Fixing intensional logic

One of my favorite hobbies is reading about intensional logic. This refers to any form of logic that allows us to refer somehow to the "sense" of a logical entity, in addition to the extension. I have never contributed any of my own research to this topic, but I believe that we can never have a useful computational theory of natural semantics without some kind of decent intensional logic.

It seems like every proposed form of intensional logic through the years has got something wrong with it. Quite a bit of recent literature is still devoted to the seemingly endless task of "fixing" some variety of intensional logic.

As far as I can see, there are two dichotomies that can help classify various approaches to intensional logic. You can do things with type theory, or without it. You can do things with possible worlds, or without them. I have already posted about my dislike for possible worlds. In fact, I doubt there is a possible world in which I like them! But kidding aside, for a working intensional logic I favor Church's formalization of Frege, the Logic of Sense and Denotation---type theory, but no worlds. This logic was published in several incarnations over many decades. In fact, Church's own publications on the subject span from 1946 to 1993! This might qualify as the longest period spent by one scholar publishing papers on something that remained unsatisfactory, even to the scholar himself. I certainly hope to spend 47 years publishing, pretty well anything, if you want to know the truth.

But there is hope for Church's method, apparently. Quite a few scholars followed up on Church's flawed logic in the past decade (including some leading names like Terence Parsons), and the most recent entry in the game to fix Church is a paper by Kevin Klement in the June issue of the Bulletin of Symbolic Logic. I'll fill in the details in another post in the near future, but I direct your attention to Klement's fine paper in the meantime.

Quantum mechanics and the spectrogram

For this post, I wanted to show a little mathematics, so I used html with MathML.
I think this blogger is not enabled for that, so I'm redirecting to this link for the post:

Quantum mechanics and the spectrogram

This should display seamlessly in Firefox; not sure about IE yet.

Philosophy of phonology

OK, so I failed to post numerous things last month. Evidently the summer is a challenging time for my blogging abilities. But, perhaps less is more.

Getting to the topic, I've been reading Anderson's (1985) Phonology in the Twentieth Century, and I'd like to highlight his description of the philosophical conflict between J. R. Firth's conception of phonology and that offered in the American generative paradigm spearheaded by Jakobson, and later Chomsky and Halle. Anderson characterizes the conflict as one between a realist philosophy of the American school, against a nominalist philosophy of the Firthian school.

Now, I don't want to specifically advocate for the Firthian paradigm in phonology, but when it comes to distinctive features in phonology, consider me a nominalist. Anderson describes nominalism as regarding the "idealizations of scientific theories simply as names for the analytic categories of the scientist," and to the extent that distinctive features are idealizations dreamt up by phonologists, I think the nominalist view is plainly correct.

On the other hand, according to Anderson "most empirical scientists, including linguists, tend to take a strongly realist attitude toward the essential elements of their theories." I don't know how uncontroversial such a claim about scientists is, but I personally feel it would be ludicrous to assert without specific evidence that a particular distinctive feature was in any sense "real." As a matter of fact, first you would have to specify in what way it was supposedly real, and then you would have to somehow prove that it was real in that sense (let's say, in a cognitive sense). Try it sometime! I don't believe there is any result that shows that any particular distinctive feature (not a phonetic feature mind you, but an abstract distinctive feature, let's say [+consonantal]) is cognitively real, and binary-valued, in the human mind.

So let's face it, when we discuss phonology, it makes more sense to be nominalists instead of realists. Realism is a kind of science-cum-religion that requires suspension of disbelief. Nominalism is purely analytical, without the extra baggage offered by realism. Why should a linguist care? Because whether you are a nominalist or a realist can affect what you would say about a variety of research projects. A nominalist would wholeheartedly support a project to, e.g., formulate a (possibly mathematical) theory of how binary or privative (i.e. unary) distinctive features appear to emerge from more specified phonetic features in the phonological systems of languages. Such a project would specifically examine how our analytical categories are founded in the detailed reality (as they ultimately must be). A realist, on the other hand, may be inclined to dismiss such a project because, after all, distinctive features aren't emergent properties, they're real, and so exist independently in a Platonic realm or something. Householder and Firth both caricatured this position as the "God's truth" view, that somehow if we could magically hit upon the right system of distinctive features, we would unearth God's truth.

But I also suggest that linguists keep the philosophizing to a bare minimum, just enough to sort out one's methodological position before deciding which projects might be worthwhile.

Features in phonology and phonetics

OK, first let me apologize for the long absence. This was owing to many factors within a busy summer including poor internet access, but I am planning several posts for this month.

I'm going to discuss phonology and phonetics in the next few posts. I'll begin with the problem of features.

The theory of features has never been worked out to anyone's satisfaction, and there are many competing accounts. I am engaged in a project on "phonetic features" with my colleague Darin Flynn of the University of Calgary, and we are trying to sort things out. First of all, since Jakobson we have had "distinctive features" in phonology. These are generally assumed to be binary or unary-valued, either present or absent in some sense. They operate at the abstract level of phonological or phonemic representation (whatever you take that to be). They are not precisely indicative of phonetic properties, but are in some way derived from the phonetic properties of speech sounds. Their purpose is to include every feature whose presence/absence "can serve to differentiate one utterance from another in any language" (Anderson 1974). This is why they are called distinctive features.

So far, so good. The more problematic topic concerns whatever is "below" the abstract phonology, in the phonetic realm. Some models essentially propose that there are such things as "phonetic features" which specify sounds at some phonetic level. This dates back at least to Chomsky and Halle's SPE model. It is a pretty good idea, but there are problems with their proposal that the phonetic features are merely the "other side of the same coin", wherein phonetic features are scalar-valued objects that are linked with or somehow identical to the distinctive features of the phonology. Historically this subject has become terribly confused, because much literature has called the binary distinctive features of phonology by the name of "phonetic features" and has implied that these provide phonetic specifications.

Anderson (1974) gives some nice reasons why the phonetic features should really be different from the distinctive features. He states that phonetic features should include "every feature in terms of which one language differs systematically from another," and these may include features which are never distinctive, such as whether final stops are released. Anderson proposes that going from phonology to phonetics one uses "detail rules" which specify the scalar values of the phonetic features on a language-specific basis. This is really the phonetic component of the grammar, as against "universal phonetic implementation" which applies to all languages because it is derived from the human speech production mechanism.

Another reason (to allow phonetic features different from distinctive features) that Flynn and I have come up with comes from models of sound change in diachronic linguistics. We noticed that numerous sound changes are well-described as resulting from auditory confusions among sounds with similar values of acoustic phonetic features such as [grave] and [flat], while these features construed as binary-valued are not useful in distinctive feature theory for phonology.

A remaining question is whether the set of phonetic features should be disjoint from the set of distinctive features, or could they be somehow linked when they overlap in a useful way?

Type theory in cosmology?

I ran across something interesting in the book Cosmology by Liebscher (2005). A conundrum known as Olbers's paradox has provided motivation for cosmological considerations. This paradox demonstrates that, if there were an infinity of potentially observable stars, the sky should have infinite brightness. Modern cosmology takes this as a proof that the universe has a history in which there were no stars, and thus the number of stars we can see by looking far enough away from Earth is effectively limited by time.

There is an alternative attempt to resolve this paradox, credited to Lambert (1761), which assumes the universe consists of systems of different order. As Liebscher describes, "a system of order n contains g_n systems of order n-1, where g_n is the multiplicity. . . now every integral over the universe may converge." This naturally reminded me of type theory being invoked to resolve paradoxes of semantics etc. Lambert's "ordered universe" cosmology is known to be inadequate because it predicts an inhomogeneous universe at every distance scale, which is contrary to observation. Many would also say that type theory is an inadequate solution for our problems in logic and linguistics.

Modeling the learning of grammar

In mathematical approaches to linguistics, the results are usually solid because they are derived mathematically. But there always remains the question whether the mathematical model chosen does actually serve as a satisfactory model of something. Carnap pointed out a long time ago that this question does not have a formal answer, it is really quite subjective. He used the term explicatum to refer to the formal model of an informally expressed explicandum.

When it comes to the learning of grammar, even restricted to syntax of natural language, it is not even so clear what the explicandum is. The only obvious fact is that children do indeed learn a language. Suppose it has a grammar---then what? Researchers disagree on how to model the learning of grammar. There is a canon of literature in algorithmic learning theory that models grammatical inference using only text strings as input data. Alex Clark is a contributor to this literature, and he has derived interesting results concerning the identifiability of substitutable context-free languages. I noticed in the 2009 Algorithmic Learning Theory proceedings that this result has recently been extended to substitutable mildly context-sensitive languages, the latter being a ubiquitous class that turns up again and again in mathematical linguistics.

A substitutable language is, roughly, one where any two substrings sharing the same context (construed linearly, not structurally) also share all their contexts. I.e. the language allows them to substitute for one another. It seems to me that this is too strong a property to be true of natural languages.

Aside from that, it is quite unlikely that children learn language from word sequences only, divorced from meanings. For these reasons, I have pursued a different modeling approach in my work on type-logical grammar induction (see my new paper in the Journal of Logic, Language and Information for a full account). I hold that learning should be modeled using sentences annotated with structural information. I think it is possible that children could use a kind of corpus analysis, together with their meaningful experience, to glean this information without benefit of a complete grammar.

But you see, the matter of modeling is subjective. I had a (friendly) argument about these very points with Alex Clark a few years ago. He was equally committed to pursuing the pure string learning model. He said my assumptions were too rich, the extra information was too plentiful. We weren't arguing about mathematical points, but rather about what is the best explicatum for the right explicandum. We can't even agree about the explicandum.

Mathematical linguistics will benefit, I believe, from the parallel pursuit of different explicata for different explicanda. Then when we know more, due to advances in cognitive science or whatever, a wide range of different mathematical results will already be known.

Why did Gentzen use sequents?

Here's a question that I've long thought to be a puzzle in the history of logic. Perhaps I've been wrong about this, but there doesn't seem to be any reason for Gerhard Gentzen's sequent calculus to make use of sequences of formulae.

Let me try to clarify what I mean here. A sequent, for Gentzen, is a special inference expression connecting a (possibly empty) sequence of antecedent formulae with at least one consequent formula. The system for intuitionistic logic requires just one consequent formula, while the classical system allows a sequence in the consequent. The sequent calculus is then a way of defining classical or intuitionistic logic as a scheme of inferences among sequent expressions rather than the formulae within. Baez and Stay (2009) nicely explain that Gentzen used sequents so he could develop a logical calculus that has the subformula property, which has in turn led to so many useful developments. But that only requires sequent expressions connecting sets of formulae, not sequences of formulae.

I feel there remains the question, why did Gentzen insist that the antecedent and consequent of a sequent be a sequence of formulae? He immediately had to compensate for this by invoking the structural rules, special rules that eliminate the sensitivity of his systems to the sequence structure. He did this, of course, because classical and intuitionistic logic aren't sensitive to the structure of formulae occurring in a sequence. In fact, later authors relying on Gentzen to present classical or intuitionistic logic have frequently just gone ahead and redefined sequents as special expressions connecting sets of formulae, thus doing away with the structural rules. Logicians more recently have played extensively with the logics that result from not using one or more of the structural rules---these are the substructural logics, which are used in type-logical grammar, among other applications. But Gentzen didn't suggest doing away with any structural rules, so far as I know. So then, why did he insist that formulae came in sequences, only to turn around and insert provisions that eliminated this sensitivity? We certainly have to thank him for his odd bit of foresight, given how much fruitful research has been done using substructural logics in recent times.