Tag Archives: logic

Unknown knowns

It’s opening the door to meet a stranger and realizing you know the person already. It’s sitting down to do a thing for the first time and finding that somehow you know how to do it. It’s trying long and hard to figure something out and at last realizing that you always knew the answer, but just didn’t know you knew. Sometimes it’s because you didn’t have the chance to see that you had been seeing it; sometimes it’s because you had chosen not to see it sooner.

What am I going on about? Let’s look at what can be called a Rumsfeld square or Rumsfeld matrix. It’s named after Donald Rumsfeld, who famously said, “there are known knowns; there are things we know we know. We also know there are known unknowns; that is to say we know there are some things we do not know. But there are also unknown unknowns—the ones we don’t know we don’t know.” (Some people have characterized this as incomprehensible, but I have no idea what they’re talking about.) If we diagram it out in Greimas style, we see there’s a fourth square he doesn’t mention:

The existence of known knowns, known unknowns, and unknown unknowns implies the existence of unknown knowns. It’s right there. Yet people don’t seem to talk about them* – perhaps that’s why they’re unknown.

But the fact we have a space for them in the matrix doesn’t mean that they actually exist. Putting things into tidy diagrams and taxonomies can be rewarding, but it doesn’t necessarily add information any more than organizing your bookshelf adds information about what’s in the books. Mainly, it tells you about how your mind – and the structures it has learned – views and organizes things. But inasmuch as it describes aspects of reality, it may also be a heuristic for discovering things we aren’t aware of yet, or at least for knowing where to look for them – or, as the case may be, for becoming consciously aware of them.

Linguistics is a great place to look for this kind of example, and I mean that in several ways. Here’s a table of the International Phonetic Alphabet symbols for consonants (thanks to Kwamikagami on Wikimedia Commons):

I won’t explain all the terminology because we don’t have all night. But you’ll see that this chart has grey areas: those are sounds that are considered impossible. A pharyngeal lateral fricative, for instance, would require having your something like the tip of your tongue stuck back deep in your throat – which, even if it were physically possible, would stimulate your gag reflex. So the grey areas help us confirm aspects of reality. They also force us to clearly define what we mean by our terms. For instance, “lateral” it means going around the side(s) of the tongue, like the sound [l]; a labiodental lateral fricative is impossible because it would require going around the sides of your tongue without using your tongue, because “labiodental” means using a lip and the teeth (and not the tongue). If “lateral” could mean the side of something other than the tongue, such as air going out the sides of the mouth while biting your lip, the labiodental lateral fricative box would not be grey.

But it still might be empty, like the labiodental trill, which is considered possible (I think it would take practice!) but no known language uses it as a speech sound. But look right above that empty box, and you’ll see the symbol ⱱ, which represents a labiodental flap: a sound you make by flipping your lower lip out brushing it past your upper teeth. That box was empty until fairly recently, when the people who agree on the chart were made aware of an African language that uses the sound as a distinct speech sound. 

In a way, an empty box is a challenge to fill it – just as a grey area is a challenge to prove it wrong, or to scrutinize the definitions. So these taxonomies help turn unknown unknowns into known unknowns, and sometimes eventually to known knowns, and they also help us understand how we know – but they don’t produce the data themselves; you still have to go find speakers of real languages for that. 

But our choice of what to include in the grid – what questions to ask – can still leave unknown unknowns unknown and unknown, and it can also divert attention away from known knowns. For example, there is no column for linguolabials (tongue and lip), which is altogether possible (you could do it right now: touch your tongue to your upper lip). In this case, it’s not because they’re judged impossible, nonexistent, or unimportant; it’s just that they’re treated as variants on bilabials and coronals, and so are represented with a mark under a letter: [n̼] and [l̼], for instance. They’re known knowns but more easily overlooked because they’re not given equal weight in the taxonomy – which effectively makes them lesser-known knowns.

But there’s one more thing that linguistics tells us about all of this. Every linguistics student comes in feeling sure they know all about the sounds we make in our mouths and how we make them, and every linguistics student comes to realize they were doing things they weren’t aware that they were doing. For instance, before I encountered phonetics, I knew how to say “pot” and “spot” like a usual English speaker, but I didn’t realize that, like a usual English speaker, I was making a puff of air after the p in “pot” but not so much after the sp in “spot.” Every introductory linguistics course has one class where the students are all holding their hands, or pieces of paper, in front of their mouths and discovering that they’re doing something that they know to do but don’t know they’re doing. This is was what is often called tacit knowledge.

In fact, most of language is tacit knowledge for most speakers. We know how to put together a sentence, but we aren’t really aware of how we do it or why some things sound right and others sound wrong. And we learn rules in school and think that anything that doesn’t follow those rules doesn’t follow any rules, when in fact “nonstandard” varieties of languages have grammars that are every bit as developed and constraining as “standard” varieties. 

Some of the things we learn don’t just block knowledge, they put in false belief in place of accurate knowledge. If you say “doin’” instead of “doing,” for instance, we typically say you’re “dropping the g,” but there is no g. The difference between those two consonant sounds – [n] versus [ŋ] – is only a matter of where they’re said in the mouth; it just happens that we don’t have a separate letter for [ŋ] so we write it as ng, which also sometimes stands for [ŋg].

That’s not just an unknown known; it adds a whole new dimension to the Rumsfeld square. As the popular saying goes (seen in many versions, and often inaccurately attributed to Mark Twain), “It ain’t so much the things that people don’t know that makes trouble in this world, as it is the things that people know that ain’t so.” But I’m not going to redraw the diagram with veracity versus delusion as another dimension right here and now, because that would be a very mentally taxing digression.

Beyond linguistics, and beyond tacit knowledge and knowledge blocked by falsehood, there are also other unknown knowns in life. I think of the time more than 20 years ago when I was looking for a job and a friend said a friend of his needed someone to do some proofing corrections on HTML. So I phoned the friend-of-a-friend and we chatted a bit and he told me to come in. When I walked in the door to the office, this person I had chatted with already on the phone looked at me, and I looked at him, and we both said, “I know you!” We had had a great long conversation at our mutual friend’s place at a party some months earlier, but had not learned each other’s full name, so when we were talking on the phone we didn’t know we knew each other. The knowledge wasn’t tacit, and it wasn’t blocked; it was transiently and accidentally obscured. (He remains one of my closest friends.)

And then there’s the Karate Kid kind of moment. In the first Karate Kid movie, the hero wants to learn karate, so he apprentices himself to an old Japanese man who makes him do menial tasks such as washing and waxing the car according to very specific instructions and painting the fence with exact strokes. When at length the hero complains that he hasn’t learned anything and is just being used for free labour, the master throws some punches at him, which by reflex he blocks using the muscle moves he had internalized through doing the scut work. He knew how to do it, but he didn’t know he knew. This, too, is tacit knowledge, but not one that had already been demonstrated, like speech knowledge; it was first manifest at the point of awareness.

That’s also how I learned how to do structural editing. I picked it up through researching and writing essays and through evaluating and grading very large numbers of student essays as a grad student and instructor. I wasn’t fully conscious of the sense of flow and structure and the intuitions I was developing, but when I first sat down to actually edit articles and books, I realized that I knew how to do something I hadn’t known I knew. Of course, once it’s a known known, it can be further developed – but I have to watch out that I don’t start misleading myself into “knowing” things that aren’t so!

And then there’s the ultimate unknown known: the “enlightenment” (satori, kensho) of Zen practice. If my sense of it from accounts I have read is accurate, it involves seeing the world and realizing that you always knew its true nature, but you just didn’t know you knew… because you were too busy putting it into boxes and matrices and categories and words. Which reminds us again that while logical deductions and categorizations can lead us to discoveries, they can also lead us away from them.

Unknown knowns are some of life’s greatest pleasures, its greatest serendipities. There are also, yes, great discoveries of unknown things you either suspected might exist (as a known unknown) or had no idea would be there like that (unknown unknown). But as T.S. Eliot wrote in “Little Gidding,” 

We shall not cease from exploration 
And the end of all our exploring 
Will be to arrive where we started 
And know the place for the first time.

In an important way, our lives are a course of coming to know ourselves and our worlds – of coming to know the things we had always known but had not been aware we knew. The unknown knowns.

* Well, Slavoj Žižek has – he has used the term for “the disavowed beliefs, suppositions and obscene practices we pretend not to know about, even though they form the background of our public values.” These are not so much unknown as “unknown” – we agree to pretend not to know them so as to avoid cognitive dissonance. He was responding to Rumsfeld’s use of the idea of unknown unknowns to justify attacking Iraq.

Authority? What authority?

Look, I know what I’m talking about.

Have you ever said that? And has anyone ever said that to you? It’s an appeal to authority, and, according to some people, it’s an instant fail: the argumentum ab auctoritate (argument from authority) – a famous logical fallacy!

Except when it’s not. Because if appeals to authority were always fallacious, our entire legal and educational systems would be voided. Among many other things. 

I’ll explain.

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What’s logical about English?

A common complaint about English – by those who are inclined to complain about English – is that it’s not logical enough. Whatever that means. Words aren’t premises and sentences aren’t syllogisms, after all.

If you inspect the targets of their opprobrium, you find soon enough that what they mean is that English isn’t tidy enough for them. It’s inconsistent. Lacking in symmetry. Their experience has led them to believe that for every up there should be a down, for every in an out; when they see an over, they think “therefore under,” and if there is no under, they are… underwhelmed.

They’ve condemned themselves to a lifetime of disappointment. English does not satisfy their need for an overarching tidiness. It is not a Zen garden; it is a forested mountain, every tree grown unplanned in its place and conditions, every rock where the ineluctable complexities of physics left it. It is not an edifice of modernist design with proportions based on the golden mean; it is a Winchester House of a language, a veritable Heathrow Airport of accretions (for those who have not been to Heathrow, let me just say I suspect that J.K. Rowling based Hogwarts on it). Like any natural language, English has been built up by habit, need, association, and analogy. It does have structure – in fact, it has some inflexible syntactic requirements. We have slots to fill, and fill them we do. We just sometimes grab whatever’s ready to hand to fill them.

Let’s consider a few examples. One case where a desire for logic has actually prevailed is “double negatives.” Anyone who has studied logic will tell you that in “not not” the second not undoes the first one. “There will not be cake” is disappointing; “There will not not be cake” is affirming. Thus, the reasoning goes, “I do have nothing” and “I don’t have nothing” are opposite. But anyone who has learned a Romance language ought to know ça ne vaut rien, no vale nada – that ain’t worth nothing.

Nothing, you see, is not not. It’s a noun, not an operator. And one thing languages like is agreement. Concord. Adjectives tend to take the same gender as the nouns they modify, for instance. In English, we use concord with tenses in some contexts: “Should we expect them tomorrow?” “They said they weren’t coming.” Notice how we use weren’t even though we’re talking about the future? We even let negative concord pass unremarked in some contexts: “They won’t be coming, I don’t think.” This doesn’t mean I don’t think they won’t be coming; it just retains the negative aspect.

But since it’s possible, with shifting emphasis, to make “There ain’t no one here” and “There ain’t no one here” mean opposite things, an argument can be made for disallowing negative concord for the sake of unambiguity. So the proscription stuck, defended by pleas for logic – although “if negative noun, then negative verb” is perfectly reasonable if that’s the rule in the language.

Syntax has its requirements – as linguists would say, there are principles and parameters that specify how it functions in a given language. Negative concord is one parameter we have managed to turn off. Others are not so easily disabled. It’s necessary to have an explicit subject (except in imperatives), for instance; I can’t write “Is necessary to have an explicit subject,” so I stuff in an it that has no meaning. It may not seem logical to have a pronoun with no referent, but consider that, from the view of our syntax, “if it has a sentence then it has a subject” is solid. Sometimes we grab and stuff on the fly – we may jam a word in the place where a word like it normally goes, even if in this case it’s a whole nother thing and what even were we thinking? This, too, comes from a simple if-then – just a little simpler than it might have been.

Another plea for logic comes when a word is pressed into service in a way that seems untidy. One I saw recently was an objection to using disconnect as a noun, as in “There is a real disconnect between the labourers and the management.” We don’t say “There is a connect between them,” we say connection, so it’s illogical not to say disconnection. Indeed, this is untidy, in the same way as it’s untidy that when my wife is at home I heat two servings of food and pour two glasses of wine, but when she’s not at home I heat one serving and open a beer (or go out for sushi). But our little untidinesses have reasons: my wife doesn’t drink much beer and doesn’t like sushi. And disconnect is an allusive use borrowed from electronics and telephony.

A line of communication is expected to remain connected, so there is no instance where we would say that it has experienced a connect. We grabbed a bit and stuck it where it fit, and in so doing made a metaphorical connection. There’s no need to construct a symmetrical positive use any more than there is a need for a 33-storey building to have 33 levels of basement. And there’s no need to disallow allusions just for the sake of tidiness – we don’t forbid lights on Christmas trees just because there are none on the house plants. If you want to make a connection, you make it; if you don’t, you don’t. That’s logical, no?

Some people also like to laugh at how “illogical” English words are. “Why do our noses run and our feet smell? Why do we park in a driveway and drive in a parkway? Why do we say a bandage was wound around a wound? How come you can object to an object?” OK, now tell me why these are illogical.

Every one of them comes from a well-motivated historical development founded on consistent principles: metaphor, ergativity, historical sense developments and standard compounding rules, phonological shifts, stress-based differentiation of nouns from verbs. In every case there was an if-then judgement based on analogy. It just happened not to be exactly analogous to some other if-then judgements, and it produced results that seem inconsistent when juxtaposed. I think that’s fine – why not have funny things? But more than that, it’s not even illogical. In every case, we got to it from “if A → A´, then B → B´.” They just happened to be local judgements made in the context of a big, multifarious, inconsistent world.

But it would be illogical to treat a multifarious, inconsistent world as though it were elegant and pervasively consistent, wouldn’t it? It certainly wouldn’t be well adapted. It would be like laying down a strict grid street plan for a very hilly city (and San Francisco knows how well that worked out). It wouldn’t be as much fun, either. And it might do real harm.

This statement is false

Last weekend my brother and I were discussing the statement “This statement is false.” Today a colleague mentioned a similar statement, “The following statement is true. The previous statement is false.” Another colleague likened this kind of pure self-contradiction to the Cretan paradox, also known as the Epimenidean paradox: the statement “All Cretans are liars” said by a Cretan, which would seem to be a false if it’s true and true if it’s false.

But the difference between the Cretan paradox and pure self-contradiction is that the Cretan paradox has a real-world referent. It makes a statement about something external to the assertion. Pure self-contradiction has no real-world referent. It makes an assertion about nothing other than itself and thus has no truth value ascertainable.

As it happens, the source of the Cretan paradox is something Epimenides wrote in support of the immortality of Zeus:

They fashioned a tomb for thee, O holy and high one
The Cretans, always liars, evil beasts, idle bellies!
But thou art not dead: thou livest and abidest forever,
For in thee we live and move and have our being.

Epimenides was himself a Cretan. Thus we know through simple pragmatics that he must have been excluding himself without saying so. To treat it as a paradox is to be disingenuous. It’s fun sport, but in the end it just shows one of the things you can’t do in logical reasoning.

Statements such as the Cretan paradox are an illusion caused by conflation of one level of analysis with a higher level of analysis: an evaluation of the members of a set cannot itself be a member of the set evaluated; evaluation is a comparison of something against one or more criteria from an external perspective – what is being analysed is subsumed within its perspective. Once we acknowledge that the statement “All Cretans are liars” cannot be part of the set of statements evaluated (making it thus a simple problem in pragmatics rather than a trick of logic), we identify an unstated assumption that makes it function, without which we get a sort of Escher staircase illusion, something that can’t exist in the real world.

But with mutually evaluative statements such as the pure self-contradictions, each must be on an evaluative level above the other – each must subsume the other within its perspective. And at the same time each has no further reference; it has no claim to truth or falsehood as the set of all other statements by Cretans does (and as that set’s members individually do).

Analyzing an utterance or set of utterances is like weighing an object. In order to weigh an object, you have to lift it (or anyway support it) and you have to be resting on something that is not part of what you are weighing. In the Cretan paradox, we see that the statement that pretends to be part of the set of Cretan statements is actually weighing them and so cannot be part of them; it is evaluating them against their real-world references – that’s what it’s resting on. In the mutual contradiction case we’re looking at, each is weighing the other, and neither rests on anything else, because neither is being evaluated against anything external to itself. It’s like two dudes trying to lift each other simultaneously. In empty space.

Meaning in human communication, ultimately, is not a question first of all of logic; it is a question first of all of pragmatics. All communication is behaviour; when you utter something, you are doing something with the aim of producing a certain effect. The person hearing you will be conjecturing what effect you are trying to produce and responding accordingly. Logic helps serve this function, but pragmatics is the true basis. And the pragmatic value of things such as paradoxes is sport – mental play, fun. And a demonstration of the invalidity of certain kinds of reasoning.