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.

Wittgenstein, Tractatus Logico-Philosophicus #6.54

My Propositions serve as elucidations in the following way: anyone who understands me eventually recognizes them as nonsensical, when he has used them — as steps — to climb beyond them.  (He must, so to speak, throw away the ladder after he has climbed up it.)

He must overcome these propositions, and then he will see the world aright.

Sun Tzu, The Art of War, Chapter XI #38

At the critical moment, the leader of an army acts like one who has climbed up a height and then kicks away the ladder behind him.  He carries his men deep into hostile territory before he shows his hand.

I haven’t heard or seen too many uses of the concept of “throwing away the ladder.”  It seems interesting, though coincidental, that it shows up in these two places.

Wittgenstein is discussing the end of philosophy, how once you understand his statements in the Tractatus, you will understand how to move beyond thinking in those terms.  And then everything will be solved.

Sun Tzu, on the other hand, is discussing how a leader can get the most out of those under her command by preventing retreat.  The famous examples are of Hsiang Yu, and later Cortez, who burnt their ships behind them to prevent mutiny and ensure that their troops would fight as if their lives depended upon it (because they did).

Sun Tzu and Wittgenstein may be two of the most commented upon authors of all time.  However, I don’t think either could have the other’s meaning in these passages, or at least I’ve never seen any commentary to that effect.  However, this does not mean there is nothing to be learned:

For Wittgenstein, the recognition of the nonsensical is what is doing the work.  His words are nonsensical and the realization of this is what allows you to move beyond them, to something better (says he).  So by doing as he says, by recognizing his words as nonsensical, your retreat is prevented, because no one, save a mad man, would willingly return to a nonsensical philosophy when a better one exists.  By climbing the ladder, you also discard it.

Compare this to Philosophical Investigations #309:

What is the aim in philosophy?– To shew the fly the way out of the fly-bottle.

The fly-bottle, a supposedly one way process, Wittgenstein is trying to walk back…  In the Philosophical Investigations he’s trying to climb down the discarded ladder.

Now, Angle I, , is just like her friends ∀ and ∃.  She can be used in a formula such as ∀x∃yz(Pxyz).

But how should we understand what is going on with the failure of the quantified tertium non datur?  With that advent of a third quantifier, what’s to stop us from having a fourth, fifth or n quantifiers?

The Fregean tradition of quantifiers states that the upside down A means ‘for any” and the backwards E mean ‘there exists some’.  So ‘∀x∃yPxy’ means ‘for any x, there exists some y, such that x and y are related by property P’.  For instance we could say that for any rational number x there exists some other rational number y such that y=x/2.

If we, however, follow closer to the game-theoretic tradition of logic, then the quantifiers no longer need take on their traditional role.  The two quantifiers act like players in a game, in which the object is to make the total statement true or false.  In our above example, we would say that backwards E would win the game, because no matter what number upside down A picks, there is always some number that ∃ could find that is twice the number ∀ chose.

Under this view of quantifiers, quantifiers acting as players in a game, there is no reason why there can’t be any number of players.  (Personally, I like the idea of continuing down the list of vowels: upside-down A, backwards E, angle I, then inverted O, O, maybe angle U? Go historical with Abelard, hEloise, and then Fulbert? Suggestions?)

Now, what is it good for? Let’s play a game of Agent Logic!

The purpose of a game of Agent Logic is to determine the loyalties of the agents in that game, i.e. discover any secret agents. A game consists of a particular logical situation, as given by formulae of independence friendly logic, with at least three different agents, each of which is represented by a quantifier: ∀, ∃, angle I, inverted O, etc. Each agent has an associated ‘domain’, and for the game to be non-trivial the intersection of the domains must have at least one element.

A game of Agent Logic is played by determining the information dependencies required to derive the target formulae from the premise formulae. Once the required information dependencies are known, then the strategies and loyalties of the agents have used may be inferred. The simplest solution to a game is one in which an information dependence indicates a loyalty: if an agent has access to certain information, then that agent must have a specific loyalty.

The person running the game is the Intelligence Director, given by the quantifier angle-I. This is you! All other agents are possible opposing Intelligence Directors or secret agents of the opposing Intelligence Directors. It is your job to figure out who has given who access to information and how that agent has acted upon it. Any information or strategy that is not derivable from the premises are considered acts of treason against you, the Intelligence Director. If the target premise (conclusion) is derivable from the premises alone, no determination of loyalty can be made.

The ‘domain’ of angle-I consists of what you depend upon, i.e. what you believe to exist and what you believe the other agent’s believe to exist. (Though it is a premise itself.)  Recall that the backslash, \, means ‘is dependent upon’ and the forward slash, /, means ‘is independent of’.

premise:

1. \ (
∀\ (a, b, c),
∀/∃,
∃\ (a, b, c, d),
a, b, c, d
)

In this ‘domain’ of angle-I, the Intelligence Director is dependent upon ∀ depending upon the existence of a, b and c, and being independent of  ∃, that ∃ depends on the existence of a, b, c and d, and the director herself depends upon the existence of a, b, c, and d.

premise:

2. ∀xPx

target (conclusion):

3. Pd

Now, since angle-I depends upon ∀ not depending upon d, there is no way to derive the target from the premises. However, since ∃ does depend upon d, if ∀ depends upon ∃, then agent ∀ has access to d.

Therefore, given treason,

4. ∀\ (∃\(d))               [premise of treason - ∀ receives information from ∃, specifically d ]

5. Pd                                  [instantiation from 2, 4]

This shows that the conclusion can be reached if ∀ is treasonous, a secret agent of ∃, i.e. ∀ is loyal to ∃ and not angle-I.

However, if we want to describe the game of Rock Paper Scissors – not just the payoff scheme – how are we to do it?

Ordinary logics have no mechanism for representing simultaneous play.  Therefore Rock Paper Scissors is problematic because there is no way to codify the simultaneous revelation of the players’ choices.

However, let’s treat the simultaneous revelation of the players’ choices as a device to prevent one player from knowing the choice of the other.  If one player were to know the choice of the other, then that player would always have a winning strategy by selecting the option that beats the opponent’s selection.  For example, if Player 1 knew (with absolute certainty) that Player 2 was going to play rock, then Player 1 would play paper, and similarly for the other options.  Since certain knowledge of the opponent’s play trivializes and ruins the game, it is this knowledge that must be prevented.

Knowledge – or lack thereof – of moves can be represented within certain logics.  Ordinarily all previous moves within logic are known, but if we declare certain moves to be independent from others, then those moves can be treated as unknown.  This can be done in Independence Friendly Logic, which allows for explicit dependence relations to be stated.

So, let’s assume our 2 players, Abelard (∀) and Eloise (∃) each decide which of the three options he or she will play out of the Domain {r, p, s} .  These decisions are made without knowledge of what the other has chosen, i.e. independently of each other.

∀x ∃y/∀x

This means that Abelard chooses a value for x first and then Eloise chooses a value for y.  The /∀x next to y means that the choice of y is made independently from, without knowledge of the value of, x.

R-P-S: ∀x ∃y/∀x (Vxy)

The decisions are then evaluated according to V, which is some encoding of the above rubric like this:

T means Abelard wins; F means Eloise wins.  R-P-S means play more Rock Paper Scissors!

–

Johan van Benthem, Sujata Ghosh and Fenrong Liu put together a sophisticated and generalized logic for concurrent action:
http://www.illc.uva.nl/Publications/ResearchReports/PP-2007-26.text-Jun-2007.pdf

[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.

The answer is that it is better to switch because the probability of winning after switching is two out of three, whereas sticking with the original selection leaves the contestant with the original winning probability of one out of three. Why?

The trick to understanding why this occurs is to view the situation not from the contestant’s viewpoint, but from Monty Hall’s. At the outset, from Monty’s point of view, the contestant has a one out of three chance of guessing the correct door. In the likely situation (two out of three) that the contestant chose wrongly, Monty then has to know where the prize is among the two remaining doors in order to open a door that does not reveal the prize. So Monty opens a door not revealing the prize and asks the contestant whether he or she would like to switch or not.

However, the contestant knows that in the likely (two out of three) situation that the initial choice was wrong, Monty had to know where the prize was in order to open the door that did not contain the prize. Since the contestant knows that Monty has to know where the prize is to make the correct choice, the contestant can (in this likely case) place him or herself in Monty’s shoes. At this point Monty knows that the remaining door is the one that contains the prize, and hence the contestant should switch.

If we consider the unlikely situation in which the contestant initially chose the door with the prize behind it, then this line of reasoning will not work. Imagine that Monty forgets the location of the prize every time the contestant guesses correctly. In this situation he can still open either of the remaining doors without ever ruining the game. From his perspective the location of the prize is unrelated to his actions; it played no part in his decision to open one door or another (he merely chose a door the contestant hadn’t).

So, in the one out of three case where the contestant initially selected the correct door, there is no way to deduce whether switching is beneficial based upon placing oneself in Monty’s shoes:  the situation where Monty has forgotten the prize’s location is indistinguishable from a situation in which he has not forgotten. Without any way to further analyze the situation and tilt the odds to over one out of three, the contestant should always assume that he or she is in the previous, more likely, situation and take the opportunity to switch.¹

.

¹Imagine that the contestant has a guardian angel that will let the game run its course if the contestant switches doors, but will change the location of the prize such that if the contestant sticks with the original door the angel will make sure that the contestant wins four out of five times. Then the probability of winning while switching will stay at 2/3 but the probability of winning while sticking will be 4/5. If the contestant had some way of divining that this was happening, this would be a case in which further analysis would be of benefit.

Now say you are a single celled organism that has the option to reproduce sexually.  To reproduce you need to increase yourself to 3/2 your original size and find a similar mate.  Then you both contribute 1/2 to the new organism and repeat indefinitely.

Asexual reproduction requires you to double in size; sexual reproduction requires only a 3/2 increase.  Therefore the turn-around time for sexual reproduction is inherently shorter than for asexual reproduction (assuming there are viable mates readily available).

Is there a selective benefit to a shorter turn around time for reproduction?  If the species must constantly be adapting to a changing environment (that would be everyone), then having a higher rate at which new mutations (and thence adaptations) are introduced into the population is critical.

Secondly, given that there is enough food but it takes time to collect, I count more offspring for sexual reproduction:

In sexual reproduction, there is an additional child from the first generation of children (as compared to asexual splitting) created in the same amount of time: At the +50% mark #1 & #2 mate to create #5, and #3 & #4 mate to create #6.  Then, at the 100% mark (or plus an additional 50%) #1 & #2 mate to create #7, #3 & #4 mate to create #8, and, at the same time, the initial children #5 & #6 mate to create #9.  #9 is also one generation ahead of the offspring of asexual replication.

Now, to be honest, I’m confused.  I don’t think that anything above is particularly complicated.  However, Wikipedia does not note this as a benefit of sexual reproduction.  It actually says that asexual reproduction is much faster.  This makes me think that I must have made a mistake or else someone would have added it.

The going theory appears to be that since every organism in an asexually reproducing species can give off children, then there is twice the potential for offspring.  This completely ignores any struggle that an organism might have that would prevent it from reproducing, or that work can be split with a mate making it easier to reproduce.

My main assumptions are, among others, that there already is a significant population of organisms, the organisms are not too fussy about their mates (no significant waste of energy searching for a mate),  energy / work is being split with the mate, and that the limiting factor has to do with gathering food.  I can’t see how, if these (reasonable?) assumptions hold, sexual reproduction isn’t the dominant, winning strategy.

The short version is that I’ve made computers try to survive the real world.  By real world, I mean my program contains lots of little files that make decisions, and these decisions are about buying and selling stocks, based upon actual real-time data available on the internet.  The decision engines (or ‘orgs’, as I like to call them) that correctly predict the movement of the stocks make money and eventually replicate.  Those orgs that are unsuccessful at predicting stock movements lose money and die off.  The replication process is governed by genetic algorithms that include various mutations.

The short short version is that the program is a cross between a stock market program and a tomagotchi (digital pet).  You host a colony of organisms that survive by ‘eating’ (buy and selling) stocks; it acts as your own personal hedge fund.

Anyway, I could use a tester or two, so if anyone here wants to participate, send me an email.  I’ll get around to writing up more details about the program soon too.

—————————————————————–

In other news,  I’ve finally gotten around to updating the NYC Area Philosophy Calendar.  Someone even sent me a nice email asking if I was still going to do it (before I got around to it.. busy busy) and another person even asked if they could start adding events.

Hmmm, interest in the calendar (it only took 2 years).  An actual object (program) that came from studying philosophy (original theory of biology, 2004.).  It’s taken some time but I feel like I must be moving up in the world.

But is there any gamesmanship to the ‘Deal or No Deal’ gameshow?  The short answer is: No.

The show begins with the contestant choosing a briefcase that contains a number that represents a real monetary amount.  The case is chosen from a group of 26 cases, with the monetary amounts ranging from a penny to a million dollars.  Recently, to up the suspense, the show has removed some of the lower amounts of money and replaced them with more million dollar cases.

The show I saw had 8 of the 26 cases carrying the million dollar value.  So when the contestant makes the initial selection, there is a slightly less than 1/3 chance of picking a million dollar case.  This case is then set aside.

The contestant then proceeds to pick other cases which are immediately opened, revealing the monetary amount they represent.  These cases are removed from the pool of cases.  After a few cases have been removed, the contestant is offered a sum of money to stop playing.  If many of the cases that have been removed were low in value, i.e. most of the million (and other high value) cases remain, then the offer will be closer to the high value cases.  If many of the high value cases have been removed, then the offer will be closer to the lower values.  Usually the value is somewhere in the middle.

These offers are made periodically when there are many cases remaining and are made after every case for the last few.  If you go all the way to the end, then you receive whatever value is in the case you initially selected.

If winning the big prize is the goal, however, all the offers are completely irrelevant.  At the outset the case the contestant chooses has a 1/3 chance of containing the big prize.  This doesn’t change throughout the game.  Let me explain why:

The rest of the cases have the same approximate ratio of million dollar values to non-million dollar values, which the contestant chooses to open randomly.  Therefore most of the time (logically speaking and whenever I watched) this ratio stays constant all the way to the end of the game.  2 cases out of the last 6 were million dollar cases in the episode I just saw.

Of course the possibility exists that the contestant will choose all of the lower value cases such that only million dollar cases remain and hence the case he or she initially chose will necessarily be a million dollar case.

However, imagine this analogous situation.  Try to pick all the cards other than Jack, Queen, King and Ace out of a shuffled deck without looking.  What will happen is that a selection of cards will be chosen irrespective of value, randomly, leaving approximately the same ratio of face cards to non-face cards remaining  (Go try it if you don’t believe me).  The chances of picking only the low values are very small.  Deal of No Deal has been on for years here in the USA and this has never happened.  The recent, and only, million dollar winner still had to choose on the last remaining case. So this part of the game has little ultimate impact upon knowing whether or not you have selected a million dollar case.

Secondly, since the cases are opened randomly during the show, no Monty Hall-like insight can be gained as to whether or not a winning case was initially selected.  Therefore the initial probability of 1/3 remains unchanged throughout the show and all the song and dance of selecting and opening the cases is a red herring (though it is top notch song and dance provided by Mr. H. Mandel and models).

This leaves the contestant in the position of deciding whether or not to accept the offer made to stop playing part way through the game without any new information.  Since the ratio of remaining monetary values remains somewhat constant, the offer made to buy the contestant out of playing should remain somewhat stable for most of the game.  It appears however, according to Wikipedia, that the initial offers are kept artificially low to build suspense, but at the end the offers are where the mathematicians say they should be.

The decision then comes down to how badly the contestant wants/ needs the money.  If the money offered to stop playing becomes large enough to significantly, to the contestant’s mind, make a big difference, he or she will likely take the money rather than take the 2/3 chance of winning significantly less.  This is what happened during the episode today: after it was made known late in the game that a sponsor was going to make a matching donation to a national charity the lady supported, she became too afraid of losing the large amount of money that was already offered, even though she said she wanted to go till the end.

In the end, the deal with ‘Deal or No Deal’ is that it is a great deal for those who get to play.  However, it is not much of a game.  The only trick is to get yourself on the show and after that how much you take home is up to luck.

There are people with blue eyes

this statement can be represented as (with the domain restricted to people):

∃x(Bx).

This statement says that there is at least one person with property B, blue eyes. So the ‘Ex’ is doing the work of looking at the people in the domain (all people) and picking out one with blue eyes.  Without this ‘∃x’ we would just have Bx, or x has blue eyes.

This concept of ‘ranging over’ and selecting an individual with a specific property out of the whole group works in the vast majority of applications.  However, I’ve pointed out a few instances in which it makes no sense to think of the domain as a predetermined group of objects, such as in natural language and relativistic situations.  In these cases the domain cannot be defined until something about the people involved are known, if at all; people may have a stock set of responses to questions but can also make new ones up.

So, since the problem resides with a static domain being linked to specific people, I suggest that we find a way to link quantifiers to those people.  This means that if two people are playing a logic game, each person will have their own quantifiers linked to their own domain.  The domains will be associated with the knowledge (or other relevant property) of the people playing the game.

We could index individual quantifiers to show which domain they belong to, but game theory has a mechanism for showing which player is making a move by using negation.  When a negation is reached in a logic game, it signals that it is the other player’s turn to make a move.  I suggest negation should also signal a change in domains, as to mirror the other player’s knowledge.

Using negation to switch the domain that the quantifiers reference is more realistic/ natural treatment of logic: when two people are playing a game, one may know certain things to exist that the other does not.  So using one domain is an unrealistic view of the world because it is only in special instances that two people believe the exact same objects to exist in the world.  Of course there needs to be much overlap for two people to be playing the same game, but having individual domains to represent individual intelligences makes for a more realistic model of reality.

Now that each player in a game has his or her own domain, what is the activity of the quantifier?  It still seems to be ranging over a domain, even if the domain is separate, so the problem raised above has not yet been dealt with.

Besides knowing different things, people think differently too.  The different ways people deal with situations can be described as unique strategies.  Between the strategies people have and their knowledge we have an approximate representation of a person playing a logic game.

If we now consider how quantifiers are used in logic games, whenever we encounter one we have to choose an element of the domain according to a strategy.  This strategy is a set of instructions that will yield a specified result and are separate from the domain. So quantifiers are calls to use a strategy as informed by your domain, your knowledge.  They do not ‘range over’ the domain; it is the strategies a person uses that take the domain and game (perhaps “game-state” is more accurate at this point) as inputs and returns an individual.

The main problem mentioned above can now be addressed: Instead of predetermining sets objects in domains, what we need to predetermine are the players in the game. The players may be defined by a domain of objects and strategies that will be used to play the game, but this only becomes relevant when a quantifier is reached in the game.  Specifying the players is sufficient because each brings his or her own domain and strategies to the game, so nothing is lost, and the domain and strategies do no have to be predefined because they are initially called upon within the game, not before.

I don’t expect this discussion to cause major revisions to the way people go about practicing logic, but I do hope that it provides a more natural way to think about what is going on when dealing with quantifiers and domains, especially when dealing with relativistic or natural language situations.