Vectorizing the Western Tradition for use in Autonomous Systems

Nathan Simpson

Introduction

This primer explores the concept of vectorizing elements of the Western legal tradition—legislation, jurisprudence, and sovereign function—into a unified continuum. By representing these elements as vectors, we can quantitatively assess their relative importance over time, providing a framework for analyzing the validity of government records. This approach allows for a nuanced understanding of the shifting significance of different legal and governmental documents across various levels of governance: national, state, and local.

The Western Tradition Continuum: Conceptual Framework

The Western legal tradition is characterized by the interplay between three primary components:

• Legislation ($\mathbf{L}$): The codification of laws and regulations by legislative bodies.
• Jurisprudence ($\mathbf{J}$): The body of judicial decisions and interpretations that shape and refine the law.
• Sovereign Function ($\mathbf{S}$): The execution of authority by the state, including the enforcement of laws and administration of governance.

These components can be conceptualized as vectors within a multidimensional legal space. The Western tradition continuum $\mathbf{T}(t)$ at any given time $t$ can be expressed as a linear combination of these vectors:

where:

• $\alpha(t)$, $\beta(t)$, and $\gamma(t)$ are time-dependent coefficients representing the relative influence or importance of each component at time $t$.

Explanation of the Formula and Base Components

• Legislation ($\mathbf{L}$): This vector represents the body of formal laws that govern society. The coefficient $\alpha(t)$ reflects how dominant or influential legislation is within the continuum at a specific time. For instance, the U.S. Constitution would be assigned a high and constant $\alpha(t)$ value, such as “100,” due to its foundational and enduring significance.
• Jurisprudence ($\mathbf{J}$): This vector captures the interpretive and adjudicative processes. The coefficient $\beta(t)$ reflects how judicial decisions influence the overall legal landscape at a given time. Jurisprudence is often responsive to changes in society, technology, and culture, so $\beta(t)$ may vary significantly over time.
• Sovereign Function ($\mathbf{S}$): This vector encompasses the exercise of state authority, including administrative actions and law enforcement. The coefficient $\gamma(t)$ reflects the degree to which sovereign power influences the legal system. For localized matters, such as the as-built plan of a sewer line, $\gamma(t)$ might be small and diminish over time as the document’s relevance fades.

Validity as a Function of Impact and Time

In analyzing government records, the validity $V(t)$ of a particular document is a function of its impact $I(t)$ over time. This relationship can be expressed as:

where:

• $y(t)$ represents the impact of the document (plotted on the y-axis).
• $t$ represents time.
• $f$ is the function that models the relationship between impact and time.

For example:

• The U.S. Constitution’s validity would be represented as $V_{\text{Constitution}}(t) = f(100, t)$, indicating a high and constant impact over time.
• A less significant document, like a sewer line plan, would have a validity represented as $V_{\text{Sewer Line}}(t) = f(0.01, t)$, where $f(0.01, t)$ trends towards zero after the line’s removal in 1990.

Expanding the Formula for Different Levels of Governance

To expand the framework for national, state, and local relevance, we can introduce additional indices or modify the coefficients $\alpha(t)$, $\beta(t)$, and $\gamma(t)$ to reflect the varying importance of legislation, jurisprudence, and sovereign function at each level:

• National Relevance:
  $$\mathbf{T}_{\text{national}}(t) = \alpha_{\text{national}}(t) \mathbf{L} + \beta_{\text{national}}(t) \mathbf{J} + \gamma_{\text{national}}(t) \mathbf{S}$$

  Here, $\alpha_{\text{national}}(t)$ might be very high for federal legislation like the Constitution, while $\gamma_{\text{national}}(t)$ reflects the influence of federal executive actions.

• State Relevance:
  $$\mathbf{T}_{\text{state}}(t) = \alpha_{\text{state}}(t) \mathbf{L} + \beta_{\text{state}}(t) \mathbf{J} + \gamma_{\text{state}}(t) \mathbf{S}$$

  For state-level relevance, $\alpha_{\text{state}}(t)$ and $\gamma_{\text{state}}(t)$ would reflect the importance of state laws and actions, while $\beta_{\text{state}}(t)$ would represent the influence of state court decisions.

• Local Relevance:
  $$\mathbf{T}_{\text{local}}(t) = \alpha_{\text{local}}(t) \mathbf{L} + \beta_{\text{local}}(t) \mathbf{J} + \gamma_{\text{local}}(t) \mathbf{S}$$

  At the local level, $\alpha_{\text{local}}(t)$ might reflect the importance of municipal ordinances, while $\gamma_{\text{local}}(t)$ would capture the impact of local government actions and decisions.

Example: Relevance Score of a Sewer Line As-Built Plan

Let’s consider an example of determining the relevance score for a sewer line as-built plan. The sewer line was put into service in 1940 and taken out of service / replaced in 1990.

We can model the relevance score R(t) as a function of time t:

Where:

• R(t) is the relevance score at time t
• k is the decay constant (we’ll use k = 0.1 for this example)
• t is the time in years

This function maintains a constant relevance of 1 while the sewer line is in service, then decays exponentially after it’s replaced in 1990.

The graph shows the relevance score over time. We can observe:

• From 1940 to 1990, the relevance score remains at 1, indicating full relevance while the sewer line is in service.
• After 1990, when the line is replaced, the relevance score begins to decay exponentially.
• By 2020 (30 years after replacement), the relevance score has decreased to approximately 0.05, indicating that the old as-built plan has very little relevance to current operations.
• The relevance continues to approach zero as time progresses, but never quite reaches it, reflecting that historical records may always retain some minimal value.

This example demonstrates how the relevance of a document like an as-built plan can be quantified and modeled over time. The sharp drop in relevance after the sewer line’s replacement reflects the diminishing importance of the original plan for current operations, while the gradual decay accounts for potential historical or archival value.

Example: Relevance Score of Economic Development Authority Members List from 1997

Now, let’s consider another example: the relevance score for a list of economic development authority members from 1997. This type of document typically has a slower decay in relevance compared to technical documents like as-built plans.

We can model the relevance score R(t) as a function of time t:

Where:

• R(t) is the relevance score at time t
• k is the decay constant (we’ll use k = 0.02 for this example, which is slower than the previous one)
• t is the time in years

This function starts with full relevance in 1997 and then decays exponentially, but at a slower rate than the sewer line example.

The graph shows the relevance score over time for the economic development authority members list. We can observe:

• The relevance score starts at 1 in 1997, indicating full relevance when the list was created.
• The relevance decays more slowly compared to the sewer line example, reflecting the longer-term importance of historical governance information.
• By 2020 (23 years after creation), the relevance score has decreased to approximately 0.63, indicating that the document still retains significant historical value.
• The relevance continues to decrease but at a much slower rate, suggesting that this type of document maintains its importance for historical and research purposes for a longer period.

This example illustrates how different types of government records can have varying rates of relevance decay. While technical documents like as-built plans may quickly lose relevance after infrastructure changes, administrative records like lists of authority members retain their importance for much longer, serving as valuable historical data for understanding past governance structures and decision-making processes.

Example: Legislative Supremacy and Town Council Governance

This example demonstrates the hierarchy of legislative supremacy, focusing on what governs a town council’s actions. We’ll examine the relative importance of various legal documents from the federal level down to local bylaws, using the Town of Appomattox, Virginia as a case study.

The following chart illustrates the relevance order of importance regarding what a town council can and cannot do:

Explanation of the hierarchy:

1. U.S. Constitution: The supreme law of the land, it sets the fundamental framework for all levels of government.
2. Federal Code of Federal Regulations (CFR): Federal regulations that implement and interpret federal laws.
3. Commonwealth of Virginia Constitution: The highest law within Virginia, subject to the U.S. Constitution.
4. Code of Virginia: State laws that govern all localities within Virginia.
5. Charter of the Town of Appomattox: The foundational document that establishes the town’s government structure and powers.
6. Town Code of Appomattox: Local laws and ordinances specific to the town.
7. Town Bylaws: Rules adopted by the town council to govern its own proceedings.
8. Robert’s Rules of Order: A widely used manual of parliamentary procedure, often adopted by local governments to guide meeting procedures.

This hierarchy demonstrates that:

• The town council’s actions must comply with all higher levels of law, from federal down to local.
• Federal and state constitutions and laws have the highest relevance and authority over town council actions.
• The town’s charter and code provide the specific framework within which the council operates.
• Bylaws and Robert’s Rules of Order, while important for day-to-day operations, have the least authority and can be more easily changed or overridden by higher laws.

Understanding this hierarchy is crucial for town council members and citizens alike to ensure that local governance adheres to all applicable laws and regulations.

Conclusion

This vectorized and indexed approach to the Western legal tradition provides a robust framework for analyzing the relative importance and validity of government records across different levels of governance. By adjusting the coefficients $\alpha(t)$, $\beta(t)$, and $\gamma(t)$ according to national, state, or local contexts, scholars and practitioners can gain a more precise understanding of how various documents and legal principles interact within the broader legal continuum. This methodology also opens up opportunities for further refinement and application in comparative legal studies, historical legal analysis, and the development of predictive models for legal and governmental evolution.