Citation

Formal definition

A Citation is a four-tuple $\mathcal{C} = (C, W, L, \rho)$:

$$\mathcal{C} = (C \in H_t,\; W \in H^*_{\tau_W},\; L \subseteq W,\; \rho \in \{\mathrm{quot}, \mathrm{par}, \mathrm{ref}, \mathrm{see}\})$$

where:

• $C$ is the citing claim — the proposition in the current fiber that the citation is designed to ground; $C \in H_t$ is present in the fiber but may not be in $H^*_t$ (may not yet be meaning-settled); the citation is the act that moves $C$ toward $H^*_t$ by connecting it to the settled record $W$
• $W$ is the cited work — a committed, persistent record: $W \in H^*_{\tau_W}$, meaning $W$ is fully settled at the time of its publication $\tau_W$; $W$ is a Record triple $(K_W, V_W, \tau_W)$ — key (author, title, venue, year), value (content), and history marker (publication date); the requirement $W \in H^*_{\tau_W}$ captures the commitment of scholarly publication: the work is meaning-settled and execution-settled at its publication point
• $L$ is the locus — the specific element within $W$ at which the supporting content resides; $L \subseteq V_W$ (a sub-element of the cited work’s content); $L$ is what narrows the citation from “W supports C” to “this specific part of W supports C”; without $L$, the citation is too vague to be verified
• $\rho$ is the grounding relation type — how $L$ supports $C$:
  • Quotation ($\rho = \mathrm{quot}$): $L$ is reproduced verbatim in $C$; strongest support; $L$ appears as a sub-element of $C$’s content
  • Paraphrase ($\rho = \mathrm{par}$): $C$ reproduces the ideas of $L$ in different words; strong support; the propositional content is preserved
  • Reference ($\rho = \mathrm{ref}$): $W$ establishes the claim $C$ makes; $C$ attributes the claim to $W$ without reproducing content; medium support
  • See-also ($\rho = \mathrm{see}$): $W$ is related but does not directly establish $C$; weak support; associative rather than evidentiary

The grounding claim: $C$ is grounded in $W$ at $L$ via $\rho$. The citation is valid iff $L$ actually supports $C$ via the stated $\rho$ — the cited locus genuinely provides the stated kind of support for the citing claim.

Citation as four speech acts simultaneously

Using Austin’s performative speech act analysis:

1. Assertive: “this claim $C$ is supported by evidence at $W$:$L$” — the citation constitutes an assertion of the grounding relation between $C$ and $W$; it claims that $C$ is epistemically connected to $W$

2. Expressive/Acknowledging: crediting $W$’s author with priority — “this idea/finding originates with the author of $W$, not with me”; the citation performs the act of intellectual acknowledgment; without citation, one appropriates the credited idea without acknowledgment (plagiarism)

3. Locating: placing $C$ within the discourse network that contains $W$ — “this paper is in conversation with $W$ and its intellectual neighbors”; citation creates the network structure of scholarship

4. Transparency-providing: enabling verification — “the reader can check $W$:$L$ to confirm that $C$ is supported”; citation makes knowledge claims intersubjectively accessible; it is the mechanism by which private knowledge becomes public, verifiable knowledge

The fourth function is the epistemically fundamental one: without citation, knowledge claims are opaque. With citation, they are in principle verifiable by anyone with access to $W$.

The citation graph must be a DAG

A valid citation network is a directed acyclic graph (DAG): $C_1 \to W_1, C_2 \to W_2, \ldots$ where $\to$ denotes “is grounded in” and there are no directed cycles. The acyclicity condition formalizes the requirement that grounding must be non-circular:

Temporal acyclicity: $\tau_W < t$ — the cited work must have been committed ($W \in H^*_{\tau_W}$) before the citing claim was made ($C \in H_t$ at $t > \tau_W$). An author cannot cite a work published after their own; the time ordering of the history monoid enforces partial acyclicity.

Logical acyclicity: even among works published at different times, there must be no cycle in the grounding relation. A grounding cycle $C \to W \to C$ (or through intermediaries) is citogenesis.

Citogenesis: cycles in the grounding DAG

Citogenesis (Munroe, xkcd 978, 2011): the pathological cycle in which:

1. An unsourced claim $C$ is introduced into a secondary source $W$ (e.g., added to a reference document)
2. A new paper cites $W$ as the source of $C$
3. $W$ is updated to cite the new paper as the source of $C$
4. Now $C$ appears to have a citation chain, but the chain is a cycle: $C \to W \to C$ (possibly through additional nodes)

Formally: $C \in H_t$ is grounded via $\rho$ in $W$, but $W$ derived $C$ from $C$ (or from claims transitively grounded in $C$). The grounding graph has a directed cycle. The citation satisfies the syntactic form of grounding while providing no independent epistemic support — the cycle traces back to the original unsourced $C$.

This is the analog of circular definition ($X$ defined by $Y$, $Y$ defined by $X$): it satisfies the form of definition while failing to provide content. In fiber terms: citogenesis creates the appearance of a grounding chain toward $H^*$ while actually running in a loop that never exits $H_t \setminus H^*_t$.

The valid citation network is acyclic precisely because valid grounding is non-circular: the chain must eventually terminate in observations, experiments, or derivations that are independently settled (in $H^*$) without reference back to the citing claim.

Citation and the Record structure

The cited work $W \in H^*_{\tau_W}$ is a Record triple $(K_W, V_W, \tau_W)$ where:

• $K_W$ is the key: author(s), title, venue (journal/conference/book), year — enough to uniquely identify the work
• $V_W$ is the value: the content of the work — the text, figures, proofs, data
• $\tau_W$ is the history marker: the commitment event (publication date)

The citation’s locus $L \subseteq V_W$ specifies which part of the record’s value provides the support — page, section, equation, figure. The combination $(K_W, L)$ gives a complete, locatable pointer.

The temporal persistence of records ($W \in H^*_{\tau_W}$ persists forward: $W \in H^*_t$ for all $t > \tau_W$) is what makes citation reliable: the cited work does not change between the time of citing and the time of verification. This is why publication — commitment to a persistent, immutable record — is a prerequisite for valid citation.

Academic vs. legal citation

The structural contrast matters for reasoning in this system:

	Academic Citation	Legal Citation
Register	Epistemic: reasons for belief	Normative: reasons for decision
Force	Persuasive: the reader may evaluate and disagree	Binding (within jurisdiction): the court must follow
Settlement	$C$ moves toward $H^*_t$ (meaning-settled belief)	$\alpha$ moves toward $H^*_t$ (execution-settled action)
Cycle consequence	Citogenesis: false authority, no epistemic support	Circular precedent: legally void (no independent ground)

The academic citation settles meaning: $\sigma_t(C) \to C$ (the saturation nucleus processes the grounding and raises $C$ toward $\mathrm{Fix}(\sigma_t)$). The legal citation settles execution: $\Delta_t(J) \to J$ where $J$ is the judgment; the precedent carries the judgment forward through the transfer nucleus into obligatory future decisions.

Open questions

• Whether the four grounding relation types (quotation, paraphrase, reference, see-also) form a total order by evidentiary strength, and whether the lattice structure of $H_t$ gives a formal strength ordering: $\rho_\mathrm{quot} > \rho_\mathrm{par} > \rho_\mathrm{ref} > \rho_\mathrm{see}$ in terms of how much the citation advances $C$ toward $H^*_t$.
• Whether the citation DAG has a natural fiber structure: each node $C$ has a “citation depth” measuring its distance from claims grounded only in first-person observations or axiomatic commitments, and whether this depth corresponds to the depth filtration of the relational history fiber.
• Whether the valid-grounding condition (locus $L$ actually supports $C$ via $\rho$) can be formalized as a morphism condition in the fiber: a citation is valid iff there exists a morphism $f_\rho: L \to C$ in the fiber’s category that witnesses the grounding type $\rho$.
• Whether peer review — the institutional process by which citation-worthy claims are evaluated before publication — corresponds to the transition from $H_t \setminus H^*_t$ (submitted, under review: present but not settled) to $H^*_t$ (published: settled); and whether the nucleus that governs this transition is the saturation nucleus (meaning closure: “the field has recognized this as established”) or the transfer nucleus (execution closure: “this can be relied upon for future work”).