Eclecticismo en la práctica lingüística

"La filosofía -decía Galileo (en Il Saggitario, 1963)- está escrita en ese grandísimo libro que continuamete está abierto ante nuestros ojos (a saber, el universo), pero no puede entenderse si antes no se procura entender su lenguaje y conocer los caracteres en que está escrito. Este libro está escrito en lenguaje matemático, y sus caracteres son triángulos, círculos, ...".

Para Galileo, la naturaleza es un libro escrito en lenguaje matemático. Como él mismo dice, si los hombres no la habían entendido hasta entonces es que "no conocían los caracteres en que estaba escrita". Pero la naturaleza no es un libro ni está escrita en lenguaje alguno. Lo que sí es un "libro" es una teoría física y ésta es la que puede estar escrita en lenguaje matemático.

El éxito de la física moderna que Galileo inaugura se debe -al menos en parte- a que, para hablar de la naturaleza, los físicos modernos dejan de hablar en el lenguaje metafísico en que habían hablado los antiguos y se ponen a hablar en un nuevo lenguaje: el lenguaje matemático.

Lo nuevo no es aquí -como la cita de Galileo parecería implicar- el conocimiento pasivo del lenguaje matemático, sino la aplicación activa de ese lenguaje a la descripción de la naturaleza.

A partir de Galileo, la ciencia deja de hablar del mundo en el lenguaje ordinario para pasar a hablar en el lenguaje de la matemática.

La ciencia física progresó rápidamente durante los siglos XVII, XVIII y XIX y el lenguaje matemático de la época de Galileo muy pronto se le hubiera quedado pequeño. Pero la matemática fue creciendo junto con la física, creando nuevos lenguajes matemáticos en los que formular las nuevas teorías físicas.

Al llegar el siglo XIX, los físicos disponían de una amplia gama de lenguajes matemáticos en que formular sus teorías, pero la estructura de estos lenguajes no estaba nada clara. La gran labor que se propusieron y en gran parte realizaron los matemáticos del siglo XIX fue la clarificación de esos lenguajes.

:26 Y la gran obra de Cantor fue la creación de un nuevo y extraordinariamente expresivo lenguaje: el lenguaje de los conjuntos, la teoría de los conjuntos. Resultaba que todos los otros lenguajes matemáticos eran traducibles al lenguaje conjuntista y que todos los otros conceptos matemáticos eran fácilmente definibles en este lenguaje. Con ello, la tarea de clarificar la estructura de los diversos lenguajes matemáticos de que disponía el físico se había convertido en la tarea de clarificar el lenguaje de la teoría de conjuntos.

Cantor creía haber descubierto los conjuntos que ya desde siempre estaban en el mundo. Pero en realidad no era un descubridor, sino un inventor. Lo que había hecho es inventar un nuevo modo de hablar.

Es una concepción ingenua la de que el mundo está dado de antemano, estructurado en cosas determinadas relacionadas entre sí de modo únivoco, y que el lenguaje viene después, reflejando con mayor o menor perfección la estructura de ese mundo.

Podemos aproximarnos al mundo con distintos lenguajes y habrá tantas estructuraciones distintas del mundo como lenguajes diferentes usemos para describirlo.

:34 La matemática se basa en una gran ilusión: en la ilusión de suponer acabado lo inacabable, ya terminado lo que no se puede terminar. Así se forman totalidades ilusorias, infinitos actuales que a su vez sirven de punto de partida a nuevos procesos semejantes, cada vez más alejados de la realidad.

Sin embargo, estas clases infinitas de las que nos hacemos la ilusión de que hablamos no existen en ningún sentido fuerte de la palabra existir. No están dadas en este mundo, ni pueden estarlo nunca. Postular otro mundo en que estén dadas es hacer mitología.

Las teorías matemáticas -en la medida en que hablen de algo- hablan unas de otras y encuentran sus modelos unas en otras. Y la posición central de la teoría de conjuntos entre ellas se debe a que todas las demás son en ella interpretables. En efecto, la relación semántica de satisfacibilidad de una teoría (algo lingüístico) en un modelo (algo extralingüístico) fácilmente puede ser substituida por la de interpretabilidad de una teoría en otra, adoptando la siguiente definición, adaptada de Tarski y Sxhonfield: [...]

:35 Esta noción de interpretabilidad de una teoría en otra cubre tanto la satisfacibilidad en modelos internos como en modelos externos.

Conclusión

A principios de siglo decía Russell que las matemáticas es una disciplina en la que no sabemos sobre lo que hablamos ni si lo que decimos es verdadero.

Hoy podríamos decir que al menos la teoría de conjuntos es una disciplina en la que sabemos que no hablamos acerca de nada y que lo que decimos no es verdadero ni falso.

¿Es esto una razón para dejar de hacer teoría de conjuntos, o, en general, para abandonar la matemática clásica, como piensan los intuicionistas?

Si el único sentido de una teoría o un lenguaje consistiese en decir verdades acerca del mundo, entonces, evidentemente, habría que abandonar la teoría de conjuntos.

Si, como dice Quine, el uso de variables cuantificadas en la teoría de conjuntos nos comprometiese "ontológicamente" a aceptar la existencia de esos fantasmas que son las totalidades infinitas actuales, y no creemos en ellos, también entonces tendríamos que abandonar la teoría de conjuntos, so pena de inconsecuencia.

Pero ni la única posible función de un lenguaje consiste en hablar acerca del mundo real ni los cuantificadores son amuletos mágicos cuyo uso nos compromete a vivir entre fantasmas.

En definitiva, la teoría de conjuntos es el foco central de ese universo lingüístico en expansión que constituyen las teorías matemáticas. La teoría de conjuntos ofrece un lenguaje común al que traducir los diversos lenguajes matemáticos y un arsenal de nociones y principios indispensables para el desarrollo de múltiples teorías formales.

Muchos de estos lenguajes, muchas de estas teorías -y otras que están aún por surgir- no han tenido hasta hoy aplicación alguna en la ciencia empírica. Pero otras teorías han encontrado interpretaciones empíricas fecundas y han servido para formular y desarrollar las ciencias de la naturaleza y de la sociedad.

En la medida en que el conocimiento y dominio del mundo real en que vivimos nos interese, nos interesarán también las teorías matemáticas que a través de su contribución a la ciencia empírica posibiliten ese conocimiento y ese dominio, y, por tanto, al menos indirectamente, nos interesará la teoría de conjuntos.

Richard Montague and Ludwig Wittgenstein represent views which are diametrically opposed on the key issue of formalization: of whether, and in what ways, formalisms to express the content of natural language (NL) should be constructed. "I shall argue, too, that the influence of Wittgenstein has been largely beneficial while that of Montague has been largely malign." The contrast between their views comes down to two issues:

• Is there a hidden structure to NL?
  • Montague: Yes; clearly he belives there is a simple logic as the hidden structure of NL.
  • Wittgenstein: Yes, but not one to be revealed by simple techniques, like logic. (He refers to "deep grammatical structures", but it is not clear what exactly he ment for them).
• Is NL itself and its use the final source of judgment or evidence?
  • Montague: No; his choice of examples, far from the concern of ordianry speakers ("Every man loves some woman") suggests that he is more concerd with logic and formal semantics than with NL "properly".
  • Wittgenstein: Yes. The different structures of "Every man loves some woman" in Predicate Calculus interest the logician, but the ordinary speaker rarely, if ever, sees that there is a "second interpretation" of the sentence.

Two very broad trends in the philosophy of language, as regards the role and importance of formalization can be distinguished:

• One group of philosophers has been for it, and for as much of it as possible. (? Leibniz, Russell'25, Tarski'35, Carnap'37, Montague'70)
• While the other group has been uncompromisigly against it. (? Popper'45, Wittgenstein'53, Hanson'58, Quine'60, Cohen'62, Dreyfus'72)

The "?" indicates a tentative list of philosophers

Leibniz, in the 17th century, believed that the formalism he proposed was the real structure of ordinary language (and, indeed, of the physical world), though without its awkward ambiguities, vaguenesses, and fallacies. In his most fantastic moments he envisaged the replacement of ordinary language by this "Universal Characteristic", to the general improvement of clarity and precision.

Whitehead and Russell's 1925 Principia Mathematica was a decisive contribution to the definition of propositional and general logic and the investigation of their properties. They were applied to the formalization of the notion of mathematical proof, but already Russel at least was setting out the ways in which this approach to logic was also, for him, a formalization of natural language. Like Leibniz, he wished to clear away what he thought of as the confusions of ordinary language. He was much concerned with the grammatical similarity of sentences like "Tigers are existent" and "Tigers are fierce" and how, in this view, this similarity had led philosophers into the error of thinking that tigers therefore had a property of existence as well as one of fierceness.

Wittgenstein was closely associated with Russell during the period between 1910 and 1920, and was developing what is now thought of as his early philosophy. This was set out in a curious early work called the Tractatus Logico-Philosophicus (1922) where he proposed what is now called the "picture theory of truth", according to which sentences in ordinary language signify or mean because their structures reflect or exhibit the same relation as that which holds between the things mentioned by the sentences. His theory of meaning at this point was more obscure than that of Russell, or Russell's predecessor, Frege.

Frege, in the late 19th century, had proposed a "dualist" theory of meaning in which each word signified in two ways (his famous example of "the Morning Star" and "the Evening Star"):

1. One way (Bedeutung), referred to some entity, or REFERENT (i.e. Venus)
2. The other (Sinn) referred to the "sense" of the word. The two phrases of the example mean something different in that they have different SENSES.

These meaning theories are referential, even when they talk of "sense". For, whatever the status or nature of the thing that is "the meaning", it is an entity that is somehow pointed at by the word. In Frege's case the word points in two different ways to two quite different sorts of entity, whereas in Wittgenstein more obscure "picture theory" there was only one kind of pointing.

[The truth of expressions can be established by truth tables and is demostrated as semantic. This is different from the method referred as proof theoretic (that comes from the sequential relation of structures in a proof)]. The word "semantic" is so troublesome that it cannot be introduced into the discussion without a word of warning, because on its ambiguities rests much of the difficulty of our whole subject.[...] It will be written as "L-semantic" when used in connection with formal logic.

Formal logic of the Twenties rests in two fundamental ideas, namely:

1. Significant combination
2. L-semantic demostration

Montague and Wittgenstein represent respectively a reaction to, and a extension of that logic. In the hands of Tarski'35 and Carnap'37 the above concepts took on new names and properties:

1. Significant combination
2. Logical syntax (replacing "L-semantic demonstration")
3. Meta-language

Carnap in his "Logical Syntax of Language" developed the notion of ill formed expression (as "Mortality is Socrates"). Carnap distinguished: (Chomsky was many years later a student of Carnap)

• Rules of formation: determined what were, and were not, well formed expressions in the logical language. Thus, in the Propositional Calculus, (P IMPLIES (OR Q)) was not, while (P OR Q) was a well formed expression.
• Rules of transformation: operated on those well formed expressions that were also thrue so as to produce theorems in logic.

Carnap was a formalizer in the Leibnizian sense since, to him, these distinctios applied equally well to proper formalization of natural language.

The distinction of language and meta-language was due to Tarski who proposed it to solve an apparently intractable problem of logic that he had inherited from Russell, who had discovered certain logical paradoxes. Tarski's statement of the problem of the paradoxes was usually in terms of the example:

The sentence in this rectangle is false

where any attempts to assign a truth-value to the sentence led to trouble. Tarski thought that this problem would be solved if we only used "true" in a meta-language and never in an object -language. Thus "That John is happy is true" would be a sentence with level confusion, for the proper name would be the meta-language sentence "'John is happy' is true", where the sentence "John is happy" is mentioned in the meta-language, but not used in it.

[Truth theory of meaning: 212]

Tarski's fundamental achievement was a theory of truth and logical consequence for formalized languages (a L-semantic theory).

Montague extended it to sentences in natural languages: a system in which the truth conditions of complex statements could be seen to be composed from the truth conditions of simple ones. Making even stronger claims than Tarski's, because Tarski did not think that such a theory could be constructed for a natural, non-formal, language. Otherwise the object-language and the meta-language would be the same, and Tarski thought that was bound to lead to trouble.

Wilks'76 and others in linguistics and AI do not share the assumption that the stating of truth conditions is explaining the meaning of a sentence. Those in the Tarskian tradition are highly critical of much work in linguistics and AI because meanings are not stated in that way, and it is an interesting open question whether AI work on NL could be so stated in general. Wilks does not criticize formal analysis of NL as such. What is being opposed is pointless formalizations of NL [such as Montague's], and it is suggested that Wittgenstein is on of the few philosophers who can provide insights into what a fruitful approach to NL migh be.

Logical positivists also had a thesis about the dependence of meaning on truth but should not be confused with the "truth-condition" approach followed by Tarski. Their principle was called the Principle of Verification, and it said "The meaning of a statement is the procedures we would carry out to establish its truth of falsehood". However wrong this principle might be (and it is wrong), it was at least serious, in a way that the modern truth condition approach is not.

Blackburn'9? (for one in a large school of logicians) holds general principles concerning the applicability of Tarski-like theories to NL. The thesis that they share is that of the "truth theory of meaning": namely that the meaning of a sentence is determined by its truth conditions, in the Tarskian sense of that phrase. It would be quite possible to reject Montague's detailed "Semantics for NL" and still accept the general tenents of the school about meaning and truth, namely that truth conditions determine the meaning of the senteces of NL in just the way they can be said to do so for the sentences of logic and methematics.

Dreyfus, a full-time opponent of the very possibility of AI, has made much use of Wittgenstein's arguments against formal logic in his own arguments against AI. However, the general structure of this chapter is that AI has real philosophical importance because it has completely changed the old debate between formalists and anti-formalists. For to handle language "formally" on a computer it is in no way necessary to accept the tenets of Tarski, Montague or any other approach based on formal logic. On the contrary, the most fruitful approaches to understanding natural language are precisely those not subservient to a powerful logical or L-semantic theory.

AI has provided a sense to the notion of the precise manipulation of language that is not necessarily open to the attacks of the anti-formalists like Wittgenstein. And a large part of the credit for breaking the old formalist/anti-formalist opposition in a new way must go to Chomsky.

Chomsky's theory of grammar is a precise theory of language: it has the form of a logic, but not the content. Chomsky took the structure of proof-theoretic (what Carnap called transformation rules), namely the repeated derivation of structures from other structures by means of rules of inference, but he let its content be no more than what Carnap had meant by Formation Rules, namely the separation of the well-formed from the ill-formed. Thus, in transformational grammar the inference (transformational) rules were to apply to axioms and theorems, but to produce not new theorems but well-formed English sentences.

Thus, Chomsky has a precise system of handling language, but with no semantic definition of truth, and not even a syntactic, proof-theoretic one either. Thus, Chomsky's was the first concrete proposal to breach the wall between formal and anti-formal approaches.

AI has gone further, and indeed Chomsky's paradigm still shares many of the drawbacks to formal approaches: in particular the rigid "derivational" structure common to logical proofs and to Chomskyan transformational derivations.

[ Wittgenstein'58 ]

1. Reference. Thesis: words do not in general have meaning in virtue of pointing at objects in the real world (or "conceptual objects" either). We could have a language based on the referential notion, but it would be a language more primitive than what we call NL.
2. Mini languages and language games. Thesis: we can construct mini-languages obeying any rules we like, and we can think of them as games. The important question is whether these games are sufficiently like the "whole game" of NL.
3. Family resemblances and boundaries. Thesis: the conventional notion of concept is wrong: namely the view that a concept relates in some way to the qualities or characteristics that all things falling under the concept have. There are not firm boundaries to concepts, nor are there to linguistic usage, nor to the application of linguistic rules.
4. The linguistic whole and confronting the world. Thesis: a language is a whole and does not confront the world sentence by sentence for the testing of the truth or falsity of each individual part.
5. Logicians have a false picture of how language is. Thesis: logicians think that language is like their favorite calculus, but they are quite wrong. Moreover, it is language itself and its use that is the standard for testing disputes that arise, not what logicians dictate.
6. Understanding is not a feeling. Thesis: we have the idea that "understanding" something involves, or is associated with, a special feeling of being right. But the tests of our being right are quite different from the feeling.
7. Application justifies our structure. Thesis: the significance of a representational structure cannot be divorced from the process of its application to actual language.
8. Real world knowledge and forms of life. Thesis: language understanding is not independent of very general inductive truths about our human experience

The feature constraints associated with phrase structure rules in PC-PATR consist of a set of unification expressions. Each expression has three parts, in this order:

1. a feature path, the first element of which is one of the symbols from the phrase structure rule
2. an equal sign (=)
3. either a simple value, or another feature path that also starts with a symbol from the phrase structure rule

As an example, consider the following PC-PATR rules:

Rule  S -> NP VP (SubCl)
        <NP head agr>  = <VP head agr>
        <NP head case> = NOM
        <S subj> = <NP>
        <S pred> = <VP>

Rule  NP -> {(Det) (AdjP) N (PrepP)} / PR
        <Det head number> = <N head number>
        <NP head> = <N head>
        <NP head> = <PR head>

Rule  Det -> DT / PR
        <PR head case> = GEN
        <Det head> = <DT head>
        <Det head> = <PR head>

PC-PATR output with feature structure: