@cite{almog-2014}: Referential Mechanics — Synthesis #
The central thesis of @cite{almog-2014}: the three mechanisms of direct reference — designation, singular propositions, and referential use — are logically independent. An expression can exhibit any subset of the three.
This module provides canonical referential profiles for each expression type, proves pairwise independence of the three dimensions, chains the key argumentation from across the Reference module, and bridges the reference theory to the rest of Linglib.
Canonical Profiles (from @cite{almog-2014}) #
| Expression | Designation | Singular Prop | Referential Use |
|---|---|---|---|
| True demonstrative | ✓ | ✓ | ✓ |
| Proper name | ✓ | ✓ | ✗ |
| dthat[the φ] | ✓ | ✗ | ✗ |
| "The φ" (ref.) | ✗ | ✗ | ✓ |
| "The φ" (attr.) | ✗ | ✗ | ✗ |
The ⟨F,T,F⟩ combination (singularity without designation) is witnessed by de re scope: a description taking wide scope over a modal contributes a singular proposition without itself being rigid. This is not an "expression type" per se but a scope configuration that demonstrates the logical independence of designation from singularity.
Independence #
The three dimensions are pairwise independent:
- Designation ⊥ singularProp: name [T,T] vs dthat [T,F] vs deReScope [F,T] vs attrDesc [F,F]
- Designation ⊥ referentialUse: demo [T,T] vs name [T,F] vs refDesc [F,T] vs attrDesc [F,F]
- SingularProp ⊥ referentialUse: demo [T,T] vs name [T,F] vs refDesc [F,T] vs attrDesc [F,F]
End-to-End Argumentation #
The central chain from @cite{almog-2014} Ch 1–2:
- dthat is rigid (KaplanLD.dthatW_isRigid)
- dthat is scope-inert (Kripke.rigid_iff_scope_invariant, forward direction)
- But dthat does NOT produce singular propositions (dthat_not_singular)
- Therefore designation ≠ singularity (designation_indep_singularProp)
- Singular propositions solve the Frege puzzle (frege_puzzle)
- So dthat alone cannot solve the Frege puzzle
Canonical Referential Profiles #
Proper name: designation + singularity, no referential use.
"Aristotle" rigidly designates an individual (@cite{kripke-1980}) and expresses a singular proposition ⟨Aristotle, property⟩ (@cite{kaplan-1989}), but does not require the speaker to have a cognitive fix on any particular occasion.
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dthat[the φ]: designation only.
sharpest separation (Ch 1): dthat[the tallest spy] rigidly
designates whoever is actually the tallest spy (dthatW_isRigid), but its
content is a general proposition (a rigid intension), NOT a structured
⟨individual, property⟩ pair. This distinguishes rigidity from direct
referentiality in Kaplan's sense.
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Referentially-used description: referential use only.
"The man drinking a martini" used referentially — the speaker has Jones in mind, using the description to identify him. Per Ch 3 §§2.2–2.12, this is a cognitive mechanism: the speaker's mind is already "loaded" with the referent. The description itself is non-rigid (designation = false), and per reading of Donnellan (§2.12), the propositional content is NOT singular — Donnellan gives a "proposition-free account, rather de re (de object coming in) in its form." Only the cognitive fix is present.
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Attributively-used description: no mechanism of direct reference.
"The man drinking a martini" used attributively — "whoever uniquely satisfies the description." No rigidity, no singular proposition, no cognitive fix.
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True demonstrative: all three mechanisms.
"That [pointing]" — rigid designation (the demonstrated object is fixed), singular propositional content (⟨demonstrated object, property⟩), and referential use (the speaker has the object in mind via the demonstration). The demonstrative is the paradigm case where all three of mechanisms converge.
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De re scope reading: singularity without designation.
"Lois believes [the man at the door]₁ is tall" (de re) — the description takes wide scope, contributing a singular proposition about the actual man at the door. But the description itself is not rigid (at different worlds, different people might be at the door).
Note: this is a scope configuration, not an expression type per se. We include it to witness the logical independence of designation from singularity — all four combinations of (designation, singularProp) must be attested for independence, and this is the only natural witness for ⟨F,T,F⟩.
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Pairwise Independence #
For each pair of dimensions (X, Y), we exhibit profiles demonstrating that X can vary independently of Y: both (X=T, Y=T) and (X=T, Y=F) are attested, as are (X=F, Y=T) and (X=F, Y=F).
Designation is independent of singular propositional content.
All four combinations of (designation, singularProp) are attested: name [T,T], dthat [T,F], deReScope [F,T], attrDesc [F,F].
Designation is independent of referential use.
All four combinations of (designation, referentialUse) are attested: demo [T,T], name [T,F], refDesc [F,T], attrDesc [F,F].
Singular propositional content is independent of referential use.
All four combinations of (singularProp, referentialUse) are attested: demo [T,T], name [T,F], refDesc [F,T], attrDesc [F,F].
Dthat: Designation Without Singularity #
central argument against conflating rigidity with direct
referentiality. dthat[the φ] is rigid by mechanism — KaplanLD.dthatW_isRigid
proves this — but its content is not a structured ⟨individual, property⟩ pair.
It is a general proposition that happens to be world-invariant.
A dthat-expression as a ReferringExpression: rigid character, profile
records designation without singularity or referential use.
Equations
- Semantics.Reference.Almog2014.dthatExpression desc cW = { character := fun (x : C) => Semantics.Reference.KaplanLD.dthatW desc cW, profile := Semantics.Reference.Almog2014.dthatProfile }
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dthat-expressions are de jure rigid: their character produces rigid content and their profile records the designation mechanism.
dthat-expressions do NOT have singular propositional content. This is the formal content of separation thesis.
End-to-End Argumentation Chain #
central argument in formal steps:
- dthat rigidifies descriptions →
dthatW_isRigid - Rigid designators are scope-inert →
rigid_iff_scope_invariant(fwd) - But dthat does NOT produce singular propositions →
dthat_not_singular - Singular propositions solve Frege puzzles →
frege_puzzle - Therefore dthat CANNOT solve the Frege puzzle — rigidity alone is insufficient; we need singularity (structured content) to distinguish ⟨Hesperus, bright⟩ from ⟨Phosphorus, bright⟩.
This is the formal core of argument that designation and singularity are independent mechanisms with different explanatory roles.
Step 1–2: dthat is scope-inert. Since dthat is rigid (dthatW_isRigid),
scope invariance follows from rigid_iff_scope_invariant.
Step 3–5: dthat cannot solve the Frege puzzle.
Given two descriptions that happen to co-denote at the actual world, dthat rigidifies both to the same individual. Their dthat-contents are identical (same rigid intension). But the structured singular propositions would distinguish them — except dthat doesn't produce structured content at all. So dthat eliminates scope ambiguity (step 2) but cannot explain informativeness.
The Frege Puzzle (Cross-Module Bridge) #
The Frege puzzle: unstructured propositions conflate what structured propositions distinguish.
Given two proper names for distinct individuals that happen to satisfy the same property at every world, their unstructured propositions are identical but their singular propositions are distinct.
Bridge to Kaplan.SingularProposition.structured_distinguishes_unstructured.
The "No Entailments" Thesis (Ch 2, §2.1) #
central metatheoretic claim: direct reference theory proper — whether via designation, singular propositions, or referential use — produces NO entailments about either modal or attitudinal questions.
"The simple logical point is this: There are no entailments from direct reference theory proper regarding either modal or attitudinal questions."
The theory assigns the same proposition to "Cicero = Cicero" and "Cicero = Tully." But that is all. Whether that proposition is necessary requires the independent metaphysical doctrine of modal haecceitism. Whether an agent believes it requires an independent theory of attitude verb semantics. The reference theory is silent on both.
Direct reference theory is silent on opacity.
Distinct individuals produce distinct singular propositions — this is a
structural fact about ⟨individual, property⟩ pairs, not an explanation of
attitude opacity. Per (Ch 2, §2.1), no doctrine regarding
modal or cognitive matters follows from direct reference theory proper.
Substitution failure in attitude reports requires an independent theory
of attitudinal verb semantics (see Attitudes.Doxastic.substitutionMayFail
for the formal framework).
Cross-Module Bridges #
Proper names are directly referential: bridges the Reference/Basic.lean
proper name definition through IsDeJureRigid to the broader system.
This is the formal content of @cite{kripke-1980}'s thesis as formalized via designation mechanism.
Bridge: PLA and the Frege Puzzle #
@cite{dekker-2012}'s cover-relative belief framework (Semantics.Dynamic.PLA) gives a
formal mechanism for Frege puzzles: two concepts can co-refer at the actual
world but diverge in belief-accessible worlds. This is exactly the scenario
where proper names have the same referent but different cognitive significance.
The bridge is informal: PLA uses cover-relative assignment functions while framework uses mechanism-based analysis. A formal connection would require unifying "mode of presentation" across both.
KDthat: Outside-In Reference (Ch 3, §2.13) #
alternative to Kaplan's dthat. In KDthat, the reference
is already fixed by an incoming signal (outside-in) before any linguistic
expression is deployed. The description in parentheses serves only as a
communicative guide — it helps the audience identify the referent the speaker
already has in mind, but does not determine it.
Contrast with dthatW, which rigidifies by evaluating the description
at the actual world (inside-out).
KDthat encodes the three-stage model of referential use (Ch 3, §1.2):
- Object-contact (outside-in):
loaded— the object that came to mind - Predicative characterization: the speaker forms (possibly false) beliefs
- Communication:
guide— the linguistic device deployed to direct the audience
KDthat: a point-demonstrative whose reference is fixed outside-in.
loaded is the individual from stage 1 (object-contact). guide is the
description used to communicate (stage 3). The guide plays no role in
determining reference.
Equations
- Semantics.Reference.Almog2014.kdthat loaded _guide = Core.Intension.rigid loaded
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Changing the communicative guide does not change the referent. This is the formal content of outside-in thesis: reference is fixed at object-contact (stage 1), not at communication (stage 3).
KDthat vs Dthat diverge when the description misfits. KDthat: "that [pointing at Jones] — the man with the martini" → Jones. Dthat: "dthat[the man with the martini]" → whoever actually satisfies it. When Jones is drinking water, these pick out different individuals.
This is the formal core of argument that Donnellan's referential use is a genuinely different mechanism from Kaplan's rigidification-by-description.
A KDthat-expression as a ReferringExpression: referential use only.
The profile is refDescProfile ⟨F, F, T⟩ because in's
framework, the expression (description) is not de jure rigid — the rigidity
comes from the speaker's cognitive fix on the loaded referent, not from
the expression's linguistic type. Per §2.12, Donnellan gives a
"proposition-free account" (no singular proposition content).
Contrast with dthatExpression, which has dthatProfile ⟨T, F, F⟩:
dthat is de jure rigid by linguistic mechanism, without referential use.
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KDthat-expressions are de facto rigid: their content is rigid (because the loaded referent is fixed), but the designation mechanism is not what secures rigidity — the cognitive fix does.
Contrast with dthat_deJureRigid: dthat is de jure rigid (rigid by
linguistic mechanism + designation=true). KDthat is de facto rigid
(rigid content + designation=false). This formalizes the core of
distinction between designation and referential use
as independent sources of world-invariance.
Informativeness Is Not Semantic (Ch 4, §2) #
dissolution of the Frege puzzle. The informativeness of "Cicero = Tully" is NOT a semantic fact — it is a cognitive/relational fact, depending on the thinker's partial information database. Semantically, the two names contribute identical content (proven below). Any difference in cognitive significance must be extra-semantic: relative to the thinker's historically incomplete knowledge of which loaded names track back to which objects.
Co-referential rigid designators have identical content. Since rigid intensions are constant functions, co-reference at any one world entails identity everywhere. The informativeness of "Hesperus = Phosphorus" cannot come from the semantics (the contents are the same); it must come from the cognitive relation between the thinker and the two names.
Direct reference assigns the same proposition to "P(a)" and "P(b)" when a and b are co-referential rigid designators.
This is the "full stop" of what direct reference theory delivers (Ch 2, §2.1). Whether the proposition is necessary requires the independent metaphysical doctrine of modal haecceitism. Whether an agent believes it requires an independent theory of attitude verb semantics. The reference theory is silent on both.
The Dual Semantic Function of Nominals (Ch 4, §4.1) #
extends @cite{donnellan-1966}'s referential/attributive distinction beyond definite descriptions to ALL nominals. Every nominal — proper name, bare plural, Det+CN phrase — has two potential semantic functions: pre-nominal (reference already established) and nominal (the noun originates the reference). The duality is semantic, not pragmatic — it is "written into the very conventional rules governing these phrases" ().
The dual semantic function of nominals.
preNominal: Reference preceded the nominal. The speaker already has the referent in mind; the noun merely expresses/conveys the link. "Bono" (when I've already seen him at the cheese section).nominal: The nominal originates the reference. It is essential to establishing the subject for subsequent predication. "A musician I met at Whole Foods" (introducing a new subject).
- preNominal : NominalFunction
- nominal : NominalFunction
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Donnellan's UseMode maps into the broader NominalFunction.
Referential use = pre-nominal (reference already established by
object-contact). Attributive use = nominal (the description originates
the reference).
Equations
- Semantics.Reference.Almog2014.nominalFunctionOf Semantics.Reference.Donnellan.UseMode.referential = Semantics.Reference.Almog2014.NominalFunction.preNominal
- Semantics.Reference.Almog2014.nominalFunctionOf Semantics.Reference.Donnellan.UseMode.attributive = Semantics.Reference.Almog2014.NominalFunction.nominal
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KDthat encodes pre-nominal function: the loaded parameter records
that the referent was fixed before any nominal was deployed.
The Orthogonality of Mechanism and Content #
deepest structural claim: the mechanism by which reference is secured (designation, cognitive fix, etc.) and the content that results (the intension, the proposition expressed) are orthogonal. Same content can arise from different mechanisms; same mechanism can produce different content.
Categorically, this is a commutativity diagram. Both dthat and kdthat
factor through rigid : E → Intension W E:
eval
Desc × World ——————————→ E
| |
| dthat | rigid
| |
v v
Content ════════════ Content
^ ^
| kdthat | rigid
| |
E × Guide ———— π₁ ————→ E
Both squares commute by construction (dthat_factors, kdthat_factors).
The outer rectangle commutes when eval(desc, cW) = π₁(loaded, guide),
i.e., when desc cW = loaded — the hMatch hypothesis.
The content projection is a coequalizer of the two mechanism paths.
But the diagram does NOT lift to ReferringExpression (which carries
the profile): the profile projection distinguishes what the content
projection conflates. This non-liftability is mechanism_content_orthogonality.
Dthat factors through rigid: dthat = rigid ∘ eval.
This is the top square of the commutativity diagram. The inside-out mechanism first evaluates the description at the actual world to obtain an entity, then rigidifies that entity.
KDthat factors through rigid: kdthat = rigid ∘ π₁.
This is the bottom square of the commutativity diagram. The outside-in
mechanism projects the loaded entity (ignoring the guide), then
rigidifies. The factorization is trivial because kdthat is defined
as rigid loaded.
The Separation Theorem. The content square commutes but the profile square does not — mechanism and content are orthogonal.
When the inside-out evaluation (desc cW) and the outside-in cognitive
fix (loaded) converge on the same entity, both paths through rigid
produce the same content. But the referential profiles — which record
how reference was secured — differ: dthat has ⟨T, F, F⟩ (designation),
KDthat has ⟨F, F, T⟩ (referential use).
Categorically: the forgetful functor π_content : ReferringExpression → Content
is a coequalizer of the two mechanism paths, but the projection
π_profile : ReferringExpression → ReferentialProfile is NOT — it
distinguishes the two paths. The profile is genuinely new information
not recoverable from the content.
The Flow Diagram Reversal (Ch 1, §2.3) #
recurring structural observation: in all four founding fathers' work, the classical Fregean direction of semantic determination is reversed. Classically, meaning (intension) determines denotation (extension) — symbol → satisfaction → object. In the historical/referential account, the object determines the reference — object → signal → loaded symbol.
The mathematical content of this reversal is a section-retraction pair:
rigid : E → Intension W E is a section (right inverse) of world-evaluation
evalAt · w : Intension W E → E.
evalAt w ∘ rigid = id(rigid_section): The historical chain (object → loaded name → evaluation) is lossless.rigid ∘ evalAt w ≠ idon non-rigid intensions (rigid_evalAt_lossy): The Fregean direction (intension → evaluate → re-embed) is lossy — it discards world-variation.
This makes E a retract of Intension W E. The image of rigid —
the rigid intensions — is isomorphic to E. Non-rigid intensions (descriptions)
live in the ambient space but collapse under the retraction.
Consequences already formalized in this module:
dthat_factors/kdthat_factors: Both paths factor through the sectionrigidinformativeness_not_semantic: The retraction annihilates modal information, so informativeness cannot be a semantic property of the retracted contentmechanism_content_orthogonality: The retraction coequalizes the two mechanism paths, but the profile is not in the retracted image
The flow diagram reversal as a retraction: evalAt w ∘ rigid = id.
Ch 1, §2.3: "this reversal of the flow diagram is a pattern that recurs in all four founding fathers' works on direct reference." Kripke's reversal for names (Ch 1), Donnellan's for descriptions (Ch 3), Kaplan's for demonstratives (Ch 2), Putnam's for common nouns (Ch 4).
The Fregean direction is lossy: non-rigid intensions (descriptions) cannot survive the round-trip through entity.
This is the mathematical content of "no entailments"
thesis: once you project to the retracted image (rigid intensions / entities),
the modal information that lived in the ambient intension space is gone.
"The man drinking a martini" varies across worlds; rigid (desc w) does not.
The retraction annihilates the very information that would distinguish
de re from de dicto readings.
The Russell-Partee-Kaplan Challenge (Ch 4, §3) #
formulation of the RPK impossibility: three natural desiderata for a global semantics of nominals cannot all be satisfied simultaneously. Each of the three historical waves of logical reformers sacrifices exactly one.
The three desiderata:
- Syntactic faithfulness: Preserve the visible subject-predicate grammar. "John is wise" and "Every philosopher is wise" have the same form.
- Semantic faithfulness: Subject terms refer to entities (type
E), not to constructed denotations (sets-of-properties, GQs, etc.). - Uniform composition: The same compositional mechanism handles all subject-predicate constructions.
The impossibility arises because referential subjects (type E) and
quantificational subjects (type (E → Prop) → Prop) have incompatible
types. Uniform composition requires a single subject type; semantic
faithfulness requires it to be E; but quantifiers cannot be entities.
The three waves each sacrifice one desideratum:
- Russell 1905: Drops syntactic faithfulness (invisible logical form)
- Montague 1970: Drops semantic faithfulness (type-lifts
Eto GQ) - Direct reference (Almog): Drops uniform composition (dual semantic function)
The three desiderata of the Russell-Partee-Kaplan challenge.
- syntacticFaith : RPKDesideratum
Preserve visible subject-predicate grammar
- semanticFaith : RPKDesideratum
Subject terms refer to entities (not constructed denotations)
- uniformComp : RPKDesideratum
Same composition for all subject-predicate structures
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- Semantics.Reference.Almog2014.instBEqRPKDesideratum.beq x✝ y✝ = (x✝.ctorIdx == y✝.ctorIdx)
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A semantic approach: satisfies some subset of the three desiderata.
- satisfies : RPKDesideratum → Bool
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Russell 1905: Drop visible grammar, keep referential semantics +
uniform composition. "Every philosopher is wise" gets invisible logical
form ∀x(P(x) → W(x)).
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Montague 1970: Keep visible grammar + uniform composition, sacrifice
referential semantics. "John" is type-lifted from E to (E → Prop) → Prop
via the Montague lift (= TypeShifting.lift).
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Direct reference (): Keep visible grammar + referential semantics, sacrifice uniform composition. "John" and "every philosopher" have different semantic functions (pre-nominal vs nominal).
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Each of the three historical approaches sacrifices exactly one desideratum. No approach satisfies all three.
Each approach satisfies the other two desiderata.
The type-theoretic content of the RPK impossibility: referential subjects
have type E, quantificational subjects have type (E → Prop) → Prop.
The Montague lift fun e P => P e embeds the former into the latter, but
it is injective-not-surjective: not every GQ arises from an entity.
This means the lift is a genuine sacrifice — "John" is no longer type E
but a constructed object of type (E → Prop) → Prop.
Bridge to TypeShifting.lift: this is the same operation as Partee's
type-raising lift(j) = λP. P(j).
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The RPK lift is injective: distinct entities give distinct GQs.
The RPK lift is NOT surjective in general: the universal quantifier
fun P => ∀ x, P x is a GQ that is not in the image of the lift
(assuming E has at least 2 elements).
This is the type-theoretic witness of the RPK impossibility: Montague's
uniform composition requires all subjects to be GQs, but not all GQs
are referential. The gap between E and (E → Prop) → Prop is real.
The RPK Impossibility Theorem. If:
(1) Subject terms denote entities (type E) — semantic faithfulness
(2) Composition is uniform: sentence meaning = P(subject) — uniform composition
then no entity can serve as the subject of "Every φ is ψ" — syntactic
faithfulness fails for quantificational sentences.
The proof: if the subject is e : E and composition is function application,
the sentence meaning is fun P => P e. But by rpkLift_not_surjective, the
universal quantifier fun P => ∀ x, P x is not in the image of this map
(assuming E has ≥ 2 elements). Therefore "Every philosopher is wise" cannot
be given a subject-predicate semantics in this framework.
This is the formal core of RPK challenge (Ch 4, §3): all three desiderata cannot be satisfied simultaneously.