Reinhart (1976) @cite{reinhart-1976}

The Syntactic Domain of Anaphora. PhD dissertation, MIT.

Key Contributions

1. C-command (def. 36, p. 32): replaces @cite{langacker-1969}'s S-node-based "command" with a branching-node-based relation
2. C-command domain (def. 38, p. 33): the subtree dominated by the first branching node dominating A — always a constituent
3. Coreference restriction (10b, p. 14): domain-based, dispensing with "precede"
4. Claim (49) (p. 40): c-command ⊆ command (= kCommand ⊆ sCommand in B&P)
5. The irrelevance of precede (§1.4): linear order is epiphenomenal for coreference

Connection to @cite{barker-pullum-1990}

Reinhart's c-command is exactly B&P's K-command (parameterized by branching nodes). @cite{langacker-1969}'s command is B&P's S-command (parameterized by S-nodes). Theorem 49 follows from B&P's antitone map: since {S-nodes} ⊆ {branching nodes}, we get C_{branching} ⊆ C_{S}.

Connection to Address-Based cCommand

The address-based cCommand in Compare.lean computes K-command for binary trees: in a binary tree every non-leaf node branches, so the "first branching node dominating A" is A's parent, and A's parent dominates B iff A's sister dominates B — which is exactly what cCommand tests.

Definition 1 (p. 8) — Langacker's "command"

A node A commands a node B if neither A nor B dominates the other and the S node most immediately dominating A also dominates B.

This is B&P's S-command, parameterized by S-nodes. Already formalized as sCommand in Compare.lean.

Definition 36 (p. 32) — C-command

Node A c(onstituent)-commands node B if neither A nor B dominates the other and the first branching node which dominates A dominates B.

This is B&P's K-command, parameterized by branching nodes. Already formalized as kCommand in Compare.lean.

Reinhart explicitly contrasts this with Langacker's command (p. 33): "The difference between the relations of command and of c-command is that while the first mentions cyclic nodes the second does not — all branching nodes can be relevant."

Definition 38 (p. 33) — C-command domain

The domain of a node A consists of A together with all and only the nodes c-commanded by A. (OR: The domain of a node A is the subtree dominated by the first branching node which dominates A.)

A key observation (p. 34): c-command domains are always constituents (subtrees), while precede-and-command domains may not be.

def Phenomena.Anaphora.Studies.Reinhart1976.cCommandDomain {Node : Type} (T : Compare.AbstractTree Node) (a : Node) : Set Node

The c-command domain of a node a: the set of nodes that a c-commands, plus a itself.

In B&P terms: {b | (a, b) ∈ kCommand T} ∪ {a}.

Equations

• Phenomena.Anaphora.Studies.Reinhart1976.cCommandDomain T a = {b : Node | (a, b) ∈ Phenomena.Anaphora.Compare.kCommand T} ∪ {a}

Claim (49) (p. 40)

A c-commands B ⟶ A commands B A does not command B ⟶ A does not c-command B

In B&P terms: kCommand T ⊆ sCommand T, provided every S-node is also a branching node — a universally accepted structural assumption (S-nodes always dominate both a subject and a predicate).

theorem Phenomena.Anaphora.Studies.Reinhart1976.cCommand_implies_command {Node : Type} (T : Compare.LabeledTree Node) (h_S_branch : Compare.sNodes T ⊆ Compare.branchingNodes T.toAbstractTree) : Compare.kCommand T.toAbstractTree ⊆ Compare.sCommand T

Claim (49): C-command implies command.

Every S-node is a branching node (S-nodes dominate ≥2 children), so {S-nodes} ⊆ {branching nodes}, and by B&P's antitone map (command_antitone), C_{branching} ⊆ C_{S}.

Restriction 10b (p. 14)

Two NP's in a non strict reflexive environment can be coreferential just in case if either is in the domain of the other, the one in the domain is a pronoun.

Reinhart argues (§1.4) that the earlier formulation using precede-and-command is both empirically wrong (fails for preposed PPs) and theoretically unnatural (c-command domains are constituents; precede-and-command domains are not).

@[reducible, inline]

abbrev Phenomena.Anaphora.Studies.Reinhart1976.IsPronoun (Node : Type) : Type

Whether node a is a pronoun.

Equations

• Phenomena.Anaphora.Studies.Reinhart1976.IsPronoun Node = (Node → Prop)

def Phenomena.Anaphora.Studies.Reinhart1976.corefPermitted {Node : Type} (isPron : IsPronoun Node) (T : Compare.AbstractTree Node) (a b : Node) : Prop

Reinhart's Coreference Restriction (10b).

Two nodes can corefer unless one is in the c-command domain of the other and is not a pronoun.

corefPermitted isPron T a b holds iff:

• neither is in the other's domain, OR
• whichever is in the other's domain is a pronoun.

Equations

• One or more equations did not get rendered due to their size.

The irrelevance of precede (§1.4, §1.5)

Reinhart's central argumentative contribution: the relation precede plays no role in determining anaphora options. Two key observations:

1. Preposed PPs (§1.5.2): In "Near him, Dan saw a snake" (45), the pronoun precedes the antecedent yet coreference is fine — because "him" (PP-internal) does not c-command "Dan" (the subject). Meanwhile, "*Near Dan, he saw a snake" (43a) is correctly blocked: "Dan" is in the c-command domain of "he" and is not a pronoun.

2. VOS languages (p. 41): In Malagasy, the pronoun precedes and commands the antecedent (by the precede-and-command definition) yet coreference is permitted — because the pronoun does not c-command the antecedent.

Both facts follow automatically from the c-command restriction (10b) without mentioning linear order.

theorem Phenomena.Anaphora.Studies.Reinhart1976.command_ignores_precedence {Node : Type} (T : Compare.AbstractTree Node) (P : Set Node) (a b : Node) : (a, b) ∈ Compare.commandRelation T P ↔ ∀ x ∈ Compare.upperBounds T a P, T.dom x b

Precede is irrelevant: c-command domains are symmetric with respect to linear order.

In B&P terms: the command relation C_P is defined purely in terms of dominance (upperBounds, dom), with no reference to precedence.

This is a structural fact about how commandRelation is defined — it uses only vertical (dominance) relations, never horizontal (precedence) relations.

Preposed PP example (§1.5.2)

@cite{reinhart-1976}'s structures (41)/(42) are ternary branching (S → PP NP₁ VP), but the key half of the argument is tree-shape-independent:

• NP₃ (PP-internal, e.g., "Dan" inside "near Dan") does NOT c-command NP₁ (the subject), because the first branching node dominating NP₃ is PP, which does not dominate NP₁.

This is verified below in a binary encoding. It is the fact that the precede-and-command approach fails to capture: the precede-and-command rule incorrectly blocks backward pronominalization in "Near him, Dan saw a snake" (45) — where "him" precedes and is S-commanded by "Dan" — while the c-command restriction correctly permits it (since "him" does not c-command "Dan").

Binary tree limitation: In @cite{reinhart-1976}'s ternary tree (41), the subject NP₁ DOES c-command NP₃ (since the first branching node above NP₁ is S, which dominates NP₃). Our binary encoding places NP₁ inside a VP shell, so the subject does NOT c-command into PP. This changes the c-command facts for that direction but does not affect the key structural observation that NP₃ cannot c-command NP₁.

theorem Phenomena.Anaphora.Studies.Reinhart1976.preposed_pp_no_ccommand : Compare.cCommand [Compare.Dir.L, Compare.Dir.R] [Compare.Dir.R, Compare.Dir.L] = false

Preposed PP: the PP-internal NP does not c-command the subject.

Binary encoding: [S [PP [P near] [NP Dan]] [VP' [NP he] [VP ...]]] NP_Dan at [L, R], NP_he at [R, L].

This holds in both binary and n-ary trees: PP is the first branching node above Dan, and PP does not dominate the subject.

Why address-based cCommand = B&P kCommand

In a binary branching tree, every non-leaf node has exactly two children, so every non-leaf node is a branching node. Therefore:

• The "first branching node dominating A" = A's parent
• A's parent dominates B iff A's sister subtree contains B
• This is exactly what cCommand a b tests: sister a is a prefix of b

So for binary trees, address-based cCommand computes the same relation as commandRelation T (branchingNodes T) = kCommand T.

We verify this on concrete examples below.

theorem Phenomena.Anaphora.Studies.Reinhart1976.subject_ccommands_into_vp : Compare.cCommand [Compare.Dir.L] [Compare.Dir.R] = true ∧ Compare.cCommand [Compare.Dir.L] [Compare.Dir.R, Compare.Dir.L] = true ∧ Compare.cCommand [Compare.Dir.L] [Compare.Dir.R, Compare.Dir.R] = true

In a binary tree, the subject [L] c-commands everything in [R, ...]

theorem Phenomena.Anaphora.Studies.Reinhart1976.object_does_not_ccommand_subject : Compare.cCommand [Compare.Dir.R, Compare.Dir.R] [Compare.Dir.L] = false

The object [R, R] does NOT c-command the subject [L]

theorem Phenomena.Anaphora.Studies.Reinhart1976.object_ccommands_verb : Compare.cCommand [Compare.Dir.R, Compare.Dir.R] [Compare.Dir.R, Compare.Dir.L] = true

The object [R, R] c-commands the verb [R, L] (mutual c-command within VP)

theorem Phenomena.Anaphora.Studies.Reinhart1976.ccommand_asymmetric_example : Compare.cCommand [Compare.Dir.L] [Compare.Dir.R, Compare.Dir.R] = true ∧ Compare.cCommand [Compare.Dir.R, Compare.Dir.R] [Compare.Dir.L] = false

C-command is NOT symmetric in general: subject c-commands object but object does not c-command subject.

theorem Phenomena.Anaphora.Studies.Reinhart1976.subject_object_asymmetry : Compare.cCommand [Compare.Dir.L] [Compare.Dir.R, Compare.Dir.R, Compare.Dir.L] = true ∧ Compare.cCommand [Compare.Dir.L, Compare.Dir.R, Compare.Dir.R, Compare.Dir.R, Compare.Dir.R] [Compare.Dir.R, Compare.Dir.R, Compare.Dir.L] = false

Subject–object asymmetry for coreference (@cite{reinhart-1976}'s key prediction):

• "She denied that Rosa met the Shah" — she c-commands Rosa, blocks coref
• "The man who traveled with her denied that Rosa met the Shah" — her does NOT c-command Rosa, coref permitted

In the second sentence, "her" is deeply embedded inside the subject NP (inside a relative clause PP). Any address under [L, ...] has a sister that also starts with [L, ...], so it can never c-command [R, ...].

Address-based coreference permission

A decidable version of corefPermitted for address-based binary trees. Given two addresses and their pronoun status, tests whether coreference is permitted under restriction (10b).

def Phenomena.Anaphora.Studies.Reinhart1976.corefPermittedAddr (isPronA isPronB : Bool) (addrA addrB : Compare.Address) : Bool

Address-based coreference permission (decidable).

Two NPs can corefer unless one c-commands the other and the c-commanded one is not a pronoun. Restriction (10b) applied to address-based cCommand.

Note: like corefPermitted, this omits the "non strict reflexive environment" qualification — it governs non-reflexive coreference only.

Equations

• One or more equations did not get rendered due to their size.

The (11) paradigm (pp. 14-15)

The key empirical test for restriction (10b). Given the structure:

`[S NP₁ [VP denied [S' NP₂ [VP' has met the Shah]]]]`

with NP₁ at [L] and NP₂ at [R, R, L], four sentences are generated by varying the pronoun status of each NP:

• (11a) Rosa denied that Rosa has met the Shah. — coref blocked
• (11b) She denied that Rosa has met the Shah. — coref blocked
• (11c) Rosa denied that she has met the Shah. — coref permitted
• (11d) She denied that she has met the Shah. — coref permitted

The matrix subject c-commands the embedded subject (not vice versa), so NP₂ is always in the c-command domain of NP₁. Therefore (10b) requires NP₂ to be a pronoun for coreference to be permitted.

theorem Phenomena.Anaphora.Studies.Reinhart1976.paradigm_11a_blocked : corefPermittedAddr false false [Compare.Dir.L] [Compare.Dir.R, Compare.Dir.R, Compare.Dir.L] = false

(11a) Rosa₁ denied that Rosa₂ has met the Shah: coref blocked.

Rosa₂ is in the domain of Rosa₁ and is not a pronoun.

theorem Phenomena.Anaphora.Studies.Reinhart1976.paradigm_11b_blocked : corefPermittedAddr true false [Compare.Dir.L] [Compare.Dir.R, Compare.Dir.R, Compare.Dir.L] = false

(11b) She₁ denied that Rosa₂ has met the Shah: coref blocked.

Rosa₂ is in the domain of She₁ and is not a pronoun.

theorem Phenomena.Anaphora.Studies.Reinhart1976.paradigm_11c_permitted : corefPermittedAddr false true [Compare.Dir.L] [Compare.Dir.R, Compare.Dir.R, Compare.Dir.L] = true

(11c) Rosa₁ denied that she₂ has met the Shah: coref permitted.

she₂ is in the domain of Rosa₁ but IS a pronoun.

theorem Phenomena.Anaphora.Studies.Reinhart1976.paradigm_11d_permitted : corefPermittedAddr true true [Compare.Dir.L] [Compare.Dir.R, Compare.Dir.R, Compare.Dir.L] = true

(11d) She₁ denied that she₂ has met the Shah: coref permitted.

she₂ is in the domain of She₁ but IS a pronoun.

Restriction (10a) — the pronoun-specific formulation (p. 14)

Two NP's in a non strict reflexive environment can be coreferential just in case one is a pronoun, the other is not and the non-pronoun is not in the domain of the pronoun.

(10a) applies only to pairs consisting of a pronoun and a full NP. It says nothing about pairs of two full NPs or two pronouns.

@cite{reinhart-1976} argues (pp. 14-17) that (10b) is strictly superior: (10a) fails to block coreference between two full NPs when one is in the domain of the other (the (11a) case).

def Phenomena.Anaphora.Studies.Reinhart1976.corefPermitted_10a (isPronA isPronB : Bool) (addrA addrB : Compare.Address) : Bool

Restriction (10a): applies only to pronoun–full NP pairs.

When exactly one NP is a pronoun, the non-pronoun must not be in the c-command domain of the pronoun. When both are pronouns or both are full NPs, (10a) does not apply (returns true).

Equations

• One or more equations did not get rendered due to their size.

theorem Phenomena.Anaphora.Studies.Reinhart1976.restriction_10a_vs_10b : corefPermitted_10a false false [Compare.Dir.L] [Compare.Dir.R, Compare.Dir.R, Compare.Dir.L] = true ∧ corefPermittedAddr false false [Compare.Dir.L] [Compare.Dir.R, Compare.Dir.R, Compare.Dir.L] = false

Non-equivalence of (10a) and (10b): (10a) fails on the (11a) case.

(10a) cannot block coreference between two full NPs (Rosa₁ ... Rosa₂), because (10a) only applies to pronoun–full NP pairs. (10b) correctly blocks it because Rosa₂ is in the domain of Rosa₁ and is not a pronoun.

theorem Phenomena.Anaphora.Studies.Reinhart1976.restriction_10b_subsumes_10a (isPronA isPronB : Bool) (addrA addrB : Compare.Address) : corefPermitted_10a isPronA isPronB addrA addrB = false → corefPermittedAddr isPronA isPronB addrA addrB = false

(10b) subsumes (10a): whenever (10a) blocks coreference, (10b) does too.

The converse fails (as restriction_10a_vs_10b shows), so (10b) is strictly more restrictive than (10a).