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Analysis of stalnaker’s and lewis’ counterfactual

theory

Chunmei xu

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Chapter one: What is counterfactual conditional?

• Indicative & subjunctive conditional

(1) If it rains, then it will be wet.

(2) If the Chinese enter the Vietnam conflict, the United States will use nuclear weapons.

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In general, The characters of subjunctive condition are as follows:

1) Antecedent is imaginary (or unlikely to turn out true)

2) Connections between antecedent and consequent

3) If antecedent is satisfied (such satisfaction may be a supposition)

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Counterfactuals vs subjunctive conditions

In actual practice, little distinction is made between counterfactuals and subjunctive conditionals which have true antecedents or consequents.

Authors frequently refer to conditionals in the subjunctive mood as counterfactuals regardless of whether their antecedents or consequents are true or not.

____from Handbook of philosophical logic(2nd edition)

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stalnaker ：

A theory of conditionals , w.L.Harper, R.Stalnker, and G.Pearce(eds),Ifs,41-55

Lewis:

Counterfactuals, David Lewis ， reissued by Blackwell publishers 2001

The title “Subjunctive conditionals” would not have delineated my subject properly

Ex: No Hitler, no A-bomb.

Besides, Some subjunctive conditionals pertaining to the future.

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(1) If it rains, then it will be wet.

二、 How to identify the truth of the counterfactual

1. Material implication

But it fails in the case of counterfactuals.

(2) If the Chinese enter the Vietnam conflict, the United States will use nuclear weapons.

(2`) If the Chinese enter the Vietnam conflict, the United States will not use nuclear weapons.

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Further:

the counterexample of transitivity

1) If Carter had died in 1979, he would not have lost the election in 1980.

2) If Carter had not lost the election in 1980,Reagan would not have been president in 1981.

If Carter had died in 1979,Reagan would not have been President in 1981.

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the counterexample of strengthening antecedents?

1) The match were struck, it would light.

If this match had been soaked in water overnight and it were struck, it would light.

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What about contraposition?

1) If it were to rain heavily at noon, the farmer would not irrigate his field at noon.

If the farmer were to irrigate his field at noon, it would not rain heavily at noon.

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二。 The works of Stalnaker:

the Ramsey test

First, hypothetically make the minimal revision of you stock of beliefs required to assume the antecedent. Then, evaluate the acceptability of the consequent on the basis of this revised body of beliefs.

Stock of beliefs world

Minimal revisionthe closest world

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consider a possible world in which A is true ,and which otherwise differs minimally from the actual world .”if A, then B” is true (false) just in case B is true (false) in that possible world

Approximately:

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2.1 the semantics

<I 、 R 、 K 、 f,[]>

I: a set of possible worlds

R: the relation of relative possibility which defines the structure.

Rab: b is possible from a.

R is reflexive relation.

K ： an absurd world where everything is true

[]: a function. [A] : a set of possible worlds where A is true.

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f: a world-selection function from statement and possible world to possible world. f(A,a)is a possible world

The restrictions of f:

（ S1 ） f （ A ， a ）∈ [A]

（ S2 ） <a, f (A ,a)> R∈

（ S3 ） for any b I∈ ， if <a ,b> R,∈ 且 b not belongs to [A] ， then f (A ,a) is undefined ， that is f (A ,a)=K

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（ S4 ） if a [A]∈ ， then f (A ,a)= a

（ S5 ） if f( A ,a) [C] and f (C ,a) [A]∈ ∈ ， then f (A ,a)= f (C ,a)

（ S6 ） a [A>C]∈ ， if and only if ， f (A ,a) [C] or f (A ,a) is undefined. ∈

∴ A>C is true at a iff C holds at the closest (accessible) A-world to a,

if there is one

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2.2 the syntax

Primitive symbols:

1) propositional variables: P1 、 P2 、 P3 、 P4……

3) dyadic operator: → 、 > 、

2) monadic operator:

~ 、 □

4) brackets: “(” 、” )”

Formation rules:

1) A propositional variable is a wff.

2) if A is a wff, so are ～ A and □A

3) if A and B are wff, so are (A →B) and (A > B)

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Axioms:

1) Any tautologous wff is an axiom.

2) □ (A→C) → □ A→ □ C

3) □ (A→C) → （ A>C ）

4) ◇ A→ （（ A>C ）→ ~ （ A>~C ））

5) A> （ B C∨ ）→（（ A>B ）∨（ A>C ）） 6) （ A>C ）→（ A→C ）

7) (A C)→ （（ A>B ） → （ C>B ））

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Transformation rules:

1) MP: if A and A → B are theorems, so is B.

2) N: if A is a theorem, so is □A.

I shall point out three unusual features of the conditional connective:

From （ A>C) ,we can’t get （ A&B>C ）

From （ A>B) and （ B>C), we can’t infer （ A>C).

From （ A>C) ,we can’t gain （ ~C> ~ A)

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3. Lewis’ works

The counterfactual cannot be any strict conditional.

But a variably strict conditional based on similarity of worlds.

David Lewis ， Counterfactuals, reissued by Blackwell publishers 2001

David Lewis, Counterfactuals and comparative possibility, Journal of philosophical logic 2(1973)418-446

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1) entailment:

To say that A entails C is to say that A leads to C by sheer unaided logic; or that it is absolutely impossible that A& ¬ C; or that every A-world is a C-world.

2) causal implication:

I use 'A causally implies C' to mean that A leads to C by logic conjoined with causal laws; or that it is causally impossible that A& ¬ C; or that every causally possible A-world is a C-world. (A causally possible world is one that conforms to the causal laws of the actual world.)

3) material implication: To say that A C is to say that it is not the case that A& ¬ C;

or—though it seems a funny thing to say—that every actual A-world is a C-world.

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extreme realism

Worlds

abstract realism

fictionalism

representationalism

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3.1 two questions in the theory of stalnaker’s

unique assumption CEM&

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Ex: A: Bizet and Verdi are compatriots

F: Bizet and Verdi are French.

I: Bizet and Verdi are Italian.

If Bizet and Verdi had been compatriots they would have been Ukrainian.

Not one , but more ..

A>C is true at a iff C holds at every closest (accessible) A-world to a, if there are any. ∴

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3.1.1 the Lewis’ semantics

<I , f , [] >

I and [] are defined the same as stalnaker does.

f: a set-selection function

the restrictions of f:

（ CS1 ） if b f∈ （ A ， a ）， then b∈ ［ A ］

（ CS2 ） if a f∈ （ A ， a ）， then f （ A ， a ） = ｛ a ｝

（ CS3） if f（ A， a） =， then f（ C， a ∩） ［ A］ =

（ CS4 ） if f （ A ， a ） ［C］

and f （ C ， a ） [A]

then f （ A ， a ） = f （ C ， a ）

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（ CS5 ） if f （ A ， a ）∩［ C ］ , then f （ A C∧ ， a ） f （ A ，a ）

（ CS6 ） a [ A>C] if and only if f∈ （ A ， a ） ［ C ］

the inference rules and axiom schema are the same as stalnaker’s except CEM.

while including CS which is （ A C∧ ）→ (A>C)

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the common between the two:

1) the ordering of worlds

2) world is the most closest of worlds to itself

the difference between the two:

the unique assumption the limit assumption

The limit assumption is justified ?

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a line appears more than one inch.(which is shorter than one inch in the actual world)

Ex:

1+m , <1+m

A>C is true at a iff some (accessible) AC-world is closer to a than any AC-world, if there are any (accessible) A-worlds

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3.1.1` stalnaker’s defense

On “unique assumption is illegal to make” and “limit assumption is unjustified”

Lewis’ theory +unique assumption +limit assumption =stalnaker’s

Between the two assumptions, in application ,one is not necessary, while the other is needed.

Unique assumption an abstract semantic theory.

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A general theory of vagueness supervaluation theory

(van fraassen,Bas.)

V1 V2 S

Fa T T T

~Fa F F F

Fb T F -

~ Fb F T -

Fb v ~ Fb T T T

(x)Fx T T T

(x)Fx→ Fb T F -

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to deny limit assumption is to get an absurd consequence.

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3.1.2 the question to CEM

CEM: (A>~C) v (A>C)

1) from (A> (B v C)) ,to infer (A>B) v (A>C) (distribution)

Which seems that:

2) from (A> (~C v C)) ,to infer (A>~C) v (A>C) (substitute ~C for B)

A> (~C v C) ??????????

Distribution?????????

the rule of uniform substitution???????

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as for (A> (~C v C)), it is necessarily true

5) A> （ B C∨ ）→（（ A>B ）∨（ A>C ））

CEM is Ok, in stalnaker’s theory.

But in Lewis’, it is not valid.

w

w1

w2(A>~C) v (A>C)

C

~C

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“might” operator and ”would” operator

A□ C =~ A ◇ ~ C

If it were the case that_______, then it would be the case that……□ ：

◇ : If it were the case that_______, then it might be the case that ……

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矛 矛 盾 盾

关 关 系 系

从

属

关

系

从

属

关

系

A□ C A□ ~ C

A ◇ C A ◇ ~ C

上反对关系

下反对关系

two kinds of conditionals

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1 ） (A □ ~C) v (A □ C) 假设 CEM 成立

2 ） ~ （ A □ C ） (A □ ~ C) 1 ）用 def[ ] ∨

3 ） (A □ ~C) （ A ◇ ~ C ） 刘易斯定理

4 ）（ A□ C ） ~ （ A ◇ ~ C ） “□ ” 与“◇ ”互定义

5 ） (A □ ~C) ~ （ A □ C ） 3 ）与 4 ）等值置换律

6 ） ~ （ A □ C ） （ A □ ~C ） 2 ）与 5 ）等值构成律

7 ） （ A ◇ ~ C ） （ A □ ~C ） 2) 与 4) 等值置换律

‘would’ counterfactuals and ‘might’ counterfactuals are the same.

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on “stalnaker can’t give an explain of ‘might’ conditional”

epistemic possibility & non-epistemic possibility.

John might come to the party’ & ‘John might have come to the party

the use of “might” in the context of the condition is the same that of out of the condition

two kinds of the use of ”might”

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might is possible modal operator

the definition of “might” operator and ”would” operator given by Lewis is misleading.

ex:

suppose there is in fact no penny in my pocket, although I do not know it since I did not look.

then

“ if I had looked, I might have found a penny”

Lewis: plainly false.

stalnaker: no, but it is true that it might be, for all I know, that I would have found a penny if I had looked.

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Human’s common tendency.

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some comments

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References

[1] Bas C. van Fraassen, , Singular terms, truth-value gaps, and free logic, Journal of philosophy volume LxIII, No.17(1966), 481-495

[2] David Lewis, On the plurality of worlds. Oxford, UK; New York, NY, USA: B. Blackwell, 1986

[3] David Lewis, Counterfactuals and comparative possibility, Journal of philosophical logic 2(1973)418-446

[4] David Lewis ， Counterfactuals, reissued by Blackwell publishers 2001.

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[5] WL Harper, R Stalnaker, G Pearce, ifs: conditionals, belief, decision, chance and time, edited by D.Reidel publishing company, Holland, 1981

[6] Robert C. Stalnaker, Possible Worlds, Noûs, Vol. 10, No. 1(1976) pp. 65-75

[7] Jonathan Bennett, A philosophical guide to conditionals, Oxford university press. 2003.

[6] 李小五 , 《条件句逻辑》北京：人民出版社， 2003.8.

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