Counterfactuals

Introduction

content/counterfactuals/introduction/introduction.tex

% Part: counterfactuals% Chapter: introduction\documentclass[../../../include/open-logic-chapter]{subfiles}\begin{document}\olchapter{cnt}{int}{Introduction}\olimport{material-conditional}\olimport{paradoxes-material}\olimport{strict-conditional}\olimport{counterfactuals}\OLEndChapterHook\end{document}

content/counterfactuals/introduction/material-conditional.tex

% Part: counterfactuals% Chapter: introduction% Section: material-conditional\documentclass[../../../include/open-logic-section]{subfiles}\begin{document}\olfileid{cnt}{int}{mat}\olsection{The Material Conditional}In its simplest form in English, a conditional is a sentence of theform ``If \dots then \dots,'' where the \dots{} are themselvessentences, such as ``If the butler did it, then the gardener isinnocent.'' In introductory logic courses, we earn to symbolizeconditionals using the $\lif$ connective: symbolize the partsindicated by \dots, e.g., by !!{formula}s $!A$ and~$!B$,and the entire conditional is symbolized by $!A \lif !B$.The connective $\lif$ is \emph{truth-functional}, i.e., the truthvalue---$\True$ or $\False$---of $!A \lif !B$ is determined by the truthvalues of $!A$ and~$!B$: $!A \lif !B$ is true iff $!A$~is false or$!B$~is true, and false otherwise. Relative to a truth valueassignment~$\pAssign v$, we define $\pSat{v}{!A \lif !B}$ iff$\pSat/{v}{!A}$ or $\pSat{v}{!B}$. The connective $\lif$ with thissemantics is called the \emph{material conditional}.This definition results in a number of elementary logical facts. Firstof all, the deduction theorem holds for the material conditional:\begin{align}  \text{If } \Gamma, !A \Entails !B \text{ then } \Gamma \Entails !A \lif !B\end{align}It is truth-functional: $!A \lif !B$ and $\lnot !A \lor !B$ are equivalent:\begin{align}  !A \lif !B & \Entails \lnot !A \lor !B\\  \lnot !A \lor !B & \Entails !A \lif !B  \intertext{A material conditional is entailed by its consequent and    by the negation of its antecedent:}  !B & \Entails !A \lif !B\\  \lnot !A & \Entails !A \lif !B  \intertext{A false material conditional is equivalent to the    conjunction of its antecedent and the negation of its consequent:    if $!A \lif !B$ is false, $!A \land \lnot !B$ is true, and vice versa:}  \lnot(!A \lif !B) & \Entails !A \land \lnot !B\\  !A \land \lnot !B & \Entails \lnot(!A \lif !B)  \intertext{The material conditional supports modus ponens:}  !A, !A \lif !B & \Entails !B  \intertext{The material conditional agglomerates:}  !A \lif !B,  !A \lif !C & \Entails !A \lif (!B \land !C)  \intertext{We can always strengthen the antecedent, i.e., the    conditional is \emph{monotonic}:}  !A \lif !B & \Entails (!A \land !C) \lif !B  \intertext{The material conditional is transitive, i.e., the chain    rule is valid:}  !A \lif !B, !B \lif !C & \Entails !A \lif !C  \intertext{The material conditional is equivalent to its    contrapositive:}  !A \lif !B & \Entails \lnot !B \lif \lnot !A\\  \lnot !B \lif \lnot !A & \Entails !A \lif !B\end{align}These are all useful and unproblematic inferences in mathematicalreasoning. However, the philosophical and linguistic literature isreplete with purported counterexamples to the equivalent inferences innon-mathematical contexts. These suggest that the materialconditional~$\lif$ is not---or at least not always---the appropriateconnective to use when symbolizing English ``if \dots then \dots''statements.\end{document}

content/counterfactuals/introduction/paradoxes-material.tex

% Part: counterfactuals% Chapter: introduction% Section: paradoxes-material\documentclass[../../../include/open-logic-section]{subfiles}\begin{document}\olfileid{cnt}{int}{par}\olsection{Paradoxes of the Material Conditional}One of the first to criticize the use of $!A \lif !B$ as a way tosymbolize ``if \dots then \dots'' statements of English wasC.~I. Lewis. Lewis was criticizing the use of the material conditionin Whitehead and Russell's \emph{Principia Mathematica}, whopronounced $\lif$ as ``implies.''  Lewis rightly complained that if$\lif$ meant ``implies,'' then any false proposition~$p$ implies that$p$ implies~$q$, since $p \lif (p \lif q)$ is true if~$p$ is false,and that any true proposition~$q$ implies that $p$ implies~$q$, since$q \lif (p \lif q)$ is true if $q$~is true.Logicians of course know that \emph{implication}, i.e., logicalentailment, is not a connective but a relation between !!{formula}s orstatements. So we should just not read $\lif$ as ``implies'' to avoidconfusion.\footnote{Reading ``$\lif$'' as ``implies'' is still widely  practised by mathematicians and computer scientists, although  philosophers try to avoid the confusions Lewis highlighted by  pronouncing it as ``only if.''} As long as we don't, the particularworry that Lewis had simply does not arise: $p$ does not ``imply'' $q$even if we think of $p$ as standing for a false English sentence. Todetermine if $p \Entails q$ we must consider \emph{all}!!{valuation}s, and $p \Entails/ q$ even when we use $p$ to symbolizea sentence which happens to be false.But there is still something odd about ``if \dots then\dots''statements such as Lewis's\begin{quote}If the moon is made of green cheese, then $2+2=4$.\end{quote}and about the inferences\begin{quote}  The moon is not made of green cheese. Therefore, if the moon is made  of green cheese, then $2+2=4$.  $2+2 = 4$. Therefore, if the moon is made  of green cheese, then $2+2=4$.\end{quote}Yet, if ``if \dots then \dots'' were just $\lif$, the sentence wouldbe unproblematically true, and the inferences unproblematically valid.Another example of concerns the tautology $(!A \lif !B) \lor (!B \lif!A)$.  This would suggest that if you take two indicativesentences~$S$ and $T$ from the newspaper at random, the sentence ``If$S$ then $T$, or if $T$ then~$S$'' should be true.\end{document}

content/counterfactuals/introduction/strict-conditional.tex

% Part: counterfactuals% Chapter: introduction% Section: strict-conditional\documentclass[../../../include/open-logic-section]{subfiles}\begin{document}\olfileid{cnt}{int}{str}\olsection{The Strict Conditional}Lewis introduced the \emph{strict conditional}~$\strictif$ and arguedthat it, not the material conditional, corresponds to implication.  Inalethic modal logic, $!A \strictif !B$ can be defined as $\Box(!A \lif!B)$. A strict conditional is thus true (at a world) iff thecorresponding material conditional is necessary.How does the strict conditional fare vis-a-vis the paradoxes of thematerial conditional?  A strict conditional with a false antecedentand one with a true consequent, may be true, or it may befalse. Moreover, $(!A \strictif !B) \lor (!B \strictif !A)$ is notvalid. The strict conditional $!A \strictif !B$ is also not equivalentto $\lnot !A \lor !B$, so it is not truth functional.We have:\begin{align}  !A \strictif !B & \Entails \lnot !A \lor !B  \text{ but:}\\  \lnot !A \lor !B & \Entails/ !A \strictif !B\\  !B & \Entails/ !A \strictif !B\\  \lnot !A & \Entails/ !A \strictif !B\\  \lnot(!A \lif !B) & \Entails/ !A \land \lnot !B  \text{ but:}\\  !A \land \lnot !B & \Entails \lnot(!A \strictif !B)  \intertext{However, the strict conditional still supports modus ponens:}  !A, !A \strictif !B & \Entails !B  \intertext{The strict conditional agglomerates:}  !A \strictif !B,  !A \strictif !C & \Entails !A \strictif (!B \land !C)  \intertext{Antecedent strengthening holds for the strict conditional:}  !A \strictif !B & \Entails (!A \land !C) \strictif !B  \intertext{The strict conditional is also transitive:}  !A \strictif !B, !B \strictif !C & \Entails !A \strictif !C  \intertext{Finally, the strict conditional is equivalent to its    contrapositive:}  !A \strictif !B & \Entails \lnot !B \strictif \lnot !A\\  \lnot !B \strictif \lnot !A & \Entails !A \strictif !B\end{align}\begin{prob}  Give \Log{S5}-counterexamples to the entailment relations which do not hold  for the strict conditional, i.e., for:  \begin{enumerate}  \item $\lnot p \Entails/ \Box(p \lif q)$  \item $q \Entails/ \Box(p \lif q)$  \item $\lnot\Box(p \lif q) \Entails/ p \land \lnot q$  \item $\Entails/ \Box(p \lif q) \lor \Box(q \lif p)$  \end{enumerate}\end{prob}\begin{prob}  Show that the valid entailment relations hold for the strict  conditional by giving \Log{S5}-proofs of:  \begin{enumerate}  \item  $\Box(!A \lif !B) \Entails \lnot !A \lor !B$  \item $!A \land \lnot !B \Entails \lnot\Box(!A \lif !B)$  \item $!A, \Box(!A \lif !B) \Entails !B$  \item $\Box(!A \lif !B), \Box(!A \lif !C) \Entails \Box(!A \lif (!B    \land !C))$  \item $\Box(!A \lif !B) \Entails \Box((!A \land !C) \lif !B)$  \item $\Box(!A \lif !B), \Box(!B \lif !C) \Entails \Box(!A    \lif !C)$  \item $\Box(!A \lif !B) \Entails \Box(\lnot !B \lif \lnot !A)$  \item $\Box(\lnot !B \lif \lnot !A) \Entails \Box(!A \lif !B)$  \end{enumerate}  \end{prob}However, the strict conditional still has its own ``paradoxes.'' Justas a material conditional with a false antecedent or a true consequentis true, a strict conditional with a \emph{necessarily} falseantecedent or a necessarily true consequent is true. Moreover, anytrue strict conditional is necessarily true, and any false strictconditional is necessarily false. In other words, we have\begin{align}  \Box \lnot !A & \Entails !A \strictif !B\\  \Box !B & \Entails !A \strictif !B\\  !A \strictif !B& \Entails \Box(!A \strictif !B)\\  \lnot(!A \strictif !B) & \Entails \Box\lnot(!A \strictif !B)\end{align}These are not problems if you think of $\strictif$ as ``implies.''Logical entailment relationships are, after all, mathematical facts and socan't be contingent. But they do raise issues if you want to use$\strictif$ as a logical connective that is supposed to capture ``if\dots then \dots,'' especially the last two. For surely there are ``if\dots then \dots'' statements that are contingently true orcontingently false---in fact, they generally are neither necessary norimpossible.\begin{prob}  Give proofs in \Log{S5} of:  \begin{enumerate}    \item  $\Box \lnot !A \Entails !A \strictif !B$    \item $!A \strictif !B \Entails \Box(!A \strictif !B)$    \item $\lnot(!A \strictif !B)  \Entails \Box\lnot(!A \strictif !B)$  \end{enumerate}  Use the definition of $\strictif$ to do so.\end{prob}\end{document}

content/counterfactuals/introduction/counterfactuals.tex

% Part: counterfactuals% Chapter: introduction% Section: counterfactuals\documentclass[../../../include/open-logic-section]{subfiles}\begin{document}\olfileid{cnt}{int}{cnt}\olsection{Counterfactuals}A very common and important form of ``if \dots then \dots''constructions in English are built using the past subjunctive form of\emph{to be}: ``if it were the case that \dots then it would be thecase that \dots'' Because usually the antecedent of such a conditionalis false, i.e., counter to fact, they are called \emph{counterfactual  conditionals} (and because they use the subjunctive form of \emph{to  be}, also \emph{subjunctive conditionals}. They are distinguishedfrom \emph{indicative} conditionals which take the form of ``if it isthe case that \dots then it is the case that \dots'' Counterfactual andindicative conditionals differ in truth conditions. Consider Adams'sfamous example:\begin{quote}  If Oswald didn't kill Kennedy, someone else did.    If Oswald hadn't killed Kennedy, someone else would have.\end{quote}The first is indicative, the second counterfactual. The first isclearly true: we know President John F. Kennedy was killed by \emph{someone}, and if that someonewasn't (contrary to the Warren Report) Lee Harvey Oswald, then someoneelse killed Kennedy. The second one says something different. It claimsthat if Oswald hadn't killed Kennedy, i.e., if the Dallas shooting hadbeen avoided or had been unsuccessful, history would have subsequentlyunfolded in such a way that another assassination would have beensuccessful. In order for it to be true, it would have to be the casethat powerful forces had conspired to ensure JFK's death (as many JFKconspiracy theorists believe).It is a live debate whether the \emph{indicative} conditional iscorrectly captured by the material conditional, in particular, whetherthe paradoxes of the material conditional can be ``explained'' in away that is compatible with it giving the truth conditions for Englishindicative conditionals. By contrast, it is uncontroversial thatcounterfactual conditionals cannot be symbolized correctly by thematerial conditionals. That is clear because, even though generallythe antecedents of counterfactuals are false, not all counterfactualswith false antecedents are true---for instance, if you believe theWarren Report, and there was no conspiracy to assassinate JFK, thenAdams's counterfactual conditional is an example.Counterfactual conditionals play an important role in causalreasoning: a prime example of the use of counterfactuals is to expresscausal relationships. E.g., striking a match causes it to light, andyou can express this by saying ``if this match were struck, it wouldlight.''  Material, and generally indicative conditionals, cannot beused to express this: ``the match is struck $\lif$ the match lights''is true if the match is never struck, regardless of what would happenif it were. Even worse, ``the match is struck $\lif$ the match turnsinto a bouquet of flowers'' is also true if it is never struck, butthe match would certainly not turn into a bouquet of flowers if itwere struck.It is still debated What exactly the correct logic of counterfactualsis. An influential analysis of counterfactuals was given by Stalnakerand Lewis.  According to them, a counterfactual ``if it were the casethat~$S$ then it would be the case that~$T$'' is true iff $T$ is truein the counterfactual situation (``possible world'') that is closestto the way the actual world is and where~$S$ is true. This is calledan ``ontic'' analysis, since it makes reference to an ontology ofpossible worlds. Other analyses make use of conditional probabilitiesor theories of belief revision.  There is a proliferation of differentproposed logics of counterfactuals. There isn't even a singleLewis--Stalnaker logic of counterfactuals: even though Stalnaker andLewis proposed accounts along similar lines with reference to closestpossible worlds, the assumptions they made result in different validinferences.\end{document}