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The impossibility of perfect forgeries?

Imagine that Banksy, (or J.S.G. Boggs, or some other artist whose name starts with “B”, and who is known for making fake money) creates a perfectly accurate counterfeit dollar bill – that is, he creates a piece of paper that is indistinguishable from actual dollar bills visually, chemically, and in every other relevant physical way. Imagine, further, that our artist looks at his creation and realizes that he has succeeded in creating a perfect forgery. There doesn’t seem to be anything mysterious about such a scenario at first glance – creating a perfect forgery, and knowing one has done so, although extremely difficult (and legally controversial), seems perfectly possible. But is it?

In order for an object to be a perfect forgery, it seems like two criteria must be met. First of all, the object must be a forgery – that is, the object cannot be a genuine instance of the category in question. In this case, our object, which we shall call X, must not be an actual dollar bill:

(1) X is not a dollar bill.

Second, the object must be perfect (as a forgery) – that is, it can’t be distinguished from actual instances of the category in question. We can express this thought as follows:

(2) We cannot know that X is not a dollar bill.

Now, there is nothing that prevents both (1) and (2) from being simultaneously true of some object X (say, our imagined fake dollar bill). But there is an obstacle that seemingly prevents us from knowing that both (1) and (2) are true – that is, from knowing that X is a perfect forgery.

Imagine that we know that (1) is true, and in addition we know that (2) is true. In other words, the following claims hold:

(3) We know that X is not a dollar bill.

(4) We know that we cannot know that X is not a dollar bill.

Knowledge is factive – in other words, if we know a claim is true, then that claim must, in fact, be true. Applying this to the case at hand, this means that claim (4) entails claim (2). But claim (2) and claim (3) are incompatible with each other: (2) says we cannot know that X isn’t a dollar, while (3) says we know it isn’t. Thus, (3) and (4) can’t both be true, since if they were, then a contradiction would also be true (and contradictions can’t be true).

‘Dollars’ by 401(K), 2012, CC-BY-SA 2.0 via Flickr.

Thus, we have proven that, although perfect forgeries might well be possible, we can never know, of a particular object, that it is a perfect forgery. But an important question remains: If this is right, then what, exactly, is going on in the story with which we began? How is it that our imagined artist doesn’t know that he has created a perfect forgery?

In order to answer this question, it will help to flesh out the story a bit more. So, once again imagine that our artist creates the piece of paper that is visually, chemically, and in every other physical way indistinguishable from a real dollar bill.  Call this Stage 1. Now, after admiring his work for a while, imagine that the artist then pulls eight genuine, mint-condition dollar bills out of his wallet, throws them on the table, and then places the forgery he created into the pile, shuffling and mixing until he can no longer identify which of the pieces of paper is the one he created, and which are the ones created by the Mint. Let’s call this Stage 2. How do Stage 1 and Stage 2 differ?

At Stage 1 we do not, strictly speaking, have a case of a perfect forgery. Although the piece of paper the artist created is physically indistinguishable from a dollar bill, the artist can nevertheless know it is not a dollar bill because he knows that he created this particular object. In other words, at Stage 1 he can tell that the forgery is a forgery because he knows the history, and in particular the origin, of the object in question.

Stage 2 is different, however. Now the fake is a perfect forgery, since it still isn’t a dollar, but we can’t know that it isn’t a dollar, since we can no longer distinguish it from the genuine dollars in the pile. So in some sense we know that the fake dollar in the pile is a perfect forgery. But we can’t point to any particular piece of paper and know that it, rather than one of the other eight pieces of paper, is the perfect forgery. In other words, in Stage 2 the following is true:

  • We know there is an object in the pile that is a perfect forgery.

But the following, initially similar looking claim, is false:

  • There is an object in the pile that we know is a perfect forgery.

We can sum all this up as follows: We can know that perfect forgeries exist – that is, we can know claims of the form “One of those is a perfect forgery”. But we can’t know, of a particular object, that it is a perfect forgery – that is, we can never know claims of the form “That is a perfect forgery”. And it is this latter sort of claim – that we know, of a particular object, that it is a perfect forgery – that leads to the contradiction.

Recent Comments

  1. SimpleSimon

    Prof: Check the bills in your wallet.
    You will see some of them have scribbles on them. The scribbles do not detract from the intrinsic worth off a bill yet they can identify a particular bill and mark it as distinct from all the others.

    Banksy scribbles his signature on the bill but can still use his perfect forgery to buy popcorn and pharmaceuticals at CVS.

    I once worked at a bookie’s so I know all about those mysterious scribbles you can sometimes find on an otherwise pristine bill.

  2. Richard Stevens

    SimpleSimon has the essence of why the article is incorrect.

    Another example – imagine that all proper dollar bills lie within a certain range (they cannot be identical). I make multiple dollar bills that are exactly identical (digital signature). When I mix the two, I can separate out all the forgeries because they are all identical, whereas the real ones are all slightly different.

    The professor needs a little less philosophy and a bit more number theory.

  3. Greg

    This article seems to conflate an object’s definition with the set of its physical attributes. What makes a dollar a dollar (and thus legal tender) isn’t just its arrangement of atoms; it must be printed by an agent with the proper legal authority. What makes art emotionally meaningful is often more than just the aesthetic; it’s the relationship between the artist and the world that’s represented iN the art. I think this makes it fairly easy to identify a “perfect” counterfeit as long as its provenance is known or demonstrable. If I watch you print a “dollar” that is physically indistinguishable from a “real” dollar, I can still objectively determine that your dollar is counterfeit because you are not the Bureau of Engraving and Printing.

  4. Dave

    The claim that “we can never know claims of the form ‘That is a perfect forgery’ ” would seem to me to be false or relying on artificially restricting the scope of the ‘we’ quantifier.

    Imagine an artist creates a dollar bill which is qualitatively indistinguishable from an actual dollar bill and then places it into one of two envelopes marked ‘A’ or ‘B’. He then takes a real dollar and places it into the other envelope. He then writes down what envelope the non-dollar is in and puts it in envelope ‘C’. He is then hit by a bus. To know what envelope contains the forgery all we have to do is open the third envelope. This isn’t dissimilar to your scenario 1 but I think it demonstrates a different point.

    To wit, I think a perfect forgery implies qualitative indetermination but not ignorance about its being a forgery. Meta-information about that causal history can be transmitted and, importantly, the causal history of the forgery is at least part of what makes it a forgery. Nonetheless, inspection of the properties of the object will not yield a determination as to which is a perfect forgery even when it can be known that something is a perfect forgery.

  5. Roy T Cook

    The first three comments miss the point I was trying to make (which doesn’t mean they aren’t making good points about the actual details of how currency, and forgery of currency, works).
    The point was meant to be this: It seems impossible, in a certain sense, to know we have made a perfect forgery of anything – that is, to make an object that we know is not an X, but where we also know that it is in every way indistinguishable from a genuine X. Of course it is the case that in real life there are all sorts of ways to distinguish real dollar bills from fakes, including marking the fakes with scribbles (SimpleSimon), or noting that they are similar to each other in a way that they are not similar to actual dollars (Richard), or by knowing its provenance (Greg) or marking its identity in some other way (Dave). The point I was trying to make with the example is that if the piece of paper is indistinguishable from real dollar bills in all of these ways – that is, it is a perfect forgery in the sense in which I introduced the term – then we can never know of that piece of paper that it is a perfect forgery.

    The point of the essay was meant to be a logical one, not a lesson about money. But perhaps I wasn’t clear about that.

  6. john schroeder

    Reminds me of the wine and poison scene from the Princess Bride!

  7. ac

    Your argument is circular. To break the circle one need only acknowledge that (2) does not apply to the forger himself. He knows it’s not a dollar bill.

  8. Dave

    I think your concept of forgery is doing all the heavy lifting in the argument. Forgery is a complex concept (relying as it does on concepts such as ‘original’ and ‘sanctioned instance’ which are hard to pin down). If Da Vinci had made two qualitatively identical versions of the Mona Lisa we would say they are copies but if I make a qualitatively identical version it is a forgery (making it even more complicated, it is probably only a forgery if I try to pass it off as Da Vinci’s I bet). Furthermore, some things probably can’t be forged (if I make a qualitatively identical copy of the fork on my desk, I just have two forks, not an original fork and a fork forgery).

    I think a plausible concept of a perfect forgery is that A is a forgery of B if, with the exception of causal-historical-relational properties, A is qualitatively identical to B and B is the kind of the thing that can be forged (whatever that means). This account of forgery is plausible for a blog post but does not have the epistemic implication that drives the rest of your argument.

  9. Faraday

    For those who are citing provenance, context, facts about historical origin, and the like: that’s exactly what’s cool about this paradox. Take the following statements:

    1. It is possible to know that an object is a forgery, if it is a forgery.

    2. If (1) is true, there must be some KNOWABLE characteristic that distinguishes a forgery from the real thing.

    3. In many cases of perfect forgeries, the historical or social circumstances of the origin of the object can be rendered UNKNOWABLE, as in the case of shuffling the bills in your pocket in the article.

    4. In such cases, the real thing and the forgery become indistinguishable, because the distinguishing characteristics have become unknowable.

    5. Thus, in many cases of perfect forgeries, it is knowable that a particular object is a perfect forgery, and it is unknowable that that object is a perfect forgery.

    You could try to resolve the paradox by saying that it’s still possible to tell a story about the forgery that distinguishes it from the real thing; that the causal histories of the objects distinguish them. But it’s part of the paradox that the circumstances described in (3) render that causal history inaccessible to us.

    What’s cool about that is, we WANT to say the forgery IS still a forgery, that there is still something **about the forgery** that we can know that makes it a forgery. But step 3 means we can’t – a perfect forgery means that, once we’ve done the shuffling or whatever, there is now no knowable property that distinguishes the forgery from the real thing.

  10. Chase

    The forger themselves would still know that one particular object in a pile would be a forgery, they may no longer know which bill it is, but the forged bill contributes to the number of bills in the pile. And this misses the point of a commonly accepted definition of forgery, someone might ask,”how can this be identical to a real dollar bill, when it is not administered by the proper entity?”

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