<!doctype html public "-//w3c//dtd html 4.0 transitional//en"> <html> <head> <meta http-equiv="Content-Type" content="text/html; charset=iso-8859-1"> <meta name="GENERATOR" content="Mozilla/4.51 [en] (Win95; U) [Netscape]"> <title>Modality and Computational Metaphysics</title> <!-- This document was created from RTF source by rtftohtml version 2.7.5 --> </head> <body> <font face="Times New Roman,Times"><font size=-1>http://www.allanrandall.ca/Modality.html</font></font> <br><font face="Times New Roman,Times"><font size=-1><a href="copyright1997.html">Copyright</a> &copy; 1997, Allan Randall</font></font> <br>&nbsp; <br>&nbsp; <center> <h2> <font face="Times New Roman,Times"><font size=-1>&nbsp;</font> </font><b>Modality in Computational Metaphysics</b></h2></center> <center><b><font face="Times New Roman,Times">Allan F. Randall</font></b> <br><b><font face="Times New Roman,Times">Toronto, Ontario, Canada</font></b> <br><b><font face="Times New Roman,Times"><a href="mailto:research@allanrandall.ca">randall@allanrandall.ca</a>, <a href="http://www.allanrandall.ca/">http://www.allanrandall.ca/</a></font></b></center> <p><br> <center> <h3> <b><font face="Times New Roman,Times">Abstract</font></b></h3></center> <blockquote><b><font size=-1>The many worlds anthropic principle is explored here from the <i>a priori </i>perspective of rationalist metaphysics, within the framework of modal logic. It is shown how the apparent contradictions of quantum superposition can be thought of in terms of different levels of world models. The framework of modal logic is used, but given the rationalist assumption that all possible worlds exist. There is thus no <i>absolute </i>distinction between possibility and necessity. To take the point of view of a conscious being in a world, however, is to adopt some such distinction--something we must do in order to do physics. This paper is intended to lay the groundwork for future attempts to develop theories of what is necessarily true in a world with conscious entities. It also contains some tentative speculations on the difficult issue of death and quantum mechanics.</font></b></blockquote> <center> <h3> <b>I. Introduction</b></h3></center> This essay attempts to lay some rough conceptual groundwork, using modal logic, for the development of a rationalist, idealist metaphysics. The ultimate goal is to find a way of characterizing those worlds, of all logically possible worlds, that contain conscious life, thereby explaining the order we see in our own empirical world without going beyond the bounds of mathematics, logic and rationality. I will make use of the standard rationalist assumptions, such as the Principle of the Identity of Indiscernibles and the Principle of Sufficient Reason. <p>Of course, this is an immense task, and this essay will only take a few faltering steps towards this goal. It will help if the reader has some understanding of modal logic<a href="modality_fn.html#fn0">[1]</a>, the lambda-calculus<a href="modality_fn.html#fn1">[2,3]</a> and rationalist/idealist metaphysics.<a href="modality_fn.html#fn3">[4]</a> My metaphysics draws primarily from Parmenides,<a href="modality_fn.html#fn4">[5]</a> Descartes,<a href="modality_fn.html#fn5">[6]</a> Leibniz,<a href="modality_fn.html#fn6">[7]</a> Bradley,<a href="modality_fn.html#fn7">[8,9]</a> Everett<a href="modality_fn.html#fn9">[10,11]</a> and Carter.<a href="modality_fn.html#fn11">[12,13]</a> I will provide only a brief overview of these ideas. See the references for more details. <p>I will lay our groundwork by describing, in terms of possible worlds, one of the primary conceptual problems with idealism. We will see how modality allows us to think more clearly about it. I will also show that modern physics gives us good empirical reasons to believe that this is actually how the world works. <center> <h3> II. Rationalism</h3></center> My rationalist metaphysics is based on two fundamental first principles. The first is the Cogito of Descartes.<a href="modality_fn.html#fn5">[6]</a> This principle tells me to think of the world around me--the world of my experience--as something derivable directly from my current, present state of consciousness. "I think, therefore I am." The only empirical data that I can know with certainty is my current state of mind. A rationalist metaphysics will build everything up from that, using reason alone. <p>The other fundamental principle of rationalism was introduced by Parmenides:<a href="modality_fn.html#fn13">[14]</a> "Whatever can be spoken or thought of necessarily <i>is</i>, since it is possible for it to be, but it is not possible for nothing to be." Parmenides sees two ways of viewing the world, one objective and one subjective. From the objective stance, all possibilities are equally real, and so the everyday world of our experience has no <i>a priori</i> claim to be more real than any other logical possibility. From the subjective point of view, however, it is only natural for us to see our own world as special and to seek the best explanation for it we can find. <p>In Leibnizian terminology, this means that all possible worlds exist objectively. We, as inhabitants of one of these worlds in particular, would like to explain the nature of our conscious experience. In the past couple of decades, the Anthropic Principle has become a popular way of doing this.[11,12,13] Since all possible worlds are equally valid from the most objective stance, we need a selection principle--some <i>a priori</i> reason to place our world in a characterizable subset of all possible worlds. If we can build a model describing this subset, that model should provide us with an explanation for much of the order we see around us. The version of the Anthropic Principle I will use starts with Descartes' Cogito: "I think, therefore I am." The <i>a priori</i> selection principle we are looking for selects out only those worlds that contain conscious life. This is the only feature of our world that we already know <i>must </i>exist, for if it did not, we would not be here to ask the question in the first place. This principle--that the universe must have features that allow consciousness or we would not be here to be conscious of it-- is called the "Anthropic Principle"; the version of it described here in terms of a plurality of worlds is the "Many Worlds Anthropic Principle", or MWAP. <center> <h3> <b>III. Modality</b></h3></center> Figure 1 illustrates the notation I will use to describe a model. <font face="Symbol"><font size=+1>W</font></font> is the "raw material" from which we draw: a set of arbitrary symbols, each of which is given membership in a subset of the worlds in W. W is our universe. For our purposes, we will define the universe as the set of all worlds in the model. In figure 1, the atom <b>B</b> is in world w<sub>2</sub>, so we say that <b>B</b> is true in that world, or: <ul><sup>&nbsp;</sup>M<img SRC="modality3.gif" height=12 width=12><font size=-1>w<sub>2&nbsp; </sub></font><b><font size=+1>B</font></b> <br>&nbsp;where M is the model, consisting of V, W and R.</ul> <center><img SRC="modality2.gif" height=122 width=184> <br><b>Fig. 1: A universe of possible worlds and their relations.</b></center> <p>V is the valuation function that defines which symbols appear in which worlds. R is the accessibility relation, which defines which worlds are possibilities from the perspective of any particular world. Recall that from Parmenides' objective stance, all possible worlds are real. However, within a particular model, only the worlds in W are considered possible. So to work within a model is automatically to take a subjective stance and intentionally exclude some worlds from the universe (for instance, figure 1 arbitrarily excludes the world in which both <b>A</b> and <b>B</b> are true). <p>But not only is W limited to a subset of all the worlds we could possibly build from <font face="Symbol"><font size=+1>W</font></font>, but R limits each world as being possible only from the point of view of certain other worlds. In figure 1, for instance, w<sub><font size=-1>1</font></sub> is a possibility from the perspective of world w<sub><font size=-1>3</font></sub>, but not vice versa. So, since <b>A</b> is true in w<sub><font size=-1>1</font></sub>, we say that <b>A</b> is <i>possible</i> in w<sub><font size=-1>3</font></sub>. But from w<sub><font size=-1>3</font></sub>'s perspective, there are only two possible worlds: w<sub><font size=-1>1</font></sub> and w<sub><font size=-1>3</font></sub>. Since <b>A</b> is true in both, we can also say <b>A</b> is <i>necessary</i> in w<sub><font size=-1>3</font></sub>, since it is true in all possible worlds (from w<sub><font size=-1>3</font></sub>'s point of view). <b>C</b>, however, is true in w<sub><font size=-1>3</font></sub> but not w<sub><font size=-1>1</font></sub>, so it is possible but not necessary in w<sub><font size=-1>3</font></sub>. In more formal notation, we say: <ul>M<img SRC="modality3.gif" height=12 width=12>w<sub>3&nbsp;&nbsp;<img SRC="modality0.gif" height=17 width=16></sub><b><font size=+1>A</font></b> (where&nbsp;<img SRC="modality0.gif" height=17 width=16> means "it is possible that"). <br>M<img SRC="modality3.gif" height=12 width=12>w<sub>3&nbsp;&nbsp;<img SRC="modality5.gif" ></sub><b><font size=+1>A</font></b> (where&nbsp;<img SRC="modality5.gif" height=17 width=16> means "it is necessary that"). <br>M<img SRC="modality3.gif" height=12 width=12>w<sub>3&nbsp;&nbsp;<img SRC="modality0.gif" height=17 width=16></sub><b><font size=+1>C</font></b> <br>M<img SRC="modality3.gif" height=12 width=12>w<sub>3&nbsp; </sub><b><font size=+1>~<img SRC="modality5.gif" height=17 width=16>C</font></b></ul> <center> <h3> <b>IV. Anthropic Models</b></h3></center> What is necessary from the objective stance is simply all logical validities (that which is true in all possible worlds in all models). There are degrees of objectivity, however. In developing the Anthropic Principle, we hope to find what is true in all possible worlds within a certain class of models: those that include all worlds with conscious life and exclude all those without conscious life. Such truths could be said to be valid within a very broad perspective, if not an absolutely objective one. In fact, some might consider any world not contained in a model of this class to be hardly a world at all. The idea of conscious inhabitants is part of the ordinary, most mundane sense of the word "world". <p>A narrower subjective stance would be that class of models containing any conscious life that also happens to fit my own personal criterion for self-identity. I might call this "my universe": the set of all possible worlds that contain an individual I identify enough with to call myself. So the world in which I just scratched my elbow, even though I did <i>not </i>scratch my elbow in <i>this </i>world, contains a conscious life form that is not literally me here now, but is close enough that I consider that person to be an alternative version of myself. So I say it is "possible" that I "could have" scratched my elbow just now. If this other individual was in my precise situation <i>except </i>that he did not have the appropriate conscious state and memories necessary for me to identify with him as being an "other" me, I would not call it a "possible world" (not, that is, from the perspective of "my universe"). I will call this "my universe" perspective the "personal identity class of models". <p>Many other perspectives are possible. A still narrower perspective, but a metaphysically very important one, is the view that sees only things that happen to my actual self as possible. Only one stream of consciousness is allowed to define the world. This view simply eliminates other selves as possibilities (by this, I mean other selves that I identity as "me", not the existence of other conscious beings, which is still allowed). The individual who scratched his elbow just now in that other world is now considered to be no more <i>me </i>than any other consciousness. This is similar to the personal-identity view, but with a narrower view of personal identity, restricted to only one stream of consciousness. I will thus call this the "strong personal-identity class of models" or the "stream of consciousness models". The earlier example, which allowed more than one stream of consciousness into my definition of self, I will sometimes call the "weak personal-identity class of models". It is clear that many different definitions of self are possible, depending on how weak we make the definition. I will generally take the stream of consciousness models as the strongest possible, but in fact there is an even stronger, narrower way to define oneself, one which seeks to go no further in its viewpoint than the Cogito of Descartes. This is the view that associates the self only with one's current state of consciousness. While this maximally sceptical view of self is philosophically important as a starting point, few people ever really adopt this as a serious notion of personal identity, so I will consider it a personal identity model only in the degenerate sense, choosing to more generally call it simply the "conscious-state-identity class of models". <p>Stream of consciousness models and conscious-state models are closely related. The latter provide us with our philosophical starting point, but we will almost always want to extend our view to the weaker stream of consciousness model, allowing that we are the same person today as yesterday. It is the stream of consciousness model, then, that is the most metaphysically important, as this is the one that selects out the world that we generally consider to be our everyday "world", the observable universe, past, present and future. One might think that this leaves us with just one world in our model, accessible to itself, but unable to "see" any other worlds. But this is only one kind of model in this class. Stream of consciousness models include only worlds that can be identified with a particular stream of consciousness, in which that stream is "true". This gives us <i>at least</i> one "world", but a model in this class could break things down into further "worlds", based on some additional criteria, giving us a sort of hierarchy of models, with only the root or top model containing a solitary world. For instance, a common metaphysical position to take is that each time slice of our universe is its own "world". There is no absolute justification for connecting up the world as it existed yesterday and the world of today and calling them a single world. From this viewpoint, the world at time<i> t</i> is considered a separate world (but in the same universe) as the world at time <i>t</i>+1 . The fact that they follow each other in time is a consequence of the model. So, in a stream of consciousness model, world w<i>'</i> is accessible to world w if either world contains a conscious being that can remember being in the other. Perhaps this definition needs tightening, but we will leave it at that for now. The point is that the world of yesterday is "accessible" to me only because I can call it to conscious memory. Note that if we "time-slice" according to conscious states, then the stronger conscious-state model can be subsumed lower in the hierarchy, underneath the higher level stream of consciousness model. We will see more explicitly how this works later on. <p>But what of the world of tomorrow? The stream of consciousness view seems to require that tomorrow be accessible, which may seem counter-intuitive at first. But recall that accessibility merely defines relations between worlds, and does not necessarily refer to cognitive or perceptual accessibility. You cannot <i>remember</i> tomorrow, but in this class of models we are saying that it is accessible to you because you-in-the-future remembers being you-now. So future time slices can be accessible to past slices,<i> in this model</i>. For any given time slice, t<i><sub><font size=-1>k</font></sub></i>, there are many possible worlds at t<sub><font size=-1><i>k</i>+1</font></sub>, but only one at t<sub><font size=-1><i>k</i>-1</font></sub>. In other words, I can only remember one past, but there are many futures with persons that can remember being me. This means that which time-slices get included in the stream of consciousness model depends on one's <i>current</i> conscious state, even though the stream of consciousness model was supposed to transcend that particular state! In other words, it turns out to be impossible to form a concept of oneself based on a <i>single</i> stream of consciousness that includes past, present and future, unless one is dead and can categorically say that one <i>has</i> no future, in <i>any</i> world. Otherwise, we are compelled to allow future time slices to be accessible. So we will clarify our definition of stream of consciousness models, so that they are always defined <i>relative to</i> a conscious-state model, <i>or</i> are given relative to an entire lifetime from birth to death, if any (since, in a way, a "true" stream of consciousness model is of the latter variety and cannot be built until one has lived one's life to the end and there are no more possible selves that one could yet turn into). <center> <h3> V. Idealism--Some Historical Context</h3></center> The rationalism that takes all possible worlds as objectively real is an "idealist" rationalism. The term "idealist" has, of course, meant different things to different people. It is sometimes defined as the belief that the external world is somehow mental in character. This meaning concentrates on the mental connotations of "idea" rather than the more abstract, even mathematical, connotations of "ideal". I will use the word "idea" in the most general sense of "that which is thought of", without specifying whether this requires a mind to think it. Many different philosophies have been labelled "idealist", and they vary greatly in the extent to which they consider the world to be ultimately mental or not. All idealism, however, somehow equates objective reality with "what can be thought of", a la Parmenides. "Subjective idealists" will tend to believe this means that the objects of the world require an actual mind to think them in order to exist, and so they will add to the above "plus a thinker to think them". Subjective idealism is not a rationalist idealism, and indeed not really a "pure" form of idealism at all, since it adds to the system "the thinker" as something other than just another idea that can be thought of. Thus, subjective idealists, such as Berkeley, explain the ordinary external world of chairs and rabbits in terms of ideas, but still believe in some ultimate reality (a mind or minds, perhaps that of God) that cannot be reduced to ideas, being that which <i>has</i> the ideas. So subjective idealism is not ultimately idealist at all, no more so than the materialism which it supposedly refutes. In my view, it is more properly called "mentalism". "Objective idealists", on the other hand, such as Hegel, F.H. Bradley, Plato or Parmenides himself, believe that the universe exists logically prior to any particular mind, and is perhaps not in itself mental in character at all. Bradley was an objective idealist who nonetheless thought that objective reality <i>was</i> ultimately mental, although he was adamant that this did not mean there was an absolute mind thinking it all. This differs somewhat from the idealism of Parmenides, who seems to have believed that mind was merely an emergent property of various body parts working in harmony. Objective reality as "everything that can be thought of" did not necessarily mean for him that the things to be thought of were mental. Whether there is really any effective difference between these two positions is perhaps debatable. <p>I will use the term "idealist" to mean all of these various philosophers from the most objective nonmentalist to the most subjective mentalist. What these views all have in common is a rejection of "material realism", the notion that the everyday "matter" of the world has some kind of objective reality of its own, <i>as</i> matter, and apart from its status as a possible idea or its status as an actual idea in a mind. Even those idealists who believe that ultimate reality is nonmental will nonetheless usually agree with the subjective idealists that everyday material objects like chairs and rabbits <i>are</i> somehow mental in character, since they are highly dependent on the viewpoint of the observer, and are only "partial aspects" of a larger nonmental reality. It is this kind of nonmentalist idealism that I will argue for. The "larger" reality is simply the totality of all possible ideas, including in it all possible worlds, of which ours is but one. Thus, the "mental" nature of material objects comes in simply as the fact that they cannot be defined or picked out without bringing in oneself. They are the objects in "my world", as opposed to all other worlds. While this is certainly not "material realism", it is still a brand of "realism", since there <i>is</i> still an external reality--it just is not equivalent to our everyday notion of material objects in space and time. However, almost all forms of idealism have in some sense been realist. Even Berkeley believed in an external reality, but he equated it with God rather than matter. Arguably, the only idealists that are true anti-realists are solipsists, who believe only in their <i>own</i> mind, but it is very difficult to find any genuine solipsists, living or dead. It is thus more correct to contrast idealism with materialism rather than realism, although in the twentieth century, "realist" usually means "material realist". <p>By historical accident, material realism is generally associated with "analytic" philosophy. But in fact, the whole notion of some possible worlds existing and some others not existing is thoroughly unanalyzable--I would claim even mystical. A wholly "analytic" philosophy would consider only that which can be rationally conceived of, and thus logically analyzed--anything else would be passed over in silence. It is ironic that the term "analytic philosophy" can thus be taken as virtually synonymous with "objective idealism", a brand of philosophy that real-world analytic philosophers considered it part of their founding mandate to eradicate. The reason for this is that modern idealism began with Berkeley, firmly entrenched in the self-contradictory subjective camp. It became, over the years, much more objective and rational. Bradley, who represents perhaps more than anyone the brand of idealism the analytics were rejecting, had brought things almost, but not quite, back to the objectivism of ancient idealists like Parmenides and Plato, but he was still ultimately a mentalist. The analytics, like Bertrand Russell and G.E. Moore, were largely rejecting this mentalism--I think justifiably. However, they threw the baby out with the bath-water, going whole-hog for a thoroughly mystical naive realism, instead of continuing the evolution of idealism, a la Bradley, into a nonmentalist and fully analytic variety. <center> <h3> VI. Ideal Strangeness</h3></center> Some (including myself) would say that idealism is a natural consequence of rationalism. Still, many philosophers continue to believe that there is something special about our ordinary world of experience that makes it "real" in an absolute objective sense--above and beyond any other merely possible worlds. One reason why idealism, in all its forms, is rejected by so many is its perceived inability to account for the order and structure in the world. Why is it that when I see a table, then look away and look back a few seconds later, the table is still there, time after time? Surely there is a possible world where the table just disappeared. So if all possible worlds exist, why do I only experience a world in which tables are stable and don't just disappear at random or turn into giant pink bunny-rabbits? If these "weird" events are possible conscious states for me to have, and thus parts of possible worlds, then why don't I ever experience them? <p>Instead of answering this right away, I would like to look first at an example. I will pretend that my conscious-identity model contains only two worlds. This model is defined from the perspective of time t<sub><font size=-1>0</font></sub>. In world w<sub><font size=-1>H</font></sub>, I have a hat. In world w<sub><font size=-1>~H</font></sub>, there is no hat at all. At time t<sub><font size=-1>0</font></sub>, I am approaching a table. I believe that I have the hat on my head. The hat has no perceivable effects on my conscious state, even when it is on my head. I am utterly oblivious to it. But I know this. I am perfectly aware that this particular hat can sit on my head without being felt at all (perhaps it is a very light hat, or I have such a thick head of hair, I cannot tell it's there). So it happens that at time t<sub><font size=-1>0</font></sub>, in both worlds, I am under the belief that the hat really is on my head. In w<sub><font size=-1>H</font></sub>, I am correct, while in w<sub><font size=-1>~H</font></sub>, my belief is false. When I get to the table at time t<sub><font size=-1>1</font></sub>, I reach to take off the hat. In world w<sub><font size=-1>H</font></sub> I find the hat, and place it on the table. But in world w<sub><font size=-1>~H</font></sub> I find to my surprise that I am not wearing a hat after all. <p>Now assume, for simplicity, that my conscious state consists only in knowing what time it is, and whether I believe there is a hat in the world or not. Assume for now that I have only two possible beliefs: either I believe the hat is there or I believe it is not. I am not capable of reserving judgment or being uncertain about it. We will abbreviate these states of affairs as follows: <ul><b>H</b>: there is a hat in the world <br>&nbsp;<b>~H</b>: there is no hat in the world <br>&nbsp;<b>B</b>: I believe there is a hat in the world <br>&nbsp;<b>~B</b>: I believe there is no hat in the world <br>&nbsp;<b>B</b>+t<sub><font size=-1>k</font></sub>: A possible conscious state at time <i>k</i>.</ul> For our purposes, <b>H</b>,<b> ~H</b>,<b> B </b>and <b>~B</b> will be considered, for the time being, as arbitrary atomic symbols in <font face="Symbol"><font size=+1>W</font></font>. In the future, however, we will want to think of <b>H</b> and <b>~H </b>as basic atoms, while <b>B</b> and <b>~B </b>will be inferred from these atoms. This is because our basic atoms are the "raw material" of our universe. <b>H</b> and <b>~H </b>are both facts that are true in one and only one personal- consciousness universe (in this particular class of models). They are things "in themselves", unlike <b>B</b> and <b>~B</b>, which are facts derivable from our universe, but insufficient in themselves to define the universe; they are "partial aspects" of the universe. <p>All of this remains to be worked out in detail, but for those familiar with computational systems such as the lambda-calculus, here is a hint as to how to think about it: <b>H</b> and <b>~H </b>can be thought of as self-contained computer programs, with no inputs and no outputs. As such, each is a complete "universe" in its own right: they are metaphysical things-in-themselves. But <b>B</b> and <b>~B </b>refer to qualities of w<sub><font size=-1>H</font></sub> and w<sub><font size=-1><b>~</b>H</font></sub>. As such, they are like test-functions that can be applied to <b>H</b> or <b>~H</b> to determine if the relevant partial aspect is there. Equivalently, they are facts derivable from <b>H</b> or <b>~H</b>, but not facts from which <b>H</b> or <b>~H </b>can be derived. I will use the Leibnizian terminology, and call <b>H</b> and <b>~H "</b>monads". But to even view <b>H</b> and <b>~H </b>as corresponding to computer programs, we need to develop the role that inference rules play in our system, which will not be addressed in this essay. So, for current purposes, consider all four symbols as atoms. The negation sign cannot at this time be considered a symbol in its own right. <b>~H </b>is simply an atom, contained in some worlds and not others. There is as yet no concept of negation in our system. <p>Because <b>H</b> is a monad, we want to say that it <i>is </i>the world, not merely an aspect of the world, like <b>B</b>. <b>H </b>has this characteristic for <i>some</i> levels of analysis, but not when we consider a "world" to be a time slice. For then, the "world" is not a metaphysical thing-in-itself, but is itself only a partial aspect. It is the whole universe of worlds--the model--that is then considered analogous to a monad. So I will apply the word "monad" to the model-level, rather than the world-level. An atom that is in all worlds of a model could be said to be monadic with respect to that model. "Monadic" can be read either as "a consciousness-containing program in the lambda-calculus with no inputs or outputs", or "true in all worlds in a model". Unless otherwise specified, I will use the term in the former sense, since that is the more metaphysically important (and Leibnizian) sense of the word. The latter I will use only if it is specified what model the "monadicity" is with respect to. <p>You might object that these atoms cannot even be said to exist in more than one model, since they are defined by their model, with no structure of their own. However, if on a higher level of analysis, we can find a one-to-one mapping between the role of an atom in one model and an atom in another model, then they could actually be called the same atom. <center><img SRC="modality10.gif" height=216 width=402> <br>&nbsp;<b>Fig. 2: Two worlds with identical consciousness at t<sub><font size=-1>0</font></sub>, but not at t<sub><font size=-1>1</font></sub>.</b></center> <p>Figure 2 shows the two worlds starting at t<sub><font size=-1>0</font></sub>: one in which I have on a hat, and one in which I don't. So far, we have been thinking of these two selves as two different versions of me (like the me that scratches his elbow and the me who doesn't). But since my consciousness "<b>B </b>+ t<sub><font size=-1>0</font></sub>" is identical in both worlds, one could equally well say that at t<sub><font size=-1>0</font></sub>, the <i>same</i> person is occupying two different worlds simultaneously! When I reach the table at time t<sub><font size=-1>1</font></sub>, I find the hat in world w<sub><font size=-1>H</font></sub>, and now perceive it to be on the table. But in w<sub><font size=-1>~H</font></sub>, I find that, much to my chagrin, the hat doesn't exist at all! The result is that my belief changes from <b>B</b> to <b>~B</b> and now our model contains two different people with different conscious states: consciousness "<b>B + </b>t<sub><font size=-1>1</font></sub>" occupies w<sub><font size=-1>H</font></sub> and consciousness "<b>~B </b>+ t<sub><font size=-1>1</font></sub>" occupies world w<sub><font size=-1>~H</font></sub> (see figure 2). <p>This is a consciousness-based analysis, since we are defining two worlds with individual streams of consciousness. But other models are possible. Since w<sub><font size=-1>H</font></sub>(t<sub><font size=-1>0</font></sub>) and w<sub><font size=-1>~H</font></sub>(t<sub><font size=-1>0</font></sub>) are identical, if we subscribe to the Principle of the Identity of Indiscernibles<a href="modality_fn.html#fn5">[6]</a>, we can equate the two and construct a model with a single world at t<sub><font size=-1>0</font></sub>: w<sub><font size=-1>H~H</font></sub> = w(t<sub><font size=-1>0</font></sub>) in figure 3. <center><img SRC="modality11.gif" height=216 width=433> <br>&nbsp;<b>Fig. 3: One "contradictory" world at t<sub><font size=-1>0</font></sub> which splits into two at t<sub><font size=-1>1</font></sub>.</b></center> <p>Here, time t<sub><font size=-1>0</font></sub> has a peculiar status. <b>H</b> and <b>~H</b> are both to be found here. Let's call this state of affairs <b>H~H</b>. But does that mean (<b>H</b> v <b>~H</b>) or (<b>H</b> &amp; <b>~H</b>)? The answer is either or neither. Recall that the logical connectives do not yet have any meaning in our system. In figure 2, we would likely think of it as a disjunction across worlds (<b>H</b> v <b>~H</b>), so it might make sense to think of figure 3 as containing a conjunction in one world (<b>H</b> &amp; <b>~H</b>). The former is a tautology, the latter a contradiction. Yet the actual state of affairs described is supposed to be the same! <p>But there is not as yet any notion of contradiction or tautology in our system. Certainly, it will be possible to define figure 3 as contradictory, but that depends on how we develop our logic. For now, consider w<sub><font size=-1>H~H</font></sub> a single world with both "a hat on my head" and "no hat on my head" somehow "superimposed" on each other. There is nothing contradictory about this in itself. It is only when developed in a particular logic that this might be considered contradictory. For now, just consider that we have a new atomic symbol: <b>H~H.</b> <p>Is<b> H~H</b> a new monad, to replace both <b>H</b> and <b>~H</b>? Not in the important sense of conscious-identity, since it only holds at t<sub><font size=-1>0</font></sub> and not t<sub><font size=-1>1</font></sub>. So <b>H~H </b>is only a partial aspect of the universe; <b>H</b> and <b>~H </b>are<b> </b>still the monads. But we <i>could</i> take <b>H~H </b>as a monad with respect to a model that considered some partial aspect of both the w(t<sub><font size=-1>1</font></sub>) worlds to be important enough to justify unifying the two worlds into one. In figure 3, we could focus on the fact that both worlds at t<sub>1</sub> contain some version of <i>me. </i>This is a more objective stance, that of "personal identity" rather than consciousness. Now things are switched around; <b>H~H </b>is the monad, and it is <b>H</b> and <b>~H </b>that are partial aspects! So the extent to which something is considered a thing-in-itself depends on the model in which we are working. To make something a monad, eliminate all worlds in which it is not true from the model. If the resulting worlds all contain a single stream of consciousness, then this is a monad in the important metaphysical sense. <center><img SRC="modality12.gif" height=156 width=396> <br>&nbsp;<b>Fig. 4: Three levels of analysis; three classes of models.</b></center> <p>Figure 4 shows the three levels of analysis we have talked about. Individual worlds contain more and more of reality as we go up the figure from the bottom to the top. A traditional idealist goal is to show how it can all be unified into one Absolute world of which all other worlds, including ours, are partial aspects. <a href="modality_fn.html#fn7">[8]</a> <center> <h3> VII. Self-Cause</h3></center> The Absolute, however, would have to be a "world" independent of any model. It would be monadic, or a thing-in-itself, all on its own--<i>sui generis </i>or self-caused. A single universe containing one self-subsistent world--this has always sounded vaguely mystical to many opponents of idealism. But one of my goals in this work is to show that, if each atom is somehow, when embedded in a model, an abstract computer program, then if the Absolute Atom represents the Universal Turing Machine, it <i>can </i>be said to be self-caused. One of the results of computational theory is that any Turing Machine (such as the lambda-calculus) can be translated into any other, indeed can itself simulate any other Turing Machine.<i><sup>3</sup></i> So the Universal Turing Machine is a theoretical beast that can be independent of the model in which it is described. Combining this with the Church-Turing Thesis, which states that any possibility can be expressed by this Universal Machine, we have the characteristics of a self-caused thing. This Ontological Version of the Church-Turing Thesis could be said to be an expression of Parmenides' Principle, where "what can be spoken or thought of" is equated with "computable". The above is just a rough sketch of the idea. The full details remain to be worked out. <p>Figure 5 expands the analysis from figure 4 into the larger picture. Below the temporal level, the universe can be further analyzed into individual subatomic particles (the symbols shown in the diagram stand for photons, electrons and quarks). At the very bottom of the diagram is the level of individual particles, where we have taken the extreme step of considering, not just each time slice, but each individual particle, as its own "world". <center> <p><img SRC="modality13.gif" height=399 width=533> <br><b>&nbsp;Fig.5: The big picture.</b></center> <p>Earlier, we used the Principle of the Identity of Indiscernibles to unify two worlds into one, based on the fact that they contained the same consciousness. Since it is a well established fact of quantum physics that any two electrons are indiscernible, we can do the same sort of thing here. This gives another level of analysis where all individual electrons in the universe are considered as the same electron (the level of particle types). The universe now contains a single electron, zipping backward and forward in time. In fact, at this level of analysis, the time slices have no meaning. Movement "backwards in time" is no different than movement to the left or right. Richard Feynman <a href="modality_fn.html#fn15">[15]</a> gives a very accessible description of this slightly twisted way of looking at electrons. It may be twisted, but it is in complete accord with the quantum equations. <p>Higher up in the diagram, we find the level of our observable universe--the level of conscious-identity (or strong personal-identity). Higher still, we find levels of analysis that, one might say, "transcend" our own particular universe. This transcendence is not some mystical notion of a rationally inaccessible higher plane of reality, like that imagined by the subjective mentalists or the material realists. The "higher reality" here is nothing more than Turing Computability or Universal Language--"whatever can be spoken or thought of". Ultimately, everything underneath this in the diagram is simply an enumeration of all possible computer programs. So the Absolute Monad would be a computer program that produces and runs every possible computer program, including itself. Such a program would, of course, never stop running. <a href="modality_fn.html#fn15.5">[16]</a> <p>Now let's go back to our primitive stick-man example, and look at the model we get when we flatten out figure 4 into a single model (figure 6). We could (fancifully) add in our Absolute Monad, and it would be true in all worlds in figure 6. If our system were the extent of all possible analysis of anything, then this would be the final goal of idealist metaphysics achieved. But of course, our system falls far short of that! So I have not included the hypothetical Absolute Monad, but the model is nonetheless highly objective compared to the more subjective models that came before. The absence of an absolute monad means that nothing is true in all worlds--this "universe" is just a collection of partial aspects. This makes it metaphysically incomplete, but that is as it should be--a simplistic little model of a stick-man with a top-hat <i>ought</i> to be inadequate, obviously. In fact, the model we have constructed does not really include a stick-man <i>or</i> a hat. These were just external interpretations, introduced as aids to the imagination--fanciful ways of talking about the very simple mathematical system shown in figure 6. In any case, complete and total objectivity is rarely desired. Usually, we want to analyze something in terms of some particular restricted model, which is inevitably incomplete from the point of view of some more objective stance. <center> <p><img SRC="modality14.gif" height=212 width=416> <br><b>&nbsp;Fig.6: A relatively objective view (Leibnizian monads are checked).</b></center> <center> <h3> <b>VIII. Quantum Idealism</b></h3></center> Finally, we must return to our question about the problem of idealism. If hats can both be and not be... if the universe can consist of two worlds suddenly "collapsing" into one... if a person can be said to inhabit different worlds simultaneously... how can there be any order to the universe? Our early example of the table turning into a pink bunny-rabbit would be another example. How can this be, when we <i>know</i> from experience that the world is ordered? <p>The method of rationalism has always been to accept reason and question experience. Descartes' Cogito asks us to take only the most basic given (consciousness) as empirical fact. Although the universe of the stick-man and his hat is strange, modern physics has revealed something the nineteenth century idealists would not dared have hoped for: empirical evidence that the world really does work this way! In fact, quantum theory's infamous Schr&ouml;dinger's Cat thought experiment <a href="modality_fn.html#fn9">[11]</a><a href="modality_fn.html#fn16">[17]</a> is identical to the hat example in all the essentials. Quantum theory allows any possibility to occur (within certain arbitrary boundary conditions--for us, the conscious-identity version of the Anthropic Principle). In the hat example, only two possibilities exist at t<sub><font size=-1>1</font></sub>, so at t<sub><font size=-1>0</font></sub> we would say the "experiment" had a 50% chance of one result and a 50% chance of the other. At t<sub><font size=-1>1</font></sub>, we would experience a "collapse" of this "superposition of states" into one state. But from a more objective stance, this would still be <i>one</i> state where <i>both</i> results have occurred. I do not experience this more objective stance, since it is above the level of individual consciousness and includes two <i>differently </i>conscious versions of me. <p>In real life, of course, a hat's description is made up of many sub-programs; it is a complex beast, inter-related with my brain and the rest of the world in a complex way. So it takes only a tiny change in the hat to affect my consciousness, and the superposition of states would be retained for only a very short time (such systems are called "chaotic").<a href="modality_fn.html#fn17">[18]</a> In our thought experiment, the hat behaves in a weird quantum manner because we <i>pretended</i> that a hat <i>could</i> exist at the level of the base resolution, or granularity, of the universe. The hat is a "quantum" thing. It has no inner complexity. It is connected up with the rest of the world in a very simple, nonchaotic way. So a great disturbance can be made to the hat (such as its disappearing) without affecting consciousness (except, of course, that it was just a fantasy that consciousness could exist in such a simplified universe in the first place!). <p>In the real world, however, there <i>are </i>objects close to the fine resolution of the universe, which <i>can</i> be modified in significant ways without affecting a nearby human's state of mind. Electrons, for instance, have a common habit of being in multiple places at the same time (or, according to the Many-Worlds interpretation,<a href="modality_fn.html#fn9">[10]</a> occupying more than one world at the same time), as well as disappearing from one location and reappearing at another, without travelling the distance in between. The quantum world is very much like the idealist's universe. Even the table's turning into a pink bunny-rabbit is allowed in quantum theory. In fact, such a possibility is automatically part of the quantum wave equation simply because it <i>is</i> a possibility. Tables turn into rabbits in only a vanishingly small percentage of worlds, however, so (although it happens in <i>some </i>worlds) we never observe it in practise. <p>Perhaps the kind of modal analysis sketched out in this essay could eventually show that all the strangeness of quantum mechanics really <i>do</i>es follow directly from a rationalist metaphysics. Much work remains to be done, however. <center> <h3> IX. Death and Immortality</h3></center> The reality of a vast plurality of worlds leads inevitably to the sticky side-issue of death and immortality, which I will only briefly address here (readers offended by interesting but half-baked speculation can skip this section). If all possible worlds exist objectively, then an appropriate stream of consciousness model would include <i>all</i> possible selves into the indefinite future. It is hard to see in this case how one could die. Is this a proof for immortality? I do not know. Certainly, if all possible worlds objectively exist, then one's "death" in one world does not necessarily mean subjective death, since one survives in others. Death would have to be a <i>logical</i> necessity to be brought into the model. There would have to be <i>no</i> possible worlds at all in which you survive whatever calamity befalls you to justifiably say that you <i>have</i> died, from the stream of consciousness viewpoint (which is, after all, roughly our everyday view of ourselves). Certainly this seems to mean that we will survive disasters like car accidents. There are surely possible worlds were the accident did not occur. But what about old age--is survival of that not at least a <i>logical</i> possibility? Surely, one might suggest, surviving old age is not like a square-circle, something even in principle inconceivable... or is it? It is impossible to answer this without knowing exactly <i>what </i>our consciousness and its environment are, and we still have only the vaguest grasp of this. Even if there are a small percentage of worlds where one somehow survives old age, it would seem that most of these would have us in a near-death state, becoming forever more and more decrepit and diseased (or until medical science can help us out), but always somehow surviving because of some fluky logical possibility. This is not a form of immortality many of us would hope for (which, of course, has nothing to do with whether or not it is true). Such a view would also have the usual laws of thermodynamics suddenly being violated around the age of one hundred, which seems a little silly (which, again, has nothing to do with whether or not it is true). <p>But what if we could <i>a priori</i> eliminate worlds in which we die as improbable (or worlds in which we <i>prematurely</i> die, if some kind of death is at some point a logical necessity)? One might argue for this on the basis that such short-life-span worlds will <i>eventually </i>be weeded out at the moment of "death" anyway. In this theory, the only worlds that have much probability in the first place are those in which we avoid death "naturally", without the need for fluky, improbable "miracles". This would be to take the "true" stream of consciousness view, only considering streams that are complete from the beginning of life to the end (if any). If that could somehow work, we could posit some kind of "maximum life-span" principle--that longer lives are more probable, in the space of all possible lives, than shorter lives. If so, we can expect medical science to provide us with indefinite life extension <i>before</i> we reach the age where miraculous flukes would be required to keep us alive. This might mean that the most probable time to be born in the history of the universe would be right around the technological cusp where science becomes advanced enough to extend life indefinitely. Someone who was born much before the cusp would have a life too short to be very probable. Someone born much later would experience a shorter life solely by virtue of having been born later (at least, if eventual death is a logical necessity--if instead life were literally eternal, the person born later would always and forever be younger than the person born earlier, but the total life-spans of both would be infinite, making it unclear, to me at least, what the probabilities would be). If the universe eventually ends, it would mean that eternal life is a logical impossibility, but that we can expect to survive to the bitter end (billions of years instead of mere decades). The life-span of the universe would thus be roughly the maximum possible life-span of a conscious being. The maximum life-span principle would explain why we seem to have been born at such a fantastically pivotal time in human history. This theory would also allow a proof for immortality (or more precisely, maximum <i>logically possible</i> life-span), one that does <i>not</i> require thermodynamics to suddenly work differently when we get old, and would allow us to live indefinitely without being in a decrepit state for perhaps vast periods of time. <p>On the other hand, I'm not sure how such a theory could really work. If we base our philosophy on the <i>Cogito</i>, we can only take our <i>present</i> consciousness as our starting point to reason about such things. To take the more objective, but not fully objective, view of true stream of consciousness seems arbitrary. It simply provides us with the most comforting, easy-to-swallow view of reality (which, of course, has nothing to do with whether it is true). The probabilities, being subjective, must surely be based on one's <i>current</i> conscious state. That leads us back to either the "eons of barely-alive decrepitude" theory or the "it's logically impossible, given our current condition, to live much past one hundred" theory. The first seems suspicious because it would violate the laws of thermodynamics (at least under their usual scientific interpretation). The second seems implausible considering the truly enormous variety of logical possibilities under consideration. Yet the true stream of consciousness model seems to be arbitrary wishful thinking. <p>On the other hand,&nbsp; it may be completely fallacious to extropolate our current consciousness into the future like this. So far, my reasoning has gone something like this: "the only futures that are relevant to me-now are those for which me-now is a valid past state. Therefore, those are the ones I will use to talk about my future probabilities, even if they are all wildly improbable." However, there is no <i>metaphysical </i>principle tying you to these futures as opposed to other futures of selves that are close to being future yous but not quite, or for that matter selves that are nothing like you. If we are as strict as we possibly can be about applying the <i>Cogito, </i>all we can really do is presume that our current conscious state is a reasonably probable one out of the set of <i>all </i>conscious states (including ours and all other possible creatures). So it is not really legitimate to declare quantum miracles highly probable, just because they are required to maintain a future self that is literally <i>our </i>future. All we can really say is that certain consciousnesses are more probable than others, and one in which quantum miracles had to occur is <i>highly </i>improbable, and so one should not expect to ever find oneself in such a state. So a more realistic view of the future is simply to realize that it involves many possible consciousnesses, and that some of these will satisfy our arbitrary concept of "me" and some won't. But whether we focus on those that do, or not, it is not sensible to view our future as consisting of quantum-miraculous selves. If we choose to focus on future consciousnesses that are very much like "me-now", we are free to do that, of course, so long as we choose relatively probable ones. It is thus more reasonable to extropolate oneself into the future by imagining a self that is almost but not quite consistent with one's current self, but that has high probability, than by imagining--as we were earlier--a more strictly consistent future self that has extremely low probability. <p>This gives us reason not to worry over being faced with an incredibly improbable eternal state of decrepitude as opposed to being saved by medical science, but still does not tell us how to view the future if there are <i>no</i> probable futures containing us, no futures where medical science saves us without any miracles. Am I then to conclude that the most probable future either contains a future me or a future <i>somebody else? </i>There seems to be something wrong with that. While it may seem reasonable to extropolate into the future towards a self that is <i>almost </i>a future me-now, extropolating into someone who is just a different person altogether is plain silly. So are we not then stuck with extrapolating into the highly improbable quantum-miraculous future-me, if we are to extrapolate at all? Perhaps we <i>shouldn't </i>extrapolate at all, if that means a future that is miraculous. A miraculous future is also quite apt to be very unstable, and unpredictable. Such a future might be more like a half-conscious dream-state with no clear linear direction of time, than anything like a solidly envisionable future in something we would call a "world" , and what sense does it make to extrapolate into such a state and call it the "future"? In the end, of course, one must realize that it is an arbitrary decision what one chooses to extrapolate into in the first place, and all that science really need provide us is a reasonable explanation for who we are right now. In any case, it seems likely to me that a miraculous future self would lack many of the features required for a stable self identity. I also currently see no way to really prefer longer life-spans as more probable than shorter ones, an assumption that it seems to me would be simply wishful thinking, since it would require calculating probabilities from the viewpoint of an entire life-span, and that seems to violate the spirit of the <i>Cogito. </i>Nonetheless, the idea may still have some merit, and is an intriquing one. Still, I see no reason to currently make any bold claims as to the mortality <i>or</i> immortality of the soul based on quantum physics. We may be immortal--in which case it will probably be medical science that saves us, not a mystical spiritual soul <i>or </i>quantum miracles--or we may be quite mortal and destined to die. A more complete understanding of quantum mechanics as a rational theory, which we have only just begun to explore in this paper, may help in all this; it may well be that the tools to properly answer the death question are just not at hand in our current scientific arsenal. <center> <h3> <b>X. Conclusion</b></h3></center> There are many things that can be said about the example of the stick-man and the hat, within the language of modal logic. For instance, two trivial facts are: <ul><img SRC="modality3.gif" height=12 width=12>w(t<sub>0</sub>)&nbsp;&nbsp;&nbsp;<img SRC="modality0.gif" height=15 width=14><b><font size=+1>A</font></b> <br><sup>&nbsp;<b><font size=+1>~<img SRC="modality3.gif" height=12 width=12></font></b></sup>w<sub>~H</sub>(t<sub>0</sub>) &nbsp; <b><font size=+1>H~H</font></b></ul> These are quite straightforward, but more complex schemata could be constructed, and we could ask whether they are valid in our class of models. We could relate these schemata to the accessibility relations between worlds in a class of models. This sort of analysis could be metaphysically very important. However, if we are actually to do real metaphysics, as opposed to simply laying the groundwork for a metaphysics, any valid schemata we come up with must reflect the real constraints on worlds containing conscious entities, not those in some little toy world like the stick-man example. Any schema from something as simplistic as the stick-man example is greatly suspect, due to the unrealistic nature of the example. That is not to say we could not find out anything useful from such toy worlds, but truly interesting metaphysical results require a model that takes into account a reasonable computational theory of what this thing we have been calling "consciousness" <i>is </i>in the first place. In the meantime, the role of logical inference and computation must be integrated in a consistent and useful fashion within the system. Then the real work can begin. <center> <h3> <b><a href="modality_fn.html">Notes &amp; References</a></b></h3></center> <hr WIDTH="80%"> <center> <address> <i>Go Back to <a href="http://www.allanrandall.ca/">Allan Randall's Home Page</a>.</i></address></center> </body> </html> <!-- FILE ARCHIVED ON 12:36:53 Jun 9, 2007 AND RETRIEVED FROM THE INTERNET ARCHIVE ON 15:48:38 Nov 21, 2012. JAVASCRIPT APPENDED BY WAYBACK MACHINE, COPYRIGHT INTERNET ARCHIVE. ALL OTHER CONTENT MAY ALSO BE PROTECTED BY COPYRIGHT (17 U.S.C. SECTION 108(a)(3)). -->