Documentation

Linglib.Studies.Siloni2012

Siloni (2012): Reciprocal verbs and symmetry #

Beyond periphrastic reciprocals (each other), languages have reciprocal verbs — predicates that express reciprocity without an anaphoric object, carried as Verb.Reciprocal entries. [Sil12] (building on [Sil08]) splits them by where reciprocalization applies — the Formation parameter of Syntax/Reciprocal.lean, an instance of [RS05]'s lex-syn parameter — and derives nine clustering properties from that split alone: "the distinctions all follow from the fact that the two types of verbs are formed in different components of the grammar."

Reciprocalization acts on argument structure: reciprocalize takes a transitive RoleList and returns an intransitive one whose subject carries both base profiles' entailments — as [RS05]'s bundled complex [θᵢ · θⱼ] role in the lexicon ((35)–(36)), or as two separately assigned roles via last-resort parasitic assignment in the syntax ((43), the Merger ledger, which also derives the raising/passive ban and the ECM derivation (63)). Event-semantically the two loci share one mechanism, [Lan00b]'s group operators (Plurality.GroupStructure) in both domains: a symmetric verb is a set of atomic events whose complex role assigns the group atom over an unordered pair, related to its base by fn. 17's meaning postulate (Verb.SymmetricDenotation, in Syntax/Category/Verb/Symmetric.lean), from which (41) — the underlying directional events — is derived (underlying_of_symmetric); the syntactic reciprocal has the accumulation reading ((46b), AccumulationReading) and the up-packed singular reading ((47), GroupEventReading), whose content provably coincides with the symmetric verb's (GroupEventReading.underlyingReading). Since the "I" reading of embedded reciprocals ([Hig80]; the LF-decomposed periphrastic arm is Studies/HeimLasnikMay1991.lean) requires a sub-event reading plus a sole-role subject, both verb types lack it (I_reading_iff_periphrastic).

The nine surface properties derive from four affordances of the locus of application (pluralEventsAvailable per generalization (29), storesOutputs, spansPredicates, reducesAccusative): the paper's concluding table is a theorem (predictedProperties_lexical), as is the perfect complementarity of the two clusters. The paper's eleven-language sample is recorded per-language (hebrewbulgarian).

The §3 case against an object-clitic analysis of se is formalized as three diagnostics — functions from the verb's grid to an observable — on which the rival transitive grid and the reciprocalized grid diverge and the data side with reciprocalization: Czech depictive case agreement (depictive_diagnostic), comparative-ellipsis remnants (ellipsis_diagnostic), and [Kay75]'s faire-causative causee marking (causative_diagnostic).

Three further pieces of the paper's argument are worked out against the substrate: the periphrastic sub-event reading is derived from [HLM91]'s each∘other LF rather than stipulated (accumulation_of_eachOtherLF); the (15)/(16) count-adverbial contrast is modeled — five atomic mutual events with inaccessible parts versus five accumulation events with ten accessible directional parts (card_mutualEvent, mutualEvent_not_subEventReading, accumulated_subEventReading); and the discontinuous construction is the dyadic realization of a symmetric entry (§7.5), commutative in its two arguments and entailing the crossed directional events (dyadicReading_comm, DyadicReading.underlying).

Three-way reciprocal classification (§2.4) #

The three classes of reciprocal constructions (§2.4).

  • periphrastic : Construction

    Reciprocal anaphor in object position (each other, l'un l'autre); the subject bears one θ-role.

  • lexicalVerb : Construction

    Reciprocal verb formed in the lexicon by θ-role bundling (§4.1); the subject bears a complex [Agent-Theme] role and the verb is symmetric.

  • syntacticVerb : Construction

    Reciprocal verb formed in the syntax by a clitic (§4.2); the subject bears two θ-roles via parasitic assignment.

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      The lex-syn architecture (§3.5, §§5–6) #

      Four affordances of a locus of application, each carrying one of the paper's mechanisms; the nine surface properties are computed from them in predictedProperties.

      Generalization (29): "plural events are not part of the lexicon's inventory" (§3.5) — sub-event accumulation is available only in the syntactic derivation, so lexicon-formed reciprocals denote atomic events (Verb.SymmetricDenotation), whose underlying events are recovered only by dissolution.

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        Whether the operation's outputs are stored entries. Applying within the inventory, the lexical operation may also take frozen entries as input (Verb.Reciprocal.IsFrozen); stored outputs can drift, head idioms, and feed lexical nominalization, and listing caps productivity (§5.3, §6).

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          Whether the operation sees the θ-roles of two distinct predicates, as in ECM configurations (§5.2); the lexical operation manipulates a single entry.

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            Whether the operation must reduce accusative case whichever argument it suppresses ([RS05]); the syntactic clitic instead absorbs the suppressed argument's own case (§5.1).

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              The locus where a construction's reciprocity is composed: in the lexicon only for lexical reciprocal verbs; periphrastic reciprocity is composed by the anaphor's plural operator in the syntax (§2.4).

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                Whether reciprocity is built from sub-events: plural events must be available where it is composed (§2.2, generalization (29)).

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                  Symmetric verbs and reciprocal readings (§2.2, §4.1–4.2) #

                  Neo-Davidsonian format after [Lan00b]: verbs are sets of events, θ-roles are functions from events to individuals, and both the individual and the event domain are semilattices carrying the group operators. fn. 17's lexical format of V_SYM is the substrate contract Verb.SymmetricDenotation; here it meets Siloni's numbered readings and the derivation of (41).

                  def Siloni2012.CrossedPair {D : Type u_1} {E : Type u_2} (V : Set E) (ag th : ED) (d₁ d₂ : D) (e₁ e₂ : E) :

                  Two base-verb events realizing the two directions between d₁ and d₂ — the kernel shared by (41b), (46b), and (47).

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                  • Siloni2012.CrossedPair V ag th d₁ d₂ e₁ e₂ = (e₁ V e₂ V ag e₁ = d₁ th e₁ = d₂ ag e₂ = d₂ th e₂ = d₁)
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                    def Siloni2012.AccumulationReading {D : Type u_1} {E : Type u_2} [SemilatticeSup E] (V : Set E) (ag th : ED) (d₁ d₂ : D) (e : E) :

                    The accumulation reading of a syntactic reciprocal ((46b)): the event is the plain sum of the two directional sub-events.

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                      def Siloni2012.GroupEventReading {D : Type u_1} {E : Type u_2} [SemilatticeSup E] (GE : Semantics.Plurality.GroupStructure E) (V : Set E) (ag th : ED) (d₁ d₂ : D) (e : E) :

                      The singular-event reading of a syntactic reciprocal ((47)): up packs the sum of the directional sub-events into an atomic group event.

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                        def Siloni2012.UnderlyingReading {D : Type u_1} {E : Type u_2} [SemilatticeSup E] (GE : Semantics.Plurality.GroupStructure E) (V : Set E) (ag th : ED) (d₁ d₂ : D) (e : E) :

                        The content of (41b): the event's dissolution is the sum of two directional base events.

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                          def Siloni2012.SubEventReading {D : Type u_1} {E : Type u_2} [SemilatticeSup E] (V : Set E) (ag th : ED) (d₁ d₂ : D) (e : E) :

                          The sub-event reading (§2.2): the event decomposes into two proper parts realizing the two directions.

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                            theorem Siloni2012.SubEventReading.not_atom {D : Type u_1} {E : Type u_2} [SemilatticeSup E] {V : Set E} {ag th : ED} {d₁ d₂ : D} {e : E} :
                            SubEventReading V ag th d₁ d₂ e¬Mereology.Atom e

                            An event with the sub-event reading has proper parts, so it is not atomic.

                            theorem Siloni2012.subEventReading_of_crossedPair {D : Type u_1} {E : Type u_2} [SemilatticeSup E] {V : Set E} {ag th : ED} {d₁ d₂ : D} {e₁ e₂ : E} (h₁ : ¬e₂ e₁) (h₂ : ¬e₁ e₂) (hc : CrossedPair V ag th d₁ d₂ e₁ e₂) :
                            SubEventReading V ag th d₁ d₂ (e₁e₂)

                            Incomparable crossed sub-events yield the sub-event reading — the accumulation reading makes sub-events visible ((46b)).

                            theorem Siloni2012.GroupEventReading.atom {D : Type u_1} {E : Type u_2} [SemilatticeSup E] {V : Set E} {ag th : ED} {d₁ d₂ : D} {e : E} {GE : Semantics.Plurality.GroupStructure E} (h : GroupEventReading GE V ag th d₁ d₂ e) :

                            On the singular-event reading too, counting sees one event: group events are atoms.

                            theorem Siloni2012.GroupEventReading.underlyingReading {D : Type u_1} {E : Type u_2} [SemilatticeSup E] {V : Set E} {ag th : ED} {d₁ d₂ : D} {e : E} {GE : Semantics.Plurality.GroupStructure E} (h : GroupEventReading GE V ag th d₁ d₂ e) :
                            UnderlyingReading GE V ag th d₁ d₂ e

                            "The interpretation in (47) is identical to that of kiss_SYM (41b)": the group-event reading's dissolution content coincides with what the symmetric verb's postulate delivers.

                            theorem Siloni2012.underlying_of_symmetric {D : Type u_1} {E : Type u_2} [SemilatticeSup E] {V : Set E} {ag th : ED} {d₁ d₂ : D} {e : E} {GE : Semantics.Plurality.GroupStructure E} [SemilatticeSup D] {GD : Semantics.Plurality.GroupStructure D} {Vsym : Set E} {agTh : ED} [h : Verb.SymmetricDenotation GD GE V ag th Vsym agTh] (he : e Vsym) (hne : d₁ d₂) (hrole : agTh e = GD.up (d₁d₂)) :
                            UnderlyingReading GE V ag th d₁ d₂ e

                            (41): ∃e [kiss_SYM(e) ∧ [Ag-Th](e, John and Mary)] entails two underlying directional kissings — Verb.SymmetricDenotation's meaning postulate ((36)'s SYM marking, fn. 17) applied.

                            theorem Siloni2012.not_subEventReading_of_symmetric {D : Type u_1} {E : Type u_2} [SemilatticeSup E] {V : Set E} {ag th : ED} {e : E} {GE : Semantics.Plurality.GroupStructure E} [SemilatticeSup D] {GD : Semantics.Plurality.GroupStructure D} {Vsym : Set E} {agTh : ED} [h : Verb.SymmetricDenotation GD GE V ag th Vsym agTh] (he : e Vsym) {V' : Set E} {ag' th' : ED} {d₁' d₂' : D} :
                            ¬SubEventReading V' ag' th' d₁' d₂' e

                            A symmetric verb's events admit no sub-event reading for any predicate and roles whatsoever: the underlying events are invisible to counting and modification (§2.2).

                            Reciprocalization on argument structure (§4.1–4.2) #

                            Reciprocalization eliminates the internal argument position and leaves the subject carrying both profiles' entailments — a bundled complex role in the lexicon ((35)–(36)), two separately assigned roles in the syntax ((43)). On the Dowty substrate the profile content is EntailmentProfile's join over a transitive RoleList; realizes [Cre25]'s Voice.reciprocalization coding-frame operation (both core roles cumulated, derived construction intransitive).

                            Reciprocalization ((35)): the internal position is eliminated and its profile bundled onto the subject; "the bundle [θᵢ - θⱼ] retains the thematic properties of [θᵢ] and [θⱼ]" (reciprocalize_dominates).

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                              Reciprocalization denucleativizes: no internal argument survives.

                              The reciprocalized subject retains the base subject's Proto-Agent entailments and inherits the object's Proto-Patient entailments — the Czech depictive (§3.1) and comparative-ellipsis (§3.2) diagnostics.

                              For any transitive base with an agentive subject and an affected object, the reciprocalized subject bears a complex role (§4.1).

                              theorem Siloni2012.bundledSubjectProfile_eq_reciprocalize (v : Verb.Reciprocal) {b : Verb} {ps po : ArgumentStructure.EntailmentProfile} (hb : v.base = some b) (hs : b.subjectEntailments = some ps) (ho : b.objectEntailments = some po) :
                              v.bundledSubjectProfile = some (reciprocalize { subjectProfile := ps, objectProfile := some po }).subjectProfile

                              A reciprocal verb entry's bundled subject role is the subject of its reciprocalized base grid: the entry-level and grid-level operations agree.

                              Against an object-clitic analysis of se (§3) #

                              The rival analysis keeps the base transitive grid, with se realizing the internal argument; reciprocalization instead removes the internal position and leaves the subject with both roles. Three diagnostics decide, each a function from the verb's grid to an observable. Czech depictives agree in case with the argument they are predicated of ((17)): with se, a depictive construing with the Theme is nominative ((18)) — the Theme sits on the subject. The comparative-ellipsis remnant construes with a case-marked argument position: transitives allow subject and object readings ((19a)), se-verbs only the subject reading ((19b)), and an accusative remnant is ungrammatical ((20)) because no accusative argument exists. And [Kay75]'s causative generalization introduces the causee of a transitive with dative à and leaves an intransitive's causee bare ((25a–b)): se-verbs take the bare causee ((25d)), patterning with intransitives even though a genuine object clitic patterns transitively ((25c)). The fourth diagnostic — derived subjects never reciprocalize (§3.4, (26)) — is Merger.eq_of_assigned_nil below.

                              [Kay75]'s causative generalization (§3.3): under faire, the causee of an object-taking predicate is introduced by dative à; an intransitive's causee stays bare accusative.

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                                (25a) vs (25d): keeping the transitive grid predicts the dative à-causee; reciprocalization predicts the bare causee. Pierre a fait s'embrasser Jean et Marie decides for reciprocalization.

                                The case of a depictive construing with the Theme (§3.1): accusative if the internal role sits on an object ((17)), nominative if it sits on the subject.

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                                  (18): with se, the theme-oriented depictive is nominative, and licitly so — the reciprocalized subject bears the object's Proto-Patient entailments; the object-clitic grid predicts accusative.

                                  theorem Siloni2012.ellipsis_diagnostic {base : ArgumentStructure.RoleList} {o : ArgumentStructure.EntailmentProfile} (ho : base.objectProfile = some o) :
                                  (reciprocalize base).args.length = 1 base.args.length = 2

                                  The comparative remnant construes with a grid position (§3.2): the transitive offers two readings ((19a)), the reciprocalized grid one ((19b)); with no object there is no accusative remnant ((20)–(21)).

                                  Parasitic assignment (§4.2, (43); the ECM derivation (63)) #

                                  The syntactic operation forms no complex role: se reduces case (43a), the internal role is retained on the verbal projection, and a retained role is assigned upon merger of another θ-argument — a last-resort step (43b). Raising and passive subjects arrive by internal merge, which assigns no role, so a retained role can never discharge on them: derived subjects never reciprocalize ((45), §3.4). In the ECM derivation (63), the embedded verb's unassigned Agent survives the EPP-deficient TP and rides the matrix external merger: Jean ends θk+θi.

                                  inductive Siloni2012.Merger (seMarked : Bool) (before assigned after : List ThetaRole) :

                                  Merger against the verbal projection's retained-role ledger ((43), (63)): canonical θ-merger discharges one role; se-marked parasitic merger discharges the merging role together with every retained role (last resort); internal merge (movement) discharges nothing.

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                                    theorem Siloni2012.Merger.length_le_one {before assigned after : List ThetaRole} (h : Merger false before assigned after) :
                                    assigned.length 1

                                    Without the morphology there is no parasitic assignment: no argument receives two roles — the marking requirement "rules out the sole potential case of overgeneration" (§4.3 end).

                                    theorem Siloni2012.Merger.marked_of_two_roles {m : Bool} {before assigned after : List ThetaRole} (h : Merger m before assigned after) (h2 : 2 assigned.length) :
                                    m = true

                                    Two θ-roles land on one argument only via the marked last resort — §4.3's sole-role diagnosis, covering (63)'s ECM case, where both roles are agentive and profile mixture would not show.

                                    theorem Siloni2012.Merger.eq_of_assigned_nil {m : Bool} {before after : List ThetaRole} (h : Merger m before [] after) :
                                    after = before

                                    A merger that assigns no role is movement and leaves the ledger intact: derived subjects never discharge a retained role ((45), §3.4).

                                    The nine-property cluster (§§2–7) #

                                    The nine empirical properties that distinguish lexical from syntactic reciprocal verbs, in the order of the paper's concluding enumeration. Each Bool records whether the property holds for a formation type.

                                    • singularEvent : Bool

                                      (i) Reciprocity involves a singular atomic event: count adverbials (five times) yield N mutual events, not 2N directional sub-events (§2.2–2.3).

                                    • productive : Bool

                                      (ii) The reciprocalization operation applies productively to transitive verbs (§3.5, §5).

                                    • ecmReciprocals : Bool

                                      (iii) ECM reciprocal verbs are possible — reciprocalization spans a clause boundary (§5.2).

                                    • frozenInput : Bool

                                      (iv) Can be formed from a frozen lexical entry, one without a transitive alternate in the vocabulary (§5.3).

                                    • semanticDrift : Bool

                                      (v) Can undergo semantic drift — acquire a meaning not shared by the transitive alternate (§5.3).

                                    • phrasalIdioms : Bool

                                      (vi) Can head phrasal idioms unavailable for the transitive alternate (§5.3).

                                    • retainsAccOnDativeSuppression : Bool

                                      (vii) Can retain an accusative argument when reciprocalization suppresses the dative rather than the accusative (§5.1).

                                    • eventNominals : Bool

                                      (viii) Can derive reciprocal event nominals (§6). Exception: Czech reciprocal event nominals exist despite the syntactic setting — Czech nominalization is a lexical operation ([Hro05]), and the nominals arise by syntactic reciprocalization of the transitive nominal, available because the Czech clitic, unlike the strictly verbal Romance clitic, attaches to nouns.

                                    • discontinuous : Bool

                                      (ix) Allows the discontinuous reciprocal construction — subject plus comitative with-phrase (§7). A property of symmetric verbs (§7.5), with verb-level exceptions in both directions: English kiss and hug resist it (fn. 32), while listed lexical entries in syntax-setting languages (French se battre 'fight') allow it (§7.2).

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                                      def Siloni2012.instDecidableEqPropertyCluster.decEq (x✝ x✝¹ : PropertyCluster) :
                                      Decidable (x✝ = x✝¹)
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                                          Pointwise complement of a property cluster.

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                                            The property cluster of a formation locus, computed from its four affordances: the paper's reduction of the nine properties to the locus of formation. [Nor23]'s later per-language profiles check the discontinuity prediction against attested judgments in Studies/Nordlinger2023.lean.

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                                              theorem Siloni2012.predictedProperties_lexical :
                                              predictedProperties Reciprocal.Formation.lexical = { singularEvent := true, productive := false, ecmReciprocals := false, frozenInput := true, semanticDrift := true, phrasalIdioms := true, retainsAccOnDativeSuppression := false, eventNominals := true, discontinuous := true }

                                              The derived lexical cluster matches the paper's concluding table.

                                              The two predicted clusters are complementary on every property: the two loci differ on every affordance and every property is a literal of one affordance.

                                              The derived discontinuity prediction agrees with the substrate classifier Formation.allowsDiscontinuous.

                                              Symmetric verbs (§2.3, §7.5) #

                                              Lexically-formed reciprocal verbs are symmetric verbs: both participants are identically involved in a singular atomic event (§2.3), as with intransitive kiss and collide.

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                                                A formation yields symmetric verbs iff it predicts a singular event.

                                                "The discontinuous construction is a property of symmetric verbs" (§7.5).

                                                The "I" reading (§2.1, §4.3) #

                                                In "John and Paul said they defeated each other in the final" ([Hig80]), the "I" reading — John said he defeated Paul, and Paul said he defeated John — requires both that the embedded verb allow the sub-event reading and that its subject bear exactly one θ-role (§4.3, p. 287). The LF decomposition of the periphrastic side and the full grain problem live in Studies/HeimLasnikMay1991.lean.

                                                The subject's entailment profile over a transitive base. The periphrastic subject bears the base subject role; both verb types accumulate both profiles' entailments — as one bundled complex role in the lexicon ((36)), as two separately assigned roles in the syntax ((43), Merger) — "via bundling or parasitic assignment respectively" (§4.3).

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                                                  The "I" reading over a transitive base (§4.3, p. 287): the sub-event reading must be available and the subject must bear a sole role — a profile that is not complex.

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                                                    Sub-event availability is necessary for the "I" reading (§4.3).

                                                    A sole θ-role on the subject is necessary for the "I" reading (§4.3).

                                                    A periphrastic reciprocal over a non-complex subject role allows the "I" reading: sub-events from the anaphor's plural operator, sole role on the subject.

                                                    Lexical derivation: no plural events in the lexicon → no sub-events → no "I" reading (§3.5 → §2.2 → §4.3), for any base.

                                                    Syntactic derivation: the bundled subject is a complex role → no "I" reading (§4.2 → §4.3), despite the sub-event reading being available.

                                                    Neither reciprocal verb type allows the "I" reading, over any transitive base with an agentive subject and an affected object.

                                                    Over any transitive base whose subject role is agentive but not complex and whose object is affected, the "I" reading is available in the periphrastic construction and only there.

                                                    Singular-event verbs lack the sub-event reading: both are the availability of plural events at the composition locus (§2.2–2.3).

                                                    The periphrastic source of sub-events (§2.1) #

                                                    The sub-event reading of periphrastic reciprocals comes from the anaphor's plural operator: on a two-membered antecedent, [HLM91]'s each∘other LF delivers exactly the two directional relations, which sum to the accumulation reading.

                                                    theorem Siloni2012.eachOtherLF_pair {D : Type u_1} [DecidableEq D] {d₁ d₂ : D} (R : DDProp) (hne : d₁ d₂) :
                                                    HeimLasnikMay1991.eachOtherLF {d₁, d₂} R R d₁ d₂ R d₂ d₁

                                                    On a pair antecedent, the each∘other LF is the crossed directional pattern: Strong Reciprocity at two members.

                                                    theorem Siloni2012.accumulation_of_eachOtherLF {D : Type u_1} {E : Type u_2} [SemilatticeSup E] [DecidableEq D] {V : Set E} {ag th : ED} {d₁ d₂ : D} (hne : d₁ d₂) (h : HeimLasnikMay1991.eachOtherLF {d₁, d₂} fun (a b : D) => eV, ag e = a th e = b) :
                                                    ∃ (e : E), AccumulationReading V ag th d₁ d₂ e

                                                    The periphrastic construction has the accumulation reading: the LF's two directional relations, read off the base verb's events, sum to a plural reciprocal event (§2.1–2.2).

                                                    Count adverbials: five mutual versus ten directional events (§2.2) #

                                                    A model of the (15)/(16) contrast. Five rounds of kissing between Dan and Rina yield ten directional events. The Hebrew symmetric verb packs each round into a group atom: hitnašku xameš pe'amim counts five mutual events (card_mutualEvent), the round's directional events are recovered by dissolution (mutualEvent_down) yet are not parts of the counted event (mutualEvent_not_subEventReading) — which is also why *hitnašku al ha-mecax 'kissed on the forehead' fails (15d). The French accumulation reading instead sums the two directional events: the same five rounds (card_accumulated) keep both directional events accessible as proper parts (directional_lt_accumulated), so s'embrassèrent cinq fois can also count ten (16b).

                                                    The two participants.

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                                                      def Siloni2012.instReprKisser.repr :
                                                      KisserStd.Format
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                                                        The event domain: nonempty finite sets of atomic event markers, the left summand for directional kissings and the right for group atoms.

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                                                          The i-th directional kissing event.

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                                                            Round j kisses in both directions: Dan's is event 2j, Rina's is event 2j + 1.

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                                                              noncomputable def Siloni2012.kissAg (e : KissEvent) :

                                                              The agent of a directional kissing: Dan for the even events.

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                                                                noncomputable def Siloni2012.kissTh (e : KissEvent) :

                                                                The theme of a directional kissing: Rina for the even events.

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                                                                  Round j's two directional events form a crossed pair.

                                                                  theorem Siloni2012.directional_not_le {i k : } (h : i k) :
                                                                  noncomputable def Siloni2012.mutualEvent (j : Fin 5) :

                                                                  The j-th mutual kissing event: the round's two directional events, packed as a group atom — the Hebrew hitnašku denotation.

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                                                                    theorem Siloni2012.card_mutualEvent :
                                                                    (Finset.image mutualEvent Finset.univ).card = 5

                                                                    (15): five mutual kissing events.

                                                                    theorem Siloni2012.card_directional :
                                                                    (Finset.image directional (Finset.range 10)).card = 10

                                                                    Ten directional kissing events underlie them.

                                                                    theorem Siloni2012.mutualEvent_down (j : Fin 5) :
                                                                    kissGroups.down (mutualEvent j) = directional (2 * j)directional (2 * j + 1)

                                                                    Each mutual event dissolves to its round's directional events.

                                                                    Yet the mutual event has no accessible parts: the count adverbial and locative modification see one atomic event per round ((15a), (15d)).

                                                                    (16): the French accumulation event of round j is the plain sum.

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                                                                      theorem Siloni2012.card_accumulated :
                                                                      (Finset.image accumulated Finset.univ).card = 5

                                                                      Five accumulation events — the five-count of (16b).

                                                                      The accumulation event keeps the sub-event reading: both directional events are accessible ((16b)'s ten-count, (16c)'s modification).

                                                                      theorem Siloni2012.directional_lt_accumulated (j : Fin 5) :
                                                                      directional (2 * j) < accumulated j directional (2 * j + 1) < accumulated j

                                                                      Both directional events are proper parts of the accumulation event.

                                                                      The discontinuous construction: dyadic symmetric verbs (§7.4–7.5) #

                                                                      The discontinuous phrase is an argument (§7.4), so lexical reciprocal verbs have two realizations of one entry: monadic, with the pair-group subject, and dyadic, with subject and comitative each contributing one member of the pair (§7.5). Syntactic reciprocal verbs have no listed symmetric entry to realize dyadically — only listed symmetric entries do, which is why drifted se battre 'fight' allows discontinuity in syntax-set French (§7.2) while English kiss and hug resist it lexically (fn. 32).

                                                                      def Siloni2012.DyadicReading {D : Type u_1} {E : Type u_2} [SemilatticeSup D] (GD : Semantics.Plurality.GroupStructure D) (Vsym : Set E) (agTh : ED) (d₁ d₂ : D) (e : E) :

                                                                      The dyadic realization (§7.5): the verb's events and complex role are those of the monadic symmetric entry, the role taken at the subject and comitative's pair-group.

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                                                                        theorem Siloni2012.dyadicReading_comm {D : Type u_1} {E : Type u_2} [SemilatticeSup D] {GD : Semantics.Plurality.GroupStructure D} {Vsym : Set E} {d₁ d₂ : D} {e : E} {agTh : ED} :
                                                                        DyadicReading GD Vsym agTh d₁ d₂ e DyadicReading GD Vsym agTh d₂ d₁ e

                                                                        Discontinuity is symmetric: Dan corresponded with Rina and Rina corresponded with Dan describe the same events — the pair-group is unordered (§7.1).

                                                                        theorem Siloni2012.DyadicReading.underlying {D : Type u_1} {E : Type u_2} [SemilatticeSup D] {GD : Semantics.Plurality.GroupStructure D} {Vsym : Set E} {d₁ d₂ : D} {e : E} [SemilatticeSup E] {GE : Semantics.Plurality.GroupStructure E} {V : Set E} {ag th agTh : ED} [h : Verb.SymmetricDenotation GD GE V ag th Vsym agTh] (hne : d₁ d₂) (hd : DyadicReading GD Vsym agTh d₁ d₂ e) :
                                                                        UnderlyingReading GE V ag th d₁ d₂ e

                                                                        The discontinuous construction entails reciprocity between subject and oblique (§7.1): the dyadic reading of a symmetric verb yields the crossed directional events in both directions.

                                                                        Cross-linguistic sample (§2.4, §5, §7) #

                                                                        The formation locus of a language's reciprocal verbs.

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                                                                          def Siloni2012.instDecidableEqLangRecipVerb.decEq (x✝ x✝¹ : LangRecipVerb) :
                                                                          Decidable (x✝ = x✝¹)
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                                                                                    English reciprocal verbs (intransitive kiss, meet, collide) are lexicon-formed symmetric verbs, though the language's primary reciprocal strategy is the periphrastic each other (hence the strategy-level classification in Studies/Nordlinger2023.lean differs); fn. 32 notes that kiss and hug nonetheless resist the discontinuous construction.

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                                                                                                    The paper's language sample. The abstract announces ten languages; eleven are named with data across the paper, so all are recorded.

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