Taxonomy of unions, intersections, and complements

Answers are below the fold:

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True or false?

Or just clickbait?

Half of the pandemic’s unemployment money may have been stolen.” That is the title of this recent report by Felix Salmon for Axios, but the only evidence in support of this increduble claim is a statement by Blake Hall, the CEO of, who says that “America has lost more than $400 billion to fraudulent claims” and that “as much as 50% of all unemployment monies might have been stolen.” I would not be surprised if unemployment fraud were rampant, but this is pretty weak evidence.

shutterstock 1470571964
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Facebook just censored me

You cannot make this shit up! I found the following message waiting for me in surprise (see screenshot below) when I recently re-activated my Facebook page in order to get into my Expedia account:

So, Facebook’s algorithms are censoring legal history papers now? My paper “Domestic Constitutional Violence,” available here via SSRN and which I presented in 2018 at a special symposium at the University of Arkansas at Little Rock, surveys a series of laws enacted by Congress, beginning with the Militia Acts of the mid 1790s, authorizing the president to use military force inside the United States. Suffice it to say I not only closed my Facebook account for good; I went ahead and deleted it!

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Tyler Cowen commits the genetic fallacy

One week ago, Tyler Cowen, the prolific co-author of the popular “Marginal Revolution” blog, published this odd post on “The IRS tax data leak.” Instead of engaging the merits of these damning allegations — the fact that many of the most wealthiest Americans are self-righteous hypocrites who game the tax system in order to avoid paying their fair share of federal taxes — Cowen decided to dwell on the “ethics” of the data leak. Among other things, Cowen concluded that “ProPublica acted unethically, and in fact nothing fundamentally new or interesting or surprising was learned from their act as accessory.”

Let us assume for the sake of argument that Cowen is right, that ProPublica somehow acted unethically by publishing private tax data and that nothing new was learned from these sordid revelations. (Put aside the fact that Cowen, like most commentators calling for greater ethics, never bothers to share with his readers what theory of ethics was in play here. Virtue ethics? Duty Ethics? Consequentialism?) Even so, Cowen is committing a version of the genetic fallacy. Instead of questioning the ethics of our perverse tax system itself, Cowen chooses to focus on the illegal provenance of ProPublica’s data. Sorry, but no dice …

On the contrary, a strong case can be made that ProPublica’s exposé, by revealing how little U.S. billionaires pay in taxes and by exposing their rank hypocrisy, was highly ethical–under whatever theory of ethics you prefer. After all, what ProPublica did was to reveal the truth, and how can transparency and truth ever be inconsistent with ethics? Furthermore, although I already knew that companies like Amazon pay little in taxes, I did not know, contra Cowen, that Jeff Bezos–the richest man in the world–pays little in taxes. Also, thanks to the N.Y. Times, I knew Donald Trump paid only $750.00 in federal income taxes in 2016; I did not know that Elon Musk paid $0 in 2018! So, don’t lecture me about “ethics”!

Understanding Genetic Fallacy With Examples
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On this day (June 14) in 1959, the Disneyland Monorail System, the first daily operating monorail system in the Western Hemisphere, opened to the public in Anaheim, California. Via Wikipedia, here are some facts and figures about Disneyland’s original monorail system:

  • Grand opening: June 14, 1959
  • Designer: WED Enterprises
  • Three trains: Mark VII Red, Mark VII Blue, and Mark VII Orange
  • Max train cars on beam: 3
  • Track length: 2.5 miles (4 kilometers)
  • Ride duration: 10 minutes, 30 seconds
  • Coupon required (originally): “E”
  • Ride system: Each train was propelled by six DC electric motors mounted in articulated powered trucks shared between cars, not just the lead car.
  • 1959 construction costs of the entire system averaged over $1 million per mile (>$620,000/kilometer)
  • Oversized ceremonial scissors failed to cut the ribbon during the televised opening ceremony, so Walt simply tore it
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Loved it!

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Today (June 12) marks the five-year anniversary of the Pulse Nightclub Massacre in Orlando, Florida. I still break down in tears every time I think about the 49 beautiful souls (pictured below) who lost their lives that night. In their honor, I want to dedicate Ariana Grande’s version of “Over the Rainbow” from June 2017. #KeepDancingOrlando

Remembrance Events Friday For Victims Of Pulse Nightclub Massacre Four  Years Ago – CBS Miami

prior probability

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Probabilistic Truth

Alternative Title: Review of Robert Sanger, “Gettier in a Court of Law” (Concluding Post)

Alternative Title #2: The Irrelevance of Gettier Problems (with Apologies to Linda Zagzebski)

I want to conclude my review of Sanger’s Gettier paper with the following observation: Truth is not some binary value — i.e. a belief is not either true or false — but rather truth comes in degrees. In a word, truth is often probabilistic.

This unorthodox idea is often referred to as “degrees of belief” or “credences” and is associated with Bayesian epistemology a/k/a “subjective probability”–a topic that I have explored in some of my previous posts; see, for example, here and here. (This approach to truth can be traced back to Frank Ramsey and Bruno de Finetti; more recently, one of the leading contemporary exponents of this approach was Richard Jeffrey; see here.) Stated simply, a credence or degree of belief formally represents the strength with which we believe the truth of various propositions. Stated simply, the higher one’s degree of belief for a particular proposition, the higher one’s confidence in the truth of that proposition. In other words, beliefs may vary in degrees of strength or weakness–beliefs may come in shades of grey–or put another way, beliefs are not binary, are not all or nothing. Instead, my belief in a given conspiracy theory, for example, may range anywhere from 0 to 1.

Before proceeding, I now want to pose two further questions about the idea of degrees of belief. One is definitional. To the point: what is the difference between a plain and simple “belief” and a Bayesian “degree of belief”? In particular, is there some threshold or cut-off point (say, .9 or .95 or .99) above which a degree of belief acts like a full-fledged belief? The other question is logistical in nature. Specifically, when we are engaged in human reasoning, are our degrees of belief “infinitely precise real numbers”–e.g., exact numerical values ranging from 0 to 1–or “something less precise”–e.g., high, medium, and low? In other words, can a degree of belief really be expressed in precise numerical terms, and if so, how? (This second question is especially delicate because, if it turns out that, for whatever reason, we cannot assign a precise numerical value to a degree of belief, how can we transpose the axioms of probability into the Bayesian “subjective probability” framework? See also the entry for “Imprecise Probabilities” in The Stanford Encyclopedia of Philosophy.)

Putting aside these subsidiary questions (for now), the main question is, What do “degrees of belief” have to do with the things we have been talking about in our previous posts in this series, such as Gettier problems and photographic evidence, such as the picture of the sheep-dog in Sanger’s case of the negligent shepherd or the Zapruder film of the JFK assassination? In brief, two key conclusions or points flow from the idea of subjective probability or degrees of belief. One is that Gettier problems are often an irrelevant sideshow. Why? Because we cannot always determine whether a belief is true or false, except in the simplest or most trivial cases–like the number of coins in someone’s pocket (Gettier’s example) or the presence of a sheep in a field (Sanger’s example). In difficult cases, like whether Lee Harvey Oswald acted alone, or acted at all, in JFK’s murder, our beliefs are probabilistic.

The second conclusion is this: even if Errol Morris is correct to conclude the “photographs [and film clips] are neither true nor false” or “have no truth-value,” my Bayesian reply is, “So what?” Truth is often probabilistic anyways, not a binary or all-or-nothing value. Whether the photograph or film clip weakens or even falsifies our beliefs–shows that our beliefs might be or are, in fact, false, as in Sanger’s sheep-dog example–or whether such evidence supports our pre-existing beliefs, either through confirmation bias or because our beliefs are indeed likely to be true, the main point is that a photograph or a film clip can change, either up or down, our degrees of belief.

What is Probability?. Understanding the interpretations of… | by Devin Soni  👑 | Towards Data Science
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Friday funnies: death, taxes, and emails

I am interrupting my series on Sanger’s Gettier paper to share this New Yorker cartoon. Benjamin Franklin, among others, is reported to have said, “… in this world nothing can be said to be certain, except death and taxes.” Shall we now add “emails” to this list of unavoidable horribles?

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Postscript: the problem of photographic evidence

Alternative Title: Review of Robert Sanger, “Gettier in a Court of Law” (Part 4)

If you go back to my previous posts in this series (see here and here, for example), you will notice that they all involve photographic evidence of some sort–the picture of the sheep-dog in Sanger’s Gettier paper or the Zapruder film of the JFK assassination. This observation, in turn, poses a new question: is a photograph or a film clip, standing alone, able to generate knowledge in the sense of a “justified true belief” or is such evidence especially vulnerable to the Gettier problem, i.e. the problem of a justified belief that turns out to be false.

As I thought about this question, I stumbled upon this beautiful book Believing Is Seeing: Observations on the Mysteries of Photography, available here, by Errol Morris, an award-winning film director. Morris’s book is about the nature of truth and belief in the domain of photography, and he later summarized the main points of his book with a series of thought-provoking tweets. (Morris posted his tweets on October 1 and 2, 2011; see here.) Of the ten tweets in his series, the ones that I found most relevant for my analysis of the Zapruder film were #3, 4, 6, 8 and 10. Below, I will restate Morris’s tweets in reverse order (also, for clarity, each of his points appears in italics, followed by my commentary in brackets):

#10. “Photographs provide evidence. (The question is of what?)” [I am starting with Morris’s last tweet in his series because it is the most relevant one for our analysis of the Zapruder film. A photograph or film clip might provide crucial evidence of “what happened?” at a given time and place.]

#8. “The more famous a photograph is, the more likely it is that people will claim it has been posed or faked.” [The Zapruder film is a case in point, but that said, sometimes photos are, in fact, fake or manipulated. Think of Ansel Adams’ famous “falling soldier” photograph during the Spanish Civil War, a picture that was probably staged.]

#6. “Uncovering the relationship between a photograph and reality is no easy matter.” [As I mentioned above, photographic evidence can purport tell us “what happened?” at a given time and place. The problem, however, is that that the photo might be staged, or even more epistemologically troubling, the photograph or film clip might be consistent with different good faith interpretations of “what happened?”]

#4. “False beliefs adhere to photographs like flies on wallpaper.” [Again, the Zapruder film is a case in point, but how do we know when a belief is true or false? How do we know which JFK conspiracy theory to believe in? This observation is the crux of the Gettier problem: a picture or movie clip might generate a narrow or wide variety of beliefs about “what happened?,” and some or many of these beliefs might even be justified, but–and here’s the rub–most of these beliefs might turn out to be false!]

#3. “Photographs are neither true nor false. (They have no truth-value.)” [This last point is Morris’s most controversial one, but is he right? Read on …]

Morris elaborated on point #3 in this fascinating essay, which was published in the New York Times on July 10, 2007 under the heading “Liar, liar, pants on fire.” Among other things, Morris makes the following keen observation (emphasis added):

“The idea that photographs hand us an objective piece of reality, that they by themselves provide us with the truth, is an idea that has been with us since the beginnings of photography. But photographs are neither true nor false in and of themselves. They are only true or false with respect to statements that we make about them or the questions we might ask of them.”

In other words, a bare photo or film clip has no truth-value when it is standing alone–i.e., without any words, like a caption or some form of narrative explanation. Stated in Gettier terms, a photograph of a sheep-dog (like the one Sanger refers to in his Gettier paper) or a film clip (like the infamous Zapruder film) might generate a number of beliefs, and some of these beliefs might even be justified, but most of these beliefs–even justified ones–might turn out to be false.

I will now conclude with some final observations. In Sanger’s example, the picture of the sheep-dog was designed to dispel the shepherd’s justified belief that he saw a sheep in his field. Sanger’s picture of the sheep-dog (assuming the enlargement did not produce a fuzzy image) ended up rebutting or falsifying the shepherd’s belief that he saw a sheep. But the case of the Zapruder film is different. Instead of providing us a true picture of what really happened on November 22, 1963, the Zapruder film generates any number of competing hypotheses or beliefs about what happened, e.g. the lone-gunman theory versus the Grass Knoll conspiracy.

What is to be done, then? I think the idea of subjective probability–the bayesian notion that there are “degrees of belief”–might be the best way out of this Zapruder film conundrum and out of other Gettier problems more generally. I will elaborate on this observation and conclude my review of Sanger in my next post.

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