In duly enacted laws, bureaucratic regulations, judicial rulings, contracts, international treaties, technical manuals, specifications, product disclaimers, and many other areas, language should be clear, precise, and devoid of ambiguity. In such areas, language needs to be univocal. That is, it needs to have only one meaning and not be open to multiple interpretations.

There is an error of logic that is so pervasive and so little understood that it has found its way into these arenas where expression must be exact. I have not discovered whether there is a formal name for the error, but I call it the “not-or error.” The error occurs when disjunction (i.e. “or”) is used but the intent is demonstrably not disjunction. Hence: not-or.

Some attorneys are aware of this error, but their explanations of it are unnecessarily complicated. They bring out heavy artillery from the arsenal of formal logic to shoot down illogical gnats, when something much simpler would suffice.

In “Sex-change surgery and universal grammar” at Language Log, attorney Neal Goldfarb reported on a 2010 case before the U.S. Tax Court in which Judge James S. Halpern, in a concurring opinion, cited the arcana of formal logic, namely De Morgan’s laws. It’s hard to believe, but the U.S. Tax Code has some ambiguity in it. Goldfarb opined:

Judge Halpern is right about what the statute’s language means, but in framing his explanation in terms of formal logic, he’s on somewhat shaky ground (and not just because he opened himself up to the dissent’s glib putdown). The problem is that natural languages don’t necessarily follow the rules of formal logic. Indeed, the whole point of formal logic is to provide an artificial language that avoids the messiness of natural language.

So we’re going to actually allow “natural language” in the Tax Code and in federal court rulings? C’mon. (In “Illogic in American Law,” I recently touched on this same tax case and suggested that judges, lawmakers, and regulators should “have logicians look over their handiwork before burdening the public with it.”)

In addition to the legal case, Goldfarb’s article concerns logic and the social science of linguistics. Goldfarb dilates on “the dispute about how to interpret disjunction under negation -- i.e., how to interpret expressions such as I don’t know anything about linguistics or tax law (with don’t signaling negation and or signaling disjunction).”

After referencing the online Oxford English Dictionary, I must conclude that Mr. Goldfarb created a nifty neologism for his article -- “Morganian.” He explores both the “default (non-Morganian)” interpretations as well as the “Morganian” interpretations:

The sentence I’m not free this week or next week is generally interpreted to mean ‘I’m not free this week and I’m not free next week.’ But it can also be interpreted to mean ‘Either this week or next week -- I can’t remember which one -- I’m not free.’

So, the use of “or” “under negation” can result in conflicting interpretations. Goldfarb also writes that were we to substitute “and” for “or” in his problem sentence that it would be “generally interpreted” the same way as “or” is generally interpreted, but “again the Morganian interpretation is also possible” if “and” is vocally stressed. (What’s a mother to do?)

I’m not buying Goldfarb’s application of De Morgan’s laws in any of this. If they really were involved, where are the parentheses? De Morgan’s laws are transformational rules in formal logic, and the transformations require parentheses, which distribute negation like this: NOT (A OR B) is the same as (NOT A) AND (NOT B).

Perhaps Goldfarb expects folks to erect mental parentheses around everything in which negation is to be distributed so that “or” will be converted into “and,” (at least by those familiar with De Morgan). Or perhaps he expects speakers to employ “air parentheses” in the manner of “air quotes” to signal that what’s being said is, you know, “Morganian.”

De Morgan is not needed to rescue Goldfarb’s problem sentence because in the English language negation is “naturally” distributed, unlike in formal logic. What’s needed is the ability to select the correct conjunction. A one-sentence comment on Goldfarb’s article by T-Rex is especially apropos: “Surely this problem would not exist if the word ‘nor’ were still widely used?” (You’re the man, T-Rex.)

De Morgan is also not needed to explain the illogic in the tax case that Goldfarb cites: O’Donnabhain v. Commissioner of Internal Revenue (2010); see pages 77-81, “The Plain Language of the Provision.” Nevertheless, Judge Halpern was quite correct in noting that the wrong conjunction was used, but De Morgan wasn’t necessary to explain that. One might hazard that Halpern invoked De Morgan to give added weight to his argument. If he had merely pointed out that the statute had used the wrong conjunction, the other 14 judges might have taken his observations even less seriously. But when Halpern writes that “Judge Foley simply disregards the rules of grammar and logic,” he’s mistaken. For generations, the rules of English grammar have sanctioned the illogic he points out.

However, in “Breaking the Law — DeMorgan’s Law,” law professor and blogger Eugene Volokh did make an appropriate application of De Morgan in the Supreme Court case of Texas Monthly v. Bullock (1989). The text of the ruling was at odds with a footnote, which was a quasi-“Morganian” transformation of the text, except that it used the wrong conjunction. The opinion is therefore “internally inconsistent” and “reduced to incoherence.”

Unlike Texas Monthly, in most “Morganian” interpretations, the conditions (e.g. “free this week” and “free next week”) remain untransformed. In fact, the only change happening is when some poor soul interprets “or” as “and.” But do these two words stipulate the same thing? Not where I’m from.

English statements like Goldfarb’s example can be so indefinite that those receiving the statements must plug in another word for it to make any sense. So they transpose, like musicians reading in one key and playing in another, and “or” becomes “and.” Sadly, this nonsense has been justified and even facilitated by language experts.

Even though in his trotting out of De Morgan, Goldfarb is bringing in irrelevancies and over-intellectualizing these matters, I still recommend his article. The most interesting part of it is how children learn disjunction. They seem to understand disjunction just fine until they’re “corrupted” (my word) by adult abusers of language and logic. Goldfarb even sidles up to the big question: Does “universal grammar” exist? A question perhaps best left for those mapping the human genome. (Unfortunately, the links to the two papers by linguist Stephen Crain have expired. However, at his university webpage I found 17 iterations of “disjunction” embedded in the titles of Crain’s publications.)

It is inexcusable that those who would presume to foist laws, regulations, and legal judgments on the rest of us don’t know the difference between “and” and “or.” So know this: If your intent is to deny, disallow, exclude, or otherwise negate all the items on a list, then “or” isn’t the word for you.

If you’re still confused about conjunctions, I’ll be glad to tutor you, but you must wait for a fortnight. You see, I’m not free this week nor next week.

Jon N. Hall is a programmer/analyst from Kansas City.