As I seem to be the only one in this discussion who agrees with me, maybe it would be useful to say what I mean directly, as I suspect my responses to other emails have been misunderstood.
All these semantic issues seem to me to be quite simple and fairly obvious when one gets them clear, and should not require the use of words that honest engineers feel obliged to put into scare quotes. I think there is a sense among non-philosophers that when words like 'reality' or 'meaning' are used in technical discussions, that they must hide some mysterious, deep, even occult notions. But in fact, they mean just what they mean everywhere else. Good philosophy isn't trying to make things obscure or mysterious, its just trying to get things clear. So this message is a kind of beginners guide to my view of the general topic of semantics, and an attempt to shed some light on this apparently difficult topic; one which is, in fact, quite simple and even kind of obvious.
BTW, nothing here is original. This is all absolutely standard textbook stuff. I learned it largely by reading Quine, but YMMV.
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1. First, we are talking about semantics, which is a fairly tightly circumscribed topic. Semantics is about the relationship between descriptions/languages/models/representations of something, and the thing or things they are supposed to be about: what they are descriptions/etc/ of. (I'll just use 'representation' and its cognates as a general word for the thing whose semantics we are talking about, everything from images to formal ontologies, though most of this is about the semantics of formal descriptions, in fact.) Semantics isn't about knowledge capture, or database management, or the Semantic Web, or about the future of Western Science, or any of that grand stuff, though it might well be relevant to those topics. Its just trying to analyze the relationship(s) between representations and what they represent. Or, as we say in the vulgar tongue, its about meaning.
2. In order to be useful, a semantic theory has to be able to cover the useful case, which is usually when the representation is supposed to be saying something about the actual world. That is, it isn't being used poetically or metaphorically or fictionally or whatever, just to represent something factual. Most representations are used in this way, and those that aren't seem to be parasitic on those that are (as when 'realistic' language is used in a novel to describe a fictional world), and certainly this is the interesting case for ontology engineering. So when I say that semantics relates a representation to reality, this is all I mean: that it relates the representation to what it represents, which in this case is, indeed, something real; which in turn simply means, not fictional or imaginary. I will call this the 'real case'; that is, where the representation is about something real, and the semantic theory needs to deal with this.
3. Semantics talks about the relationship between two things: the representation itself, and whatever the representation is a representation of. But now, already, we have an issue, since we apparently need to say, in our semantic theory, what that second 'thing' is, and to do that in the real case, we apparently need a way to talk about real things in our semantic theory itself. But we have to use a language to do this talking in; and doesn't that beg the question? Well, yes, in a sense it does. If semantic theory were supposed to be able to reduce meaning to something else, something different, something non-meaning-like, then we would seem to be stuck at this point in an intellectual trap. But semantics has a less ambitious goal. It is not reductive in this way: it doesn't set out to eliminate representation and meaning in terms of something else, only to describe them in ways that might be useful. (note1) And of course it uses language to do the describing - what else could it do? - so in a way might be said to presume its own ideas in order to state them. Someone might adopt a skeptical position and say, but your semantic theory can itself only be a description - a representation - which purports to describe reality, and so itself needs a semantics in order to be understood, which is circular. And indeed, to repeat, if the goal were reductive this accusation of circularity might be a valid objection: but as the goal is only analytical, it can be simply accepted. Indeed, any semantic description - any description at all - must in a sense presume a semantics, in order to be understood. That is inevitable, and should not be seen as a 'problem' or an 'issue' to be solved or avoided.
4. What it does mean, however, is that a semantic theory can be held to account in several ways.
4a. First, it should not presume solutions to any semantic problems it itself purports to analyze. For example, to say that the symbol "is-a" means what "is a" means in English might be put forward as a small part of a semantic theory. But this is immediately subject to the objection that it tells us nothing, precisely because we don't have a satisfactory semantic analysis of the English phrase "is a". If someone feels that we do, or that none is needed, then they might find this kind of semantic analysis perfectly satisfactory, as Dick McCullough apparently does. But it does not meet the standards normally expected of a semantic theory precisely because of its immediate circularity: it presumes what it sets out to describe, and so provides no useful analysis at all. Apply it to English to see why. It would be a semantics for English which said: "is a" in English means what "is a" means in English. Which is not exactly false, but also isn't very useful.
4b. Second, and a related point, the semantics should use a way to talk about the second half of its topic - the reality described by the representation - in terms which are as semantically neutral as possible. That is, the semantic theory itself should make as few presumptions as possible about the nature of the things being described by the representation. (note 2) This requires the semantic theory itself - not the representation it is analyzing - to be minimal, which means to be very general, in its description of reality. It needs to make some assumptions in order to be a theory at all, but its own descriptive apparatus needs to be always cut to the very bone, as it were, and only use those terms or ideas which are absolutely necessary for the theory itself to function, no more. Part of the basic insight of semantics is knowing how to say enough but absolutely no more, how to get by with an absolutely minimal description of the reality-half of the relationship it is analyzing. Notice that being described simply does not mean that the thing described must itself be simple. If I describe someone by saying they are tall, I haven't said very much about them. But that does not mean that they do not have any other attributes.
4c. Third, the semantic theory should be correct when applied to itself, to the language in which it itself is written. This is the benign side of the circularity objection: at the very least, a semantic theory of how representations describe reality should be applicable to its own descriptions of reality (which, as I hope I have explained, it must somehow use, in order to be a semantic theory at all.) It should not be a counterexample to itself. Of course such a self-application will be circular, so would be subject to the first kind of objection, if put forward as an analysis of the theory's own ideas; but that is not the point. Rather, this self-application criterion is simply a basic test of the its own inner coherence, not a doomed attempt at an explanatory reduction of it to some simpler or more basic framework.
5. As I hope is now reasonably clear, any semantic theory must of necessity use a representation (a description) to refer to reality. It - the theory - must be couched in a language, and so is itself a representation, of the kind that it itself purports to analyze. At this point we have rather a lot of representations to keep track of, so must speak carefully. There is the original representation that the semantic theory is about, and I will use this word "representation" to refer to this. It might for example be a formal notation of some kind, or an image, or a set of communication protocols, etc.. Then there is the descriptive framework in which the semantic theory itself is couched, which I will call the metatheory. The metatheory is the representational framework in which the original representation itself, the reality it describes, and the relation between them, is described.
5a. The question arises, how complex must the metatheory be? Must it of necessity be more complicated than the representation, since it has to be able to describe all the reality that the representation does, and then some? But if the metatheory is more complicated than the original representation, have we not simply got ourselves into an endless loop requiring more and more complicated metatheories? No. Remember point (3) above: we are not trying to reduce meaning to something else, only to analyze it. Our analysis can be couched in the same terms, and held to the same standards of precision and usefulness, as any other analysis; but that is all. It is not required to explain itself away - clearly an impossible task - only to be useful. In this case, being useful means only providing some useful insight into the semantic relationships between representation and represented.
6. Given all the above, then, let me try to give a very quick sketch of how "Tarskian" model theory, the standard logical semantical framework, works. Here, the representation is a formal logical language, and we might as well take FOL as the example, since its pretty much the standard by which all others are judged. The basic idea is to say, for each 'meaningful' aspect of the formal representation language, what the world has to be like in order for this aspect to make exact sense; then to say exactly how the arrangement of the world determines the meanings of the expressions of the logic. The key idea here is that in our semantic theory we must be careful to say as little as we possibly can about the reality: only what we need to say in order to make the semantics work, no more. This requires a little care.
6a. Look at the syntax. FOL expressions are sentences, which are supposed to be true or false. Sentences are made up from atomic sentences, boolean expressions (and, or, not, implies) which just tinker around with truth-values of smaller sentences; and from quantifiers (forall, exists). Take the forall quantifier. Its supposed, intuitively, to say that something is true of anything. Any thing. What does this imply about the world being described? At the least it seems to assume that there are things, and that there might be more than one of them, and that a truth can be relativized to them, and it makes sense to talk about all of them. It doesn't seem to say or presume anything about what they are, about their essential nature; only that it makes sense to speak of all of them. They can be individuated from one another: if finite, they can be counted. Something might be true of this one but not that one. They comprise a collection rather than a kind of amorphous cloud. The lot of them, together, can be described as a set, in fact. Set theory is just the tool we need here: it seems to capture exactly the necessary assumptions and no more. So our world must comprise at least a set of things. Let us be modest and call this set the universe.
6b. Now consider an atomic sentence, say the sentence "P(a)". This is supposed, intuitively, to say that P is true of a; and (without going into too much detail) it is clear from the syntax that this 'a' has to be the name of one of those things in the universe-set. What does 'being true of' something really mean? Well, frankly, I don't really know; but in order to make a semantic theory I don't really need to know. All I need is a list of the things that P is true of, and then I can say that "P(a)" is 'true' if a is on the list and 'false' if not. So, in the spirit of minimizing our ontological commitment, we will say that a predicate like 'P' has to "mean" at least this list. It might have a much more complicated and mysterious other meaning, perhaps a 'real' meaning: no matter. As long as that other meaning can be used to extract this list, that is all our semantics needs; and indeed, it really does need this: it cannot make do with less, for then we would not know how to evaluate "for all". So this list it is. The real world must provide this much structure, and it need not (for our economic semantic purposes) provide any more: this is enough.
So, here is our meta-description, in our semantic theory, of the reality represented by a FOL representation. It consists of a 'universe' set; a mapping from each logical individual name (like 'a') to something in the set; and of each predicate name (like 'P') to a list - a subset, as order isn't important - of the main set; and, although I havnt gone into it in detail, from each name denoting a binary relation to a subset of pairs of members of the universe, and so on: in general, a relational extension - a set of tuples of things in the universe - for each relation symbol. (note 3) Put another way, the FOL semantic metatheory describes the real world as a relational structure; and for each world so described, it tells how to determine the truth-value (true or false) of any sentence in the FOL representation. It may be referred to as a model theory (the relational structure here being the 'model') or a logical semantics or a truth-functional semantics or a Tarskian semantics, after its inventor.
6c. Before proceeding, lets mention some of things this kind of semantics does not do. It does not give any account of the meaning of anything outside strict FO logic. As an analysis of English meanings it is woefully incomplete and even plain wrong, since the English words 'or' and 'implies' and 'exists' have richer meanings in English than they do in FOL. It is entirely tailored to the syntax of this particular formal representational notation. None of this should be understood as a criticism of model theory, in my view. It does what it is intended to do. That it does not do more, is hardly good grounds for attacking it.
7. So, is the real, actual, world in fact a relational structure? Or is this just yet another model, another representation which itself needs to be related to the real real world by yet another semantic theory, this time one that applies to mathematical abstractions like relational structures instead of to formal logic? John Sowa has argued for this position many times. Let me call this idea the semantic black hole, for reasons which will become clear.
7a. Let us take this objection and ask how to overcome it. We would need a semantic theory for relational structures, one which relates them to the reality they represent. This second semantic theory will, in its turn, need to be able to describe this reality in a meta-meta-representation. Surely, then the same objection will apply to this second semantic theory: it must describe the world somehow, but whatever terms it uses to do so - whatever ontological framework it adopts - can be called yet another representation, not the real, real, real reality itself. We will never get to the ultimate quiddity, the essential rock-bottom reality, this way. To insist that any description of reality must really be a description of another representation, amounts to a rejection of the entire semantic enterprise. To adopt this stance makes it impossible for any semantic theory to ever be stated: for as I hope I have made clear, any semantic theory must use some representation to describe the reality which is at the pointy end of the semantic arrow the theory purports to analyze. (Note 4)
7b. In any case, the semantic black hole objection is simply wrong. Model theory is stated using a metalanguage, of course, and the reality that the FOL representation is about is described in this metalanguage, of course, one that makes certain minimal ontological assumptions about reality, of course. But this inevitable fact does not imply that the reality so described is not real. Mathematical language is used all the time to describe real things: engineering and science would be impossible else, not to mention such mundane activities as sharing out a packet of candy between a group of children. So, to say that the real world being described by FOL is a relational structure, is not to say that it is some kind of Platonic abstraction, only that it is presumed by this description to have a certain minimal amount of structure, and that this structure is what is semantically relevant for the purposes in hand. So, consider the following example of a semantic relational structure for a first-order language containing the predicate 'P': the members of the universe are four pennies lying in a row on a table, and 'P' denotes the property of being heads-up. This is a relational structure made of pennies. Or consider, more ambitiously, the universe containing all sodium atoms in a person's body at a certain time, and 'P' is the subset of these that are sodium-24. Or, the universe comprising all the rivets in the Eiffel Tower, and 'P' the property of having been replaced at some time after the end of the second world war. You get the idea. All of these are relational structures in the required sense. The general notion of relational structure is defined mathematically, using the language of sets, but examples of such structures - pieces of the world that satisfy the meta-description used by the FOL semantic theory - are all around us. The semantic theory does not relate FOL to another kind of representation: it relates FOL to something real, using a (meta)language to describe that reality (note 5).
7c. Replies to some common objections.
7c1. "This reply fails to address the main issue, which is that one can always think of the relational structure as a mere abstraction, without changing the truth-values assigned by the semantics at all. This semantic theory fails therefore to some to terms with something basic in reality-talk. According to this semantic theory, all of FOL might as well be about pure abstractions. And this is revealed when people give examples of Tarskian models purely abstractly, which they often do."
A: Indeed, they do, and one always can. However, this tells you something important about FOL: there is no way in FOL itself to be certain that it is about something definite in the real world. In particular, FOL does not support proper names: symbols which are taken to be 'fixed' to something actual thing in the world and always identify that particular thing, like place-names or personal names. No amount of FOL axioms will ensure that "Cymri" means Wales, since if those axioms are satisfiable at all, they are also satisfiable in an isomorphic universe which contains no countries at all, only symbols. And this is not a weakness or limitation of the semantic theory, but rather a very basic weakness of FOL itself, one which is revealed by the semantic theory, and which it establishes beyond all possible doubt or argument. This is one major success of the semantic theory, in fact, to reveal the expressive limitations of the formalism. (BTW, in case you think this point is trivial, just check out the ongoing explosion of confusion surrounding the so-called "http-range-14" issue in the W3C Web Architecture (TAG) Group archives.)
7c2. "This semantic theory is vacuous and useless because it so obvious. It just says things like 'and means and' and 'forall means for all', but dressed up in fancy mathematical language."
A: Well, yes, it is kind of simple, I agree. Quine discusses this point at length in his wonderfully written introduction to logic, where he treats logical semantics as a kind of careful translation into the meta-language. The semantics only works because we know what 'and' and 'for all' mean already and can use them, or some re-phrasing of them, in the meta-language itself. But still, it is not quite so circular as a direct translation, and it has some surprising consequences as a result. Logical beginners are often taken aback by things like DeMorgan's laws or material implication, which follow quite simply from the semantics but are hard to understand if all one has to fall back on are intuitive translations. And certainly, important results such as the compactness and completeness theorems are not at all obvious from a mere re-rendering of the logic into simplified English. The fact is, that FOL is kind of half-intuitive: it rests of course on an intuitive foundation, but it has surprises as well, and can extend the scope of reasoning far beyond what any human brain can achieve with unaided intuition. The simplicity and precision of logical semantics is a vital support in following these extended logical riffs: one simply cannot do it by trying to 'read' the logic directly. Four or five layers of nested quantification is beyond the mental attention span of the best human thinkers.
7c3. "Since the meta-theory uses mathematical language to describe 'reality', it is saying that the world is a mathematical abstraction, which is wrong. Mathematics only describes Platonic abstractions, not real things."
A: (See earlier posts for more detail): No, the use of mathematical language does not, fortunately, imply that one must be talking only about mathematical abstractions.
7c4. (same as 7c3 but expressed differently): "But that semantics says that the real world is a relational structure (a 3-tuple/ an algebra/ etc.). Surely you can't really mean to claim that reality really is a relational structure (a tuple/algebra/etc/)??"
A: Yes, I do really mean that. But it doesn't have the dire implications that you seem to think it has. Look at the four-pennies example again.
7c5. "There is something more basically wrong with your model theory, because it assumes that the actual world can be described as a relational structure. Maybe this is just metaphysically wrong: maybe reality isn't comprised of separate 'things' which stand in definite 'relations' to one another. What justifies this assumption, and what happens to the semantics if it is false? "
A: Good question, and thank you for not mentioning quantum theory. If that assumption really is false, then this semantic theory simply does not fit the real world, as you say. But the moral would not be that the semantic theory is wrong, but that FOL itself is wrong. We began with FOL. These assumptions about the nature of reality were the minimal assumptions we were obliged to make in order to make sense of the FOL representational formalism itself. It is the representation, not the semantic meta-theory, which makes this very fundamental ontological assumption about the basic metaphysical structures of the real world. If reality really isn't made up of things standing in relationships, then FOL simply cannot be used to describe it. The semantics is the slave of the formalism at this point. The job of the semantic theory is to expound how the representation can describe a world. If it can't, and if this is revealed by the semantic theory, then that theory has done its job. Don't shoot the messenger.
8. Notice how the semantics was constructed by trying to capture minimally what the reality had to be assumed to be like, in order for the logical sentences to make sense, given their intended intuitive meanings. It begins with these intuitive meanings and uses them to guide the semantic construction; and the meanings it then provides are indeed very close to those original intuitions; perhaps, some feel, too close (cf 7c2 above). This means, though, that the 'formal' semantical meaning, and the 'intuitive' meaning, of a FOL _expression_ are not sharply at odds with one another. Detractors of model theory often seem to assume that it gives a different, alien sense of meaning which is best ignored by ordinary users of logic, who simply 'read' the logic and understand it intuitively using plain old English tags or intuitions rather than this graduate-school-mathematics stuff. The graduate-school and the great-unwashed versions of logical understanding should mostly match up, however, and indeed they mostly do. The exceptions are always cases where the English-based intuition presumes too much of the relatively weak FOL expressivity (hence the puzzlement about the 'paradoxes' of material implication, such as a contradiction implying everything). Logical introductory text books are replete with exercises and cautions against these cases, and users of logic-based formalisms all acquire a grasp of meaning which corresponds more closely to the model theory, which is in fact provably the appropriate one for FOL reasoning (note 6). But my point here is only to emphasize that formal semantics and intuitive semantics are not engaged in a battle for the soul of FO logic. The former is simply a more carefully expressed re-statement of the latter in terms which allow for exact analysis to be done and theorems to be proved.
9. OK, that's enough for now. TGIF.
Pat
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Notes
(1) The fact that logical semantics ("model theory") uses set-language, and has often been involved in questions arising in the foundations of mathematics, seems to have produced the idea that logical semantics is motivated by the same reductive concerns as those that exercise the foundations of mathematics (FOM). This impression is completely false. There has indeed been a huge effort in FOM, lasting nearly a century, to reduce all of mathematical language to pure set theory, where the whole universe (in this case the whole Platonic pure-mathematical universe, starting with the natural numbers) is re-built using constructions in set theory, so that the number three is defined to be the set {{} {{}} {{}{{}}}}, following a clever recursion in which zero is the empty set and every number is the union of the immediately preceding number with the union set of itself. (FOM is nothing if not ingeniously clever.) The ultimate purpose of such a reduction is to achieve some kind of guarantee of internal consistency for the mathematical edifice; to show that if at least set theory is consistent, then so is everything else. But this desperate search for some guarantee of mathematical consistency is not intrinsic to set theory, which is simply a very general theory of abstract collections, and can be applied to almost anything. It is useful precisely because it is so very general and so simple. Model theory is one (rather elementary) application of set theory, and building a secure foundation for all of mathematics is another. They have virtually nothing to do with one another, apart from both using the same tool-kit. In particular, the fact that logical semantics is phrased using the language of sets does not imply that it is a reduction of reality to some pure-set-theoretic construction. This common fallacy is often repeated, even by philosophers such as Barry Smith who should know better.
(2) Actually this should be stated more carefully, as some representations are intended to have limited scope and only used to refer to particular kinds of thing or aspects of reality, e.g. Labanotation, and in such cases it is obviously OK for the semantics to make matching presumptions. But for general-purpose representations like most natural and logical languages, the point holds.
(3) This is classical FOL, not Common Logic, which because of its more liberal syntax requires some modifications to the way the real world is described in the semantic metatheory.
(4) There is one possible way to stop this regress, by insisting that the 'ultimate' semantics is stated entirely using demonstratives rather than descriptions. Instead of saying, <syntax> means <metadescription>, one says <syntax> means this, accompanied by a gesture towards an actual thing or part of the real (physical) world. For one the most telling critiques of this idea, see Gulliver's Travels, Book 3 (the Island of Laputa).
(5) In the real case, that is. To be more exact: to something that can be real. It can also be an abstraction or imaginary, of course, which is appropriate since the formalism FOL itself can be used to describe abstract or imaginary things just as well as real ones.
(6) The famous Goedel completeness theorem.
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