Reciprocals are definites

Formalization of [Bec01] (NLS 9). Each other uniformly denotes "the other ones among them" — an anaphoric plural definite recasting [HLM91] — and a reciprocal sentence is a special kind of relational plural. The paper admits exactly four semantic readings — collective, Strong Reciprocity, Weak Reciprocity, and situation-based WR — tracking the readings of relational plurals. The other interpretations of the [DKK+98] survey are not semantic: Partitioned SR and One-way WR are cover effects ([Sch96c] subgroups and exceptions), Intermediate Reciprocity is situation-based WR under a subsituation context (§5), and Inclusive Alternative Ordering is left underived, a lexical process limited to spatio-temporal relations (§6.3).

Main definitions

• otherOnesAmongThem — the HLM definite ((76)), in its singularity-parts case.
• CollectiveReading — predication of the reciprocal group itself ((84)).
• SituationWeakReciprocity — WR with cumulation over the situation argument ((193), bivalent, singleton covers).

Main results

• strongReciprocity_iff_otherOnes, collectiveReading_iff_strong — SR is double distribution over the definite ((81)); with a distributive relation the collective schema collapses into SR.
• situationWR_imp_weakReciprocity — under persistence, situation-based WR entails WR: the right edge of the entailment chain SR → sitWR → WR feeding the Strongest Meaning Hypothesis ((219), fn. 15).

The six-scheme entailment lattice ((28)) and the WR-as-cumulation identity ((120), weakReciprocity_iff_liftRel) live in Plurality/Reciprocal.lean and are consumed here. The convergence with [HD20] on presuppositional distinctness is housed in Studies/HaugDalrymple2020.lean (the later paper draws the comparison); the trivalent divergence from [Ste98] is noted at SituationWeakReciprocity.

References

• [Bec01] — the paper; [HLM91] — the recast analysis.
• [DKK+98] — the reading survey and the Strongest Meaning Hypothesis; [Lan78] — reciprocity as relational-plural predication; [FL73] — Partitioned SR; [Kan87] — IAO.
• [Lin83], [Sch96c], [Sha80], [BS00] — the plural-predication substrate: *, covers, maximality, **.
• [Ste98] — the WR-by-cumulation analysis §4 builds on and improves.

The HLM definite

def Beck2001.otherOnesAmongThem {α : Type u_1} [DecidableEq α] (A : Finset α) (x : α) : Finset α

The reciprocal's denotation ((76)): "the other ones among them" — the maximal subgroup of the antecedent A not overlapping the contrast argument x. In the singularity-parts case (distinctness = non-identity, p. 92) this is A.erase x; the full (76) routes through Sharvy maximality and Link *, with non-overlap replacing non-identity when covers contain genuine subgroups ((78)).

Equations

• Beck2001.otherOnesAmongThem A x = A.erase x

theorem Beck2001.otherOnesAmongThem_excludes {α : Type u_1} [DecidableEq α] (A : Finset α) (x : α) : x ∉ otherOnesAmongThem A x

The definite excludes the contrast argument.

theorem Beck2001.otherOnesAmongThem_subset {α : Type u_1} [DecidableEq α] (A : Finset α) (x : α) : otherOnesAmongThem A x ⊆ A

The definite is a subgroup of the antecedent.

theorem Beck2001.otherOnesAmongThem_nonempty {α : Type u_1} [DecidableEq α] (A : Finset α) (x : α) (hne : 1 < A.card) : (otherOnesAmongThem A x).Nonempty

On a plural antecedent the definite is defined (nonempty) at every member — the plurality presupposition ((115), §4.3.1).

The semantic readings, read off the definite

theorem Beck2001.strongReciprocity_iff_otherOnes {α : Type u_1} [DecidableEq α] (R : α → α → Prop) {A : Finset α} (hA : 1 < A.card) : Reciprocal.StrongReciprocity R A ↔ ∀ x ∈ A, ∀ y ∈ otherOnesAmongThem A x, R x y

On a plural antecedent, SR is double distribution over the definite ((81)): distribute over the antecedent, then over "the other ones among them".

def Beck2001.CollectiveReading {α : Type u_1} [DecidableEq α] (R : α → Finset α → Prop) (A : Finset α) : Prop

The collective reading ((84)): the relation holds of each member and the reciprocal group itself, with no distribution over that group — "the forks are propped against each other".

Equations

• Beck2001.CollectiveReading R A = ∀ x ∈ A, R x (Beck2001.otherOnesAmongThem A x)

theorem Beck2001.collectiveReading_iff_strong {α : Type u_1} [DecidableEq α] (R : α → α → Prop) {A : Finset α} (hA : 1 < A.card) : CollectiveReading (fun (x : α) (Y : Finset α) => ∀ y ∈ Y, R x y) A ↔ Reciprocal.StrongReciprocity R A

On a plural antecedent and with a relation that distributes to members, the collective schema is exactly SR: (81) and (84) differ only in distribution over the definite.

Situation-based Weak Reciprocity

(120)'s bivalent collapse — WR is ** of the strict-distinct relation — is the substrate theorem weakReciprocity_iff_liftRel, shared with [Ste98]'s (26b); the two analyses differ only trivalently, where [Ste98] asserts distinctness and [Bec01] presupposes it ((113), the distinct-subgroups effect of §4.3.2). What remains of the paper's own reading inventory is the situation layer.

def Beck2001.SituationWeakReciprocity {α : Type u_1} {σ : Type u_2} (R : α → α → σ → Prop) (A : Finset α) (subs : Finset σ) : Prop

Situation-based WR ((193), bivalent, singleton covers): every relevant subsituation contains a distinct pair in the relation, and every member participates in some relevant subsituation, in each direction.

Equations

• Beck2001.SituationWeakReciprocity R A subs = ((∀ s ∈ subs, ∃ x ∈ A, ∃ y ∈ A, x ≠ y ∧ R x y s) ∧ (∀ x ∈ A, ∃ s ∈ subs, ∃ y ∈ A, x ≠ y ∧ R x y s) ∧ ∀ y ∈ A, ∃ s ∈ subs, ∃ x ∈ A, x ≠ y ∧ R x y s)

theorem Beck2001.situationWR_imp_weakReciprocity {α : Type u_1} {σ : Type u_2} (R : α → α → σ → Prop) (A : Finset α) (subs : Finset σ) (s : σ) (hpers : ∀ (x y : α), ∀ s' ∈ subs, R x y s' → R x y s) (h : SituationWeakReciprocity R A subs) : Reciprocal.WeakReciprocity (fun (a b : α) => R a b s) A

Under persistence — what holds in a subsituation holds in the evaluation situation — situation-based WR entails WR at the evaluation situation: the right edge of the entailment chain SR → sitWR → WR that feeds the paper's Strongest Meaning Hypothesis ((219), fn. 15).