[This originally appeared on Language Log in February 2010, as a guest post.]
The United States Tax Court recently decided that payments for sex-reassignment surgery are deductible as medical expenses. Among the 15 judges, there were six separate opinions, with five of the judges dissenting. Most of the debate dealt with questions like whether Gender Identity Disorder is a “disease” (a key term in the statue) and if so whether sex-reassignment surgery, which doesn’t change the patient’s subjective sense of gender identity, constitutes a “treatment” for the disease (ditto).
Those are issues with interesting linguistic dimensions, but what I want to talk about here is a different aspect of the case: 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).
In the case decided by the Tax Court, the IRS had ruled that sex-reassignment surgery isn’t not deductible, on the theory that it amounts to cosmetic surgery, which the tax code excludes from its definition of medical care. The code defines cosmetic surgery as—
any procedure which is directed at improving the patient’s appearance and does not meaningfully promote the proper function of the body or prevent or treat illness or disease.
The language in boldface is what’s at issue here.
The Tax Court rejected the IRS’s position, with the majority of the judges concluding that sex-reassignment surgery was a treatment for the disease of Gender Identity Disorder and therefore wasn’t cosmetic surgery. (The majority didn’t decide the question whether the procedure is “directed at improving appearance”—though that’s how the dissenters characterized it—or whether it promotes the proper function of the body.)
On the majority’s interpretation, the boldfaced language above means that an appearance-improving procedure doesn’t count as cosmetic surgery if it either promotes proper bodily function or prevents or treats illness or disease. The dissenters, on the other hand, argued that for such a procedure to be excluded from the cosmetic-surgery category, it has to both promote proper bodily function and prevent or treat illness or disease. In other words, the dissenters were equating (1a) with (1b), while the majority equated it with (1c):
1a. does not [(meaningfully promote…) or (prevent or treat…)]
1b. [does not meaningfully promote…] or [does not prevent or treat]
1c. [does not meaningfully promote…] and [does not prevent or treat]
One of the dissenting opinions accused the majority of ignoring the statute’s plain meaning:
The majority’s analysis proceeds as if the statute employs “and” rather than “or” between the “meaningfully promote the proper function of the body” and “prevent or treat illness or disease” prongs. Respondent appears to agree with this interpretation in lieu of a plain reading of the statute. In essence, the majority and respondent engage in reconstruction, rather than strict construction, of section 213(d)(9).
Judge Halpern, who voted with the majority and wrote a separate concurring opinion, wasn’t buying this argument. In explaining why he didn’t buy it, he invoked principles of formal logic:
Because the second part of the test contains two expressions separated by “or”, that part of the test contains a “disjunction”; i.e., a compound proposition that is true if one of its elements is true. Importantly, however, the second part of the test contains not just a disjunction (i.e., (p or q)), but rather the negation of a disjunction (i.e., not (p or q)). Judge Foley errs because he assumes that the expression “not (p or q)” is equivalent to the expression “(not p) or (not q)”….
In formal logic, there is a set of rules, De Morgan’s laws, relating the logical operators “and” and “or” in terms of each other via negation. [Citation to Wikipedia.] The rules are:
not (p or q) = (not p) and (not q)
not (p and q) = (not p) or (not q)
The first of the rules would appear to govern the disjunction in section 213(d)(9)(B), which is of the form “not (p or q)”.
To which the dissenters responded:
Judge Halpern’s mechanical application of De Morgan’s laws is not prudent. Simply put, congressional intent is not subservient to De Morgan’s laws.
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.
Now, it just so happens that for expressions in the form not (p or q), English does in fact behave consistently with De Morgan’s law. Or at least it usually does. 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.’ (These examples are adapted from the Cambridge Grammar of English Language.) I suspect that the latter interpretation arises only in a fairly narrow range of contexts, but in any event, the point remains that De Morgan’s laws don’t invariably apply in English.
There’s an even bigger deviation from De Morgan’s laws in the case of negated conjunction (as opposed to negated disjunction). According to De Morgan’s laws, not (p and q) = (not p) or (not q). But in English, not (p and q) is more often interpreted as (not p) and (not q). So I’m not free this week and next week is generally interpreted as ‘I’m not free this week and I’m not free next week.’ But again the Morganian interpretation is also possible, especially (only?) if the and is stressed: I’m not free this week AND next week will typically be interpreted to mean ‘It is not the case that I’m free both weeks (but I am free one of them).’
Note that in the case of disjunction, stressing the coordinator (or) doesn’t have the same effect as in the case of conjunction. That is, it doesn’t cancel the default (non-Morganian) interpretation. On the contrary, it reinforces it. I’m not free this week OR next week is interpreted as ‘I’m not free this week AND I’m not free next week.’
Not all languages follow the same pattern as English. For instance, Japanese apparently doesn’t follow De Morgan’s laws in the case of negated disjunction. So in Japanese, not (p or q) = (not p) or (not q). Or at least that’s true for adult speakers of Japanese. Young Japanese-speaking children, it seems, do follow De Morgan’s laws, but then switch over when they more completely master the language. (See, for example, Andrea Gualmini & Stephen Crain, Why No Child or Adult Must Learn De Morgan’s Laws, Proceedings of the 24th Annual Boston University Conference on Language Development, Cascadilla Press, Summerville, MA (2004).)
In fact, Stephen Crain argues that in all languages disjunction is interpreted consistently with De Morgan’s law in at least some structures, and that this is evidence supporting the nativist side of the debate over whether knowledge of language is innate. Here’s the abstract to his paper The Interpretation of Disjunction in Universal Grammar:
Child and adult speakers of English have different ideas of what ‘or’ means in ordinary statements of the form ‘A or B’. Even more far-reaching differences between children and adults are found in other languages. This tells us that young children do not learn what ‘or’ means by watching how adults use ‘or’. An alternative is to suppose that children draw upon a priori knowledge of the meaning of ‘or’. This conclusion is reinforced by the observation that all languages adopt the same meaning of ‘or’ in certain structures. For example, statements of the form ‘not S[A or B]’ have the same meanings in all languages, and disjunctive statements receive a uniform interpretation in sentences that contain certain focus expressions, such as English ‘only’. These observations are relevant for the long-standing “nature versus nurture” controversy. A linguistic property that (a) emerges in child language without decisive evidence from experience, and (b) is common to all human languages, is a likely candidate for innate specification. Experience matters, of course. As child speakers grow up, they eventually learn to use ‘or’ in the same way as adults do. But, based on findings from child language and cross-linguistic research, it looks like certain aspects of language, including the interpretation of disjunction, are part of the human genome.
There are some obvious questions: Is Crain correct in saying that disjunction is interpreted consistently with classical logic in (at least some constructions in) all languages? And that adult speech in languages like Japanese really provides children learning to speak no evidence for the interpretation of disjunction that they start out with?
However, I’m pointing out Crain’s work, not as an opening to talk about whether there’s such a thing as Universal Grammar, but as an exercise in Six Degrees of Something-or-Other. Here we have a decision interpreting the United States Internal Revenue Code that turns (at least in part) on an issue that might be relevant to figuring out the fundamental nature of language. And I thought tax law was boring.