Double negatives: the big picture

In yesterday’s installment, the two kids of the Lombard family in the comic strip One Big Happy, Ruthie and Joe, advance a devious — and transparently malicious — idea about the pragmatics of conversation. As a slogan,

Two nasties make a nice.

That is, saying two nasty things about someone counts as saying a nice thing about them, yuk yuk. We-e-ell, the kids maintain, with impish speciousness, that that’s just a special case of the general principle that

Two negatives make a positive.

First thing: such a slogan is a highly abbreviated formula in ordinary language of some significant technical principle, the virtue of the slogan being that it is striking and memorable; it’s an aide-memoire. But it’s just a label, and labels are not definitions.

Second thing: the kids’ version exploits a massive ambiguity in the adjectives negative / positive, and a corresponding ambiguity in the verb make. To which I now turn.

To recap. Yesterday’s posting “Double negatives”, had two strips:

— in one, Joe, exhorted to “say one positive thing about your sister”, rails at her: “You’re a pain in the neck and you have stinky feet”, defending himself to his mother by asserting confidently, “Dad said two negatives make a positive” (Ruthie sticks her tongue out at him)

— in a later strip, Ruthie offers, as an example of a double negative: “Joe is short and funny-looking”

The sense of positive (and the contrasting negative) here, from OED3 (Dec. 2006):

A. adjective uses II. senses opposed to negative
4. d. Consisting in or characterized by constructive action or attitudes; inclined to hope for the best or to ‘look on the bright side’, optimistic; good, beneficial, advantageous.

The objects that are said to be positive or negative in the Lombard household are linguistic expressions, as used to attribute characteristics — favorable or unfavorable to people. These linguistic expressions can be combined into larger linguistic  expressions by juxtaposition. The Lombard kids cling, maliciously, to the preposterous idea that juxtaposing two unfavorable expressions yields a favorable one.

Meanwhile, in the world of logic. The objects at issue are logical operators, in particular the contradictory-forming negation ¬, and the mode of combination is the application of an operator O2 to the output of another, O1:

O2(O1(p)), abbreviable as O2 O1 p, when no confusion would result; where p is a proposition (an abstract object that is the bearer of truth and falsity)

Including the application of an operator to the output of the same operator:

O(O(p)), or simply O O p — for instance ¬ ¬ p

And now from the Stanford Encyclopedia of Philosophy article on negation by Larry Horn, section 1.8 on double negation, beginning with

contradictory-forming negation (It is not the case that they are happy, They are not happy; logical operator ¬) vs. contrary-forming negation (They are unhappy; logical operator @). The Law of Double Negation — ¬ ¬ p ≡ p — applies only to ¬ . [AZ: ≡ is logical equivalence]

In ordinary language, semantic double negation (as opposed to negative concord as in I ain’t never done nothing to nobody, an agreement phenomenon in which only one semantic negation is expressed …) tends not to cancel out completely. This is predictably the case when a semantic contrary is negated: not uncommon is weaker than common; one can be not unhappy without being happy. But even when an apparently contradictory negation is negated (from the unexceptionable it’s not impossible to the more unusual double-not of Homer Simpson’s concessive I’m not not licking toads …), the duplex negatio of A doesn’t affirm A, or at least it provides a rhetorically welcome concealment, as suggested by Frege’s metaphor of “wrapping up a thought” in double negation. The negation in such cases (impossiblenot-licking) is coerced into a virtual contrary whose negation, ¬©A, is weaker than (is unilaterally entailed by) A

The syntax of negation. It should be obvious by now that the distribution of negative elements in the syntax of actual languages  is far from a mirror of logical formulae. The divergence is especially notable in language varieties exhibiting negative concord, a spreading of negative elements throughout a negated clause — notable in many non-standard varieties of English, but also a feature of many standard languages.

So it is that in non-standard English I didn’t see nobody is logically equivalent to standard English I didn’t see anybody (and not to I saw somebody). The facts of usage are much more complex than this, but here two negatives just make a negative.

The general case. (Using two examples from my earlier discussion:

— a domain D of entities (logical operators; linguistic expressions)

— properties negative and positive defined on those entities

— and a mode M of combination of two entities to yield a third (composition of logical operators; juxtaposition of linguistic expressions)

We can then ask about the properties of M combining d1 and d2 from D, where d1 and d2 are both negative. For logical operators, in the special case where d1 and d2 are both (negative) ¬ , their combination via M is positive: two negatives make a positive. For linguistic expressions, if the positive vs. negative properties are favorable vs. unfavorable attribution, their combination via M is negative: two negatives make a negative (two nasties make a nasty, whatever the Lombard kids would like to imagine).

Thanks to the huge number of senses of positive and negative, and an accompanying multiplicity of ways of combining entities that are either positive or negative, there are quite a few other examples we can play with. Just a couple of examples.

On to arithmetic. First, the relevant sense of positive (and, correspondingly, negative) on numbers: greater than zero (vs. less than zero). There are a variety of ways to combine two numbers, but the simplest of these are by addition and by multiplication.

So, what happens when we we combine two negative numbers in these ways? The sum of two negative numbers is negative (two negatives make a negative); but the product of two negative numbers is positive (two negatives make a positive).

And photography. A positive photographic image shows the lights and shadows as in the scene photographed, while a negative image has light and dark reversed. If two images are combined by juxtaposition, then the combination of two negative images is of course another (two-part) negative image. Such combinations are not common, but they’re attested. From the Etsy seller RadicalBohemian:

Soldier saluting in uniform, two images on one antique glass plate negative (circa late-1800s)

Two negatives make a negative.

But enough for today.

5 Responses to “Double negatives: the big picture”

  1. J B Levin Says:

    The last pair of images display another characteristic not mentioned, not relevant to your topic, but easy for me to enjoy with my poorly-tracking eyes: they are a stereoscopic pair. Now that I examine them closely I can see differences between the two images; but viewed together, one per eye, the effect is marked.

  2. Mitch4 Says:

    Traditional dialogue:

    — Two wrongs do not make a right!
    — No, but sometimes three rights make a left!

  3. Mark Mandel Says:

    And two Wrights made an airplane.

    Clearly, the Lombard kids are illogical operators.

  4. MWarhol Says:

    A multiple negation overheard, a salesman complaining that a prospective customer had not purchased a single one of the items that he had for sale:

    “I ain’t never had nobody not buy nothin’ before!”

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