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 D_i(Heads, Tails),
Я_ix
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.

In The Genial Gene Joan Roughgarden seeks to replace the competitive understanding of evolution, known as sexual selection, with a cooperative one. The first sentence of her book reads, “This book is about whether selfishness and individuality, rather than kindness and cooperation, are basic to biological nature” (p. 1).

So what is the argument? Taking this first line, she wants to conclude something about basic biological nature. To do this, one can either define what basic biological nature is and then use that definition to derive conclusions, or else survey the natural world and find the best interpretation for whatever empirical results were found. She opts for the latter strategy.

To this end she first surveys and compiles examples of what people consider to be evidence for sexual selection and argues that this evidence has been misconstrued or simply does not support the theory of sexual selection. Then she offers a few logical arguments against sexual selection with the aim to highlight contradictions within the theory.

She then develops her alternative, called Social Selection. Social Selection is fundamentally based upon cooperation, not competition, and she proceeds to reinterpret the empirical research with respect to cooperation. Given the results of this reinterpretation, she concludes that the cooperative approach provides a more accurate picture of the empirical data than the competitive approach. Therefore social selection, not sexual selection, is fundamental to biological nature.

Can this argument be maintained?

Her argument fundamentally turns on the interpretation of the empirical research. (If her logical arguments were strong enough to undermine sexual selection on their own, she would have dedicated more space to them. At best, in my opinion, they could raise questions about sexual selection, but are not inherently damaging enough, even if they are accepted uncontested, to force a major revision to sexual selection.) She interprets the research in terms of cooperation and her opponents are those who interpret the research in terms of competition. Roughgarden claims her interpretation is the correct one.

Insofar as she is making an inference saying her interpretation is the best conclusion, her argument fails. She readily admits that the defenders of sexual selection are able to consistently create explanatory fixes for apparent contradictions in the empirical research. Since they are able to explain the data, the fact that she is unsatisfied by their explanations (and likes her own better) is no grounds for convincing her opponents to give up their explanations. After all, they have history and authority on their side. Her coming up with better numbers, that is, having formulas that (she says) more accurately represent the research, is not a sufficient reason for discarding a theory that has held up for many years, especially one that continues to be an area of active research. So, she has not successfully argued that social selection should replace sexual selection.

However, if we consider a more modest conclusion, then Roughgarden may be able to maintain part of her argument. She makes the point that the core Darwinian theory does not include sexual selection; it is a later contribution (ppg. 3-4). This suggest that there may be theoretical room for cooperation in addition to competition. But how much room?

Now the interpretive problem that she set up cuts the other way. Instead of her trying to convince us that her cooperative interpetation of the empirical research is the correct one, we ask the competitive interpretation why it is the best one. Empirical research alone cannot support one conclusion over another: the data must first be interpreted before a conclusion can be reached. As mentioned above, sexual selection has history and authority on its side, but age and endorsements are not arguments for being the sole fundamental methodology of biological nature. Without history and authority, sexual selection proponents only have their ability to explain bioogical research, which is no more than Roughgarden has. Therefore, advocates of sexual selection have no further theoretical resources to support their claim that sexual selection is the fundamental method working in evolution.

This means that Roughgarden does succeed in part. Based on the arguments she provides she is unable to maintain that kindness and cooperation underpin evolution, but she is able to cut sexual selection down to her size. She has shown that it is possible to reinterpret biological research in terms that do not rely upon competition and that sexual selection has no special theoretical privelege. Therefore sexual selection proponents can no longer claim to be fundamental biological reality: even though Roughgarden was unable to fell their theory, they won’t be able to down her either, and so she has established theoretical room for cooperation in Darwinian theory.

Define Biological Energy as the ability to do work, the ability to change the environment.  Then Fitness can be related to Energy because the higher the fitness the greater the ability to change the environment.

E ∝ f

If we consider an organism that lives in a place with infinite resources – a Garden of Eden – and also replicates at the speed of the chemical reaction of replication – there is no maturation process, it immediately starts to replicated as soon as it is created – then it’s life is identical to it’s replication process.  Define d to be the speed of the chemical process of replication.  Then the ability of this organism to change the environment is given by it’s fitness, the rate it replicates at and it’s life:

E = fd²

Or something.

Instead, I came up with “Death Implies Economy”.  What this means is that we are fundamentally limited in time and resources, and hence we cannot afford to waste what little we have on unnecessary complication.  DIE is a metaphysical justification of ontological parsimony:  regardless of how we come to the knowledge of death, the principle only requires that we are fundamentally limited and is agnostic as to how we come to understand this of ourselves.  [One may revise the principle to 'Demise Implies Economy' without problem or changing acronym.]

Now, the reason I wanted to figure out Occam’s Razor was because I thought it might help me understand entropy better.  Entropy seems to be this force or cause that basically is always at work and does whatever we don’t want it to.  Jerk.  Of course the universe has no reason to conform to our way of doing things, or worse, my way of viewing the world, but entropy just seems to be excessive:  why should our physical science be subject to a form of energy loss?  This makes me think it is our fault.  No, not ‘fault’, but intrinsic part of how we go about our science.  My apologies to the universe for calling it a jerk.

So, back to Occam’s Razor and DIE.  If DIE underpins Occam’s Razor, then we are metaphysically bound to proceed in a piecemeal manner.  Even our most radical theories are not developed by immortals with no care for time.  So, in some sense, our theories are also fundamentally limited and hence will always admit some unknown factors as a metaphysical consequence.

It is fair to ask if this is all just a fancy way of stating pessimistic induction, “Since we haven’t gotten theories perfect in the past, we shouldn’t expect to in the future”?  How can I make the claim that we will never succeed in this scientific endeavor?

My answer is that these questions raise legitimate issues, but the specific question at hand is not to speculate on what will happen with future theory but how we are to understand entropy and simplicity now.  And to question whether our adherence to ontological parsimony has the theoretical consequence of an unresolvable force.   Since we must believe the theories we have, at least to some extent, whatever these theories do not describe must be left in an accordingly deep mystery– as the result of an unexplained force at least as powerful as the forces we do explain. Therefore I have to conclude that, given a metaphysical understanding of Occam’s Razor such as DIE, there is a legitimate concern of inevitable unresolvable causal consequences which could manifest as various forms of entropy.

The short short short version of the theory is this general postulate: one organism’s traits are another’s environment and vice versa.  Hence all competition can be viewed as environmental phenomena.  This gives Natural Selection as a result of Fitness and an environmental factor, which I refer to as Acceleration.

If you want to see the paper as it stands now, you can access it here or below.[6in/120mm ebook formatted]

Fitness is the resistance to change in the rate of change of the species.

Acceleration is change in the rate of change of the species.

Natural Selection = Fitness × Acceleration

4.2.1 Two Kinds of Fitness

To understand Natural Selection we need to understand fitness and how to calculate its value. One way the fitness of an organism can be understood is in terms of how well it will be able to interact with its ecology to acquire what it needs to live and reproduce. The traits of the organism will be crucial as it struggles to survive: every little adaptation or edge that the organism has can be the difference between survival and death. Therefore the traits of the organism determine its fitness.

However, the fitness of an organism is dependent upon its environment. The different situations an organism finds itself in, which are determined by the ecology and chance, will determine its ability to reproduce. For example being fast is meaningless if there is no secure footing to run on. Therefore it is the situation that determines the traits that matter and hence fitness is a function of environmental selection.

At this point it can look as if there are two distinct and incompatible methods for calculating the fitness of an organism: trait based selection and environmental selection.

4.2.2 The Equality of Trait Selection and Environmental Selection

Imagine a jaguar out in the jungle. Unbeknownst to anyone, however, his welfare is being carefully monitored by stealthy scientists. Any time the jaguar might be in trouble, be it a lack of food or an unfriendly competitor, the scientists step in and protect the jaguar from harm and do it without being seen.

An independent observer, someone who doesn’t know about the scientists watching over the jaguar, might think that the jaguar has an uncanny ability to find food and avoid dangerous situations. He might suspect that the jaguar has excellent ears that can hear danger from very far away and a nose that can smell even the faintest waft of food. He would believe that in the struggle for survival the jaguar was incredibly well adapted.

Ought we to smile at the man and say that he errs in his conclusion? I do not believe we should. We could be in the very same position as the jaguar. We like to think that we have evolved the way we have by struggling and adapting. However, we may have just as easily been assisted by some benevolent but reclusive extraterrestrials. They could be the reason our species has been able to accomplish all that we have, and we would not know.

Regardless of the existence of any such extraterrestrials, the example shows that we cannot tell the difference between struggling and surviving based upon traits, and nature conforming (or disconforming) to our adaptations. It is a matter of perspective to believe either that our adaptations were the cause of our success or if it was the environment that happened to favor us.

4.2.3 The Natural Selection Field

Instead of switching back and forth between environmental and trait selection, we can say that both kinds of selection create a field. This field is ontologically as basic as the two kinds of selection and it is what interacts with the individual organisms and environment. The interactions of an organism and the field determines the course of the organism’s life, and an ecology’s total field is determined by everything in it.

Although every organism and each ecology is unique, none are alien. By looking at similar organisms and similar ecologies, we can use natural history to determine important adaptations and key environmental features. Taken together these features specify the shape of the Natural Selection field of that ecology, which informs us on how an organism or species will interact with their environment.

An organism’s overall fitness will determine how great its effect will be in the Natural Selection field. Introducing a species with high fitness into a new ecosystem can cause great changes, whereas introducing a species into an environment that it cannot survive in will barely create a change at all. For example, when humans, with our high fitness, move into a new area, we will profoundly alter that ecology. However, if we bring a flower with us that can’t survive the cold nights in our new home, then the flower will die, barely registering any change in the Natural Selection field.

4.2.4 General Relativistic Natural Selection

With the existence of the field we can say how evolution acts upon a species. At every moment an organism interacts with a natural selection field created by its surrounding ecology. The constant interaction with the field will gradually modify the species by benefiting certain individuals and by putting others at a disadvantage.

Insofar as the natural selection field is indistinguishable from the struggle for survival, we will not be able to further analyze why species change: this theory is terminal in the same way as General Relativity. If we could show that the way organisms and species benefitted or were put at a disadvantaged by the environment, without regard to the individual adaptations of the organisms, or conversely show how an adaptation increased an organism’s fitness without regard to the environment, then an investigation into these specific phenomena could yield insight into why a species changes. However, since we cannot make this distinction, the natural selection field is the final answer as to why a species changes.

Unlike the previous theory, general relativistic natural selection is wider because it is applicable during rapid ecological changes. The prior theory of natural selection relied upon trait based analysis to determine future reproductive success and hence was unable to accurately predict success during rapid change. Relativized natural selection can say that the organisms and species experiencing a disaster (or utopia) are experiencing a change in the natural selection field. This change in the natural selection field manifests as a rapid change in the lives of the organisms. Once the ecological change is finished, then we can revert back to the old notion of natural selection.

[this is an excerpt from a longer paper, which can be found here]

Patched a bunch of things together to make a nice story.  Fixed the little issue about fitness being circular.  Expanded natural selection to apply more generally.  Causal structure.  Epistemological foundations.  ooOoOO0Ooooooo.

And it’s good fun.  I swear.  Epistemology, history of physics, evolution… makes me happy.  You should really read it.

Download here. [pdf, 304kb]

[1] …(∃z) … Pz …

This says that at some point in the theory there is some object z that has property P.

After much hard work, we discover that the object z with property P can be described as the combination of two more fundamental objects w and v with properties R and S:

[2] …(∃z) … Pz … ⇒ …(∃w)(∃v) … (Rw & Sv)…

Now lets say that in our theory, any object that had property P depended upon some other objects, x and y:

[3] …(∀x)(∀y)…(∃z) … Pz …

In our revised theory we know that objects w and v must somehow depend upon x and y, but there are many more possible dependence patterns that two different objects can have as compared to z alone.  Both w and v could depend upon x and y:

[4] …(∀x)(∀y)…(∃w)(∃v) … (Rw & Sv)…

However, let’s say that w depends on x but not y, and v depends on y but not x.  Depending on the rest of the formula, it may be possible to rejigger the order of the quantifiers to reflect this, but maybe not.  If we allow ourselves to declare dependencies and independencies, arbitrary patterns of dependence can be handled.  The forward slash means to ignore the dependency of the listed quantified variable:

[5] …(∀x)(∀y)…(∃w/∀y) (∃v/∀x) … (Rw & Sv)…

Besides the convenience and being able to represent arbitrary dependence structures, I think there is another benefit for this use of the slash notation:  theoretical continuity.  In formula [2] above, there is a double right arrow which I used to represent the change from z to w and v, and P to R and S.  However, I created this use of the double right arrow for this specific purpose;  there is no way within normal logic to represent such a change.  That is, there is no method to get from formula [3] to formula [4] or [5], even though there is supposed to be some sort of continuity between these formulas.

Insofar as the slash notation from Independence Friendly Logic allows us to drop in new quantified variables without restructuring the rest of the formula, we can use this process as a logical move like modus ponens (though, perhaps, not as truth preserving).  Tentatively I’ll call it ‘Hypothesis Introduction’:

[6]

1. …(∀x)(∀y)…(∃z) … Pz …
2. …(∀x)(∀y)…(∃w/∀y) (∃v/∀x) … (Rw & Sv)…      (HI [1])

The move from line one to line two changes the formula while providing a similar sort of continuity as used in deduction.

One potential application of this would be to Ramsey Sentences.  With the addition of Hypothesis Introduction, we can generalize the Ramsey Sentence into, if you will, a Ramsey Lineage, which would chart the changes of one Ramsey Sentence to another, one theory to another.

A second application, and what got me thinking about this in the first place, was to game theory.  When playing a game against an opponent, it is mostly best to assume that they are rational.  What happens when the opponent does something apparently irrational?  Either you can play as if they are irrational or you can ignore it and continue to play as if they hadn’t made such a move.  By using Hypothesis Introduction to introduce a revision into the game structure, however, you can create a scenario that might reflect an alternate game that your opponent might be playing.  In this way you can maintain your opponent’s rationality and explain the apparently irrational move as a rational move in a different game that is similar to the one you are playing.  This alternate game could be treated as a branch off the original.  The question would then be to discover who is playing the ‘real’ game – a question of information and research, not rationality.

via BackRe(Action)