Frege, Russell, and theories of reference

Intuition Beginner

Reference is the relation between a name and the thing it names. Say "Aristotle" and you reach a man dead two thousand years; write "Venus" and you point at a planet. The puzzle is not that names reach things, but how. The marks on the page are inert; what breathes life into them so that they pick out one object rather than another, or none at all?

One answer runs through meaning: a name refers by carrying a description of its bearer, so "Aristotle" means "the pupil of Plato who wrote the Metaphysics." Another answer runs through use: speakers fix a name at a first naming and hand it along a chain of talk, and the name reaches its object through that chain, not through any description. A third answer splits the question: a name has both a reference, the object, and a sense, the route the mind takes to reach it.

This unit follows the long argument among these answers, from Frege's two-level theory, through Russell's attack on the hidden grammar of "the," to Kripke's claim that names are rigid and reach their bearers directly. Each move was a reply to a real puzzle; each reply exposed a new one.

Visual Beginner

The five positions in the table below are rival answers to one question: what fixes the reference of a name?

Theory	What fixes reference	Key figure
Sense / reference	The object, reached by a mode of presentation (the sense)	Frege
Cluster descriptivism	A description, or cluster, uniquely satisfied by the bearer	Russell, Searle
Quantificational analysis	No hidden referent; "the $F$" is a quantifier, not a name	Russell
Rigid designation	The object itself, in every world where it exists	Kripke
Causal-historical chain	A baptism plus a social chain of uses handed down	Kripke, Putnam

The diagram traces the move from a name that carries a route to a name that reaches its bearer directly.

Worked example Beginner

The reference of "Aristotle" looks settled until you ask what settles it. Trace the name through two rival theories and watch them pull apart on a single imagined case.

Step 1. On the descriptivist view, "Aristotle" means "the pupil of Plato who taught Alexander the Great." Reference is fixed by whoever fits that description.

Step 2. Now imagine that very man, Aristotle, dying as an infant and never meeting Plato. The description is then unsatisfied, or satisfied by someone else entirely.

Step 3. Yet in speaking of "Aristotle" in that imagined case, we still mean him, that very infant, not no one and not a stand-in.

Step 4. So the description cannot be what fixes the reference. What fixes it is the bearer himself, reached through the chain of uses running back to him.

What this tells us: the reference of a proper name is direct, routed through a causal-historical chain of uses, and not mediated by any description a speaker happens to associate with the name.

Check your understanding Beginner

Formal definition Intermediate+

Reference, in the philosophy of language, is the two-place relation between an expression and the object it picks out. Following the founding texts ^{[Frege 1892]}, the technical vocabulary is:

• Reference (Bedeutung): the object an expression denotes; for a singular term, its bearer; for a predicate, its extension, the set of objects it is true of.
• Sense (Sinn): the objective mode of presentation by which an expression determines its reference; senses are shareable contents that fix a reference without being identical to it.
• Definite description: a phrase of the form "the $F$," which surface grammar treats as a singular term but which, on Russell's analysis, is a quantified phrase.
• Rigid designator: a term that denotes the same object in every possible world in which that object exists; proper names and natural-kind terms are the paradigm cases ^{[Kripke 1980]}.
• Causal-historical chain: the social transmission of a name from an initial baptism through successive speakers, by which reference is fixed on the direct-reference theory.

A theory of reference states a relation $\mathrm{Ref}(e,w)=o$ between an expression $e$, a possible world $w$, and an object $o$. Frege distinguishes sense from reference so that two expressions may share $o$ yet differ in cognitive value. Russell denies that "the $F$" contributes a reference at all, analysing it as a quantifier. Kripke imposes rigidity, the condition $\mathrm{Ref}(e,w)=\mathrm{Ref}(e,w^{\prime })$ for all worlds $w,w^{\prime }$ in which the bearer exists.

The descriptivist thesis

Descriptivism holds that a proper name $N$ is semantically equivalent to a definite description, or a weighted cluster of descriptions, satisfied uniquely by the bearer ^{[Russell 1905]}. On this view the reference of $N$ in a world $w$ is whatever object in $w$ satisfies the associated description. The modal profile of the name is then inherited from the description: in worlds where the description is unsatisfied, the name fails to refer.

The direct-reference thesis

Direct reference holds that a proper name contributes only its bearer to the propositions in which it occurs ^{[Kripke 1980]}. Reference is fixed, not by a description, but by an initial baptism and a causal-historical chain of uses. Two speakers ignorant of the same facts about Aristotle may still corefer, so long as their uses trace back through the chain to him. The name's contribution to the proposition is the object itself, not a descriptive content.

Counterexamples to common slips

• Sense is not connotation. Sense is an objective, truth-relevant mode of presentation determining reference; connotation is a subjective association varying from speaker to speaker.
• Rigidity is not constancy of description. A rigid designator keeps its bearer fixed across worlds; a description keeps its descriptive content fixed, but may pick out different bearers in different worlds.
• Direct reference is not the denial of meaning. Direct-reference theorists grant that names have cognitive significance; they deny that the significance is a descriptive sense that fixes the reference.

Key argument Intermediate+

Four arguments structure the dialectic of reference. Each is a reply to a pressure exposed by its predecessor, and together they trace the move from Frege's two-level theory through Russell's quantificational analysis to Kripke's rigid designation.

Frege's argument for sense

Identity statements can be informative. "Hesperus is Hesperus" is empty; "Hesperus is Phosphorus" was a discovery, though both names denote Venus ^{[Frege 1892]}. If meaning were reference alone, the two would be indistinguishable, since the reference is identical. They are not. Therefore meaning has a second component: the sense, the mode of presentation, by which a thinker reaches the reference. Two co-referential terms share a Bedeutung and differ in Sinn.

Reconstructed. (i) "Hesperus is Hesperus" and "Hesperus is Phosphorus" differ in cognitive value. (ii) The two share their reference, Venus. (iii) If meaning were reference alone, they would not differ in cognitive value. (iv) Therefore meaning is not reference alone; sense must be admitted.

Russell's theory of descriptions

"The present King of France is bald" appears to be a subject-predicate sentence about a King ^{[Russell 1905]}. If "the present King of France" were a genuine singular term, it would have to refer, and the sentence would be true or false of that referent; but there is no King, so the sentence seems neither true nor false. Russell's resolution is that the surface grammar misleads. The description is not a singular term but a quantified phrase:

$$\text{"The }F\text{ is }G\text{"}\equiv \exists x(F(x)\wedge \forall y(F(y)\to y=x)\wedge G(x)).$$

The sentence thus asserts existence (an $F$), uniqueness (only one $F$), and predication (it is $G$), packed into one grammatical subject. With no unique King, the existence conjunct fails, and the sentence is simply false, not truth-valueless. Empty descriptions are analysed away rather than treated as failed referring expressions.

Strawson's critique

Strawson objected that Russell confuses sentences with their uses ^{[Strawson 1950]}. A definite description is not, in itself, a referring expression or a quantifier; it is a device a speaker may use to refer on a particular occasion. The sentence "The present King of France is bald" has no truth value on its own; whether a use of it is true or false depends on whether the speaker succeeds in picking out a unique King. Where Russell makes the sentence false, Strawson makes a use of it lack a truth value, and locates reference in the speech act rather than the expression. The exchange sharpened the question of whether reference is a semantic property of expressions or a pragmatic act of speakers.

Kripke's modal argument against descriptivism

Suppose, with the descriptivist, that "Aristotle" means "the pupil of Plato who wrote the Metaphysics." Then in any possible world, "Aristotle" denotes the satisfier of that description ^{[Kripke 1980]}. But there is a possible world in which Aristotle died in infancy, never meeting Plato; in that world the description denotes no one, or someone else, yet the name still refers to Aristotle. The description cannot constitute the name's meaning. Proper names are rigid designators, denoting the same object in every world in which that object exists. The consequence is the necessary a posteriori: "Hesperus is Phosphorus," if true, is necessarily true, yet known only through astronomy.

Reconstructed. (i) Names are rigid designators. (ii) If $a$ and $b$ are rigid and $a=b$, then in every world in which both exist, $a=b$. (iii) Therefore $a=b$ is necessary. (iv) Yet such identities are discovered empirically. So necessity and a-priority come apart.

The causal-historical chain fixes which object a rigid designator latches onto: an initial baptism names the bearer, and subsequent uses inherit reference by transmission along the chain ^{[Kripke 1980]}. A speaker may know almost nothing true of the bearer and still refer, provided the chain is intact.

Bridge. This argument builds toward 42.02.01 pending (formal semantics and model theory), where rigid designation is precisified as a reference function constant across possible worlds within Kripke semantics, and appears again in 20.11.01 (metaphysics), where the necessity of identity grounds modal metaphysics, and in 20.12.01 (philosophy of language foundations), which sets the broader frame. The foundational reason theories of reference recur across the curriculum is that any claim about what is said presupposes a theory of how words latch onto objects; this is exactly the load the Frege, Russell, and Kripke dialectic carries; putting these together, the bridge is that sense, the quantifier, and the rigid designator are three successive answers to one question, and the pattern generalises from natural language into the formal languages of modal logic and intensional type theory.

Exercises Intermediate+

Lean formalization Intermediate+

lean_status: none. Theories of reference are prose-first; the correctness gate is argument reconstruction and fidelity to the primary texts. The formal apparatus relevant to this material, possible-worlds intensions, rigid designation as a constant reference function, and Kripke semantics for modal logic, is developed in 42.02.01 pending (formal semantics and model theory) and the proof-theory chapters of §42. The present unit's task is to reconstruct the arguments of Frege, Russell, Strawson, Donnellan, and Kripke, not to formalise them.

Advanced results Master

Three results dominate the post-Kripkean literature, each a stance on whether reference is mediated by descriptive content. They are live, partly incompatible positions; the field has not converged.

Donnellan's referential/attributive distinction. Russell and Strawson treated definite descriptions as having a single semantic function. Donnellan argued that "the $F$" has two distinct uses ^{[Donnellan 1966]}. In the attributive use, the speaker means "whatever is the unique $F$"; in the referential use, the speaker uses "the $F$" to pick out a particular object, which may or may not be $F$. The murderer, called "the man drinking champagne," may be referred to successfully even if he is drinking cider. The referential use can secure reference to an object that fails the descriptive condition. This threatens Russell's uniform analysis, since a referentially used description can express a true predication about its target even when uniqueness or the descriptive content fails.

Kripke's three arguments against descriptivism. The modal argument above is one of three. The epistemic argument observes that if "Aristotle" meant "the author of the Metaphysics," then "Aristotle wrote the Metaphysics" would be analytic, knowable a priori; it is not, since the authorship can be disputed on empirical grounds. The semantic argument notes that speakers who share a name's reference often associate different, even incompatible, descriptions with it, yet unproblematically corefer; if the description constituted the meaning, such speakers would speak different languages. The three arguments press on different facets of descriptivism: what is possible, what is knowable, and what is shared among speakers ^{[Kripke 1980]}.

The causal theory and the problem of error. Reference fixed by a causal-historical chain is robust to ignorance but vulnerable to transmission error. The name "Madagascar" was borrowed by Marco Polo from a term denoting a region of the mainland and came, through a corrupted chain, to denote the island; the chain fixed a reference its origin did not intend. The theory treats this as a feature: reference is determined by the actual chain of uses, not by the speaker's intentions or descriptions. Critics object that a chain sustaining enough error no longer refers to the original bearer at all, and that the theory offers no clean criterion for when transmission has broken.

Two-dimensionalist and neo-descriptivist replies. Two-dimensionalism distinguishes a primary, epistemic intension from a secondary, subjunctive intension, recovering the contingency of "water is ${\mathrm{H}}_2\mathrm{O}$" at the epistemic level while granting Kripke the modal necessity. Neo-descriptivism replaces simple descriptions with rigidified or causal descriptions, attempting to preserve descriptivist resources against the modal argument. Neither programme has displaced direct reference, but both keep the descriptivist tradition live.

Synthesis. The arguments of this unit build toward a single trajectory in which sense gives way to the quantifier and the quantifier gives way to the rigid designator; this trajectory appears again in 20.11.01 (metaphysics), where necessity of identity is stated ontologically, and in 42.02.01 pending (model theory), where rigidity becomes a constant reference function across possible worlds. The foundational reason the dialectic has this shape is that each theory answers the same question, what fixes reference, under a stronger constraint than its predecessor; this is exactly why Strawson's pragmatic turn and Kripke's modal turn could not be absorbed by the descriptivist framework; putting these together, the bridge is that reference is progressively detached from descriptive content and reassigned, first to use, then to the bearer itself, and the pattern generalises into the possible-worlds semantics that grounds modal logic and intensional type theory.

Full proof set Master

Proposition (Russellian analysis entails existence and uniqueness). Under Russell's quantificational analysis, "The $F$ is $G$" is true if and only if there exists exactly one $F$ and it is $G$; in particular, if no unique $F$ exists, the sentence is false.

Reconstruction. By the analysis, "The $F$ is $G$" expresses $\exists x(F(x)\wedge \forall y(F(y)\to y=x)\wedge G(x))$. For this conjunction to hold, each conjunct must hold: $F(x)$ for some $x$ (existence), $\forall y(F(y)\to y=x)$ (uniqueness), and $G(x)$ (predication). If no object satisfies $F$, the first conjunct fails and the existential is false. If two or more objects satisfy $F$, the uniqueness conjunct fails, and the existential is again false. Therefore the sentence is false whenever there fails to be exactly one $F$, and true only when there is exactly one $F$ and it is $G$. $\square$

Remark (rigidity forces necessity of identity). A companion result concerns rigid designators: if $a$ and $b$ are rigid and $a=b$, then necessarily $a=b$. Since $a$ denotes its bearer in every world in which that bearer exists, and likewise $b$, and the bearers coincide in the actual world, $a=b$ holds in every world in which the bearer exists. This is the formal core of Kripke's necessary-a-posteriori identities, developed fully in 20.11.01.

Connections Master

• Philosophy of language — foundations 20.12.01. This unit presupposes the foundational three-part frame of reference, sense, and use set out there, and deepens it exclusively on the reference axis; the speech-act and meaning-as-use material in the prerequisite supplies the context the Frege, Russell, and Kripke dialectic requires.

• Metaphysics 20.11.01. Kripke's rigid designation and the necessity of identity are stated there in ontological terms, and this unit supplies their semantic underpinning; the necessary-a-posteriori identities for natural kinds depend on the rigid-designator semantics reconstructed here.

• Formal semantics and model theory 42.02.01 pending. The reference function $\mathrm{Ref}(e,w)$, rigidity as constancy across possible worlds, and Kripke semantics for modal logic supply the formal precision into which the prose arguments of this unit are precisified.

• Philosophy of mind 20.13.01. The content of mental states, what a thought is about, is analysed with the same reference apparatus; the causal-historical chain for names is the model for externalist theories of mental content.

Historical & philosophical context Master

Frege's "Über Sinn und Bedeutung" (1892) introduced the sense/reference distinction to solve the puzzle that identity statements can be informative, separating the mode of presentation, Sinn, from the object picked out, Bedeutung ^{[Frege 1892]}. Russell's "On Denoting" (1905) rejected Frege's treatment of empty descriptions, replacing sense with the quantificational analysis and calling it a paradigm of logical analysis ^{[Russell 1905]}.

Strawson's "On Referring" (1950) reopened the question by arguing that Russell had confused sentences with their uses, locating reference in the speech act rather than the expression ^{[Strawson 1950]}. Donnellan's "Reference and Definite Descriptions" (1966) split the descriptive uses into referential and attributive, further pressuring the uniform Russellian analysis.

The causal-historical turn came with Kripke's 1970 Princeton lectures, published as Naming and Necessity (1980), and Putnam's "The Meaning of 'Meaning'" (1975); together they replaced descriptivism with direct reference and reopened the modal questions the earlier tradition had largely closed ^{[Kripke 1980]}. The contemporary field divides among direct-reference theorists, two-dimensionalists, and neo-descriptivists, with no settled consensus.

Bibliography Master

@article{Frege1892,
  author = {Frege, Gottlob},
  title = {Über Sinn und Bedeutung},
  journal = {Zeitschrift für Philosophie und philosophische Kritik},
  year = {1892},
}

@article{Russell1905,
  author = {Russell, Bertrand},
  title = {On Denoting},
  journal = {Mind},
  volume = {14},
  year = {1905},
}

@article{Strawson1950,
  author = {Strawson, P. F.},
  title = {On Referring},
  journal = {Mind},
  volume = {59},
  year = {1950},
}

@article{Donnellan1966,
  author = {Donnellan, Keith S.},
  title = {Reference and Definite Descriptions},
  journal = {The Philosophical Review},
  volume = {75},
  year = {1966},
}

@article{Searle1958,
  author = {Searle, John R.},
  title = {Proper Names},
  journal = {Mind},
  volume = {67},
  year = {1958},
}

@book{Kripke1980,
  author = {Kripke, Saul A.},
  title = {Naming and Necessity},
  publisher = {Harvard University Press},
  year = {1980},
}

@article{Putnam1975,
  author = {Putnam, Hilary},
  title = {The Meaning of ``Meaning''},
  journal = {Minnesota Studies in the Philosophy of Science},
  year = {1975},
}

@book{Neale1990,
  author = {Neale, Stephen},
  title = {Descriptions},
  publisher = {MIT Press},
  year = {1990},
}