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CHAPTER I
INTRODUCTION
the paradox is only a conflict of what reality isAnd your feeling of what reality ought to be.
- Richard Feynman
A.Background of the StudyThe advent of quantum physics vis-a-vis Einsteins theory of
relativity took science a great leap towards obtaining a grand unified
truth, albeit it also threatened the demise of all fundamental scientific
foundations, leading to a premise a tad closer to uncertainty. What the
evidences are concluding is that the evidences themselves are
inconclusive- the world as we know it might not have been real at all,
and aside from the radical notion that physical objects exist due to our
constant consciousness, it also suggests that a single phenomenon can
branch up to infinite probable results that are as real as the actual world
we live in.
Contrary to popular belief, the notion of possible worlds traced its
origin way back to medieval philosophy, with the likes of Severinus
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Boethius postulating a Christian worldview integrated with Aristotelian
elements, specifically the theory of potency and actuality. It was not
until 400 years ago that the philosopher Wilhelm Gottfried Leibniz
utilized this notion for the existence of other possible worlds in
allusion to his metaphysical account of the monads. These worlds are
comprised of monadic elements which failed to actualize in reality- they
are, in a sense, potential substitutes for each individual monad currently
existing. Those which were deemed by the Supreme Monad to be the
best among all its alternatives were aggregated and actualized, thereby
arriving at an optimistic conclusion that this current world poses to be
the best of all possible worlds in a teleological sense. It has to be noted
that Leibniz, though managing to postulate a relativistic view of time
and space, did not elaborate further on the nature of these potentials
along their own space- time, but from the fragments of his work it is
apparent that he would most certainly deny any predication due to the
fact that their activity ceased from the moment they failed to exist.
The 20th
century marked the decline in metaphysics as logical
empiricists such as Bertrand Russell and Rudolf Carnap vehemently
attacked the discipline due o its inability of verification, thus, its being
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un- scientistic. Fortunately for the metaphysicians, the following period
witnessed what Thomas Kuhn dubbed as scientific revolution- a major
paradigmatic shift which enabled science to break free of its three-
hundred- year- old chains from the dogma of classical physics. A
greater, more encompassing theory has been conceived, and as scientists
probed into the mystery of quantum mechanics and quantum field
theory, certain realizations dawned upon them as to the truthfulness of
proposals endorsed by the modern philosophers. Leading quantum
physicists are baffled by the observations and inconsistencies
experimental results are showing. Nobel awardee for Physics Richard
Feynman was so convinced that he was said to have remarked that it is
safe to say that no one fully understands quantum mechanics. Indeed,
the phenomenon was shrouded in mystery, and as the ground
assumptions has been rebuked, scientists are left with nothing but their
instruments and speculations.
To acknowledge the gravity of the problem, one must first
understand the premise. Suppose a beam of light is fired in an array of
two cardboards, one in front of the other and each with a hole bored
through them. The first cardboard facing a transmitter beam has a hole
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exactly in the middle, while the next cardboard has holes poked on both
sides so that no two holes overlap. The beam is then fired, with light
particles (photons) being controlled so that they pass through one
photon at a time. A photosensitive screen which records the frequency
of each photons collision at the opposite side depicts a rather curious
pattern- instead of showing two neat piles of photon similar to what is
normally produced by particles, the figure exhibited interference akin to
that of a waves. This wave- particle duality of light happens to be the
crux of the story- scientists simply could not make amends to the
respective definitions of particle and wave so as to accommodate this
phenomenon. They could either accept that light as well as other
elementary substance is a particle and follow the Copenhagen
Interpretation and its implications, or they could subscribe to its wave
property and acknowledge that the world is constantly oscillating as
predicted by the Many Worlds Interpretation.
Hugh Everetts Many Worlds Interpretation (MWI) champions the
necessity of possible worlds. In the infamous Schrodingers Cat- in- the-
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Box though experiment1, the set up was modified so that instead of the
inevitable wave function collapse, all the possible realities that could
develop from the conditions are actualized, leading to proliferation of
alternate universes in each quantum state2. Instead of the perceiver
opening up the box and finding the cat either dead or alive, MWI posits
that reality branches out and produces two worlds for each of the
circumstances enumerated. Of course, these ramifications occur at the
elementary level, although for illustrations sake it was magnified to
exhibit a tangible situation.
B. Statement of the ProblemIt seemingly came to the point that, in order for us to unify the
conflicting conditions certain phenomena imply we have to adopt a
premise that there might be metaphysical elements science finds the
1The following is a brief account of the experiment: Suppose a box contains an electron. The
probability then of the electron being found inside the box is 1, ceteris paribus. Now suppose an
automatic wall divided the box into two, thereby isolating the electron on either side. The
probability now of finding the electron on either side is .5. A toxic gas (radioactive decay in someversions) was installed in the vicinity so that when the observer (which is outside the system)opens the wrong partition, it will be released and kill the only living creature in the setup, in this
case, a cat. This notion thereby implies that, unless the perceiver actually decided whether to open
a partition or not, the cat is in a state of limbo between life and death.2 A concrete representation of this with respect to our cat is the actualization of a universe in
which it is alive and another one which is otherwise. Both worlds are likely to be true.
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need to subscribe to. One such premise is the notion of possible worlds-
which, aside from its usage in semiotics and semantics, have been
necessitated to exist in order to account for the curious occurrence
depicted in the behaviour of elementary particles. Supposing that many
worlds indeed exist and it is out of our capacity to perceive them, could
we then utilize our most powerful cognitive facultythat of modality
to somehow predict the nature of these worlds? The appearance of the
word infinite might prove to be misleading for some, yet philosophers
such as David Lewis subscribe to infinite concrete possible worlds. One
is then entitled to ask how this infinity- possibility interplay works.
Should infinity also include those which are improbable? Would it be
acceptable to have elements outside infinity?
The purpose of this study is to define the boundaries of infinity in
order to provide clarity and at the same time predict the nature of laws
that govern possible worlds in the Leibnizan and Schrodengerian CI and
MWI context.
To manage the study, the researcher formulates the following
subquestions:
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1. What is the concept of rationality and how infallible does itstand?
2. What is modality and how is impossibility conceptualized inaccordance to this?
3. Given the parameters of modal logic, why is it impossible toconceive of infinitely possible worlds?
C.Significance of the StudyIt is inevitable for physics to trace back its roots to philosophy as it
faces a new facet whose own existence hinges on philosophical conceptions.
One of the purposes of this study is to recommend certain parameters that
physicists might as well consider probing into. The discipline of modal
logic, for instance, has more prevalently been used in possible world
semantics. His study in effect aims to reconcile theory and actuality as far
as the two concepts are concerned and hopes to elevate the role of
philosophy side- by- side the natural sciences one more.
One of the topics discussed therein is infinity. The researcher
hopes that the study would help clarify some misunderstandings about
infinity as comprehended when scientists pertain to infinite possible
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worlds. Again, this task requires incorporation of another discipline,
namely the Cantorian Set Theory. This aims to showcase the versatility of
philosophy in its ability to encroach other fields (as the said theorem is to
be subjected under modal logic).
Being a non- scientific person presenting the subject matter to non-
scientific audience, this study also provides a philosophical approach on
one of sciences more technical problems by discussing in a layman form,
without presumably losing its dash of scholasticism. Students of philosophy
are very often disheartened whenever faced with issues involving
disciplines outside their comfort zone. The researcher equally hopes to
sever the aforementioned mindset by providing a crude yet sufficient
attempt at presenting the topic in such a way that is comprehensible and
enticing enough to initiate further researches on the same topic. Similarly,
she expects that the research would be of further use to those who choose to
pursue the same endeavour.
D.Scope and LimitationsAs much as the topic and speculations involved are mostly
metaphysical, this study does not wish to invoke postulating a spiritual
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realm. Although it can be said that the spiritual might comprise one of the
many possible worlds dealt with in this study, we do not want to use it as an
excuse to linger on the subject longer than necessary. As a fitting and more
concrete concept in the form of consciousness is available, we would
instead direct concerns and inquiry involving the former idea to the latter
one.
For simplicitys sake, the study outline several sentences/ claims
that are said to be necessary in postulating the notion of possible worlds.
These sentences are to be manipulated and treated only in their discursive
element- any interpretation outside the formal linguistic equation is to be
disregarded. This is imposed to limit misinterpretation based on
postmodernist readings and focus the discussion on its logical formulation,
formal and non- formal, instead.
The study attempts to reconcile in a sense the possible world
semantics as governed by modal logic and the actual infinite possible
worlds as depicted by theories proposed in quantum mechanics. Similarly,
it endorses the use of Kripkean Modality System as its primary reference,
albeit only theorems K, D, T, B, S4 and S5 are to be reviewed and utilized.
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Being a philosophical paper, this study does not delve too much on
the technicalities involved in presenting the arguments. All computations
and equations the researcher deems necessary according to her discretion
can be found at the appendix and would be kept at a minimum. If perchance
certain computations are to be made to accentuate the point, the
mathematical theory incorporated in this study- the so- called Set Theory-
will not be approached mathematically.
E. Review of Related Literature and Studies
Trigg, Robert. Reality at Risk: a Defence of Realism in Philosophy and
the Sciences 2nd
ed. Harvester Press: Brighton, Sussex, 1989
The dilemma of quantum mechanics often begs to be approached
philosophically rather than scientifically; indeed, what theoretical physicists
specializing in this field had been doing all along is translating the
metaphysics and epistemology into math and trying to speculate the
methodology upon which their efforts are directed. One of the most
prominent problems under siege is the fact that the observer has now played
a significant part in determining reality. Roger Trigg in this book
summarized the argument as the inability to correctly measure results
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objectively due to the fact that the mere presence of the observer and the
instrument creates interference on the result; it has become increasingly
difficult to isolate the observer from the subject being observed. In the
same book, he quoted physicist John Wheeler to have said that observation
in this sense is not anymore a befitting word to describe the process; instead,
one has to think that the concept occurring irrevocably is participation.
Measurement is a post- product of analysis and discretion, and it involves a
great deal of consciousness to be able to apprehend. An entity does not
exist if it is incapable of being measured, reiterating Rene Descartes.
However, this presents us a trivial question: in an instance reflecting
Heisenbergs Uncertainty principle, one cannot know the location and
momentum of a particle at the same time. Does this in effect justify that, at
a certain point in time, the particle cease to exist? The researcher, as well as
Trigg, does not believe so. A perspective of this sort tends to be overly
condescending in laying the blame upon the object instead on the limitation
that blinds humanity.
Hannah, Robert. Rationality and Logic. MIT Press: Cambridge,Massachusetts, 2006
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What are these limitations? Aside from the senses (many people
would have quite agreed that it is somewhat difficult to conceive of an
additional sensible mode outside the ones enumerated), the other faculty
responsible is rationality itself. Hanna in his book exemplifies the interplay
between logic and normativity by enumerating the philosophical
perspectives with which these concepts are viewed. Arriving at a total of
four combinations by merging two sets of binary options stemming from
the assumption that logic is normative; Hanna was able to present the
common sensical view of how logic is perceived, as well as other
alternative versions of its epistemology. For instance, intrinsically
hypothetically normative logic talks of reasoning as: i) necessary, relational
or non relational; ii) conditional, instrumental; and iii) prescriptive or
evaluative. This is an important categorization for it opens up different
avenues on how logic is conceived. A more thorough discussion on this
topic is to be made on the following chapters of this study.
Plantinga, Alvin. The Nature of Necessity. Clarendon Press: Oxford, 1974
Perhaps this inconclusiveness prompted Alvin Plantinga to take an
extremist Quinesian stance and remark that some philosophers hold that no
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proposition- not even the austerest law of logic- is in principle immune
from revision. Plantingas stance on this elaborated in his book as it
discusses the seemingly susceptibility of grounded logical laws to fall into
neglect as human understanding progresses. This particular text is of utmost
importance in this study, as we shall see on the latter part.
Yap, Gilbert. Conquering the World of Images Through Awareness.
University of the Philippines, Diliman. unpublished Master's thesis:
2009
One intriguing argument is the formation of a pre- empted mental image
in lieu of the perceivers own discretion and its interference over the actual
interpretation of data. Eastern mysticism has much to say about this topic,
as Gilbert Yap elaborated in his masters thesis. His discussion on the
defects of consciousness was elucidated in his quoting Skitt (2000): When
the individual mind is occupied with deception because of the presence of
images, the individual cannot be and cannot see the actuality an image
which prevents me from looking exactly what is. Similarly, he
continued: people try to address their problems by using various strategies,
mechanisms and means. These in whatever form, originated from ideas.
And, as mentioned, these ideas are mental contents that act as images. If
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that is the case, the ways that supposedly solve the problems merely creates
another problem, that is, deception. Although the researcher agrees with
Yap in this matter, she rejects his alternative solution and overall approach
to the problem, stating that it poses inconsistency over the prevailing
methodology of this study, that is, analytical and primarily hinged upon
tangible scientific facts.
Isaeva, Elmira. Human Perception of Physical Elements and the
Simplex Interpretation of Quantum Physics in Progress inPhysics: January 2000 (Volume One) p. 47- 51
A related and consequently more appropriate study by Elmira Isaeva
reiterates a point similar to that of Yaps, albeit it a more technical manner.
She asks that, upon the brains condensation/ analyzation of facts, which
reflection- passive or active, unequivocal or multiple valued- occurs?
This proves to be a major concern especially since physicists are wary that
the perceivers supposedly unbiased judgement is marring the experiment
results due to their active consciousness (the process we have, in effect,
more or less summed up in Yaps thesis). Both Isaeva and Yap (the latter
echoing Jiddu Krishnamurti in his thesis) agree that consciousness is not
much of a reliable faculty in discerning the outside world, notwithstanding
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each utilized different premises. In addition, knowledge as classified by
Isaeva to be either usual, unusual, transcendental, or transient allows us to
delve deeper into the nature of knowledge exhibited by logic. At the end of
her paper, she concluded that our consciousness comprehends the
objective quantum world, a statement which the researcher rejects in
reasons that shall be discussed later.
Bell, John S. Speakable and Unspeakable in Quantum Mechanics.Cambridge University Press: 1987
One might wonder why scientists are making a big fuss out of the
property of light and other elementary substances being either a wave or a
particle. If that indeed is the case, why not integrate the two into a single,
unique categorization that specifically pertain to these? J. S. Bell pointed
out that this has, in fact, already been done by Louise de Broglie and David
Bohm. Known as the de Broglie- Bohm synthesis or the pilot wave, this
picture disposes of the necessity to divide the world into systems and
apparatus. Basically, it proposes a bizarre requirement that events from
one place reach other places faster than the speed of light. Since they
intellectually agreed that nothing in the known universe is faster than the
speed of light (and for the sake of maintaining the relativistic mass equation
to be sensible), the thesis has to be discarded.
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Skyrms, Bryan. Possible Worlds, Physics and Metaphysics.
Philosophical Studies: An International Journal for
Philosophy in the Analytic Tradition Vol. 30, No. 5 (Nov., 1976),pp. 323-332
Many Worlds Interpretation in this regard is a modified version of
this thesis. Simply put, it does not subscribe to wave function collapse (i.e.
does not believe that the perceiver tends to dictate what reality to actualize)
but proposes instead that upon every change in the quantum state reality
branches out and creates a space- time of its own; not just any mathematical
reality, but a concrete one which possesses almost the same characteristics
as this world. David Lewis in his Counterfactuals (1973) is convinced of
these real yet unobservable worlds, yet he also asks: how do we set the
criteria on how real these worlds could be? A suggestion is put forth by
Bryan Skyrms in Possible Worlds, Physics and Metaphysics (1975) and
was stated as such: A world is real if and only if every proposition true-
in- that- world is true (p. 2). However, as was discussed in his paper, this
assumption might lead to contradiction if discussed semantically. The
solution he came up was to assign coordinates to specific relational
conditions of the sentence and either treat them as distinct possible worlds
which are both real (but is thus inconsistent) or combine them both into a
single super- world in which the premise and its negation are true at the
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same space- time plane and posit that such inconsistent world is real. He
also provided a glimpse on what he thinks these possible worlds are with
regards to Everetts interpretation of MWI, and his assumption would be
probed further at the succeeding chapters of this paper.
Rucker, Rudy. Infinity and the Mind: the Science and Philosophy of the
Infinite. New Jersey: Princeton University Press, 1995In proceeding with the discussion, we are also obliged to look at how
infinity is viewed.Rudy Rucker in his book entitled Infinity and the Mind
(1995) identified two major categories of infinity- those that exist in the
physical world and those existing in the so- called Mindscape. Infinities in
the mindscape are comprised mainly of abstractions (such as thoughts and
ideas) and are dealt with using Cantorian Set Theory. The physical
infinities on the other hand are laid down in the form of spatial, temporal
and infinitesimals which are individually dealt with. However, as to which
of these infinities the notion of possible worlds subscribe to would be
answered shortly.
Two objections have been raised against infinity by St. Thomas
Aquinas in his bookSumma Theologiae which was quoted and answered by
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George Cantor, the father of the mathematical branch of set theory. The
following quote comprises Ruckers citation of the said quote:
The existence of actually infinite multitude is
impossible. 1) For any set of things one considers must be aspecific set. And sets of things are specified by number ofthings in them. Now no number is infinite, for number
results from counting through a set in units. So no set ofthings can actually be inherently unlimited, nor can it
happen to be unlimited. 2) Again, every set of thingsexisting in the world has been created, and anything createdis subject to some definite purpose of its creator, for causes
never act to no purpose. All created things must be subject
therefore to definite enumeration. Thus even a number of
things that happens to be unlimited cannot actually exist.
Oppy, Graham. Philosophical Perspectives on Infinity. Cambridge:Cambridge University Press, 2006
Without succumbing to Cantors qualms, one can easily say that
using this objection is by no means a futile attempt to debunk infinity as
used in the context of quantum physics. This is especially evident in the
second argument- quantum mechanics, being a non- teleological theory,
would not be able be able to ascribe itself to such requirements. Ruckers
discussions on infinity, as we shall see later on this paper, provide a
succinct solution to the objections on the impossibility of infinite possible
worlds. A similar approach was undertaken by Graham Oppy in
Philosophical Perspectives on Infinity (2006); indeed, this material shall
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serve as one of the studys primary sources due to its discussion on the
interplay of modality and infinity.
F. MethodologyRather than regarding logic and the likes an innate, universal faculty,
the researcher attempts to point out that such presupposition could have
been nothing more than a convention established by the so- called
authorities on logic. What is apparent here is that, taking into consideration
that not all people possesses the same level of reasoning capacity, they tend
to abide by the convention set by the makers of the rules without much
introspection. This then brainwashes us into thinking uniformly and
believing that the said convention indeed is true because, as a result, all
people grasp it and take it to be innate. Failure to comply implies
retardation of the said faculty, under of course the normative assumption
that all humans must be able t o reason out such- and- such way.
It is to be noted that reasoning is different from rationality, and this
differentiation poses a great deal of arguments in support to our objection.
Taking the empiricists point of view, they claimed that during birth
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humans are endowed with sense faculties geared for acquiring experiences.
Hence, the stand of rationality being innate is jeopardized, and since it is
but a potential without so much as a single defining criteria encompassing
all strokes, becomes difficult to justify. We therefore take a skeptic stand
towards its axioms in logic, mathematics and modality, seeing as these are
largely supported by the Platonic notion of forms.
Applying this notion to the epistemological possibilities/ necessities
comprising a possible world, how would the basic assumptions of modal
logic fare? Certainly, given the premises set by the quantum field theories,
we are able to find real life applications for these laws. However, it must be
known that what is out there deals with entities outside human
understanding (say, the superstring theory which is currently backed upon
only by equations coupled with speculations). In this note, scientists rely on
the pre- established conventions set by finite human understanding, hence
the study of the universe inclines to a bias perspective akin to that of the
goldfish- in- a- bowl thought experiment.
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G. Conceptual FrameworkThe study surveys several technicalities (i.e. mathematical proofs
and expositions) of the concepts presented. However, it is to be noted that
the study mainly focused on the philosophical interpretation that the
equation entails; the equations themselves are not to be made the central
point of the study. The researcher will attempt to discuss the implications of
a quantum state- bounded possible world in the context of modality.
The researcher attempts to utilize the input- process- output modelto
be able to achieve this. The input focuses on the issues of reliability of
modality with regards to h8uman rationality. It inspects the arguments
supporting the notion that logic is not the key to good reasoning3
and
integrates it to the truths presented via modality for all possible worlds.
This in turn comprises the process; examining, for instance, the limits
imposed by mathematics our conception of sets and infinities. Lastly, the
outputportion hopes to yield results based from these method and establish
these results as a legitimate source of reference for studies concerning
similar interest.
3A wordplay in lieu of Jaakko Hintikkas article, Is Logic the Key to All Good Reasoning?
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Certain parameters from the discipline were considered and given a
theoretical treatment using modal logic. These conditions were converted
into propositions and analyzed by means of first- order logic. Premises are
given, converted into its logical counterpart, and analyzed using modal
logic. The results then are presented to preserve objectivity, albeit the
researcher is still entitled to lay her interpretations on the subject matter in
the succeeding chapters.
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Figure 1 Represents the arguments the researcher puts forth. The known conditions of
possibility, necessity and impossibility comprise the whole of current human understanding;
the rest of the box corresponds to ideas rejected by these (for instance, the possibility of an
impossible world and other paradoxes).
Realm of Human Understanding
Realm outside Human Understanding
Possibilities Impossibilities
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Figure 2 Shows the flow of argument of the researcher. Following this line of argument, the
notion of impossible worlds is in a paradoxical position- the possibility of an impossible world
being an element of infinitely possible worlds.
Innate potential for rationality endowed at birth
Experiences constituting knowledge (Quine)/ Beinglimited by language (Wittgenstein)
Taking these conventions as the sole reference of
knowledge
Rejecting alternative ideas not consistent with the
conventions (e.g. possible world considerations)
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H. Definition of Terms
For the sole purpose of making the study less susceptible to
misinterpretation, the following list the definition that corresponds with the
concept as used:
Quantum mechanics here pertains to the system by which quanta (packets
of energy) are observed, measured and behaves. It is also referred to in the
introduction as the quantum state theory, although the second concept
already integrates this field with that of relativity. The distinction between
these two is not given much regard in the course of the study.
We are to combine Ruckers, Lewis and Everetts theses in order for us to
understand what a possible world is. In its strict sense, a possible world is
something out of the reality we are currently experiencing. Adopting the
classical Cartesian view, reality is that which can be measured; positing that
they indeed exist, we therefore assume that these possible worlds are
realities waiting to be realized.
The concept of necessity is also dealt with in this paper, as well as its
counterpart, possibility. Here the researcher would resort into using the
typical representation of these two quantifiers: impossibility is expressed
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as ~ (not possible); applying modal negation to the proposition yields ~
(necessarily not). Emphasis was given on the word order.
I. Division of the ThesisThe thesis is to be divided into five parts; the first part serves
as the introduction to the problem and the approaches undertaken in order
to address it; Chapter II deals with the notion of necessity alongside
impossibility and the concept of possible worlds as advocated by the
Leibnizian metaphysical perspective and the Many Worlds/ perceiver-
centered interpretation as subscribed by Bohm, Wheeler and others.
Chapter III includes a thorough discussion on modal logic, and Chapter IV
presents arguments that hope to elucidate the nature of possible worlds with
respect to modal logic. The final chapter, Chapter V, contains a synthesis of
the previous discussions and presents the authors conclusion regarding the
subject matter.
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CHAPTER II
RATIONALITY, LANGUAGE AND THE WORLD
In everything one thing is impossible: rationality.
Friedrich Nietzsche
How does one know that he knows what he know? One of the many
epistemological problems which philosophy took care to delve in and
answer, the subject of human rationality perplexed philosophers and
psychologists as they have come to realize that they could but provide a
biased perception, inevitably due to their inability to detach themselves
from the subject. Personal discretions and the likes often mar ones own
judgment, hence the inconclusivity of conclusions especially in the fields of
social and non- science. While natural science, logic included, seems to
enjoy immunity from these attacks, certain factions of philosophers have
raised contentions on the alleged formidable foundation of the said
disciplines. This chapter examines the reliability of cogency as humans
strongest faculty alongside understanding pegged to language and how it
would fare under scrutiny from certain schools of thought, namely,
constructivism and pragmatism.
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A. Rationality as a Universal Human TraitAristotles claim in De Anima that man is a rational creature stands
uncontested for millennia at hand, serving as a rule a self- evident premise
for the consequentialist schools who established ideologies with this
allegation as their foundation without even bothering on the truthfulness of
the claim. Albeit the rationalist- empiricist debate have struck a number of
avid supporters as early as the 16th
century, the argument had only taken a
paradigmatic shift when Willard Van Orman Quine wrote his Two
Dogmas of Empiricism, which in effect triggered a movement which
advocated a whole new perspective of scepticism directed towards the basic
assumption aforementioned.
It is evident and rather unfortunate that the renowned thinker Immanuel
Kant subscribed to this view in formulating his Categorical Imperatives
without justifying as to how he arrived at such a decision. Taking into
consideration that reason in itself is innate, he proceeded to berate the
rationalist- empiricist dichotomy without so much as a second glance as to
what comprises understanding, neglecting, for the most part, the role of
language in the discussion (footnote- metaphysical discussion on
rationality).
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B. Nature of RationalityThe term rationality is often interchangeably used with reasoning, and
as is conveniently done by some philosophers. However, it is the authors
initiative to firstly distinguish the two and afterwards treat them as one, as
she reckons the need to identify their relationship. Rationality and
reasoning are, upon closer inspection, hailing from two different
viewpoints; reasoning is a system commencing from a specified convention,
while rationality is bent on maintaining a normative- reflective perspective.
Thus rationality is the umbrella term to which reasoning falls underthis
assumption is not to be taken as an explanatory, causally necessitated
relation, for it could be such the case as these two are not causal but
correlated. A prime example would be a computing machine-- although it
has the capability to regress its procedure and pinpoint its premises which
prompted it to arrive at a systematically sound solution, still it lacks the
attribute of being rational and reflect outside its given set of norms.
Determining this distinction, however, is not included in the scope of this
discussion. It is clear thus by far that rationality invokes aspects other than
reasoning, yet for arguments sake we will narrow down our discussion to
that of rationality as exhibited in reasoning.
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Robert Hannas definition of rational creatures involves them being
conscious, rule- following, intentional, volitional, self- evaluating, self-
justifying, self- legislating, reasons- giving, reasons- sensitive and
reflectively self- consciousof, for short, normative reflective animals.
Amidst these many criteria, the author would like to focus upon the
conscious, rule- following, reason- sensitive and reason- giving aspect and
how one actually justify that this is the case without invoking self- evident
causes.
For humans, Stein (1996) offered three possible ways in which one is
deemed rational a) the Aristotelian context of man legalistically being a
rational animal; b) fallibility of human reasoning system attributed to
external forces; and c) the Freudian conception of humans as irrational
creatures.
C. Categories and DistinctionsAn elaboration is needed to further emphasize the point. Hanna (2006)
proceeded to distinguish rationality as conceived and appropriated by
rationality specialists:
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1. Mentalistic vs Procedural Sense- deals with the consciousdeliberation of invoking rationality by the person. The first argues
that every rational act undertaken is always conscious; the latter
allows some deviations from consciousness without denying that
these deviations are still rational.
Subcategories under mentalistic sense include i) rationality of
animals; ii) rationality of mental episodes or acts; iii) rationality of
mental states; iv) an animals mental capacity for rationality; v)
occurrent rationality (with respect to mental episodes); vi) occurrent
rationality of state types; and vii) occurrent rationality of state
tokens; with the last two distinctions classified under category v).
2. Meeting- the- minimal- standardvsMeeting- the- maximum/ideal- standard Sense- this distinction deals with the limits of
considering an entity to be rational by setting conditions; perchance
they reach the basic requirements of the first distinction or they fulfil
all the criteria of the second one. This distinction is often
controversial and is often used for the dignity debate; the standards
imposed by both sides however are
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3. Principled vs Holistic vs Instrumental Sense- these threefolddistinction differ from their foundation on which rationality is
anchored. Principled sense grounds its arguments on the capacity of
the individual to recognize a priori principles (the Kantian
conception of rationality) subscribing to rigidly normative moral
claims. The second one on the other hand refers to the individuals
capacity to cohere (or even attempt to cohere) an intricate web of
quasi- rational considerations such as belief, emotions or will which
mutually intertwined hence dubbed as the Hegelian conception of
rationality. Instrumental sense also has its share of prominence, as
its alternative name Humean conception of rationality echoes its
devotion to the philosophers perspective, namely, that rationality is
possessing faculties which basically champions the empiricist
viewpoint by asserting that the rules are imbibed conditionally as
channelled by experience.
What these distinctions suggest is that one can, by combining
options from different levels, arrive at a specific conception of rationality
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exhibiting two or more of the following senses. However, for simplicitys
sake this study would delve more on the first and third distinction as -
D. Limits to Rationality (Priest)In the field of game theory, an alternative assumption is made perchance
the individual fails to comply with the established norms utilitarianism
imposes. This concept is called bounded rationality, and up to recently has
yet to be diagrammed for economists to be able to create satisfactory model
describing the behaviour of individuals in this shape. Mathematics and
logic, on the other hand, are hailed as two of the most formidable reasons
why rationality holds true- for instance, no answer other than four would fit
the description of two plus two. Basking in their a priori limelight, it seems
almost impossible to question self- evident truths such as these whose
truthfulness transcends not only this world, but also including other
possible worlds. These disciplines bank on the Platonic theory of forms for
their truthfulness- since apparently they are intuitively true and coincide
greatly with our perception of reality, they are regarded as axioms and are
repeatedly able to justify themselves simply through tautology. Stein (1996)
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laid down the foundations for the so- called Standard Picture ofRationality,
which depicts good reasoning as adhering to the laws established by rules
of mathematics and logic. It is not therefore surprising that this principle is
normative in character for it imposes how one ought to reason.
However, as he pointed out, there seems to be some anomaly in
adopting this perception. Firstly, if one ought to reason according to the
norms, one ought to have access to it at least. It has been answered in the
previous discussions that some approach considers these laws to be self-
evident and considers them the end of regress. It is a given that all human
beings are capable of reasoning, yet not all has the capability to maximize
their reasoning faculty and actually adhere to it every time.
An important implication of the limits of human rationality lies on the
so- called Godel Incompleteness Theorem, an evidence for the inability of
humans to construct a satisfactory proof for the 'completeness' (i. e.
describe coherently) of mathematics, specifically number theory or the
study of pure mathematics. The second theorem appears to be more radical
than the first; paradoxically, a formal system (such as mathematics or logic)
can prove its own consistency if and only if it is inconsistent (quote this).
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E. Understanding and Benacerraf's DilemmaGiven the speculations aforementioned, one is now inclined to ask
the nature of systematized knowledge humans concieve and utilize for
reasoning. Hintikka (2001) hinted that logic indeed is the key to good
reasoning, albeit philosophers tend to attribute its usefulness wrongly. He
identified logical rules as an interplay between definitory andstrategic rules,
the former dealing with the actual individual rule as utilized in proving
conclusions, whilst the latter is more concerned with how one should
proceed in manipulating these rules so as to garner desired outcome/
conclusion. Undue credit is given to definitory rules, (Hintikka)
Such was also the objections faced in the discipline of mathematics,
which for the meantime would be put under scrutiny due to its profession of
being a tad akin to logic in methodology and origin. Logic as is
mathematics professes to be a 'body of truths and knowledge' whose rules
are grounded on axiomatic assumptions which exude metaphysical airs due
to its subtle platonic implication. The truthfulness of propositions is rigidly
objective regardless of whether man has the ability to discern it or not; in
other words, its truth value stands true outside human understanding and
language. Benacerraf used the term 'abstract' to describe these objects.
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On the other hand, knowledge imposes the direct involvement of
understanding, and Hart deems that "perception is [the] preferred basic
mode of contact with the objects required for the truth of our knowledge" (p.
3 introduction- Philosophy of Mathematics). Integrating thus the reasons
previously stated, it would consequently result to a black- white dichotomy
between platonism and empiricism, two thoughts whose reconcilability
appears to lie out of the question, as the necessary elements for
mathematical truths are the very same ones which impede mathematical
knowledge. This is in a way related but not similar to the mathematical
dispute between platonism and formalism, where the latter deems the said
discipline to be a mere series of rules applied to symbols, thus creating
'axioms' and proving 'theorems' (Infinity and the Mind, p. 169).
Two elements constitute logic- truth and grammar (Quine- Philosophy
of Logic) hence it is inevitable to consider semantics in dealing with it.
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F. The World Outside Language- An Appeal to Pragmatism andRule Following
A common dispute is the conceivability, or rather the lack of it, of
the world beyond human language. Wittgenstein (1974) echoed this
sentiment on his book Remarks on the Foundations of Mathematics
wherein he cited an example of a perfectly rational human being unable to
participate in the conventional language game. Certain assumptions even
accuse Wittgenstein of being a full- blooded conventionalist for his
extending linguistic norms and conventions to that of mathematics. He was
quoted into saying as follows:
59 ...Now if I say this to somebody it is surely supposed to mean:"just try, these bits, properly arranged, really do yield figure." I
want to encourage him to do something and I forecast that hewill succeed. And the forecast is founded on the ease in which
we can construct the figure from the pieces as soon as we know
how. (footnote- RFM p. 20e)
It is indubitable that for the most part, human understanding is in
part hinged upon language, as was championed by Wittgenstein (and
consequently by echoed by other philosophers of mathematics such as
Dummett and Pinker).
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CHAPTER III
MODALITY, NECESSITY AND INFINITY
Necessity is not an established fact, but an interpretation.
y Friedrich Nietzsche
Friedrich Nietzsche, father of postmodernism and one of the most
prominent philosophers of the 19
th
century, perhaps is not speaking per se
of the logical interpretation of necessity. Nevertheless it did not disqualify
him from legitimately uttering this claim, albeit the premise from which
this is derived would probably lie in an entirely different discipline with a
correspondingly different context. Is necessity indeed interdependent on
interpretation? The last chapter attempts to champion the fallibility of
human reasoning and its tendency to be conventionally- bounded, as
opposed to a more acceptable claim that it gears towards a more naturalistic
disposition. This chapter now focuses on one aspect of logic, that is,
modality and its powers, and determine if its following suit is as
questionable as the previous topic (since apparently it is a manifestation of
ones reasoning prowess). For the meantime, one is asked to suspend the
prejudices invoked by the aforementioned exposition and try to maintain a
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neutral stance in addressing modality.
A. Modal Logic and Modal Worlds
Modal logic is the discussion of the modes in which a certain
proposition is deemed possibly, necessarily or impossible true or false. One
of the methodologies in which modal logicians employ is the practice of
speculating using the 'possible worlds' argument. To wit, there are three
prevalent natures of possible worlds used to argue this case as was
enumerated by Priest (2001):
1. Modal Realism- endorses the view that other possible worlds are
quite as tangible as the objects of reality in our current world, though
situated at a different spacetime coordinate. This belief is advocated
by David Lewis (1975) and is used especially in his theory of
counterfactuals. The normal parameters of a possible world is its
differing space, time and causation from this universe (cite Priest).
2. Modal Actualism- takes on the belief that other abstract worlds do
exist and that the composition of such worlds need not to fulfill the
physicality criteria as opposed to the first one. Instead of physical
elements, the objects in these worlds are comprised of propositions
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which are deemed to be true. A certain shortcoming of this view
however is the necessity of the criterion of validity in which the
elements are to be subjected under and the inability to establish those
criteria without hinging them upon human rationality.
Combinatorialism, a subcategory of actualism, was
able to avoid this objection by stating that a possible world is indeed
comprised of things in this world, yet arranged at a relationally
different way.
3. Meinongianism- this belief takes the notion of possible world to be
comprised of non- existent objects such as unicorns, fairies and other
imaginary entities. Most of the objects in this universe are
epistemically stipulated to exist.
B. David Lewis' Modal Realism
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c. Historical Background
Just as we opened the last chapter surveying Aristotles conceptions
regarding rationality, this chapter similarly traces back to the great
philosopher to defend its roots and determine its ramifications. Necessity
per se was used in this sense as an alternative to the term essence, as
Aristotle is shown to be a staunch supporter of essentiality in nature.
Essentiality in the Aristotelian context talks of the features of an entity
which makes it essentially one; Aristotle for instance attributed rationality
to humans as its essential characteristic.
His discussion elicits different interpretations from current readers,
as was evidenced by the two worldviews associated with the interpretation
from his writings. Analytic a priori faction champions the main concept of
rationalists perception (as opposed to empiricists) by arguing that some
knowledge is more accessible though reasoning and lesser by sense
experience.
The next interpretation focuses more on the semantic rules the
interpretation implies and proves to be more applicable to our subject of
inquiry.De re and de dicto necessity deals with qualifying the statements in
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the notions of possibility and necessity, albeit they differ in terms of
indication; de dicto necessity qualifies the entire sentence to be either
necessary or possibly (Necessarily bachelors are unmarried men) while
de re functions more on the predicate portion of the statement, hence
Bachelors are necessarily unmarried men. These two distinctions as
introduced by Abelard accounts for most of the debates currently existing
in the study of necessity. Saul Kripke, to which the main bulk of readings
on modality is to be referenced, apparently follows among the list of
philosophers who have undertaken this view.
D. Kinds of NecessityIt is pertinent to first discern types of necessity before proceeding to
address it directly. Conditionals, for instance, enjoys two types of necessity:
1. Conceptual necessity- the lesser common view, this kind deniesthe Leibnizian context of necessity and possibility by totally
incorporating properties with the object itself, hence blurring the
distinction between the premise and conclusion (cause and
effect).
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2. Causal necessity- more commonly assimilated to that ofcounterfactuals, this type defends the false- therefore- true
pattern of conditionals and states that it is not necessarily the case
that the premise needs to be true in order for the conclusion to
occur.
Priest (2001 ibid) was able to provide a more distinct
conceptualization of necessity depending on the discipline it subscribes to.
For instance, necessity in terms of modal logic is characterized in the S5
rule (it being the most powerful theorem to date, see Priest 2001). Physical
necessity on the other hand deems the laws of nature and physics to be the
indispensable factor; laws of metaphysics such as accidents and essences on
the other hand dictates what is necessary in the metaphysical realm. The
truths subsumed under these necessities falls under the T- theorem of
reflexivity, at the very least. Epistemic and moral necessity meanwhile have
yet to be identified under which theorem their respective necessities falls
under, as it appears to be qiote problematic to immediately subsume them
under the existing rules of modal logic.
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E. Possible Worlds- The Kripkean Context in LanguageOne of the uncanny revolters from the Kantian school was Saul
Kripke, who insisted the existence of necessary a posteriori statements.
This might seem counterintuitive to some, especially since it has been
established that analytic statements are purported to be self- evident and
hence should not be dependent on sensible experience. Kripke, in his
Naming and Necessity (1971, 1980) rejects so, saying that we simply
cannot automatically know self- evidently that such- and- such is necessary.
One requires knowing through experience that prime numbers, for instance,
only have one and itself as factors (following from Quine). Albeit the fact is
considered a mathematical necessity, he pointed out that it is not rigidly
normative to equate 'can' and 'must'; consequently, 'necessary' is not
correspondingly synonymous to 'a priori' (footnote, page 38). One might
then be led to ask what the relationship between necessity and a prioricity is,
in which suggestions as binding as causality (again, we are faced with the
problem of determining which entails which) or a totally ambiguous claim
as correlativity might surface. Kripke waived the
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F. An Alternative View- Logic of CounterfactualsA possible world, or 'possible, alternative history' as Kripke fondly
calls it, exists only in his mind as a semantic wordplay in which truths are
established as to whether or not they could encompass all probable existent
world. He does not, the author believes, consider the idea of an 'actual'
possible world coexisting alongside the 'real' world. What he subscribes to
is the idea of a 'transworld identification'; in relation to his concept of
names being rigid designators, the name only points out to a single entity
all throughout the alternative worlds. David Lewis, on the other hand,
presents a more concrete application of modality as he proposes his
Counterfactuals theory. It is interesting to remark that the MWI follows
closely this line of reasoning as it advocates the validity of both options
actually existing. Kripke
E. The "Infinite" Argument
The nearest yet misleading stance this thesis adheres to is that of
classical finitism, the belief that infinity per se is a topic beyond human
cognition. Albeit the author wishes to point out, contrary to this view, this
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thesis accounts for potential infinities as actual ones; again, integrating
MWI, would be an appropriate stance given the scientific circumstances.
Aside from the potentiality/ actuality discussion, the thesis nevertheless
does not wish to support the finitist stand. Furthermore, it is not the notion
per se of the infinite/ finite divide that the researcher attacks; it is the limit
of our perception regarding the notion of infinity.
Tiles (1989) listed three arguments as to why classical finitists as
expropriated by empiricists do not consider the talk of infinity to be
sensible (footnote):
1. An assumption on the finitude of the universe to which mathematics is
applied.
2. An assumption that mathematics is only applied to this universe via
processes of measurement.
3. An assumption that meaning is to be equated with empirical meaning.
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CHAPTER IV
INFINITY AND QUANTUM POSSIBILITIES
"It is sometimes said that if infinitely many planets existed,then every possible planet would have to exist,
including, for instance, a planet exactly like ours,
except with unicorns. Is this necessarily true?"
Perhaps the query echoed by Rudy Rucker above comprises
the common perception regarding the realm of possible worlds, more so in
the conception of its attributed infinite property. So far the concept of
possibility has been identified through their linguisitc and mathematical
considerations. The question now lingers as to what would be the case if the
considerations in which quantum physics adheres to are stretched? Are we
correct in determining that these causal necessities, if ever they are indeed
causal, are but bounded still by human rationality? This chapter discusses
how the self- imposed and apparently delimiting modality hinders our
notion of the concepts of both "possibility/ impossibility" and "infinity" and
why Rucker's answer to the aforementioned statement is a clear, crisp "no".
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A. A Mathematically Impossible World
Albeit a common conception that Math is governed by universal
laws nevertheless necessary, one appears to conveniently forget the fact that
these considerations are hinged to conventions, oftently overlooked and
considered to be self- evident. An example of such is the field of base
arithmetic using the powers of ten. Cognitively, the equation "four times
five" necessitates the answer "twenty" and nothing else. But suppose one
considers the possibility of human beings being standardly polydactyl and
began counting in the standard base twelve. According to mathematicians,
had we unfortunately decided to adopt this paradigm shift and settled on
base twelve, our entire system of counting would change, garnering an
answer of eighteen to the above equation.
B. Actual Possible World
This idea might venture a paradox, for how can something be actual
andpossible at the same time? What, then, does it mean for something to
be actually possible? In the preceeding discussions the theory of MWI was
introduced, stating, also, that it corresponds to Lewis' notion of
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counterfactuals. This view in fact is dubbed modal realism and could be
equated modally using the S5 (box) rule.
C. Delimitations and Problematic Suppositions
It is clear perhaps the need to distinguish between the actual and the
possible for us to be able to discern the states accounted for our notion of
possible states. However, an indispensible concern hovers as to its actual
boundaries- taking into considerations the current scientific breakthrough,
where do we draw the line between actuality and possibility? Take, for
instance, Stephen Hawking's proposition of the Balck Hole Information
Paradox which seemingly violated the fundamental assumption of the Law
of Conservation of Energy with relation to the matter being engulfed into a
black hole. Hawking sees the need to propose a supporting argument, that is,
a white hole wherein the debris from a black hole is channelled to, in order
to be consistent with the previous assumptions. Correspondingly, it
suggests that the spacetime containing the white hole necessitates an actual
parallel universe more or less like our own (or at least composed of the
same matter, assuming that it does not obliterate itself upon contact of
matter to anti- matter). Would we then account for its being possible or
being actual?
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D. Many Worlds Interpretation ofQuantum Physics
Hugh Everett's postulation of this interpretation differs from, albeit
subtly, Richard Feynman's Multiple history and H. Dieter Zeh's many
minds interpretation. Leaving all technicalities behind, it endorses the
notion that an event, after splitting at a quantum level, proceeds to branch
out and create their own histories independent of the previous
circumstances and dependent on the number of probabilities that could have
occured given the number of options present. It explicitly endorses the
existence of 'doppelgangers' (i. e. object counterparts) whose composition
perhaps is that of the same origin, given that only chronological branching
out is acknowledged in this view. Hence, the possibility of one being born
to different parents perhaps is out of the question for it does not follow
chronologically from the set of events induced at one's being born.
[Illustration of branching histories]
What then are the nature of these branching worlds? Suppose
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E. Branching to Infinity
One of the safe hypothesis to make is that the parallel world
immediately branching out from this world subscribes to same laws of
physics as this world, albeit the occurences and history of that world would
take a different turn upon its deviation.
F. Infinite Individuals and Set Theory
Suppose one hypothetically decides to gather all his counterparts
from all branched- out possible worlds. Depending on the life span and
perhaps circumstantial decisions done by that individual (and consequently
the people around him) in his world, there could be infinitely many
individuals in the set of all branched out possible worlds. However, adding
the time coordinate would then induce the individual to limit the world
from which to access 'himself' in to that which he himself currently exists
(excluding those worlds which he ceased to exist/ not yet existing). Would
the set created by those elements contain an infinite or finite elements? It is
arguable that at this instance, a finite set is acceptable, provided of course
that time is continuously flowing in one direction chronologically.
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Difficulties would arise if, for instance, time is taken to be a dependent
variable whose property is not bounded one- way, hence allowing the
possibility of time- travel. In this hypothetical model, the probability of
individual A going to specific period T is accounted for, hence undoubtedly
implying an infinite number of individuals at Set S, time T because time is,
in essence, not constrained (Godel, me thinks it possible).
Would it, in the spirit of the Schrodinger inconsistency, allow to contain an
entity which is both dead and alive (presupposing for instance that being
dead or alive is comparable to one's existence/ inexistence)? Would the law
of non- contradiction hold true in this particular set?
G. Modalities in Possible Worlds
Taking into consideration that conceptually, our 'reality' is taken to
be that which is perceptually accessible to us, it is by definition arguable
that there is only one 'reality'-- that which is comprised of the world
measurable by our senses.