}[t]{lccc}
			\textit{Causation:} & antecedent role & Object     & subsequent role \\
													& $\Uparrow$      & $\Uparrow$ & $\Uparrow$ \\
			\textit{Space:}     & ablative/origin & locative   & allative/goal \\
		\end{tabular}
		\trailingcitation{\citep[225]{Croft2012}}
	\end{exe}
\end{step}

%\begin{step}[subsection]
%	\subsection{Formal accounts of Agent PPs}
%	There are two main theories of Agent PPs.
%	In adjunct strategies, Agent PPs are adjuncts to Voice~\citep{Bruening2013,Legate2014,IngasonEtAl2016}:
%
%	\begin{exe}
%		\ex
%		\begin{xlist}
%			\ex
%				\gls{The senator was bribed by the lobbyist.}
%				\trailingcitation{(\citealp[25]{Bruening2013}, types added)}
%
%				{\tikzset{%
%					level 1+/.style={level distance=3\baselineskip},
%					level 1/.style={sibling distance=-60pt},
%					level 2/.style={sibling distance=-30pt},
%					level 3/.style={sibling distance=-20pt}}
%				\setstretch{1}
%				\Tree
%					[.{Pass \\ $st$}
%						[.{Pass[S:Voice(S:N)] \\ $(e)st,st$} ]
%						[.{Voice[S:N] \\ $st$}
%							[.{Voice[S:N] \\ $est$}
%								[.{Voice[S:V, S:N] \\ $st,est$} ]
%								[.{V \\ $st$}
%									[.{V[S:N] \\ $est$ \\ bribe} ]
%									[.{N \\ $e$ \\ the senator} ]
%								]
%							]
%							[.{P[S$_a$:Voice(S:N)] \\ $est,st$}
%								[.{P[S:N, S$_a$:Voice(S:N)] \\ \imp{$e,est,st$} \\ by} ]
%								[.{N \\ $e$ \\ the lobbyist} ]
%							]
%						]
%					]
%				}
%			\ex
%				$\sv{\mathit{by}} = \lambda x \lambda f_{est} \lambda e . f(e,x)$
%		\end{xlist}
%	\end{exe}
%\end{step}

\begin{step}[subsection]
	\subsection{A formal account of Agent PPs}
	In the formalization that we will adopt,
		Agent PPs merge in the same place as the external argument in the active voice~\citep{AngelopoulosEtAl2020}.
	This is preferable over an adjunct position because the Agent PP can bind a reflexive pronoun:

	\begin{exe}
		\ex
			\gls{The packages were sent by the children$_i$ to themselves$_i$.}
			\trailingcitation{(\cite[14]{AngelopoulosEtAl2020}, types added)}

			{\tikzset{%
				level 1+/.style={level distance=3\baselineskip},
				level 2/.style={sibling distance=30pt},
				level 3/.style={sibling distance=10pt}}
			\setstretch{1}
			\Tree
				[.{VoiceP \\ $st$}
					[.{Voice \\ $st,st$} ]
					[.{\textit{v}P \\ $st$}
						[.{PP$_i$ \\ $e$}
							[.{P \\ \imp{$ee$} \\ \gls{by}} ]
							[.{DP \\ $e$} \edge[roof]; {\gls{the children}$_i$} ]
						]
						[.{\textit{v}' \\ $est$}
							[.{\vphantom{P}\textit{v} \\ $st,est$} ]
							[.{VP \\ $st$} \edge[roof]; {send the packages to \gls{themselves}$_i$} ]
						]
					]
				]
			}
	\end{exe}

	But note that the Agent preposition is seen as a purely functional element with type $ee$,
		denoting the identity function: $\sv{\textit{by}} = \lambda x.x$.
	It thus assumes homonymy of the Agent preposition \gls{by} with spatial \gls{by}, as in \gls{the house by the lake}.
	There is no room for semantic content of the Agent preposition.
\end{step}

\begin{step}[subsection]
	\subsection{Multiple Agent prepositions: the case of French}
	In French passives the Agent can be introduced by both \gls{de} \enquote{from} and \gls{par} \enquote{by, via}.
	Which prepositions are allowed varies:

	\begin{exe}
		\ex \label{basic}
		\begin{xlist}
			\ex \label{chien}
				\gls{Le chien est lavé par/*de Marie.}
				\trailingcitation{\citep[584]{Straub1974}} \\
				`The dog was washed by Mary.'
			\ex \label{mois}
				\gls{Le mois de février est précédé du/*par le mois de janvier.}
				\trailingcitation{\citep[591]{Straub1974}} \\
				`February is preceded by January.'
			\ex \label{etudiantes}
				\gls{Les étudiantes sont accompagnées par/de leurs familles.}
				\trailingcitation{(after \citealp[200]{Gaatone1998})} \\
				`The students are accompanied by their families.'
			\ex \label{detenu}
				\gls{Le détenu est accompagné par le/*du policier.} \\
				`The inmate is accompanied by the policeman.'
		\end{xlist}
	\end{exe}
\end{step}

\begin{step}
	We will argue that the difference between \gls{de} and \gls{par} has to do with {\sc influence}:
	In (\ref{detenu}), the inmate is somehow \enquote{influenced} by the policeman,
		whereas the students in (\ref{etudiantes}) are not necessarily \enquote{influenced} by their parents.
	Hence \gls{de} marks a lack of influence.
	This can also be seen in (\ref{chien})--(\ref{mois}).
\end{step}

\begin{step}
	Although the French data confirm \citeauthor{Croft2012}'s (\citeyear{Croft2012}) Space $\Rightarrow$ Causation metaphor,
		his theory cannot explain the difference between \gls{de} and \gls{par}.
	Both mark an antecedent role, but \citeauthor{Croft2012} does not make further distinctions:

	\begin{quote}
		Although \imp{one cannot predict which participant roles a specific Oblique case marking will subsume}---%
			case markers are usually quite polysemous---%
			one can predict that a specific Oblique case marking will subsume only antecedent roles or only subsequent roles.
		That is, \imp{one can generally categorize Oblique morphosyntactic markers as either Antecedent or Subsequent}, as in (\ref{antecedent})--(\ref{subsequent}).
		\trailingcitation{\citep[223, emphasis added]{Croft2012}}
	\end{quote}
\end{step}

\begin{step}
	The formal approach also has trouble handling the French data.
	If the Agent preposition denotes the identity function,
		how can we differentiate between \gls{de} and \gls{par}?
	%Binding data shows that we cannot use an adjunct strategy:

	%\begin{exe}
	%	\ex
	%	\begin{xlist}
	%		\ex
	%			\gls{Les étudiantes$_i$ sont accompagnées par/de leurs parents$_j$ de leur$_{i\text{/}j}$ propre gré.}
	%			\trailingcitation{(after \citealp[200]{Gaatone1998})} \\
	%			`The students were accompanied by their parents out of their free will.'
	%		\ex
	%			\gls{Les étudiantes$_i$ étaient venues sans leurs parents$_j$ de leur$_{i\text{/*}j}$ propre gré.} \\
	%			`The students came without their parents out of their free will.'
	%	\end{xlist}
	%\end{exe}

	%But in argument strategies the preposition is taken to denote the identity function;
	%	with type $ee$, it has no access to the eventuality influenced by the complement.
\end{step}

\begin{step}
	We need a formalism that remains closer to intuitive thinking about causation.
	To be able to implement the ideas from \citet{Croft2012} we want a single denotation for each preposition,
		from which can derive both spatial and causal meanings.
	Force dynamics~\citep{Talmy1988,CopleyEtAl2015,CopleyEtAl2020} will help us with this.
\end{step}

\begin{step}[subsection]
	\subsection{\texorpdfstring{A polymorphic denotation of \gls{de} and \gls{par}}{A polymorphic denotation of de and par}}
	The use of \gls{de} and \gls{par} outside passives first suggests that force dynamics might be the right approach.
	\gls{De} is used to name Causes that are situations (\ref{cause-de}),
		while \gls{par} is used to name Causes that are forces (\ref{cause-par}).

	\begin{exe}
		\ex \label{cause-de}
		\begin{xlist}
			\ex
				\gls{Jean est mort de/*par} \{\gls{faim / vieillesse / la maladie de Parkinson}\}. \\
				`Jean died of hunger / old age / Parkinson's disease.'
			\ex
				\gls{Marie} \{\gls{s'est écriée / a gémi}\} \gls{de/*par} \{\gls{douleur / admiration / plaisir}\} / ??\gls{de l'impact du ballon}. \\
				`Marie cried out / groaned from pain / admiration / pleasure / the impact of the ball.'
		\end{xlist}
		\ex \label{cause-par}
			\gls{La fenêtre s'est cassée *de/par} \{\gls{un tremblement de terre / l'impact du ballon}\}. \\
			`The window broke due to an earthquake / the impact of the ball.'
	\end{exe}

	This is similar to what \citet[139--142]{CopleyEtAl2015} describe for English \gls{from},
		which can mark forces but not entities:

	\begin{exe}
		\ex
		\begin{xlist}
			\ex \gls{The floor broke from the *(weight of the) elephant.}
				\trailingcitation{\citep[141]{CopleyEtAl2015}}
			\ex \gls{The window broke from John*('s hitting it).}
		\end{xlist}
	\end{exe}

	With a distinction between entities (type $e$), forces (type $f$), and situations (type $s$)
		we can obtain a formalization that derives the correct facts for (\ref{cause-de})--(\ref{cause-par}),
		and the behaviour of \gls{de} and \gls{par} in passives as well.
\end{step}

\begin{step}
	We will give \gls{de} and \gls{par} an abstract definition based on its basic spatial meaning.
	The type is $\eta\theta t$, where $\eta$ and $\theta$ can be any type
		as long as the abstract spatial meaning has a reasonable interpretation for that type~(cf.~\cite[162]{Morrill1994}).

	\begin{exe}
		\ex
		\begin{xlist}
			\ex $\sv{\mathit{de}}_{\eta\theta t} = \lambda x_\eta \lambda y_\theta . \FROM(x,y)$, for any types $\eta, \theta$
			\ex $\sv{\mathit{par}}_{\eta\theta t} = \lambda x_\eta \lambda y_\theta . \THROUGH(x,y)$, for any types $\eta, \theta$
		\end{xlist}
	\end{exe}
\end{step}

\begin{step}
	For instance:

	\begin{exe}
		\ex
		\begin{xlist}
			\ex
				\gls{un médecin de Paris} \\
				`a doctor from Paris'
			\ex $\sv{\mathit{de}}_{eet} = \lambda x_e \lambda y_e . \FROM(x,y)$, interpreted as ``$y$ comes from $x$''.
		\end{xlist}
		\ex
		\begin{xlist}
			\ex
				\gls{le train à Lyon par Dijon} \\
				`the train to Lyon via Dijon'
			\ex $\sv{\mathit{par}}_{eet} = \lambda x_e \lambda y_e . \THROUGH(x,y)$, interpreted as ``$y$ goes through $x$''.
		\end{xlist}
	\end{exe}
\end{step}

\begin{step}
	For the Cause markers \gls{de} and \gls{par} we get:

	\begin{exe}
		\ex
		\begin{xlist}
			\ex
				\gls{Jean est mort de/*par faim.} \\
				`Jean died of hunger.'
			\ex $\sv{\mathit{de}}_{sst} = \lambda s_0 \lambda s_1 . \FROM(s_0, s_1)$, interpreted as $(\Net(s_0))(s_0) = s_1$.
		\end{xlist}
	\end{exe}
\end{step}

\begin{step}[list]
	\begin{exe}
		\ex
		\begin{xlist}
			\ex
				\gls{La fenêtre s'est cassée *de/par un tremblement de terre.} \\
				`The window broke due to an earthquake.'
			\ex $\sv{\mathit{par}}_{fst} = \lambda f \lambda s_1 . \THROUGH(f, s_1)$, interpreted as $\exists s_0 : \Net(s_0) = f \land f(s_0) = s_1$.
		\end{xlist}
	\end{exe}
\end{step}

\begin{step}
	Mapping the concepts of force dynamics onto \citeauthor{Croft2012}'s (\citeyear{Croft2012}) Space $\Rightarrow$ Causation metaphor,
		we can explain why \gls{de} has type $sst$ but not $fst$,
		and why \gls{par} has type $fst$ but not $sst$.

	\begin{exe}
		\ex
			\begin{tikzpicture}
				\node[circle,draw] (s0) at (0,0) {$s_0$};
				\node[circle,draw] (s1) at (6,0) {$s_1$};
				\draw[->] (s0) -- (s1) node[midway,above] {$f$};
				\node at (0,-1) {Ablative};
				\node at (0,-1.7) {\gls{de} `from'};
				\node at (3,-1) {Perlative};
				\node at (3,-1.7) {\gls{par} `through'};
				\node at (6,-1) {Object};
				\node at (6,-1.7) {passive subject};
			\end{tikzpicture}
	\end{exe}
\end{step}

\begin{step}[subsection]
	\subsection{Formalizing the Agent prepositions}
	We assume an intermediate projection which we will call OriginP to implement the argument strategy~\citep{AngelopoulosEtAl2020} in the framework of \citet{CopleyEtAl2020}:

	\begin{exe}
		\ex
			\gls{the soup was heated by John}

			{\tikzset{level 1+/.style={level distance=3\baselineskip}}
			\setstretch{1}
			\Tree
				[.{VoiceP \\ $fd$}
					[.{Voice$_{\text{PASS}}$ \\ $efd,fd$} ]
					[.{OriginP \\ $efd$}
						[.{PP \\ $ft$}
							[.{P \\ $eft$ \\ \gls{by}} ]
							[.{DP \\ $e$} \edge[roof]; {\gls{John}} ]
						]
						[.{Origin' \\ $efd$}
							[.{Origin \\ $fd,efd$} ]
							[.{\textit{v}P \\ $fd$} \edge[roof]; {\gls{heat the soup}} ]
						]
					]
				]
			}

		\ex
		\begin{xlist}
			\ex \label{denotation-origin}
				$\sv{\text{Origin}} = \lambda p_{fd} \lambda e \lambda f, \Origin(f,e) . p(f)$
			\ex \label{denotation-by}
				$\sv{\textit{by}} = \lambda e \lambda f . \Origin(f,e)$
			\ex \label{denotation-passive}
				$\sv{\text{Voice}_{\text{PASS}}} = \lambda p_{efd} \lambda f . \exists x : p(x,f)$
		\end{xlist}

		\ex
			$\sv{\text{VoiceP}}$ \\
			%$= (\lambda p_{efd} \lambda f . \exists x : p(x,f)) (\sv{\text{OriginP}})$
				%\hfill (\ref{denotation-passive}) \\
			\null\qquad$= \lambda f . \exists x : \sv{\text{OriginP}}(x,f)$
				\hfill (\ref{denotation-passive}) \\
				%\hfill (Function Application) \\
			\null\qquad$= \lambda f . \exists x, \sv{\text{PP}}(f) : \sv{\text{Origin'}}(x,f)$
				\hfill (Predicate Restriction) \\
			%$= \lambda f . \exists x, \sv{\text{PP}}(f) : (\lambda p_{fd} \lambda e \lambda f , \Origin(f,e) . p(f))(\sv{\text{\textit{v}P}})(x,f)$
				%\hfill (\ref{denotation-origin}) \\
			\null\qquad$= \lambda f . \exists x, \sv{\text{PP}}(f), \Origin(f,x) . \sv{\text{\textit{v}P}}(f)$
				\hfill (\ref{denotation-origin}) \\
				%\hfill (Function Application)
			\null\qquad$= \lambda f . \exists x, \Origin(f, \sv{\text{John}}), \Origin(f,x) . \sv{\text{\textit{v}P}}(f)$
				\hfill (\ref{denotation-by}) \\
			\null\qquad$= \lambda f, \Origin(f, \sv{\text{John}}) . \sv{\text{\textit{v}P}}(f)$
				\hfill (redundant existential bind)
	\end{exe}
\end{step}

\begin{step}
	So we're looking for an interpretation of \gls{de} and \gls{par} with type $eft$:

	\begin{exe}
		\exr{etudiantes}
			\gls{Les étudiantes sont accompagnées par/de leurs familles.}
			\trailingcitation{(after \citealp[200]{Gaatone1998})} \\
			`The students are accompanied by their families.'
		\ex \label{de-passive}
			$\sv{\mathit{de}}_{eft} = \lambda e \lambda f . \FROM(e, f)$, interpreted as $\Origin(e, f)$.
		\ex \label{par-passive}
			$\sv{\mathit{par}}_{eft} = \lambda e \lambda f . \THROUGH(e, f)$, interpreted as $\Agent(e,f)$.
	\end{exe}

	We hypothesize, based on \citet{Croft2012}, that spatial distance is related to distance to the Patient in the causal chain.
	A greater distance ($\FROM$) is interpreted as a general $\Origin$.
	The origin of the causal chain is used not only for Agents,
		but for anything that can be seen as the ultimate starting point of a force:
		volitional agents, but also natural phenomena or reasons, for example.

	An entity that is more proximal to the Patient ($\THROUGH$) more likely manipulates the Patient directly.
	We describe that here with $\Agent$, for lack of a better term.

	This link between proximity to the Patient and direct manipulation is not surprising.
	We see it also in the distinction between the English Agent marker \gls{by} and Instrument marker \gls{with}.
	Instruments manipulate the Patient more directly,
		and \gls{with} is more proximal than \gls{by}: cf.\ \gls{the girl by/with the bike}.
\end{step}

\begin{step}[subsection]
	\subsection{Corollaries} 

	The fact that \gls{par} is especially suited to mark direct manipulators (Agents)
		has an impact on the further pragmatic interpretation of \gls{de} and \gls{par}.
	This allows us to explain differences in the distribution of \gls{de} and \gls{par} that were previously not well-described.
	What we called {\sc influence} above can be derived from Proto-Agent (or Proto-Patient) properties (cf.\ \cite{Dowty1991}).
	Consider (\ref{detenu})--(\ref{detenu-no-influence}):
	
	\begin{exe}
		\exr{detenu}
			\gls{Le détenu est accompagné par le/*du policier.} \\
			`The inmate is accompanied by the policeman.'
		\ex \label{detenu-no-influence}
			\gls{L'ex-détenu est apparu devant le tribunal, accompagné par le/du policier qui l'avait alors arrêté.} \\
			`The former inmate appeared in front of the courthouse, accompanied by the policeman that had previously arrested him.'
	\end{exe}

	These examples establish a different relationship between the policeman and the inmate:
		only in (\ref{detenu}) do we expect the policeman to restrain the inmate in his movements.
	This difference can be described in terms of Proto-Agent / Proto-Patient properties:

	\begin{itemize}
		\item
			In (\ref{detenu}), the inmate is more \imp{affected},
				because he is less free to move around.
		\item
			In (\ref{detenu}), the policeman is more \imp{goal-oriented},
				because he has the concrete goal to prevent the inmate from escaping.
	\end{itemize}

	Example (\ref{detenu}) thus has a more prototypical Agent than (\ref{detenu-no-influence}),
		which explains why only \gls{par} is allowed.
\end{step}

\begin{step}
	This also works for examples with inanimate arguments:

	\begin{exe}
		\ex
		\begin{xlist}
			\ex \label{village-rainer}
				\gls{Rainer participe pour la première fois à une course de slalom en montagne. Cette course a eu lieu à Reitnau, petit village situé au coeur de la Suisse et surplombé \imp{par} une chaîne de montagne.}%
				\footnote{\url{http://www.kueschall.ch/fr/Archiv\_978.aspx}, retrieved November 13, 2010 by \url{http://web.archive.org}.} \\
				`Rainer takes part in a mountain slalom for the first time. This race took place in Reitnau, a small village located in the heart of Switzerland and overlooked \imp{by} a mountain range.'
			\ex \label{village-tourism}
				\gls{La première image offerte aux visiteurs est un village paroi surplombé \imp{de} ses deux tours et de son église auxquels on accède par un réseau de ruelles ou d'escaliers.}%
				\footnote{\url{https://www.saintmartinlevieil.fr/}, retrieved September 16, 2022.} \\
				`The first image offered to visitors is a walled village overlooked \imp{by} its two towers and its church which are accessed by a network of alleys or stairs.'
		\end{xlist}
	\end{exe}

	In (\ref{village-rainer}) our mental image of the village changes because of the mention of the mountain range:
		that is what makes it a suitable place for the mountain slalom.
	This is a more abstract reinterpretation of affectedness and goal-orientedness.
\end{step}

\begin{step}
	This can become as abstract as visual dominance:

	\begin{exe}
		\ex \label{surplomber-de}
		\begin{xlist}
			\ex
				\gls{de \dots\ hauts plateaux désolés surplombés \imp{de} sommets déchiquetés}%
				\footnote{\url{http://lesdeuxvoyageurs.com/Inde/Ladakh2005/Accueil\_Ladakh/Accueil\_Ladakh2005.html}, retrieved December 8, 2021.} \\
				`desolate highlands overlooked \imp{by} jagged mountain tops'
			\ex
				\gls{un mur d'enceinte surplombé \imp{de} barbelé}%
				\footnote{\url{http://www.haitiministries.com/www/nouvelles/}, retrieved December 8, 2021.} \\
				`a compound wall surmounted \imp{by} barbed wire'
		\end{xlist}

		\ex \label{surplomber-par}
		\begin{xlist}
			\ex
				\gls{un sentier surplombé \imp{par} des rochers de grès rose tout le long du parcours}%
				\footnote{\url{http://www.netrando.com/fr/direct/PHALDAB012.htm}, captured by Linguee.com, retrieved December 8, 2021.} \\
				`a small path that winds \imp{at the feet of} pink sandstone cliffs all along the way'
			\ex
				\gls{un stade surplombé \imp{par} un tremplin de saut à ski}%
				\footnote{\url{http://fr.fifa.com/tournaments/archive/tournament=102/edition=6946/news/newsid=88409.html}, captured by Linguee.com, retrieved December 8, 2021.} \\
				`a [soccer] stadium overlooked \imp{by} a ski jump'
		\end{xlist}
	\end{exe}

	In the examples in (\ref{surplomber-de}), neither participant is foregrounded with respect to the other,
		a balance emphasised by modifiers (\gls{désolés \dots\ déchiquetés}; \gls{d'enceinte}).
	This is not the case in (\ref{surplomber-par}), where the prepositional object really changes the way we see the subject.
\end{step}

\begin{step}
	By focusing on the force/Agent, \gls{par} can pick out a resultative interpretation of a verb
		when \gls{de} leads to a stative interpretation:

	\begin{exe}
		\ex
		\begin{xlist}
			\ex
				\gls{Notre-Dame délaissé \imp{de} ses touristes en plein confinement}%
				\footnote{\url{https://twitter.com/chouettephoto/status/1344600099113074691}, retrieved March 18, 2022.} \\
				`the Notre-Dame, abandoned \imp{of} its tourists in full lockdown'
			\ex
				\gls{En cas d'enfant délaissé \imp{par} ses parents (art. 501 de la charia), \dots}%
				\footnote{\url{http://www.arabhumanrights.org/publications/countries/lebanon/crc/lebanon3-05f.pdf}, captured by Linguee.com, retrieved March 18, 2022.} \\
				`In case of a child neglected \imp{by} its parents (art. 501 of the sharia), \dots'
		\end{xlist}
	\end{exe}
\end{step}

\begin{step}
	In this way, the spatial meaning of \gls{de} and \gls{par} still resonates in causal uses,
		and from there it can have various effects, depending on context.
\end{step}

\begin{step}[section]
	\section{Different perspectives on the causal chain}

	\subsection{Biblical Hebrew}
	The Biblical Hebrew prepositions \gls{min} `from, out of' and \gls{b\schwa} `in, by, against'
		present a similar problem as French \gls{de} and \gls{par}.
	Both are said to be Agent markers in the grammars, but we look here at (more frequent) other causal functions.

	\gls{B\schwa} is the default preposition for Instruments (\ref{instrument-be}), but \gls{min} occurs as well (\ref{instrument-min}).

	\begin{exe}
		\ex
		\begin{xlist}
			\ex \label{instrument-be}
				\heb{רַבִּ֗ים אֲשֶׁר־מֵ֨תוּ֙ בְּאַבְנֵ֣י הַבָּרָ֔ד מֵאֲשֶׁ֥ר הָרְג֛וּ בְּנֵ֥י יִשְׂרָאֵ֖ל בֶּחָֽרֶב׃}
				\trailingcitation{(Joshua 10:11 ESV)}
				\gll rabbîm \glot{}ăšer mē\b{t}û \imp{b\schwa{}}=\glot{}a\b{b}nē hab=bārā\b{d} mē=\glot{}ăšer hār\schwa\=gû b\schwa{}nē yiśrā\glot\=el \imp{b\schwa}=ḥāre\b{b} \\
				many {\sc rel} died {\imp{in}=stones of} the=hail from={\sc rel} killed {sons of} Israel \imp{in}=sword \\
				\glt `There were more who died \imp{because of} the hailstones than the sons of Israel killed \imp{with} the sword.'
			\ex \label{instrument-min}
				\heb{וְעַתָּ֗ה הוֹאֵל֙ וּבָרֵךְ֙ אֶת־בֵּ֣ית עַבְדְּךָ֔ ... וּמִבִּרְכָ֣תְךָ֔ יְבֹרַ֥ךְ בֵּֽית־עַבְדְּךָ֖ לְעוֹלָֽם׃}
				\trailingcitation{(2 Samuel 7:29 ESV)}
				\gll w\schwa=\phar{}attâ hô\glot\=el û=\b{b}ārē\b{k} \glot{}e\b{t} bê\b{t} \phar{}a\b{b}d-\schwa\b{k}ā \dots{} û=\imp{mib}=bir\b{k}ā\b{t}-\schwa\b{k}ā y\schwa\b{b}ōra\b{k} bêt \phar{}a\b{b}d-\schwa\b{k}ā l\schwa=\phar{}ôlām \\
				and=now want and=bless {\sc obj} {house of} servant-your \dots{} and=\imp{from}=blessing-your {will be blessed} house servant-your to=eternity \\
				\glt `Now therefore may it please you to bless the house of your servant, \dots, and \imp{with} your blessing shall the house of your servant be blessed forever.'
		\end{xlist}
	\end{exe}
\end{step}

\begin{step}
	Conversely, \gls{min} is the default preposition for Reasons (\ref{reason-min}), but \gls{b\schwa} occurs as well (\ref{reason-be}).

	\begin{exe}
		\ex
		\begin{xlist}
			\ex \label{reason-min}
				\heb{וַיְמַהֵ֣ר שָׁא֗וּל וַיִּפֹּ֤ל מְלֹא־קֽוֹמָתוֹ֙ אַ֔רְצָה וַיִּרָ֥א מְאֹ֖ד מִדִּבְרֵ֣י שְׁמוּאֵ֑ל}
				\trailingcitation{(1 Samuel 28:20 ESV)}
				\gll way\schwa{}mahēr šā\glot{}ûl wayyippōl m\schwa{}lō\glot{} qômā\b{t}-ô \glot{}arṣ-â wayyīrā\glot{} m\schwa\glot{}ō\b{d} \imp{mid}=di\b{b}rê š\schwa{}mû\glot{}ēl \\
				{and hastened} Saul {and fell} {full of} height-his earth-wards {and he feared} very {\imp{from}=words of} Samuel \\
				\glt `Then Saul fell at once full length on the ground, filled with fear \imp{because of} the words of Samuel.'
			\ex \label{reason-be}
				\heb{וְלֹֽא־תִכָּרֵ֥ת הָאָ֖רֶץ בָּרָעָֽב׃}
				\trailingcitation{(Gen 41:36 ESV)}
				\gll w\schwa=lō\glot{} \b{t}ikkārē\b{t} hā=\glot{}āreṣ \imp{b}=ā=rā\phar{}ā\b{b} \\
				and=not {shall be cut off ({\sc mid})} the=land \imp{in}=the=famine \\
				\glt `(That food shall be a reserve \dots), so that the land may not perish \imp{through} the famine.'
		\end{xlist}
	\end{exe}
\end{step}

\begin{step}
	We argue that \gls{min} `from' marks {\sc dominance} or {\sc full control},
		while \gls{b\schwa} `in, by, against' is used for entities that are not in full control:

	\begin{itemize}
		\item In (\ref{instrument-be}), the stones / sword are Instruments of other entities (God and the Israelites): no full control.
		\item In (\ref{instrument-min}), the blessing is so powerful that it lasts forever, hence it is dominant.
		\item In (\ref{reason-min}), ``at once'', ``full length'', and ``very'' indicate the complete fear that overcomes Saul: full control.
		\item In (\ref{reason-be}), the famine is not a dominant factor since the land has built up reserves.
	\end{itemize}
\end{step}

\begin{step}[subsection]
	\subsection{Biblical Hebrew: more examples}

	When a reason for joy is given, \gls{b\schwa} `in' is often used,
		while \gls{min} is used for reasons for fear.
	This is understandable, since fear is usually something you are overcome by,
		while joy is something that is more easily controlled.

	\begin{exe}
		\ex
		\begin{xlist}
			\ex
				\heb{שִׂמְחוּ֙ בַּאֲבִימֶ֔לֶךְ וְיִשְׂמַ֥ח גַּם־ה֖וּא בָּכֶֽם׃}
				\trailingcitation{(Judges 9:19 ESV)}
				\gll śimḥû \imp{ba}=\glot\upa\b{b}îmele\b{k} w\schwa=yiśmaḥ gam hû\glot{} \imp{bā}=\b{k}em \\
				rejoice \imp{in}=Abimelech and={let rejoice} also he \imp{in}=you \\
				\glt `rejoice \imp{in} Abimelech, and let him also rejoice \imp{in} you.'
			\ex
				\heb{וְלִשְׂמֹ֖חַ בַּעֲמָל֑וֹ זֹ֕ה מַתַּ֥ת אֱלֹהִ֖ים הִֽיא׃}
				\trailingcitation{(Ecclesiastes 5:19 ESV)}
				\gll w\schwa=li=śmōaḥ \imp{ba}=\phar\upa{}māl-ô zô mattat \glot\upe{}lōhîm hî\glot{} \\
				and=to=rejoice \imp{in}=toil-his this {gift of} God she \\
				\glt `(\dots) and [to] rejoice \imp{in} his toil---this is the gift of God.'
		\end{xlist}

		\ex
		\begin{xlist}
			\ex
				\heb{וַיַּסְתֵּ֤ר מֹשֶׁה֙ פָּנָ֔יו כִּ֣י יָרֵ֔א מֵהַבִּ֖יט אֶל־הָאֱלֹהִֽים׃}
				\trailingcitation{(Exodus 3:6 ESV)}
				\gll wayyastēr mōš\=ę pān-āw kî yārē\glot{} \imp{mē}=habbîṭ \glot{}el hā=\glot\upe{}lōhîm \\
				{and hid} Moses face-his because feared \imp{from}=look.{\sc inf} to the=God \\
				\glt `And Moses hid his face, for he was afraid to look at God.'
			\ex
				\heb{וְהִנֵּ֥ה קָרַ֖ן ע֣וֹר פָּנָ֑יו וַיִּֽירְא֖וּ מִגֶּ֥שֶׁת אֵלָֽיו׃}
				\trailingcitation{(Exodus 34:30 ESV)}
				\gll w\schwa=hinnēh qāran \phar{}ôr pān-āw wayyîr\schwa\glot{}û \imp{mig}=geše\b{t} \glot\=el-āw \\
				and=look shone skin face-his {and they feared} \imp{from}=approach.{\sc inf} to-him \\
				\glt `and behold, the skin of [Moses's] face shone, and they were afraid to come near him.'
		\end{xlist}
	\end{exe}

	Only \gls{min} `from' can render something impossible.
	This is understandable if only \gls{min} has enough control to completely rule something out:

	\begin{exe}
		\ex
			\heb{הַרְבָּ֥ה אַרְבֶּ֖ה אֶת־זַרְעֵ֑ךְ וְלֹ֥א יִסָּפֵ֖ר מֵרֹֽב׃}
			\trailingcitation{(Genesis 16:10 ESV)}
			\gll harbâ \glot{}arb\=ę \glot{}e\b{t} zar\phar\=e-\b{k} w\schwa=lō\glot{} yissā\=pēr \imp{mē}=rō\b{b} \\
			multiply.{\sc inf-abs} {I will multiply} {\sc obj} seed-yours and=not {it will be countable} \imp{from}=multitude \\
			\glt `I will surely multiply your offspring so that they cannot be numbered \imp{for} multitude.'
	\end{exe}

	And in Isaiah 28:7, \gls{b\schwa} and \gls{min} are used,
		together with \gls{yayin} `wine' / \gls{šē\b{k}ār} `strong drink' and various verbs,
		to build a climax:

	\begin{exe}
		\ex
		\begin{xlist}
			\ex \gls{ŠGH \imp{b\schwa{}} yayin} `go astray in wine';
			\ex \gls{T\phar{}H \imp{b\schwa{}} šē\b{k}ār} `stagger in strong drink';
			\ex \gls{ŠGH \imp{b\schwa{}} šē\b{k}ār} `go astray in strong drink';
			\ex \gls{BL\phar{} \imp{min} yayin} `be numbed from wine';
			\ex \gls{T\phar{}H \imp{min} šē\b{k}ār} `stagger from strong drink'.
		\end{xlist}
	\end{exe}
\end{step}

\begin{step}[subsection]
	\subsection{Comparison with French}

	At first, these results seem to conflict with what we said for French.
	In Hebrew the more distal preposition marks higher dominance / control,
		while in French the more distal preposition marked less influence:

	\begin{exe}
		\ex
			Biblical Hebrew:

			\begin{tabular}[t]{llllll}
				\emph{Causation:} & Dominance / full control & $\rightarrow$ & Less control & $\rightarrow$ & Patient\\
				                  & \gls{min} `from' &                       & \gls{b\schwa} `in, by, against' \\
				\emph{Space:}     & Ablative &                               & Locative/Proximative \\
			\end{tabular}
	\end{exe}
\end{step}

\begin{step}[list]
	\begin{exe}
		\ex
			French:

			\begin{tabular}[t]{llllll}
				\emph{Causation:} & No influence & $\rightarrow$ & Influence & $\rightarrow$ & Patient\\
				                  & \gls{de} `from' &            & \gls{par} `through' \\
				\emph{Space:}     & Ablative &                   & Perlative \\
			\end{tabular}
	\end{exe}
\end{step}

\begin{step}
	Still, both languages can be understood using \citeauthor{Croft2012}'s (\citeyear{Croft2012}) Space $\Rightarrow$ Causation metaphor.
	It is logical if the Ablative marks more control,
		because the entity at the start of the causal chain is not controlled itself (Biblical Hebrew).
	But it is also logical if the Ablative marks less influence,
		because the entity is not necessarily in direct contact with the Patient (French).
\end{step}

\begin{step}
	We can explain these seemingly contradictory results with different perspectives on the causal chain:

	\begin{itemize}
		\item
			French has a {\sc Patient-oriented perspective}:
				in this language, \gls{de} and \gls{par} developed causal semantics
				through recycling the distance they express to the Patient.
			It is not crucial that \gls{de} marks the absolute beginning of the causal chain,
				but the relative proximity to the Patient is important.

		\item
			Biblical Hebrew has a {\sc Origin-oriented perspective}:
				\gls{b\schwa} and \gls{min} developed causal semantics
				based on the distance to the beginning of the causal chain, rather than the Patient.
			It is crucial that \gls{min} points to the absolute beginning of the causal chain
				for it to express dominance or full control.
	\end{itemize}
\end{step}

\begin{step}[section]
	\section{Conclusion}

	\subsection{Formalizing Agent prepositions}
	Common formal theories of Agent prepositions take the Agent preposition as a purely functional element~(e.g.\ \cite{AngelopoulosEtAl2020}).
	They have trouble accounting for languages with multiple Agent prepositions,
		where the spatial meaning of these prepositions has not been lost entirely~(cf.\ \cite{Croft2012}).

	Force dynamics~\citep{CopleyEtAl2020} provides a formalism that remains closer to intuitive thinking about causation.
	The distinctions it makes between entity, force, and situation are useful to formalize differences between Agent prepositions.
	We have proposed a way to implement Agent prepositions à la \citet{AngelopoulosEtAl2020} in the framework of \citet{CopleyEtAl2020}.
\end{step}

\begin{step}[subsection]
	\subsection{\texorpdfstring{Making the Space $\Rightarrow$ Causation metaphor more fine-grained}{Making the Space => Causation metaphor more fine-grained}}
	\citeauthor{Croft2012} only distinguishes Antecedent and Subsequent Obliques,
		and actively warns against going any further:

	\begin{quote}
		Although \imp{one cannot predict which participant roles a specific Oblique case marking will subsume}---%
			case markers are usually quite polysemous---%
			one can predict that a specific Oblique case marking will subsume only antecedent roles or only subsequent roles.
		That is, \imp{one can generally categorize Oblique morphosyntactic markers as either Antecedent or Subsequent}, as in (\ref{antecedent})--(\ref{subsequent}).
		\trailingcitation{\citep[223, emphasis added]{Croft2012}}
	\end{quote}

	There is indeed no universal mapping between spatial categories and causal concepts.
	However, we \emph{can} make the metaphor more fine-grained when looking at a concrete language:
		\imp{relative position in space} mirrors \imp{relative position in the causal chain}.
	Thus we can describe the system of French as ``proximity to Patient is influence'',
		and the system of Biblical Hebrew as ``proximity to Origin is control''.
\end{step}

\begin{step}
	We can do the same for English:

	\begin{exe}
		\exr{antecedent}
		\begin{xlist}
			\ex Cause: \gls{The rabbit died \imp{from}/\imp{of} thirst.}
				\trailingcitation{\citep[223]{Croft2012}}
			\ex Agent: \gls{The cat food was eaten \imp{by} raccoons.}
			\ex Means: \gls{I went downtown \imp{by} bus.}
			\ex Instrument: \gls{Sue broke the coconut \imp{with} a hammer.}
			\ex Comitative: \gls{I went to the park \imp{with} Carol.}
		\end{xlist}

		\exr{subsequent}
		\begin{xlist}
			\ex Result: \gls{They smashed the statue \imp{to} pieces.}
				\trailingcitation{\citep[223]{Croft2012}}
			\ex Result: \gls{The boy carved the stick \imp{into} a knife.}
			\ex Beneficiary: \gls{Sue broke the coconut \imp{for} Greg.}
		\end{xlist}

		\ex
		\begin{tabular}[t]{llllll}
			Cause $\>\rightarrow$ & Agent $\>\rightarrow$ & Comitative/Instrument $\>\rightarrow$ & Patient $\>\rightarrow$ & Result $\>\rightarrow$ & Beneficiary \\
			\gls{of, from}        & \gls{by}              & \gls{by, with}                        &                         & \gls{to, into}         & \gls{for} \\
			Ablative              & Proximative           & Closer proximative                    &                         & Allative               & Closer allative \\
		\end{tabular}
	\end{exe}

	{\footnotesize
		(For \enquote{closer proximative}, consider the difference between \gls{the house by/with the lake}.
		For \enquote{closer allative}, consider that \gls{(be)for(e)} is the endpoint of the direction expressed by \gls{to}.)}
\end{step}

\begin{step}[subsection]
	\subsection{Perspectives on the causal chain}
	Furthermore, we suggested that languages may have different perspectives on the causal chain.
	We identified the Patient-oriented perspective of French and the Origin-oriented perspective of Biblical Hebrew.
	This makes \citeauthor{Croft2012}'s~(\citeyear{Croft2012}) theory more predictive while remaining falsifiable.
\end{step}

\begin{step}
	Natural follow-up questions are:

	\begin{enumerate}
		\item Are there other perspectives besides Origin-oriented and Patient-oriented?
		\item Is the perspective a language-wide parameter or can different prepositions in the same language take a different perspective?
		\item What determines the perspective of a language (or of a preposition)?
	\end{enumerate}
\end{step}

\clearpage
\begin{step}[section]
	\vspace{.8em}
	\setstretch{1}
	\def\enotesize{\normalsize}
	\theendnotes

	\printbibliography
\end{step}

\end{document}