Before we delve into the question of how relevant information may be embedded in language, it is fruitful to elucidate what is syntax and what is semantics from the view point of both linguistics and the study of logic.
In linguistics, syntax is the arrangement of words in sentences, clauses, and phrases, and the study of the formation of sentences and the relationship of their component parts. Although whether ``words'' should be taken as the proper building block is arguable, any atomic entities can be replaced in the definition above. Syntax is a property taken to be intra-lingual. Classically speaking, one does not have to resort to the outside world (i.e. to any extra-lingual) factors for the study of syntax. On the other hand, semantics is the study of the meaning of linguistic constituent parts, and this has to resort to extra-lingual properties.
There are other ways of seeing syntax and semantics. The most interesting one is how most mathematicians or logicians see ``syntax'' (as a formal property) and ``semantics'' (as truthfulness). For them, syntax is the study of the well-formed formulas of a logical system. In the study of mathematical logic, syntax is associated with a deductive system and denotes how a formula (a sentence) can be formally derived from a set of axioms and very parsimonious ``meta-rules'' (Modus Ponens, cut, etc.). This is called a deduction. On the other hand, semantics5.2 is the study of the truthfulness of a deductive system. In modern mathematical logic, semantics is an algebraic structure which is outside and independent of the deductive system. If all the formally (i.e. syntactically) deducible formulas of a deductive system are also true (corresponding to the semantic structure), the system is said to be sound. On the other hand, if all the true formulas (corresponding to the semantic structure) can be formally deduced, the system is said to be complete. In the literature of mathematical logic, syntax and semantics validity are expressed by 
and 
respectively.
For mathematicians, there is not much room to dispute whether such a separation of syntax and semantics is desirable or whether completeness or soundness is of any value. Given the strong Platonist (objective idealist) or intuitionist (subjective idealist) position of most mathematicians, it is rather a matter of the rules of the game. The rules (here the analytic, axiomatic, and abstract ones) are simply given by years and years of hard training.
In linguistics, because grammar is traditionally put at the center of many main-stream linguistic studies since Chomsky's influential works [34], grammar rules are treated pretty much the same way as ``syntax'' is treated in mathematical logic and grammaticality in formal language. In comparison with syntax, which studies the relationship of language constituent parts, semantics has inevitably become peripheral, and is, strictly speaking, an ``extra-linguistic'' study.
But this dichotomy is problematic. Let us see why. A closer look at language in light of mainstream cognitive science (using the computer as a metaphor of mind) reveals that syntactic properties may not be qualitatively difference from what are conventionally called semantic. If someone knows that ravens and sparrows are similar -- called birds, it is hard to imagine that he cannot know `go' and `give' are similar -- called verbs. On the other hand, if the fact that `ravens and sparrows are birds' is to be represented in a classical cognitive system, this fact becomes purely syntactical. This is because the relations between the representation of `raven', `sparrow' and `bird' are nothing but the relations of representations in a formal language system.
In fact, even in language itself it is hardly noticed that syntactical constituents (such as `verb', `noun' etc.) may also have `semantics' -- in that they refer to something that is not in the language being used but only in the scholarly discourse of linguists. To begin with, if we say
we do mean something, don't we? In fact, the subject matter (`the verb conjugation') lies in an utterance of someone else in a specific language environment. It can only be outside of the speaker when she says `This verb conjugation is not correct.' For example, if she says (the purposeful mistake is marked with the asterisks),
are not correct.the utterance is obscure if not illegible. In this sense, roughly speaking, syntax and its constituents are something that carry `linguistic meaning' according to a specific linguistic school. As a consequence, something supposed to be syntactic (therefore without meaning) suddenly acquires its meaning at a `higher level.'
This issue cannot bother a conventional mathematician. For him, it is a matter of hierarchical thinking (which is therefore a view from without). An honest linguist cannot afford to accept this habitual hierarchical thinking without criticism, however. For one thing, one of the greatest emancipations of modern linguistics was to get out of a normative paradigm and become descriptive. The difficulty lies in that the describing happens to be the described as well. This forces us to take a view from within. As far as this kind of self-reference is concerned, the situation in linguistics is similar to that of mathematical logic. But mathematical logicians are much better off, for they do not have to shy away from being `normative.' What they are interested in is the soundness of their proofs. On the other hand, if language is a scheme that is not mathematical, we need a persuasive account that it is appropriate to use a normative mathematical ``model'' to describe language. Specifically, what good is it to attribute some features to semantic and some others to syntactic if they turn out to be mixed up?
In any case, we have to clear up our understanding of syntax a bit. It seems that context is a moderately complicated and syntactic property that we should begin with. So let us begin our endeavor by posing the following question: ``How does context influence the use of language?'' Consider the following example (cf. Barwise [35]):
At first sight, the information this utterance conveys seems very clear. Unfortunately, it is not the case if we analyze it further. For one thing, it is a short-hand form of a detailed description, such as ``It is here (e.g. 
and daylight saving time) and now 4:00 p.m. (with the sun light making a particular angle to the meridian, etc.)'' So, though a little tedious, it is all right if one says,
In this regard, it seems that these two sentences convey the same information (so they have the same meaning)5.3. But this is not always the case. This can be seen by putting Example 3 and Example 4 in different situations (e.g. as answers to different questions). It turns out that if the question is ``What time is it?'', they do mean the same. However, if the asker poses the question: ``Do you know what time is it now in Tokyo?'' -- she may want to make a phone call to Tokyo -- the second sentence becomes meaningless or irrelevant. In the latter scenario, the asker probably already knows it is 4:00 p.m. here. The information she actually wants is the time difference between here and Tokyo. This simple example shows how context may play a part in the information of everyday life. Information is, roughly speaking, context-sensitive.
One may come up with a theory in order to get rid of context sensitivity. For example, one can transform the whole scenario into ``grids'' in an informational framework and come up with a static context independent picture. (In this example, by extending any sentences to its fullest form.) But this is misleading. There is something more than naive information deeply buried in context.
First, the apparent ``information'' conveyed in the above dialog is, by and large, also context sensitive. For example, it is implicitly assumed that cooperation and courtesy are desirable (this can be falsified if the parties involved here are foes). If the latter assumption did not stand, the ``information'' conveyed could become purposeful ``misinformation.''
Second, ``information'' is taken here as something which has well-defined properties. While it is the case in many situations5.4, it is also very often not so. For the sake of argument, consider what would happen if the dialog took place in a fictitious world where watches never tell the correct time? Artificial as it may look, a similar scenario is common in this world. For example, we can replace the watch in the above argument with a ``virtual oracle'' which defines ``degree of beauty.'' An informational approach has a particularly hard time to accommodate situations that are about beliefs (and misbeliefs). Unfortunately, these are not rare in natural language and common sense.
Before going into the third difficulty of an informational approach, we have to realize that any information is physical (information has to be grounded in ``reality'') and modern physics is obscure, as far as reality is concerned. (This is a recurrent theme.) This brings up the third difficulty of an informational approach, which is much more profound. The concept of physical ``information'' itself could turn out to be context-sensitive in the sense that it may depend on quasi-linguistic contexts (here mathematics) that are extra-physical and therefore automatically become extra-informational. The whole informational framework, aiming at eliminating context-sensitivity, suffers from the same crisis of context-sensitivity. If the postulates delineated in Section 4.3 are correct, the brain may amplify all these quantum effects into ill-defined ``information.'' We are using an deceitful yardstick to measure the land.
The difficulty of a naive informational approach to meaning may encourage us to think over the alternatives. In this sense, it is interesting to mention a seemingly Chomskyan-opposite Whorfian view on language and thought:
His thinking itself is in a language -- in English, in Sanskrit, in Chinese. And every language is a vast pattern system, different from others, in which are culturally ordained the forms and categories by which the personality not only communicates, but also analyzes nature, notices or neglects types of relationships and phenomena, channels his reasoning, and builds the house of his consciousness. -- Benjamin L. Whorf (p. 252 [23])
Despite numerous controversies and critics of the so-called ``Whorfian Hypothesis,''5.5 it seems to be compatible with the world view of modern physics. If symbols are to be treated as eigenstates of a formulation operator in a Hilbert space, these symbols are stripped of any semantic content once they are measured. The relationship between these symbols is purely syntactic. Consequently, syntax alone, the way of using language in language without resorting to outside world, is enough to weave the web of meaning. In this regard, syntax and semantics merge into a unified whole. We can call the whole the context or the situation. It is upon this unification that an active mind takes each piece of ``information'' into account and finds meaning embedded in a natural language utterance or representations.
As stated in the beginning of this chapter, ``information'' is in fact a misleading idea, for what matters is the active extraction of ``information'' out of a unity of context5.6. This is how information can become relevant. Taking the mental world as a quantum system in which the observing instrument and the observed cannot be separated, the whole experience and thoughts of a person contribute to this person as a whole. Likewise, this can not be separated from his access to linguistic context. In this sense, human reasoning, including rigorous logical reasoning and common sense, is strictly context-sensitive.
There are several implications of this view. For one thing, a quantum mechanical view of language lends itself to the belief that translation between languages can only be achieved with limited success. It should be noticed that the picture to which quantum linguistics subscribes is not only untranslatability but also the intangibility of the language usage of a person per se. This is the problem of private language à la Ludwig Wittgenstein [37]. However, quantum mechanics offers a much brighter picture. Since quantum mechanics is holistic, the parties involved in a discourse can form a unified system. The entanglement between the parties then makes error-free communication possible5.7. But this cannot be linguistic, for the Uncertainty Principle of language forbids even an error-free formulation of ``private'' language. It must also be noticed that since actively aligning the measuring devices of the parties plays a crucial role in establishing the quantum wholeness, so does the active participation (in its everyday sense) of parties in a discourse contribute to the understanding. This seems to have been an ancient wisdom of interpersonal communication. It falls out automatically from a quantum mechanical approach to language.
There are also practical implications of this approach to linguistic context -- for example in machine translation in which context is an immediate problem. Words in one language are seldom related to words in other languages in a one-to-one fashion. Translation therefore heavily depends on the correct identification of context in the source language in order to distinguish among these many-to-many mappings. Any moderate machine translation system has to take context into account. Here is where language à la quantum mechanics may provide a niche. A quantum machine translator can be designed right from the beginning based on the idea of wholeness and entanglement. One only has to employ the vocabularies of the source and target language respectively as two complete bases of a common mental state, and then train the machine to achieve corresponding reasoning (a unitary transformation) in translating between the two. This idea is pursued in Chapter 7.
To sum up, quantum theory can offer an account of relevant information. In the following sections, we will apply this approach to several thorny logical problems in common sense reasoning -- non-monotonicity, counterfactual conditionals, and causal explanation.