On the Holographic Principle and the Fractal Model of the Universe

This is a response prompted by +rare avis asking if I had seen this article and was asked in this post. I am posting my response to Drifting Labyrinths for the sake of posterity.

No, I haven't seen this particular article yet, but it seems to misrepresent the "holographic principle," which is common in popular articles on the matter. The Holographic Principle does not say that we "live in a hologram." And every time someone says this it makes baby Jesus cry and I throw up--just a little--in my mouth.

The "holographic principle" is defined as: ...a mathematical principle that the total information contained in a volume of space corresponds to an equal amount of information contained on the boundary of that space (source).

Here's a thing: we seem to live in at least a four dimensional universe: the three spatial dimensions that define things with volume + the time dimension = spacetime (of Relativity) = 4 dimensions.

So, there is a sense in which our experiences are occurring in a "fractal dimension" where a "fractal dimension" is defined as: a measure of how "complicated" a self-similar figure is. In a rough sense, it measures "how many points" lie in a given set. A plane is "larger" than a line, while S sits somewhere in between these two sets (source).

Put differently, a "fractal dimension" is not a whole number but some number between two whole numbers. The plane in the above definition has 2 dimensions, and the line is described in a single dimension, and the fractal dimension of the set of points that define the fractal S in that plane is greater than 1 but less than 2.

So, similarly, our experience of time can be seen as occurring on the boundary of a fractal that exists in a four dimensional space. So, the fractal dimension, then, is greater than 3 but less than 4, and that is kind of like our experience of the possibly four dimensions of spacetime: we readily experience the 3 dimensions of volume in their completeness (as whole numbers, say) but not so with the fourth dimension of time, which we only experience in increments--as some fraction of the whole--and, really, only one "point" at a time.

So, what the "holographic principle" implies in such a model, is that the information in the four dimensional volume of spacetime can be encoded on the surface of that volume, and this surface is the 3 dimensions of volume we readily experience + the "partial" experience of the fourth dimension of time because existence, as is my conjecture, happens on the fractal boundary of a n-dimensional volume (in this case we are discussing, n = 4).

As I put forward months ago (but stated somewhat improperly at the time with respect to the current experiment to test for such a property) and now restated here with more precision and clarity:

The fractal model of the universe is a possible explanation of the structure of the universe if and only if the holographic principle is a property of our universe.

That is a prediction of the model, in other words, and one that can be verified scientifically by experiment.

Put differently, by my understanding of the fractal model that I am endorsing as a possible model of the structure of the universe this model necessarily requires that the holographic principle be true of our universe, or so it seems to me.

And this is precisely because the experiences that we have seem to be of a fractal dimension that is greater than 3 but less than 4 (if we posit the universe is a 4d structure). In other words, we exist on or within the lesser space that describes the surface or boundary of a larger volume, and in order to have experiences that are derived from that larger volume the information contained in that volume must be available on its surface, which is what the holographic principle says.

Put differently still, in a "block universe" conception of our universe (to use a current model that "fits" the following description), the whole universe already exists as whole and complete--from beginning to end--and is a four dimensional object, which is to say a volume inscribed in four dimensional spacetime. But we don't experience that object and its volume directly. We experience parts of the total information that the 4d volume contains, and we do so exactly because such information is available in total on the greater than 3 but less than 4 fractal "surface" of this 4d volume. That is our experience of the total information as divided in time and is part of the turbulent boundary of the fractal.

A little differently still, the fractal itself is "timeless" and it is defined exactly because it is all possible manifestation of or in the total 4D universe in contrast to all the other possible manifestations that did not occur, and the boundary between these two sets of points define the reality of our experiences: it is the turbulent boundary of a fractal or can be adequately modeled that way, anyway. All the information of the universe is found in that 4D volume and the fractal boundary that is the surface of that volume mirrors this information as the boundary goes towards infinity--which is to say takes on a sort of volume itself that is greater than 3 and less than 4, but is always approaching 4.

---

It is interesting, in a synchronicity sort of way, for the following reason:

I believe this model--or one very much like it--will become part of a future scientific understanding of our universe. This is also a prediction, but not a scientific one--it is an intuitive prediction--although this prediction will be shown true or false in time, so it is a "testable" prediction. I am not aware of anyone who has constructed a similar model as this one, but I am sure it is coming. It only makes sense.

Now, earlier tonight I was on the telephone with an old friend, +Lisa Mizeri, and we hadn't talked for awhile, so we were catching up on all sorts of things, and I mentioned that since I did not get accepted into the Masters of Philosophy program here at the University of Victoria that I was seriously considering starting a BSc in physics.

Part of our earlier conversation was about how, at least in theory, if we live in some odd folding 4D "hyperwhatever" type shape, then we ought to be able to do things, as was her suggestion, like "hiding rows in Excel." We were talking specifically about making travel easier, and in particular travel across Canada. She was saying we should just be able to "click" and hide Alberta, Saskatchewan, Manitoba, and Ontario, and then it would be easier to get from BC, where I live, to Quebec where she and her family currently live. And then we'd simply "click" again, and those provinces would all be revealed once again.

Later, in the conversation, when I mentioned, among other things, that I might go do this BSc in physics, then I joked that perhaps I would figure out how to fold space in such a way that I could open my apartment door, step through it, and step into her and her family's apartment in Montreal and close the door behind me. I have not made it out their way for a real live visit and Montreal would be a pretty cool place to visit in addition to seeing my old friend.

Anyway, I think that an understanding of the universe in terms of a fractal model is going to be at least a step in that direction. And we were joking earlier about my having written a voluminous tome that would somehow survive into the future--as an actual bound book of paper pages--about my various "conjectures" about reality (some of which we discussed peppered throughout the conversation) that would turn out to be true, and folks, some of them anyway, would wear buttons or t-shirts (or whatever the future equivalent of such things would be) that read "Hydomako Was Right."

And of course we laughed about this. And I joked "What the fuck is a 'Hydomako'?" as I am sure most, if they were bothered to, would be prone to ask, heh.

Now, I think it is much more likely that some ingenious person already in the field of physics with a PhD will publish something that is very much like this model, and is more rigorous in terms of its formalization, that is, it will have equations that show such and such and the math will be present to justify the ideas. So people in the future will, if they do at all, wear buttons and t-shirts that say "[Ingenious Physicist with PhD's Name Here] Was Right."

So, I'd be lying if I said I didn't want my name attached to a revolutionary scientific idea. That would be awesome. I think this or a similar fractal model is that revolutionary scientific idea or at least part of it, but I am not sure if I will end up being the one to publish that more rigorous and mathematically sound paper that sparks or is part of such a revolution. I think by the time I earn a PhD in physics (at least five to seven years and maybe more, if my plan to earn a PhD in Philosophy is any indicator of success (going on seventeen years now and still sitting pretty with a BA), yup)--if indeed I ever accomplish such an endeavor--well, someone is likely going to write that paper before I do.

On the other hand, it is more about the idea--this idea, I feel, needs to get out there--and if I have this idea, others do too because that is how ideas and the "collective consciousness" works. So, if you have this idea too, dear reader, and are in a position to make it more rigorous and have also been prompted by some of my "rough work," then, hey, give me a footnote or something, ok?

See also this thread for further work on the fractal model or at least further explication of it in terms of the A & ~A "theory" about the world. With thanks to +sreejith s, of course.

For a whole lot more on A & ~A see my philosophy website.

And if you really want the hardcore stuff, then read this.

E = mc^2

In my less than scientifically rigourous and not necessarily physical formulation of a particular variation on "Relativity Theory," well, the formula in the title can be interpreted as:

Entertainment = misanthropy (times) cynicism (squared).

With that in mind, I present unto you:

The Wall

or

The Philosopher King (For a Day)

If I had my way, I'd have all of ya' shot.

Now, I don't care what "race" ya' are, what the colour of yer skin is, or what yer appearance and features are in general: we're all human beings.

And I don't care what religion you practice (or not): we all have ways to make sense of the world, our place in it, and a means to create meaning from an otherwise seemingly meaningless existence.

And I don't care what you do to yerself: smoke pot; take drugs; drink; do yoga; eat only vegetables; drink too much coffee, man; or whatever else. We are each free--if indeed we are in any sense "free"--to make our own choices regarding our own particular lifestyles insofar as those choices effect (mostly) our own being (and here the qualification of "mostly" because simply through relating our lifestyle choices may effect others, but that is a whole 'nother "can of worms"--let's allow a degree of simplification here--for entertainment value--thanks).

And I don't care what yer sexual orientation is or what kinks you have: see above regarding "what you do to yerself." So long as what we are doing is consensual, well, everyone else can mind their own fucking business and stay the fuck out of another's undies (unless, of course, there's mutual consent there, then, go on, dive right in).

And I don't care what yer position is in the social hierarchy: good, decent people can be found within all the socially defined (however so) "castes."

What I do care about is how each and any of us relate to other people, and this can be often expressed specifically via our "profession," or "career choices," or "jobs" in general.

And I do, more or less, agree with Shakespeare:

Let's kill all the lawyers first.

But of course, we can't simply generalize over all the lawyers, 'cause at least some of them are going to be good, decent human beings who use their profession as a means to help and assist others, so:

Let's kill some but not all the lawyers first.

Get 'em up against the wall.

And then there's the politicians. Again, it's going to be some but not all.

'Gainst. The. Wall.

And some but not all religious leaders.

Get 'em up against the wall.

And some but not all (but probably most) bankers.

'Gainst. The. Wall

Actually, let's do the bankers second--right after the lawyers.

And the landlords: some but not all.

Get 'em up against the wall.

CEOs and other board members, well, most of you, I'd reckon:

'Gainst. The. Wall

War mongers, military chiefs of staff, generals, and people who command large groups of other people to kill yet other people:

Get 'em up against the wall.

My, it's getting pretty bloody in here, isn't it? The air is thick with the smell of gunpowder--let's make sure we are all wearing respirators--and I do hope you are all also wearing protective gear for yer ears.

And what are we gonna' do with all these corpses? I suppose incineration will be the most practical solution: fire up the furnaces, comrades!

I am sure I've left out some other general categories of people as defined by their professions. This is merely an initial sort of modest proposal and we can hash out the fine details as we incinerate the bodies, OK?

Now, the important part, obviously, is coming up with a fair and reasonable criteria that can--much more often than not--separate the "some" from the "all."

How do we do this? It is a tricky affair to be sure, and, well, we're likely gonna' have to break a few eggs, now and then, if we want to make an omelet. Sorry families and friends of the wrongly put down: we'll make sure you are well compensated by the redistribution of wealth this endeavour will certainly entail. Simply think of it in the same light as when you mash a mosquito that hasn't even bitten you yet: there'll always be more of them along that will bite you.

So, the criteria.

I say, as yer pretend philosopher king fer the day, we let people draw the line themselves and then wait for their own hypocrisy to make them cross it.

Yes, each of us is free to be as helpful or not helpful with regards to everyone else as we so chose, but when we take direct action and willfully engage in behaviours that harm others, well, that's gotta' be a strike against us, yes?

You want freedom of speech? You want freedom of thought? You want freedom to express yourself? Have it. Have it all. Stand on a soapbox in the market place and cry out your hate, your hardship, your pain, your frustration, and whatever else: I don't care.

Act on those same things in ways which harm others, well, that's gotta' be a strike.

Paint pictures, make movies, build monuments, write manifestos and tracts, create whatever you want: I don't care.

Take advantage of others via whatever means in order to exploit them, well that's a strike.

Three strikes?

Up against the wall.

It's simple, really: "treat others as you would have them treat yourself, and do not treat others as you would not have them treat yourself."

And, yes, that itself is likely too simple, but, hey, everything looks better on paper--even digital representations of paper--then it does when put into practice; regardless, this is all only entertainment anyway: I have little interest in cleaning up the mess--hypothetical or otherwise.

That said, let's start with the lawyers: get 'em up against the wall.

On the Mistakes of AI Supporters and Detractors

This is a response to a post on Google+ by +Singularity 2045. The original post can currently be viewed here. This post (if no longer available at the time of reading my response below) was based on a link to this page, apparently.

When it comes to pro or con arguments with respect to "machine intelligence," or "AI," the logic on both sides of the argument is frequently lacking.

In the OP Singularity 2045 references a statement of Mark Tegmark, which is presented as:

One thing is certain, and that is that the reason we humans have more power on this planet than tigers is not because we have sharper claws than tigers, or stronger muscles. It’s because we’re smarter. So if we create machines that are smarter than us there’s absolutely no guarantee that we’re going to stay in control,

and Singularity 2045 subsequently critisizes it in the following:

Analogies regarding tigers are only valid if tigers had created the human race via intelligent AI engineering of human brains, or AI design of precursor human brains. The point is our intelligent engineering of AI makes humans utterly different to any unintelligent species below us unable to create higher intelligence.

It is easy to see how, regardless of Tegmark's use of analogy or whether Tegmark is pro or con AI, this statement is possibly true:

If we create machines that are smarter than us there’s absolutely no guarantee that we’re going to stay in control.

The thing here is we need to understand what we mean by "artificial intelligence."

If we merely mean really smart machines that are still machines, that is, they are still entirely constrained by their programing and what we command them to do via that programing, then we can not really "lose control over them" in any meaningful sense.

Things could go wrong with the programing, sure, there could be glitches and bugs, sure, but ultimately by definition, and in virtue of, our programming of these machines, we are always going to be "in control" of them even if our control of them goes sideways due to human error (via glitches and bugs in programming). Put differently, when a machine (as we currently understand them) breaks down or malfunctions or whatever else, our temporary "loss of control" is metaphorical. It's not that we suddenly found ourselves facing a machine that chose to go "out of control," rather, the degree of our control over the machine simply reached a low we would rather it had not.

Put differently still, when we "lose control" over a given machine, we do not suddenly think we are living in the world of Maximum Overdrive.

By means of a simple analogy: we can literally "lose control" of our pet dog, for example, but we cannot literally "lose control" of our pet rock.

However, if by "artificial intelligence" we mean the creation of a machine that can also think for itself, which entails, it can choose how to act and respond to its environment, and is unconstrained by its programing--that is, it is free to alter its programming and is not dependent on our commands--then it is entirely possible that we will "lose control" over it. Indeed, by definition we have little to no control over any "free agent," and if we can exercise control over a "free agent," then such an agent is not truly "free."

Now here's an analogy that is actually workable. We have no idea about other human beings. We think we might "know" them, and we think we might be able to predict their behaviours based on our knowledge of them with respect to previous interactions with them under such and such a circumstance or another.

However, people can be unpredictable. They can surprise us. This is because either:

1) They have something known as "free will" and can choose to act or respond differently with respect to how we predict they might respond, or

2) There is no such thing as "free will," but our universe, in its complexity, is a nonlinear deterministic system, which entails that we can not always predict results; put differently, a nonlinear deterministic system is unpredictable, by definition.

(There is, of course, a third option here in that the universe is a deterministic linear system: this is highly unlikely, in my opinion, but if that is the case, then no one is "controlling" anything, and everything in the universe is merely the playing out of predetermined results based on previous causes, which we could, in theory, anyway, predict if we could figure out what those initial conditions, in fact, were, but this is an aside.)

Now, if we are unable to always and accurately predict the behaviour of other human beings--whose general biological and psychological framework are similar and familiar to our own--then how much less so will we be able to predict the behaviour of a machine intelligence that either has "free will" or is itself able to act, as we do, along the lines of a participant in a nonlinear deterministic system?

In other words, we have a difficult enough time trying to control and predict the behaviours of complex systems (human beings) that are reasonably similar and familiar to ourselves, so there is no reason to think that we can have any greater success in controlling or predicting a machine entity that we have never encountered before.

Therefore, the only logical conclusion to the debate as to whether or not a machine intelligence (of the "free" or "nonlinear" variety) is going to be beneficial to humankind or malevolent is that currently such a question is undecidable.

Logically speaking, we have no good reasons to suppose it is either. Taking a side at this point in time (where we have ZERO experience with such a "machine intelligence") is simply speculative navel gazing based on our own particular prejudices and biases. We have no actual empirical evidence on which to base our conclusions.

We can guess and speculate all we want, but until such an entity is actually in the world, there is nothing upon which to ground our guesses and speculations. It is exactly like trying to decide if alien visitors to the planet would be a boon or a misfortune to humankind: until we actually have alien visitors (assuming we have none already and assuming that if we do, their impact on the world has been hidden from us and, therefore, most of us have no way to adequately assess the matter), there is simply no way to predict the outcome.

Always Forget

I was out at the local market to do some hunting and gathering--OK, mostly gathering--when I spied none other than Kurt Cobain on the cover of this month's Rolling Stone. Some story about little Frances Bean (who, I suppose, ain't that little no more) trying to find her long dead father, whatever that is supposed to mean. I reckon you can just listen to his records, Frances, he seemed to put a lot of himself into them, or so it seems to me.

"Ah yes," I thought to myself, "it is that month, isn't it?"

I always forget the day and I came home to look it up and the anniversary of your death was on April 5th and that was--can you believe it--twenty-one years ago. Wow. I remember when that Nevermind record was the shit. Spoke to and for a whole generation of slackers and Gen Xers and post-punk lost souls wandering in the wasteland of a confusing, unfriendly, and uncaring modern society of idiot fucktards. What's changed, really?

Grunge the media called it. And it invaded the airwaves and help transform the idiot jock fucks into "alternative" cool folks ("he's the one who likes all our pretty songs, and he likes to sing along, but he knows not what it means"--oh the sarcastic irony of it all, Kurt), and brought the hip subculture--or some faction of a hip subculture, anyway--into the pop culture, and the rest is a tragic sort of history. Spawned a shitload of shitty fuckin' bands that sound like Theory of a Dead Nickel Creed.

And I hate that band. Well, "hate" is a pretty strong word. I guess I'd simply be much happier--or, perhaps better, wouldn't have to be bothered to care at all--if they all died in a tragic bus crash or something. Maybe all the masters and copies of all their songs could simply vanish from the face of the Earth along with them. All those bands that are that band--y'know what I mean.

You know, Kurt, that record isn't even all that good. I mean, I happened to put it on the other month and, having not heard it in years, well, it simply didn't do it for me anymore. Sure, there's still some good songs on it, yeah, but it doesn't have the same rawness of Bleach, say, and it certainly lacks the maturity of In Utero, and it's just not as fun as some of the tracks on Incesticide. And like I've ever bothered to listen to the "unplugged album" because, really, what's the fuckin' point? So, in this humble listener's opinion, well, Nevermind is the worst Nirvana album. But it does have few songs that were still pretty good to hear. Well, maybe it is less that it's not a good record and maybe it's more that it was a record that needed the historical context to make it as great as it was for a time. Still the worst of the bunch, if you ask me though.

Maybe Courtney had ya' killed, as some people still figure, or maybe you really did kill yourself, or maybe you wanted to kill yourself but were too scared to end it by your own hand--a lot of people with suicidal ideations are: it is a pretty difficult step to actually take, after all--and you hired someone else to do it for you and maybe some of these people actually know this to be true.

Who knows? I mean, I sure don't, anyway. And I reckon perhaps like the assassination of, say, JFK or like the actual events of 9/11, well, most of us will simply never learn the truth of the matter and the truth will fade into mythology over time. To quote a different Kurt, also no longer with us, who also died in April, and also tragically--but with much less mystery--"and so it goes."

And so you're gone now, and have been for twenty-one years. Crazy. And almost every year I forget that April is the month that marks the anniversary of your death. And every year it simply doesn't seem to matter, anyway, if I remember you or not. And maybe that's what you'd prefer, really. "Oh well, whatever, never mind."

But this year, thanks to a cursory glance across the magazine rack at the local market, well, here's a little tribute to you, Kurt. All in all is, indeed, all we all are. And if that's the case, then we'll see you again in the Western Lands.

What Is an Interpretation?

I've been meaning to write on this subject for awhile now, but recent conversations seem to expedite the necessity of getting clear on what I mean when I use the word 'interpretation'. So, without further fan fare, hand waving, or other particularly useless and time wasting linguistic devices, let's go!

We all interpret. Simple right? We all have a way of looking at the world that frames our experiences along certain lines, which then enables us to express things about how our experiences and observations are related to one and other, and we generally express ourselves using some "natural language" or another. Our natural languages tend to obscure the formal language which underlies our statements. When we express ourselves in a natural language, we are actually also using an underlying layer of reasoning and logic which is necessarily about our interpretative strategies. Put differently, if we are making any sort of statements beyond mere logical formalisms, then reason and logic apply only within our interpretations, but we are unable to use reason or logic in the absence of an interpretation with respect to our use of a natural language about the world.

So, what do we mean by an interpretation? Well, we sort of mean the way we use the word 'interpretation' colloquially: like, for example, when we say things such as "That's only your interpretation" or "Let me offer my interpretation of this painting," and, in some sense, we also are using an interpretative structure when we translate languages from one specific language to another specific language. This latter, though, is more of a "meta" interpretive structure: an interpretation as a  function that assigns elements in one interpretation to elements in another interpretation--we are less concerned about this sort of interpretation than we are about specific interpretations, like those we most often refer to when we use the word 'interpretation' in common speech.

Referring to my notes from Logic I with Dr. Mark Gardiner taken many, many moons ago, we will set out on a more than less thorough presentation of the formal and rigorous definition of an interpretation. Don't worry too much though, I will do my best to translate along the way from the formal definition into a more natural language understanding of such a thing; so, simply plow through the formalisms as best as you are able, and it is my hope that once we come out the other side of the natural language interpretation of the formalisms, the reader will be able to go back to the formal definition with a deeper sense of understanding. So, fasten the seat belts on your brains, then, we're in for a bit of a bumpy ride.

In logic, an interpretation is defined in the following way:

1. An interpretation requires a set of linguistic symbols.

In logic these are referred to as 'wffs' or, as I pronounce it, and I think others do as well, "woofs." This is simply an abbreviation for "well formed formulas." Now, without trying to teach or instruct the reader from scratch, what we mean by a "woof" is simply a string of linguistic symbols that also satisfies the rules of their construction. In symbolic logic, for example 'A & B' is a "woof" but 'AB &' is not a "woof." Another example, 'A -> B' is a "woof" but '->AB' is not a "woof." What we mean, then, by a "well formed formula," is a string of linguistic symbols that conform to specific rules in their construction in such a way that they are able to represent a clear and precise statement that we are then able to use within our system of logic. And what we mean by "a set of linguistic symbols" is all and only those strings of linguistic symbols that conform to the rules governing their construction.

Now, to be entirely as clear as I am able, let's look at the same idea as applied to English.

The sentence, "Sally had a nice day at the beach," would count as a "woof" in English. The sentence "Beach a the at Sally nice had day" is not a "woof" in English. Another example, "Seven is the number that occurs between eight and nine," is a "woof" in English (even though we might observe it is a formula that, when interpreted in terms of English qua mathematics, we see is not true--more on that in a bit). The sentence "Eight the is seven nine occurs and number between that," is not a "woof." So what we mean by a "well formed formula" is that it follows the rules of a grammar such that the sentence is sensible within our structure of interpretation; that is, the string of linguistic symbols occur with a specific sort of order that allows them to be interpreted.

Getting the picture? Good. Let's continue.

2. An interpretation requires a structure.

What the heck do we mean by that?

Here is the formal definition of a structure, U:

U must consist of precisely two things:

1. a nonempty set of individuals or elements, which together constitute a Universe of Discourse also known as a Domain.

2. one or more relations defined on this Universe of Discourse.

Any clearer? No, not really, right? And in order to present this formally and in such a way that is understandable, the reader really needs to have sat through a formal logic class. And it is not my intention to subject anyone to that against his or her will. So, I will refer again to my notes taken a few days later, where we discuss these formal matters in a somewhat less formal way. These notes, by the way, were taken on April Fools day in 1999 (probably the right year, anyway--I can't be particularly bothered to figure it out exactly): make of that what you will. This section of my notes bears the title "Metaphysical Musings."

And I quote:

A structure consists of only two types of things:

1. objects, and

2. sets of objects.

End quote.

So, in reference to the formal definitions in (1) and (2) above, the "nonempty set of elements" is simply the things that are counted as things within our Universe of Discourse. That is, it is the scope of things that our Universe of Discourse is able to "talk" about. In a natural language such as English, this is all the objects that our words refer to: dogs, cats, stars, protons, people, places, black holes, clouds, the weather, colours, and on, and on, and on, etc..

Now with regards to (2), what we mean by "one or more relations defined on this Universe of Discourse" are the relations that can be expressed in such a way that all and only those things which bear a specific property occur in some specific set of objects; in other words, the definitions of the relations that establish that the set of objects actually have some specific property.

To return to the formal definitions in terms of Predicate Logic, which is a more complex and developed version of Symbolic Logic, we define these sets of objects in two ways:

(I) By Intension, and

(II) By Extension.

"By intention" is simply a list of a set of necessary and sufficient conditions which any object in our Universe of Discourse must satisfy in order to have the property in question.

"By extension" is simply the set of every object that has the property in question.

Now, to return to some simplified versions of these things with respect to the natural language of English, we can consider the following as ways to understand what we mean by intensional and extensional definitions of objects that are in turn collected in some set of things.

Let's use the property of "blueness," as in, things (objects) that have the property of being blue.

Now, a definition by Intension (and this is by no means to be considered as a complete definition of the Intension regarding the property of blueness, but merely to give an idea of what such a definition would consist of) might include things like:

(A) The objects in question must reflect light at wavelengths that fall between 450–495 nm and at a frequency of 606–668 THz that have a photon energy of 2.50–2.75 eV.

(B) The objects in question must conform to what we mean by "being blue."

(C) The objects in question are in fact coloured blue.

We see that by using this list we offer a set of conditions which any object must satisfy in order to be a member of the set of all things that have the property of blueness. The objects that do not satisfy the definition by intension are not to be taken as members of the set of all things which have blueness.

A definition by extension, then, is simply the list of all the objects in our domain that also have the property of blueness. This set includes, for example (and again, not to give a complete list, but only to give the reader an idea of what is meant by "definition by extension"): bluejays, blue balloons, the billiard balls typically numbered with either a '2' or a '10', cars that are blue, the sky (sometimes), lakes (sometimes, but more and more less frequently, perhaps), garments that are blue, and so on and so on, etc..

So, it is my hope that the reader has a bit more sense of what we mean by the formal definition of a structure. We mean, then, the set of all objects that our Universe of Discourse is able to discuss and all the subsets of this set that are defined in terms of the objects having specific properties. We can note, then, that the subsets are a way of dividing up the objects in the Universe of Discourse in many different ways, which will necessarily entail that some objects are members of several subsets, but not all objects that are members of a given subset will necessarily occur in all the same subsets as one and other. For example, and returning to our subset of things with blueness: our blue billiard balls occur in the same subset of blueness that bluejays occur in, but bluejays do not occur in the set of all round things in the same way that billiard balls do not occur in the set of all birds. Some of the elements in the set of balloons that are blue will occur in the set of all round things along with the set of all billiard balls that are blue; but, the set of all balloons that are blue will not occur in the set of all things we use to play billiards, where, obviously, the set of all billiard balls, regardless of their colourings, will occur as a member of that set.

Getting the hang of this? Good.

Finally, but certainly not least:

3. An interpretation requires a set of rules for associating symbols with things.

This is not nearly as mystifying as it might seem, although when we look at it formally below, it will seem quite mystifying indeed.

All this really means is that there is some function that, for a given structure U, takes the objects and sets of objects and maps them to the strings of symbols which make up our language.

Formally, this is called an "interpretation function," and here is its formal definition:

For a given structure U, an interpretation function assigns:

1. to each constant of Predicate Logic an individual belonging to the Universe of Discourse of U, and

2. to each k-ary predicate symbol of Predicate Logic, a set of k-tuples consisting of individuals belonging to the Universe of Discourse of U.

Yes, because that is so clear now, right? Again, this is the formal definition and without knowing what the heck we mean by things like "k-ary predicates," "constants" with respect to PL, and "k-tuples," well, it's pretty much senseless. Again, without actually having sat through a formal Logic class, it is difficult to really get a grasp on these things formally.

So, let's turn back to the section we called "Metaphysical Musings" and see what the notes have to say on this matter.

I quote:

An interpretation bridges "language" and "the world" by:

1. Associating single objects with constants, and

2. Associating sets of objects with predicates.

End quote.

So, what we are getting at by the formal definition of (1) above regarding the constants of PL and the individual members of the Domain of Discourse of U, is simply a symbolic representation in language that picks out only the given object we are considering.

In the natural language of English we might, from the general set of all bluejays in our Domain of Discourse, give a specific name to a specific member of that set, let's say 'Bob,' so that when we discuss "Bob the bluejay" we know that we are referring to the same and only that particular bluejay we've called Bob.

Back to the informal talk about the formal language, an interpretation function assigns a name, say 'c1' to one and only one object from the Domain of Discourse so that there is no confusion as to what we are referring to when we come to evaluate the statements that are made in our formal language by way of operators acting in conjunction with predicates and the constants associated with these predicates. And this is, essentially, related to our association of sets of objects with specific predicates, which, more formally, is about the "k-ary predicate symbols" and the set of "k-tuples."

To go back to English, then, what we mean is that when we say something like:

"Bob the blue bluejay is a pretty bird and he is sitting on the thick branch of that tall tree."

We have a series of words that correspond to not only objects or sets of objects (Bob, bluejays, birds, branches, trees) in our Domain of Discourse, but also words that correspond to predicates, which are properties, (blueness, prettiness, sitting, thickness, tallness) in our Domain of Discourse. We also have "logical operators" which link up various individual claims or statements regarding the objects and their properties. This gets quite complex when we go from a formal language to a natural language. Indeed, some logicians spend some of their time trying to translate philosophical statements from the natural language in which they occur into a formal Predicate Logic version in order to accurately and mechanically evaluate their truth claims. Most efforts are difficult, slow going, and open for debate regarding the translations--especially with respect to which logical operators might best suit a natural language sentence.

In regards to the sentence above, there is basically only the repetitive occurrence of the logical operator & (or at least we will agree to look at it this way for ease of presentation and understanding), which we in natural language, funnily enough, call "and." Taking apart the above sentence and representing all the occurrences of & in it, is, given the simplicity of the statement, pretty much straight forward, so here it is:

1) Bob is a bird & he is blue.
2) Bob is a bird & he is blue & he is a bluejay.

3) Bob is a bird & he is blue & he is a bluejay & he is pretty.

4) Bob is a bird & he is blue & he is a bluejay & he is pretty & he is sitting.

5) Bob is a bird & he is blue & he is a bluejay & he is pretty & he is sitting & he is sitting on a branch.

6) Bob is a bird & he is blue & he is a bluejay & he is pretty & he is sitting & he is sitting on a branch & that branch is thick.

7) Bob is a bird & he is blue & he is a bluejay & he is pretty & he is sitting & he is sitting on a branch & that branch is thick & that thick branch is part of a tree.

8) Bob is a bird & he is blue & he is a bluejay & he is pretty & he is sitting & he is sitting on a branch & that branch is thick & that thick branch is part of a tree & that tree is tall.

See how much is contained in a relatively simple statement? Bet you had no idea that when you utter a simple phrase there is a whole bunch of background formalism present in terms of what we mean by logic or logical structure.

We are now, I feel, in a position that we can, in some sense, understand what we mean by an interpretation and how an interpretation functions in terms of things like: making claims, assessing truth, discussing objects, and etc..

An Interpretation that would allow us to make sense of the statement:

"Bob the blue bluejay is a pretty bird and he is sitting on the thick branch of that tall tree,"

needs the following objects to be present in its Domain of Discourse:

{the set of birds, the set of trees}.

The Interpretation must also have the following predicates:

{blueness, prettiness, thickness, tallness}

The Interpretation must have a function which assigns strings of symbols to these objects and the set of objects that have any given specific property. Let's call this function in this example E.

Now, E takes things from our set of objects and maps them to a set of words.

E, in our example, maps from the set of all birds that are also bluejays, to the set of words that make up our English language such that:

'Bob' picks out a specific bird, that is also a specific bluejay, from the set of all birds that are also bluejays.

E would also pick out the predicates, or properties, associated with Bob, namely:

That Bob is a bird, that Bob is a bluejay, that Bob is pretty, and that Bob is sitting.

Notice the interplay between the name "Bob," how it is we identify Bob from the set of all birds, and how the predicates as they relate to Bob help us in deciding which bird is meant by 'Bob'. Taken altogether, we hope we have an Interpretation that is rich enough to point to that specific bird that we are calling, in this instance, "Bob." In other words, our "interpretation function" is such that it allows us to pick out what we have named "Bob" as distinct from all other things named "Bob," in such a way that we understand some things about what Bob actually is and how Bob actually is, namely, Bob is pretty, Bob is sitting, and etc..

OK, let's turn to what we mean by Truth and we shall see how (1) truth only exists inside an interpretation and (2) what is taken to be a "fact" only exists inside an Interpretation relative to the truth-value that the Interpretation establishes in regards to statements about objects that are recognized in the Domain of Discourse of the Interpretation.

Back to our relatively simple statement:

"Bob the blue bluejay is a pretty bird and he is sitting on the thick branch of that tall tree."

We have seen how this statement breaks down into a series of conjuncts, the totality of which is reasonably captured by:

Bob is a bird & he is a blue & he is a bluejay & he is pretty & he is sitting & he is sitting on a branch & that branch is thick & that thick branch is part of a tree & that tree is tall.

A "conjunct" is simply any of the statements which occur on either side of any given occurrence of the logical operator &. So, the conjuncts of our statement are:

1) Bob is a bird.

2) Bob is blue.

3) Bob is a bluejay.
4) Bob is pretty.

5) Bob is sitting.

6) Bob is sitting on a branch.

7) That branch is thick.

8) That branch is part of a tree.

9) That tree is tall.

In order for our statement to be true in our Interpretation, each of the conjuncts also has to be true. If any one of the conjuncts is false, then the whole statement is false by the rules of logic. The interpretation, then, requires that we are able to consider each term with respect to the conditions that make it either true or false. This is to say, that for each of the conjuncts we need to be able to understand what makes any one of them true or what makes any one of them false. Let's simply look at one of the conjuncts, namely, "Bob is a bird."

In our Interpretation there are two possibilities: either "Bob is a bird" is a true statement or "Bob is a bird" is a false statement. So, if our Interpretation correctly assigns the word 'Bob' to the object that is a bird, and this bird is exactly the one we mean by the string of symbols 'Bob', then on our interpretation the statement "Bob is a bird" is true. However, if our Interpretation instead assigns the word 'Bob' to our uncle, then "Bob is a bird" is false in our Interpretation, and so, with regards to our whole statement:

"Bob the blue bluejay is a pretty bird and he is sitting on the thick branch of that tall tree,"

our Interpretation would evaluate it as a false statement, since Bob is a human being and not a bird. Now, every other conjunct about Bob our uncle could be true: he could be blue, he could be pretty, he could be sitting on a branch of a tall tree, but since he is not a bird and specifically not a bluejay, our claim about Bob as a blue bird that is also a bluejay that is sitting, pretty, and etc., is in fact false.

Without an Interpretation which assigns words to things and establishes the relationships about the things in terms of their properties, well, there simply isn't any sense of "truth" to the matter. We'd have only words and only things but no bridge between them; that is, without an interpretation, we have no way to make claims about anything because we can't talk about the things and we can't associate things with their properties in any sensible, as in utterable, manner.

We can easily see how "facts" fall out of this. If there is a correspondence between our language and the objects in our language's Domain of Discourse and the statements we make about these objects are also true, then that is what we call a "fact." If Bob really is a bird, and if by the word 'Bob' and 'bird' our Interpretation establishes a relationship between those things such that is is true that "Bob is a bird" corresponds to Bob actually being a bird, then we say that it is a fact that Bob is a bird.

Again, without an Interpretation, then, there are no facts of the matter, there are only objects that have properties which are not in any way sensibly able to be discussed; put differently, the facts of the matter are uninterpreted without an association between our language and its objects of Discourse. What we mean by "fact," then, is something that only comes to be when considering true statements within an Interpretation.

To go way back to the beginning, consider the statement:

"Seven is the number that occurs between eight and nine,"

We know this to be false simply because we have an Interpretation that establishes that the object referred to by the word 'seven' occurs in a mathematical structure such that the object seven has its place between the objects referred to as "six" and "eight". Thus, it is our Interpretation that establishes that the fact of the matter does not correspond to our statement about seven in terms of the property of it occurring between the numbers we call "eight" and "nine". Without the Interpretation we have no facts of the matter because we have no way to understand the properties corresponding to the object seven, and we have no way to assess the statement about that object in its relation to other objects.

Thus, we see clearly how truth and facts are entirely dependent upon an Interpretation.

So, turning to reason and the use of logic, we can also see clearly that we are only able to reason about things insofar as these things correspond to some Interpretation which allows us to understand things about the objects in the Universe of Discourse in such a way that we are, in fact, able to reason about these things in terms of the referents and the properties that these objects have as picked out by our Interpretation. Logic is simply the rules by which we understand how the relationship between statements and the statements' objects and properties are evaluated in terms of being either true or false. Without some set of sentences and rules by which we can logically manipulate and assess those sentences, well, there's no logic to be had at all.

An important thing to note is that all interpretations are open to the use of reason and logic, but we are also able to use reason and logic on an uninterpreted set of sentences whereby the "woofs" do not correspond to any objects of a domain of discourse other than the "woofs" themselves. Such practices are called "pure logic" in the sense that the analysis and results that come out of such a study are about Logic itself. This, however, is a purely formal exercise and is not associated with any natural language; that is, when we reason about logic and use logic to analyze itself we must necessarily not be referencing anything that has to do with objects beyond the objects of logic itself.

So, when I make the claim that "reason and logic apply only within our interpretations, but we are unable to use reason or logic in the absence of an interpretation," that is not quite the case, it is, in a sense, a colloquial way of speaking, of being a bit loose with the language. What I mean is that any time we are reasoning about things in the world or using logic with regards to our statements about things in the world, then we must necessarily do so with respect to an interpretation. Without an interpretation reason and logic can only ever be about themselves.

To put this more concretely, then, when I make a claim like:

 "I do tend towards having a distaste for certain 'breeds' of atheists who declare logic and reason as their allies and claim these support their lack of belief. They are mistaken. Logic and reason apply only within interpretations and their interpretations assume their lack of belief..."

 What I mean is that whenever a person is using logic and reason to assess their beliefs, he or she can only use it insofar as to reason from the assumptions within their Interpretation and use logic to assess the truth-value claims of how these assumptions are interconnected via logical operators (such as 'and', 'or', 'not', 'if...then...', and 'if and only if') with respect to the assumptions of a given interpretation. Logic and reasoning alone cannot establish anything about any belief or lack thereof as without an interpretation there simply is no such thing as belief or lack thereof.

In order to preempt atheists who might wish to say, due to their misunderstanding of the above presentation, that since lack of belief entails that there is no belief, we can thus use pure reason and pure logic alone to support that lack of belief since you (being me) asserted that belief does not exist in terms of reasoning about pure logic alone. I will simply point out that reasoning about logic with logic has nothing to do with any of the other beliefs or facts that exist only in their Interpretations by which they have derived or otherwise assumed the conclusion regarding their lack of belief.

If you, the reader, actually made it to the end of this article, congratulations: you now probably have a better grasp on logic, reasoning, and interpretation than most of your peers, at least, those of your peers who have never taken a course on Logic and have not been trained in ways of reasoning in a formal sense (i.e, mathematically, for example). I hope you found the ride not too bumpy and that your brain made it through without too many bruises. Now, amidst a short burst of fanfare with maybe an accompanying drum roll and perhaps some hand waving, I bid you adieu.