Counterfactuals

Minimal Change Semantics

Equation form expr-00c1e20c7784eecf

M=⟨W,O,V⟩\mModel{M} = \tuple{W, O, V}

Read as: the model capital M equals the ordered triple W, O, V

Means: the model capital M equals the ordered triple W, O, V

3 occurrences in this chapter

Equation form expr-0251db30f8e57014

p□→q,q□→r⊭p□→rp \cif q, q \cif r \Entails/ p \cif r

Read as: if it were p then it would be q, together with if it were q then it would be r, does not entail if it were p then it would be r

Means: if it were p then it would be q, together with if it were q then it would be r, does not entail if it were p then it would be r

1 occurrence in this chapter

Equation form expr-05c2d5f9bd74bcd1

M⊩A□→B[w]\mSat{M}{!A \cif !B}[w]

Read as: in model capital M at world w, the counterfactual if it were A then it would be B is satisfied

Means: in model capital M at world w, the counterfactual if it were A then it would be B is satisfied

1 occurrence in this chapter

Equation form expr-0af2ffeac988b1cc

w7w_7

Read as: w sub seven

Means: w sub seven

2 occurrences in this chapter

Equation form expr-0bfe935e70c321c7

uu

Read as: u

Means: u

2 occurrences in this chapter

Equation form expr-0f45d9fdad45f090

M⊮(p∧q)→r[w2]\mSat/{M}{(p \land q) \lif r}[w_2]

Read as: in model capital M at world w sub two, it is not the case that if both p and q then r is satisfied

Means: in model capital M at world w sub two, it is not the case that if both p and q then r is satisfied

1 occurrence in this chapter

Equation form expr-123965755500cecc

A□→¬B!A \cif \lnot !B

Read as: if it were A then it would be not B

Means: if it were A then it would be not B

2 occurrences in this chapter

Equation form expr-148de9c5a7a44d19

pp

Read as: p

Means: p

11 occurrences in this chapter

Equation form expr-189d66db574b71b1

S1⊋S2⊋…⊋{w}S_1 \supsetneq S_2 \supsetneq \dots \supsetneq \{w\}

Read as: S sub one strictly contains S sub two, which strictly contains the continuing sequence of spheres, which strictly contains the singleton set containing w

Means: S sub one strictly contains S sub two, which strictly contains the continuing sequence of spheres, which strictly contains the singleton set containing w

1 occurrence in this chapter

Equation form expr-18fbc71be6dc930c

A→B,B→C⊨A→C!A \lif !B, !B \lif !C \Entails !A \lif !C

Read as: if A then B together with if B then C entails if A then C

Means: if A then B together with if B then C entails if A then C

1 occurrence in this chapter

Equation form expr-192a3e93a81739ba

M⊩B□→C[w]\mSat{M}{!B \cif !C}[w]

Read as: in model capital M at world w, the counterfactual if it were B then it would be C is satisfied

Means: in model capital M at world w, the counterfactual if it were B then it would be C is satisfied

3 occurrences in this chapter

Equation form expr-1beefd8beceeb265

w6w_6

Read as: w sub six

Means: w sub six

3 occurrences in this chapter

Equation form expr-207a39a9c7240f95

V(p)={w5,w6,w7}V(p) = \{w_5, w_6, w_7\}

Read as: V of p equals the set containing w sub five, w sub six, and w sub seven

Means: V of p equals the set containing w sub five, w sub six, and w sub seven

1 occurrence in this chapter

Equation form expr-20a3bd7b3d0a81cc

V(p)={w2}V(p) = \{w_2\}

Read as: V of p equals the singleton set containing w sub two

Means: V of p equals the singleton set containing w sub two

1 occurrence in this chapter

Equation form expr-211265509db5a6e6

M⊩q□→r[w]\mSat{M}{q \cif r}[w]

Read as: in model capital M at world w, the counterfactual if it were q then it would be r is satisfied

Means: in model capital M at world w, the counterfactual if it were q then it would be r is satisfied

1 occurrence in this chapter

Equation form expr-227bd74fe0ab9bfe

□→\boxright

Read as: Lewis's box-right counterfactual symbol

Means: Lewis's box-right counterfactual symbol

1 occurrence in this chapter

Equation form expr-22fcce974c4bdb33

M\mModel M

Read as: the model capital M

Means: the model capital M

1 occurrence in this chapter

Equation form expr-240381337f804e9f

A→B!A \lif !B

Read as: if A then B

Means: if A then B

1 occurrence in this chapter

Equation form expr-241307f534cace2c

V(p)={w1,w2}V(p) = \{w_1, w_2\}

Read as: V of p equals the set containing w sub one and w sub two

Means: V of p equals the set containing w sub one and w sub two

2 occurrences in this chapter

Equation form expr-2632b13f61dc812c

M⊮A□→B[v]\mSat/{M}{!A \cif !B}[v]

Read as: in model capital M at world v, the counterfactual if it were A then it would be B is not satisfied

Means: in model capital M at world v, the counterfactual if it were A then it would be B is not satisfied

1 occurrence in this chapter

Equation form expr-28e51fe3865b29fe

Ow={S1,S2,S3}O_w = \{S_1, S_2, S_3\}

Read as: O sub w equals the set containing S sub one, S sub two, and S sub three

Means: O sub w equals the set containing S sub one, S sub two, and S sub three

1 occurrence in this chapter

Equation form expr-2a05fbee69d682f7

V(q)={w1,w2}V(q) = \{w_1, w_2\}

Read as: V of q equals the set containing w sub one and w sub two

Means: V of q equals the set containing w sub one and w sub two

1 occurrence in this chapter

Equation form expr-3106fac3e3c8f992

w2w_2

Read as: w sub two

Means: w sub two

8 occurrences in this chapter

Equation form expr-36e4d0d8a8262c01

p□→r⊭(p∧q)□→rp \cif r \Entails/ (p \land q) \cif r

Read as: if it were p then it would be r does not entail if it were both p and q then it would be r

Means: if it were p then it would be r does not entail if it were both p and q then it would be r

1 occurrence in this chapter

Equation form expr-39c3d5a5ec4b06af

⊆\subseteq

Read as: is a subset of or equal to

Means: is a subset of or equal to

1 occurrence in this chapter

Equation form expr-3f4b7d2b1b570acc

A\formula{A}

Read as: A

Means: A

6 occurrences in this chapter

Equation form expr-41a68720a6849e03

B\formula{B}

Read as: B

Means: B

6 occurrences in this chapter

Equation form expr-438757f12dd8c3fc

w1w_1

Read as: w sub one

Means: w sub one

6 occurrences in this chapter

Equation form expr-454349e422f05297

rr

Read as: r

Means: r

1 occurrence in this chapter

Equation form expr-461d8f2f2223ec1f

W={w,w1,…,w7}W = \{w, w_1, \dots, w_7\}

Read as: W equals the set containing w, w sub one, continuing through w sub seven

Means: W equals the set containing w, w sub one, continuing through w sub seven

1 occurrence in this chapter

Equation form expr-4794277b410dfc51

¬p\lnot p

Read as: not p

Means: not p

1 occurrence in this chapter

Equation form expr-4c94485e0c21ae6c

vv

Read as: v

Means: v

2 occurrences in this chapter

Equation form expr-4ddd9a2680586f8f

M⊩C[v]\mSat{M}{!C}[v]

Read as: in model capital M at world v, C is satisfied

Means: in model capital M at world v, C is satisfied

1 occurrence in this chapter

Equation form expr-50e721e49c013f00

ww

Read as: w

Means: w

31 occurrences in this chapter

Equation form expr-52eb11ddfa24980c

V(r)={w1}V(r) = \{w_1\}

Read as: V of r equals the singleton set containing w sub one

Means: V of r equals the singleton set containing w sub one

2 occurrences in this chapter

Equation form expr-536ae23e1ad19805

{w}∈Ow\{w\} \in O_w

Read as: the singleton set containing w is a member of O sub w

Means: the singleton set containing w is a member of O sub w

1 occurrence in this chapter

Equation form expr-5604c95d023face1

w1,w2,w3w_1, w_2, w_3

Read as: w sub one, w sub two, and w sub three

Means: w sub one, w sub two, and w sub three

1 occurrence in this chapter

Equation form expr-5857d55749449d15

V(q)={w,w1}V(q) = \{w, w_1\}

Read as: V of q equals the set containing w and w sub one

Means: V of q equals the set containing w and w sub one

1 occurrence in this chapter

Equation form expr-594963fb341972fa

O:W→℘(℘(W))O\colon W \to \Pow{\Pow{W}}

Read as: O is a function from W to the power set of the power set of W

Means: O is a function from W to the power set of the power set of W

1 occurrence in this chapter

Equation form expr-5af5d83af9d1d3cc

OwO_w

Read as: O sub w

Means: O sub w

8 occurrences in this chapter

Equation form expr-6017291547d45908

B□→C!B \cif !C

Read as: if it were B then it would be C

Means: if it were B then it would be C

1 occurrence in this chapter

Equation form expr-614c21994adfa8b6

S′={w,w1,w2}S' = \{w, w_1, w_2\}

Read as: S prime equals the set containing w, w sub one, and w sub two

Means: S prime equals the set containing w, w sub one, and w sub two

1 occurrence in this chapter

Equation form expr-62b67e1f685b7fef

>>

Read as: Stalnaker's greater-than counterfactual symbol

Means: Stalnaker's greater-than counterfactual symbol

1 occurrence in this chapter

Equation form expr-6899808994f71ef2

p□→q⊭¬q□→¬pp \cif q \Entails/ \lnot q \cif \lnot p

Read as: if it were p then it would be q does not entail if it were not q then it would be not p

Means: if it were p then it would be q does not entail if it were not q then it would be not p

1 occurrence in this chapter

Equation form expr-698cd4222a287d75

SiS_i

Read as: S sub i

Means: S sub i

1 occurrence in this chapter

Equation form expr-69ef9ee2c45eb3b7

M⊩A[w′]\mSat{M}{!A}[w']

Read as: in model capital M at world w prime, A is satisfied

Means: in model capital M at world w prime, A is satisfied

2 occurrences in this chapter

Equation form expr-6d82acc81495ce5c

M⊮(p∧q)□→r[w]\mSat/{M}{(p \land q) \cif r}[w]

Read as: in model capital M at world w, the counterfactual if it were both p and q then it would be r is not satisfied

Means: in model capital M at world w, the counterfactual if it were both p and q then it would be r is not satisfied

1 occurrence in this chapter

Equation form expr-6f19307cf53b8de7

S1={w}S_1 = \{w\}

Read as: S sub one equals the singleton set containing w

Means: S sub one equals the singleton set containing w

1 occurrence in this chapter

Equation form expr-7230056d6fa6dea8

O={{w},{w,w1},{w,w1,w2}}O = \{\{w\}, \{w, w_1\}, \{w, w_1, w_2\}\}

Read as: O equals the set of three spheres: the singleton w; the set containing w and w sub one; and the set containing w, w sub one, and w sub two

Means: O equals the set of three spheres: the singleton w; the set containing w and w sub one; and the set containing w, w sub one, and w sub two

1 occurrence in this chapter

Equation form expr-74af6e6e3971777c

M1=⟨W,O,V⟩\mModel{M_1} = \tuple{W, O, V}

Read as: the model capital M sub one equals the ordered triple W, O, V

Means: the model capital M sub one equals the ordered triple W, O, V

1 occurrence in this chapter

Equation form expr-7d16ee82599976fd

S={w,w1}S = \{w, w_1\}

Read as: S equals the set containing w and w sub one

Means: S equals the set containing w and w sub one

2 occurrences in this chapter

Equation form expr-7f37d047e0568165

S1S_1

Read as: S sub one

Means: S sub one

1 occurrence in this chapter

Equation form expr-7f6cf40b81199c63

u∈Su \in S

Read as: u is a member of S

Means: u is a member of S

1 occurrence in this chapter

Equation form expr-822b488cc527e9ff

¬q\lnot q

Read as: not q

Means: not q

2 occurrences in this chapter

Equation form expr-8238c028f61fc0f7

A!A

Read as: A

Means: A

11 occurrences in this chapter

Equation form expr-858e88771bf016c5

S′∖SS' \setminus S

Read as: S prime minus S

Means: S prime minus S

2 occurrences in this chapter

Equation form expr-890370556cd21de5

S′={w,w1}S' = \{w, w_1\}

Read as: S prime equals the set containing w and w sub one

Means: S prime equals the set containing w and w sub one

1 occurrence in this chapter

Equation form expr-89cd02eacd5a9ed0

Ow={{w},{w,w1},{w,w1,w2}}O_w = \{\{w\}, \{w, w_1\}, \{w, w_1, w_2\}\}

Read as: O sub w equals the set of three spheres: the singleton w; the set containing w and w sub one; and the set containing w, w sub one, and w sub two

Means: O sub w equals the set of three spheres: the singleton w; the set containing w and w sub one; and the set containing w, w sub one, and w sub two

2 occurrences in this chapter

Equation form expr-8a97403a309cea3b

v∈Sv \in S

Read as: v is a member of S

Means: v is a member of S

1 occurrence in this chapter

Equation form expr-8cf9f0ef3fca196b

Si+1S_{i+1}

Read as: S sub i plus one

Means: S sub i plus one

1 occurrence in this chapter

Equation form expr-8de0b3c47f112c59

SS

Read as: S

Means: S

6 occurrences in this chapter

Equation form expr-8e35c2cd3bf6641b

qq

Read as: q

Means: q

5 occurrences in this chapter

Equation form expr-9763a163e7ff6591

w3w_3

Read as: w sub three

Means: w sub three

1 occurrence in this chapter

Equation form expr-9954b8d1073d15e2

(p∧q)(p \land q)

Read as: both p and q

Means: both p and q

1 occurrence in this chapter

Equation form expr-9bce9b18cabace59

M⊮q→r\mSat/{M}{q \lif r}

Read as: model capital M does not satisfy if q then r

Means: model capital M does not satisfy if q then r

1 occurrence in this chapter

Equation form expr-9dd4e4b646218a71

M⊩B[w′]\mSat{M}{!B}[w']

Read as: in model capital M at world w prime, B is satisfied

Means: in model capital M at world w prime, B is satisfied

1 occurrence in this chapter

Equation form expr-9fb3053a4827ef20

S2∈OwS_2 \in O_w

Read as: S sub two is a member of O sub w

Means: S sub two is a member of O sub w

1 occurrence in this chapter

Equation form expr-a08f60fda3f22844

M⊩A□→B[u]\mSat{M}{!A \cif !B}[u]

Read as: in model capital M at world u, the counterfactual if it were A then it would be B is satisfied

Means: in model capital M at world u, the counterfactual if it were A then it would be B is satisfied

1 occurrence in this chapter

Equation form expr-a43e0d8c06476226

S′S'

Read as: S prime

Means: S prime

1 occurrence in this chapter

Equation form expr-a6613ea09a4716da

M⊮B[u]\mSat/{M}{!B}[u]

Read as: in model capital M at world u, B is not satisfied

Means: in model capital M at world u, B is not satisfied

1 occurrence in this chapter

Equation form expr-a983a9f5438c641c

S2⊆S1S_2 \subseteq S_1

Read as: S sub two is a subset of or equal to S sub one

Means: S sub two is a subset of or equal to S sub one

1 occurrence in this chapter

Equation form expr-abb2c7d2e90d1c0c

W={w,w1,w2}W = \{w, w_1, w_2\}

Read as: W equals the set containing w, w sub one, and w sub two

Means: W equals the set containing w, w sub one, and w sub two

3 occurrences in this chapter

Equation form expr-b4c1a8ce5b685884

S2={w,w1,w2,w3}S_2 = \{w, w_1, w_2, w_3\}

Read as: S sub two equals the set containing w, w sub one, w sub two, and w sub three

Means: S sub two equals the set containing w, w sub one, w sub two, and w sub three

1 occurrence in this chapter

Equation form expr-b5b446abb4d0323b

M⊩A[w]\mSat{M}{!A}[w]

Read as: in model capital M at world w, A is satisfied

Means: in model capital M at world w, A is satisfied

1 occurrence in this chapter

Equation form expr-b7b96c57fe5a9d1c

M⊩B[u]\mSat{M}{!B}[u]

Read as: in model capital M at world u, B is satisfied

Means: in model capital M at world u, B is satisfied

1 occurrence in this chapter

Equation form expr-b99d2e5f37ae7e17

w′w'

Read as: w prime

Means: w prime

3 occurrences in this chapter

Equation form expr-bde413ec3eb41d19

u∈⋃Owu \in \bigcup O_w

Read as: u is a member of the union of O sub w

Means: u is a member of the union of O sub w

1 occurrence in this chapter

Equation form expr-c457dbcdd3b49987

w4w_4

Read as: w sub four

Means: w sub four

2 occurrences in this chapter

Equation form expr-c49d2ec94024e19d

S1⊆S2S_1 \subseteq S_2

Read as: S sub one is a subset of or equal to S sub two

Means: S sub one is a subset of or equal to S sub two

1 occurrence in this chapter

Equation form expr-c5bf73cb03229b8b

p→qp \lif q

Read as: if p then q

Means: if p then q

2 occurrences in this chapter

Equation form expr-c655e97f84b9ea86

M⊮¬q→¬p[w2]\mSat/{M}{\lnot q \lif \lnot p}[w_2]

Read as: in model capital M at world w sub two, if not q then not p is not satisfied

Means: in model capital M at world w sub two, if not q then not p is not satisfied

1 occurrence in this chapter

Equation form expr-cb6e9e560afb4ede

{w}\{w\}

Read as: the singleton set containing w

Means: the singleton set containing w

1 occurrence in this chapter

Equation form expr-ccd37b1c10ab75cf

A□→B!A \cif !B

Read as: if it were A then it would be B

Means: if it were A then it would be B

4 occurrences in this chapter

Equation form expr-cd615a60e0e9e377

M⊮p→r[w2]\mSat/{M}{p \lif r}[w_2]

Read as: in model capital M at world w sub two, if p then r is not satisfied

Means: in model capital M at world w sub two, if p then r is not satisfied

1 occurrence in this chapter

Equation form expr-cdc525abf7a300b5

S3S_3

Read as: S sub three

Means: S sub three

1 occurrence in this chapter

Equation form expr-d04658a0d836a4c4

p→rp \lif r

Read as: if p then r

Means: if p then r

1 occurrence in this chapter

Equation form expr-d055ee4dbcdd0c8b

B!B

Read as: B

Means: B

13 occurrences in this chapter

Equation form expr-d0a918b38a513fd7

w5w_5

Read as: w sub five

Means: w sub five

3 occurrences in this chapter

Equation form expr-d17c79af91cd9cfc

S∈OwS \in O_w

Read as: S is a member of O sub w

Means: S is a member of O sub w

1 occurrence in this chapter

Equation form expr-d61252d38c5eba2f

B→C!B \lif !C

Read as: if B then C

Means: if B then C

2 occurrences in this chapter

Equation form expr-d926cf57c0e537a8

M⊮B[v]\mSat/{M}{!B}[v]

Read as: in model capital M at world v, B is not satisfied

Means: in model capital M at world v, B is not satisfied

1 occurrence in this chapter

Equation form expr-d963f46ee5546d2e

{w,w1,w2}\{w, w_1, w_2\}

Read as: the set containing w, w sub one, and w sub two

Means: the set containing w, w sub one, and w sub two

2 occurrences in this chapter

Equation form expr-dc064a096251ec8f

□→\cif

Read as: the counterfactual conditional connective

Means: the counterfactual conditional connective

1 occurrence in this chapter

Equation form expr-de7d1b721a1e0632

ii

Read as: i

Means: i

1 occurrence in this chapter

Equation form expr-e3212229d200318d

¬B\lnot !B

Read as: not B

Means: not B

1 occurrence in this chapter

Equation form expr-e3fa6a55b84ca1b5

M⊩p□→r[w]\mSat{M}{p \cif r}[w]

Read as: in model capital M at world w, the counterfactual if it were p then it would be r is satisfied

Means: in model capital M at world w, the counterfactual if it were p then it would be r is satisfied

1 occurrence in this chapter

Equation form expr-e999586c3d48ee28

S={w,w1,w2}S = \{w, w_1, w_2\}

Read as: S equals the set containing w, w sub one, and w sub two

Means: S equals the set containing w, w sub one, and w sub two

1 occurrence in this chapter

Equation form expr-ee5f174b0f9497a5

A→C⊨(A∧B)→C!A \lif !C \Entails (!A \land !B) \lif !C

Read as: if A then C entails if both A and B then C

Means: if A then C entails if both A and B then C

1 occurrence in this chapter

Equation form expr-ef636ba8b6892981

V(q)={w2}V(q) = \{w_2\}

Read as: V of q equals the singleton set containing w sub two

Means: V of q equals the singleton set containing w sub two

1 occurrence in this chapter

Equation form expr-f74e33870c5cf950

S⊊S′S \subsetneq S'

Read as: S is a proper subset of S prime

Means: S is a proper subset of S prime

1 occurrence in this chapter

Equation form expr-f8a56090efa9857e

S3={w,w1,…,w6}S_3 = \{w, w_1, \dots, w_6\}

Read as: S sub three equals the set containing w, w sub one, continuing through w sub six

Means: S sub three equals the set containing w, w sub one, continuing through w sub six

1 occurrence in this chapter

Equation form expr-fcb5f40df9be6bae

WW

Read as: W

Means: W

1 occurrence in this chapter

Equation form expr-fd9031b91e56c996

M⊩p□→q[w]\mSat{M}{p \cif q}[w]

Read as: in model capital M at world w, the counterfactual if it were p then it would be q is satisfied

Means: in model capital M at world w, the counterfactual if it were p then it would be q is satisfied

2 occurrences in this chapter

Equation form expr-fe0701bc4d1e66b6

V:At0→℘(W)V\colon \PVar \to \Pow{W}

Read as: V is a function from propositional variables to the power set of W

Means: V is a function from propositional variables to the power set of W

1 occurrence in this chapter

Closest-world sphere illustration

Five nested spheres are centered on w. The A and B proposition regions show that every closest A world is a B world, illustrating the truth condition for if it were A then it would be B.

Source

Definition of a sphere model

A sphere model is a triple W, O, V. W is nonempty, V is a propositional valuation, and O assigns each world a centered, nested system of spheres closed under nonempty unions and intersections.

Source

Figure: diagram of a sphere model

The figure contains three nested spheres centered on w, worlds w through w sub seven, and the region where p holds. Its complete world and region structure is linearized in the enclosed diagram.

Source

Diagram of a sphere model

The inner sphere contains w; the second contains w through w sub three; the outer contains w through w sub six; w sub seven lies outside every sphere. The p region contains w sub five, w sub six, and w sub seven.

Source

Sphere-model satisfaction clause for counterfactuals

If it were B then it would be C is true at w either vacuously, when B is false throughout all spheres around w, or nonvacuously, when some B-admitting sphere makes if B then C true throughout.

Source

Figure: nonvacuously true counterfactual

The closest A worlds around w all lie in the B region. The complete nested-region relationship is read in the enclosed diagram.

Source

Nonvacuously true counterfactual diagram

Five spheres surround w; the closest A-admitting part lies wholly within B, so if it were A then it would be B is nonvacuously true.

Source

Figure: vacuously true counterfactual

The A region is outside every sphere around w. The complete nested-region relationship is read in the enclosed diagram.

Source

Vacuously true counterfactual diagram

Five spheres surround w, while the A region does not meet any sphere. There is no A-admitting sphere, so the counterfactual is vacuously true.

Source

Figure: false counterfactual and false opposite

Some but not all closest A worlds are B worlds, so both the counterfactual and its opposite are false. The enclosed diagram is fully linearized.

Source

False counterfactual, false opposite diagram

Five spheres surround w. The closest A worlds overlap B only partly, making both if it were A then B and if it were A then not B false.

Source

Figure: false counterfactual and true opposite

No closest A world lies in B, so the counterfactual is false and its opposite is true. The enclosed diagram is fully linearized.

Source

False counterfactual, true opposite diagram

Five spheres surround w. The closest A worlds are disjoint from B, so if it were A then B is false and if it were A then not B is true.

Source

Figure: contingent counterfactual

Overlapping sphere systems centered on u and v show that the closest A worlds to u are all B worlds, while some closest A worlds to v are not B worlds.

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Contingent counterfactual diagram

A solid sphere system is centered on u and a dashed system on v. The A and B regions make the counterfactual true at u and false at v.

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Counterexample to antecedent strengthening

A three-world sphere model makes if it were p then r true at w but if it were both p and q then r false at w. The enclosed diagram preserves the spheres and valuations.

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Figure: counterexample to antecedent strengthening

The figure depicts the three-world model used by the surrounding example. Its exact sphere and proposition-region structure is linearized in the enclosed diagram.

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Antecedent-strengthening countermodel diagram

The spheres are the singleton w, then w with w sub one, then all three worlds. p holds at w sub one and w sub two, q only at w sub two, and r only at w sub one.

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Exercise finding a failure of counterfactual transitivity

Find a convincing intuitive example showing that counterfactual conditionals are not transitive. No solution is supplied.

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Three-world counterexample to transitivity

The example defines a sphere model in which if it were p then q and if it were q then r are true at w, while if it were p then r fails.

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Exercise drawing the transitivity countermodel

Draw the sphere diagram for the preceding three-world counterexample to transitivity. No solution is supplied.

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Exercise varying the Hoover countermodel

Determine whether the relative closeness of the two described Hoover worlds is necessary and construct a counterexample without that assumption. No solution is supplied.

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Counterexample to contraposition

A three-world sphere model makes if it were p then q true at w while if it were not q then not p fails. The enclosed diagram records the exact worlds, spheres, and proposition regions.

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Figure: counterexample to contraposition

The figure depicts the three-world Goethe model from the surrounding example. Its complete sphere and region structure is linearized in the enclosed diagram.

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Contraposition countermodel diagram

The spheres are the singleton w, then w with w sub one, then all three worlds. p holds at w sub one and w sub two; q holds at w and w sub one.

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Cross-reference reference-001164

the diagram of a sphere model

Source occurrence

Cross-reference reference-001165

the nonvacuous sphere-model satisfaction condition

Source occurrence

Cross-reference reference-001166

the nonvacuously true counterfactual figure

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Cross-reference reference-001167

the vacuously true counterfactual figure

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Cross-reference reference-001168

the false counterfactual and false opposite figure

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Cross-reference reference-001169

the false counterfactual and true opposite figure

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Cross-reference reference-001170

the contingent counterfactual figure

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Cross-reference reference-001171

the counterexample to antecedent strengthening figure

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Cross-reference reference-001172

the three-world counterexample to counterfactual transitivity

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Cross-reference reference-001173

the three-world counterexample to counterfactual transitivity

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Source disclosures

Ordered structures

Closest-world sphere illustration

Structure: diagram tikz.

Five nested spheres are centered on w. The proposition region B contains all worlds in the closest A region reached from the center; the A region also extends farther out. Thus every closest A world is a B world. End of diagram.

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Diagram of a sphere model

Structure: diagram tikz.

Three nested spheres are centered on w. The second sphere adds w sub two, w sub three, and w sub one. The outer sphere adds w sub five, w sub six, and w sub four. w sub seven is outside every sphere. The proposition region p contains w sub five, w sub six, and w sub seven. End of diagram.

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Nonvacuously true counterfactual diagram

Structure: diagram tikz.

Five nested spheres are centered on w. The closest portion of proposition region A lies entirely within proposition region B. Therefore the counterfactual is nonvacuously true. End of diagram.

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Vacuously true counterfactual diagram

Structure: diagram tikz.

Five nested spheres are centered on w. Proposition region B meets the sphere system, but proposition region A lies outside all five spheres. There is no A-admitting sphere, so the counterfactual is vacuously true. End of diagram.

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False counterfactual, false opposite diagram

Structure: diagram tikz.

Five nested spheres are centered on w. At the closest A worlds, proposition region B overlaps only part of the A region. Some closest A worlds satisfy B and some do not, so both the counterfactual and its opposite are false. End of diagram.

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False counterfactual, true opposite diagram

Structure: diagram tikz.

Five nested spheres are centered on w. The closest A worlds do not overlap proposition region B at all. Thus the original counterfactual is false and its opposite is true. End of diagram.

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Contingent counterfactual diagram

Structure: diagram tikz.

A solid system of seven spheres is centered on u, and a dashed system of four spheres is centered on v. Proposition regions A and B cross both systems. All A worlds closest to u are B worlds, while some A worlds closest to v are not B worlds. The counterfactual is true at u and false at v. End of diagram.

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Antecedent-strengthening countermodel diagram

Structure: diagram tikz.

Three nested spheres are centered on w. The middle sphere adds w sub one, and the outer sphere adds w sub two. Proposition q holds only at w sub two; r holds only at w sub one; and p holds at both w sub one and w sub two. End of countermodel diagram.

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Contraposition countermodel diagram

Structure: diagram tikz.

Three nested spheres are centered on w. The middle sphere adds w sub one, and the outer sphere adds w sub two. Region labels show q at w and w sub one, not q at w sub two, p at w sub one and w sub two, and not p at w. End of countermodel diagram.

Read the source-bound structure in context