content/counterfactuals/minimal-change-semantics/minimal-change-semantics.tex
1% Part: counterfactuals2% Chapter: minimal-change-semantics3\documentclass[../../../include/open-logic-chapter]{subfiles}5\begin{document}7\olchapter{cnt}{min}{Minimal Change Semantics}9\olimport{introduction}11\olimport{sphere-models}13\olimport{true-false}15\olimport{antecedent-strengthening}17\olimport{transitivity}19\olimport{contraposition}21\OLEndChapterHook23\end{document}25
content/counterfactuals/minimal-change-semantics/introduction.tex
1% Part: counterfactuals2% Chapter: minimal-change-semantics3% Section: introduction4\documentclass[../../../include/open-logic-section]{subfiles}6\begin{document}8\olfileid{cnt}{min}{int}10\olsection{Introduction}12Stalnaker and Lewis proposed accounts of counterfactual conditionals14such as ``If the match were struck, it would light.'' Their accounts15were proposals for how to properly understand the truth conditions for16such sentences. The idea behind both proposals is this: to evaluate17whether a counterfactual conditional is true, we have to consider18those possible worlds which are minimally different from the way the19world actually is to make the antecedent true. If the consequent is20true in these possible worlds, then the counterfactual is true. For21instance, suppose I hold a match and a matchbook in my hand. In the22actual world I only look at them and ponder what would happen if I23were to strike the match. The minimal change from the actual world24where I strike the match is that where I decide to act and strike the25match. It is minimal in that nothing else changes: I don't also jump26in the air, striking the match doesn't also light my hair on fire, I27don't suddenly lose all strength in my fingers, I am not simultaneously28doused with water in a SuperSoaker ambush, etc. In that alternative29possibility, the match lights. Hence, it's true that if I were to30strike the match, it would light.31This intuitive account can be paired with formal semantics for logics33of counterfactuals. Lewis introduced the symbol ``$\boxright$'' for34the counterfactual while Stalnaker used the symbol~``$>$''. We'll35use~$\cif$, and add it as a binary connective to propositional36logic. So, we have, in addition to !!{formula}s of the form $!A \lif37!B$ also !!{formula}s of the form~$!A \cif !B$. The formal semantics,38like the relational semantics for modal logic, is based on models in39which !!{formula}s are evaluated at worlds, and the satisfaction40condition defining $\mSat{M}{!A \cif !B}[w]$ is given in terms of41$\mSat{M}{!A}[w']$ and $\mSat{M}{!B}[w']$ for some (other)42worlds~$w'$. Which $w'$? Intuitively, the one(s) closest to~$w$ for43which it holds that~$\mSat{M}{!A}[w']$. This requires that a relation44of ``closeness'' has to be included in the model as well.45Lewis introduced an instructive way of representing counterfactual47situations graphically. Each possible world is at the center of a set48of nested spheres containing other worlds---we draw these spheres as49concentric circles. The worlds between two spheres are equally close50to the world at the center as each other, those contained in a nested51sphere are closer, and those in a surrounding sphere further away.52\begin{center}53\begin{tikzpicture}[scale=.7]\small54 \spheresystem{5}55 \draw (0,0) node {$w$};56 \propositionintersect{0}{4}{40}{3.5}57 {\tikzset{proposition/.append={smooth,tension=1.4}}58 \proposition[shift={(1.6,-1)}]{90}{1}{30}{4}}59 \path (1.5,3) node[below] {$\formula{B}$};60 \path (3,0) node {$\formula{A}$};61\end{tikzpicture}62\end{center}63The closest $!A$-worlds are those worlds $w'$ where~$!A$ is satisfied64which lie in the smallest sphere around the center world~$w$ (the gray65area). Intuitively, $!A \cif !B$ is satisfied at~$w$ if $!B$ is true66at all closest $!A$-worlds.67\end{document}
content/counterfactuals/minimal-change-semantics/sphere-models.tex
1% Part: conditional-logics2% Chapter: minimal-change-semantics3% Section: sphere-models4\documentclass[../../../include/open-logic-section]{subfiles}6\begin{document}8\olfileid{con}{min}{sph}10\olsection{Sphere Models}12One way of providing a formal semantics for counterfactuals is to turn14Lewis's informal account into a mathematical structure. The spheres15around a world~$w$ then are sets of worlds. Since the spheres are16nested, the sets of worlds around~$w$ have to be linearly ordered by17the subset relation.18\begin{defn}20 A \emph{sphere model} is a triple~$\mModel{M} = \tuple{W, O, V}$21 where $W$ is a non-empty set of worlds, $V\colon \PVar \to \Pow{W}$22 is a valuation, and $O\colon W \to \Pow{\Pow{W}}$ assigns to each23 world~$w$ a \emph{system of spheres}~$O_w$. For each $w$, $O_w$ is a24 set of sets of worlds, and must satisfy:25 \begin{enumerate}26 \item $O_w$ is \emph{centered} on~$w$: $\{w\} \in O_w$.27 \item $O_w$ is \emph{nested}: whenever $S_1$, $S_2 \in O_w$, $S_128 \subseteq S_2$ or $S_2 \subseteq S_1$, i.e., $O_w$ is linearly29 ordered by~$\subseteq$.30 \item $O_w$ is closed under non-empty unions.31 \item $O_w$ is closed under non-empty intersections.32 \end{enumerate}33\end{defn}34The intuition behind $O_w$ is that the worlds ``around'' $w$ are36stratified according to how far away they are from~$w$. The innermost37sphere is just $w$ by itself, i.e., the set~$\{w\}$: $w$ is closer38to~$w$ than the worlds in any other sphere. If $S \subsetneq S'$, then39the worlds in $S' \setminus S$ are further way from $w$ than the40worlds in~$S$: $S' \setminus S$ is the ``layer'' between the $S$ and41the worlds outside of~$S'$. In particular, we have to think of the42spheres as containing all the worlds within their outer surface; they43are not just the individual layers.44\begin{figure}46\begin{center}47\begin{tikzpicture}[layerwidth=1.5,scale=.6]\tiny48 \spheresystem{3}49 \propositionintersect{0}{3}{60}{6}50 \path (0,0) node[world] {$w$};51 \spherepos{-30}{2}{node[world] {$w_2$}}52 \spherepos{220}{2}{node[world] {$w_3$}}53 \spherepos{90}{2}{node[world] {$w_1$}}54 \spherepos[shift={(.1,0)}]{10}{3}{node[world] {$w_5$}}55 \spherepos[shift={(.1,0)}]{-10}{3}{node[world] {$w_6$}}56 \spherepos{225}{3}{node[world] {$w_4$}}57 \spherepos{20}{4}{node[world] {$w_7$}}58 \path (5,0) node[left] {$p$};59\end{tikzpicture}60\caption{Diagram of a sphere model}61\ollabel{fig:sphere-model}62\end{center}63\end{figure}64The diagram in \olref{fig:sphere-model} corresponds to the sphere66model with $W = \{w, w_1, \dots, w_7\}$, $V(p) = \{w_5, w_6,67w_7\}$. The innermost sphere $S_1 = \{w\}$. The closest worlds to $w$68are $w_1, w_2, w_3$, so the next larger sphere is $S_2 = \{w, w_1,69w_2, w_3\}$. The worlds further out are $w_4$, $w_5$, $w_6$, so the70outermost sphere is $S_3 = \{w, w_1, \dots, w_6\}$. The system of71spheres around $w$ is $O_w = \{S_1, S_2, S_3\}$. The world~$w_7$ is72not in any sphere around~$w$. The closest worlds in which $p$~is true73are $w_5$ and~$w_6$, and so the smallest $p$-admitting sphere74is~$S_3$.75To define satisfaction of a formula~$!A$ at world~$w$ in a sphere77model~$\mModel M$, $\mSat{M}{!A}[w]$, we expand the definition for78modal !!{formula}s to include a clause for $!B \cif !C$:79\begin{defn}81 $\mSat{M}{!B \cif !C}[w]$ iff either82 \begin{enumerate}83 \item\ollabel{sphere-vac} For all $u \in \bigcup O_w$, $\mSat/{M}{!B}[u]$, or84 \item\ollabel{sphere-nonvac} For some $S \in O_w$,85 \begin{enumerate}86 \item $\mSat{M}{!B}[u]$ for some $u \in87 S$, and88 \item for all $v \in S$, either89 $\mSat/{M}{!B}[v]$ or $\mSat{M}{!C}[v]$.90 \end{enumerate}91 \end{enumerate}92\end{defn}93According to this definition, $\mSat{M}{!B \cif !C}[w]$ iff either the95antecedent~$!B$ is false everywhere in the spheres around~$w$, or96there is a sphere~$S$ where $!B$ is true, and the material conditional97$!B \lif !C$ is true at all worlds in that ``$!B$-admitting''98sphere. Note that we didn't require in the definition that $S$ is the99\emph{innermost} $!B$-admitting sphere, contrary to what one might100expect from the intuitive explanation. But if the condition101in~\olref{sphere-nonvac} is satisfied for some sphere~$S$, then it is102also satisfied for all spheres~$S$ contains, and hence in particular103for the innermost sphere.104Note also that the definition of sphere models does not require that106there \emph{is} an innermost $!B$-admitting sphere: we may have an107infinite sequence $S_1 \supsetneq S_2 \supsetneq \dots \supsetneq108\{w\}$ of $!B$-admitting spheres, and hence no innermost109$!B$-admitting spheres. In that case, $\mSat{M}{!B \cif !C}[w]$ iff110$!B \lif !C$ holds throughout the spheres $S_i$, $S_{i+1}$, \dots, for111some~$i$.112\end{document}
content/counterfactuals/minimal-change-semantics/true-false.tex
1% Part: conditional-logics2% Chapter: minimal-change-semantics3% Section: true-false4\documentclass[../../../include/open-logic-section]{subfiles}6\begin{document}8\olfileid{con}{min}{tf}10\olsection{Truth and Falsity of Counterfactuals}12A counterfactual $!A \cif !B$ is (non-vacuously) true if the closest14$!A$-worlds are all $!B$-worlds, as depicted in \olref{fig:true}.15\begin{figure}16\begin{center}17\begin{tikzpicture}[scale=.7]\small18 \spheresystem{5}19 \draw (0,0) node {$w$};20 \propositionintersect{0}{4}{40}{3.5}21 {\tikzset{proposition/.append={smooth,tension=1.4}}22 \proposition[shift={(1.6,-1)}]{90}{1}{30}{4}}23 \path (1.5,3) node[below] {$\formula{B}$};24 \path (3,0) node {$\formula{A}$};25\end{tikzpicture}26\caption{Non-vacuously true counterfactual}27\ollabel{fig:true}28\end{center}29\end{figure}30A counterfactual is also true at $w$ if the system of spheres31around~$w$ has no $!A$-admitting spheres at all. In that case it is32\emph{vacuously} true (see \olref{fig:vacuous}).33\begin{figure}34\begin{center}35\begin{tikzpicture}[scale=.7]\small36 \spheresystem{5}37 \draw (0,0) node {$w$};38 \proposition{0}{6.5}{40}{3.5}39 {\tikzset{proposition/.append={smooth,tension=1.4}}40 \proposition[shift={(1.2,-1)}]{90}{1}{30}{4}}41 \path (1.5,3) node[below] {$\formula{B}$};42 \spherepos{0}{7}{node {$\formula{A}$}}43\end{tikzpicture}44\caption{Vacuously true counterfactual}45\ollabel{fig:vacuous}46\end{center}47\end{figure}48It can be false in two ways. One way is if the closest $!A$-worlds are50not all $!B$-worlds, but some of them are. In this case, $!A \cif51\lnot !B$ is also false (see \olref{fig:false}).52\begin{figure}53\begin{center}54\begin{tikzpicture}[scale=.7]\small55 \spheresystem{5}56 \draw (0,0) node {$w$};57 \propositionintersect{0}{4}{40}{3.5}58 {\tikzset{proposition/.append={smooth,tension=1.4}}59 \proposition[shift={(1.6,0)}]{90}{1}{30}{3}}60 \path (1.5,3) node[below] {$\formula{B}$};61 \path (3,0) node {$\formula{A}$};62\end{tikzpicture}63\caption{False counterfactual, false opposite}64\ollabel{fig:false}65\end{center}66\end{figure}67If the closest $!A$-worlds do not overlap with the $!B$-worlds at all,68then $!A \cif !B$ is false. But, in this case all the closest $!A$-worlds are69$\lnot !B$-worlds, and so $!A \cif \lnot !B$ is true (see70\olref{fig:false-opposite}).71\begin{figure}72\begin{center}73\begin{tikzpicture}[scale=.7]\small74 \spheresystem{5}75 \draw (0,0) node {$w$};76 \propositionintersect{0}{4}{40}{3.5}77 {\tikzset{proposition/.append={smooth,tension=1.4}}78 \proposition[shift={(-1,0)}]{90}{1}{30}{3}}79 \path (-1,3) node[below] {$\formula{B}$};80 \path (3,0) node {$\formula{A}$};81\end{tikzpicture}82\caption{False counterfactual, true opposite}83\ollabel{fig:false-opposite}84\end{center}85\end{figure}86In contrast to the strict conditional, counterfactuals may be88contingent. Consider the sphere model in \olref{fig:contingent}. The89$!A$-worlds closest to~$u$ are all $!B$-worlds, so $\mSat{M}{!A \cif90 !B}[u]$. But there are $!A$-worlds closest to~$v$ which are not91$!B$-worlds, so $\mSat/{M}{!A \cif !B}[v]$.92\begin{figure}93\begin{center}94\begin{tikzpicture}[scale=.6]\small95 \spheresystem{7}96 \spheresystem[shift={(0:3.1)},dashed]{4}97 \propositionintersect{45}{4}{40}{4.5}98 \begin{scope}99 \clip \propositionplot{45}{4}{40}{4.5} ;100 \spherelayer[shift={(0:3.1)}]{3}101 \end{scope}102 \proposition[shift={(1.4,-.5)}]{90}{1}{35}{5}103 \path (0,0) node {$u$};104 \path (0:3.1) node {$v$};105 \spherepos{35}{9}{node {$\formula{A}$}}106 \spherepos{80}{9}{node {$\formula{B}$}}107\end{tikzpicture}108\end{center}109\caption{Contingent counterfactual}110\ollabel{fig:contingent}111\end{figure}112\end{document}
content/counterfactuals/minimal-change-semantics/antecedent-strengthening.tex
1% Part: counterfactuals2% Chapter: minimal-change-semantics3% Section: antecedent-strengthening4\documentclass[../../../include/open-logic-section]{subfiles}6\begin{document}8\olfileid{cnt}{min}{agg}10\olsection{Antecedent Strengthening}12``Strengthening the antecedent'' refers to the14inference $!A \lif !C \Entails (!A \land !B) \lif !C$. It is valid15for the material conditional, but invalid for counterfactuals. Suppose16it is true that if I were to strike this match, it would light. (That17means, there is nothing wrong with the match or the matchbook surface,18I will not break the match, etc.) But it is not true that if I were to19light this match in outer space, it would light. So the following20inference is invalid:21\begin{quote}22 If the match were struck, it would light.23 Therefore, if the match were struck in outer space, it would light.25\end{quote}26The Lewis--Stalnaker account of conditionals explains this: the closest28world where I light the match and I do so in outer space is much29further removed from the actual world than the closest world where I30light the match is. So although it's true that the match lights in the31latter, it is not in the former. And that is as it should be.32\begin{ex}34 The sphere semantics invalidates the inference, i.e., we have $p35 \cif r \Entails/ (p \land q) \cif r$. Consider the model $\mModel{M}36 = \tuple{W, O, V}$ where $W = \{w, w_1, w_2\}$, $O_w = \{\{w\}, \{w,37 w_1\}, \{w, w_1, w_2\}\}$, $V(p) = \{w_1, w_2\}$, $V(q) = \{w_2\}$,38 and $V(r) = \{w_1\}$. There is a $p$-admitting sphere $S = \{w,39 w_1\}$ and $p \lif r$ is true at all worlds in it, so $\mSat{M}{p40 \cif r}[w]$. There is also a $(p \land q)$-admitting sphere $S' =41 \{w, w_1, w_2\}$ but $\mSat/{M}{(p \land q) \lif r}[w_2]$, so42 $\mSat/{M}{(p \land q) \cif r}[w]$ (see \olref{fig:antecedent-strengthening}).43\begin{figure}45\begin{center}46\begin{tikzpicture}[layerwidth=1.5,scale=.6]\tiny47 \spheresystem{3}48 \propositionintersect{-10}{3}{30}{5}49 \begin{scope}50 \clip plot[smooth,tension=1] coordinates51 {(0,4.5) (30:.95) (4.5,-3.5)} -- (4.5,4.5) ;52 \spherefill{2}53 \end{scope}54 \draw[proposition] plot[dashed,smooth,tension=1] coordinates55 {(0,4.5) (30:.95) (4.5,-3.5)} ;56 \proposition{30}{2}{30}{5}57 \path (0,0) node[world] {$w$};58 \spherepos{30}{2}{node[world] {$w_1$}}59 \spherepos{-10}{3}{node[world] {$w_2$}}60 \spherepos{-10}{4}{node {$q$}}61 \spherepos{30}{4}{node {$r$}}62 \draw (1,4.5) node[below] {$p$};63\end{tikzpicture}64\caption{Counterexample to antecedent strengthening}65\ollabel{fig:antecedent-strengthening}66\end{center}67\end{figure}68\end{ex}69\end{document}
content/counterfactuals/minimal-change-semantics/transitivity.tex
1% Part: counterfactuals2% Chapter: minimal-change-semantics3% Section: transitivity4\documentclass[../../../include/open-logic-section]{subfiles}6\begin{document}8\olfileid{cnt}{min}{tra}10\olsection{Transitivity}12For the material conditional, the chain rule holds: $!A \lif !B, !B14\lif !C \Entails !A \lif !C$. In other words, the material conditional15is transitive. Is the same true for counterfactuals? Consider the16following example due to Stalnaker.17\begin{quote}18 If J.~Edgar Hoover had been born a Russian, he would have been a Communist.19 If J.~Edgar Hoover were a Communist, he would have been be a traitor.21 Therefore, If J.~Edgar Hoover had been born a Russian, he would have23 been be a traitor.24\end{quote}25If Hoover had been born (at the same time he actually did), not in the26United States, but in Russia, he would have grown up in the Soviet27Union and become a Communist (let's assume). So the first premise is28true. Likewise, the second premise, considered in isolation is29true. The conclusion, however, is false: in all likelihood, Hoover30would have been a fervent Communist if he had been born in the USSR,31and not been a traitor (to his country). The intuitive assignment of32truth values is borne out by the Stalnaker--Lewis account. The closest33possible world to ours with the only change being Hoover's place of34birth is the one where Hoover grows up to be a good citizen of the35USSR. This is the closest possible world where the antecedent of the36first premise and of the conclusion is true, and in that world Hoover37is a loyal member of the Communist party, and so not a traitor. To38evaluate the second premise, we have to look at a different world,39however: the closest world where Hoover is a Communist, which is one40where he was born in the United States, turned, and thus became a41traitor.\footnote{Of course, to appreciate the force of the example we42 have to take on board some metaphysical and political assumptions,43 e.g., that it is possible that Hoover could have been born to44 Russian parents, or that Communists in the US of the 1950s were45 traitors to their country.}46\begin{prob}48 Find a convincing, intuitive example for the failure of transitivity49 of counterfactuals.50\end{prob}51\begin{ex}\ollabel{ex:trans-counterex}53 The sphere semantics invalidates the inference, i.e., we have $p54 \cif q, q \cif r \Entails/ p \cif r$. Consider the model $\mModel{M}55 = \tuple{W, O, V}$ where $W = \{w, w_1, w_2\}$, $O_w = \{\{w\}, \{w,56 w_1\}, \{w, w_1, w_2\}\}$, $V(p) = \{w_2\}$, $V(q) = \{w_1, w_2\}$,57 and $V(r) = \{w_1\}$. There is a $p$-admitting sphere $S = \{w, w_1,58 w_2\}$ and $p \lif q$ is true at all worlds in it, so $\mSat{M}{p59 \cif q}[w]$. There is also a $q$-admitting sphere $S' = \{w,60 w_1\}$ and $\mSat/{M}{q \lif r}$ is true at all worlds in it, so61 $\mSat{M}{q \cif r}[w]$. However, the $p$-admitting sphere $\{w,62 w_1, w_2\}$ contains a world, namely~$w_2$, where $\mSat/{M}{p \lif63 r}[w_2]$.64\end{ex}65\begin{prob}67Draw the sphere diagram corresponding to the counterexample in68\olref[cnt][min][tra]{ex:trans-counterex}.69\end{prob}70\begin{prob}72 In \olref[cnt][min][tra]{ex:trans-counterex}, world $w_2$ is where73 Hoover is born in Russia, is a communist, and not a traitor, and74 $w_1$ is the world where Hoover is born in the US, is a communist,75 and a traitor. In this model, $w_1$ is closer to~$w$ than $w_2$76 is. Is this necessary? Can you give a counterexample that does77 not assume that Hoover's being born in Russia is a more remote78 possibility than him being a Communist?79\end{prob}80\end{document}
content/counterfactuals/minimal-change-semantics/contraposition.tex
1% Part: counterfactuals2% Chapter: minimal-change-semantics3% Section: transitivity4\documentclass[../../../include/open-logic-section]{subfiles}6\begin{document}8\olfileid{cnt}{min}{cpo}10\olsection{Contraposition}12Material and strict conditionals are equivalent to their14contrapositives. Counterfactuals are not. Here is an example due to15Kratzer:16\begin{quote}17 If Goethe hadn't died in 1832, he would (still) be dead now.18 If Goethe weren't dead now, he would have died in 1832.20\end{quote}21The first sentence is true: humans don't live hundreds of years. The22second is clearly false: if Goethe weren't dead now, he would be still23alive, and so couldn't have died in 1832.24\begin{ex}\ollabel{ex:contraposition-counterex}26 The sphere semantics invalidates contraposition, i.e., we have $p27 \cif q \Entails/ \lnot q \cif \lnot p$. Think of $p$ as ``Goethe28 didn't die in 1832'' and $q$ as ``Goethe is dead now.'' We can29 capture this in a model $\mModel{M_1} = \tuple{W, O, V}$ with $W =30 \{w, w_1, w_2\}$, $O = \{\{w\}, \{w, w_1\}, \{w, w_1, w_2\}\}$,31 $V(p) = \{w_1, w_2\}$ and $V(q) = \{w, w_1\}$. So $w$ is the actual32 world where Goethe died in 1832 and is still dead; $w_1$ is the33 (close) world where Goethe died in, say, 1833, and is still dead;34 and $w_2$ is a (remote) world where Goethe is still alive.35\begin{figure}36\begin{center}37\begin{tikzpicture}[scale=.6,layerwidth=1.5]\tiny38 \spheresystem{3}39 \begin{scope}40 \clip \propositionplot{0}{.8}{50}{4.5} -- (4.5,-4.5) -- (-4.5,-4.5) -- (-4.5,4.5) -- (4.5,4.5);41 \spherelayer{3}42 \end{scope}43 \propositionintersect{0}{2}{110}{5.5}44 \proposition{0}{.8}{50}{4.5}45 \path (0,0) node[world] {$w$};46 \spherepos{0}{2}{node[world] {$w_1$}}47 \spherepos{-50}{3}{node[world] {$w_2$}}48 \spherepos{28}{3}{node {$q$}}49 \spherepos{42}{3}{node {$\lnot q$}}50 \spherepos{-55}{4}{node {$p$}}51 \spherepos{-67}{4}{node {$\lnot p$}}52\end{tikzpicture}53\caption{Counterexample to contraposition}54\ollabel{fig:contraposition}55\end{center}56\end{figure}57There is a $p$-admitting sphere $S =58\{w, w_1\}$ and $p \lif q$ is true at all worlds in it, so $\mSat{M}{p59 \cif q}[w]$. However, the $\lnot q$-admitting sphere $\{w, w_1,60w_2\}$ contains a world, namely~$w_2$, where $q$ is false and $p$ is61true, so $\mSat/{M}{\lnot q \lif \lnot p}[w_2]$.62\end{ex}63\end{document}