... translation^1.1

The original sentence in German is the grand conclusion of Wittgenstein's Tractatus Logico-Philosophicus [2]. The English counterpart is carefully translated by C.K. Ogden, presumably with the translator's thorough understanding of Wittgenstein's original text.

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... system^1.2

See http://www.systransoft.com/.

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...``ihm''^1.3

Dative masculine or neutral pronoun in German, and indeed there is no way for a machine translation system to know which gender a pronoun should be without looking into context.

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... Nature.^1.4

This emphasis of science on natural phenomena, however, is mostly an Anglo-Saxon tradition. In German, for example, the concept of science is much broader. There are Geisteswissenschaften (humanities, literally sciences of mind) -- Sprachwissenschaften (philology, linguistics), Literaturwissenschaft (literature, literature studies), and even Rechtswissenschaft (jurisprudence, law) and Betriebswissenschaft (business management). All these disciplines are seen as sciences. However, at least in the Western civilization, Nature is often taken as an antithesis of Humanity, in which the human will is transcendental to natural laws.

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... start^1.5

There was, and perhaps still is, a substantial trend in the disciplines of humanities in which natural sciences, such as physics or biology, are taken as shining examples of their own discipline. A salient example is the so-called social science. The trend started with Auguste Comte (1798-1857), who coined the word ``sociology'' and is taken as the founder of positivism. In a sense, the modern school of cognitive science and various endeavors to reduce human psychology to neuronal activities (classical bio-chemistry) can be seen as microscopic versions of positivism.

Nevertheless, one should not ignore the fact that there are also significant critics of positivistic philosophy -- its modern form can be traced back to Karl Marx (1818-1883). It is, with justification, termed as ``negative philosophy.'' (In the social theory context, see, for example, [8]). In a sense, the dialog and dispute of what is positive (in Nature) and what is negative (human will and critics) comprise a centerpiece of the Western civilization.

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... Nature^1.6

In general, a naturalist does not have to be a scientist (unless she believes there is an intelligible account of Nature) and a scientist does not have to be a naturalist (unless she believes there are no supernatural accounts).

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... law.''^1.7

It is arguable whether all living beings are able to ``break the law'' in its everyday sense as well. Nevertheless, following instincts is, at least for most conventional natural scientists, following the law. But knowing what instincts are and overcoming them consciously -- to sleep on a bed of nails and be hurt, for instance, poses a more profound question about what the ``law'' really is.

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... view^1.8

Monism, for one thing, sees matter and mind to something unified.

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... reality^2.1

In fact, the argument can be turned around with equal validity that the physical reality (whatsoever it may be) points to language. Remember a hierarchical thinking (from without) can not be genuine monistic.

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... computation).^2.2

Rheomode is a concept coined by David Bohm [1] -- `rheo' comes from a Greek verb, meaning `to flow.'

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...mind^2.3

Indeed, even in quantum theory this view is often taken by physicists. There are similar but serious arguments to get one out of the difficulty of quantum mechanics as provided by the Copenhagen interpretation (for a summary of interpretations of quantum mechanics see [10]) by resorting to somewhat mysterious consciousness and by rendering the most subtle physical ``reality'' (in its everyday sense) as ``meaningless'' -- this is by no means something to which physicalists might seriously subscribe.

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...proved.^2.4

Strictly speaking, the mathematicians referred to here are those who are trained by the same logical method. For example, a mathematician trained in constructive school [14] (e.g. with intuitionist logic which accepts $p\to \neg \neg p$ but not $\neg \neg p\to p$) might refuse to accept a theorem proved by another ``traditional'' mathematician using an ad absurdum argument.

In fact, the refutation of exclusive middle is a consequence of the philosophical view of constructive mathematics. In short, a constructive mathematician does not accept that there is objective mathematical reality. Consider the following proof which is not accepted by constructive mathematicians:

Theorem 1   There exist two irrational numbers a and b such that $a^b$ is rational.

Proof: Now $({\sqrt 2})^{\sqrt 2}$ is either rational or irrational. In the first case, we may take $a=b=\sqrt 2$; in the second case, we may take $a=({\sqrt 2})^{\sqrt 2}$ and $b={\sqrt 2}$, since then $a^b=2$ is rational.

However, there is no known contradiction between the theorems proved by intuitionist mathematicians and those proved by traditional mathematicians given the same set of axioms. The controversy is rather on ``acceptable'' proofs. Interestingly, it is perhaps the belief in universality of mathematics that has driven constructive mathematicians to prove ``existing'' theorems again.

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...Barwise77.^2.5

For example, the Compactness Theorem and the Completeness Theorem of (classical) first order logic.

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... process?^2.6

In fact, the modern digital computer works on a principle of approximation. For example, if the voltage across a junction in a CMOS memory chip is higher than a threshold value, a register is interpreted as ``1,'' otherwise ``0.'' The tension between computation and physics can be seen more clearly on an analog computer. Consider a scale, for example. For a scale to be balanced, the weight on the left arm times the length of the left arm should be equal to the weight on the right arm times the length of the right arm. It is hard to see any obvious and compulsory reason that an abstract multiplication operation should have a physical embodiment.

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... electromagnetism^2.7

For the sake of pure mathematical aesthetics, even four equations are redundant. In fact, two of the four equations can be deduced from the other two with the help of the Theory of Special Relativity.

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... physics^2.8

This is not to say that a computer cannot be simulated by hardware obeying classical physics. In fact, by carefully squeezing the transient state of classical electromagnetic circuitry, clever engineers can build computers that simulate discrete computation.

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... effect^2.9

Perhaps the most significant computation is evolution in Nature. Not surprisingly, the chemical reactions and mutations on which evolution is based are all quantum effects.

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... objects^2.10

See [20] ``[I]t is in principle impossible, however, to formulate the basic concepts of quantum mechanics without using classical mechanics...The possibility of a quantitative description of the motion of an electron requires the presence also of physical objects which obey classical mechanics to a sufficient degree of accuracy.''

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... language^2.11

As in quantum mechanics, what can be observed directly are only the properties of particles, in language what can be observed directly are only words or morphemes. A wave is a patterned relation of particles and can be understood only indirectly. A wave is influenced by the experimental setup as a whole.

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... sciences^3.1

Strictly speaking, this cannot be correct, for at least at the present time the Theory of General Relativity is still not unified with quantum mechanics. However, there are already several candidates that might offer a unified framework for quantum mechanics and the Theory of General Relativity (e.g. String theory). In any case, quantum mechanics will probably be subject to only minor modification and the formalism will remain largely valid.

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... synchronically.^4.1

If an apparent scrabbling delivered to us from the late Renaissance or a noisy recording of an utterance is deciphered as ``love,'' it is the symbol of ``love,'' which, strictly speaking, has become time-independent.

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... frequency-characteristics^4.2

Strictly speaking, it also depends on the orientation of the listener. While the majority of native English speakers may agree on the ending consonant is /v/, a native Chinese speaker may identify it as /b/ in one case and /f/ in another. In fact, this example shows that an invariant inventory of symbols across the language border is not realistic. In quantum mechanical framework of natural language, the different interpretation of /v/, /b/, or /f/ can easily be understood as orthogonal eigenstates of different languages.

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... something^4.3

It is tempting to call this something information, but as information (in Shanon's sense) is objective representation, at best we can argue that it is the representation at a ``higher'' level. The hierarchy is here clear but may be misleading.

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... defined^4.4

Strictly speaking, a classical object is nevertheless a quantum object with a huge number of quanta. The stability is to be understood as an extremely high probability ($\approx 1$) of finding a particular property of the object.

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... English^4.5

There are many cases where a specific state (and the associated symbol) is an eigenstate of both linguistic formulation operators. For instance, there is little room to dispute that an eigenstate associated with ``I'' in English is also an eigenstate associated with ``ich'' in German. Nevertheless, if there is an eigenstate which is not that of two languages at the same time, there is an uncertainty relation between the two. This is because what matters is the representation of the states of affairs, not single symbols.

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... communication^4.6

In a sense, a classical mistake of linguistics is to confuse what is perfect with what is effective.

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... meaning^5.1

This is not to say that there cannot be an ``effective'' account. But that would be an engineering concern, not a scientific one.

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...semantics^5.2

Modern mathematical logicians prefer to call semantics a model.

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... meaning)^5.3

For someone who has minimal knowledge of the General Theory of Relativity, it could be immediately pointed out that the first sentence is actually meaningless, while the second is OK. This is because any time statement is attached to a specific framework. However, this is not the point of this example. We are discussing common sense here.

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... situations^5.4

Strictly speaking, well-definedness is a stochastic (a quantitative rather than qualitative) property in quantum mechanics.

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... Hypothesis,''^5.5

While the so-called ``Whorfian Hypothesis,'' or linguistic relativity, is mostly interpreted as a language-determinism, it seems that Whorf espouses a theory in which the relation between culture and language is bidirectional.

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... context^5.6

David Bohm called this active information [36]. In my opinion, instead of active information, meaning might be a more suitable word as far as language is concerned.

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... possible^5.7

The two parties may understand each other. However, they may not be able to formulate their understanding.

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... models^5.8

Most probabalistic models are weak at explaining highly structural and abstract cognition, for probability is a real number between zero and one (thus continuous), while structural theory and symbols themselves are discrete (either zero or one).

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... well^5.9

Technically speaking, a wave function in quantum mechanics is a superposition of an ensemble of all possible states. A statement talking about all possibilities (so-called God's view) inevitably introduces a sort of Cartesian dualism.

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... established^5.10

In fact, there is no way to claim with certainty that a projection (the complex coefficient) is zero. So there is no way to establish absolute causality. One has to repeatedly perform actual measurements until an average probability can be obtained .

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... based^6.1

Exclusive OR is a very primitive logical operation in decision making. In the early phase of development, a child has to discern what is desirable and what is not in order to learn anything at all. XOR may be the most primitive judgement. On the other hand, AND is a primitive logical operation to juxtapose two concepts and make a judgement.

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... eigenstates,^6.2

It is a convenient rule that if we can tell the difference between two symbols at first sight, they should be represented by different eigenstates. This is the case if we draw a diagram of an XOR-gate on paper and look at the input. The examples discussed in this chapter have a parallel structure, so it is natural to use different symbols based on their spatial position. This can be different if the input is a sequence of symbols, in which case time plays a crucial role. These examples are discussed in Chapter 7

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... eigenstates^6.3

Readers who are familiar with quantum computation should notice that we ``flatten'' the two input bits and one output bit as eigenstates pertaining to one single operator, not three operators each of which is oriented in different bit-positions (so the result is a direct product of three two-dimensional Hilbert spaces). In the latter case, the dimensions of the resulting Hilbert space is $2^3=8$.

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... vector^6.4

Generally speaking, the choice of minimization method has to take both the computational resources and the quality of result into consideration. In this preliminary study, however, these issues are not addressed very sophisticatedly. Conjugate gradient method is chosen largely because it is a commonly used minimization algorithms that can deliver very accurate minima. It nevertheless requires more computational resources than, for example, the random walk algorithm.

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...122^6.5

A random walk algorithm can search the domain of optimization more thoroughly. It is a global minimization method. Given the relatively few parameters, it can be achieved in reasonably short time. Nevertheless, the minimization efficiency is not an important issue in this preliminary study.

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... situations^6.6

A coherent situation is a situation in which there is no ambiguity in classical logic. It should not be confused with a coherent state in quantum mechanics, which is a pure state.

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... set^6.7

It should be noticed that once a quantum system is measured, the outcome becomes classical. That is, the outcomes of a quantum measurement are always well defined. We assume that the relationship between classical mechanics and quantum mechanics is similar to that between classical logic and quantum computation.

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... point)^6.8

Few, if any, multi-valued logic endeavors have multi-valued logic as their meta-logic (the logic at a higher level). In fact, when we assert that the logic value of a certain statement is $\zeta$ (which can be other than $\mathbb T$ or $\mathbb F$), we assert at the same time that it is not not-$\zeta$. In this sense, the meta-logic is still two-valued.

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... framework^7.1

For example, the disagreement in the so-called argument structure -- the verb to give normally requires a direct object and an indirect object, but in certain contexts these can be omitted: Do you plan to give money to UNICEF? No, I gave last week.

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... speakers^7.2

For example, consider the difference between American English and British English; or the different speaking habits of an attorney and a teenager.

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... factors^7.3

For example the polite form of a Japanese utterance is usually determined by the different social status of the speakers and listeners.

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... sense^7.4

For example, statistical machine translation is based on communication theory on a noisy channel. The translation is a procedure to ``recover'' the original signal (sentence in the target language) given a deteriorated version (sentence in the source language.) It is hard to imagine that the parameters of such a model would have any linguistic meanings in the conventional sense.

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... language^7.5

For example in the use of idioms or collocations, etc. E.g. sore throat or pain in the neck.

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... train^7.6

The number of free parameters of a generating operator is proportional to the square of the size of the vocabulary. The computational complexity is proportional to the length of the sentence to be generated. Thus the total complexity of a generating operator may be 5-10 times of that in the reasoning process of the following section. Furthermore, there must be an additional status to indicate the end of an utterance. Since the main purpose of this preliminary study is the transformation of states of affairs, a full-scale generating scheme is dropped and reserved for further study.

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... affairs^7.7

This is a weakness of most connectionist approaches to NLP in which the referred state of affairs is remote to its claimed grounding. In fact, it is the implementation of symbolic tasks by a connectionist architecture that is mostly being studied. In other words, symbols are already abstracted by the designer of the architecture and the actual forming of symbols from the bottom up is seldom addressed.

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... corpus^7.8

The prefix of a German separable verb is placed at the end of the sentence and represented by two independent symbols in conjugated form.

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... mathematics^7.9

Other good examples are perhaps games such as go. Although the rules of go are very simple and the playing algorithm is well-defined, there is no computer program that can beat a moderate human go-player. The search space is simply too large even for today's best super computer. This may change, of course. But what is really fascinating is that human beings do not seem to search the problem space as a computer does, and so are able to beat a sophisticated computer.

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... well^8.1

On the other hand, evolution may offer an account of why we are as we are now. But an adequate theory of evolution probably also has to include quantum mechanics [19].

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... quantities^8.2

Modern physics is overwhelmingly quantitative. An extreme case is String Theory which promises a unified account for all four fundamental forces and elementary particles. In String Theory, fundamental physical objects are ``notes'' that are played on tiny strings of the Planck scale. They are all just quantitative aspects of the genuine ``atoms'' -- strings [85]

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...proof^8.3

Gödel's Incomplete Theorem also implies that there is truth that is unaccessible through any formal proof system.

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... user-friendly.^8.4

In fact, we can only say that the engineers who design the NLP-system are willing to help or friendly, but not the program or hardware itself.

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