What’s really fun is that, with a few certain assumptions, the sim is vastly more likely!
(The assumptions are that sims are possible – indeed practical – in the first place, and that there are many technological civilizations capable of producing them.)
In a kind of Bayesian way. Since we don’t know that those assumptions are true, then, sure, the likelihood drops. But does it drop all the way to par? Does Occam really help? If this is a sim, we lack any knowledge about the simulators. And the probability of an event about which we have zero knowledge falls to 50% by default.
What she’s looking at is neither an AND gate, nor a NAND gate, nor an OR gate, but can, via a suitable mapping from voltage patterns to logical values, be interpreted as either.
I ask you again: what is there to computing the NAND-function other than having a device which reliably gives you the correct answer if you put a certain input to it? What’s C doing ‘wrong’ by using the interpretation I described above? Is the ‘design police’ going to come out of the woodwork to say, sorry ma’am, but this is an illegal usage of implementation maps?
Really, what I’m saying is not any more controversial than ‘the same string of symbols can be mapped to different meanings’ (because I am saying exactly that).
I’m honestly at a loss how you can think my stance would lead to this. It’s a prerequisite of the argument I’m making that the observations of A, B, and C are exactly identical; that there is an objective level of physical facts to be observed, and that every observer agrees on them.
No, of course not. The argument perfectly well includes designed artifacts; I merely think that your focus on design clouds your view towards the facts, and hence, urged you to consider more general physical systems.
And people describe my physical device consistently; I’ve pointed this out like half a dozen times now. The problems is merely that its computational properties are not among its objective properties—that there is no sense in which its physical state ‘just is’ a given computational state, which is again the same as saying that the word ‘dog’ in no sense is a dog.
OK, so maybe then do me the courtesy and describe what, following the procedure I outlined above, you believe C should obtain if she input the following sequence of logical values: first, 0 and 0; then, 0 and 1; then, 1 and 0; and finally, 1 and 1?
And what, precisely, do you require to be the case in order for somebody to have performed a computation?
No. What I’m doing here is to point out that the mapping between physical states of a system and logical states of a computation is arbitrary. Which it is—it’s just the same kind of mapping as that between symbols and their meaning. There’s nothing that says symbol x must mean y; or that symbol a can’t mean b. And that’s all there’s to it.
Again, then tell me what you think C should see.
Close, actually; but it’s not the meaning of the bits that can be interpreted arbitrarily, but rather, what bits to map to physical inputs.
Note that the Humpty Dumpty situation is only made silly by the fact that in such a case, there exists a pre-defined code, and communication means that all must play the same game; but again, there is no pre-defined code, no design spec, when it comes to natural artifacts.
I think it’s possible, but unlikely. An interesting but probably useless speculation.
Believing that the universe is real has the advantage of matching what we perceive, and of being theoretically falsifiable. An often unspoken assumption of most versions of the simulation hypothesis is that the creators are being intentionally deceptive; otherwise after all either we’d likely find flaws or they’d just tell us. But that doesn’t have to be true; the “reality hypothesis”* could *be falsified tomorrow by the simulator revealing themselves with giant letters in the sky announcing themselves and apologizing for not noticing we’d popped up in part of his galactic physics simulator.
On the other hand the simulation hypothesis is unfalsifiable, since you can never be sure if what you think is real is just a simulation good enough to fool you. Even if you “broke out of the simulation into reality”, you couldn’t be sure that wasn’t just *another *simulation.
The “you couldn’t accurately simulate a universe” claim doesn’t really work though, since a deceptive simulator would have access to our minds and be able to make us simply not notice any flaws or limitations in the simulation. In such a case the universe could be vastly simpler and less consistent that it looks, and our minds would be filling in the missing details and ignoring the contradictions. Of course that is another reason why the simulation hypothesis is unfalsifiable; you can’t falsify something when someone else can just stick a subroutine into your mind commanding you to never falsify it.
Actually each civilization would have to produce many such simulations in order for the probability of us being a simulation to be high, or almost all of the simulations they do produce results in other simulations. Neither of these seem likely to me.
No, no, no! The probability of something you don’t know anything about is unknown. Say you know nothing about something, and think about the probability of event X happening. 50%? Now how about Y? 50% also. And Z? 50% too.
Oops.
I’ve seen the 50% fallacy used by creationists to explain why god is such a likely cause of a universe whose origin is unknown.
Sometimes the unknown is your friend.
Hell yes. A instances of a certain symbol in a truth table mean the same thing. If she is looking at the same physical object as A and B, and using the same symbols as A and B, she needs to consistently map the 0 from their observation to something and the 1 to something else. What she is doing, and you are agreeing with, is inconsistently mapping 0s and 1s to 0s and 1s.
If she wants to map 0s at the input to Rs and 1s at the input to Ss, she might get a consistent mapping. But it still won’t be a NAND gate.
No, you are saying it is legitimate for the same symbols in the same context to more or less be mapped to different meanings. I’ve already said in describing the basics of Boolean algebras that you can map the 0s and 1s to anything you please - consistently. But you can’t decide that based on a roll of a 12 sided die that this signal will have a 0 mapped to a 0 or that one will have it mapped to a 1.
But they do not. Say they are observing the output of gate at the same time. A might point out to B that he has inverted the meaning of the signals - and get agreement - and then they would agree. No such thing could be said of C. There is no single consistent mapping from what they see to what she sees. You even admit this.
Now, if A and B look at the circuit diagram they will agree if it is an AND gate or OR gate, which are equivalent under the transformation I’ve mentioned above. A NAND gate is not. You cannot look at the truth table of an AND gate and correctly interpret it as a NAND.
To justify your lack of belief in machine intelligence, you are requiring an incorrect and absurd observation be considered as correct.
If random inconsistent interpretations are correct, I repeat my observation that science will not work. Nor will much of anything else. "I paid you $14 for a $500 tab? I just interpreted $500 as $14. And Half Man Half Wit says it’s okay!
Maybe you observed yourself demonstrating this, but that does not make it true. I’ve given you proofs that you cannot correctly interpret an AND as a NAND, but you just ignore them.
I’ve read all 800 or whatever pages of Science and Sanity - hardly anyone on the planet understands that the tag dog is not the object dog better than I do.
Sure. Say it is an AND gate in real life (no matter what she interprets it as) and she is using 0 as 0 volts and 1 as 5 volts (a very old gate.)
She will get 0 0 -> 0, 0 1 -> 0, 1 0 -> 0 and 1 1 -> 1.
Now, if instead she interprets a 0 as 5 volts and a 1 as 0 volts, she will get
0 0 -> 0, 0 1 -> 1, 1 0 -> 1 and 1 1 -> 1.
Your example seems to assume she is interpreting 0 at the input as 0 volts and 0 at the output as 5 volts. That is inconsistent. If she does this, she is a blockhead.
It is very easy for the first line of the second case to say the input is 0 (1) and 0 (1) so the output of 1 and 1 is 1. But they you are forgetting to interpret the output signal in the same way as the input signal - which makes it a 0.
Some result based on inputs? Saying 1 + 1 = 3 is a computation, just an incorrect one. My programs do incorrect computations all the time. If you require correctness in all cases, there are few computations performed by any machine anywhere. All processors come with lists of known bugs so if you require perfection none of their results are computations.
You are going beyond this, to say they don’t even have to be consistent in one mapping. That’s the problem. I have no problem with the arbitrary labeling of anything. As long as it is a consistent arbitrary labeling.
No, but there is an agreed upon set of tags for the various physical objects we see and their characteristics.
What you are proposing is that if we agree that a rock is called hard and fur is called soft, it is also okay to say that a rock is soft and fur is soft. Just an interpretation, right?
BTW, can you give us one example of where this inconsistent mapping you claim is correct is actually used and accepted as correct?
No question that it is unfalsifiable. But as I’ve mentioned above, it would be far easier for the simulator writers to get it correct than to individually hack the simulated minds of people, dogs, cats, elephants etc., etc. And pretty much every simulation would have to do this.
Or, as we can say, “it’s simulators all the way down.”
Given some of the effects of brain damage and such I actually doubt that. A human mind is quite good at re-writing its own perceptions even without intentional hacking.
Plus, it would much more likely just be a matter of a “don’t notice it’s a simulation” subroutine being automatically attached to every mind upon creation, not one of hacking each person by hand.
Yes, fifty per cent. Because we don’t have any reason to suppose it is more likely, or less likely. Any other number would imply a bias for or against.
Yes, it is also unknown, but the default number is 50%, because it can’t be any other number.
This is valid, whether or not it is abused by creationists. They abuse the laws of thermodynamics too, but that doesn’t impugn the validity of those laws.
There’s nothing inconsistent about it. Yes, the mapping differs for inputs and outputs; but your only argument against that seems to be that you don’t like it, hence, you want it elevated to a metaphysical principle that such things are disallowed; I think you overestimate your clout there.
No, the context differs: one mapping holds in the context of inputs, the other in the context of outputs.
There’s no die rolling. The way the voltage patterns are interpreted is given by a fixed, consistent rule in every case.
No; I’ve pointed out that it’s completely unproblematic for A or B to map their interpretation to C. If somebody has both A’s and C’s ledger page, then they can predict what C obtains, based on what A has obtained. This is very simple.
Then how come you have such problems with understanding that the same symbol can mean different things in different contexts?
So, how do you tell whether it’s an AND gate ‘in real life’?
For which, again, you offer nothing but your say-so.
The mapping is consistent; it just differs between inputs and outputs. You might not like it, but the fact of the matter is that C can use it to implement, reliably, the NAND gate, and whenever you ask her the NAND of two logical values, she will provide the right answer based on her mapping. Just arbitrarily requiring that the mapping must be the same for all the voltages is like requiring that the meaning of a word must be the same everywhere in a sentence—i.e., nonsense. ‘Buffalo buffalo Buffalo buffalo buffalo buffalo Buffalo buffalo’ is a perfectly correct English sentence after all; but I suppose you would have the skies open to shatter everybody trying to illegally have the same symbol mean more than one thing.
Sure, there are many examples. Take the buffalo-sentence above, for instance. But the problem still is, of course, that you somehow still believe that ‘accepted’ has any meaning in this discussion. It doesn’t; nobody needs to ‘accept’ whether the interpretation C uses is a correct one for her to be able to correctly implement the NAND-operation; there need be no seal of approval, no deed by the bureau of how things are supposed to work, no design specification locked away in a file drawer in the cellar of the institute of standards. She simply uses that interpretation, and happily computes the NAND, your disapproval notwithstanding.
But anyway. Even if you were right on the above ‘inconsistent’ interpretation (which, again, is anything but), it’s still a fact that a given physical system admits multiple interpretations in terms of computations. Changing all the ANDs to ORs (and vice versa) is an example where it seems you already agree—and certainly, that means you can’t tell which of the two computations is being carried out by a system. So, even that is enough to prove my point—if, say, only the computation using one labeling instantiates a mind, and the other doesn’t, you can’t tell without an arbitrary act of interpretation whether a given physical system instantiates a mind. Even if both were miraculously to instantiate a mind, you’d still be unable to tell which mind is being instantiated.
The only way out here would be to claim that these two distinct functions actually implemented the same mind—which puts a large burden of proof in your court.
But it’s easy to find other examples of the underdetermination of the computational by the physical. Take two inputless finite state automata—one with two states, and another with three states. They both loop—i.e. the execution trace of one is 121212…, and the execution trace of the other is 123123123…
Now, take a physical system with four states available to it, which likewise repeat. Consider two mappings: one which takes the first two states to state 1 of the first FSA, and the second two states to its state 2; and another which takes its first two states to the state 1 of the second FSA, the third state to 2, and the fourth state to 3.
Which FSA is implemented by the physical system? Either, of course, if any is. And the choice of FSA, system, and mapping was, of course, arbitrary; hence, many other choices are possible.
So you see: looking at the physical system does not tell you what it computes.
Since you don’t know anything about this area, you are forgetting something - if this gate is being used in a circuit its outputs will be the input of another gate, and its inputs will be the outputs of other gates. So in this “consistent” mapping 0s magically change to 1s and 1s to 0s on a wire. Now, in many cases flip-flop outputs are fed back to an input. So the value of this line will change even without going anywhere else.
And this is consistency in your world?
The input or output is not the context - the proper context is the wire which carries the signal. Where along the wire does she interpret the value to change? Halfway? A micron from the input?
It isn’t me who is enforcing this. It is the entire industry. I could write a column using your quotes verbatim and people would think it is one of my funniest ones - even better than base 1 arithmetic.
You also said that it is okay for C to change interpretation across inputs - the for three real 0s, she’d see 0 1 0. Now the order of inputs for these gates does not matter. If she had X Y Z = 010 and then got a version of the gate which is XZY, would she interpret it as 010 still or 001?
Because in this case it is the same context, and you don’t know enough about the subject to understand this.
Look at the schematic. Which has inputs of transistors tied to ground or the power supply line. Ground is not a symbol - it represents a line tied to physical ground in the system, along with many other lines and even a ground plane designed so that it does not pick up crosstalk. It all eventually goes to literal ground. Now, if you are going to tell me that you can interpret the earth as emitting a five volt signal …
See above for my justification. Plus I’ve been working in this area long enough with enough credits that I could say my post is my cite. Not that I’d have to - anyone with the slightest knowledge of this area wouldn’t need a cite. It is like HariSeldon who used to be a Math professor justifying why 1 + 1 = 2. I think I’d trust him on that.
Actually your example doesn’t have her implement anything. Since with the implementation of a gate the power and ground lines are shared across all transistors, she would have a bit of a problem implementing the gate with inconsistent uses of them.
And if she is interpreting a gate that other people interpret correctly as an AND, please sketch how all logic functions can be implemented with it. (It is your contention, after all.) I think you might find this a bit difficult. You’ve ignored this point - I don’t know if it is because you know it causes problems or if it is because you don’t understand it.
You have an odd idea of correct English sentence. Colorless green ideas sleep furiously is a more correct English sentence. Semantically it makes more sense than yours (not much) because at least the parts of speech are right and it makes sense syntactically.
Your sentence, by the way, would only make sense if it came with some interpretation of which buffalo means what. You don’t get to add more information to a gate. Remember the joke about the prisoners yelling out numbers standing for jokes? In your world, did they never actually relate the numbers to jokes?
If C turns in her “NAND” truth table with her interpretation, then A and B can map it into the correct truth table for an AND. Like I said, she’s a blockhead. But if she says “here is the mapping that makes this truth table that looks like a NAND look like an AND, and then I’ll ignore the mapping and actually call it a NAND” she has gone beyond being a blockhead into something worse. That is yet another instance of how she is being inconsistent.
Follow me? She reports the mapping to show her interpretation is consistent with the others, and then ignores the mapping in describing what the gate is.
That’s the best you got? Sure, we hear sentences like that all the time in the real world? I kind of meant something that you didn’t make up to support your argument.
She either used her truth table to implement a NAND, and thus did not use the mapping you describe, or she implemented it with the mapping you described and did not implement a NAND. Your thinking is quite muddles. BTW there are standards for this kind of thing, but I don’t need to appeal to them. Your descriptions is internally inconsistent, Try stepping back and thinking about it.
Incorrect again. Remember, implementing an AND with an OR requires complementing the inputs and outputs. If you draw a box around the OR and the complemented inputs and outputs you have something equivalent to an AND, but if you look inside and see the inverters you’ll know the gate is an OR. If however you draw the box inside the not gates complementing the inputs and outputs you’ll get something which is not an AND. We didn’t just relabel things - we did a logical manipulation.
Since I’ve not admitted that arbitrary acts of interpretation are correct, wrong. Now, since mind is poorly defined, and since we don’t have any foolproof way of determining if something has a mind, I can agree that it would take a lot of effort figuring this out. But interpretation has nothing to do with it.
Oh good grief. This is just an example of an FSA with redundant states which can be mapped into an equivalent FSA with fewer states. This is well understood. If states 3 and 4 (in the real world we start with state 0) output a 3 and 4 then they would not be redundant, but since you have them outputing a 1 and 2 they are.
Sometimes circuits are implemented with several identical gates in parallel, with their input and outputs tied together. This is done to increase the drive. They are clearly logically redundant, and can be removed without affecting the logic of the circuit. If you accept redundancy you can clearly have an infinite number of identical implementations. That is why all proofs mention irredundant circuits.
IIRC you can prove two FSMs with the same number of states equivalent or not. This handles labeling the states different things. But if you have inconsistent labelings you have non-identical FSMs - unless you think FSMs with n states are all identical. Since you don’t seem to understand gates, I don’t want to get into FSMs.
Only under the assumption that the observer doles a random mapping of physical outputs to symbols and doesn’t tell anyone what they are.
To repeat the important thing here - is C using her mapping when calling it a NAND gate or not?
You’re still missing the point. Since only in- and output states matter to the computation’s user, are under their control, that’s where the interpretive mapping is applied. So let’s maybe use a different example. Think about the set of all functions from n binary variables to one binary variable, such that exactly one input yields output 1. To compute such a function, you need a device with 2[sup]n[/sup] input states, and two output states.
Now, each of those input states is associated to a binary number. One thing I’ll sincerely hope you agree with me on is that this association is arbitrary: I can associate state S[sub]i[/sub] of my system to all-0 just as well as to all-1, and to every other n-bit number.
But then, it’s immediately clear that I need only re-label the input states to perform any such function. If I want the number k to be accepted, then I change the labeling such that the physical evolution of the device if initialized in the state I map to k ultimately ends in the state I map to the output 1, or yes, or accept, or what have you.
The same thing can obviously be done with n input bits and m output bits: I need only permute the association of inputs to binary numbers, and the association of outputs to binary numbers, in order to implement any function taking n-bit strings to m-bit strings.
The order of inputs still is different physically—it’s a different situation whether you apply 0 volts to input 1 and 5 volts to input 2 than vice versa. So again, no problem.
Well, are you really going to tell me that only devices with transistors, a connection to ground, and so on can be used to instantiate logical gates?
This is your biggest misunderstanding—that we’re in your area of expertise. We’re not; this isn’t about chip design, testing, programming or whatever. It’s philosophy of mind. Now, I’m not going to claim absolute expertise on the subject, and I’m certainly not going to claim absolute authority; but I have contributed to the peer-reviewed literature on the subject.
There’s no need to worry about transistors, ground lines, or anything of that sort. All she does is apply input voltages, and measure output voltages. As long as you grant me that she can distinguish between those, she can implement the mapping I gave above—or even use measuring/input devices that implement that mapping for her—and will be fine. Whenever she’s interested in computing the NAND of two inputs, she can do so, and even if you’re standing tut-tut-tutting right next to her, that won’t make her result any less correct.
Neither. It’s just besides the point.
The sentence is perfectly semantically meaningful and doesn’t require anything but an understanding of the English language to parse.
Actually, that’s not a bad example. In your world, apparently, they wouldn’t have had to first agree on a given mapping of jokes to numbers—some jokes just mean those numbers, in an objective way. But of course, that’s not how things in the real world work (not even if there’s somewhere an industry specification of how jokes are supposed to map to numbers). The connection between symbols and their meanings is arbitrary; and yes, one and the same symbol can have more than one meaning.
Why AND? Why not OR?
But that’s not at all what’s happening. She has a mapping which, unambiguously, takes input and output states to logical values; so do the others. Thus, there exists a mapping between her logical states and those either A and B uses. I could draw up the mapping, but you have a habit of just ignoring it when I do so, so I won’t go through the trouble again.
I didn’t; you might have at least googled. And yes, we do hear sentences where the same word means different things all the time.
No. It merely requires changing whether you consider 5 volts to be a logical 1 or a logical 0. Neither of these two options is any more correct than the latter. Otherwise, you’re left arguing that there’s something irreducibly logical-1-y about 5 volts that 0 volts lack, which I really hope isn’t what you believe.
Again, you completely misunderstood the point I was making. But let’s drop that, it’d take us too far afield; I first need you to understand the difference between ‘outputting’ a value and ‘being mapped to’ something. So, again, take the AND- and OR-interpretation: are you seriously going to hold that only one of those is objectively right?
Of course she is. Only using a mapping is anything a computational machine at all—after all, there are no logical states out there in the world, anymore than there are numbers, or concepts.
Wrong on many, many counts. First, terminology. State has a specific meaning, which is the value of the memory elements of an FSM. You can talk about input values, not input states. You are just revealing your ignorance here.
Second, representing an input as a binary (or octal or hex) number is just a short hand for the ordering of values of those inputs. It is not arbitrary, and each table of inputs has a header showing the assignment of the values to particular inputs. The order of the columns may be arbitrary (big endian vs little endian for example) but that does not change the mapping of a column to an input.
You can call the assignment of values to each row after a Star Trek character for all I care - but there had better be a mapping back to the actual binary values. And anyone deciding to use binary numbers for the names and using different ones (oh, I’ll call 010 110) is an idiot. But if there is a consistent mapping you can still get back to the real one.
Third, your notion of input and output also shows your ignorance of logic design. For debug and for testing we observe the internal values of wires all the time. Not only that, we can FIB a part and observe the internal values of signals without bringing them to a primary output. We can also do it with ebeam probing. (Which shows the actual voltages.) So your assumption that internal signals can be treated somehow differently from external signals is incorrect.
Now, clearly the subcircuit defined by observing an internal signal does a computation also - after all you can build a circuit with traditional inputs and outputs with exactly the gates of the cone of logic of the subcircuit.
As for relabeling the inputs, maybe this is why you seem to think that no one understands the difference between the tag and the thing. You can call a 001 a 101 all you want - but when you apply real voltages you get 001, and calling it a 101 does not change the function computed one bit. When we get an incorrect output we don’t get to relabel it as the correct one and then move on.
How do you correlate the logical mapping to the physical mapping then? If you have a clear mapping, no problem - but you might as well be using Spanish. Or more likely Mandarin. And I assure you that naming signal values in Mandarin does not turn AND gates into NAND gates.
Of course not - my adviser worked on the IAS machine and Illiac-2. But I was assuming these people were looking at a gate in a standard technology, which does use transistors and ground. I believe vacuum tubes used power and ground - the switch was different but not the basic principle. Google “schematic of NAND gate” and you’ll see what I mean.
If you used this example in your papers, I’d bet anything that the reviewers had no knowledge of the field of digital electronics and therefore could not see the flaws in your argument. I’ve been program chair of many conferences, have been on lots of Ed boards and have edited special issues of journals. It is hard to find reviewers for something outside of the field of the journal.
If you are using a logic truth table as an example you had better understand it. You don’t.
You said a 0 would be 0 volts at the input and 5 volts at the output. What I was saying is that you can’t build something like this. Remember, it is the same part as the thing A called an AND gate - so you don’t get to add extra power supply and ground lines.
It is exactly the point. An AND gate is provably not equivalent to a NAND gate. If C takes an AND gate and interprets it into a NAND gate she is wrong. I’m sorry you don’t understand this, but it is true nonetheless.
Of course one symbol can have different meanings - and we rely on the context to disambiguate them. But you seem to be saying that we cannot determine what computation any system is doing because some idiot uses the same symbol to mean two different things in the same context. We obviously can’t do all possible computations - we can’t always compute if a program will halt - but we can do lots of them.
I chose the AND since it was your first truth table. You cannot get from OR to NAND either.
The mapping is obvious - but irrelevant. When you take the AND and apply the mapping you get what looks like a NAND. But those aren’t the real inputs to the gate, they are just a label for the real inputs. When you reverse the mapping to get the real inputs, which you can do of course, you go right back to the AND.
Let me introduce you to person D. He takes a 32 bit ALU (arithmetic logic unit) and breaks the inputs into blocks of 4 bits, and assigns a letter to each. He then runs it and looks at the output using the same mapping.
If he is consistent about the mapping, he gets letters at the output which he maps back to a binary string and all is okay. If however he puts some words into the input and then interprets the output as actual words - not as mappings of letters to bits - and claims he has implemented some sort of word processing system, he is deluded. Or do you think he has just done something useful? Because this is basically what C is doing.
That is exactly complementing the inputs. For a specific implementation there is something 1y about 5 volts. You can design a part using 0 volts as a logic 1 (I believe) but you can’t redefine an existing part to do so.
An OR is logically equivalent to an AND\ if you complement both inputs and outputs. Which is basically what you are doing here. And I just said they are equivalent if you draw the box outside the complementation. Which you are more or less doing with labeling but not gates, but for the moment let’s call it okay. They are not equivalent if you complement the inputs and not the output.
I can prove the equivalence (if I look it up - it has been a while.) I challenge you to prove the equivalence of either with a NAND gate. Since ANDs can be proven to not be able to compute all functions, and NANDs can be, I think you will have a problem. And mathematical proofs don’t let you call something something different at step 5 and proceed as if no change was made.
You mean logical values. Which are voltages in an implementation. Which do exist unless you don’t believe in electrons. Not the mathematical representation does not exist per se, but it is rigorously defined by the laws of Boolean Algebra - which laws you are violating by mixing voltages and logical entities.
If you could rigorously define what you claim C is doing (which does not involve saying she just is changing values willy nilly) I think you’d find that the result is no longer a Boolean Algebra, and is thus outside the scope of the discussion.
In words of one syllable.
You have an AND (or an OR). You want a NAND. You look at the two truth tables and say “I can just rename these inputs to make the AND truth table look like a NAND. Done. So I see the new truth table, ignore the mapping I just did, and claim I now have an actual NAND gate in this package.”
My friend, that is not engineering, it is not theory of mind, it is not philosophy. It is magic.
You have no reason to believe that one is more likely than another, but you also have no reason to believe that they are equally likely.
In sampling theory if you are computing sampling error there is a term where, for a binomial distribution, you estimate the probability of each possibility, p and pbar. In this case if you don’t know them you use 50%, but that is not because that is the right distribution, but because that gives the most conservative estimate.
And I’m certainly not arguing that any other number is better. The point is that if you use any number at all you are going beyond what you claim you know.
If you choose to use 50% as a hypothesis, that is something different - and you are admitting that you can be way off. But a hypothesis is not an assumption.
No, you’re still making the unwarranted assumption that in some way, arbitrary conventions have an influence on what goes on. I’m using terminology in the hopes of finally getting you away from the idea that it matters what you call things, what the design specifics are, and so on. Again, you should pretend that you’ve never heard the word ‘chip’, or anything connected to it, before—the point is a conceptual one, not one that can’t be answered in terms of established practice.
Again, this is only true if you assume we’re in a situation where you have some sort of idea how things are supposed to work, some convention that says, e.g., ‘0 volts is logical 0’ and the like; but that’s not the situation we’re in. Rather, we have found, say adrift in space, a physical device, knowing nothing about any design principles, or indeed, whether it was designed at all.
Logic design just has nothing to do with it. Really, this is a dead end you keep trapping yourself in.
So let’s maybe be a little more concrete. You have a physical system that can be prepared in one of four states—I[sub]1[/sub], I[sub]2[/sub], I[sub]3[/sub] and I[sub]4[/sub]. Then, it evolves according to whatever laws govern it (which are also completely irrelevant to the discussion) into one of two possible states—O[sub]1[/sub] and O[sub]2[/sub]. Note that in identifying what these states are, lots of interpretation has already been put into the device—such as which level of resolution to view it at, etc. But I’m giving you this for free.
Now, the full characterization of this system at the level we’ve chosen to view it is this:
An interpretation is now something that assigns logical values (or values out of any alphabet you care to use for your computation; but we can stick to logical values because they’re, as you well know, sufficient to implement any computation whatsoever) to physical states. One interpretation might look as follows:
Under this interpretation, the device implements the AND-gate. I trust I need not write down the interpretation under which it implements the OR-gate, but I note that the fact that I could do so already implies that merely studying that system will not fix whether it ‘actually’ implements an AND or an OR—which already suffices for my argument.
Which gate is now being implemented? That’s right: NAND. So, what do you think makes one of those interpretations any better than the other? Which one is the right one?
If it helps, don’t think of the device as something with transistors and voltages. Think of a set of billard balls, for example: one configuration, if struck (we fix the initial momentum), will end up with the ball in pocket a; the others will leave it in pocket b. There’s no design spec to help, no industry standard, not even a ground, any other voltages, or wires. (And of course, one can do universal computation with billard balls just as well as with transistors.)
Remember, the question is what computation an arbitrary physical system implements. This must have a definite answer for the computational theory of mind to make sense.
So, the mapping is as above, and there’s absolutely no foreign language skills needed to utilize it.
You’d loose that bet, though. But of course, digital electronics is again a complete red herring. In fact, I happen to know who one of the reviewers was, and he’s a noted expert in formal systems and the theory of computation.
But of course, I didn’t need to make the argument so explicitly in the paper, but could just reference it—the original is due to Hilary Putnam, whom you might recognize as not exactly being a slouch when it comes to the theory of computation either (and who was perhaps the first to promote an explicit computational theory of mind, only to later disavow it based on exactly this argument). If you want to take a look for yourself, it’s in the appendix of Representation and Reality, where he proposes, and proves, the following theorem:
Every ordinary open system is a realization of every abstract
finite automaton.
The idea here is basically the same as one John Searle has articulated independently, known as Searle’s Wall, in which he argues that one can take the wall of his office to implement the WordStar-program (the argument is a little dated). Many other versions of this argument exist, such as John Bishop’s ‘Dancing with Pixies’-reductio.
All of these arguments, however, are really just a simple corollary to what’s known as Newman’s Problem: namely, that knowledge of structure—that is, the relationships in which parts of a system stand to one another—allows one to solve only cardinality questions; that is, since the structure (and hence, the computation) we consider a system to implement is arbitrary, which is what I meant above by the modeling-relation, all we can ever say about a system is of how many entities it is composed—which obviously does not suffice to solve questions regarding what is being computed.
So, are you claiming that a physical system like the one I proposed above can’t exist?
Which one in the example above is the idiot, then?
The real inputs are just voltages. Any logical value is just a label for the input. Certainly you don’t believe that any physical system’s state ever just is a given logical value—that’s the same as saying the word ‘dog’ is a dog.
So, what is that? What makes 5 volts 1-y? What makes ‘dog’ doggy?
So, can you say whether a physical system implements an AND or an OR, or can’t you? Because in your last post, it seemed like you were saying you can—which just isn’t right. But if you can’t, then my argument goes through, whether you buy the NAND-mapping or not.
No. No voltage ever is a logical value. That’s just a category error, on the same level as saying the word ‘dog’ is a dog. Certainly, ‘dog’ maps to a dog under a given understanding—but that understanding is not fixed by anything either about the word, the sequence of graphemes or phonemes, or about the dog itself.
No—I use the mapping to compute the NAND of two inputs. Just like you use a different mapping to compute the AND.
He’s simply using an uninformative prior—which is provably the most sensible thing to do in situations of complete ignorance. Anything else would amount to assuming knowledge you have no grounds to assume.
This whole megillah is me attempting to demonstrate to you that a right answer does exist. C calling it a NAND gate is the wrong answer. You are also talking about implementation here. If you want to make this about abstract entities adrift in space about which nothing is known, then I’ll agree that it is indeed possible or even likely to not know the computation it performs.
But nowhere in this discussion, even when we were talking about simulators, were we talking about that.
If you prove that hummingbirds can’t fly, and I direct your attention to a flying hummingbird at my flowers, don’t blame me. And don’t say “prove it is a hummingbird and not some other entity you are just interpreting as a hummingbird.”
You are the one that is saying that an AND gate is a NAND gate if you use an arbitrary convention to rename its inputs. I’m the one saying that the gate inside that package is an AND no matter what you call its inputs. Mapping from symbols to voltages is indeed arbitrary, but once you’ve instantiated the mapping you can’t change it and expect to get a correct result.
Your A, B and C have sent inputs to this part. And they interpret the outputs. We are not dealing in the abstract world here. Your contention must hold for both designed and undesigned things, so them not knowing does not matter if I can show that you can tell what gate it is given basic knowledge of gate truth tables and specs. Remember, our simulator writer who is trying to see of there is a consciousness inside the simulation knows plenty about it.
This makes no sense. What is the value of O2 for S1, S2, S3? What is the value of O1 for S4? This example isn’t even wrong.
This question, by the way, has nothing to do with the implementation but only the specification of this function.
If you told me what O2 is for S1,S2 and S3, say then I can tell you. Say it is 1 for S1 and S2, and 0 for S3. Then you have the inversion of the first input, not a NAND. And if it is 1 for S1, 0 for S2 and 1 for S3, you have an inverter for the second input. In the first case the second input is a don’t care, in the second the first is. Or it could be a NAND if the values are 1 for S1 - S3. You don’t provide enough information for us to know for sure. Way back in the beginning I said that if you provide only one row of a truth table, there are many possible explanations. This is an example of that. No voltages or implementations involved here, so you should be happy.
Can balls be in both pocket a and b? If not, this is not a good analogy.
Exactly. And I’ve shown you over and over again that we both can discover this and do. Though I’m not sure what you mean by arbitrary. We have something that we who hold to the computational theory of mind say implements the mind - the brain. This is hardly arbitrary. It started as a black box but is getting to be more and more a gray box, and our discoveries about how the brain works supports a computational theory. It certainly does not support the contention that the brain has a von Neumann architecture, but that is a strawman argument and always has been.
I don’t have time now, but I do want to read proofs that things we do every day are logically impossible.
The physical system you proposed above is incomplete and incoherent. Try again.
I described, at great length, the complexity of mapping voltages to logical values, and how actual circuits involve more than just 1s and 0s. You clearly understood none of this, based on this accusation.
What part of AND is logically equivalent to OR with inputs and outputs inverted don’t you get? You can’t tell for sure that A’s result represents an AND or an AND with buffers at each input and output or an AND with an even number of inverters at each input and output or two ANDs in parallel with their outputs tied together. All these things are logically equivalent and perform the same calculation. You are not even responding to this, just ignoring it.
Mappings don’t compute anything. They are mappings. You clearly don’t understand the difference.
Here’s what I get when I searched for uninformative prior
In summary, there is no such thing as a prior with “truly no information”.
Ok. So, once more, here is the physical system (l: low voltage—say 0 volts; h: high voltage—say 5 volts; both to be taken with appropriate tolerances, and we’re only talking about normal operation):
I1 | I2 | O
-------------------------
l | l | l
l | h | l
h | l | l
h | h | h
So then, tell me: which is the right answer? AND? OR? Something else entirely?
What we are talking about is whether what computation a physical system—in the most general case—performs is an objective fact associated with the system. If you now agree that it isn’t, then this whole discussion is moot—then, there’s no general way to tell whether a given physical system instantiates a mind.
There’s no changing of the mapping occurring; I don’t know where you see that being the case. The mapping is fixed beforehand, and using this mapping, computations are being performed.
This is logically fallacious: the set of designed things is a subset of all physical things; hence, what you prove about the former, need not hold for all elements of the latter. In fact, it suffices to show one arbitrary system for which it doesn’t hold to falsify your assertion; and it likewise suffices to exhibit such a system to demonstrate my claim that in the general case, you can’t tell what computation a physical system instantiates.
I have no idea what you’re asking for here, I’m sorry. ‘The value of O2 for S1’? O2 is not, and does not have, a value; it’s a state, one of the two states the system can end up in at the end of its evolution. S1, conversely, is a state in which we can set it up. An example—and note here that I’m again specializing—would be that gate that gives us so much trouble: S1 might be the state ‘0 volts at I1, 0 volts at I2’, and O2 might be the state ‘5 volts at O’.
If it helps, maybe think of the states as messages sent according to a pre-agreed upon code (but please, do remember that just because two people have agreed on a code, doesn’t make that code natural law, and doesn’t mean that two other—or indeed, the same two—people couldn’t just switch to a different code whenever they fancy)—so if the system is set up in state S1, that’s equivalent to using a signal out of four possible ones that is interpreted as ‘00’ (four signals, of course, being enough to transmit every two-bit message). The same goes for the output states. The only added difficulty is now that we haven’t just an information channel between input and output, but something that carries out certain transformations on our signal states, taking input-signals to output-signals.
Otherwise, I have told you which output state each input state evolves to, and have given you two different mappings of input and output states to logical values.
So maybe let’s do an example? You initialize the system in the state S1, and you use the first of the two mappings I indicated—that is, you interpret S1 as ‘00’. That’s, so to speak, the program you feed to your computing device. Now, you let the device perform its evolution. As I defined things above, the evolution takes the state S1 to O1. Recall we’re using the first mapping—call that M1. So, O1 is mapped to ‘logical 0’. Hence, the value of your operation, if fed the input ‘00’, is ‘0’.
Then, you set up the state S2. Continue using the first mapping. S2 maps to ‘01’. It evolves into O2. O2 maps to ‘0’. And so on: S3 yields, under M1, ‘10’ being evaluated to ‘0’, S4 yields ‘11’ being evaluated to ‘1’. You’ll have no difficulty recognizing the truth table of the logical AND here.
Now, we change the mapping, using instead the second one—M2. Again, we set up the state S1, which we still interpret as ‘00’; then, the system evolves to O1. Under M2, O1 is interpreted as ‘logical 1’. Hence, the operation we’re now implementing takes ‘00’ to ‘1’. Continue as above, with states S2, S3, and S4. You should have no trouble realizing that now, we are constructing the truth table of the NAND-operation.
Finally, consider the following mapping M3:
S[sub]1[/sub]: 11
S[sub]2[/sub]: 10
S[sub]1[/sub]: 01
S[sub]1[/sub]: 00
O[sub]1[/sub]: 1
O[sub]2[/sub]: 0
It shouldn’t be too difficult to see, now, that this mapping implements the OR-operation, on the very same physical system.
What you’re saying here is simply not true. I have provided everything needed to derive the complete truth table. You can take any two bit input string, follow what I said above, and find uniquely what it evaluates to, under a given mapping.
Err, huh? Can the output of a gate be at two different voltages simultaneously?
The computational theory of mind states that any physical system instantiating the right sort of computation instantiates a mind—hence, we can’t limit us to some particular system, or class of systems; the question of which computation a given physical system instantiates must possess a unique answer in every case, if we are to avoid the problem of whether a given system implements a mind being undetermined (and hence, needing some arbitrary choice to be determined).
It’s not. I’m not sure what you don’t understand, honestly. I’ve given a complete specification of what states the system can be in, and to which states it will evolve. This completely defines the system.
I understood well enough. The problem, however, is still that you think how things are done in ‘actual circuits’ have any bearing on the issue at hand.
This, too, I understand well enough. What I haven’t been able to figure out, however, is if you hold that a physical system is either definitely an AND-gate or an OR-gate, or whether you believe that the same physical system can be used to perform both AND- and OR-operations, without making any changes to its physical form.
That is, once again—is the system described by the following input/output voltage measurements:
I1 | I2 | O
-------------------------
l | l | l
l | h | l
h | l | l
h | h | h
either an AND- or an OR-gate? And if so, which one of the two is it?
Yes, yes, I get it, I ‘clearly’ don’t understand a lot of things, you can stop telling me, and instead bring arguments rather than empty accusations of ignorance, it’s frankly just tiresome. However, what I’m trying to say—what I have been trying to say all along through this ever more unhappy series of posts—is merely that you need a mapping from physical states to logical, computational states before you can compute anything. Thus, no, I’m not saying that the mapping itself computes anything, but rather, that you need to use some specific mapping in order to perform some specific computation.
This much should be clear: there are no logical values out there in the world that you could input into a computer. Believing so is a category error. Hence, one indeed uses a mapping in order to compute something.
[
[QUOTE=wikipedia]
An uninformative prior or diffuse prior expresses vague or general information about a variable.
[…]
The simplest and oldest rule for determining a non-informative prior is the principle of indifference, which assigns equal probabilities to all possibilities.
[/QUOTE]
](Prior probability - Wikipedia)
Sure, but the point is that x AND y =/= x OR y; that is, they both implement different logical functions of their logical inputs, i.e., different computations. On the computational theory of mind, a given mind is associated with a given computation. Say, mind M is implemented using computation C[sub]M[/sub]. But if I can now make an arbitrary decision regarding whether a given physical system implements C[sub]M[/sub] or !C[sub]M[/sub], where the latter is obtained from the former by exchanging all ANDs with ORs and vice versa, then it’s not at all clear whether !C[sub]M[/sub] implements the same mind—and indeed, it seems very unlikely, as the latter yields different (logical) outputs for the same (logical) inputs as compared to C[sub]M[/sub].
So if this is right, then it’s not an objective fact of the matter, under the computational theory of mind, that my physical system instantiates a mind—i.e. that it implements C[sub]M[/sub], rather than !C[sub]M[/sub]. But this is fatal to the theory.
Worse, even if one can somehow argue that both computations yield the same mind, one would still have to contend with the fact that similar changes of mapping suffice to make the system compute the NAND, or, in fact, any other binary function of two variables. And certainly, there can be no question then that what a system computes ultimately comes down to a choice of mapping by the user—which leads to vicious circular regress for the computational theory of mind.