Interpreting AND as OR

Previously, I wrote that I’m skeptical of interpretation as an analytic tool. In physical reality, generally speaking, I think there is a single correct interpretation (more of a true account than an interpretation). Every other interpretation is a fiction, usually made obvious by complexity and entropy.

I recently encountered an argument for interpretation that involved the truth table for the boolean logical AND being seen — if one inverts the interpretation of all the values — as the truth table for the logical OR.

It turns out to be a tautology. A logical AND mirrors a logical OR.

The argument is due to John Mark Bishop from his 2009 paper, Why Computers Can’t Feel Pain. While I quite agree with Bishop’s conclusion, I disagree with his response to the two objections — primarily the latter, which is the main topic of this post.

I think I also disagree with his general thesis that computational states are observer-relative. I think they’re far more obvious.

The objection is stated: “Computational states are not observer-relative but are intrinsic properties of any genuine computational system.”

I think the objection is correct.

Simply put: Genuine computational systems are obviously computational systems on the account of their entropy, complexity, and intentionality.

Put another way: There is no obvious physical argument favoring the Pixies (where are they?). There is an obvious physical argument favoring the computation (just look at it).

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Bishop uses, as an example of observer-relative computing, the truth table for an AND. He suggests that interpreting the signal values the other way around makes the table table for an AND something else entirely.

But it really doesn’t.

Seeing them as different at all depends on seeing logical AND as something rather different from logical OR.

But that’s not the case.

They are intimately connected, somewhat like +4 is connected to -4.

That is, they’re both 4, one is a kind of “mirror image” of the other, and we can go back and forth freely, from one to the other, just by inverting the value.[1]

The same thing is true of AND and OR.

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To see how, we need these two important logical identities:

  1. NOT (a AND b) = (NOT a OR NOT b)
  2. NOT (a OR b) = (NOT a AND NOT b)

The first expression is known as a NAND (not-and); the second is known as a NOR (not-or). As you see, they are equivalent to their mirror partner with inverted inputs.

As a side note: The built-in NOT in the NAND and NOR gates, combined with the possibilities inherent in the identities above, make those gates the more common and useful gates in logic circuits.

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Here’s one way to illustrate their mirror identity:

  1. Given: (a AND b) = x
  2. Mirror: (NOT(a) AND NOT(b)) = NOT(x)
  3. Rule #2: NOT(a OR b) = NOT(x)
  4. NOTs cancel: (a OR b) = x

Or we can do it without involving x:

  1. Given: (a AND b)
  2. Mirror: NOT(NOT(a) AND NOT(b))
  3. Rule #2: NOT(NOT(a OR b))
  4. NOTs cancel: (a OR b)

Either way, the magic happens in step 2, when we invert all the logic.

It’s the equivalent of multiplying +4×-1 to get -4. (Or -4×-1 to get +4.) Effectively, we’ve multiplied logical AND×NOT and gotten OR.

And, of course, it works the other way around. We can multiply OR×NOT to get back the AND.

In a sense, when we say, “interpret all the voltages the opposite way,” we’re actually performing the multiplication that inverts the logic.

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But perhaps the reply is: Well, the logic still goes from AND logic to OR logic.

Well, no it doesn’t (I reply), on two counts.

Firstly, when we invert the values and see an OR, we’re also inverting the table top and bottom. (See the tables shown up top. The horizontal orange is the same inputs.)

Looking at the actual gate, nothing changes. The inputs both being +5 volts still causes +5 volts on the output, and the other three combinations of inputs still result in 0 volts on the output.

Secondly, you’d have to look at the entire circuit in the mirror, and that might demonstrate the mirror interpretation wrong. (I have to think about that.)

I very much suspect that, especially with regard to inputs and outputs, the one true account interpretation of the circuit would become clear.

Even if replacing an AND with an OR turns out to make sense in the entire inverted circuit, the nature of the logic is still super obvious.

For example, you can’t interpret the gate as an XOR or a half-adder. It’s clearly not performing those functions.

It’s performing the functions described by the truth table. The over all circuit (not the observers) determines what that truth table means.

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Something to consider as well is that logical truth tables in general are invented mathematical abstractions.

The operations are basic enough that there are some vague analogues in nature, but, generally speaking, logic gates are a creation of intelligence.

I think that’s an important point to keep in mind when we talk about algorithms. Just because we can model nature with an algorithm, that doesn’t mean nature uses them.[2]

More to the point here, the putative physical logic gates and their voltages are a reification of that invented abstraction. Any analysis of such physical instances has to trace back to the abstract origins.

The point is the intentionality of an algorithmic implementation. I’m saying it always exists, and it’s always ultimately pretty obvious.

Unlike eyes, algorithms really are watches made by a watchmaker.

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Bishop cites a second example to illustrate that computation is relative. This one involves a chess-playing computer.

In the first case, the computer displays a board and interacts with the user to play an obvious game of chess. Clearly a chess computer, right? Right.

In the second case, the computer displays the board as a one-dimensional strip of lights where colors indicates pieces. The user can still interact, but now the computer is… what? A work of art?

No, it’s still a chess computer. It just has a really weird display. But imagine someone learning to play chess according to the new system. They’d still be playing the same game offered by the underlying algorithm.

Which is clear and obvious.

I’m not necessarily suggesting an observer would identify it as chess (although I think they would after careful observation).

I think they would identify it as a system with complex rules right away.

Once they noticed the number of colored lights decreasing, and certain states ending the interaction, they might guess it’s a game.

If someone who knows chess studied it for a long time, they might well recognize it. (Especially if its opening moves were predictable.)

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The first objection Bishop handles involves counterfactuals in the flow of states that are assumed to represent consciousness.

That’s actually an interesting enough topic on its own that I think I’ll get into it another time.

Briefly, the argument involves selection in algorithms (e.g. If-Then-Else). The branches taken represent a given set of states. The debate is whether the branches not taken, and the states they represent, matter in whether the actual states experience consciousness or not.

The discussion of states gets us into the Rock Wall of Dancing Pixies (with Clocks), and that’s definitely a topic for other posts.

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The the extent Bishop argues against computationalism on the account that computation is relative, I don’t agree. I don’t think it is.

I think a correct computational interpretation is obvious and clear from its complexity, low entropy, and intention.

Stay logical, my friends!


[1] The general idea is there is a closed group under a “multiplication” operation. The integers are closed under (actual) multiplication — multiplying two integers always results in an integer.

A requirement is a multiplicative inverse: a value that multiplied against a member of the group returns that member. For integer multiplication, that value is 1.

Logic gates form a closed group of many operations on boolean values (the result of a logical operation is always a boolean). We can consider the boolean multiplicative inverse to be a NOP (“no op”) gate — a gate that outputs its input.

That makes the NOT (or INV) gate the equivalent of -1 for the integers. The thing you can “multiply” a thing with to get the mirror thing.

[2] Reality comes first; intelligence comes later. Physical processes come first and are primary; algorithms are secondary models created by intelligence.

About Wyrd Smythe

The canonical fool on the hill watching the sunset and the rotation of the planet and thinking what he imagines are large thoughts. View all posts by Wyrd Smythe

8 responses to “Interpreting AND as OR

  • SelfAwarePatterns

    It’s funny. You disagree with Bishop’s reasoning but agree with his conclusions. I agree with (some) of his reasoning but disagree with his conclusion. I think Bishop’s main mistake is in assuming that consciousness is objective. Both computation and consciousness are matters of interpretation.

    That said, I do agree that not all interpretations are equal. For me, it’s a matter of how much energy has to be invested. In the case of both computers and nervous systems, engineering and evolution coevolved the core systems and interpretational (I/O) systems to keep the energy at a minimum.

    But there will always be cases on the broad blurry boundary where there won’t be a fact of the matter answer. Interpretations of rocks and walls doing computation require profoundly sophisticated mapping systems. But to me, interpretations of something like nervous systems don’t. (Although I’m open to the possibility that at some point a more productive interpretation might be found.)

    Obviousness seems like an…obviously bad metric. It’s a quick heuristic we all use, but the history of science seems to be a history of previously obvious things turning out to be flat wrong. (Geocentrism, the immutability of species, a universal now, etc.) Reality seems to delight in showing us how absurd it is, at least in human terms. So obviousness can be a starting point, but I think we have to use more than that for our arguments.

    • Wyrd Smythe

      “Both computation and consciousness are matters of interpretation.”

      Right. Whereas I don’t see it that way.

      “For me, it’s a matter of how much energy has to be invested.”

      Yes, absolutely.

      “But there will always be cases on the broad blurry boundary where there won’t be a fact of the matter answer.”

      Can you name such a system?

      “Obviousness seems like an…obviously bad metric.”

      🙂 Yet for it’s it’s essentially the same metric you just named: “it’s a matter of how much energy has to be invested.”

      Yes, absolutely. Something obvious doesn’t take much energy to recognize. The less obvious something is, the more energy it takes to discern it.

      I’m not talking about conceptually obvious. (Give me some credit. 🙂 ) I’m talking about how easy it is to see the putative computation against the background noise of all the Pixies.

      (FWIW, The previous post explores this in more detail. This post builds on it.)

    • Wyrd Smythe

      “You disagree with Bishop’s reasoning but agree with his conclusions. I agree with (some) of his reasoning but disagree with his conclusion.”

      I disagree with some of his reasoning — mainly that computation is observer-relative. I see computation as intentional and thus easy to spot. Nevertheless, I do agree with his main point, which is that computational states alone can’t have phenomenal experience.

      “Both computation and consciousness are matters of interpretation.”

      That is the part of his account you agree with, I take it, the argument that computation is relative?

      Then what of the analysis I made here that debunks Bishop’s assertions about the relativeness of truth tables and chess computers?

      I think I’ve made a good argument that computation is “obvious” (i.e. genuine computation doesn’t take much energy to recognize or extract). What is the counter-argument?

      • SelfAwarePatterns

        “Can you name such a system?”

        The example that springs to mind are cellular systems and DNA. Or the interaction of unicellular organisms between their sensorium and motorium. I could see arguments either way on whether these systems are doing computation.

        “Then what of the analysis I made here that debunks Bishop’s assertions about the relativeness of truth tables and chess computers?”

        I’m not enamored of Bishop’s examples, but I do think he made his point with the AND/OR example. Your criticism is that that’s all it is, that it can’t as easily be reinterpreted to other be things, which I think is valid, but is beside the specific point he made. More exotic interpretations will require reconfiguring a lot of stuff around the mechanism, which I think we agree that, except for a relatively small number of interpretations, become increasingly unproductive.

        “What is the counter-argument?”

        I’m not prepared to defend Bishop’s unlimited pancomputationalism. I think we’re agreed that there are a limited number of actually productive interpretations, although again I don’t see the sharp objective breaks you do.

      • Wyrd Smythe

        “I could see arguments either way on whether these systems are doing computation.”

        I can understand why. At that level, it’s much easier to talk about the symbols of discrete “symbolic processing” — especially when we get down to the DNA or cellular level. These things certainly look a lot like machines.

        How much does it affect the debate that every instance of algorithmic systems for which there is no debate is a human creation? Or that it takes getting down to such a low level for there to really even be a debate?

        Here’s a curve ball: If all the algorithms we know are deliberate intellectual creations, and if we do see apparent algorithms operating at nature’s lowest levels,… does that suggest the existence of The Programmer? 😀

        (If I could only convince everyone to stick with the CS definition, most of this debate would go away and we could focus on other debates. 🙂 🙂 )

        “Your criticism is that that’s all it is, that it can’t as easily be reinterpreted to other be things, which I think is valid, but is beside the specific point he made.”

        The overall point he made in the paper (that computers don’t feel pain)? I quite agree. I don’t think interpretational matters matter here.

        “More exotic interpretations will require reconfiguring a lot of stuff around the mechanism,”

        That’s an interesting point. Interpreting Pixies requires significant computation, but interpreting the AND/OR truth table is just a matter of perspective.

        I agree those are different, but at least some of that difference comes, does it not, from looking at the AND/OR truth table in isolation? As I wrote in the post, looking at the circuit in toto makes its interpretation a lot less ambiguous.

        I can’t help but think looking at the whole circuit would make the reversed voltage view untenable.

        “I’m not prepared to defend Bishop’s unlimited pancomputationalism.”

        I thought he was arguing against it? His Pixies are a reductio ad absurdum argument, or did I misunderstand?

      • SelfAwarePatterns

        My understanding of his argument is that because computation is ultimately an interpretation, but obviously consciousness isn’t, that the computational theory of mind can’t be true, that it requires we accept all those pixies. At least that’s the impression he gave me when he jumped into our discussion a while back. (I actually misunderstood his point the first time I commented on his ideas.)

        As we’ve discussed, while I accept that there’s no sharp line that can be drawn, the interpretation to accept pixies is light years past even the blurry boundary. But in any case, I accept that consciousness itself has the same interpretive issues as computation, although similar to it, I find many interpretations so extreme as to be uninteresting.

      • Wyrd Smythe

        “My understanding of his argument is that because computation is ultimately an interpretation, but obviously consciousness isn’t, that the computational theory of mind can’t be true, that it requires we accept all those pixies.”

        That’s how I see it, too. He’s arguing against computationalism.

        I’m guessing you take some issue with the idea that “obviously consciousness isn’t [an interpretation].” 🙂

  • Wyrd Smythe

    Another way to look at this is that both ways of seeing the truth table (or the gate) in isolation are equally valid because they are essentially equivalent. Both are low-energy (“obvious”) interpretations.

    What would take ridiculously high energy would be to interpret the table as a half-adder or an XOR gate. Or anything other than an AND/OR gate.

    The bottom line for me is that the more light-weight an interpretive map is, the more that map is essentially a one-to-one bijection between a basic clear concept (e.g. “AND Truth Table”) and the system in question, the more likely that map is a true account.

    In terms of the table, or the gate, ex vivo, both maps, to AND and OR, are essentially bijections — one-to-one maps.

    [My contention has been both map to the same thing (just as +4 and -4 both map to the quantity 4), so of course they can be seen either way.]

    [[I also contend that examining the gate in vivo very likely makes it clear whether the gate is functioning as a union or a disjunction. At least, that’s been my experience examining unknown circuits. If you’ve ever done that, you already know that, just because the component is an AND gate, it may — in the circuit context — be operating as an OR. (NAND and NOR in most real-world circuits, but same thing.)]]

    So I think this does not at all change my point that interpretations amount to, on the one hand, fictions that can be largely ignored, and on the other hand, a true account of what the system is actually doing.

    The simple fact is, the interpretations of the real world “computing” tend all to require high energy and all to have convoluted maps between the physical thing and the abstract computational concept.

    In fairness, a valid counter-argument is that maybe nature has found completely different paths to computing (yes, computing) than we can recognize. I’m not sure how strong this argument is, but it’s certainly a valid point that needs to be considered.

    An extension of that argument is that the CS definition of computing is simply too restrictive. I think it’s a valid and useful “razor” for seeing what I believe are different classes of things clearly, but it’s well worth exploring ideas about alternate forms of computing.

    (I’ve just never yet seen one that made any sense.)

    But absent strong argument or evidence, I’m not at all persuaded by such hard-working interpretations. I’m a simple guy. Their maps are far too complex for my liking.

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