Rational Attitude Semantics @cite{fusco-sgrizzi-2026} #
@cite{dowty-1979}
Unified semantics for attitude verbs that support both belief and intention readings. The key insight: these are not two separate verb types but a single verb whose interpretation is determined by complement structure.
Core Idea #
A rational attitude is a mental state that can be either:
- Belief: a propositional attitude evaluated via CONTENT (doxastic modal base)
- Intention: a sub-propositional attitude evaluated via INERTIA (inertial modal base)
The difference is determined by complement size:
- CP complement → CLOSURE applies → propositional content → belief
- Sub-CP complement → event variable open → intention
Denotation (@cite{fusco-sgrizzi-2026}, ex. 24) #
⟦convincere⟧ = λP.λx.λy.λe. ∃e'. Convince(e) ∧ Agent(e,y) ∧ Patient(e,x) ∧ CAUSE(e,e') ∧ RATIONAL-ATTITUDE(e') ∧ Experiencer(x,e') ∧ P(e')
The parameter P is determined by complement size:
- di-infinitive (CP): P = CLOSURE(λe. VP(e)) — existentially closed
- a-infinitive (aP): P = λe. VP(e) — event variable open
The two readings of a rational attitude verb, determined by complement size.
belief: complement is propositional (existentially closed by CLOSURE); evaluated via CONTENT modal base (doxastic alternatives)intention: complement is sub-propositional (event variable open); evaluated via INERTIA modal base (normal continuation)
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- Semantics.Attitudes.RationalAttitude.instBEqReading.beq x✝ y✝ = (x✝.ctorIdx == y✝.ctorIdx)
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Map complement size to rational attitude reading.
CP-sized complements (fLevel ≥ 6) trigger CLOSURE, yielding a propositional content suitable for belief evaluation. Sub-CP complements leave the event variable unsaturated, producing an intention reading via INERTIA.
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Complement size determines reading: CP boundary is the threshold. This formalizes Fusco & Sgrizzi's structural size hypothesis.
CLOSURE: existential closure of the event variable at the CP level.
For CP complements, CLOSURE converts an event predicate P(e) into a proposition ∃e. P(e), yielding a belief-compatible propositional content.
This is existsClosureW from event semantics, re-exported under the
name used in @cite{fusco-sgrizzi-2026}.
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A causative attitude verb: Agent causes Experiencer to enter a rational attitude state whose content is determined by complement P.
⟦convincere⟧ = λP.λx.λy.λe. ∃e'. VerbPred(e) ∧ Agent(e,y) ∧ Patient(e,x) ∧ CAUSE(e,e') ∧ RATIONAL-ATTITUDE(e') ∧ Experiencer(x,e') ∧ P(e')
The parameter P is supplied by the complement:
- CP (di): CLOSURE applied → P is propositional (belief)
- Sub-CP (a): P is an event predicate (intention)
The verb's descriptive predicate (e.g., Convince)
- agent : E
The agent of the matrix event
- experiencer : E
The patient/experiencer who enters the attitude
Agent thematic role
Patient thematic role
Experiencer thematic role (on the attitude event)
CAUSE(e, e'): the matrix event e causally brings about the attitude state e'. Abstract relation; instantiated per verb/scenario.
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Semantics of a causative attitude verb applied to complement P.
Returns a proposition existentially closed over both the matrix event and the resulting attitude event.
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Belief reading: CLOSURE applies to the embedded VP, yielding a proposition. The attitude is evaluated via CONTENT (doxastic alternatives).
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- v.beliefReading embeddedVP = v.denote fun (x : Semantics.Events.Ev Time) => ∃ (e'' : Semantics.Events.Ev Time), embeddedVP e''
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Intention reading: no CLOSURE — the embedded VP is applied directly as an event predicate. The attitude is evaluated via INERTIA.
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- v.intentionReading embeddedVP = v.denote embeddedVP
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Both readings are instances of the same denote applied to different P.
This is the paper's central formal claim (ex. 24): convincere has ONE
denotation; the belief/intention split is compositional, arising from
complement size (CP triggers CLOSURE, sub-CP does not).
CAUSE*(s, e, w): causal self-reference relation (@cite{grano-2024}, (79); @cite{searle-1983}).
The attitude state s causally brings about event e in world w
"in the right way." Distinguished from plain CAUSE by requiring that
the causation proceed via the agent's intention-in-action, not via
a deviant causal chain (Harman 1976; Chisholm 1966).
Example (Harman): Betty aims her gun intending to kill the target. Her intention makes her nervous; nervousness causes her to pull the trigger; the target is killed. The outcome was caused by the intention, but not "in the right way" — the causal chain was deviant. CAUSE* would not hold, correctly predicting that Betty did not carry out her intention.
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- Semantics.Attitudes.RationalAttitude.CauseStar W Time = (Semantics.Events.Ev Time → Semantics.Events.Ev Time → W → Prop)
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Semantics for intention reports with causal self-reference (@cite{grano-2024}, version 4, (79)).
⟦Kim intends to leave⟧ᵂ·ᵗ = ∃s. INTENTION(s,w) ∧ HOLDER(k,s,w) ∧ ∀⟨w',x⟩ ∈ CONTENT(s): ∃e. CAUSE*(s,e,w') ∧ P(x,w',e)
The complement P has type (E → W → Ev Time → Prop) — an event
predicate with an open eventuality argument. This is the formal
correlate of @cite{grano-2024}'s Premise 3: IND would existentially
close the event argument, yielding (E → W → Prop), which is
type-incompatible with CAUSE*. The type system enforces that
intention reports require eventuality abstraction.
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Plain belief reports do NOT require CAUSE*: the complement is a proposition (event argument already closed by IND), so there is no event for CAUSE* to bind.
⟦Kim believes that Sandy left⟧ᵂ = ∀w' ∈ DOX(k,w): P(w')
The contrast in complement type — (W → Prop) for belief vs
(E → W → Ev Time → Prop) for intention — is what makes 'believe'
indicative-selecting and 'intend' subjunctive-selecting.
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- Semantics.Attitudes.RationalAttitude.beliefHolds dox agent P w = ∀ w' ∈ dox agent w, P w'
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Premise 3: Type-Level Enforcement (@cite{grano-2024}, §4.3) #
Intention reports require eventuality abstraction: intentionHolds
demands a complement with an open event argument (P : E → W → Ev Time → Prop),
while beliefHolds takes a closed proposition (P : W → Prop).
Indicative mood existentially closes the event argument ((87)),
yielding W → Prop, which is type-incompatible with intentionHolds.
Only subjunctive/nonfinite clauses leave the event argument open,
enabling the E → W → Ev Time → Prop type that CAUSE* requires.
The distinction is enforced by construction — no theorem is needed
because you literally cannot pass a W → Prop where
E → W → Ev Time → Prop is expected. The Lean type checker is the proof.
Truth assessment is available for belief readings but not intention readings. "It's true/false" can felicitously evaluate a belief but not an intention.
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Modal auxiliaries can appear in CP complements (belief) but not in sub-CP complements (intention). The CP provides the structural space to host modal heads (Mod, F-level 2).
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Intention readings are obligatorily future-oriented: the intended event is projected into inertia worlds (future continuation). Belief readings have no temporal constraint (they can be about past, present, or future).
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Object control is obligatory for intention readings: the experiencer must be the agent of the intended event. Belief readings allow both subject and object control configurations.
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Map rational attitude readings to @cite{searle-1983}'s direction of fit.
Belief readings have mind-to-world fit: the propositional content must match independently existing reality. Intention readings have world-to-mind fit: reality must be changed to match the content.
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Map rational attitude readings to @cite{searle-1983}'s psychological mode.
This connects @cite{fusco-sgrizzi-2026}'s complement-size analysis to Searle's theory of Intentional states: the same verb produces different psychological modes depending on syntactic complement structure.
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The direction of fit derived from the reading matches the direction derived from the corresponding psychological mode.
Belief readings are not causally self-referential; intention readings are. This is the formal correlate of @cite{fusco-sgrizzi-2026}'s CONTENT vs INERTIA modal base distinction: INERTIA worlds are those where the agent's intentions cause the events to come about.