Counterfactuals

Introduction

Equation form expr-00af0727b662b041

□¬A⊨A⥽B□B⊨A⥽BA⥽B⊨□(A⥽B)¬(A⥽B)⊨□¬(A⥽B)\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)

Read as: Four entailments concerning strict conditionals. First, necessarily not A entails that A strictly implies B. Second, necessarily B entails that A strictly implies B. Third, A strictly implies B entails that necessarily, A strictly implies B. Fourth, not A strictly implies B entails that necessarily, A does not strictly imply B. End of four entailments.

Means: Four entailments concerning strict conditionals. First, necessarily not A entails that A strictly implies B. Second, necessarily B entails that A strictly implies B. Third, A strictly implies B entails that necessarily, A strictly implies B. Fourth, not A strictly implies B entails that necessarily, A does not strictly imply B. End of four entailments.

1 occurrence in this chapter

Equation form expr-043b130790fae68b

¬A∨B\lnot !A \lor !B

Read as: not A or B

Means: not A or B

2 occurrences in this chapter

Equation form expr-06b50a0a3c67572f

p⊨qp \Entails q

Read as: p entails q

Means: p entails q

1 occurrence in this chapter

Equation form expr-148de9c5a7a44d19

pp

Read as: p

Means: p

7 occurrences in this chapter

Equation form expr-240381337f804e9f

A→B!A \lif !B

Read as: if A then B

Means: if A then B

5 occurrences in this chapter

Equation form expr-4383919132a678ce

p→(p→q)p \lif (p \lif q)

Read as: if p, then if p then q

Means: if p, then if p then q

1 occurrence in this chapter

Equation form expr-4b1b85797430946d

q⊭□(p→q)q \Entails/ \Box(p \lif q)

Read as: q does not entail that necessarily, if p then q

Means: q does not entail that necessarily, if p then q

1 occurrence in this chapter

Equation form expr-4d71c5573892d6ef

q→(p→q)q \lif (p \lif q)

Read as: if q, then if p then q

Means: if q, then if p then q

1 occurrence in this chapter

Equation form expr-4f377505378a2af0

□(A→B)⊨□(¬B→¬A)\Box(!A \lif !B) \Entails \Box(\lnot !B \lif \lnot !A)

Read as: necessarily, if A then B, entails necessarily, if not B then not A

Means: necessarily, if A then B, entails necessarily, if not B then not A

1 occurrence in this chapter

Equation form expr-58e0615bc97d9a4d

If Γ,A⊨B then Γ⊨A→B\text{If } \Gamma, !A \Entails !B \text{ then } \Gamma \Entails !A \lif !B

Read as: if Gamma together with A entails B, then Gamma entails if A then B

Means: if Gamma together with A entails B, then Gamma entails if A then B

1 occurrence in this chapter

Equation form expr-59cf7cd4fdb9ff5b

T\True

Read as: true

Means: true

1 occurrence in this chapter

Equation form expr-5ef8e0446555d2d2

A,□(A→B)⊨B!A, \Box(!A \lif !B) \Entails !B

Read as: A together with necessarily, if A then B, entails B

Means: A together with necessarily, if A then B, entails B

1 occurrence in this chapter

Equation form expr-672eb44f56c4f5e1

A⥽B!A \strictif !B

Read as: A strictly implies B

Means: A strictly implies B

2 occurrences in this chapter

Equation form expr-685c907cf7129ace

p⊭qp \Entails/ q

Read as: p does not entail q

Means: p does not entail q

1 occurrence in this chapter

Equation form expr-6e2be378761c0112

¬(A⥽B)⊨□¬(A⥽B)\lnot(!A \strictif !B) \Entails \Box\lnot(!A \strictif !B)

Read as: A does not strictly imply B entails that necessarily, A does not strictly imply B

Means: A does not strictly imply B entails that necessarily, A does not strictly imply B

1 occurrence in this chapter

Equation form expr-745f84f9fc9a9708

A⥽B⊨□(A⥽B)!A \strictif !B \Entails \Box(!A \strictif !B)

Read as: A strictly implies B entails that necessarily, A strictly implies B

Means: A strictly implies B entails that necessarily, A strictly implies B

1 occurrence in this chapter

Equation form expr-74c262d70dd4b774

□(¬B→¬A)⊨□(A→B)\Box(\lnot !B \lif \lnot !A) \Entails \Box(!A \lif !B)

Read as: necessarily, if not B then not A, entails necessarily, if A then B

Means: necessarily, if not B then not A, entails necessarily, if A then B

1 occurrence in this chapter

Equation form expr-8238c028f61fc0f7

A!A

Read as: A

Means: A

3 occurrences in this chapter

Equation form expr-8de0b3c47f112c59

SS

Read as: S

Means: S

5 occurrences in this chapter

Equation form expr-8e35c2cd3bf6641b

qq

Read as: q

Means: q

5 occurrences in this chapter

Equation form expr-9203f9c813d3f68f

A⥽B⊨¬A∨B but:¬A∨B⊭A⥽BB⊭A⥽B¬A⊭A⥽B¬(A→B)⊭A∧¬B but:A∧¬B⊨¬(A⥽B)However, the strict conditional still supports modus ponens:A,A⥽B⊨BThe strict conditional agglomerates:A⥽B,A⥽C⊨A⥽(B∧C)Antecedent strengthening holds for the strict conditional:A⥽B⊨(A∧C)⥽BThe strict conditional is also transitive:A⥽B,B⥽C⊨A⥽CFinally, the strict conditional is equivalent to its contrapositive:A⥽B⊨¬B⥽¬A¬B⥽¬A⊨A⥽B!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

Read as: Strict-conditional entailments. A strictly implies B entails not A or B, but not A or B does not entail that A strictly implies B. B does not entail that A strictly implies B. Not A does not entail that A strictly implies B. The next source line says that not if A then B does not entail A and not B; this source formula is retained with a caveat. Conversely, A and not B entails that A does not strictly imply B. Modus ponens: A together with A strictly implies B entails B. Agglomeration: A strictly implies B together with A strictly implies C entails that A strictly implies both B and C. Antecedent strengthening: A strictly implies B entails that A and C strictly implies B. Transitivity: A strictly implies B together with B strictly implies C entails that A strictly implies C. Contraposition holds in both directions between A strictly implies B and not B strictly implies not A. End of entailments.

Means: Strict-conditional entailments. A strictly implies B entails not A or B, but not A or B does not entail that A strictly implies B. B does not entail that A strictly implies B. Not A does not entail that A strictly implies B. The next source line says that not if A then B does not entail A and not B; this source formula is retained with a caveat. Conversely, A and not B entails that A does not strictly imply B. Modus ponens: A together with A strictly implies B entails B. Agglomeration: A strictly implies B together with A strictly implies C entails that A strictly implies both B and C. Antecedent strengthening: A strictly implies B entails that A and C strictly implies B. Transitivity: A strictly implies B together with B strictly implies C entails that A strictly implies C. Contraposition holds in both directions between A strictly implies B and not B strictly implies not A. End of entailments.

1 occurrence in this chapter

Equation form expr-985ed720e1b91a04

⥽\strictif

Read as: the strict conditional connective

Means: the strict conditional connective

4 occurrences in this chapter

Equation form expr-a49f9a63917c5a59

¬p⊭□(p→q)\lnot p \Entails/ \Box(p \lif q)

Read as: not p does not entail that necessarily, if p then q

Means: not p does not entail that necessarily, if p then q

1 occurrence in this chapter

Equation form expr-a7e0ebb9bbda1f55

□(A→B),□(B→C)⊨□(A→C)\Box(!A \lif !B), \Box(!B \lif !C) \Entails \Box(!A \lif !C)

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

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

1 occurrence in this chapter

Equation form expr-ad61a1016cc4192b

v⊨B\pSat{v}{!B}

Read as: v satisfies B

Means: v satisfies B

1 occurrence in this chapter

Equation form expr-ad9293b54c2b636f

(A→B)∨(B→A)(!A \lif !B) \lor (!B \lif !A)

Read as: either if A then B, or if B then A

Means: either if A then B, or if B then A

1 occurrence in this chapter

Equation form expr-b2dc39f093dd2bea

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

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

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

1 occurrence in this chapter

Equation form expr-b376ba81938b1751

v⊨A→B\pSat{v}{!A \lif !B}

Read as: v satisfies if A then B

Means: v satisfies if A then B

1 occurrence in this chapter

Equation form expr-b7339d1bd119bb61

□(A→B)⊨¬A∨B\Box(!A \lif !B) \Entails \lnot !A \lor !B

Read as: necessarily, if A then B, entails not A or B

Means: necessarily, if A then B, entails not A or B

1 occurrence in this chapter

Equation form expr-bdcb7192fe841cb0

v\pAssign v

Read as: the truth-value assignment v

Means: the truth-value assignment v

1 occurrence in this chapter

Equation form expr-c96c7728a5ba3e4a

A→B⊨¬A∨B¬A∨B⊨A→BA material conditional is entailed by its consequent and by the negation of its antecedent:B⊨A→B¬A⊨A→BA false material conditional is equivalent to the conjunction of its antecedent and the negation of its consequent: if A→B is false, A∧¬B is true, and vice versa:¬(A→B)⊨A∧¬BA∧¬B⊨¬(A→B)The material conditional supports modus ponens:A,A→B⊨BThe material conditional agglomerates:A→B,A→C⊨A→(B∧C)We can always strengthen the antecedent, i.e., the conditional is monotonic:A→B⊨(A∧C)→BThe material conditional is transitive, i.e., the chain rule is valid:A→B,B→C⊨A→CThe material conditional is equivalent to its contrapositive:A→B⊨¬B→¬A¬B→¬A⊨A→B!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

Read as: Material-conditional entailments. If A then B is equivalent, in both directions, to not A or B. B entails if A then B, and not A entails if A then B. Not if A then B is equivalent, in both directions, to A and not B. Modus ponens: A together with if A then B entails B. Agglomeration: if A then B together with if A then C entails if A then both B and C. Antecedent strengthening: if A then B entails if both A and C then B. Transitivity: if A then B together with if B then C entails if A then C. Contraposition holds in both directions between if A then B and if not B then not A. End of entailments.

Means: Material-conditional entailments. If A then B is equivalent, in both directions, to not A or B. B entails if A then B, and not A entails if A then B. Not if A then B is equivalent, in both directions, to A and not B. Modus ponens: A together with if A then B entails B. Agglomeration: if A then B together with if A then C entails if A then both B and C. Antecedent strengthening: if A then B entails if both A and C then B. Transitivity: if A then B together with if B then C entails if A then C. Contraposition holds in both directions between if A then B and if not B then not A. End of entailments.

1 occurrence in this chapter

Equation form expr-ccb87154d21d3d96

2+2=42+2=4

Read as: two plus two equals four

Means: two plus two equals four

3 occurrences in this chapter

Equation form expr-d04ff80d9f6dc462

→\lif

Read as: the material conditional connective

Means: the material conditional connective

11 occurrences in this chapter

Equation form expr-d055ee4dbcdd0c8b

B!B

Read as: B

Means: B

3 occurrences in this chapter

Equation form expr-d094fd9bd1c9f12d

□¬A⊨A⥽B\Box \lnot !A \Entails !A \strictif !B

Read as: necessarily not A entails that A strictly implies B

Means: necessarily not A entails that A strictly implies B

1 occurrence in this chapter

Equation form expr-d9992ac8b50ce917

⊭□(p→q)∨□(q→p)\Entails/ \Box(p \lif q) \lor \Box(q \lif p)

Read as: it is not valid that either necessarily, if p then q, or necessarily, if q then p

Means: it is not valid that either necessarily, if p then q, or necessarily, if q then p

1 occurrence in this chapter

Equation form expr-dbaee93a8c7c5677

A∧¬B⊨¬□(A→B)!A \land \lnot !B \Entails \lnot\Box(!A \lif !B)

Read as: A and not B entails that it is not necessary that if A then B

Means: A and not B entails that it is not necessary that if A then B

1 occurrence in this chapter

Equation form expr-e1a81bc531d3e71c

¬□(p→q)⊭p∧¬q\lnot\Box(p \lif q) \Entails/ p \land \lnot q

Read as: not necessarily if p then q does not entail p and not q

Means: not necessarily if p then q does not entail p and not q

1 occurrence in this chapter

Equation form expr-e632b7095b0bf32c

TT

Read as: T

Means: T

5 occurrences in this chapter

Equation form expr-e64d3df81ed8cfb3

(A⥽B)∨(B⥽A)(!A \strictif !B) \lor (!B \strictif !A)

Read as: either A strictly implies B, or B strictly implies A

Means: either A strictly implies B, or B strictly implies A

1 occurrence in this chapter

Equation form expr-ea151b1ce772ba33

□(A→B)\Box(!A \lif !B)

Read as: necessarily, if A then B

Means: necessarily, if A then B

1 occurrence in this chapter

Equation form expr-ed371311a33fe53c

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

Read as: necessarily, if A then B, together with necessarily, if A then C, entails necessarily, if A then both B and C

Means: necessarily, if A then B, together with necessarily, if A then C, entails necessarily, if A then both B and C

1 occurrence in this chapter

Equation form expr-ed7a029f72f172a6

2+2=42+2 = 4

Read as: two plus two equals four

Means: two plus two equals four

1 occurrence in this chapter

Equation form expr-f9baaf9f77711629

F\False

Read as: false

Means: false

1 occurrence in this chapter

Equation form expr-febc2d0a7ade256c

v⊭A\pSat/{v}{!A}

Read as: v does not satisfy A

Means: v does not satisfy A

1 occurrence in this chapter

Deduction theorem for the material conditional

If Gamma together with A entails B, then Gamma entails the material conditional from A to B.

Source

Material-conditional entailment laws

A source-ordered display records equivalence with not A or B, consequent and false-antecedent entailments, falsity conditions, modus ponens, agglomeration, antecedent strengthening, transitivity, and contraposition.

Source

Strict-conditional entailment laws

The display contrasts failed material-style inferences with valid modus ponens, agglomeration, antecedent strengthening, transitivity, and contraposition. One anomalous source formula is retained and disclosed.

Source

Exercise constructing strict-conditional countermodels

Give S five counterexamples to four invalid entailments involving necessity and the material conditional. No solution is supplied.

Source

Exercise proving valid strict-conditional entailments

Give S five proofs of eight listed valid entailments: distribution to the material consequence, incompatibility, modus ponens, agglomeration, antecedent strengthening, transitivity, and both contraposition directions. No solution is supplied.

Source

Four modal consequences for strict conditionals

Necessary falsity of the antecedent and necessary truth of the consequent each suffice for the strict conditional; a true strict conditional is necessarily true, and a false one is necessarily false.

Source

Exercise proving the strict-conditional paradoxes

Using the definition of the strict conditional, give S five proofs of the three displayed necessity entailments. No solution is supplied.

Source

Source disclosures