Authored by: Douglas Reay.

Hypothetical Definitions

In mathematics we define most things precisely, and usually the solutions to problems are presented as a step by step progression starting from clearly defined things and finishing with an answer that is proven to be a correct solution.

But it is sometimes useful to jump straight to consideration of a final solution, asking “Suppose there were a solution to this problem, let us call this solution X, what properties must X have?”

This can be a useful approach and, even for problems that turn out to have no valid solutions, we would still consider the signifier “X” to be something that we have succesfully ascribed meaning to: we could talk about “X” to other mathematicians and they would respond back to us in a manner that we can predict based upon the supposition that they share with us an understanding of what “X” is intended to refer to.

The same applies when we consider the meaning of words, rather than the meaning of mathematical symbols.  Words can acquire meaning inductively, based upon sense data, the way a child learns language.  But words can also be defined deductively, as solutions to problems set up by applying logical conjunctions to propositions or by analogy.

And, as with “x”, just hypothetical definitions can be useful and should be considered meaningful, even if there is no actual thing to which they can refer, or it is unknowable as to whether the referent exists.

Brain in a Vat

Putnam (Putnam, 1981) argues that a Brain in a Vat couldn’t think or reason about Brains or Vats because the thing it referred to as “Brain” could only be the simulation of a Brain fed to it by the supercomputer maintaining its illusory world, rather than the actually piece of flesh sitting in the jar.   However suppose the Brain engaged in thoughts upon the following lines:

I am going to refer to the world accessible to my sense data as “World 1″, and use the phrase “Type 1 Brain” to refer to the sort of physical thinking organ that my senses tell me exist in World 1.

I am going to refer to the virtual world running on the computer on my desk as “World 2″, which is a subset of World 1, and use the phrase “Type 2 Brain” to refer to the locus of thought of any being whose perceptual universe is limited to World 2.

I am going to refer to any hypothetical superset of World 1 that is larger than World 1 as “World 0″, and use the phrase “Type 0 Brain” to refer to a hypothetical type of thing that bears the same relationship to Type 1 Brains as Type 1 Brains do to Type 2 Brains.

The phrase “Type 0 Brain” does indeed contain the word “Brain” which the Type 1 Brain originally defined, as a child, by reference to parts of World 1 that it could sense.  But the phrase itself is more than the sum of its constituent parts.  It is a hypothetical definition, and as such can be said to meaningfully refer to things the Type 1 Brain will never directly sense.

If, for instance, the Type 1 Brain wrote down in on a piece of World 1 paper “Hello World 0, please reply”, a computer programmer in World 0 examining the bits of data in the computer running the World 1 simulation could decode this message and program the simulation to make appear a piece of paper in response, carrying the message “Hi, this is Frank, how are you enjoying my simulation, would you like me to make you a pet unicorns ?”.   They could engage in  a meaningful dialog, even though the Type 1 Brain would still be limited to its Type 1 senses.

And, just as a mathematician can consider what properties a valid solution would have to have, and perhaps elliminate some from consideration by demonstrating that two contradictory properties would both be required, by engaging in thought about our hypothetical World 0, and then looking around our World 1 and noticing the lack of pet unicorns, we can start to place constraints upon what World 0 is likely to contain.

Doubt all that can be doubted?

Descartes (Descartes, 1641) advocated starting by doubting all that can be doubted.   But that takes us nowhere, because everything can be doubted.

The Ken Thompson Hack (Thompson, 1984) explains why, once the security on a machine has been fully breached, you can not necessarily detect this by using the compromised tools.  The same applies to reason.  We may think that valid syllogisms always produce true conclusions if given true premises.  Indeed, that may even be how we define what “valid” means.  But our only means of deciding which syllogisms are valid depend upon our memory and those same syllogisms.

Suppose Frank, a World 0 computer programmer, in order to write a paper on the nature of logic, decided to see if he could create a World 1 simulation in which all the intelligent inhabitants thought that “If A can see B, and B can see C, then A can see C” was a valid syllogism.  Could he do it?  Well, he might have to ensure that evolution gave the beings in the simulation visual telepathy so that, in most cases, the reasoning did work in practice and the beings didn’t die off.  But yes, there is no intrinsic reason why he could put a routine in the simulation that ensure that whenever a particularly philosophically inclined World 1 being started trying to reason it out, the conclusion was places in his thought stream that the syllogism was valid.

Indeed, there is no intrinsic difficulty preventing Frank from including in the simulation a method for the beings to arrive at conclusions that was more reliable than reason, such as sacrificing an animal while chanting certain words, then meditating for a few minutes and awaiting an internal emotion or insight inclining the being towards or away from certain options they were deciding between.

Don’t doubt just for the sake of doubting

So, if doubting everything gains us nothing, can we gain anything by doubting some things but not others?

To answer that it is helpful to consider our purpose.  What Descartes was seeking, that he hoped to gain by his method of doubt, was truth and certainty.  That, alas, is overly ambitious, but suppose we think of reason as a tool with many uses.  People can try to use the tool to find pure truth, but it can also be used in everyday life to helping us avoid mistakes and delusions.  So, setting aside the possible existence of other tools, let us restrict ourselves to the purpose of honing the tool of reason, and ask ourselves how we may use doubt to increase the effectiveness and reliability of this tool.

Reason

The first thing we can allow ourselves to stop doubting is reason itself, because doubting it has no predictable advantage for our purpose.  Our purpose allows us no outside check on the effectiveness of  reason compared to outside tools, and, if we are being deluded by some higher agency into thinking that various invalid syllogisms are valid, there is nothing that reason can tell us about the consequences of that so, as far as reason is concerned, following the syllogisms we believe to be valid might be just as effective at reaching true conclusions as rejecting them.  And, since we have no basis to decide which syllogisms to reject as pranks by Frank, and Frank could just as easily be fooling us into thinking invalid some syllogisms that actually are valid, we do not reduce the false-positive rate of our tool by not making the assumption.  So:

WORKING ASSUMPTION 1 : First-order logic is sound

Probability

Deductive reasoning, by itself, can’t take us far enough to affect our decisions.  It needs something to work upon, and for that we need inductive reasoning (of the non-mathematical kind) and, for that we need probability.  The minimum set of axioms needed is Zermelo and Fraenkel’s extensions of Paeno, with the addition of the axiom of choice, which mathematicians refer to as ZFC.

WORKING ASSUMPTION 2 : ZFC

Occam’s razor

Duns Scotus wrote (translated) “Plurality is not to be posited without necessity” which is a dictum that can be traced back to much earlier philosophers, and is best known in the version attributed to Occam.  The reason why this is wise advice has now be demonstrated by Hutter (Hutter, 2005) who has proven that, other things being equal, the simpler theory is actually more likely to be true (Reay, 1988).   This is a key result however, since it is a direct consequence of our first two working assumptions, we don’t need to list it as an additional one.

Reality

So we now we have a way to make decisions, we have a decision to make.  Consider the standard mental model of how ‘reality’ functions.   There is an external world which our body perceives through our senses that translate into electrical and chemical changes in the physical brain, some of which our conscious ‘mind’ is aware of and experiences subjectively as qualia and these, plus what we think of as our direct perception of our memories and thought processes constitute our internal worldview based upon which our conscious (and subconscious) mind reacts or makes decisions that direct our body and our further thoughts and emotions, which get stored as further memories.

Which parts of this model must we choose not to doubt, for our purpose, or are there at this point several possibilities that still leave reasoning as an effective tool?

Some would argue that we might be brains in a vat, so we should definitely doubt the reliability of an apparent regularities in the external (World 1) reality, and so the logical starting point is to accept our own existance and the qualia as that ‘we’ perceives them.

Others argue that various psychiatric ailments are known to interfere with memory and sense of self; and that the body’s perception system has known defects (eg optical illusions) which makes assuming an objective shared external reality a more reliable starting point.

If we craft the wording of our next working assumption carefully enough, we can remain agnostic on this point, accepting both possibilities, and using Occam’s Razor to allocate probabilities to them.

WORKING ASSUMPTION 3 : There is sufficient regularity to the reality that can be accessed or modeled by the reasoning process we identify as our own being, that we can apply inductive reasoning to it with better than random results.

Time

A corollary of adding this third working assumption is the nature of time, because it assumes the concept of results – of testability.  The concept of having a model of how reality works, using reasoning to form a prediction based upon that model, then later getting a result (an action guided by that prediction achieving or failing to achieve the anticiapted result) at a later time.

It is logically possible that Frank only started his simulation running 5 seconds ago, and everything we think happened before that time is merely what we were programmed to think by the initial starting state.  It is also possible that there are regularities in the simulation, but that at midnight Frank is going to upload a software patch that changes the laws of nature in the simulation, and that everything will work differently from that point on.

This is known as The Problem of Induction.  Just because induction has worked in the past, that’s no evidence that induction will work in the future, because that would be using induction to support induction which is circular reasoning.

However we can bootstrap induction supporting induction if we can find an initial reason to think that induction is even just slightly more likely to be correct than not.  And Occam’s Razor gives us that reason, because the theory that Frank will upload a software patch a midnight contains an additional piece of information (the time of the change), which increases its Chaitin-Kolmogorov complexity (Chaitin, 1987).

Our third working assumption isn’t quite the same as ontological naturalism or Hume’s Principle of Uniformity of Nature, but it leans sufficiently far in that direct that we can perform physics, while still leaving open the technical possibility of a World 0 and supernatural intervention in World 1.

And alternative formulation of our third working assumption might be “There are, in our lives, at least some decisions (choices between possible options for an action to take) where reason can help us predict what some of the likely consequences would be of choosing each option, with better than random odds.”

From ‘Is’ to ‘Ought’

These first three working assumptions are sufficient to let us function in reality, and act towards goals.  But they tell us nothing about what goals to set.  For that we need one final step:

WORKING ASSUMPTION 4 : At least some of the decisions in our life will matter.

This a safe assumption to make because, if none of the decisions matter, then our decision (to make a false working assumption) doesn’t matter (because it is one of the decisions we made in our life).

Once we have this fourth working assumption, we may apply Occam’s Razor to the various competing theories of what matters, to find the simplest that has at least as much predictive power as any of the others.

References

Chaitin, G. J. [1987] “Algorithmic Information Theory“ Cambridge University Press ISBN: 0521343062

Descartes, René [1641] “Meditations on First Philosophy” http://www.wright.edu/cola/descartes/

Hutter, Marcus [2005] “Universal Artificial Intelligence: Sequential Decisions Based On Algorithmic Probability” ISBN 3-540-22139-5 http://www.hutter1.net/ai/uaibook.htm

Putnam, Hilary Whitehall [1981] “Reason, Truth, and History” ISBN-10: 0521297761

Reay, Douglas William Windle [1988] “Reay’s Lemma” http://www.toothycat.net/wiki/wiki.pl?ReaysLemma

Thompson, Ken [1984] “Reflections On Trusting Trust” http://cm.bell-labs.com/who/ken/trust.html