Breaking the Law -- DeMorgan's Law:

Lots of court opinions are derided as illogical, but there's one that's illogical in a pretty technical way. I was just teaching it today, and thought I'd blog briefly about it.

The opinion is Justice Brennan's opinion, joined by Justices Marshall and Stevens, in Texas Monthly v. Bullock (1989). In that case, the Court struck down a Texas sales tax exemption for religious books and magazines; Justice Brennan's opinion concluded the law violated the Establishment Clause, because it impermissibly discriminated in favor of religion.

But the Court had before then upheld some religion-specific exemptions, so Justice Brennan had to explain when such exemptions were permissible and when they weren't. Here is what he said:

[W]hen government directs a subsidy exclusively to religious organizations that is not required by the Free Exercise Clause and that [(1)] either burdens nonbeneficiaries markedly or [(2)] cannot reasonably be seen as removing a significant state-imposed deterrent to the free exercise of religion, as Texas has done, it [violates the Establishment Clause.

(We can set aside the "is not required by the Free Exercise Clause" phrase because of later developments.)

Yet that's what he said in the text; here's what he said in footnote 9:

[W]e in no way suggest that all benefits conferred exclusively upon religious groups or upon individuals on account of their religious beliefs are forbidden by the Establishment Clause unless they are mandated by the Free Exercise Clause. [Case discussion omitted.-EV] All of these [constitutionally permissible religion-specific exemptions], however, involve legislative exemptions that [(1)] did not, or would not, impose substantial burdens on nonbeneficiaries while allowing others to act according to their religious beliefs, or [(2)] that were designed to alleviate government intrusions that might significantly deter adherents of a particular faith from conduct protected by the Free Exercise Clause.

So the text essentially says: Unconstitutional if (1) substantial burden on nonbeneficiaries or (2) no removal of state-imposed deterrent to religious exercise.

The footnote essentially says: Constitutional if (1) no substantial burden on nonbeneficiaries or (2) removal of state-imposed deterrent to religious exercise.

Sounds congruent, because not (A or not B) sounds equivalent to not A or B. But of course the two aren't equivalent; DeMorgan's Law says not (A or not B) is equivalent to not A and B.

The opinion is thus essentially internally inconsistent. It's theoretically possible to reconcile the two provisions, for instance by saying that the test in the text is limited to subsidies (including tax exemptions) and the test in the footnote is more general, or by reading the elements of each test subtly differently. But I doubt this makes sense; I think the Justices were trying to make the tests cover the same territory, use the same elements, and be functionally identical. Unfortunately, they broke DeMorgan's Law, and are being punished by having their opinion reduced to incoherence.

Professor Volokh, I think not (A or not B) sounds equivalent to not A or not not B, but that's a pointless quibble. My real point is, I think there's a bit of a problem here in the difference between "not" as a logical operator and "not" as an element of how people think. One is strictly "not" means "everything other than", whereas the other allows for "not" to mean "everything other than" or to mean "the opposite of". Likewise, "or" in most people's minds can have an exclusive element that it lacks in formal logic: not so much "A union B" as "A, alternatively B".

Anyway, I'm gonna say that legally, it probably doesn't matter, since Stevens went on to make it pretty clear in his City of Boerne v. Flores concurrence that they were going for what the text says, not what the footnote says.
3.11.2008 8:33pm
Bruce Hayden (mail) (www):
This is the problem of having someone with a math and CS degree teaching law. I would suspect that if Justice Brennan had been faced with DeMorgan's Law, he likely respond "Huh?"
3.11.2008 8:42pm
That's why normal-speaking people hate hardcore mathematicians.
3.11.2008 8:55pm
I suppose that Prof. Volokh would also have us believe that all unicorns have two horns. (a logically true statement)
3.11.2008 8:59pm
Eugene Volokh (www):
Westie: But my point is that if the normal-speaking people pay more attention to mathematicians, they'd avoid problems like this. After all, Brennan's error made his opinion logically inconsistent, as a practical matter and not just as a theoretical one.
3.11.2008 9:01pm
drewsil (mail):
While DeMorgan's Law seems a bit esoteric, if you think about it a little bit you can convince yourself that it must be true. Just consider a law that allows nonbelievers to be killed (presumably by a religion that advocates the murder of heathens). It imposes substantial burdens on nonbelievers (death), but also removes a state imposed deterrent to religious exercise (law against murder). Hence it would be constitutional according to the footnote but unconstitutional according to the text.

This is generally the case with most laws of logic, they seem weird when you hear them, and text that violates them often appears reasonable; however, when you consider specific examples the wisdom of the laws becomes evident.
3.11.2008 9:01pm
a sign at a bar states that "alcohol will *not* be served to anyone who is either (1) under 21, or (2) intoxicated.

question for Steve2, Bruce Hayden, & Westie (choose the letter that corresponds to the most accurate statement):

according to the bar's sign, alcohol *may* be served to:

(A) Steve, who is 23 and drunk off his ass;

(B) Jamal, who is 19 and sober;

(C) Alyosha, who is 22 and sober;

(D) Steve and Jamal; or

(E) Alyosha and Steve.
3.11.2008 9:13pm
Archit (www):
The problem isn't just esoteric. What if a benefit (1) imposed substantial burdens on nonbeneficiaries and (2) had been designed to alleviate government intrusions that deter free exercise? The footnote says it is constitutional because of (2) and the text says it is unconstitional because of (1). And the opinion doesn't make it easy to figure out which or should be read as an and.
3.11.2008 9:19pm
Bruce Hayden: DeMorgan's law is not a difficult concept to grasp and it as well as most other logical identities are quite intuitive once one is able to make the distinction between "or" and "exclusive or" (a distinction that doesn't always exist in common parlance). We expect those interested in obtaining law degrees to understand it, by way of a number of LSAT questions that implicate DeMorgan's law. I don't see why we can't expect Supreme Court Justices to also have a firm grasp on those simple rules.
3.11.2008 9:20pm
Skyler (mail) (www):
I have to admit to being a bit flumoxed. Did you ask if Brennan was illogical?

I don't know of a time when he was logical.
3.11.2008 9:28pm
b. gives a good illustration, but to make it even clearer, here it is more closely tracked:

Let A = intoxicated and B = at least 21

X will be served if and only if: -(A or -B), i.e., if you are: intoxicated OR not at least 21: no service.

Equivalent to:

X will be served if and only if: -A and B, i.e., if you are: not intoxicated AND at least 21: service.

(I hope I didn't mess that up)
3.11.2008 9:33pm
CEB: follow up, if you believe that -A or B is equivalent, the bar would serve persons who are under 21 but not intoxicated and persons who are at least 21 but intoxicated.

I realize now that this might not really make it clearer 0_o
3.11.2008 9:39pm
It should be noted that the phrase "either A or B" could be interpreted to mean "exclusive or", that is, "(A or B) and not (A and B)" However, this would still result in a logical fallacy because the negation of " A exclusive or B" is:

Not ((A or B) and not (a and B))

= Not(A or B) or (A and B)

= (Not-A and Not-B) or (A and B)

i.e. either they're both true or neither is true.
3.11.2008 9:44pm
My point is that if a Supreme Court Justice can screw this up, how do we expect normal people to keep track of it, since they're the ones who are supposed to be violating the law? I would bet that if I asked 100 people to explain the test, at least 95 of them wouldn't come up with your reading, technically correct though it may be.
3.11.2008 9:48pm
Bruce Hayden (mail) (www):

If you want to have DeMorgan's Law beat into you, take a digital circuitry class (if you haven't already). This was one of those things that I knew with my math and CS background, but didn't really KNOW it until I spent months simplifying circuits with it.

Of course, it makes logical sense - we are talking logic after all. My point was that for most of the population, it is probably something that you pick up somewhere in HS in passing, and then promptly forget. EV has the background that he caught this. But I doubt that very many others in the legal community did, without someone like him pointing it out. And now we can all go, of course.
3.11.2008 10:07pm
CDU (mail) (www):
My point is that if a Supreme Court Justice can screw this up, how do we expect normal people to keep track of it, since they're the ones who are supposed to be violating the law?
I doubt any "normal people" would run afoul of this opinion. It seems to be directed at state and federal legislators who (regardless of whether or not they are "normal") have lawyers to look these things over before the law is passed.
3.11.2008 10:08pm
Tyson Stanek (mail):
I had an argument with a professor about this in law school. I had to answer a true/false question on a quiz that, in my mind, depended upon whether the "or" was inclusive or exclusive. She didn't understand that concept, so I assumed "inclusive".

A similar pet peeve of mine is when a professor used words like "every", "all", "always", "never", etc. in a question. In my mind, these words have powerful meanings. "Always" means from the Big Bang until the end of time. "Never" means under no circumstance that can, did, does, or will exist.

Law school and mathematics don't mix, IMHO.
3.11.2008 10:12pm
Nathan_M (mail):
I think there's an argument you are mistranslating the text.

Let C equal "is constitutional"
let B equal "substantial burden on nonbeneficiaries", and
let R equal "removal of state-imposed deterrent to religious exercise"

You say Brennan is saying (I'll use "~" for "not", but no other symbols aside from parentheses):

Text: ~C is equivalent to (B or ~R)
Footnote: C is equivalent to (~B or R)

That is obviously inconsistent. But I think what Brennan is actually saying is:

Text: if (B or ~R) then ~C
Footnote: if C then (~B or R)

There is no inconsistency with this. The problem with Brennan's logic only arises if you translate his text as being equivalence statements rather than "if ... then" statements.

The problem with my translation, of course, is it not clear about the constitutionality of (~B or R) laws. But based only on what you have excerpted from the judgement, and especially for the footnote, I think it is more accurate to read Brennan as making a conditional statement.
3.11.2008 10:22pm
Nathan_M, I believe Eugene covered this when he stated the footnote could perhaps be read to cover something more general. In any case, while your conditionals are not inconsistent, they are certainly not equivalent, in which case Brennan would be advocating two different tests of Constitutionality: (B or ~R) or (B and ~R).

Bruce Hayden, my point is that lawyers are expected to be able to engage in deductive reasoning like Eugene did (but perhaps not nearly as mathematically rigorous). Justices likewise.
3.11.2008 10:30pm
Nathan_M, thanks for putting into words what I was groping at so horribly.

For that reason, in your example b., because the sign states, "If ((age >= 21) &(intoxicated = false)) THEN (More_Booze)", alcohol may be served to Alyosha.
3.11.2008 10:31pm
And, embarrassingly, I just realized ((B or ~R) or (B and ~R)) is equivalent to (B or ~R), so perhaps Nathan_M is correct.
3.11.2008 10:34pm
Duffy Pratt (mail):
Nathan_M has it right. Brennan points out that all the cases that were constitutional were either not burdensome, or alleviated government intrusions on free exercise. He did not, however, say that the two were equivilent.

In law, it's hard to hold writers too closely to the "logical" interpretations of basic words like "or", "and", "if ... then", and even "not". For an illustration, see how Marshall reads "necessary AND proper" in McCullogh.
3.11.2008 10:36pm
Gilbert (mail):
Thanks Nathan_M. I think the confusion arises when using "not" both to negative propositions A and B as well as to negative the status of constitutionality.
3.11.2008 10:51pm
Richard Aubrey (mail):
Who is it being punished?
Are their High and Mightinesses being fined, spanked or held up to ridicule? Um. Forget that last.
3.12.2008 12:25am
Not JG:
3.12.2008 12:39am
ReaderY wrote at 3.11.2008 8:44pm:
It should be noted that the phrase "either A or B" could be interpreted to mean "exclusive or", ...
I was told we wouldn't be discussing Eliot Spitzer's trysts in this thread.
3.12.2008 1:01am
Dave Hardy (mail) (www):
I never heard of DeMorgan's Law, but I do know that I just filed a motion for reconsideration on an (unpublished) appellate ruling that "get off my land" equals false imprisonment since it is a restraint on liberty of movement.

I don't have terribly high expectations of logic from courts, sorry to say.
3.12.2008 1:49am
Joseph Bottum (mail):
Text: if (B or ~R) then ~C
Footnote: if C then (~B or R)

There is no inconsistency with this.

Yes, there is, if one is supposed to think that the footnote follows from or simply restates the text. The recasting of or conjunctions into if-then hypotheticals still preserves the logical relations of the terms. (See, for instance, Jan Łukasiewicz's brilliant 1950s Aristotle's Syllogistic from the Standpoint of Modern Formal Logic, which translates the syllogism into an if-then system.)

What the footnote seems to be proposing, phrased as you do as hypotheticals, is modus tollens, the inversion and negation of the elements: If X then Y gives If ~Y then ~X.

In this case, the text's X is (B or ~R), and the question remains, What is ~X? De Morgan's Law makes it (~B and R), and that's right, for even as hypotheticals, the two propositions produce different results:

Text: If (drunk or underage) then no liquor-->The drunk 21-year-old gets no more booze.
Footnote: If liquor then (sober or of age)-->The drunk 21-year-old can booze it up some more.

The only way out of this is to read the or as some kind of bastard and--which it occasionally is, in ordinary language.
3.12.2008 1:56am
Nathan_M (mail):

Yes, there is, if one is supposed to think that the footnote follows from or simply restates the text.

Well, obviously the footnote doesn't logically follow from the text, I don't think anyone claimed it did. I'm certainly not.

I was just saying that if you translated Justice Brennan's comments my way, as opposed to Professor Volokh's way, there is no logical inconsistency. And there isn't, which you can prove quite easily be making a truth table.

Footnote: If liquor then (sober or of age)-->The drunk 21-year-old can booze it up some more.

That's wrong, it's the "affirming the consequence" fallacy. To steal from Wikipedia, your argument is of the form:

If Bill Gates owns Fort Knox, then he is rich.
Bill Gates is rich.
Therefore, Bill Gates owns Fort Knox.

Saying that everyone who gets liquor is sober or of age does not imply that everyone sober or of age can get liquor.
3.12.2008 2:51am
Nathan_M (mail):
"Affirming the consequent", I mean. If I didn't make silly typos then I wouldn't mind not being able to edit posts....
3.12.2008 2:53am
Joseph Bottum (mail):

Fun stuff. Thanks for pursuing this. Yes, your Gates example is affirming the consequent. But your application of it to my point is wrong. My example was that the bar would have a defense under the second hypothetical that it wouldn't under the first. Your rephrasing shifts from "gets liquor" to "can get liquor," and it's in this shift that it looks like affirming the consequent.

But using a truth table is a good idea--and, in fact, the truth table will give you different results for the two hypotheticals:

Let p be your first hypothetical, if (B or ~R) then ~C, and let q be your second, if C then (~B or R). Here's the table:

B R C p q
t t t f t
t t f t t
t f t f f
t f f t t
f t t f t
f t f t t
f f t f t
f f f t t

Those are really quite different results for p and q.
3.12.2008 5:04am
Nathan_M (mail):
I agree it's fun, but I was the only person in my course who liked symbolic logic. I just wish it wasn't so long ago now, because your post has had me going back and forth over whether I'm right or not. I think I am, though.

First, I noticed one mistake on your truth table, on the 5th line. If ~B, R, and C then p is true.

Now, the difference between us is, I think, our definition of consistence. I think two statements are consistent if, and only if, one is true the other could be true. Your definition is that they are consistent if, and only if, one is true then other is true.

I'm not 100% sure which definition is right, but I think mine is. (I did a truth table for the way Eugene phrased it, and it is inconsistent under my laxer definition of consistency.) You are requiring that the two statements follow logically from each other, and I think that is a higher requirement than consistency. I stand to be corrected on that, though. Like I said, it's been too long since undergrad.

Even as I write this I'm going back and forth over whether my definition of consistency is correct, though. I wish I hadn't left that damn logic textbook in Australia....
3.12.2008 6:08am
Joseph Bottum (mail):

The bad fifth line is what I get for trying to do the table in my head while I typed.

Anyway, I think consistent in this context would mean "had the same truth table," which strikes me as reasonable enough, given that Justice Brennan seemed to be saying that he was just rephrasing in his footnote.

Anyway, it's the first line of the table that everyone was choosing as an example, where all three elements are true: the drinker is (B) drunk and (R) of age and (C) gets served booze.

Under the first hypothetical, if (B or ~R) then ~C, the bar has broken the law (true antecedent, false consequent).

Under the second hypothetical, if C then (~B or R), the bar hasn't broken the law (true antecedent, true consequent).
3.12.2008 7:32am
x (mail):
An old rule of computer programming ... (k)nots and o(a)rs should only be found together on boats.
3.12.2008 12:06pm
David M (www):
The Thunder Run has linked to this post in the - Web Reconnaissance for 03/12/2008 A short recon of what’s out there that might draw your attention, updated throughout the check back often.
3.12.2008 12:21pm
Duffy Pratt (mail):
Consistent does not mean having the same truth tables. The statement A And B is consistent with the statement A Or B. All that means is that there is a state of affairs (a line in your truth table, if you will) where both statements are true. Having the same truth table is equivalence, not mere consistency.

Also, beware of forcing ordinary language into the contrapositive (modus toluns). Try it with this example from Austen: There are biscuits on the table, if you want some. Does this mean that if there are no biscuits on the table, then you don't want any?
3.12.2008 12:25pm
ctw (mail):
"I never heard of DeMorgan's Law"

You may be able to find it in FindLaw. IIRC, it was a circuit decision, either DC in 2009 or AC in 2006. Logically, it would be the DC circuit. Or not.

- Charles
3.12.2008 2:39pm
Spartacus (www):
People who think that math and law don't mix apparently don't realize that DeMorgan's is logic; do they also think that logic and law don't mix? That is the sad state of the legal profession today.
3.12.2008 3:36pm
But my point is that if the normal-speaking people pay more attention to mathematicians, they'd avoid problems like this.
3.12.2008 3:45pm
Clayton E. Cramer (mail) (www):
Unfortunately, a lot of people don't write or read very carefully--and use "and" when they really mean "or" or they use "or" when they mean "and." And then there are the speakers of bureaucratese who write "and/or" when they usually mean "or." Every once in a great while, someone uses "and/or" in the correct sense of "we might do both of these choices, or we might pick only one of these choices."

Cheer up: every time I mention DeMorgan's Law outside of work, I have to explain it, and then they look at me as though I stepped off a flying saucer.
3.12.2008 5:01pm
Suzy (mail):
Duffy, you are correct. All that is required for consistency is that both claims can be true on the same line of the truth table. For equivalence, all lines must have the same value, whether that be true or false.

I think Nathan_M is correct that we need to construe the statements differently than simply (B or not R), and (not B or R). However, Joseph is correct that we should demand equivalence, rather than simply consistency, since otherwise we'd have two different constitutional tests, and scenarios where something could be both constitutional and not-constitutional.

I still don't see a problem with what Brennan wrote, though. For example, he is obviously not trying to mark off as constitutional cases where a burden is created even though a deterrent is removed. Here, Joseph's suggestion that the "or" in the footnote is best read as some kind of bastard "and" is on the right track, not because we need charitable interpretation or a stretch of the meaning of "or", but because the two terms "B" and "R" are not necessarily bound together and Brennan needs to address those possibilities too. Could we not have something that achieves "not R" without the question of B even becoming relevant? It is only if every case of R/notR has necessary consequences in terms of B/notB, or vice versa, that the comment in the footnote becomes a problem.
3.12.2008 7:12pm
james80 (mail):

Morgan's law can be expressed in simple math, which is cool, but its usefulness depends on whether "or" is inclusive or exclusive:

Inclusive: -1= -1 or -(1), Exclusive: -1= -1 or -(-1)
Inclusive: 1= -(-1) and 1, Exclusive: 1= -(-1) only/or -1

1. Correct: To turn off the light (break the circuit) disconnect the red or the blue wire.
2. Correct: To turn off the light (break the circuit) disconnect the red and the blue wire.
3. Incorrect: To turn on the light connect the red or the blue wire.
4. Correct: To turn on the light connect the red and the blue wire.

Statement (3) is incorrect only because statement (1) and (2) are both correct.

1. Correct: Unconstitutional if substantial burden on nonbeneficiaries or no removal of state-imposed deterrent to religious exercise.
2. Correct: Unconstitutional if substantial burden on nonbeneficiaries and no removal of state-imposed deterrent to religious exercise.
3. Incorrect: Constitutional if no substantial burden on nonbeneficiaries or removal of state-imposed deterrent to religious exercise.
4. Correct: Constitutional if no substantial burden on nonbeneficiaries and removal of state-imposed deterrent to religious exercise.

Again statement (3) is only incorrect because statement (1) and (2) are both correct. In other words: if not(A or B) and not(A and B) then (A and B). Of course, if the statement read "either or" or "and or" this would be immediately obvious.

However, these same rules do not necessarily apply to linguistics where the meaning of "or" may shift between exclusive and inclusive depending on common usage.

An apple is a fruit or a vegetable.
It is not an apple because it isn't a fruit or a vegetable.

The second proposition sounds incorrect because it seems to follow from an inclusive "or"--if it is not a vegetable it can still be an apple. However, in context with the leading proposition's exclusive use of "or" it is still accurate. Although, it would sound better if it read: It's not an apple if it is not a fruit or it is not a vegetable. What really sets conversational linguistics apart from math and logic, though, is the question mark.

Is an apple a fruit or a vegetable?

A) fruit
B) vegetable

If an apple = (A or B) here Morgan's law doesn't apply, because, apple = not B. Even Wittgenstein eventually got away from trying to conform language to a system of logic and, at least in his later years, described language as a type of game.
3.13.2008 7:18am