Generics

The following is a tutorial essay for Philosophy of Logic and Language with James Kirkpatrick. In effect, it is a quick agenda for an hour-long discussion; for this week, I chose generics, because that’s where James specialises. A free-standing and much more readable version of some of the material may eventually go up on my Substack.

Generic expressions include (particular readings of) the following.

(1) Tigers have stripes.

(2) A duck lays eggs.

(3) The mosquito carries malaria.

I argue that (the relevant readings of) (1), (2), and (3) are true under the same conditions as (4), (5), and (6) respectively.

(4) All tigers have stripes.

(5) All ducks lay eggs.

(6) All mosquitos carry malaria.

I. The naive case

First, consider statements of the form ‘Q₁(Fs are Gs), or at least Q₂(Fs are Gs)’, which we can call ‘weakenings’. Weakenings could also be in dialogue form, where one speaker says ‘Q₁(Fs are Gs)’ and the other says ‘No, but Q₂(Fs are Gs)’. We should expect that weakenings are unacceptable just in case Q₂ is at least as strong as Q₁ on the assumption that there are many Fs. (Note that every quantifier is at least as strong as itself, and no quantifier is at least as strong as one to which it is incomparable; I use the sentence ‘cats are nice’ to avoid problems induced by consistent common knowledge about prevalence in our beginning examples.)

(7) [All/#Most] cats are nice, or at least [most / #all] cats are nice.

(8) [Most/#Many] cats are nice, or at least [many / #most] cats are nice.

(9) [Many/#Some] cats are nice, or at least [some / #many] cats are.

The awakenings (7), (8), and (9) suggest respectively that ‘all’ is stronger than ‘most’, which is stronger than ‘many’, which is stronger than ‘some’. Indeed, one can check using the same method that, at least on these four quantifiers, the relation of relative strength is transitive and asymmetric.

Consider also the following weakenings.

(10) [All/#Normal] cats are nice, or at least [normal / #all] cats are nice.

(11) [Most/?Normal] cats are nice, or at least [normal/?most] cats are nice.

(12) [?Many/Normal] cats are nice, or at least [?normal / many] cats are nice.

(13) [#Some/Normal] cats are nice, or at least [#normal/some] cats are nice.

The comparison can be run not only with ‘normal(ly)’, but also with ‘usual(ly)’, ‘(proto)typical(ly)’, ‘prototypical(ly)’, and ‘generally’. Indeed, each of these special quantifiers seem to be of either equivalent or incomparable strength.

Given a background stock of explicit quantifiers with which to compare, let us naively try to do the same thing with the generic. We will use the bare plural form of the generic because it is the smoothest to put into the same format; but it is easy to substitute indefinite or definite generics as well (perhaps with ‘is a friendly animal’ / ‘are friendly animals’, because the generic readings in those forms is harder to access in the more informal register associated with ‘are nice’).

(14) [#All] cats are nice, or at least [#all] cats are nice.

(15) [#Most] cats are nice, or at least [most] cats are nice.

(16) [#Normal] cats are nice, or at least [normal] cats are nice.

In particular, (15) and (16) show respectively that the generic is stronger than ‘most’ and that it is stronger than ‘normal(ly)’. More importantly, (14) shows that the generic is at least as strong as ‘all’, and that ‘all’ is at least as strong as the generic. That is, the generic is equivalent to ‘all’.

Of course, this initial consideration is by no means conclusive; there may well be something wrong with the assumptions made about the behaviour of weakenings. Nevertheless, it is the natural starting point of inquiry; the truth-conditional equivalence of generics and universals should be the default hypothesis.

II. The psychological case

The main reason to depart from the view that generics are universals is the patterns of inference that we see with generics: firstly, they are more readily accepted than one would expect universals to be; secondly, they permit more exceptions than one would expect universals to permit.

However, both of these expectations run into both theoretical and empirical trouble. It is generally useful to generalise liberally from past experience: we should expect to have some method of verbal reasoning that readily facilitates induction, without it being clear to us explicitly that such reasoning is invalid. That is, when reasoning with more or less natural categories—especially potentially dangerous ones—we should expect to take evidence which supports an existential generalisation to implicitly support a universal one. For instance, upon learning that someone was mauled to death by a polar bear, it is natural to exercise caution around black bears as well.

In the first instance, the inference may proceed directly from ‘that polar bear was dangerous’ to ‘this black bear is dangerous’, unmediated by any quantified intermediate statement. Nevertheless, it may be easier to remember two properties of bears in general than to remember one property of one particular bear and another property of another particular bear. Thus, generic statements (‘bears are dangerous’) are a natural candidate to facilitate such a habit in verbal reasoning. Nevertheless, even existential (‘some bears are dangerous’) and universal statements (‘all bears are dangerous’) could play the same role. Whichever one of these would be more useful is the one that we should expect generic statements to pattern more like. With existentials, the key (deductively invalid but nevertheless heuristically useful) inference is made at run-time; with universals, it is made upon storage of the generalisation, which is ready then to apply immediately at run-time. This is a significant advantage.

From the armchair, given considerations about the benefits of inductive reasoning, we should expect universal generalisations, like generic generalisations, to be accepted on the basis of comparatively weak evidence. Empirically, we do indeed see about half of respondents accept false universal generalisations which are nevertheless characteristic of a category, like ‘all ducks lay eggs’ (see, among others, Khemlani et al. 2007). Further, such a precautionary disposition should not be easily overridden by a single counterexample (say, a peaceful encounter with a black bear). Overly brittle generalisations may be dangerous, and in particular may prevent one from accessing the knowledge that particular bears that one encounters in the future are dangerous. Empirically, we do indeed see that many subjects are willing to accept universal generalisations like ‘all lambs are friendly’ while accepting anomalies by also accepting ‘all dirty German lambs are friendly’. Thus, we already know that ordinary speakers are inconsistent with regard to universal claims, accepting them while admitting exceptions. One should definitely not respond to these empirical findings (of easy generalisation and exception compatibility) by imposing restrictions on ‘all’, out of a misguided attempt to be more charitable to native speakers. We simply make mistakes when reasoning with universal generalisations, because even extremely useful cognitive patterns can sometimes go wrong.

III. Generics in formal discourse

In some contexts, the quick reasoning facilitated by generics may not be worth the concomitant unreliability. Nevertheless, to the extent that generic generalisations are used, it is worth investigating their usage when more care is required. Of course, some conventions may arise that depart from ordinary English meanings: often, as in everyday speak, definitions in mathematics formulated in terms of ‘if’ are understood to really mean ‘if and only if’. Nevertheless, it is striking that bare plurals in mathematical discourse are more or less uniformly understood to be equivalent to universal quantification. Even ordinary discourse about familiar mathematical objects seems prone to this. Consider, for instance:

(17) # Real numbers are irrational.

(18) ?? Prime integers are odd.

(19) ?? Sets have elements.

None of (17), (18), and (19) are true; they would be marked incorrect on any exam, for they admit exceptions (the rationals, two, and the empty set, respectively).

Nevertheless, there is some room for worry here. Firstly, a generic like (18) might be useful in reasoning.

(20) The integer 78 is even, (?? but primes are odd,) so it is not a prime.

These are clearly bad arguments by mathematical standards. Indeed, removing ‘but primes are odd’ makes (21) sound less bad: one can readily supplement the auxiliary fact that primes other than 2 are not even, and 78 is not 2. Further, (21) is certainly true.

(21) If primes are odd, and 78 is not odd, then 78 is not prime.

So, the problem with (20) lies with the generic ‘primes are odd’. But ‘primes are odd’ is as good a candidate for a true generic about a mathematical object as one can hope for. In (17), the probability that a real selected uniformly at random is irrational is exactly 100%; but perhaps the rationals are among the prototypes of real numbers, which might matter. On the other hand, with (18), it is certainly true that normal or prototypical primes are odd, and that 2 and 2 are simply pathological cases. Thus, (17) suggests that generics are stronger than ‘most’, and (18) suggests that they are stronger than ‘normal’.

So, when generic generalisations are put under stress—that is, when they are used in domains where the ordinary ampliative inferences that they facilitate are unsafe—they seem to pattern like universals.

IV. Generics and contextual domain restriction

With all this being said, our ordinary practice with generics still seems more or less well-behaved, but in a way that outstrips the stricter truth-conditions posited here: we will assert false generic generalisations over a domain which is broad enough to admit exceptions, but ordinary communication proceeds smoothly, with salient narrower domains being automatically substituted often without either speaker or audience noticing. But exactly the same is true of universals, like ‘every beer is in the fridge’. So, this is not a dissimilarity between generics and universals, and in particular between the three generics and the three universals with which we started.

Whatever mechanism accounts for the looser behaviour of universals should be a strong default candidate as the mechanism for generating the looser behaviour of generics. For instance, on the theory that the universal quantifier takes a contextual parameter restricting its scope, it is natural to hold that generics operate similarly. Alternatively, given the (previously defended) view that contextual domain restrictions are really pragmatic repair (usually, insertion of ‘relevant’), it is natural to hold that generics operate similarly.

Nevertheless, there do seem to be some differences between (1), (2), and (3) on the one hand, and (4), (5), and (6). Defenders of the contextual parameter view may have a hard time explaining this without either giving up the attractive unification of generic generalisations and universal generalisations or appealing to pragmatic considerations for generics and risking making somewhat redundant the posited contextual parameter. However, given the view of pragmatic repair, a natural place to look for the relative acceptability of generics over universals is in the pragmatics. What could make this difference?

A simple story is the following: given that generics are easier to produce and easier to process, one should default to a generic formulation over an explicitly universal formulation of a given generalisation. So, if the universal is made explicit, then the standards are raised: in particular, ad hoc exceptions should not be tolerated, and otherwise ordinary instances dismissed as irrelevant. This is strong enough to generate the contrast between the generics and the universals, but weak enough to preserve the potential for universals to be pragmatically restricted.

V. Generics as existentials?

One last worry: given that (1), (2), and (3) uncontroversially have existential readings, why shouldn’t the assimilation of generics to existentials be privileged? That is, why not deny the ambiguity in the first place? At the very least, the bare plurals and indefinites clearly have existential readings; the definites must be treated separately, perhaps as kind or collective predication; and it is independently plausible that a kind or collective does something if one of its members does, and so this also patterns with existentials.

With this reading in mind, one is even able to utter the following truly, suggesting that the bare plurals are as weak as existentials.

(22) (Not many / Few) cats are nice, but cats are nice.

(23) Many cats are nice, or at least cats are nice.

Further, the ‘run-time’ inference being the key one (from our psychological considerations) was not a very decisive point; and, in the other direction, that bears are dangerous is an item of knowledge on the existential view, but merely an extremely useful false belief on the universal view (which is a somewhat worse fit with a broader knowledge-first approach to epistemology). Further, given the patterns of reasoning (the liberal elimination rules) associated with generics, existentials still secure the same particular inferred knowledge; so, given a knowledge-maximisation approach to charity, the existential view seems to come out far ahead by doing as well on the knowledge generated by generics and much better on whether the believed generics themselves count as knowledge. Finally, there are indeed true readings of (17), (18), and (19), and there is a false reading of (21)—namely, the existential reading of the bare plural.

However, these positions might not be irreconcilable. In particular, given the ready appeal to pragmatic repair in the above, one might hold that (i) the bare generics are existentials, but (ii) particular uses of generics are often pragmatically repaired, with ‘every’ or ‘every relevant’ supplemented. Thus, our opening generics were in some sense (pragmatically if not semantically) ambiguous, between a literal (existential) reading and a pragmatically reinterpreted (universal) reading. To the extent that generic generalisations are what are communicated by the latter readings of the strings (1), (2), and (3), it turns out in the end that generic generalisations are universals.