Minimal Change Semantics

Source file content/counterfactuals/minimal-change-semantics/minimal-change-semantics.tex

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Introduction

Stalnaker and Lewis proposed accounts of counterfactual conditionals such as “If the match were struck, it would light.” Their accounts were proposals for how to properly understand the truth conditions for such sentences. The idea behind both proposals is this: to evaluate whether a counterfactual conditional is true, we have to consider those possible worlds which are minimally different from the way the world actually is to make the antecedent true. If the consequent is true in these possible worlds, then the counterfactual is true. For instance, suppose I hold a match and a matchbook in my hand. In the actual world I only look at them and ponder what would happen if I were to strike the match. The minimal change from the actual world where I strike the match is that where I decide to act and strike the match. It is minimal in that nothing else changes: I don't also jump in the air, striking the match doesn't also light my hair on fire, I don't suddenly lose all strength in my fingers, I am not simultaneously doused with water in a SuperSoaker ambush, etc. In that alternative possibility, the match lights. Hence, it's true that if I were to strike the match, it would light.

This intuitive account can be paired with formal semantics for logics of counterfactuals. Lewis introduced the symbol “$\boxright$source” for the counterfactual while Stalnaker used the symbol “$>$source”. We'll use $\cif$source, and add it as a binary connective to propositional logic. So, we have, in addition to formulas of the form $!A \lif !B$source also formulas of the form $!A \cif !B$source. The formal semantics, like the relational semantics for modal logic, is based on models in which formulas are evaluated at worlds, and the satisfaction condition defining $\mSat{M}{!A \cif !B}[w]$source is given in terms of $\mSat{M}{!A}[w']$source and $\mSat{M}{!B}[w']$source for some (other) worlds $w'$source. Which $w'$source? Intuitively, the one(s) closest to $w$source for which it holds that $\mSat{M}{!A}[w']$source. This requires that a relation of “closeness” has to be included in the model as well.

Lewis introduced an instructive way of representing counterfactual situations graphically. Each possible world is at the center of a set of nested spheres containing other worlds---we draw these spheres as concentric circles. The worlds between two spheres are equally close to the world at the center as each other, those contained in a nested sphere are closer, and those in a surrounding sphere further away.

The closest $!A$source-worlds are those worlds $w'$source where $!A$source is satisfied which lie in the smallest sphere around the center world $w$source (the gray area). Intuitively, $!A \cif !B$source is satisfied at $w$source if $!B$source is true at all closest $!A$source-worlds.

Source file content/counterfactuals/minimal-change-semantics/sphere-models.tex

Sphere Models

One way of providing a formal semantics for counterfactuals is to turn Lewis's informal account into a mathematical structure. The spheres around a world $w$source then are sets of worlds. Since the spheres are nested, the sets of worlds around $w$source have to be linearly ordered by the subset relation.

The intuition behind $O_w$source is that the worlds “around” $w$source are stratified according to how far away they are from $w$source. The innermost sphere is just $w$source by itself, i.e., the set $\{w\}$source: $w$source is closer to $w$source than the worlds in any other sphere. If $S \subsetneq S'$source, then the worlds in $S' \setminus S$source are further way from $w$source than the worlds in $S$source: $S' \setminus S$source is the “layer” between the $S$source and the worlds outside of $S'$source. In particular, we have to think of the spheres as containing all the worlds within their outer surface; they are not just the individual layers.

The diagram in the diagram of a sphere model corresponds to the sphere model with $W = \{w, w_1, \dots, w_7\}$source, $V(p) = \{w_5, w_6, w_7\}$source. The innermost sphere $S_1 = \{w\}$source. The closest worlds to $w$source are $w_1, w_2, w_3$source, so the next larger sphere is $S_2 = \{w, w_1, w_2, w_3\}$source. The worlds further out are $w_4$source, $w_5$source, $w_6$source, so the outermost sphere is $S_3 = \{w, w_1, \dots, w_6\}$source. The system of spheres around $w$source is $O_w = \{S_1, S_2, S_3\}$source. The world $w_7$source is not in any sphere around $w$source. The closest worlds in which $p$source is true are $w_5$source and $w_6$source, and so the smallest $p$source-admitting sphere is $S_3$source.

To define satisfaction of a formula $!A$source at world $w$source in a sphere model $\mModel M$source, $\mSat{M}{!A}[w]$source, we expand the definition for modal formulas to include a clause for $!B \cif !C$source:

According to this definition, $\mSat{M}{!B \cif !C}[w]$source iff either the antecedent $!B$source is false everywhere in the spheres around $w$source, or there is a sphere $S$source where $!B$source is true, and the material conditional $!B \lif !C$source is true at all worlds in that “$!B$source-admitting” sphere. Note that we didn't require in the definition that $S$source is the innermost $!B$source-admitting sphere, contrary to what one might expect from the intuitive explanation. But if the condition in the nonvacuous sphere-model satisfaction condition is satisfied for some sphere $S$source, then it is also satisfied for all spheres $S$source contains, and hence in particular for the innermost sphere.

Note also that the definition of sphere models does not require that there is an innermost $!B$source-admitting sphere: we may have an infinite sequence $S_1 \supsetneq S_2 \supsetneq \dots \supsetneq \{w\}$source of $!B$source-admitting spheres, and hence no innermost $!B$source-admitting spheres. In that case, $\mSat{M}{!B \cif !C}[w]$source iff $!B \lif !C$source holds throughout the spheres $S_i$source, $S_{i+1}$source, dots, for some $i$source.

Source file content/counterfactuals/minimal-change-semantics/true-false.tex

Truth and Falsity of Counterfactuals

A counterfactual $!A \cif !B$source is (non-vacuously) true if the closest $!A$source-worlds are all $!B$source-worlds, as depicted in the nonvacuously true counterfactual figure.

A counterfactual is also true at $w$source if the system of spheres around $w$source has no $!A$source-admitting spheres at all. In that case it is vacuously true (see the vacuously true counterfactual figure).

It can be false in two ways. One way is if the closest $!A$source-worlds are not all $!B$source-worlds, but some of them are. In this case, $!A \cif \lnot !B$source is also false (see the false counterfactual and false opposite figure).

If the closest $!A$source-worlds do not overlap with the $!B$source-worlds at all, then $!A \cif !B$source is false. But, in this case all the closest $!A$source-worlds are $\lnot !B$source-worlds, and so $!A \cif \lnot !B$source is true (see the false counterfactual and true opposite figure).

In contrast to the strict conditional, counterfactuals may be contingent. Consider the sphere model in the contingent counterfactual figure. The $!A$source-worlds closest to $u$source are all $!B$source-worlds, so $\mSat{M}{!A \cif !B}[u]$source. But there are $!A$source-worlds closest to $v$source which are not $!B$source-worlds, so $\mSat/{M}{!A \cif !B}[v]$source.

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Antecedent Strengthening

“Strengthening the antecedent” refers to the inference $!A \lif !C \Entails (!A \land !B) \lif !C$source. It is valid for the material conditional, but invalid for counterfactuals. Suppose it is true that if I were to strike this match, it would light. (That means, there is nothing wrong with the match or the matchbook surface, I will not break the match, etc.) But it is not true that if I were to light this match in outer space, it would light. So the following inference is invalid:

The Lewis--Stalnaker account of conditionals explains this: the closest world where I light the match and I do so in outer space is much further removed from the actual world than the closest world where I light the match is. So although it's true that the match lights in the latter, it is not in the former. And that is as it should be.

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Transitivity

For the material conditional, the chain rule holds: $!A \lif !B, !B \lif !C \Entails !A \lif !C$source. In other words, the material conditional is transitive. Is the same true for counterfactuals? Consider the following example due to Stalnaker.

If Hoover had been born (at the same time he actually did), not in the United States, but in Russia, he would have grown up in the Soviet Union and become a Communist (let's assume). So the first premise is true. Likewise, the second premise, considered in isolation is true. The conclusion, however, is false: in all likelihood, Hoover would have been a fervent Communist if he had been born in the USSR, and not been a traitor (to his country). The intuitive assignment of truth values is borne out by the Stalnaker--Lewis account. The closest possible world to ours with the only change being Hoover's place of birth is the one where Hoover grows up to be a good citizen of the USSR. This is the closest possible world where the antecedent of the first premise and of the conclusion is true, and in that world Hoover is a loyal member of the Communist party, and so not a traitor. To evaluate the second premise, we have to look at a different world, however: the closest world where Hoover is a Communist, which is one where he was born in the United States, turned, and thus became a traitor.Footnote: Of course, to appreciate the force of the example we have to take on board some metaphysical and political assumptions, e.g., that it is possible that Hoover could have been born to Russian parents, or that Communists in the US of the 1950s were traitors to their country.

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Contraposition

Material and strict conditionals are equivalent to their contrapositives. Counterfactuals are not. Here is an example due to Kratzer:

The first sentence is true: humans don't live hundreds of years. The second is clearly false: if Goethe weren't dead now, he would be still alive, and so couldn't have died in 1832.