0. INTRODUCTION
Ideas that are not entirely correct have been put forth in the forums; I hope to alleviate some confusion that posters have had about the differences between form and content and how they affect arguments. This is not an exhaustive philosophical argument or really well-researched. It's just an overview, but I hope an interesting one.
The following is a list of what can be expected from this essay, though I have tried to develop it sequentially because an honest reading of it is predicated on being kept somewhat in the dark -- suspending activities ending in '-lief?'. Since this is hard to do, I have arranged the information very linearly, but not in a very obvious order (except to readers who have read or heard isomorphic things). To facilitate this, it is divided into sections:
1. A Notational Game (NG)
2. Interpreting the NG
3. The NG and Debating
4. Prime Numbers, Bennett's Proof and the Metaphysics of Quality
5. Conclusion
(I'm not really formally trained in this subject matter; I've read a few books. I've thought quite a bit about it, though, so this essay is a very heavily filtered version of the way things are.)
~Mark Hagen, 21 July - 17 August 2004
1. The Notational Game
As its name suggests, the NG is a way to play with symbols according to rules. In particular,
the point of NG is to write and manipulate ordered sequences of symbols according to the
rules. Ideally, this is to be done in a totally accurate way, although there is significant choice
at which rules may be applied in a situation. The goal is to look at which sequences can be
had by modifying given ones according to the rules, and which can?t (there will generally be
many more of the latter for a given number of rules used.) To start, we?ll need symbols to
play with. As I?m feeling generous, we can have a countable infinity of basic symbols:
Definition. Basic Symbols
If n is a natural number, then B(n) is a basic symbol.
Note that this means B(1), B(12) are symbols (the quotes are not part of the symbol),
and B(n) and B(3n+2) are not symbols either. Also, you may find yourself introducing
symbols as you play the NG. This must be done in accordance with:
Rule 1.
If n is the largest number such that B(n) appears in an instance of NG,
and a new basic symbol is introduced, it must be B(n+1).
Rule 1 does not have the same charm that the rest of them do, and is not especially important,
but NG is a very picky game indeed.
The remaining symbols are those referred to by the rest of the rules. They are:
Definition. Regulated Symbols.
a, c, e, g, i are regulated symbols.
Now, to play, we need the rules and the procedure. Here, by convention, we use the letters w,
x, y, z to refer to sequences of symbols which are either basic symbols or which can be
formed from the procedure and the introduction of basic symbols.
?Definition?. Procedure.
Given some x, follow the rules to form a new sequence.
This gives us our concept called ?writability?. A sequence is writable relative to another
sequence or sequences if applying the procedure to them yields the sequence in question.
Rule 2. The i-rule.
Given x and y, xcy, xey, xgy must be replaced with ixcyi, ixeyi, ixgyi.
The former sequences are not allowed; the latter ones are.
Rule 3. The even-i-rule.
For any x, simply writing ix or xi is not allowed; ixi is allowed unless
other rules prohibit it.
Rule 4. The ai-rule.
ax is allowed for allowable sequences x; iaxi is not.
Rule 5. The c-rule.
If x and y are writable, then ixcyi is also writable.
Rule 6. Antics.
If ixcyi is writable, then x is writable and so is y.
Rule 7. The cg-rule.
If ixcixgyii is writable, then so is y.
Rule 8. a nonproliferation.
If x is writable, then aax is writable and vice versa.
Rule 9. a proliferation.
As in Rule 8, ixgyi and iaygaxi may be interchanged.
Rule 10. Political correctness.
iaxcayi and aixeyi are interchangeable.
Rule 11. The ea-rule.
ixeyi and iaxgyi are interchangeable.
Rule 12. The Introduction Rule.
If after writing x, one can apply other rules to write y, then ixgyi is
writable.
Rule 13. Let?s not get confused.
At most one of x and ax is writable.
Study the rules a bit first. I have intentionally provided little help?I want the first
experience with them to be somewhat mindless and rote. You will first notice a few
redundancies in the rules. See if you can point out what they are. Then play the game.
Write down a string and see where you can take it with the rules; cite which rules you
used so you don?t get lost and so you develop a feel for which NG situations ask for
which rules. Try to be as clever as you can in using them, but for now leave their
implications alone, and don?t interpret them. Instead, appreciate NG as a machine for
turning something into something else, and doing it in a predictable and orderly fashion.
I know of one ambiguity in my presentation, but it requires a bit of though to notice, and
eliminating it would make the rest of the discussion difficult, so if you do find the
problem, please do mention it and it will be discussed.
Don?t continue until you think you understand the rules quite well.
Now a puzzle: I will write a set of sequences and tell you that they are writable. I will
also supply another sequence, and you must confirm that the latter is writable given the
former. In other words, you must provide intermediate sequences along with the rules
used. I?ll give two examples.
Example 1.
There are no given sequences. Instead, you are to use the rules to confirm that
iB(1)cB(2)iggiB(2)cB(1)i is writable. By ?gg?, I mean that writablility should be
confirmed for iB(1)cB(2)igiB(2)cB(1)i and the reverse.
First, we can use the Introduction Rule to write:
1. iB(1)cB(2)i
Then we can write 2. B(1) and 3. B(2) by the Antics rule. It?s then a simple matter to
write 4. iB(2)cB(1)i by the c-Rule. To do the reverse, we just start with 5. iB(2)cB(1)i
and do the same thing.
Example 2.
Suppose the following are writable:
1. iiB(1)gB(2)igiB(3)gB(4)ii
2. B(5)gaB(4)
3. aiB(6)gB(1)i
Confirm the writability of iB(5)gaB(3)i.
This one is hard and to do it one must pick up on a very subtle trick. (It?s subtle in this context; if
you figure out what the NG might ?mean? then the trick is pretty obvious. It involves aB(1).)
Example 3.
Suppose the following are writable:
1. iiB(1)gB(2)igB(3)i
2. aB(3)
3. iiB(2)gB(4)igiB(5)gB(3)ii
Confirm the writability of aB5.
This involves the same trick. PM me if you can explain it, and feel free to discuss the questions
and share ideas.
2. Interpreting the NG
If we look quite carefully at the rules of NG, we find that they have a striking familiarity
as a whole. If we are familiar with certain ways of thinking, we may develop colloquial
meanings for each of the symbols, by way of analogy (in fact, we may even do this
subconsciously). Probably, ?i? will be the most obvious in its function, the ?a? and then ?c?. ?g?
and ?e? will take longer, perhaps, but we construct an artificial context very quickly, because our
minds work in a very analogical way.
While it should be remembered that NG is purely a set of abstract rules for manipulating
certain symbols, we notice that if for some n, B(n) is associated with a simple declarative
sentence, then aB(n) follows the same rules in NG that the negation of the corresponding sentence
does in logic. In a similar way, c is ?and?, e is ?inclusive or?, g is implication. ?i?s are brackets
to remove ambiguities (although I didn?t make the rules very explicit about where to put them?
as stated, my NG rules are not exactly complete).
It cannot be emphasized enough that ?g? does not MEAN ?if?then? ? there is merely a
correspondence between the rules of NG and the rules of the simplest form of propositional logic.
In fact, the symbols of propositional logic don?t mean anything either; they are on the same level
as the symbols of NG. Instead, there is a correspondence between the way in which propositional
logic and NG relate to some of the ways in which language is structured. Thus, while NG is in a
sense free of content, it provides a way of doing very simple deductive reasoning. For if we
assume certain statements to be true, we can write NG-strings which correspond to them. By
applying the rules of NG, we can create new strings, which we then ?translate? back to natural
language, giving new statements which are true relative to our assumptions. Two things are
worth mentioning.
First, I haven?t yet been clear on how the ?translation? works, but it is simple. Each
declarative sentence in the assumptions that does not contain a word corresponding to a
Regulated Symbol is replaced by a Basic Symbol. Compound sentences are then joined by the
appropriate Regulated Symbol. If, during the NG-manipulations, new Basic Symbols are
introduced, they necessarily correspond to new assumptions, so it is a feature of NG that the
result sequences will be expressible in terms of the assumptions. Thus the translation may be
reversed.
Second, using the above procedure rests, of course, on the assumption that the method is
valid. Now, that can easily be coded in an NG ?sequence as an assumption and operated on. A
rec for anyone who starts a meaningful discussion on where this is going.
Essentially, NG and more complicated methods can be used as a means of making
deductively valid arguments (that is tautological, because by deductively valid, I just mean ?not
violating and rules of logic?, some of which are isomorphic to NG). In general, the complicated
methods involve a more sophisticated ability to talk about objects through the use of the
quantifiers ?for all? and ?there exists?. I didn?t want to do that with NG, because there is not
really simple way to introduce quantifiers in a system without making it look that it is designed to
talk about things. In introducing NG, lack of content was the highest concern.
As far as reasoning goes, NG is slow, it must be applied with care, it can handle only
limited input, but it is always consistent. If we apply such reasoning as closely as we can, even in
situations where the assumptions are not very clear, we can be assured of getting sound
conclusions. Of course, we will not find something that is universally true in any sense, but in
view of the following example, the concept of universal truth is very unclear.
Suppose that the following are writable:
1.iB(1)giB(2)cB(3)ii
2.iB(2)gaiB(3)cB(4)ii
3.iiB(5)gB(6)igB(7)i
4.iiB(6)gB(8)igiaB(1)gB(7)ii
5.aB(7)
From your examples, you should know that B(1) is writable given the above. Using
rules, you can get iB(2)cB(3)i and thus B(2) and B(3). Ultimately, you have iaB(3)eaB(4)i.
Since you have B(3), you cannot have aB(3), from which you conclude that aB(4) is writable.
Can you write this formally in NG?
Now suppose that the second ?given? line is modified to read iB(2)gaiB(3)caB(4)ii.
Running through a similar, but simpler, derivation, it is possible to conclude that B(4) is writable.
this is a trivial example of the fact that small changes in the assumed sequences can lead to
completely different conclusions. Thus it is obvious that when we use NG to argue, are
conclusions are not ?true??they are merely ?true? in the sense that they have been validly
deduced from our assumptions: they are true relative to the assumptions. For some reason,
people forget that changing our assumptions can affect conclusions, and talk about ?logic? as
leading to a specific opinion, or about something being universally true, ?logically?. The first
type of argument sometimes leads to unfair accusations against formal reasoning; the second is
evidence of a serious misunderstanding of things.
Of course, this phenomenon is subtle in that the association between NG symbols and
statements can?t be contradictory: we can?t formalize assumptions into Ng by associating a
sentence with B(1) and its negation with B(2) and expect non-contradictory results. In this sense
the Let?s Not Get Confused Rule is also a metarule about translation.
In the next section, we?ll see how NG and NG-based arguments can be used in actual,
practical reasoning and how the choice of assumptions leads to problems which can be resolved
in practice, through other methods of reasoning.
3. The NG and Debating
With our knowledge of reasoning in NG-like ways, we are ready to ask a few questions:
1. What is sought in an argument?
2. How is an argument constructed?
3. How are assumptions to be chosen?
4. How is a debate to be conducted and what are its goals?
After discussing these, we will look at some of the attacks on NG-based reasoning that have been
made, and we will examine some examples of good arguments, though their deductive structures
reach a bit beyond NG proper.
When one presents an argument, in my view, the object is to present the logical
consequences of one's assumptions about a specific question--nothing more. By this I mean that
one either answers the question or draws some conclusion from assumptions which represent a
'reasonable' position on the subject (this will be clarified later; it should be mentioned that in
some types of argument the assumptions are largely agreed-upon conventions, as in mathematics
and some parts of science. In those cases, though, there is far less context than in an argument
about, say, ethics, so the hard part is choosing which part of the position--which axioms--to use,
rather than coming up with a sensible set of assumptions, which is the hard thing in most
arguments.) It should be pointed out that by 'argument' I mean something produced as a single
chain of reasoning and defense of assumptions, not an exchange between people.
Arguments are formed when letters, symbols, and pictures are put in a clever order (to
paraphrase Douglas Adams). The arguer first makes clear what conclusion is to be defended and
then proceeds to construct a chain of reasoning (often in colloquial language) which is guided by
the rules of some NG-like, or more complicated, system. As they are needed, assumptions are
alluded to and evidence is introduced. Other types of reasoning are also used, besides the strictly
deductive NG-based methods. Some of these are suggested by the word 'evidence'.
Inductive reasoning is the process of drawing general conclusions from specific data and
is much harder to formalize than deductive reasoning is. An example of inductive reasoning is
the thought of a person who notes that geese fly south in the autumn and north in the spring and
conclude that for some reason geese prefer to be in southern regions during the winter. Note that
a new concept , preference, has been introduced as an attempt to explain the data. A more
concrete version of inductive reasoning would be doing 'what is next in the sequence'? puzzles,
although those introduce difficulties discussed in other threads. This is by no means a complete
description of how to construct arguments, but it gives a general idea. I recall a good post by
Acolyte in a thread called 'The Art of Debating' on this subject.
Assumptions can be chosen formally or informally. The former is much easier to
understand but places important restrictions on the type of argument.
When we formally choose assumptions, we accept them on faith in a way analogous to
the assumption of writability entailed by the Introduction Rule of NG. In this way, we recognize
that any conclusions we make are, as always, relative to our assumptions. In short, we have been
formally choosing our assumptions all along; the system is closed.
Theoretically, the formal choice of axioms and definitions is the basis of all of
mathematics. This science concerns itself only with finding new deductive relationships between
the axioms and the invention of new formal assumptions to work with. The logical machinery,
however, is more powerful than that of NG and is very hard to separate from the axioms (maths is
in this way similar to a computer program or RNA synthesis in that the rules of manipulating the
parts are constructed from those parts).
Mathematics, however, is rooted in so much history that the formal definitions were
thought of to accommodate pre-existing informal concepts while paying lip service to a trend
toward formalization which emerged around 1900. For example, the concept of an 'integral
system' exists only so we can say '(N,0,'😉 is an integral system, so the natural numbers have
these properties...'. This kind of thing is part of a tremendous development, but formalizations
in practice usually have some context (which seems to settle Eugene Wigner's observation that
mathematics is 'unreasonably effective'😉. Also, it is interesting to note that at a very basic level
like NG, it must be remembered that the words 'assumption' or 'axiom' and 'rule' are different.
As described, an 'assumption' corresponds to an NG sequence which we deem writable, while a
'rule' is one of the things listed above, which tells us how Regulated Symbols or their
equivalents are to be used. It is in this sense that mathematics combines them, because
mathematical axioms usually deal with rules for manipulating some object.
This brings us to the informal choice of assumptions. The kind of sane person who does
not spew self-referential schizophrenic parody into the forums also usually argues about things
that are less formal than maths. In arguments such as these, we seek to ground our reasoning in
something outside our argument. By this I mean that we wish to choose assumptions that are
preferable to other assumptions that could be used: we want to find 'true' assumptions from
which to produce our arguments. In these cases, deduction tends to be less a feature of the
argument than clever choice of consistent and reasonable assumptions. Choosing consistent
assumptions is in principle easy, but making them 'reasonable' requires many hard-to-formalize
methods, of which inductive reasoning, the scientific method, observation etc. are examples.
Most assumptions, however, fall into three basic categories, namely assumption from faith,
assumption from observation, and assumption from necessary truth. It is important to briefly talk
about the existence and validity of these types of assumption. I hope to shed some light on how
to form reasonable positions as well as to explain why disagreement in informal arguments is
logically permissible. The ultimate goal is to show how all of this relates to debating. I shall,
however, ignore the art of rhetoric in this discussion.
Assumptions from faith are the least complex type from a formal point of view. One
school of thought argues that mathematical assumptions are in this category, but I use 'faith'
more in a religious sense here, and stipulate that these are assumptions drawn from literature,
tradition, or the mind of the assumer. The important distinction between these assumptions and
the other types is that these have no upper limit on information content, and thus arguments based
on them tend to rely lightly on logic and heavily on assumption, although such arguments are
certainly not irrational by definition and there are many exceptions to this idea.
Assumptions from observation consist if things measured or observed by the human
senses, together with the assumption that the method of observation is valid (though this may be
considered an assumption from faith). These assumption, when the observations have been made
in certain ways, are the axiomatic basis of science. Science consists largely of using NG-
influenced reasoning to explain observed phenomena. Because the observed world is very
complicated, arguments from observation tend to depend very heavily on abductive inference;
observations are used to construct possible arguments (hypotheses) which are ordered according
to likelihood. What is really going on though, is that likely assumptions are being built from the
data and their logical conclusions (the hypotheses) are drawn. So to be good, a hypothesis must
be internally deductively valid and its assumptions must be as simple and as powerful as possible
given the data. This is a delicate and clever balance. In most situations, though, nothing this
elaborate is required. Reasoning based on things you see out your window is also legitimate.
Finally, some assumptions are so 'obvious' that to disagree with them feels, in some
way, irrational. I don't know very much about how this works, and I don't know where the line
of skepticism should be drawn. There is certainly a case to be made against the existence of this
type of assumption. Relative to the amount of hammering on 'truth' being a not very useful
concept I did, this all seems like a very difficult concept. For example, the basic rules of
arithmetic could be formalized so that 3+5 = 7, and internal consistency could still be preserved.
However, this would be a radical alteration of our intuitive concept of addition, which is based on
observational assumptions and on our intuitive tendency to abstraction. In this way, I regard the
rules of arithmetic as necessary to a sensible understanding of things, even though I can conceive
of a logical world in which they are wrong (just as I can conceive of a logical world in which I
am not currently listening to Franz Ferdinand, or in which the Bible is literally true). I don't think
arithmetic is a good example, however, because I suspect that at bottom the basic rules of
arithmetic rest on observational assumptions, as much as that offends my particular sense of
mathematical aesthetics.
In general, necessary truth assumptions are identified by their consequences, which is a
fancy way of saying that I haven't figured out how to positively identify them, and am still not
clear on what they are. However, they can be approximated. If you construct an argument in
favor of some point, it is important, although in practice quite hard, to examine the assumptions
you used in a critical way. It is a good idea
(3. Continued)
ur
assumptions. As you iterate this process, you will likely find your assumptions getting simpler,
until you can't think of new ones to prove the previous bunch--you haven't found 'necessary
truth', necessarily, but you've probably come closer since it makes intuitive sense that simpler
ideas are more universally applicable. Finally, after this somewhat unsatisfying finish, note that
this 'backward construction' process is basically inductive, since conditionals are not as a rule 2-
way. By this I mean that in working from a complicated assumption to simpler ones, any set of
simple ones which imply the complicated one will do. That's all I have to say on assumptions:
in faith, choose what you want, in observation, choose what you see, and in truth choose what's
simple ('Pluralitas non est ponenda sine neccesitate'😉. Make sure you don't choose assumptions
which contradict one another. Now I want to introduce a format for a debate, and I think a chart
is the best way to do it. This is a very idealized and schematic view, and there is more to a debate
than this, but this outline or something like it is an essential feature of any debate.
I. A Question Introduced
II. A Deductively Valid Argument Presented as Discussed to Answer I.
III. Assumptions of II. Examined and Simplified
IV. Do Debaters Agree on Assumptions? If Yes, go to V. If No, go to VI.
V. Since Assumptions Agreed on, All Sides Accept II.
VI. Which Assumptions Disagreed?
VII. Why Disagreed?
VIII. Can Assumptions Be Modified to be More Reasonable? If Yes, go to IX. If No, go to X.
IX. Go to II with new Assumptions
X. Are Current Assumptions Acceptable on Second Review? If Yes go to XI. If No, go to
XII.
XI. II. Stands
XII. Modify I.
This is obviously not set in stone, but it includes essential 'checks and balances' which ensure
that the goal is to construct a set of reasonable assumptions on which to base a deductively valid
argument. Note that the point of having two debaters, in this view, is not to compete or convince
but to harness the idea that two or more varied minds can test ideas more efficiently and
sensitively than one can. (These are RHP debates we are discussing, not political slanging
matches.) If people become convinced of new points of view, this is a by-product of the testing
process and not the result of a concerted effort to convince them (RHP debates are not, for the
most part, policy debates). We shall now proceed to some examples of arguments. In some cases
(the argument I wrote myself, and the RHP-based argument due to bbarr), I will be able to
provide detail. My other two arguments are quite long and context-embedded, so I will provide
references only. While this is somewhat unfortunate, it is the surest way to get the point across.
To compensate, my own sample argument has been very carefully explained.
4. Prime Numbers, Bennett's Proof, the Alpha-Helix and the Metaphysics of Quality
First, I'm going to present a mathematical argument which is a proof of a very standard
theorem which is at the root of my favorite branch of maths, number theory. I'd imagine the
proof is pretty standard too, as it's very straightforward and tremendously illustrative of the
things we've been discussing (I chose a mathematical argument because it is, I think necessarily,
easier to explain and analyze). To start, a prime number is a positive integer divisible only by
itself (distinct from 1) and 1. This definition is itself based on the definition of divisibility: an
integer n is divisible by m if there is some integer a such that am = n. This in turn is based on the
definition of multiplication, the principle of generalization, etc. Informally speaking, the claim I
will prove is a profound consequence of some very simple ideas (like the definition of natural
numbers, multiplication, etc.). In a sense, the notion of prime numbers is one of our complicated
assumptions in that it encapsulates more fundamental facts into an easy-to-swallow chunk. In
other words, we can think about a wide range of very simple properties of the natural numbers
without even consciously admitting them into our reasoning: instead, we just let the idea of
primeness stand in for a certain relationship between these other ideas. In view of this, the claim
that will be proved may seem obvious and this fact will be addressed. The argument will be
made conversationally as you will get much less help with the others.
CLAIM (Fundamental Theorem of Arithmetic):
For every natural number n, there is exactly one collection of (not necessarily distinct) prime
numbers whose product is n.
Proof:
The claim seems to be saying two things, namely that for each integer, we can find a bunch of
primes whose product is that integer, and that once we have done that we should stop looking for
others.
In other words, the claim is of the form iB(1)cB(2)i, so we can attack each bit separately.
First, we'll just show that such a collection exists -- i.e. we'll prove it as it would be without
'exactly one'. In order to do this, we'll need another assumption which is fundamental to maths
and is equivalent to some more basic properties of the natural numbers, namely the Axiom of
Induction (in much the same way as the concept of primeness encapsulates certain properties of
multiplication, this axiom encapsulates a few facts about the ordering of the natural numbers):
The Axiom of Induction:
If a statement is true of some natural number k (translate this as 'B(k)'😉 and
iB(n)gB(n+1)i corresponds to a true implication for n > k, then B(n) for all n>=k.
In other words, a good way to prove that a statement holds for all positive integers is to
check some base case and then show that the validity of the statement for some integer can be
inferred from assuming the validity for the previous one. We can use this in our argument now.
First, notice that our claim holds for 2: there is a set of primes, in this case just
consisting of 2, whose product is 2 (the desired set, from now on, will be termed the 'prime
factorization'😉. Now suppose that some n > 2 has a prime factorization. Note that since we've
chosen n generally, this is equivalent to assuming that all integers between 2 and n have a prime
factorization (technically, this is an example of the equivalence of what are called 'mathematical'
and 'complete' induction, but that's not important). Now, what about n+1? Surely, it is either
prime or it isn't. If it is, then we're all set: it has a prime factorization, namely itself. If not,
then it is, by definition, divisible by two smaller numbers, say a and m: am = n + 1. Now a and m
are between 2 and n (inclusive), so by our assumption they have prime factorizations, and thus
their product, n+1, does as well. We're done with the first bit.
Now that we know that each positive integer has a prime factorization, how do we show
it is unique? If something is unique, then there cannot be more than one of them. So le's assume
that a number n has two prime factorizations and show that they cannot be different. In
particular, suppose there are a bunch of (not necessarily different) primes p1, p2, p3,...pk such
that n = p1p2p3...pk and also that there are primes q1, q2, ... qr such that n = q1q2q3...qr. This
tells us that q1q2q3...qr is divisible by p1 and furthermore that q1q2...qr/p1 = p2p3...pk. Thus
some combination of the 'q's is equal to p2p3...pk. More importantly, the product of some
combination of the 'q's is equal to p1. But p1 is prime, which means the 'combination' of 'q's
can consist of only one i.e. there exists l between 1 and r such that p1 = ql. there is no loss of
generality in saying l = 1, because multiplication does not care about order, so we have p1 = q1.
Now just cancel this from both sides of the equality between the prime factorizations and start
again, to obtain p2 = q2 ... pk = qr. Thus the two prime factorizations are equal, and the claim is
proved. 'Tis traditional to write 'quod erat demonstrandum' at this point.
That's a mathematical argument. It is the easiest of the four to follow and has the
simplest and most deliberate conclusion because mathematical arguments are heavily deductive
and have relatively few assumptions, although I did not exhaustively list mine and in practice the
assumptions are layered and harder to spot.
On another note, I would guess that the claim seemed obvious from the definition of a
prime and that the argument appeared more complicated than it intuitively seems it should be.
This phenomenon is a consequence, I think, of the earlier-mentioned phenomenon that there is
more to the notion of primeness than is explicit in the definition. In particular, I think the claim
is, even to a greater extent than that it is deducible from the definition of a prime, contained in the
definition.
(4 Continued)
Next, we examine an ethical argument. This has been posted at RHP before; it was
written by bbarr and it is, to me at least, the hardest to analyze of all of the arguments here. It is
certainly deductively valid, but the assumptions are not obvious to me. However, I hope its
author would like to comment on it and my brief comments that follow it, because it is a
compelling and interesting piece of reasoning.
First, let us suppose:
(1) S is throughout a fully reflective agent.
This idea here in not that we are always fully reflective, nor that given limitations on information
and time we ought to be. We are interested in what is involved in being fully reflective because
the capacity for full reflection (or autonomy) is a basic feature of our being agents, or, in other
words, creatures who form beliefs and intentions based upon reasons. Fully reflective activity is
the paradigm case of human action. My argument will be that this reflectivity is the source of the
categorical imperative.
Next, let us suppose that our fully reflective agent is confronted with some desire:
(2) S is faced with the desire, D, that favors his now performing action A. (I want now to flay
Floyd)
Since S is fully reflective he is reflectively aware that (2). We understand S's potential A-ing on
the basis of his awareness that (2) to be an exercise of his agency, and not merely the output of
some causal process. That is, we are assuming for this discussion that agent's actions are
explained by reference to their reasons for acting, and not by some third-person account of the
causal processes eventuating in their action. Anyone who is not a thorough skeptic about morality
will have to make a similar assumption. So, given that S's sees the question of whether to A in
virtue of (2) is an exercise in practical rationality, he needs to determine whether D provides him
a good reason for A-ing. That is,
(3) S is faced with the question: Should he now act on D? Should he endorse his now acting on
D? (Does wanting to flay Floyd give me a good reason to flay Floyd?)
As a fully reflective agent, S will only A because of D if he sees D as providing a good reason to
A. That is, S must either endorse A-ing in virtue of D or not endorse A-ing in virtue of D. But
what is involved in the endorsement of A-ing in virtue of D? Well, at a minimum, taking D to be
a good reason for A-ing commits S to the endorsement of a general principle, a hypothetical
imperative, that D is a good reason for A-ing. More specifically, endorsing A-ing in virtue of D
commits an agent to endorsing A-ing in virtue of D in circumstances exactly like these. So,
(4) If S reflectively endorses A-ing in virtue of D now, then S endorses a general principle P, of
hypothetical form, that endorses so acting. (Yes, my wanting to flay Floyd is a good reason to
flay Floyd, so it's a good rule that if someone (like me) wants to flay Floyd (in circumstances just
like this) they ought to flay Floyd).
Now, since S is fully reflective, he is aware of both endorsing A-ing in virtue of D and the
hypothetical imperative P. Also, since S is fully reflective, he is faced with the question of
whether he ought to endorse being a person who endorses P. If he can?t endorse being that sort of
person, the hypothetical imperative P will lose its endorsement as well, and this loss of
endorsement will iterate to the desire, D, for A-ing. So, failing to endorse being a person who
endorses P leaves one without any reason for A-ing.
The endorsement of being a person who endorses the hypothetical imperative P will commit him
to an endorsement of some practical identity, or a description of himself under which he acts,
trivially, the description of being a person who endorses the hypothetical imperative P. Notice
that the reflective demands presented thus far apply to any desire for any end. So, if our fully
reflective agent S is to ever act, he must endorse some practical identity. So,
(5) S must endorse some practical identity or other that grounds or supports the hypothetical
imperative P and thus his A-ing in virtue of D. (I endorse being the type of person who thinks
wanting to flay Floyd provides a good reason for flaying Floyd).
Since S is fully reflective, he is aware of this entire complex of reflective endorsement leading to
his endorsement of a conception of himself; an endorsement of his practical identity. But this
entire complex of reflective endorsement is itself is something that S can either endorse or reject.
So, the demands of full reflection do not stop with (5). Again, if S rejects this complex, if he
cannot endorse being the type of person who would through reflection be led to endorse being the
type of person who would act according to the hypothetical imperative P and thus find D a good
reason for A-ing, then he cannot A in virtue of D. To do so would be for S to act without a reason
he ultimately endorses, and to act without a reason one endorses is not only irrational, it is a
fundamental failure of agency. In fact, to act without a reason one ultimately endorses is to fail to
act at all, for it is the having of reasons that distinguishes an agent?s actions from the mere
moving of his body. To act, then, it is necessary for S to endorse his reflective nature as a deeper,
or more fundamental conception of his practical identity. In other words, he must endorse his
being essentially autonomous. So,
(6) S must endorse his reflectivity itself; his nature as a reflective or autonomous creature, as a
conception of his practical identity. (I endorse being, fundamentally, a reflective or autonomous
person).
Notice that (6) is not saying that S must only endorse his reflectivity on this particular occasion,
but that he must endorse it generally. The reason for this is the same reason that took us from (3)
to (4), namely that reflective endorsement as a matter of necessity involves endorsement of a
general principle. Notice also that no further question can arise for the S about whether or not to
endorse his endorsement of his reflectivity or autonomy. To raise such a question he would have
to employ the very faculty at issue, so raising this question presupposes his endorsement of
reflective endorsement itself, and hence his being a reflective or autonomous creature. So what
has this shown? It has shown that in order for S to act at all he must endorse being a reflective
creature, an autonomous creature; in short, he must endorse, and thereby value, his personhood.
Now given that S has been led to valuing his personhood, what good reason does he have for not
valuing it in that of another? How could S simultaneously value his own personhood and fail to
value the personhood of another without being inconsistent? Any supposed difference between S
and another in virtue of which one could claim that S's failing to value another's personhood was
not inconsistent would have to be a merely contingent difference. And reflection upon that
difference would lead S, via an argument similar to the one just presented, to realize that that
difference can only be valued if personhood itself if already valued. So the value of personhood
(reflectivity, autonomous agency) is fundamental, in that a fully reflective agent can value
nothing unless he also values personhood. So failing to value personhood is to fail to be fully
reflective, and failing to be fully reflective is to be, to that extent, irrational.
The assumptions are well-outlined by bbarr, but in this case it's more important to have a general
feel for what's going on. In Ben's argument, each step consists of a statement about the previous
part of the argument. Now this argument is strictly deductive, and a rough-and-ready notion of
deductive logic (though from our NG discussion we know it's not a great one) is 'reasoning from
general principles to specific conclusions'. Now what bbarr has done is to make a series of steps,
each of which is apparently more general and encompassing than the previous one, which taken
together home in on a specific conclusion. None of this is very precise, but to me this argument
has a sort of magical feel, like inflating balloons and watching them get smaller, paradoxically,
until suddenly it becomes apparent that the balloon has reached massive size. I was something of
an ethical skeptic before seeing this argument; now I essentially agree with its conclusion.
The next argument will not be reproduced here. In fact, it may look as though I've gone
out of my way to make it hard to find.
It is metaphysical and is found in Robert Pirsig?s book 'Zen and the Art of Motorcycle
Maintenance'. I 'lent' my copy to a friend about six months ago, so I'm afraid I can't even tell
what pages it is on. I will hide my shame in a puzzle: this argument is the one that uses the same
'trick' employed in Example 2 of the NG section.
5. Conclusion
We've seen how formal techniques for manipulating symbols can, in principle, be built
up into pretty powerful methods for making arguments. We've also given cursory attention to
choosing what to feed these methods and how to organize their use. What remains to be done is
to look at two of the ideas about reasoning that are found in the forums.
First, there seems to be some notion that formal reasoning is tied to a specific set of
views; this is not the case. Because one's conclusion is based on one's assumptions if one has
reasoned correctly, it is likely the two people can place equal value on the use of formal reasoning
and have completely contradictory opinions on a given subject, unless they have tapped into some
unknown wellspring of necessary truth assumptions. In general, a real-world situation of the type
discussed in the forums is so complicated that a given debater will be unlikely to recognize more
than a small fraction of the total available premises from which to argue. That is the point of
debating--incorporating things others have noticed into our own arguments to strengthen them,
rather than engaging in pointless handwringing based on an incorrect interpretation of how
arguments work.
Second, pointing out that the conclusion of person X's argument contradicts the
conclusion of a different argument offered by person X does not constitute a refutation of either
argument in general. If there is an error in reasoning (in particular, if contradictory conclusions
are reached from the same assumptions each time) then such a comment is a refutation, but in
general one person is capable of making two sets of assumptions about a question and arguing
from both. One set of assumptions may be better than the other, but neither argument is
necessary irrational. This is an example of a more general principle, however. Accusing
someone of holding contradictory viewpoints or of hypocrisy is, in situations like debates,
illogical. When dealing with things like arguments and information in an abstract form,
categorizing viewpoints (assumptions and arguments) in terms of the people holding them is
completely arbitrary. There is no contradiction in putting two internally consistent but
disagreeing ideas together, especially if they're being put in arbitrary categories. I think it is
phenomena like this, in which the person having the idea is subordinated to the idea itself, that
makes some people very uncomfortable with formal reasoning. This is an understandable view,
but it should be pointed out that certain developments, such as the rule of law taking precedence
over the rule of the ruler, are examples of this and are favored by the same people.
Lastly, I should point out how freeing this type of reasoning is. Certainly it is wise to
seek out strong sets of assumptions about a given issue, but it is interesting to argue different
points of view and free oneself from any commitment to specific beliefs, an action I like to call
'having ideas'.
Appendix 1.
1. I wanted to mention truth tables, but potential confusion between the notions of
'true' and 'writable', pointed out by bbarr, prevented me from doing this. I also
didn't want to give the impression that logic is related at all to the colloquial notion
of truth.
2. I wanted to introduce a full-blown first-order predicate calculus, but couldn't do it in
as content-free a way as I would have liked.
3. I wanted to talk about incompleteness, Hilbert's 10th problem, the halting problem,
etc. This is because such notions (Goedel's theorems in particular) are often taken to
be limitations or inadequacies in logic. However, this is akin to saying that neutrons
are 'inadequacies' in particle physics or that mountains are 'problems in geology;
these things are really just part of the picture.
Appendix 2.
The following things informed the writing of this essay to some extent:
1. 'Mathematical Reasoning', a course taught by Laura Schmidt and Kim Tanaka at the
Johns Hopkins CTY program in 2001. This course included an intro to symbolic
logic, though it was never suggested that formal logic was purely abstract--
'meaning' was introduced at the beginning.
2. 'Chapter 0: Fundamental Notions of Abstract Mathematics' by Carol Schumacher
was the text for 1.
3. 'Goedel, Escher, Bach: And Eternal Golden Braid' by Richard Hofstadter is
probably one of the most interest-dense books ever written, and one of my favorites.
This probably influenced this essay a great deal, though I like my presentation of NG
better than his of the propositional calculus because it does not give away so much at
once. 1979 if memory serves.
4. The RHP forae provided bothe the impetus for writing this and supplied one of the
sample arguments, due to Bennett Barr, late 2003.
5. 'Zen and the Art of Motorcycle Maintenance' by Robert Pirsig supplied another
sample argument, 1974.
6. An old notebook of mine has a page, dated October 2000, containing a proof of FTA
similar to the one given here, although it's nothing particularly new or clever.
Originally posted by royalchickenyou've put a lot of work into this rc.
I can't edit any further, so all of the little question marks flaoting about either indicate quotation marks, colons, or semicolons--very sorry.
it will take some time to give it due consideration.
thanks for the effort.
in friendship,
prad
