Tag Archives: measurement

Яandom Logic

If we try to represent tossing a coin or a die, or picking a card out of a deck at random, in logic, how should we do it?

Tossing a coin might look like:

Toss(coin) → (Heads or Tails)

Tossing a die might be:

Toss(die) → (1 or 2 or 3 or 4 or 5 or 6)

Picking a card:

Pick(52 card deck) → (1♣ or 2♣ or … or k♥)

This begs asking, do these statements make sense? For instance look what happens if we try to abstract:

∀x Toss(x)

such that ‘Toss’ represents a random selection of the given object.

But this is weird because Toss is a randomized function and x is not selected randomly in this formula. Perhaps if we added another variable, we could generate the right sort of function:

∀y ∃x Toss(yx)

Then x would be a function of y: we would select x with respect to y. The problem is still that a Toss involves randomness. So this setup is incorrect because treating x as a function of y is not randomized, because y is not random.

How can we represent randomness in logic?

As noted, functions alone will not work. Variables and interpreted objects cannot invoke randomness. Perhaps we can modify some part of our logic to accommodate randomness. The connectives for negation and conjunction haven’t anything to do with randomness either.

But, if we use the game theoretic interpretation of logic, then we can conceive of each quantifier as representing a player in a game. Players can be thought of as acting irrationally or randomly.

Therefore, let’s introduce a new quantifier: Я. Я is like the other quantifiers in that it instantiates a variable.

  1. Яx T(x)
  2. Tb

However, Я is out of our (or anyone’s) control. It does instantiate variables when it is it’s turn (just like other quantifiers) but it instantiates randomly. So we have three players, Abelard, Eloise and Random (or the Verifier, Falsifier and Randomizer).

But more is still needed. We need a random selection between specific options, be it between heads and tails, 1-6, cards, numbers, or anything else. One way of doing this would be to create a special domain just for the random choices. Я would only instantiate from this domain, and if there are multiple random selections, we will require multiple indexed domains.

Hence, given Di(Heads, Tails),
Яix
represents a coin flip since Я randomly instantiates out of the domain containing only Heads and Tails.

(aside:
I prefer to use an artifact of Independence Friendly logic, the dependence indicator: a forward slash, /. The dependence indicator means that the quantifier only depends on those objects, variables, quantifiers or formulas specified. Hence

Яx/(Heads, Tails)

means that the variable x is randomly instantiated to Heads or Tails, since the only things that Яx is logically aware of are Heads and Tails. Therefore this too represents a coin flip, without having multiple domains.)

Now that we have an instantiation rule for Я we also need a negation rule for it. If some object is not selected at random, then it must have been individually selected. In this case the only other players that could have selected the object are ∀ and ∃. Hence the negation rule for Я is just like the negation rule for the other quantifiers: negating a quantifier means that a different player is responsible for instantiation of the variable. If neither player is responsible, it can be considered random: ¬Яx ↔ (∀x or ∃x). We can leave the basic negation rule for ∀ and ∃ the way it is.

Therefore, given the additions of the new quantifier and domain (or slash notation), we can represent randomness within logic.

———

See “Propositional Logics for Three” by Tulenheimo and Venema in Dialogues, Logics And Other Strange Things by Cedric Degremont (Editor) College Publications 2008, for a generalized framework for logics with 3 quantifiers. Since the above logic requires either indexed domains or dependence operators, Яandom Logic is a bit different, but it is a good discussion.

Posted in game theory, logic, science. Tagged with , , , , .

Time and the Limits of Science

Measurement takes time; measurement is a process.  So the measurement of time immediately yields this theoretical issue:

Since measurement takes time, our ability to break time into ever smaller pieces will always be proportional to the method of measurement used.  The faster our measurement device that measures time, the more divisible time will be.  Insofar as there are limits to how fast a measurement process can occur (relativistic or other), there will be limits on the lengths of time we can measure. From this perspective, time is discontinuous: there will be a point at which we can no longer split time into smaller pieces.

From a different perspective, time must be continuous: we can start our measurement of time whenever.  Since there are no restrictions on when our measurement may begin, each and every instant must be just as good as every other instant, hence time is continuous.

So which is it: Is time continuous or discontinuous?

Or is the question badly formed? The discontinuity argument is based upon the ideas of measurement and relativity.  The latter argument, for continuity, is based upon what might be considered a fact of modal reality.  Perhaps the two arguments are not talking about the same thing.

I can’t give an end-all be-all answer to the questions of time, but here is my opinion:   Time is continuous, but when we start to do scientific activities, time can and will only be able to be measured discretely.  Therefore the two arguments are not using one word to describe two different phenomena.

The question then becomes how doing science limits what we can observe.

This might sound like an extremely unlikely situation, but consider the case of organized sports.  When playing a sport or game you are bound, restricted, to following certain rules.  However, by following these rules, you and the other players can demonstrate skills and abilities that you otherwise would not have been able to observe:  Lots of people may be in shape, but only a small fraction of those people are professional athletes.  Those athlete demonstrate their superior physical and mental prowess by performing on the game field by being restricted by the official rules.

Getting back to science, does it now seem so unlikely that we restrict ourselves in certain ways in order to accomplish other tasks?  For time to be scientifically useful, we need to have some sort process that has a fixed point from which to start counting from, and a unit to count.  Then we can compare an unknown process to this known process, and we have done so with much success.

This comparison could not have occurred without the introduction of an arbitrary fixed point and unit of measurement: by restricting our concept of time to these particular processes we enable ourselves to perform scientific research.  Research is not possible if we use the unrestricted modal notion: no comparison can be made because there is no inter-modal process to compare a worldly (intra-modal) phenomenon to.  But with the use of fixed points, units and processes, we also become subject to relativistic limitations.  It seems like a very small price to pay considering the success of science.

To sum up: time is subject to modal considerations, which gives it special properties such as being continuous.  Once we start to do science, though, we restrict ourselves to the non-modal aspects of time, which allows us to use it as a tool in scientific research.  This also makes time appear to have different properties, but upon closer study, these properties are artifacts of the measurement process and not time itself.

Posted in measurement, ontology, philosophy, physics, Relativity, science, time. Tagged with , , , .

something about time

There is something about time that I can’t seem to stop thinking about.

We measure time by agreeing upon an event and then counting from that point onward.  Today is October 17, 2008 AD.  It is this AD that keeps my attention.  It has been 2008 years, ten months and seventeen days since the birth of Jesus of Nazareth: AD stands for Anno Domini, or year of our lord.  Those not wanting to be explicitly Christian use CE, which stands for Common Era, which is just a nice way of saying the same thing without recognizing Jesus as the lord.  Wikipedia dates the use of this term to 525 AD, though this is how everyone has been measuring time forever. AD began to be used in 525, but before that people just used other events (like natural disasters, battles worn or lost, etc.) as starting points to count the date from.

The only result is that time is not universal but relative to whenever people agree to start counting from.  This is nothing new, but maybe like The Ring, if I pass it along, then it won’t bother me any more.  If you become similarly afflicted, I apologize, but you know what to do.

Posted in measurement, Relativity, science, time. Tagged with , , , .

Where Does Probability Come From? (and randomness to boot)

I just returned from a cruise to Alaska. It is a wonderful, beautiful place. I zip-lined in a rain forest canopy, hiked above a glacier, kayaked coastal Canada and was pulled by sled-dogs. Anywho, as on many cruises, there was a casino, which is an excellent excuse for me to discuss probability.

What is probability and where does it come from? Definitions are easy enough to find. Google returns:

a measure of how likely it is that some event will occur; a number expressing the ratio of favorable cases to the whole number of cases possible …

So it’s a measure of likelihood. What’s likelihood? Google returns:

The probability of a specified outcome.

Awesome. So ‘probability as likelihood’ is non-explanatory. What about this ‘ratio of favorable cases to the whole number of cases possible’? I’m pretty wary about the word favorable. Let’s modify this definition to read:

a number expressing the ratio of certain cases to the whole number of cases possible.

Nor do I like ‘a number expressing…’ This refers to a particular probability, not probability at large, so let’s go back to using ‘measure’:

a measure of certain cases to the whole number of cases possible.

We need to be a bit more explicit about what we are measuring:

a measure of the frequency of certain cases to the whole number of cases possible.

OK. I think this isn’t that bad. When we flip a fair coin the probability is the frequency of landing on heads compared to the total cases possible, heads + tails, so 1 out of 2. Pretty good.

But notice the addition of the word fair. Where did it come from, what’s it doing there? Something is said to be fair if that thing shows no favoritism to any person or process. In terms of things that act randomly, this means that the thing acts in a consistently random way. Being consistently random means it is always random, not sometimes random and other times not random. This means that fairness has to do with the distribution of the instances of the cases we are studying. What governs this distribution?

In the case of of a coin, the shape of the coin and the conditions under which it is measured make all the difference in the distribution of heads and tails. The two sides, heads and tails, must be distinguishable, but the coin must be flipped in a way such that no one can know which side will land facing up. The shape of the coin, even with uniform mass distribution, cannot preclude this previous condition. Therefore the source of probability is the interdependence of physical conditions (shape and motion of the coin) and an epistemic notion (independence of knowledge of which side will land up). When the physical conditions and our knowledge of the conditions are dependent upon each other then the situation becomes probabilistic because the conditions preclude our knowing the exact outcome of the situation.

It is now time to recall that people cheat at gambling all the time. A trio of people in March 2004 used a computer and lasers to successfully predict the decaying orbit of a ball spinning on a roulette wheel (and walked out with £1.3 million). This indicates that after a certain point it is possible to predict the outcome of a coin flipping or a roulette ball spinning, so the dependence mentioned above is eventually broken. However this is only possible once the coin is flipping or the roulette ball is rolling, not before the person releases the roulette ball or flips the coin.

With the suggestion that it is the person that determines the outcome we can expand the physical-epistemic dependence to an physical-epistemic-performative one. If I know that I, nor anyone else, can predict the outcome until after I perform a task, then the knowledge of the outcome is dependent upon how I perform that task.

This makes sense because magicians and scam artists train themselves to be able to perform tasks like shuffling and dealing cards in ways that most of us think is random but are not. The rest of us believe that there is a dependence between the physical setup and the outcome that precludes knowing the results, but this is merely an illusion that is exploited.

What about instances in which special training or equipment is unavailable; can we guarantee everyone’s ability to measure the thing in question to be equal? We can: light. Anyone who can see at all sees light that is indistinguishable from the light everyone else sees: it has no haecceity.

This lack of distinguishability, lack of haecceity (thisness), is not merely a property of the photon but a physical characteristic of humans. We have no biology that can distinguish one photon from another of equivalent wavelength. To distinguish something we have to use a smaller feature of the thing to tell it apart from its compatriots. Since we cannot see anything smaller, this is impossible. Nor is there a technology that we could use to augment our abilities: for us to have a technology that would see something smaller than a photon would require us to know that the technology interacted at a deeper level with reality than photons do. But we cannot know that because we are physically limited to using the photon as our minimal measurement device. The act of sight is foundational: we cannot see anything smaller than a photon nor can anything smaller exist in our world.

The way we perceive photons will always be inherently distributed because of this too. We cannot uniquely identify a single photon, and hence we can’t come back and measure the properties of a photon we have previously studied. Therefore the best we will be able to accomplish when studying photons is to measure a group of photons and use a distribution of their properties, making photons inherently probabilistic. Since the act of seeing light is a biological feature of humans, we all have equal epistemological footing in this instance. This means that the epistemic dependence mentioned above can be ignored because it adds nothing to the current discussion. Therefore we can eliminate the epistemic notion from our above dependence, reducing it to a physical-performative interdependence.

Since it is a historical/ evolutionary accident that the photon is the smallest object we can perceive, the photon really is not fundamental to this discussion. Therefore, the interdependence of the physical properties of the smallest things we can perceive and our inherent inability to tell them apart is a source of probability in nature.

This is a source of natural randomness as well: once we know the probability of some property that we cannot measure directly, the lack of haecceity means that we will not be able to predict when we will measure an individual with said property. Therefore the order in which we measure the property will inherently be random. [Assume the contradiction: the order in which we measure the property is not random, but follows some pattern. Then there exists some underlying structure that governs the appearance of the property. However, since we are already at the limit of what can be measured, no such thing can exist. Hence the order in which we measure the property is random.]

————–

If I were Wittgenstein I might have said:

Consider a situation in which someone asks, “How much light could you see?” Perhaps a detective is asking a hostage about where he was held. But then the answer is, “I didn’t look.” —— And this would make no sense.

hmmmm…. I did really mean to get back to gambling.

Posted in biology, epistemology, evolution, fitness, independence friendly logic, logic, measurement, mind, philosophy, physics, Relativity, science, Special Relativity, technology. Tagged with , , , , .