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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 spacetime, 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 spacetime 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 spacetime 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.

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