Tag Archives: logic

What are Quantifiers?

What are quantifiers?  Quantifiers have been thought of things that ‘range over’ a set of objects.  For example, if I say

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.

Posted in epistemology, game theory, logic, philosophy. Tagged with , , , , , , .

Dependence Logic vs. Independence Friendly Logic

I picked up Dependence Logic: A New Approach to Independence Friendly Logic by Jouko Väänänen. I figure I’ll write up a review when I am finished with the book, but there is one chief difference between Dependence Logic and Independence Friendly Logic that needs to be mentioned.

On pages 44-47 when describing the difference between Dependence Logic and Independence Friendly Logic Väänänen says,

The backslashed quantifier,

∃xn{xi0,…,xim-1}φ,

introduced in ref. [20], with the intuitive meaning:

“there exists xn, depending only on xi0,…,xim-1, such that φ,”

The slashed quantifier,

∃xn/{xi0,…,xim-1}φ,

used in ref. [21] has the following intuitive meaning:

“there exists xn, independently of xi0,…,xim-1, such that φ,”

which we take to mean

“there exists xn, depending only on variables other than xi0,…,xim-1, such that φ,”

The backslashed quantifier notation is part of what Väänänen calls ‘Dependence Friendly Logic’, and is equivalent to the ‘Dependence Logic’ that the rest of the book expounds. This backslash notation makes the difference between Dependence (Friendly) Logic and Independence Friendly Logic clear by showing that the former logic takes the notion of dependence to be fundamental whereas the latter takes independence to be fundamental. Väänänen takes this to be an advantage because he says that Dependence Logic avoids making

one ha[ve] to decide whether “other variable” refers to other variables actually appearing in a formula ?, or to other variables in the domain…

However, this treatment misses an important philosophical difference between Independence Friendly Logic and Dependence Logic. Dependence Logic is fundamentally based upon Wilfrid Hodges work, ‘Compositional Semantics for a language of imperfect information’ in Logic Journal of the IGPL (5:4 1997) 539-563, in which Hodges lays out a compositional semantics for languages such as Independence Friendly Logic using sets of assignments instead of individual assignments to determine satisfaction (T or F). Väänänen infers that Independence Friendly logic is just a bit unruly when it comes to specifying variables because he is working within a system that assumes sets of assignments are a useful and unproblematic way to determine satisfaction.

However the unseen problem of using sets of assignments is that something is added by assuming the domain is a set. For example, let’s take try to define a location and take the set of all the points in the universe. However, we immediately run into relativity: All locations are defined relative to each other and the people trying to figure out where things are, i.e. There is no predetermined set of all the points in the universe. The issue is that the domain of potential assignments, the objects in the universe, may be dependent upon the person or people using them (the players of the semantic game in this case). If the domain is dependent upon the players, the set cannot be constructed until after the players have begun the game. Therefore, if we postulate that the domain is a set at the outset then the players know something about the game that they are playing, namely that it does not depend upon them because it was predetermined.

Following this line of thought it seems possible to constructed a game in which the domain {Abelard, Eloise} is such that Abelard and Eloise are the actual people playing the game and the formula is ‘Someone x lost the game by instantiating this formula’ such that whoever instantiated that formula would win the game according to the rules. But then the formula would not be satisfied, so that player would have lost, but then it would be satisfied, a paradox. It is easy enough to declare that the domain must be independent of the players, but again this signals something about the game being played to the players before the formula to be is revealed.

Lastly there is something to be said about using logic to represent natural language here too: if you consider the set of all possible responses to some question, you are not ever considering all possible responses, but all the possible responses you can think of at that time. Therefore if we are using game semantics and imperfect information to represent natural language, then it is a mistake to predetermine the domain of all possible responses separate from the people involved. Again, the domain being linked to the people involved is at odds with the domain being a predetermined set.

Long story short, there is a very good reason for not always using sets of assignments to determine satisfaction. Depending on the situation, a set may offer non-trivial information about a game or misconstrue the game being played. Independence Friendly logic makes no assumptions about the type of game being played and is therefore of greater scope than logics that are based upon Hodges work. Of course one is free to use sets of assignments to determine satisfaction and derive set-theoretic results, but the compositionality gained comes at the price of limiting the types of games that can be played.

Posted in fun, game theory, independence friendly logic, internet, logic, philosophy, Relativity. Tagged with , , , , .

The Monty Hall Problem

[check out my more recent Monty Redux for, perhaps, a clearer exposition]

The Monty Hall Problem illustrates an unusual phenomenon of changing probabilities based upon someone else’s knowledge. On the game-show Let’s Make a Deal the host, Monty Hall, asks the contestant to choose one of three possibilities – Door One, Two or Three – with one door leading to a prize and the other two leading to goats. After the contestant selects a door, another door is opened, one with a goat behind it. At this point the contestant is allowed to switch the previously selected door with the remaining (unopened) door.

Common intuition is that this choice does not present any advantage because the probability of selecting the correct door is set at 1/3 at the beginning. Each door has this 1 out of 3 chance of having a prize behind it, so changing which door you select has no effect on the outcome.

In hindsight, this intuition is wrong. If you initially selected the first goat and then switch when you get a chance, you win. If you selected the second goat and switch, you win. If you selected the prize and switch, you lose. Therefore if you switch, you win 2 out of 3, whereas if you do not switch you win only 1/3 of the time.

So what has gone horribly wrong here:

  1. Why is most everyone’s intuition faulty in this situation?
  2. How does switching doors make any difference?
  3. When did the 1/3 probability turn into a 2/3 probability?

At the beginning of the game you have a 2 out of 3 chance of losing. Likewise the game show has a 2 out of 3 chance of winning (not giving you a prize) at the beginning of the game. Both of these probabilities do not depend upon which door the prize is behind, but only upon the set-up of a prize behind only one of three doors. For instance, an outside service (not the game show) could have set everything up such that both you and the game show would be kept in the dark: there would still be 2 goats and a prize, but neither you nor the game show would know which door led to the prize.

Now imagine that it is the game show that is playing the game. The game show is trying to win by selecting a goat. From this perspective, whichever door that was chosen is good: this door has a 2 out of 3 probability of being a winner (being a goat). Therefore when given the opportunity to change (after the outside service opens a door and shows a goat), there is no reason to do so.

Of course you, the contestant, are the one making the selection, and you do not want a goat. However, if you imagined yourself in the position of the game show at the beginning, as trying to select a goat, you would reasonably assume that, just as the game show did, you were successful in choosing a goat. When given the choice to switch, now that the other goat has been removed, it seemingly makes sense to change your selection.

In this case the easiest way to view the situation is in terms of how to lose, or by considering all the possible outcomes (as mentioned above). Though this is a guess, it seems that our first blush reaction to this problem is always to view it in terms of winning and this is the reason we do not immediately recognize the benefit in switching. We start out with a 1/3 chance of winning and switching doors doesn’t immediately seem to increase this percentage.

To answer how switching doors makes a difference we need to look more closely at the doors. The door that was initially selected has a 1 out of 3 chance of being a prize, and this does not change. If you were to play many times and ignore changing doors, then you would win 33.3% of the time. At the outset the other two doors each have the exact same chance of being a winner, 1 out of 3. So the other two doors combined have a 2 out of 3 chance of containing a winning door.

Now the game show changes the number of doors available from 3 to 2, with one door guaranteed to contain a prize. If you were presented this situation without knowledge of the previous process, then you would rightly put the chance of selecting the prize at 1 out of 2, 50%.

However, you know something about the setup: The door that was initially selected had a probability of having a prize behind it set at 1 out of 3. The thing behind the other door, though, has been selected from a stacked deck: Whatever is behind the door was selected from a group of objects with a 2 out of 3 chance of containing a prize (1/3 + 1/3). You know that the odds on this door are stacked in your favor because the game show knowingly reveals the goat: In the 2/3 case in which you have previously selected a goat, the prize is behind one of the other two doors. When the game-show reveals (and removes) a goat, it guarantees that the prize is behind the last door. Therefore switching doors at the end is equivalent to combining and selecting the probability associated with the two doors not initially selected.

If the game show did not knowingly reveal the goat, you would not be able to take advantage of the stacked deck. Imagine that you select the first door and then another door is opened randomly, revealing a goat. By randomly eliminating this door (and not looking behind the unselected doors) the door that was initially selected becomes unrelated to the present choice: Only by looking behind the unselected doors does the initial selection become fixed in reference to the other doors. Since no one looked behind the doors, some bored, but not malicious, demon could have come and switched whatever was behind the selected and remaining door and neither you nor the game-show would be able to tell. Therefore switching doors when a goat is randomly revealed provides no advantage because the initial selection cannot be related to the probable location of the prize.

Only when the contestant can fix the probable locations of the prize because the location of the prize is known by the game-show, is it possible to assign interdependent probabilities on the location of the prize and the previous selection made. The odds are then tilted in the contestant’s favor by switching away from the low probability initial selection to the door that has the combination of remaining probabilities.


The logic of this needs to be represented game-theoretically with the different quantifiers representing different players of a game of incomplete information. The game would run* like this:

Domain={prize, goat, goat}

Contestant Game Show
1. ∃x∃y∃z∀a/x,y,z∃b∀c/x,y,z(a=x & b=y & c=z)
2. ∃y∃z∀a/x,y,z∃b∀c/x,y,z(a=g & b=y & c=z)
3. ∃z∀a/x,y,z∃b∀c/x,y,z(a=g & b=g & c=z)
4. ∀a/x,y,z∃b∀c/x,y,z(a=g & b=g & c=p)
5. ∃b∀c/x,y,z(p=g & b=g & c=p)
6. ∀c/x,y,z(p=g & g=g & c=p)
7. ∀d∀c/x,y,z(d=g & g=g & c=p)
8. ∀c/x,y,z(g=g & g=g & c=p)
9. (g=g & g=g & p=p)

Line 1 is the initial setup of the prize game: the goal is for the contestant to make his or her placement of the prize and goats match the game show’s placement. Whatever is on the left side of an = will be what the contestant thinks is behind a door and what is on the right of an = will be what the game show puts behind the door, such that each = represents a door. If the formula is satisfied then the contestant will have successfully guessed the location of the prize.

Lines 2, 3 and 4 represent the results of the Game Show placing the prize and goats. Line 5 is the result of the first move of the contestant choosing where he or she thinks the prize is: the ‘a/x,y,z’ means that whatever placed in spot a has to be done independently, i.e. without knowledge, of what x or y or z is. Then the game show reveals a goat behind one of the doors not selected by the contestant. Line 7 represents the choice that is given to the contestant to switch his or her initial placement of where the prize is. Line 8 is the important step: since the contestant does not know what is behind the doors (c/x,y,z) it looks as if there is no advantage to switching. However, the contestant does know that when making a choice to reveal a goat in line 6 that at this point the game show had to know what was behind every door. This means that c is dependent upon b which was depended upon x, y, and z. With this knowledge the contestant can figure out that there is an advantage to switching because the selection of b in line 6 fixed the locations of the prize & goats and in doing so fixed the odds. Since the odds were intially stacked against the contestant, switching to the only remaining door flips the odds in the contestant’s favor, and is done so in this example. Line 9 shows that all the contestant’s choices match up with what the game show has placed behind the doors and hence she or he has won the prize.

* To do a better representation would require keeping the gameshow from not placing a prize anywhere by using a line like ‘x≠y or x≠z’. For graphical brevity I left it out.

Posted in game theory, independence friendly logic, logic, measurement, philosophy. Tagged with , , , .

Paradox Analysis

Apropos my earlier rant on people who think that paradoxes are meaningless, I figured I ought to take a stab at giving some meaning to paradox. To this end I reformulated a paradox in my terms. I suppose I should called it the Mirror Paradox, though ‘Looking-Glass Paradox’ seems more lyrical and has an historical nod. My apologies to whoever actually came up with this first, though I am sure I haven’t heard it before…

In my room I have a full length mirror. If I look at the man in the mirror and point to him saying, “There I am!” then where am I? If I am the man on the other side of the mirror, then I am not sitting on this side of the mirror. However, the man on the other side of the mirror has just pointed at me and said that he is not on his side of the mirror, but on mine. So I am not on my side, nor is he on his side. But then neither of us are on our side or on our mirror self side.

Now with semantic paradoxes and the like, we don’t have an agreed upon framework for analyzing what is going on in a paradox. Many times it is a paradox that signals that some such theory is unsatisfactory. However, this paradox deals with locations of people, namely me and mirror me, and we do have a general consensus on judging objects’ locations: in physics we determine some object’s location with reference to some previously agreed upon location.

Let us ignore for the moment that mirrors do not actually open up into other dimensions that you could enter if only your reflection didn’t get in the way. All that is important is that we have an exact double of yourself that at the instant you declare that you are where he or she is, that person does the exact same thing.

Declaring your location relative to your reflection is no different than declaring your location relative to anything else. Your reflection simultaneously declaring its location relative you is likewise unproblematic on its own. However, since the two non-identical perspectives are associated with only one person, we have a disconnect between perspectives and the person who holds the perspectives.

This problem of perspective is most telling. In Russell’s Paradox, there is no problem, no obvious contradiction that is, until the question “Is this set self-membered?” has been asked and answered twice. The first time through is arbitrary, let us assume no: Russell’s set is not part of itself. Now we ask, “If it is not self-membered, then is this set not self-membered?” Now we answer yes and have arrived at a contradiction. There is no problem yet, we merely have to revise our assumption: let us assume that Russell’s set is included in itself. Of course, then we ask, “If it is self-memberd, then is this set not self-membered?” and our answer is no: contradiction. At this point the paradox exists, but not before. It required us to look at the one set from two perspectives, one in which it is self-membered and one in which it is not.

The comparison of assumptions and perspectives that is drawn here is a good one. Our perspective, in a different sense, is our background assumptions. When we have contradictory perspectives on a subject we have incompatible background assumptions. The Mirror Paradox pulls our background assumption of location out of the background. We all assume that one perspective is associated with one location, but when you declare that you are someplace else and your reflection does the same, then you end up with two perspectives.

We can’t tell before hand whether we can have more than one perspective or a set that is defined by non-self-membership. Therefore, since the problem occurs with the selection of assumptions or perspective, the meaning of paradoxes, semantic or otherwise, is that your fundamental background assumptions are problematic. Sure, each paradox will only pertain to that particular system that it exists in, but for that system it will signal the most important and deep underlying problems.

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A side note: I thought of this while in bed last night and didn’t look at a mirror until this afternoon, even though I do have that full length mirror. Then I actually did point and say “There I am!” It was a bit of a strange experience because for some fraction of time I felt like me and my mirror self were in some sort of vortex with the rest of the world frozen outside. Almost needless to say I was a bit surprised if not shocked- I wasn’t expecting a reaction. When philosophy grabs you, even for an instant, it is spooky. I suggest you try this and see if you have the same reaction if only because I don’t think there are other paradoxes to actually participate in, save becoming a very methodological barber. How often do you get to participate in an experiment that isn’t prefaced by ‘thought’? Between the small mirror in my bathroom and my full length mirror, the full length elicited a better reaction, so use a full length one if you can.

Posted in logic, philosophy, Relativity. Tagged with , , , , , , .

Are Paradoxes Meaningless?

Aaron Cotnoir has suggested that people think that paradoxes are meaningless.  I think they are lucky that they hadn’t suggested that to me unless they wanted to see me freak out.

It was my good fortune to have my first real exposure to the work of Frege, Russell and Wittgenstein be from Thomas Ricketts.  I can’t remember verbatim what he said, but this is close:

No one knows how long it took Frege to understand what Russell had written in his letter (Russell’s Paradox), be it a few seconds, a minute, ten minutes or a few hours.  But we do know that at that moment his entire universe collapsed.

Only out of gross ignorance of history can anyone believe that paradoxes are meaningless.  Frege’s project up until Russell came along and spoiled everything was, at least in part, to give a firm foundation for mathematics based solely upon logic.  With just a few laws coupled with his newfound quantification he was able to provide a seemingly consistent theory and then also provide sophisticated philosophy of language to bolster his views.

There was probably a moment when Frege allowed himself to dare to think he’d solved one of the greatest mysteries of the universe.  Not only had he legitimately and demonstrably changed mathematics forever, but the ramifications of his theory were obviously far-reaching into philosophy and science.  Then Russell sent him that letter that struck at the very axioms of his theory.  It was a jugular shot and I can’t see Frege feeling other than like all the blood had been drained from his body.  Everything he had worked for was put in jeopardy.

So if anyone believes that paradoxes are meaningless, I suggest to go read some history.  Paradoxes can destroy. Any theory that comes along and says paradoxes are meaningless, is garbage.

Posted in logic, metaphysics, mind, philosophy. Tagged with , , , , , , , .

Positive and Negative Biological Time

In my biorelativity series I used mutations per generation as a measurement of distance. However, with my recent historical/generative musings, specifically the post on the logical foundations of biorelativity (the logic of which is at the foundation of how I arrived at biorelativity), I fear I may have ignored the distinction between a mutation and an adaptation.

Consider an organism with some feature. The feature can be considered both a mutation or an adaptation depending on what the organism is being compared to. If the organism is being compared to another organism, then the feature is likely to be called a mutation. If the organism is being discussed in reference to the ecosystem, then the feature will be referred to as an adaptation.

Now I am sure that there may be some technical properties/definitions having to do with genetics or whatnot that distinguish mutation and adaptation. This is not my concern, though, because in my arguments the two can be used interchangeably.

What does concern me is that there are different sets of related concepts associated with the two words. An adaptation is, to my ear, always a positive thing. A mutation can be good or bad, e.g. mutant freak. By this line of thought adaptations are useful mutations, a subset.

Since mutation is the measurement of time and adaptation is only those mutations which are useful, then we can use adaptation to signify the forward motion of biological time (and forward change of a species as adaptations per generation) which will almost always be what people are discussing (“as time marches on, as things adapt…”). Conversely, to describe biological time going backwards, we could say something like ‘unmutating’.

——

On a slightly different note it is interesting that that there is no word for adapting in the opposite direction: it’s a significant gap. Unadapting? This could imply mere stagnation; the idea here is to think of what it would mean to be adapting in a way to specifically undo previous adaptations. I think a word like this does not nor cannot meaningfully exist: the logical/grammatical structure of adaptation presupposes forward progress.

Consider, “If there were a verb meaning ‘to believe falsely’, it would not have any significant first person present indicative.” (Philosophical Investigations Part II Section x)

“The species is currently *counteradapting*” — It just makes no sense.

Posted in biology, evolution, logic, measurement, philosophy, science, time, wittgenstein. Tagged with , , , , , , , , , .

The Logic of Biological Relativity [draft]

How can we represent biological relativity in logical notation?

Organism a is adapting relative to organism b

Aab

Organism b is adapting relative to a

Aba

Organisms a and b are adapting relative to each other

Aab & Aba

This schema is unsatisfactory because it describes the situation from an indeterminate outside perspective: a and b are said to be adapting relative to each other without regard to the observer describing the situation. Relativity applies to all the perspectives in question (with special focus on any observer perspective) and hence we need a way to include the observer perspective. This means we need to take into account how the observer is adapted such that the observer(s) can be compared to the organisms in question.

To remedy this problem let quantifiers range over organisms and include witnesses to identify the specific organisms in question:

For any organism x, for any organism y, there exists an organism z and there exists an organism u such that x is adapted relative to y according to organism z, and y is adapted relative to x according to organism u.

(∀x)(∀y)(∃z)(∃u)A[xyzu]

Unfortunately this formulation is insufficient because witness z is logically dependent upon both x and y (as is u as well) and we want z to only witness x and u to only witness y: as both z and u are dependent upon both x and y, both x and y must be chosen before selecting z and u. This means that organisms x and y are selected (logically) independent of the witness organisms defeating the purpose of having those witnesses.

Getting around this difficulty is not trivial in first order logic. There is no way in first order logic to linearly order the four quantifiers such that z only depends on x and u only depends on y (Kolak & Symons p.249 [p.40 of the pdf]). Independence Friendly logic suffices though :

(∀x)(∀y)(∃z/∀y)(∃u/∀x)A[xyzu]

This statement says that for any organism x, for any organism y, there exists an organism z that does not depend on y and an organism u that does not depend on x, such that organism x as witnessed by z, and organism y as witnessed by u, are adapted relative to each other.

However, though this statement gets very close to describing biological relativity, if we consider how the witnesses witness the organisms, i.e. how z witnesses the organism x, there is a problem. By stating that z witnesses x and that z is independent of y, the statement ‘x is adapted relative to y as witnessed by z’ is nonsense: since z is independent of y it could not be a witness to ‘x adapting relative to y.’ Likewise for u.

The solution is simple enough though:

(∀x)(∀y)(∃z/∀x)(∃u/∀y)((x=z) & (y=u) & A[x,y])

By letting x=z, making z independent of x and dependent on y, z witnesses y from the perspective of x without requiring x to be chosen before z. Likewise for u: if y=u, u is logically independent of y and u is dependent on x, then u may be chosen before y, u is dependent as a witness to the choice of x and witnesses x from the perspective of y. Perhaps more prosaically: x and y are adapting relative to each other, as witnessed by organisms z and u (who have the equivalent adaptations respectively to x and y), and it is not necessary to predetermine what those adaptations are.

Posted in biology, evolution, fitness, game theory, independence friendly logic, logic, measurement, Relativity, science. Tagged with , , , , , , .

The Logic of Relativity [draft]

How can we represent relativity in logical notation?

a is moving relative to b

Mab

b is moving relative to a

Mba

a and b are moving relative to each other

Mab & Mba

This schema is unsatisfactory because it describes the situation from an indeterminate outside perspective: a and b are moving relative to each other without regard to the observer describing the situation. Relativity applies to all the perspectives in question (with special focus on any observer perspective) and hence we need a way to include the observer.

To remedy this problem let quantifiers range over perspectives and include witness individuals to identify the specific perspectives in question:

For any perspective x, for any perspective y, there exists a perspective z and there exists a perspective u such that x is moving relative to y according to witness z, and y is moving relative to x according to witness u.

(∀x)(∀y)(∃z)(∃u)M[xyzu]

Unfortunately this formulation is insufficient because witness z is logically dependent upon both x and y (as is u as well) and we want z to only witness x and u to only witness y: as both z and u are dependent upon both x and y, both x and y must be chosen before selecting z and u. This means that perspectives x and y are selected independent of the witness perspectives defeating the purpose of having those witnesses.

Getting around this difficulty is not trivial in first order logic. There is no way in first order logic to linearly order the four quantifiers such that z only depends on x and u only depends on y (Kolak & Symons p.249 [p.40 of the pdf]). Independence Friendly logic suffices though :

(∀x)(∀y)(∃z/∀y)(∃u/∀x)M[xyzu]

This statement says that for any perspective x, for any perspective y, there exists a perspective z that does not depend on y and a perspective u that does not depend on x, such that perspective x as witnessed by z, and perspective y as witnessed by u, are moving relative to each other.

However, though this statement gets very close to describing relativity, if we consider how the witnesses witness the perspectives, how z witnesses the perspective x, there is a problem. By stating that z witnesses x and that z is independent of y, the statement ‘x is moving relative to y as witnessed by z’ is nonsense: since z is independent of y it could not be a witness to ‘x moving relative to y.’ Likewise for u.

The solution is simple enough though:

(∀x)(∀y)(∃z/∀x)(∃u/∀y)((x=z) & (y=u) & M[x,y])

By letting x=z, making z independent of x and dependent on y, z witnesses y from the perspective of x without requiring x to be chosen before z. Likewise for u: if y=u, u is logically independent of y and u is dependent on x, then u may be chosen before y, u is dependent as a witness to the choice of x and witnesses x from the perspective of y. Perhaps more prosaically: x and y move relatively to each other, as witnessed by z and u (who have the equivalent perspectives, respectively to x and y), and it is not necessary to predetermine what those perspectives were.

A time variable rounds everything out nicely:

(∀t)(∀x)(∀y)(∃z/∀x)(∃u/∀y)((x=z) & (y=u) & M[t,x,y])

So, at time t (say now) let’s let u be your (the reader’s) perspective and z be my (the author’s) perspective. Then this statement describes our current motions as relative to each other because my perspective depends upon y, which is your perspective and your perspective depends on x, which is my perspective. Success!

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Posted in game theory, independence friendly logic, logic, measurement, physics, Relativity, science. Tagged with , , , .