Rumsfeld’s Logic of Known Knowns, Known Unknowns and Unknown Unknowns

 

On February the 12th, 2002, the then Secretary of Defense of the United States, Donald Rumsfeld, stated the following words (as captured in a YouTube video here):

“[A]s we know, 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.”

At first I wasn’t going to tackle this passage because some people may believe that I have some political sympathy for Donald Rumsfeld and what he said. However, since this passage was spoken some 18 years ago, and was spoken by someone who no longer has a prominent position in politics, I can’t see why that should be the case. Besides which, I shan't refer to the political context of Rumsfeld’s words at all (although I will offer a little background). Indeed I shall take his words as a short piece of logic and epistemology.

So it may seem odd — or perverse — too see these words as Rumsfeld’s attempt at logic and epistemology!… Actually, I don’t see it that way — at least not entirely.

In terms of at least a little context. Rumsfeld’s words were a response to a question about the lack of evidence linking Saddam Hussein’s regime in Iraq to the supply of “weapons of mass destruction” to various terrorist groups.

The passage above is almost always taken to be gobbledegook and/or political dissimulation. They were even awarded the Foot in Mouth Award. (Read the BBC on this here.) Indeed I’ve vague recollections of interpreting his words as political dissimulation and prevarication the first time I heard them.

Rumsfeld himself said that “the logic” of his words might have seemed “obscure” and “enigmatic”. Rumsfeld also mentioned Socrates:

“Some with an interest in philosophy have made note of a line attributed to Socrates: ‘I neither know nor think that I know.’ This has been interpreted to mean that the beginning of wisdom is the realization of how little one truly knows.”

Interestingly enough, even the Slovenian philosopher Slavoj Žižek couldn’t resist making exclusively political sense of Rumsfeld’s words. I mean that in the sense that Žižek didn’t really say anything about the purely logical force of the passage. (See Žižek’s article — from 2004 — here.) Of course there was no reason to take Rumsfeld’s words as being purely — and innocently — logical! That’s obviously the case. Then again, Žižek himself did — kinda — recognise the logic of Rumsfeld’s statements. That said, Žižek simply connected Rumsfeld’s logic to political deceit and hypocrisy. More specifically, Žižek happily accepted that there were “unknown unknowns” when it came to Saddam Hussein’s regime. However, although Žižek accepted Rumsfeld’s unknown unknowns, he also stressed the known knowns. (Žižek had the torture at Abu Ghraib in mind here.)

Having put Žižek’s negative response, there were also positive responses from some people. And, by that, I don’t mean positive responses which were purely motivated by politics. (They were political too.) For example, Mark Steyn called Rumsfeld’s words “a brilliant distillation of quite a complex matter”. In addition, Australian economist John Quiggin stated that “[a]lthough the language may be tortured, the basic point is both valid and important”..

I would simply say that the logic and epistemology underlying Rumsfeld’s words couldn’t help but be “tortured” — as my own commentary will show!

In any case, Rumsfeld’s words weren’t entirely original anyway. For example, the phrases “known unknowns” and “unknown unknowns” had already been used in strategic planning and project management. It can also be seen that they date back to 1997 (see here). (The phrase “known unknowns” has been used to refer to “risks you are aware of, such as cancelled flights”.) Rumsfeld himself wrote:

“I first heard a variant of the phrase ‘known unknowns’ in a discussion with former NASA administrator William R. Graham, when we served together on the Ballistic Missile Threat Commission in the late 1990s.”

Indeed my bet is that these phrases can probably be found many times before the 1990s. And, as I’ll attempt to show, they’ve also been featured many times in logic and philosophy — even if not in the precise way in which Rumsfeld expressed them!

But what of Rumsfeld’s peculiar way of expressing them?

The main reason that Rumsfeld’s words were seen as gobbledegook (rather than as pure political dissimulation) was the repeated use of the word “know” and its derivatives. That is, we have such phrases as “know knowns”, “know we know”, “known unknowns” and “unknown unknowns”. However, this is very similar to such well-known phrases as “love to love”, “the death of death”, “the end of the end”, “truer than true”, “bigger than big”, “life in life”, etc.

Again, at first glance, Rumsfeld’s words seem to either be gobbledegook or political dissimulation — or both! Yet they also make logical sense. This means that his words may be logical and an expression of (context-based) political dissimulation at one and the same time! Added to that is the fact that Rumsfeld’s words are… well, somewhat poetic. Indeed they’re almost like a Zen koan.

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So let’s simplify Donald Rumsfeld’s words:

As we know, 1) There are known knowns. There are things we know we know. 2) We also know there are some things we do not know. 3) But there are also unknown unknowns - The ones we don’t know we don’t know.

And all the above can be pared down further in this way:

(1) known knowns (2) known unknowns (3) unknown unknowns

Now let’s take one statement at a time:

(1) Known Knowns

Rumsfeld said that “[k]nown knowns are facts, rules, and laws that we know with certainty”. For example, “[w]e know [that] gravity is what makes an object fall to the ground”.

(1) is actually fairly problematic. It immediately hints at a possible infinite regress which has be found many times in various parts of philosophy.

Let me explain.

If we know p, and also know that we know p, then do we also need to know that we know that we know p? (This is very Rumsfeldian in tone.) And so on. What’s more, if we can stop at knowing that we know p (i.e., we don’t need to concern ourselves with knowing that we know that we know p), then perhaps we can also cut this regress short by simply settling for knowing p. Again, why do we also need to know that we know p?

At a more basic level, we can provisionally accept that there are “known knowns”. Generally, people accept that we know that we know that, say, the sun is the center of the solar system. But, again, what does knowing that we know p add to simply knowing p? What is it, exactly, to know that we know any given p?

In addition, if there can be “known knowns”, then what sense can we make of subject S not knowing that he knows p? That is, can S know p without also knowing that he knows p? If S can know p without also knowing that he knows p, then the opening clause “knowing that” (in “knowing that I know p”) may be redundant. In other words, why not stick with “S knows p”?

This is somewhat like an epistemic version of the redundancy theory of truth.

Take this compound sentence:

The sentence “Snow is white” is true.

Some philosophers have argued that the clause “is true” is “redundant”. (It may still have pragmatic force.) Similarly, in the sentence

“I know that I know that snow is white.”

the clause “I know that” may also be redundant. (Perhaps that too only has pragmatic force.)


Something similar to this epistemic regress is found in logic.

Take Lewis Carrol’s premises paradox.

In this paradox the sceptic demands a justification of the premises which lead to a specific conclusion (i.e., in a logical argument). More tellingly, the sceptic also requires a justification of the inferential links between the premises themselves, not just a justification of the premises.

Yet if such justifications were given of the premises, then these justifications would also need justifications too. Indeed this would also apply to the justifications of the links between the premises and between all the premises and the conclusion. And then those justifications would themselves require their own higher-order justifications.

The solution to this (as with simply accepting the statement “S knows that p”) is simply to argue that the way the premises lead to a conclusion simply doesn’t need a justification. All justifications are contained within the terms and statements used in the argument; as well as in the “logical rules” implicitly used. In other words, the logical argument must stand on its own if we’re to avoid an infinite regress.

(2) Known Unknowns

Rumsfeld stated that “[k]nown unknowns are gaps in our knowledge, but they are gaps that we know exist”. He continued:

“If we ask the right questions we can potentially fill this gap in our knowledge, eventually making it a known known.”

This makes sense at the same time as being problematic.

For example, we know that there are aspects of distant galaxies that we don’t know anything about. More mundanely, we know that we don’t know how many grains of sand there are on Earth. (Surely there must be an exact number.) We also know that we don’t know how many times King Henry VIII farted in his entire lifetime. (He must have farted a given number of times in his lifetime.)

However, there’s something odd about knowing that we don’t know any given x. In order to know that we don’t know about any given x, we must at least know something about that x in order to so much as mention it. We don’t know, then, the exact number of grains of sands on the Earth (or how many times King Henry VIII farted in his lifetime); but we do know that there must be a specific number.

This means that talking about “unknowns” is fine. However, also talking about “known unknowns” is problematic.

On a different tack.

The British philosopher Colin McGinn strongly claims that we will never know certain “deep truths” about consciousness (see here). But how does he know that? How does McGinn know that we will never know these deep truth about consciousness? Surely, in order to know that we don’t know anything about any given x, then that implies that we must at least know something about that given x. Of course it’s true that knowing something about x isn’t the same as knowing everything about x. However, we still know something about x. And if we know something about x, then how can we rule out our knowing everything — or at least much more — about x? So, in McGinn’s case, there are indeed (to go back to Rumsfeld’s words) “some things we do not know” about consciousness. However, can we also conclude that we will never know the deep truths about consciousness?

(3) Unknown Unknowns

Rumsfeld says that the “category of unknown unknowns is the most difficult to grasp”. Moreover, “[t]hey are gaps in our knowledge, but gaps that we don’t know exist”. He continued:

“There are many things of which we are completely unaware — in fact, there are things of which we are so unaware, we don’t even know we are unaware of them.”

Surely it’s a little problematic to say that there is a “gap[] in our knowledge” if, ostensibly, we don’t know anything about that gap or even anything about the subject of that gap. Technically, a human subject can neither know nor not know something about that which he doesn’t know about. More prosaically, a subject can’t have a position on some subject he’s never even heard of. Perhaps such an epistemic situation can’t even be described as being a lack of knowledge in that there are infinite things which any given subject will not — and cannot — know about. So, again, this is hardly an epistemological deficit.

(3) above is also Rumsfeld’s cute distinction between our not knowing about (to use my earlier examples) the supposed deep truths of consciousness, how many grains of sand there are on Earth, and how many times King Henry VIII farted in his lifetime, and our not knowing about things we don’t even know we don’t know about. (Here again things are getting very Rumsfeldian.) In Rumsfeld’s own words, these are “things of which we are so unaware, we don’t even know we are unaware of them”.

In these cases of unknown unknowns, it’s simply impossible to give any examples. No examples can be given of things we know we don’t know about. That’s because if any examples were given, then that would betray the fact that we — at the least — know something about the things we don’t know everything about. However, in Rumsfeld’s case, we’re supposed to be talking about “unknown unknowns”. That is, things “we don’t know we don’t know”.

So can’t we ask how we know that there are some things “we do not know”? To know that there are unknowns in any given area (or even unknown generally) hints at the fact that we are at the least (metaphorically) on the periphery of those unknowns. In other words, what are these unknown things we’re referring to?

So is this situation a little like Plato’s beard?

In the Plato’s beard analogy (as expressed by W.V.O Quine), the argument (at its most basic) is that the very mention of some x which is supposed not to exist confers some kind of “being” on it. Thus Pegasus, nothing and even the round square must have some kind of being. Why? Because we refer to these… things.

So if we use Plato’s beard and reapply it (if in a loose way) to this logical and epistemological case, then the very mention of unknown unknowns hints that we at least know that these things are unknown — therefore we know at least one thing about them. (Socrates knew at least one thing — that he “knew nothing”.)

To sum up.

It’s not that we need to know the things we don’t know because then we’d know them. However, do we know that (to get back to Rumsfeld’s words) “there are some things we do not know”? It is very likely that we don’t know many things. However, do we also know that we don’t know many things? How could we know that? To repeat: it’s highly probable that we don’t know many things. However, can we also know that we don’t know many things?

Professor E. Brian Davies’s Mathematical Empiricism

 

Edward Brian Davies was born in 1944. He is a Fellow of the Royal Society and was Professor of Mathematics at King’s College London (1981–2010). Davies has written papers on spectral theory, non-self-adjoint operators, operator theory/functional analysis, elliptic partial differential operators, Schrodinger operators (in quantum theory) and so on. He was awarded a Gauss Lectureship by the German Mathematical Society in 2010.

This commentary is on the relevant parts of Davies’s book, Science in the Looking Glass. The following is not a book review.

A Short Introduction to Mathematical Empiricism

E. Brian Davies’s own position on mathematics (or at least on numbers) is generally called mathematical empiricism.

The following short introduction is a basic account of such a position as it relates specifically to what Davies has to say on the subject. Nonetheless, this isn’t to say that Davies would endorse everything in this account of empiricism and its relation to mathematics.

Mathematical empiricism has it (in very broad terms) that mathematics simply can’t be known a priori. (Something that is known a priori is something that can be known by “reason alone”.) As Davies will state later (if very broadly and in a slightly different way), the mathematical empiricist believes that “mathematical facts” are discovered by empirical research. Indeed this position can be traced back to the philosopher John Stuart Mill in the 19th century — and probably before him. Davies’s own position is close to Mill’s in that the latter believed that the empirical justification for mathematics (or at least for certain types of number) comes directly from empirical objects.

In more concrete terms, “quasi-empiricists” argue that when doing their research, mathematicians “test hypotheses”; as well as prove theorems.

Mathematical empiricism can again be found in the 20th century in the works of (among many others) W. V. O. Quine and Hilary Putnam .

Without going into great detail, the most obvious argument against mathematical empiricism is that this position must have in it that literally all the results (or theorems) of mathematics are as fallible as the results in the empirical sciences. Or, alternatively put, that mathematical results are always contingent and never necessary. Of course this position is — many would argue — hugely problematic and that’s obviously so (or so it would seem). However, this isn’t the place to go into detail on this specific issue. (See the short note at the end of this piece.)

E. Brian Davies’s Mathematical Empiricism

At first glance it’s difficult to see how mathematics generally, and numbers specifically, have anything to do with what philosophers call “the empirical” (This is obviously the case for the mathematicians and philosophers whom call themselves Platonists). Nonetheless, most people are aware of the fact that mathematics is applied to the world and is an extremely useful — indeed necessary — tool for describing empirical reality. However, empiricists go one step further than this by arguing that mathematics itself is empirical in nature. Or they argue - at the least in E. Brian Davies’s case — that certain types of real number have an empirical status.

Quickly, it’s worth pointing out here that one can make a distinction between two things when it comes to empiricism and mathematics:

1. Having an empiricist (philosophical) position on mathematics itself.
2. Making one’s own philosophical empiricism more scientific by making it more mathematical.

Although these are two different positions, it can be said that both apply to Davies’s own position.

Small Real Numbers

E. Brian Davies puts his position at its most simple when he says that for a “‘counting’ number its truth is simply a matter of observation”. Here there’s a reference to “counting”; which is a psychological (or cognitive) phenomenon. By inference, it also refers to what we count. And what we count are empirical objects (i.e., objects we can experience using our sense organs). That means that empirical objects need to be experienced (or observed) in the psychological (or cognitive) act of counting.

Prima facie, it’s hard to know what Davies means when he writes that

“[s]mall numbers have strong empirical support but huge numbers do not”.

Even if that means that we can count empirical objects easily enough with numbers, does that alone give small numbers “strong empirical support”? Perhaps we’re still talking about two completely different (or separate) things: small numbers and empirical objects. Simply because numbers can be used to count objects, does that (on its own) confer some kind of empirical reality on such numbers? We’re obviously justified in using numbers for counting objects; though that may just be a matter of usefulness. Again, do small numbers themselves have the empirical nature of objects (as it were) passed onto them simply by virtue of their being used in acts of counting?

Davies then mentions Peano’s axioms in this respect.

Surely small numbers existed before “assenting to Peano’s axioms”.

Davies believes that by accepting such axioms we then have the means to create, produce or construct small numbers. That is, we firstly take the axioms; and then we derive all the small numbers from them. However, before the creation of these axioms, and the subsequent generation of small numbers as theorems, didn’t the small numbers already exist? A realist (or Platonist) would say, “Yes”. A constructivist (of some kind) would say, “No”.

Davies appears to put a set-theoretic position on numbers in that he tells us that “‘counting’ numbers [do] exist in some sense”. (I say set-theoretic in the sense that the nature of each number is determined by its one-to-one correspondence — i.e., bijection - with other members of other sets.) What sense? In the sense that

“we can point to many different collections of (say) ten objects, and see that they have something in common”.

Prima facie, I can’t see how numbers suddenly spring into existence simply because we count the members of one set and them put the members of other equal-membered sets in a relation of one-to-one correspondence with the original set. In other words, how numbers are used can’t give them an empirical status. Something is used. However, does that use entirely determine the metaphysical nature of (small) numbers? (We use pens; though the use of a pen and the pen itself are two different things.)

In any case, what these “collections” have in common (according to Davies) is that we can “see” their equivalences. So do we also see the relevant numbers rather than count them?

How do we count without using numbers? That is, even if there are equivalence classes, are numbers still surreptitiously used in the very definition of numbers?

Davies then goes on to argue a case for the empirical reality of small real numbers. There is a logical problem here, which he faces.

Davies offers a (kind of) numerical version of the sorites paradox for vague concepts (or even vague objects). Let me put his position in the following logical form; using Davies’s own words:

1. “If one is prepared to admit that 3 exists independently of human society.”
2. “Then by adding 1 to it one must believe that 4 exists independently…
3. “[Therefore] the number 1010100 must exist independently.”

This would work better if Davies hadn’t used the clause “exists independently of human society”. I say that because it’s empirically possible (or psychologically possible) that there must be a finite limit to human counting-processes. Thus counting to 4 is no problem. However, according to Davies, counting to 1010100 may not be something “human society” can do. Yet 1010100 exists even though Davies believes that mathematics tells us that

“It is not physically possible to continue repeatedly the argument in the manner stated until one reaches the number 1010100”.

Extremely Large Real Numbers

Davies begins his case for what he calls the “metaphysical” or “questionable” nature of extremely large numbers by saying that they “never refer to counting procedures”. Instead,

“they arise when one makes measurements and then infers approximate values for the numbers”.

The basic idea (again) is that there must be some kind of one-to-one correlation (or correspondence) between real numbers and empirical objects. If this isn’t forthcoming, then certain real numbers have a “questionable” (or “metaphysical”) status.

From his position on small numbers, Davies also concludes that “huge numbers have only metaphysical status”.

I don’t really understand this.

Which position in metaphysics is Davies referring to? His use of the words “metaphysical status” makes it seem like some kind of synonym for “lesser status”. However, everything has some kind of metaphysical status — from coffee cups to atoms. Numbers do as well. So it makes no sense to say that “huge numbers have only metaphysical status” until you define what status that is within metaphysics. Perhaps the statement should be: “Huge numbers only have a … metaphysical status.” In that statement, the three dots should be filled with some kind of position (or “mode of being”) within metaphysics.

Davies goes on to say similar things about “extremely small real numbers” which “have the same questionable status as extremely big ones”. I said earlier that the word “metaphysical status” (within this context) seems as if it is some kind of synonym for “lesser status”. That conclusion is backed up when Davies also uses the phrase “questionable status”. Thus a metaphysical status is also a questionable status. 

Nonetheless, I still can’t see how the words “metaphysical status” can be used in this way. Despite that, I’m happier with the latter locution (i.e., “extremely small real numbers have the same questionable status as extremely big ones”), than I am with the former (i.e., “huge numbers have only metaphysical status”).

Since Davies believes that there must be some kind of relation (or correspondence) between real numbers and empirical things (or objects), he also sees a problem with extremely small real numbers. Davies argues that physicists or philosophers may attempt to set up a relation between extremely small numbers and “lengths far smaller than the Planck length”. Thus the idea would be that Planck lengths divide up single empirical objects. Small numbers, therefore, correlate with individual empirical objects; whereas extremely small numbers correlate with the various Planck lengths of an object (i.e., rather than with objects in the plural).

Yet Davies doesn’t believe that this approach works.

He argues that this is because Planck lengths “have no physical meaning anyway”. By inference, this also means that extremely small numbers don’t have any empirical support. In other words, they have either a “questionable status” or a “metaphysical status” (perhaps both).

Models, Real Numbers and the External World

Davies’s more general position is that

“real numbers were devised by us to help us to construct models of the external world”.

As stated earlier, does this mean that numbers gain their empirical status simply because they’re “used to help us construct models of the external world”? However, even though real numbers are used in this way, that still may not give them an empirical status. Can’t numbers be abstract objects and still have a role to play in constructing models of the external world? (There is the problem — among other problems - of our causal interaction with abstract non-spatiotemporal numbers or objects.)

In terms of a vague analogy. We use cutlery to eat our breakfast. Yet breakfast and cutlery are completely different things. Nevertheless, they’re both (as empirical objects) in the same ballpark. What about using a pen to write about an event in history? A pen is an empirical object. What about an historical event? We can say that the pen which writes about an historical event exists. Can we also say the same about the historical event itself? Yet there’s still a relation between what the pen does and a historical event even though they have two very different metaphysical natures.

Non-physicists may also want to know how real numbers “help us to construct models of the external world”. Are the models literally made up of real numbers? If the answer is “Yes”, then what does that actually mean? Do real numbers help us measure the external world via the use of models? That is, do the numbered relations of a model match the unnumbered relations of an object (or bit of the external world)? Or do numbers actually (metaphorically?) belong to the external world just as much as they belong to the (mathematical) models we have of the external world? In other words, is the world already numerical (i.e., as in the Pythagorean position in which “all is number”)? Have we the philosophical right to say of the studied objects (or bits of the world) what we also say of the models of those studied objects (or bits of the world)?

Conclusion

E. Brian Davies puts the (or his) empiricist position on mathematics at its broadest by referring to philosophers and mathematicians whom he sees as being fellow empiricists. He cites John von Neumann, W.V.O. Quine, Alonzo Church and Hermann Weyl. These mathematicians and philosophers “accepted that mathematics should be regarded as semi-empirical science”. Of course saying that maths is “semi-” anything is open to many interpretations. Nonetheless, what Davies says about real numbers above (at least in part) clarifies this position.

Davies then brings the debate up to date when he tells us that contemporary mathematicians are “[c]ombining empirical methods with traditional proofs”. What’s more, “the empirical aspect [is often] leading the way”. Indeed Davies says that this empiricist position is “increasingly common even among pure mathematicians”.

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Note:

The bald empiricist position on mathematics — or on numbers — is very easy to attack.

So just to take one example (which was aimed at J.S. Mill). The English philosopher A.J. Ayer (who was himself an empiricist… of sorts) stated (in his Language, Truth, and Logic) that if we took an empiricist position on mathematics (or specifically on arithmetic) itself, then the statement “2 + 2 = 4” would need to be seen as contingent and even uncertain. Indeed we’d also need to rely on observing two pairs coming together in order to formulate the statement in the first place.

This isn’t to say that empiricists don’t have counter-arguments to this. After all, once two pairs coming together (as it were) has been observed (either collectively or by an individual), then that doesn’t mean that all further uses of arithmetic require that we keep on matching empirical objects together in such a manner. So, sure, an abacus (or an empirical equivalent) may be psychologically and historically important for (most people) when it comes to learning arithmetic. However, that’s a fact about the human process of learning arithmetic: it’s not a fact about arithmetic itself.