Logical Analysis of Supreme Court Arguments

The tools of Logic and Contemporary Analytic Philosophy can be used to evaluate the arguments set forth in Supreme Court opinions. Three tools in particular are very useful:
- Argument Reconstruction: the process of restating a real-life argument so its premises, conclusion, and reasoning are clear.
- Proof: proving that the conclusion of an argument follows from its premises by deriving the former from the latter using the rules of deductive logic.
- Refutation by Logical Analogy: showing that the conclusion of an argument doesn’t follow from its premises by setting forth an argument of the same logical form whose premises are true and conclusion false.

Cases

Bostock v Clayton County 2020 (Discrimination)

Title VII of the Civil Rights Act of 1964 makes it unlawful for an employer to discriminate against a person because of the person’s race, color, religion, sex, or national origin. The Act does not explicitly say that it is unlawful to discriminate because a person is gay, homosexual, lesbian, or transgender.
Clayton County (Georgia) fired Gerald Bostock because he was gay. Bostock filed a lawsuit alleging that the county discriminated against him for being gay, in violation of Title VII. An Appeals Court dismissed the case because, it said, Title VII does not prohibit such discrimination. The Supreme Court agreed to hear the case.
Writing for the Court, Justice Gorsuch ruled that an employer who discriminates against an employee because they are homosexual (or transgender) discriminates against the employee in part because of sex. Thus, Clayton County fired Bostock in violation of Title VII.
Justices Alito and Kavanaugh filed separate dissents.

Main Thesis

Gorsuch’s Main Thesis:
- An employer who discriminates against an employee because they are gay discriminates against the employee in part because of sex. Likewise, an employer who discriminates against an employee because they are transgender discriminates against the employee in part because of sex.
As Gorsuch says:
- “it is impossible to discriminate against a person for being homosexual or transgender without discriminating against that individual based on sex”
- “When an employer fires an employee for being homosexual or transgender, it necessarily intentionally discriminates against that individual in part because of sex.”

Qualified Thesis

Gorsuch’s sets forth a valid argument establishing what I call his Qualified Thesis:
- An employer who discriminates against an employee because he is a man and is attracted to men discriminates against the employee in part because of sex. Likewise, an employer who discriminates against an employee because they were biologically male at birth and now identify as female discriminates against the employee in part because of sex. (Same for a woman attracted to women and a person biologically female at birth who now identifies as male.)
- View Proof of Qualified Thesis.

The Problem with Gorsuch’s Argument for the Main Thesis

Gorsuch sets forth a cogent argument for the Qualified Thesis. And he seems to think that if the Qualified Thesis is true so is the Main Thesis. But the inference is questionable, at least.
- Suppose an employer discriminates against everyone who’s gay, whether male or female. Assume he fires Jones, a man, because’s he’s gay. It doesn’t follow from the fact that (a) Jones is male and was fired because he’s gay that (b) Jones was fired because he was male and attracted to men, i.e. because he was a gay man. He was fired because he was attracted to people of the same sex, not because he was a male attracted to people of the same sex. Indeed, Jones would still have been fired had he not been male. It’s thus false that Jones would not have been fired but for being male. Being male is not a but-for cause (a necessary condition) of his being fired.
Justice Alito makes the point this way:
- “Contrary to the Court’s contention, discrimination because of sexual orientation or gender identity does not in and of itself entail discrimination because of sex. We can see this because it is quite possible for an employer to discriminate on those grounds without taking the sex of an individual applicant or employee into account. An employer can have a policy that says: “We do not hire gays, lesbians, or transgender individuals.” And an employer can implement this policy without paying any attention to or even knowing the biological sex of gay, lesbian, and transgender applicants.”

Gorsuch’s Curious Reply

Gorsuch considers a hypothetical case of discrimination against people who are gay or transgender.
- “Suppose an employer asked homosexual or transgender applicants to tick a box on its application form. The employer then had someone else redact any information that could be used to discern sex. The resulting applications would disclose which individuals are homosexual or transgender without revealing whether they also happen to be men or women.”
Gorsuch argues that the employer discriminates against gays in part because of their sex, despite not knowing the sex of any particular applicant.
- “By discriminating against homosexuals, the employer intentionally penalizes men for being attracted to men and women for being attracted to women. By discriminating against transgender persons, the employer unavoidably discriminates against persons with one sex identified at birth and another today. Any way you slice it, the employer intentionally refuses to hire applicants in part because of the affected individuals’ sex, even if it never learns any applicant’s sex.”
Thus, for Gorsuch, discriminating against a person because they are gay or transgender simpliciter implies discriminating against them in part because of their sex. But it’s not clear what Gorsuch’s argument is. Here’s my best guess at reconstruction:
1. S discriminates against P because P is gay.
2. Therefore, S discriminates against P because P is attracted to people of the same sex.
  - From 1 per the definition of “gay.”
3. Therefore, S discriminates against P because P is a male and P is attracted to men or P is female and P is attracted to women.
  - From 2 because “x is attracted to people of the opposite sex” is logically equivalent to “x is male and attracted to men or x is female and attracted to women.”
4. Therefore, S discriminates against P either because P is a male and P is attracted to men or because P is female and P is attracted to women.
  - This does not follow from 3.
5. Therefore S discriminates against P in part because of P’s sex.
  - Follows from 4 and the Qualified Thesis per Constructive Dilemma:
    - S discriminates against P because P is a male and P is attracted to men or S discriminates against P because P is female and P is attracted to women.
    - If the former then, per the Qualified Thesis, S discriminates against P in part because of P’s sex.
    - If the latter then, per the Qualified Thesis, S discriminates against P in part because of P’s sex.
    - Therefore S discriminates against P in part because of P’s sex.
Line 4 doesn’t follow from line 3 because it doesn’t follow from
- S discriminates against P because P is gay (or, equivalently, because P is a male and P is attracted to men or P is female and P is attracted to women).
that
- S discriminates against P because P is a male and P is attracted to men (i.e. P is a gay man)
Nor does it follow that
- S discriminates against P because P is a female and P is attracted to women (i.e. P is a gay woman).

Conclusion

Gorsuch’s argument for his Main Thesis, that it follows from the Qualified Thesis, fails because an employer can discriminate against a person because they are gay without discriminating against them because they are a gay man (or a gay women). Gorsuch does, however, show that an employer who discriminates against a person because he is gay man (and not simply because he is gay) discriminates against the person in part because of the person’s sex. And likewise for discrimination against a person because she is gay woman.

Proof of Qualified Thesis

Employer S discriminated against P because P is male and P is attracted to men.
For any x, if S discriminates against x because x is F and x is G, then S discriminates against x in part because x is F.
Therefore S discriminated against P in part because P is male.
- From 1 and 2
P’s sex is male.
Therefore S discriminated against P in part because of P’s sex (violating Title VII).
- From 3 and 4

Kavanaugh’s Dissent

Kavanaugh’s Argument, Reconstructed
- Only the ordinary meaning of “discriminate because of sex” is relevant in determining whether a person is discriminated against because of sex. The literal meaning is irrelevant.
- Gorsuch’s argument uses the literal meaning of “discriminate because of sex” rather than its ordinary meaning.
- Therefore’s Gorsuch’s argument is irrelevant in determining whether a person is discriminated against because of sex.
Assessment
- There’s a recognized distinction between the literal use of a phrase and its figurative use. Idioms are good examples: beating around the bush, spilling the beans, getting cold feet. In such cases we can clearly state the literal and figurative meanings:
  - He has cold feet.
    - He’s apprehensive about a planned course of action
    - His feet are cold
  - He spilled the beans
    - He leaked a secret
    - He dropped edible seeds.
- According to Kavanaugh, there are literal and ordinary meanings of “discriminate because of sex.” But he never says what they are.

Trump v Barbara, 2026 (Birthright Citizenship)

In January 2025 President Trump issued an executive order directing federal agencies not to recognize anyone as a US citizen (a) whose mother at the time of their birth was an illegal alien or a lawful but temporary visitor and (b) whose father at the time of their birth was neither a US citizen nor a lawful permanent resident.
The question before the court was whether the executive order was consistent with the Citizenship Clause of the 14th Amendment:
- “All persons born or naturalized in the United States, and subject to the jurisdiction thereof, are citizens of the United States and of the State wherein they reside.”
Writing for the Court Chief Justice Roberts ruled that:
- “Children born in the United States to parents unlawfully or temporarily present are ‘subject to the jurisdiction’ of the United States and are citizens at birth under the Fourteenth Amendment’s Citizenship Clause.”
Justices Thomas and Alito filed dissents.

Roberts’ Argument

Anyone born or naturalized in the United States, and is subject to the jurisdiction thereof, is a citizen of the United States per the Citizenship Clause of the Fourteenth Amendment.
Anyone within the territory of the US is subject to its jurisdiction except foreign ministers and members of Indian tribes with their own territorial jurisdictions.
- The phrase “subject to the jurisdiction” refers to the power of the United States to govern those within its territory.
  - An American Dictionary of the English Language, Noah Webster (1828)
  - Dictionary of the English Language, Joseph E. Worcester (1820)
No one unlawfully or temporarily present in the United States is a foreign minister or a member of an Indian tribe with its own territorial jurisdiction.
Therefore any child born in the United States whose parents are unlawfully or temporarily present in the country are citizens of the United States per the Citizenship Clause of the Fourteenth Amendment.

law.cornell.edu/wex/jurisdiction
- Jurisdiction can be defined as:
  - 1. Power of a court to adjudicate cases and issue orders; or
  - 2. Territory within which a court or government agency may properly exercise its power.

Thomas’ Argument

Anyone born or naturalized in the United States, and subject to the jurisdiction thereof, is a citizen of the United States per the Citizenship Clause of the Fourteenth Amendment.
A person is subject to the jurisdiction of the United States if only if they are domiciled in the United States.
- “A person is subject to the jurisdiction of the government of his domicile.”
Therefore anyone born in the United States and domiciled in the U.S. at the time is a citizen of the United States per the Citizenship Clause of the Fourteenth Amendment.

law.cornell.edu/wex/domicile
- Domicile is the place of someone’s true, permanent home, which they have the intention of returning to, when absent. The function of a domicile is to correlate a person with a particular place.
- Domicile is determined by two concurrent elements: (1) the person’s presence in the state of domicile, and (2) the intention to stay there indefinitely.

Application to children of temporary visitors
- A foreign temporary visitor to the United States is not domiciled in the United States.
- A child assumes the domicile of their parents.
- Therefore a child born in the United States to a foreign temporary visitor is not a citizen of the United States per the Citizenship Clause of the Fourteenth Amendment.

Are illegal immigrants domiciled in the United States?
- Justice Thomas argued that the Citizenship Clause confers citizenship on children born in the U.S. only if the parents are domiciled in the United States. So, are illegal immigrants domiciled in the United States?
- Justice Thomas:
  - Trump’s executive order “is at least facially consistent with the Citizenship Clause in its exclusion of the children of illegal aliens because at least some such persons are not domiciled here, and therefore also are not citizens.”
- Justice Gorsuch:
  - “Still, I wonder: Is a child born here to parents who have long chosen to make this Nation their permanent home not a citizen under the Fourteenth Amendment solely because his parents’ presence violates statutory law? If those parents are not domiciled here, then where are they domiciled?”

Alito’s Argument

Anyone born or naturalized in the United States, and is subject to the jurisdiction thereof, is a citizen of the United States per the Citizenship Clause of the Fourteenth Amendment
A person is subject to the jurisdiction of the United States if and only if they are not subject to any foreign power.
- See the argument below.
Therefore anyone born in the United States who is not subject to any foreign power at the time is a citizen of the United States per the Citizenship Clause of the Fourteenth Amendment.

Example of Application
- A child is born in the United States to illegal immigrants from Guatemala.
- The constitution of Guatemala confers Guatemalan nationality on a person born abroad to a Guatemalan parent.
- Therefore the child’s nationality at birth is Guatemalan.
- Therefore the child is subject to a foreign power at birth.
- Therefore, the child is not a citizen of the United States at birth per the Citizenship Clause.

Argument that “subject to the jurisdiction of the United States” means “not subject to a foreign power.”
Thesis:
- The citizenship clause of the Fourteenth Amendment and the citizenship provision of the 1866 Civil Rights Act are “substantively identical:”
  - 14th Amendment
    - “All persons born or naturalized in the United States, and subject to the jurisdiction thereof, are citizens of the United States and of the State wherein they reside.
  - CRA
    - “All persons born in the United States and not subject to any foreign power, excluding Indians not taxed, are hereby declared to be citizens of the United States.”
First piece of evidence:
- “During a 6-month period, the same Congress debated two provisions that addressed the very same question and adopted them both. Those circumstances in themselves make it highly unlikely that the two provisions differed in substance.”
Second piece of evidence:
- “One of the chief reasons for the adoption of the Fourteenth Amendment was to prevent the CRA from being held unconstitutional.” So the provisions were designed so that anyone granted citizenship under the CRA was also granted citizenship under the 14th Amendment.
Third piece of evidence:
- “Following the ratification of the Fourteenth Amendment, Congress reenacted the CRA in 1870. That reenactment would have been pointless if the CRA’s test for birthright citizenship were less generous than the Fourteenth Amendment’s.” So the provisions were designed so that anyone granted citizenship under the 14th Amendment was also granted citizenship under the CRA.

A Simple Refutation?

Suppose a foreign national generates counterfeit $100 dollar bills, enters the U.S. as a tourist, and uses the fake bills to purchase thousands of dollars worth of goods. Will he be prosecuted if he’s caught? Of course. That’s because he’s subject to the jurisdiction of the United States. But he’s not domiciled in the United States. So, the second premise of Thomas’ argument is false:
- A person is subject to the jurisdiction of the United States if only if they are domiciled in the United States.
Moreover, he’s subject to a foreign power. So the second premise of Alito’s argument is also false.
- A person is subject to the jurisdiction of the United States if and only if they are not subject to any foreign power.

A Subtle Distinction?

Section 1 of the Fourteenth Amendment reads:
- “All persons born or naturalized in the United States, and subject to the jurisdiction thereof, are citizens of the United States and of the State wherein they reside. No State shall make or enforce any law which shall abridge the privileges or immunities of citizens of the United States; nor shall any State deprive any person of life, liberty, or property, without due process of law; nor deny to any person within its jurisdiction the equal protection of the laws.”
Why subject to the jurisdiction of the United States but within the jurisdiction of a state? One explanation is that “subject to a jurisdiction” suggests more than being “within a jurisdiction,” e.g. being domiciled in the jurisdiction or not being subject to a foreign power. But another explanation is simply that the writer wanted to use one word for the states and a different word for the nation as a whole.

Gamble v US, 1996 (Double Jeopardy)

Terance Gamble, a felon, was found in possession of a handgun. He was convicted under Alabama’s possession-of-firearm-by-a felon statute and given a one-year sentence. He was then prosecuted under the federal possession-of-firearm-by-a felon statute and received a 46-month sentence on top of the one-year sentence.
The Supreme Court ruled that Gamble’s two convictions did not violate the Double Jeopardy Clause of the Fifth Amendment, that no one shall be “twice put in jeopardy of life or limb” for the same offense.

Alito’s Argument

Writing for the Court, Justice Alito argued as follows:
- Offenses that violate laws of different sovereigns are different offenses.
- Alabama’s law against a felon possessing a firearm and the federal law against a felon possessing a firearm are laws of different sovereigns, Alabama and the United States.
- Therefore, Gamble’s two convictions were for different offenses.
- So, Gamble’s two convictions did not violate the Double Jeopardy Clause.
The first premise of the argument is the Dual-Sovereignty Principle:
- Offenses that violate laws of different sovereigns are different offenses.

Alito claimed that the Dual-Sovereignty Principle follows from the Double Jeopardy Clause and therefore from the Constitution. Thus he says:
- “Although the dual-sovereignty rule is often dubbed an “exception” to the double jeopardy right, it is not an exception at all. On the contrary, it follows from the text that defines that right in the first place.”
- “We see no reason to abandon the sovereign-specific reading of the phrase “same offence,” from which the dual-sovereignty rule immediately follows.”

The core of Alito’s opinion is his argument for the Dual-Sovereignty Principle:
- “As originally understood, an ‘offence’ is defined by a law, and each law is defined by a sovereign. Thus, where there are two sovereigns, there are two laws and two ‘offences.’”
That is:
- An offense is a violation of a law.
- A law is the law of a sovereign.
- Therefore, if laws are the laws of different sovereigns, then the laws are different laws and their offenses are different offenses.

Alito’s argument can be reconstructed as the following deductive argument:
1. Laws of different sovereigns are different laws.
2. Offenses that violate different laws are different offenses.
3. Therefore offenses that violate the laws of different sovereigns are different offenses.
The argument is valid, that is, the conclusion (line 3) follows from the premises (lines 1 and 2) per the rules of deductive logic, in particular, what’s known as Predicate Logic with Identity.
- See the proof below.

Being valid, Alito’s argument establishes the Principle of Dual Sovereignty only if both premises are true. But the second premise succumbs to the following counterexample:
- A person is guilty of Grand Auto Theft if they take possession of a vehicle without the owner’s consent and with the intent of keeping it permanently.
- A person is guilty of Joy Riding if they take possession of a vehicle without the owner’s consent.
- Jones steals a Ferrari with the intention of keeping it and therefore violates both statutes.
- According to Blockburger v United States:
  - An act that violates two statutory provisions constitutes two offenses if and only each provision requires a proof of fact that the other does not require.
- It’s not the case that Joy Riding requires a proof of fact that Grand Theft Auto does not require.
- So, Jones’ theft of the Ferrari is a single offense.
- It’s therefore false that offenses that violate different laws are different offenses.
Thus Alito’s argument fails to establish the Principle of Dual Sovereignty. Alito therefore fails to show that the Principle of Dual Sovereignty follows from the Double Jeopardy Clause of the Fifth Amendment and from the Constitution.

Thomas’ Argument

In his concurrence Justice Thomas sets forth a different argument that state and federal prosecutions for the same offense don’t violate the Double Jeopardy clause:
- The people who ratified the Fifth Amendment did not understand it to prohibit prosecution by a State and the Federal Government for the same offense.
- Therefore the Fifth Amendment doesn’t prohibit prosecution by a State and the Federal Government for the same offense.
But the form of argument is invalid. Here’s what’s called a Refutation by Logical Analogy:
- Suppose the Constitution included a Contagious Provision that said:
  - No person with a contagious disease shall enter the United States.
- Assume that the ratifiers mistakenly believed that tuberculosis was not contagious.
- The following is a logical analog of Thomas’ argument:
  - The ratifiers did not understand the Contagious Provision to prohibit people with tuberculosis from entering the United States.
  - Therefore the Contagious Provision does not prohibit people with tuberculosis from entering the United States.
The problem is that a law has a life and logic of its own. A law that says anyone who satisfies condition C shall be X applies to anyone who satisfies C, no matter what people believed when the law was passed.

Proof that Alito’s Core Argument is Valid

Formalized Argument

Let Lμν mean that μ is a law of sovereign ν.
Let Oμν mean that μ violates ν (i.e. μ is an offense against statute ν)
Alito’s argument can be formalized as follows:
- Premise 1: Laws of different sovereigns are different laws.
  - ∀x∀u∀y∀w(Lxu & Lyw & u ≠ w → x≠y)
    - That is, for any x, u, y and w, if x is a law of sovereign u and y is a law of sovereign w and u and w are distinct, then x and y are distinct)
- Premise 2: Offenses that violate different laws are different offenses.
  - ∀p∀x∀q∀y(Opx & ∃u(Lxu) & Oqy & ∃w(Lyw) & x≠y → p≠q)
    - That is, for any p, x, q and y, if p violates x and x is a law of some sovereign and q violates y and y is a law of some sovereign and x and y are distinct, then p and q are distinct.
- Conclusion: Offenses that violate the laws of different sovereigns are different offenses.
  - ∀p∀x∀u∀q∀y∀w(Opx & Lxu & Oqy & Lyw & u≠w → p≠q)
    - That is, for any p, x, u, q, y and w, if p violates x and x is a law of sovereign u and q violates y and y is a law of sovereign w and u and w are distinct, then p and q are distinct

Derivation of Conclusion from the Premises

∀x∀u∀y∀w(Lxu & Lyw & u ≠ w → x≠y)
- Assumption of Premise 1
∀p∀x∀q∀y(Opx & ∃u(Lxu) & Oqy & ∃w(Lyw) & x≠y → p≠q)
- Assumption of Premise 2
Opx & Lxu & Oqy & Lyw & u≠w
- Assumption for Conditional Proof
Lxu & Lyw & u ≠ w → x≠y
- Line 1 Universal Specification
Opx & Lxu & Oqy & Lyw & x≠y
- Lines 3 and 4 Tautological Inference
Opx & ∃u(Lxu) & Oqy & ∃w(Lyw) & x≠y
- Line 5 Existential Generalization
Opx & ∃u(Lxu) & Oqy & ∃w(Lyw) & x≠y → p≠q
- Line 2 Universal Specification
p≠q
- Lines 6 and 7 Modus Ponens
Opx & Lxu & Oqy & Lyw & u≠w → p≠q
- Lines 3 and 8 Conditional Proof
∀p∀x∀u∀q∀y∀w(Opx & Lxu & Oqy & Lyw & u≠w → p≠q)
- Line 9 Universal Generalization

Reynolds v. Sims, 1964 (One person, one vote)

A state voting district in Alabama, Jefferson County, had 41 times the number of eligible voters as another district in the state. David Sims of Jefferson County filed a lawsuit alleging that the large discrepancy in voters diluted his vote, violating the Equal Protection Clause of the Fourteenth Amendment.
The Court ruled that the Equal Protection Clause requires that state legislative districts have roughly equal populations.

Chief Justice Warren’s Argument, Reconstructed

The Equal Protection Clause of the Fourteenth Amendment requires the uniform treatment of persons standing in the same relation to a government action.
All voters stand in the same relation to the allocation of legislative representation.
Therefore, from 1 and 2, the Equal Protection Clause requires the uniform treatment of all voters in the allocation of legislative representation.
The uniform treatment of all voters in the allocation of legislative representation requires that everyone’s vote has the same weight.
Everyone’s vote has the same weight only if legislative districts have equal populations.
- See Warren’s Mathematical Demonstration below.
Therefore, from 4 and 5, the uniform treatment of all voters in the allocation of legislative representation requires that legislative districts have equal populations.
Therefore, from 3 and 6, the Equal Protection Clause of the Fourteenth Amendment requires that legislative districts have equal populations.

Warren’s “Mathematical Demonstration”

The weight of a person’s vote in a district of n people is 1/n.
Therefore, if one district has a smaller population than another, the weight of a vote in the first is greater than the weight of a vote in the second.
- That’s because: if n < m, then 1/n > 1/m.
Thus, as Warren puts it, more people in the larger district “must vote before the effect of their voting is equivalent to that of their favored neighbor.”

Harlan’s Dissent

Harlan’s Argument:
- People who ratified the Fourteenth Amendment did not believe that the Equal Protection Clause limited the power of the states to apportion their legislatures as they saw fit.
- Therefore, the Equal Protection Clause does not limit the power of the states to apportion their legislatures as they saw fit.

The form Harlan’s argument is invalid. Here’s a Refutation by Logical Analogy:
- Suppose the Constitution included a Contagious Provision that said:
  - No person with a contagious disease shall enter the United States.
- Assume that the ratifiers mistakenly believed that tuberculosis was not contagious.
- The following is a logical analog of Harlan’s argument:
  - The ratifiers did not believe that the Contagious Provision prohibited people with tuberculosis from entering the United States.
  - Therefore the Contagious Provision does not prohibit people with tuberculosis from entering the United States.

Wesberry v Sanders 1964 (One person, one vote)

In Wesberry v Sanders, the Court ruled that districts for the House of Representatives must have equal populations (as much as feasible).

Justice Black’s Argument, Reconstructed

The Apportionment Clause of the Fourteenth Amendment implies that the weight of a vote for a member of the House is the same for every state.
Votes have the same weight if and only if their districts have equal populations.
Therefore, the Apportionment Clause implies that districts have equal populations.

Argument for Premise 1

“Representatives shall be apportioned among the several States according to their respective numbers.”
- the Apportionment Clause
That is, a state’s number of representatives should be proportional to its population.
Therefore, there is a number c, the coefficient of proportionality, such that the number of a state’s representatives should be c times its population.
- Follows from 2 per the definition of proportionality.
The coefficient of proportionality is the same for every state.
- Follows from 3 since there is only one coefficient of proportionality.
The coefficient of proportionality is the weight of a vote for a member of the House of Representatives.
- See Example below.
Therefore the weight of a vote for a member of the House of Representatives is the same for every state.
- Follows from 4 and 5.

Example

Suppose six representatives are apportioned among three states in proportion to their populations: 3000, 2000, and 1000.
The coefficient of proportionality = 1/1000.
So the states are allocated:
- 3 representatives = 1/1000 x 3000.
- 2 representatives = 1/1000 x 2000.
- 1 representative = 1/1000 x 1000.
The coefficient of proportionality is the weight of a vote.
- The weight of a vote in a district with n members = 1/n. So the weight of a vote in a district with 1000 members = 1/1000.

Argument for Premise 2

Let the number of people in district A = n and the number of people in district B = m.
It’s a mathematical fact that 1/n = 1/m if and only if n = m.
The weight of a vote in a district with k members = 1/k.
Therefore, votes in districts A and B have the same weight (1/n and 1/m) if and only if A and B have equal populations (n and m).
That is, votes have the same weight if and only if their districts have equal populations.

Actual Apportionment

The actual allocation of representatives to states is approximately proportional.
Suppose you divide the US population by 435 to get the average district population. Then divide each state’s population by that average and round up or down to the nearest integer. You’ll get the correct number of representatives for 46 of the states, with these exceptions:
- Montana, which has 2 representatives, is calculated to have 1;
- New York, which has 26 representatives, is calculated to have 27;
- Ohio, which has 15 representatives, is calculated to have 16;
- Rhode Island, which has 2 representatives, is calculated to have 1.
Two reasons for the exceptions:
- Every state gets at least one representative, no matter its population.
- The algorithm does not allocate representatives by dividing state populations by the average district population. Rather, it uses a different calculation, awarding representatives one a time in a series of contest rounds among all 50 states.

Miranda v. Arizona, 1966 (Right Against Self-Incrimination)

Ernesto Miranda was arrested and brought to the police station where he was interrogated by police officers regarding a kidnapping and rape. Miranda was not advised that he had a right to an attorney. After two hours, Miranda signed a written confession. At his trial, the written confession was admitted into evidence over the objection of defense counsel. Miranda was found guilty of kidnapping and rape.
The Supreme Court ruled that his confession was inadmissible because “Miranda was not in any way apprised of his right to consult with an attorney and to have one present during the interrogation, nor was his right not to be compelled to incriminate himself effectively protected in any other manner.”

Justice Warren’s Argument by Analogy

There is no difference, with respect to having a right against self-incrimination, between
- being compelled to testify against oneself in court
- being psychologically manipulated to incriminate oneself in a custodial interrogation.
  - A custodial interrogation is the interrogation of a person by law enforcement after they have been taken into custody or otherwise deprived of their freedom in any significant way.
The Constitution prohibits a person from being compelled to testify against themselves in court.
- No person “shall be compelled in any criminal case to be a witness against himself.” (Fifth Amendment)
Therefore, the Constitution prohibits a person from being psychologically manipulated to incriminate themselves in a custodial interrogation.
Corollary
- The prosecution may not use a defendant’s confession that resulted from psychological manipulation in a custodial interrogation.

The Practical Problem

How should a court determine the admissibility of a defendant’s confession made during a custodial interrogation?
1. The court makes a decision based on an examination of a transcript or video of the interrogation.
2. The court determines whether the interrogators used standardized procedural safeguards (the Miranda warning) that protect the defendant’s right against self-incrimination.
The second is the simpler solution.
Therefore “the prosecution may not use statements, whether exculpatory or inculpatory, stemming from custodial interrogation of the defendant unless it demonstrates the use of procedural safeguards effective to secure the privilege against self-incrimination.”

Marbury v. Madison, 1803 (Judicial Review)

In Marbury v. Madison Justice John Marshall:
1. Inferred that a particular act of Congress was unconstitutional (“repugnant to the Constitution”);
2. Argued that the Courts have the power to declare acts and laws null and void;
3. Declared the particular act of Congress null and void.

Reconstruction of Marshall’s Reasoning

Section 13 of the Judiciary Act of 1789 grants the Supreme Court a certain power (the power to issue writs of mandamus).
- See Writ of Mandamus below.
Article III, Section 2 of the Constitution logically implies that the Supreme Court does not have that power.
- See Argument for Premise 2 below.
Therefore either (i) the Constitution takes precedence and Section 13 of the Judiciary Act is unconstitutional or (ii) the Constitution can be changed by an act of Congress.
The nature of a written constitution is that it cannot be changed by the legislature it defines and limits.
- See the Significance of a Written Constitution below.
Therefore Section 13 of the Judiciary Act is unconstitutional.
The Courts have the power to declare laws null and void.
- That’s because:
  - Unconstitutional laws must be declared null and void.
  - The Courts have the power to decide conflicts between laws.
Having that power, the Supreme Court therefore declares Section 13 of the Judiciary Act null and void.

Writ of Mandamus

A writ of mandamus is a (usually extraordinary) court order instructing a government official, agency, or lower court to perform an action it’s (already) required to perform.
In the last days of his presidency, John Adams appointed William Marbury to a circuit judgeship. When Thomas Jefferson became president, he ordered his secretary of state, James Madison, not to do the necessary paperwork. Marbury then petitioned the Supreme Court to issue a writ of mandamus, ordering Madison to submit the paperwork.

Argument for Premise 2

In all Cases affecting Ambassadors, other public Ministers and Consuls, and those in which a State shall be Party, the supreme Court shall have original Jurisdiction.”
- Article III, Section 2
“In all the other Cases before mentioned, the supreme Court shall have appellate Jurisdiction, both as to Law and Fact, with such Exceptions, and under such Regulations as the Congress shall make.”
- Article III, Section 2
Therefore the Supreme Court has only appellate jurisdiction to issue writs of mandamus.
A writ of mandamus can only be issued by a court of original jurisdiction.
- Additional premise supplied by Marshall.
Therefore, the Supreme Court does not have the power to issue writs of mandamus.

Significance of a Written Constitution

In Paragraph 146 Marshall says that the fact that the Constitution is a written constitution is a “sufficient” reason for rejecting the doctrine that the Constitution does not take precedence over laws passed by Congress. He then adds:
- “But the peculiar expressions of the constitution of the United States furnish additional arguments in favour of its rejection.”
He then goes on to set forth arguments rejecting the view based on the text of the constitution.
There’s a good reason why Marshall’s main argument is that the Constitution is a written constitution and why he doesn’t simply appeal to the Constitutional text. If his arguments were based only on the text of the Constitution, they would beg the question. He would be arguing, in effect, that the Constitution takes precedence over laws passed by Congress because the Constitution says it takes precedence. Which is like arguing that a document is true because the document says it’s true. Since it’s external to the Constitution, Marshall’s main argument avoids begging the question.