HISTORICAL THEORY OF REFERENCE AND THE CRITERIA OF IDENTITY: The Clash of Kripke's and Linnebo's Theories of ReferenceAbstract
In his development of an ontology for mathematics in line with Fregean abstractionism, Øystein Linnebo proposes criteria of identity as a mechanism for reference. This proposed mechanism appears to conflict with Saul Kripke's historical theory of names. However, Linnebo, distinguishing between semantic and metasemantic domains, asserts that no real conflict exists; while Kripke's theory is semantic, his is metasemantic. This paper posits a counter-argument, contending that Linnebo's account, contrary to his claim, indeed conflicts with Kripke's historical theory of names. The paper underscores that, in Kripke's framework, the reference relation transcends the individual, inhabiting a social domain.
Keywords:
Historical Theory of Reference; Criteria of Identity; Thin Objects; Abstraction Principle
1. Introduction
Building upon Frege's epistemological framework, Øystein Linnebo has put forth an innovative ontology for mathematics. This account acknowledges the existence of abstract objects independent from the human mind, yet it contends that their accessibility is ensured via abstraction.
Nevertheless, it is important to recognize that abstract objects such as numbers exist in a manner distinct from that of ordinary macro-objects. The existence of ordinary objects typically necessitates a greater demand on reality, whereas the existence of numbers contributes minimally. Analogously, the existence of a house doesn’t significantly add to the reality of its constituent bricks and furniture arranged in a particular manner. When the existence of an object significantly contributes to reality, it is termed a “thick object”; conversely, when its contribution is minimal, it is referred to as a “thin object”. Additionally, when the existence of the object demands almost nothing from reality, it is identified as an “ultra-thin object”.
Diverging from the view of ultra-thin objects, Linnebo introduces criteria of identity as a condition for reference. Under his conceptualization, an object's reference is made possible by this identity condition. Central to this discussion, Linnebo asserts that his concept of the criteria of identity does not clash with Kripke’s dismissal of the descriptivist theory of names.
This paper, nonetheless, takes a contrary stance. It posits that there is, indeed, a conflict between Linnebo’s approach and Kripke’s repudiation of descriptivism. We argue that Linnebo's theory, despite its claim, does not escape the ambit of Kripke's criticisms of descriptivism. The ensuing discourse will attempt to elucidate this theoretical discord, thereby casting doubt on the purported harmony between Linnebo's ontology and Kripke's theory.
2. Establishing Reference: The Role of The Criteria of Identity
The existence of abstract objects, especially those pertaining to mathematics, represents a pivotal question in philosophy. If we posit that mathematical discourse corresponds to a facet of reality, one that is only accessible through abstraction, we find ourselves echoing the school of thought initiated by Gottlob Frege. This prompts us to consider the nature of this reality and the means by which we access it.
When we say “The Illiad consists of twenty-four books,” what does “twenty-four” represent? It does not denote a property of the Illiad per se, given that the Illiad can also be viewed as a single poem. This leads to the question: Is it one or twenty-four? Frege argues that numbers do not directly relate to objects themselves but to the particular concepts under which those objects fall. Hence, “twenty-four” pertains to the conceptualization of the Illiad as a series of books (Frege, 1953, §29-33).
Contrary to viewing numbers merely as properties of concepts, Frege posits that they are self-subsistent objects. This is illustrated by stating “Jupiter has four moons” and equivalently “The number of Jupiter’s moons is four,” where “the number of Jupiter’s moons” refers to a distinct, self- subsistent object (Frege, 1953, §57). Frege then probes the accessibility of such objects, for which we lack even mental picture (Frege, 1953, §60). His response invokes the context principle, asserting that words acquire meaning only within the context of a proposition. Consequently, “our problem becomes this: To define the sense of a proposition in which a number word occurs” (Frege, 1953, §62). In seeking these propositions, Frege lays the groundwork for his abstraction principle.
Drawing on David Hume’s conceptualization of numbers, Frege further developed the abstraction principle to elaborate on this foundational concept (Frege, 1953, §63- 69). Hume’s principle indicates that talking about identity numbers is just about having a one-to-one correspondence between the extensions of two concepts, like stones and sheep. For example, saying the number of stones is the same as the number of sheep means there’s a one-to-one match between the extension of the concept stone and the extension of the concept sheep. The abstraction principle takes this idea and applies it to all abstract objects. The principle in question is articulated in a biconditional form, where the right-hand side is framed to be epistemologically more accessible, while the left-hand side serves the purpose of introducing abstract objects. For instance, once we establish that lines l1 and l2 are parallel, we are afforded the existence of these lines' direction as a separate object (Frege, 1953, §66).
dir(l1) = dir(l2) ↔ l1 // l2, where dir(l) denotes the direction of line l.
In the same vein, if there exists a one-to-one correspondence between the extensions of two concepts, we can introduce the number of objects that fall under either concept as a new object within our ontology.
#Fs = #Gs ↔ bijection(F, G), where #F denotes the number of Fs.
To solidify the foundations of Hume’s principle, Frege conceptualized numbers in a specific manner. He defined the number of objects that fall under a concept F as the extension of the concept ‘equinumerous to the concept F’ (Frege, 1953, §68). With numbers defined in this way, Hume’s principle could be demonstrated. For this proof, Frege introduced Basic Law V, which essentially states that the extensions of two concepts are identical if and only if every object that falls under one concept also falls under the other and vice versa (Frege, 2013, p. 61). This leads to the Law of Extension, which posits that an object belongs to the extension of a concept if and only if it falls under that concept (Frege, 2013, p. 75). However, applying this law to the concept comprising extensions that are not members of themselves results in a paradox, famously identified by Russell: if the extension of this concept is a member of itself, then it cannot be a member of itself. Conversely, if it is not a member of itself, it must be a member of itself (Russell, 1967).
Although Bertrand Russell's paradox led to the abandonment of the Fregean project, subsequent neo-Fregeans such as Charles Parsons, Bob Hale, and Crispin Wright rekindled the issue by accepting Hume’s principle without seeking its derivation from more fundamental truths (Parsons, 1965) (Hale, 1987) (Wright, 1983).
The resurgence of the Fregean project gave rise to what Linnebo terms the "ultra-thin" conception of objects (Linnebo, 2018, pp. 87-89). There exist two versions of this conception: the Syntactic Priority Thesis and Agustin Rayo’s Compositionalism. The former advocates that a term’s appearance in a true atomic sentential context is sufficient for it to refer to an object (Hale & Wright, 2001, pp. 31-150). The latter, Rayo’s Compositionalism, posits that a singular term ‘t’ refers to an object if t is the value of an existentially quantified variable (Rayo, 2013, pp. 13-22).
Linnebo introduces a nuanced version termed “thin objects,” to which he adds the criteria of identity as a means to explain reference. He states, “…the obtaining of an appropriate criterion of identity becomes sufficient for the existence of an object governed by this criterion. For example, two lines’ being parallel is sufficient for the existence of a direction that is shared by the two lines. A truth about parallelism is thus reconceptualized in a way that reveals a new object, namely a direction, which was not involved in the original truth” (Linnebo, 2018, p. 23). Further, Linnebo clarifies his criteria of identity through the analogy of robots that determine references to physical entities based on sensory stimuli from their environment.
What does it take for one of these robots to refer to a physical body? At the very least, the robot must receive ‘perceptual’ information from some part of the body, which thus serves as a specification of the referent. More interestingly, the robot needs some mechanism for determining when two such specifications are associated with one and the same body. I call this a unity relation. This unity relation will reflect fundamental features of physical bodies such as: bodies are three-dimensional, solid objects; bodies have natural and relatively well distinguished spatial boundaries; bodies are units of independent motion; bodies move along continuous paths; bodies have natural and relatively well-distinguished temporal boundaries. Write ‘~’ for this unity relation (which is a partial equivalence relation) and ‘B’ for the ‘body builder’ that maps a specification to the body, if any, that it determines. Then we obtain the following criterion of identity:
This criterion achieves something remarkable. It figures at the heart of an account of what it takes for the robots to refer to a physical body (Linnebo, 2023, pp. 250-251)
The crux of his argument lies in the assertion that, under the ultra-thin objects view, reference becomes inexplicable. Singular terms refer, but the specific object being referred to remains unidentifiable due to the absence of identity conditions. (Linnebo, 2018, pp. 92-93)
3. Kripke versus Linnebo: Analyzing the Role of the Criteria of Identity in Naming
The function of proper names and their relationship to the objects they denote have long been contentious topics in the philosophy of language. Saul Kripke, in his seminal work Naming and Necessity, challenges the descriptivist view according to which the meanings of singular terms correspond to definite descriptions of their referents. He maintains that descriptions do not play any role in the semantics of proper names and demonstratives, highlighting that this position holds true from metaphysical, epistemological, and semantic perspectives. From the metaphysical perspective, the object referenced by a proper name could deviate from the descriptions associated with it (Kripke, 1980, p. 75). On the epistemic front, descriptions pertaining to the objects are not given a priori (Kripke, 1980, pp. 83-84, 87). Semantically speaking, the absence or incorrectness of information does not lead to the failure of proper names to refer (Kripke, 1980, p. 85).
Kripke’s dismissal of the role of descriptions in the semantic valuation of proper names seems to be at odds with Linnebo’s “criteria of identity.” The reason is that Linnebo’s approach reduces the reference relation to certain specifications of denoted entities, as illustrated in the robot example above. However, if Kripke is right, any such specification cannot be metaphysically identified with the designated object, as these are contingent properties. Additionally, there cannot be any a priori knowledge concerning the specifications of the referred objects. Finally, the semantic argument states that such specifications are irrelevant, just as any specification an ordinary person has about Feynman is irrelevant to the reference relation between “Feynman” and Feynman.
Nevertheless, Linnebo contends that his approach does not fundamentally conflict with Kripke’s rejection of descriptivism. He supports the compatibility of his theory with Kripke’s by emphasizing the distinction between semantics and metasemantics. Thus, while Kripke challenges the descriptivist theory of names on a semantic level, Linnebo’s proposition is situated in the domain of metasemantics, differentiating it from semantics (Linnebo, 2018, pp. 40-41).
To clarify this distinction, Linnebo poses three questions that differentiate semantic and metasemantic inquiries:
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To whom does the name " ØL" refer?
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Why did my parents choose my name, and why have other speakers followed suit?
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What is involved in the relation of reference obtaining between this name and that person? (Linnebo, 2018, p. 39)
Linnebo categorizes the first question as extensional, the second as etiological, and the third as constitutive. He posits that metasemantics focuses on “constitutive questions that concern the relation of reference” rather than on the semantic value of expressions (Linnebo, 2018, p. 39). The distinction between semantics and metasemantics, especially in light of Kripke’s rejection of descriptivism, is not immediately evident. To elucidate this, let’s explore Linnebo’s abstraction principle through examples. Linnebo articulates a criterion of identity as follows: f(α) = f(β) (Linnebo, 2018, p. 35), where α and β represent specifications, and the application of f to these specifications yields new objects. Let us now examine Frege’s observations regarding the criteria of identity:
If we are to use the symbol a to signify an object, we must have a criterion for deciding in all cases whether b is the same as a, even if it is not always in our power to apply this criterion. (Frege, 1953, §62).
Let us formalize Frege’s criteria of identity:
∀x Aristotle = f(x) ↔ the author of Metaphysics ~ x, where ~denotes that descriptions correspond to the same portion of the world, and f denotes the function that turns descriptions into objects. Therefore, for Frege, in order for “Aristotle” to refer, the following function must exist: λx. the author of Metaphysics ~ x.
According to Frege, this function must be applied for the name “Aristotle” to refer. However, as previously noted, Kripke disputes this viewpoint. From a metaphysical standpoint, Aristotle’s authorship of Metaphysics is merely contingent. Epistemically, such knowledge cannot be a priori, as it’s only through a posteriori realization that Aristotle is identified as the author of Metaphysics. Semantically, the act of referring to objects does not require precise specifications. In essence, reference goes beyond any descriptions or specifications.
However, it is important to recognize a significant distinction between Frege’s and Linnebo’s approaches to the criteria of identity. In Frege’s framework, the criteria of identity are deemed necessary for a proper name to achieve reference. Conversely, Linnebo posits that identity criteria serve merely as a sufficient condition for referentiality (Linnebo, 2018, p. 40). This implies that one might use “Aristotle” to refer to Aristotle even in the absence of a specific criterion of identity. For Linnebo, having a criterion of identity for Aristotle facilitates reference, but it is not the sole method of doing so; alternative routes for reference are possible. Viewed from this angle, if the existence of a criterion of identity enables the use of the name “Aristotle”, then reference is achieved. However, this assertion might seem vacuously true within a Kripkean framework, primarily because Kripke asserts that there is no explicit criterion of identity required to use the name “Aristotle”. Thus, the conditional’s antecedent is not met.
According to Linnebo, the descriptions that descriptivists view as crucial for meaning do not pertain to the criteria of identity he discusses. For him, the criteria of identity establish the foundation for the reference relationship. Yet, I remain uncertain about how the role of specifications in Linnebo’s criteria of identity diverges from the role of descriptions in descriptivist reference determination. Viewing the semantic/metasemantic distinction from another perspective, Linnebo notes that while Kripke criticizes descriptivism at a semantic level, he simultaneously presents a positive metasemantic perspective.
Linnebo suggests, “The semantic thesis of direct reference is compatible with a variety of metasemantic views on the constitution of reference, including ones according to which this constitution involves some non-trivial structure. Indeed, when Kripke conjectures that an initial “baptism” and subsequent causal chains serve to constitute certain forms of reference, he proposes a metasemantic account with non-trivial structure” (Linnebo, 2018, p. 41).
The distinction between semantics and metasemantics in this context is not straightforward to me. This is because Kripke’s rejection of descriptivism and his preference for a historical chain of reference operate on the same level, not distinct ones. Kripke contrasts descriptivism with his historical theory of names, illustrating that naming is not isolated act but one embedded within a community’s network of interactions.
So, what does make my use of “Cicero” into a name of him? The picture which leads to the cluster-of-descriptions theory is something like this: One is isolated in a room; the entire community of other speakers, everything else, could disappear; and one determines the reference for himself by saying-“By “Gödel” I shall mean the man, whoever he is, who proved the incompleteness of arithmetic”. Now you can do this if you want to…
But that’s not what most of us do. Someone, let’s say, a baby, is born; his parents call him by a certain name. They talk about him to their friends. Other people meet him. Through various sorts of talk the name is spread from link to link as if by a chain” (Kripke, 1980, p. 91).
The fundamental distinction between Kripke’s historical theory and descriptivism lies in the former’s negation of any innate mechanism within language users for establishing reference. According to Kripke, there isn’t an internalized mechanism that facilitates reference-making within a linguistic agent. Instead, the process of reference is situated externally, in the historical lineage of language use. Kripke elucidates this concept by reflecting on an idea considered by Strawson:
The identifying description, though it must not include a reference to the speaker’s own reference to the particular in question, may include a reference to another’s reference to that particular. If a putatively identifying description is of this latter kind, then, indeed, the question, whether it is a genuinely identifying description, turns on the question, whether the reference it refers to is itself a genuinely identifying reference. So one reference may borrow its credentials, as a genuinely identifying reference, from another; and that from another (Strawson, 1959, p. 182).
Kripke articulates a response to Strawson’s integration of the chain of communication theory with descriptivist principles, emphasizing the importance of the actual communicative history over the speaker’s subjective understanding of reference sources. Kripke points out, “the point is that Strawson, trying to fit the chain of communication view into the description theory, relies on what the speaker thinks was the sources of his reference. If the speaker has forgotten his source, the description Strawson uses is unavailable to him; if he misremembers it, Strawson’s paradigm can give the wrong results. On our view, it is not how the speaker thinks he got the reference, but the actual chain of communication, which is relevant” (Kripke, 1980, pp. 92-93). This delineation clarifies that, in Kripke’s view, the historical chain of reference is not an internal construct within the language user but a function of the communal use of proper names. This distinction further highlights a critical difference between Kripke’s and Linnebo’s theories. While Linnebo suggests an intrinsic mechanism within the linguistic agent for referring to objects by name, Kripke explicitly denies such an internal mechanism. Thus, internalizing the historical chain as Strawson suggests diverges from Kripke’s foundational rejection of descriptivism, underscoring that Kripke’s historical chain is an external, communal process rather than an individual cognitive strategy.
The discussion above suggests that if we posit a semantic- metasemantic distinction in comparing Kripke’s work with Linnebo’s, then we observe a transition between semantics and metasemantics in Kripke’s framework. His disavowal of descriptivism extends into what can be termed his metasemantic approach. Conversely, Linnebo’s theory might be seen as adopting a metasemantic form of descriptivism. Yet, Linnebo himself rejects this characterization in a footnote, stating “In fact, my view is not even committed to metasemantic descriptivism, which holds that the referent is fixed by means of a definite description” (Linnebo, 2018, p. 41). He further clarifies that “the grasp of the unity relation need not be an explicit one, which is accessible to consciousness” (Linnebo, 2018, p. 41). Linnebo further illustrates this concept through his “toy model,” demonstrating how a robot refers to physical objects via specific sensory inputs.
What the toy model suggests is that these subpersonal processes must involve a criterion of identity in order for the resulting representation to refer to a physical body. I do not claim that this process is accessible to our consciousness or ever rises above the subpersonal level. On the contrary, the lack of conscious or explicit access to the process is something that I welcome. This view represents a major departure from most earlier philosophical work on criteria of identity (Linnebo, 2018, pp. 29-30).
I question whether the absence of conscious or explicit access to the referencing process truly circumvents Kripke’s critique of descriptivism. Consider a robot designed to use proper names as humans do. Under descriptivism, it would rely on descriptions to pinpoint a proper name’s referent. Linnebo might suggest that his criteria of identity are implicitly encoded within the robot. Yet, do such implicit rules suffice for accurate referent identification? Take, for instance, a robot using “Aristotle” to refer to Aristotle. Any information the robot processes about Aristotle, whether at a personal or subpersonal level, is not deemed necessary nor a priori known, echoing Kripke’s view that reference extends beyond any individual mechanism. Reference, therefore, is more than an individual’s interaction with the world-it exists externally. The term “historical theory of reference” seems to encapsulate this notion.
4. Clarifications and Caveats
Let me conclude this paper with a note to prevent two possible misunderstandings. The reader of this paper may reasonably ask whether I equate descriptivism with semantic internalism. In other words, my understanding of the concept of descriptivism might seem ambiguous. To clarify, I will remain silent on whether descriptivism is a form of semantic internalism. What “semantic internalism” means, and whether it is a semantic or metasemantic position, is beyond the scope of this paper, as I have not used the term “semantic internalism” here. Since semantic internalism can have different connotations, I will address the potential ambiguity by using different terms. The reader might associate my use of descriptivism with two different ideas. The first is psychologism, which suggests that meanings are either psychological states of language users or are at least determined by the psychological states of language users. The second is that descriptions or specifications are entities that a language user can grasp to determine the reference of a term. I use the term “descriptivism” in the second sense. In other words, I do not mean that, for descriptivism, meaning is “in the head.” Rather, language users can use descriptions or specifications to determine the reference of expressions. To be clearer, my understanding of Frege’s approach is exactly as formulated by Hilary Putnam, as follows:
Most traditional philosophers thought of concepts as something mental. Thus the doctrine that the meaning of a term (the meaning “in the sense of intension,” that is) is a concept carried the implication that meanings are mental entities. Frege and more recently Carnap and his followers, however, rebelled against this “psychologism,” as they termed it. Feeling that meanings are public property- that the same meaning can be “grasped” by more than one person and by persons at different times-they identified concepts (and hence “intensions” or meanings) with abstract entities rather than mental entities. However, “grasping” these abstract entities was still an individual psychological act. None of these philosophers doubted that understanding a word (knowing its intension) was just a matter of being in a certain psychological state (somewhat in the way in which knowing how to factor number in one’s head is just a matter of being in a certain very complex psychological state) (Putnam, 1975, p. 218).
So, I do not mean that either Frege’s descriptions, which Kripke critiques, or Linnebo’s specifications are entities in the mind of a speaker. They may be mind-independent entities. However, language users grasp them to refer to objects, as these descriptions or specifications determine the content. For example, when Kripke criticizes Strawson, he does not accuse Strawson of psychologism. In other words, Kripke is not criticizing psychologism. What he criticizes is the idea that certain descriptions or specifications determine the reference, regardless of whether those descriptions or specifications are mind-independent. What he criticizes in Strawson’s approach is that the description “the man Joe thinks proved the incompleteness of arithmetic” would determine the reference of “Gödel” (Kripke, 1980, p. 90).
Even though I do not use the term “semantic internalism” in the paper, I occasionally use the term “internalizing.” By “internalizing,” I do not mean that these specifications or descriptions are ideas in the minds of language users. Instead, I mean what Putnam refers to as “grasping” in the quote above. According to the descriptivist, a language user can grasp these to determine the referent. Even a computer can access them to determine the referent. Therefore, I do not associate descriptivism with fundamentally private and isolated language users. In descriptivism, or Linnebo’s approach, agents can still be part of a community. The descriptions or specifications they use may be accessible to others. However, they have the option to use terms (e.g., “Gödel”) in complete isolation. This is what Kripke rejects when he says, “that’s not what most of us do” (Kripke, 1980, p. 91).
Another misunderstanding might concern how I approach the distinction between semantics and metasemantics. The reader might think that my entire argument boils down to the claim that Kripke may not be aware of the semantic versus metasemantic distinction. In other words, I may appear to suggest that, since Kripke does not clearly differentiate between semantics and metasemantics, I am justified in transferring his statements from the semantic level to the metasemantic level.
Let me clarify my point. First, by being silent on the distinction, I straightforwardly apply Kripke’s arguments against descriptions to Linnebo’s specifications. In doing so, I disregard which labels might better fit the level at which Kripke argues against descriptivism or the level at which Linnebo introduces criteria of identity and specifications. I do not mean to suggest that they necessarily operate at the same level, whether metasemantic or semantic. One may be at the semantic level, while the other is at the metasemantic level. Still, if Kripke’s argument is correct, then there are no recipes or instructions-such as Frege’s descriptions or Linnebo’s specifications-for determining reference. My point is that even if Linnebo’s claim and the descriptivism that Kripke critiques are at different levels, we can still apply Kripke’s argument to Linnebo’s specifications. To counter this argument, Linnebo claims that his approach is not a form of metasemantic descriptivism. Again, I am not concerned with which label might better suit Linnebo’s theory. What interests me is whether Kripke’s arguments against descriptivism apply to Linnebo’s theory. To repeat the point I have already discussed, in order to avoid the label “metasemantic descriptivism,” Linnebo argues that his specifications are not explicit guides or rules; rather, they implicitly determine content. For Linnebo, since specifications are not accessible to consciousness, they cannot be considered descriptions. However, Kripke’s epistemological, semantic, and metaphysical arguments that I mentioned above still apply to these implicitly formulated specifications. Consider two computers: one is given explicit specifications of Aristotle, and the other is given implicit specifications of Aristotle. The first can be exemplified by symbolic computation, while the second can be exemplified by artificial neural networks. It is also worth noting that the specifications in both cases can be independent of the computer, even though the computer can access them. However, if Kripke’s claim is correct, neither the symbolic nor the neural computer can determine the reference; reference is beyond those recipes, descriptions, or instructions built into the computer. None of these entities can be used to determine the reference of “Aristotle” or “Einstein.” The specifications for “Einstein” built into the neural system of an ordinary language user might correspond to a person other than Einstein, even though “Einstein” still refers to Einstein. To sum up, my point is not based on whether Kripke makes a semantic-metasemantic distinction. It is based on the applicability of Kripke’s argument to Linnebo’s specifications in determining the reference of certain expressions.
5. Conclusion
Throughout this paper, we have sought to examine the tension between Kripke's historical theory of names and Linnebo's criteria of identity. Linnebo, echoing Frege's philosophy of mathematics, asserts that the existence of mathematical objects doesn't demand a substantial commitment from reality. This perspective, which makes abstract mathematical objects readily accessible, differs notably from other similar views due to Linnebo's addition of a crucial mechanism for reference: the criteria of identity. Linnebo proposes that the existence of a criterion for identifying the object in question ensures the establishment of reference.
Linnebo acknowledges the apparent clash between his criteria of identity and Kripke's arguments against descriptivism. However, he argues that there is no real conflict as his perspective is rooted in metasemantics, while Kripke's theory is fundamentally semantic.
In our exploration of Kripke's historical theory of names, we put forth the interpretation that, according to Kripke, the mechanism of reference is not confined within the individual. Instead, it transcends individual cognition and encapsulates broader social dynamics. This position contests conventional theories that place the reference mechanism firmly within the realm of individual cognitive processes. We argue that Kripke’s theory presents the reference relation as a historical phenomenon rather than a product of individual mental states or actions, thereby contrasting starkly with Linnebo's criteria of identity.
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