Potential Possibilities and the Limitations Involved in Formal Theories of Quantification - Essay Example

Potential Possibilities and the Limitations Involved in Formal Theories of Quantification for Natural Language Written By: There are many potential possibilities and also equally as many limitations involved in the creation and execution of formal theories in the quantification of natural language. There would also be a great many who would willingly discuss the consideration of quantifiers and their positive and negative potential in the creation of formal theories of quantification in the world of natural language. There are also many who would give reason to apply these limiters that quantify natural language. Although, there are potential possibilities for such formulation, there has been no entirely feasible method created. The theories and theorems that are involved in quantifying language fail to provide us a limitless and potentially useable formula for quantification regardless of their complexity and their rationale supporting it. Jon Barwise and John Etchemendy in Language Proof and Logic give a very understandable argument as to the reasons for quantifiers and the reasons they are not always accurate in their use. A very pertinent consideration for their argument starts out their ninth chapter in Language Proof and Logic by saying, "In English and other natural languages, basic sentences are made by combining noun phrases and verb phrases." (1. Chapter 9 page 227, Language Proof and Logic.) The consideration continues further in that Barwise and Etchemendy contend that, "Quantification takes us out of the realm of truth-functional connectives." (2. Chapter 9 page 227, Language Proof and Logic.) This gives us reason for the consideration that quantifiers are not always the most useful method for determining natural language tendencies. Quantifiers, according to Barwise and Etchemendy, have a tendency to dull the truthfulness of sentences giving them a generalisation that may not bear an ounce of truth within them. In the case of first-order logic, the process assumes that there would be an infinite list of variables so there would be no possible way to run out of these variables, regardless of a sentence's complexity. Theorists like Fitch would understand all of these separate variables involved, of which there are many, but others like Tarski's World would not, in that Tarski's World uses six in place of infinite variables as Fitch would manage. This would in fact present a rather expressive limitation in Tarski's World of language use. Expanding the set of terms of language usually means adding variables to it. At this point, only individual consonants, also known as names would be considered the sole amount of basic terms. Obviously, first-order logic, in the eyes of Barwise and Etchemendy believe a complex series of quantifiers is necessary to describe natural language. They consider universal and existential quantifiers in their equations. Universal quantifiers are those that are all encompassing and ultimately considered unconditional. Existential quantifiers are more limiting in scope in giving a value but not a limitless value toward the quantification. W. Tecumseh Fitch, from the University of St. Andrew's School of Psychology wrote a treatise titled The Evolution of Language: A Comparative Review. Fitch says in his work that the study of language evolution is often considered little more than speculative story-telling. Fitch further states that this has actually had little to do with the development of many fields which would touch upon it. Those fields include linguistics, evolutionary biology and neuroscience. Over the last fifteen years increasingly productive study of language evolution has occurred in various different quarters and there is far more collaboration and exchange in relation to this increasing study. There in fact would be three key innovations which should be explained in language evolution. The first critical step would be to distinguish among the various component abilities in languages. Mechanisms that are both specific to language and also would be considered uniquely human can be termed as what is known as the faculty of language in a narrow sense. Conversely, any mechanism involved in language is considered as the faculty of language in a broad sense. FLN as faculty of language in a narrow sense is inclusive as a subset of FLB or faculty of language in a broad sense, thus causing the contents of the FLN to be determined empirically in place of a priori. The three mechanisms of language that are agreed upon include underlying speech, syntax and semantics. They have been agreed to be unusual in our species and also crucial for language. It is the work done alongside Fitch's elaborations on language as completed by Barwise and Etchemendy and the others that would facilitate reason for including his treatise in this paper. Fitch would add to the consideration of Barwise and Etchemendy, yet would also give reason as to how and why their work at defining natural language provides limitations as such. He does so in saying, "Speech provides the standard signalling modality for the communication of language in all world cultures. Despite its reliance on an auditory and vocal apparatus that is broadly shared among most mammals, speech includes at least two components which are unusual namely our reconfigured vocal tract and vocal imitation." (3. Fitch, W. Tecumseh; The Evolution of Language page 194) Like Barwise and Etchemendy, Fitch discusses the changes in syntax and semantics, yet, Fitch includes those changes whereas Barwise and Etchemendy exclude them, this would be where the difference in study would lie. Well formed formulas as they are considered in the work done by Barwise and Etchemendy that gives a definition requiring that there be a much larger class over and above first-order logic that is called well-formed formulas. There are variables even in formula including atomic and complex formula which are built from the atomic formula by utilizing truth-functional connectives and quantifiers in accordance with the Completeness Theorem. The Completeness Theorem tells us that the introduction and elimination rules would be complete in consideration of the logic of the truth-functional connectives. An antithesis to this would be the Soundness Theorem which actually would imply a sort of incompleteness since showing that rules involving Ft would only allow us to prove tautological consequences of the premises involved in the work performed by Barwise and Etchemendy. These two theorems, the Soundness and Completeness Theorems, would have practical uses that Barwise and Etchemendy believe are worth keeping in mind. Yet, in actuality, the theories and theorems that are presented by Barwise and Etchemendy would in fact prove limitations upon the essence of natural language itself. The theories that are presented all contain a grain of truth within them, yet there is no absolute method of quantifying natural language tendencies. The reason for this would be that quantification takes away from the essence of the sentence formulation, presents absolutes and limitations that we would normally not see. Concepts of tautology and tautological consequence are considered, but, according to Barwise and Etchemendy these concepts don't get them very far in the consideration of first-order logic. The need for a much more refined method in analyzing logical truths and logically valid arguments is necessary, especially when they are dependent upon quantifiers and identity. Within this short section it is already evident that so many delimiters being placed on the rules and considerations already in existence in natural language formulation would in fact manage to make it that much more difficult to consider the work and the reality of things in this limited series of determinants. Barwise and Etchemendy continue their treatise with various charts and graphs and in chapter ten would show comparisons between Propositional Logic, First-Order Logic, and General Notion. They state that, "These concepts are meant to apply to those logical truths, consequences, and equivalences that are such solely in virtue of the truth-functional connectives, the quantifiers, and the identity symbol." (4. Chapter 10 page 267, Language Proof and Logic) As such, in regard to the factors involved in what they believe is a fair method to quantify natural language tendencies; this further appears to limit the capacity of that same natural language. Regardless of the consideration that natural language automatically would have limitations; this process as discussed by Barwise and Etchemendy would actually further limit the honesty and truthfulness of the language as a whole. FOL, in regard to English translation, would not take into account the sparsely managed availability of primitive concepts within the logic procedure. Regardless that this would actually make it easier to learn the language, it also leaves us without entirely natural ways of saying just what you want to say. Another difficulty according to Barwise and Etchemendy would be the fact that there is far too much ambiguity in the English language and first-order logic is inherently clear in concept. As a result, in place of having a variety of appropriate interpretations, we are thus limited because of this process. This is yet another reason to not be favourable of such creation of formal theories in the formulation of natural language. Barwise and Etchemendy provide still more consideration in their work by saying, "First-order languages do not in general allow us to talk directly about numbers, only about elements of our domain of discourse. The blocks language, for example, only talks about blocks, not about numbers. Still, it is possible to express these three kinds of numerical claims in FOL." (5. Chapter 14, page 366, Language Proof and Logic) Yet, in fact, the consideration that first order logic can be applied to natural language and then quantified in this method actually diminishes the truthfulness of the language as is readily seen in the equations and quantifications created by Etchemendy and Barwise. It is this ideology that many others consider to be prominent in the work toward understanding how natural language might actually be allowed the capacity for formulation of formal theories of quantification. In First-Order, Curry-Typed Logic for Natural Language Semantics by Fox, Lappin and Pollard, the formulas and formulations utilized mimic those of Barwise and Etchemendy. There is only one difference in that there would be an addition formulating absurdity within the equation, negation and side conditions. This version even contains equations for truth along with the absurdity. One interesting notation within this work would be the admission by the authors within the conclusion of the paper. "The paper presents a highly intensional theory that can emulate some key aspects of typed logics within a first-order framework. In this system, extensional equivalence does not entail intentional identity." (6. First-Order, Curry-Typed Logic for Natural Language Semantics page 15) It is for this reason that the theories and considerations presented within the work would mimic those of Barwise and Etchemendy. Again, we see that the processes and procedures that were originally outlined in the work done by Barwise and Etchemendy would be considered in the first-order language techniques, this continues to provide limitations to the capacity of the idea involved in creating formal theories of quantification in natural language. Formality in quantification of natural language would in fact require more than mere simplification. It would also require the capacity to broaden and contract based on certain key factors within the language itself. Yet, again this is impossible because every language has a different basis by which it was formed. This is the reason that there would not be a definitive method of creating a formal theory of quantification in natural language, no matter the language being studied. Geoffrey K. Pullum and Barbara C. Scholz bring up the origins of generative frameworks. They discuss how Noam Chomsky laid out a very convincing consideration for laying out goals in stating explicit grammars in 1960. Yet, even Chomsky would hardly present the idea as its first developer in the approach. Emil Post, who in fact was a mathematician, "developed a schema defining the general form of inference rules in terms of what he calls productions." (7. Contrasting Applications of Logic in Natural Language Syntactic Description; page 2) The problem with this would be that there is a finite element involved, and as has been seen, natural language appears to have an infinite source in its creation. The two go further to state that as of "today, virtually all theoretical linguistics that aims at explicitness is based on Chomsky's development of Post." (8. Contrasting Applications of Logic in Natural Language Syntactic Description; page 4) In contrast to Post's work, however, "Chomsky's early transformational grammars represented an elaboration rather than a further restriction of Post systems." (9. Contrasting Applications of Logic in Natural Language Syntactic Description; page 4) The fact that the original work of Post was, in actuality, elaborated upon by Chomsky says a great deal in the potential development of a structured and formal theory of quantification in natural language. This compounds the consideration that those theories that have been developed by Barwise and Etchemendy could in fact be considered derivations of those created by Chomsky and Post. Creating more complexity hardly fosters a true capacity at understanding natural language or even formalizing the capacity to formulate a workable framework and theory in quantifying natural language. In Patrick Doyle's Al Qual Summary he sums everything up in his overview of one of his lectures at Dartmouth. His summary states, "To understand something is to transform it from one representation into another, where this latter representation is chosen to correspond to a set of available actions that could be performed, and for which a mapping is designed so that for each event an appropriate action will be performed." (10. Natural Language; Al Qual Summary) Gottlob Frege was trained as a mathematician and his interests in logic grew out of the interests he had in the foundations of arithmetic. Frege believed in the ideal of logics, which was modest in one sense, but quite ambitious in others. Yet, again, this consideration toward language was limited by the boundaries of arithmetic. Frege agreed with Leibniz in the idea that natural language was unsuited to a task that equated it with the logics he felt existed in mathematics. In point of fact, Frege felt it necessary to actually create a language that would in essence combine the tasks of Leibniz's "calculus ratiocinator" and what Frege termed as "lingua characterica", which he considered as being a logically perspicuous language in which logical relations and possible inferences would be clear and unambiguous." Frege even had a term for this language, "Begriffsschrift", and this was more than likely a borrowed term from one of Leibniz's ideas. Frege rejected many of the ideas formulated as a result of Leibnizian tradition, considering them to be unsuitable in formula. Frege modelled logical language, in contrast to natural language, upon the international language of mathematics replacing the subject and predicate style of logical grammatical analysis with notions of function and argument. The one thing Frege had done that others did not was to expand the notion of function to allow arguments and values other than numbers. Frege, like his contemporaries, assigned values to language in mathematical logic, thus limiting the capacity of potential in gaining truthful understanding of language and its various idiosyncrasies. Diane Blakemore writes, in her text Relevance and Linguistic Meaning: The Semantics and Pragmatics of Discourse Markers, that "the view of semantics which underlies the argument that expressions such as but and well have pragmatic meaning rather than semantic meaning. This view is implicit in Gazdar's (1979) definition of pragmatics: PRAGMATICS = MEANING MINUS TRUTH CONDITIONS (Gazdar 1979:2)" (11. Blakemore, Relevance and Linguistic Meaning, 12) Ms. Blakemore continues to say that in accordance with this view, discourse connectives including but must have pragmatic meaning in place of semantic because they would fail to contribute to the truth of conditional context in the utterances which contained them. In this, Blakemore is quite different in her belief that you can segregate meaning from language as Barwise and Etchemendy have done in their work because in doing so, it would fail to gain complete comprehension. Diane Blakemore's work continues to be quite intriguing in content as she states, "according to a recent textbook on generative syntax, 'grammar is not just concerned with the principles which determine the formation of words, phrases and sentences, but also with the principles which tell us how to interpret (= assign meaning to) words, phrases and sentences' (Radford 1997:1). However, it seems that the notion of meaning that Radford had in mind is limited to 'structural aspects of meaning such as the assignment of case features" (12. Blakemore, Relevance and Linguistic Meaning, 14-15) This, in essence, gives credence to the work done by Barwise and Etchemendy with their evaluation as to how to quantify language. Although, it also places similar limitations upon quantification of language as Barwise and Etchemendy would manage with their theorems and postulations. Ms. Blakemore contends that it would be "impossible to have a view of pragmatics without having a view of semantics, or vice versa, and it is not surprising that the relevance theoretic approach to pragmatics comes with a view of semantics attached." (14. Blakemore, Relevance and Linguistic Meaning, page 59.) This is something which Barwise and Etchemendy would not touch upon, yet Patrick Doyle reflects this in his treatise, as would Pullum and Scholz. In determining whether a viable framework can be managed with the quantification and formulation of natural language, we must have all the factors considered. In truthful examination of all the processes and procedures utilized by Frege, Barwise, Etchemendy and the others that have been consulted, it is evident that there continue to be limitations in the formulas that would be created in quantifying natural language. Regardless of the fact that there are potential possibilities in developing such frameworks, that doesn't mean that they would function appropriately or that there would not be aberrations in their execution. The fact that so many of these models are mathematical and essentially logical in formula would in fact be the reason why so many of them apparently fail. The reason behind this would be that language itself has an unlimited capacity in its creation as a result of its basis changing as the society changes in formula and expression. REFERENCES CITED: 1. Barwise, Jon and John Etchemendy; Language, Proof and Logic; London: Seven Bridges Press; Copyright 2000; Chapter 9 page 227 2. Barwise, Jon and John Etchemendy; Language, Proof and Logic; London: Seven Bridges Press; Copyright 2000; Chapter 9 page 227 3. Fitch, W. Tecumseh: The Evolution of Language: A Comparative Review; Biology and Philosophy; Copyright Springer 2005 page 194 4. Barwise, Jon and John Etchemendy; Language, Proof and Logic; London: Seven Bridges Press; Copyright 2000; Chapter 10 page 267 5. Barwise, Jon and John Etchemendy; Language, Proof and Logic; London: Seven Bridges Press; Copyright 2000; Chapter 14, page 366 6. Fox, Chris; Shalom Lappin; Carl Pollard; First-Order, Curry-Typed Logic for Natural Language Semantics; Department of Computer Science, University of Essex, Colchester, UK page 15 http://www.dcs.kcl.ac.uk/staff/lappin/papers/fox-lappin-pollard-nlulp02.pdf 6. Pullum, Geoffrey K; Barbara C. Scholz; Contrasting Applications of Logic in Natural Language Syntactic Description; page 2; http://people.ucsc.edu/pullum/Oviedo.pdf 7. Pullum, Geoffrey K; Barbara C. Scholz; Contrasting Applications of Logic in Natural Language Syntactic Description; page 4; http://people.ucsc.edu/pullum/Oviedo.pdf 8. Pullum, Geoffrey K; Barbara C. Scholz; Contrasting Applications of Logic in Natural Language Syntactic Description; page 4; http://people.ucsc.edu/pullum/Oviedo.pdf 9. Doyle, Patrick; Natural Language; Al Qual Summary; May 14, 1997; http://www.cs.dartmouth.edu/brd/Teaching/AI/Lectures/Summaries/natlang.html 10. Blakemore, Diane. Relevance and Linguistic Meaning : The Semantics and Pragmatics of Discourse Markers. Cambridge, England: Cambridge University Press, 2002. Questia. 24 May 2006 page 12 . 11. Blakemore, Diane. Relevance and Linguistic Meaning : The Semantics and Pragmatics of Discourse Markers. Cambridge, England: Cambridge University Press, 2002. Questia. 24 May 2006 page 14-15 . REFERENCES: 1. Barwise, Jon and John Etchemendy; Language, Proof and Logic; London: Seven Bridges Press; Copyright 2000; Chapters 9 to 14 2. Liu, Hugo and Push Singh; Commonsense Reasoning in and over Natural Language; Media Laboratory; Massachusetts Institute of Technology; USA http://web.media.mit.edu/push/CommonsenseInOverNL.pdf 3. Boyd, Jeremy; Cognitive Limits on Natural Language Suppletion; November 14, 2003; http://ling.ucsd.edu/boyd/comps2.pdf 4. Fox, Chris; Shalom Lappin; Carl Pollard; First-Order, Curry-Typed Logic for Natural Language Semantics; Department of Computer Science, University of Essex, Colchester, UK http://www.dcs.kcl.ac.uk/staff/lappin/papers/fox-lappin-pollard-nlulp02.pdf 5. Hovy, Eduard; Survey of the State of the Art in Human Language Technology; University of Southern California, Chapter 4 Language Generation; Language Technology World http://www.lt-world.org/HLT_Survey/ltw-chapter4-all.pdf 6. Sevigny; McMaster; Humanities Department: Towards an Information-Based Procedural Grammar for Natural Language Understanding; http://www.humanities.mcmaster.ca/sevigny/sevignycjal.pdf 7. Hess, Michael; How Does Natural Language Quantify ; University of Zurich, Seminar of General Linguistics; http://acl.ldc.upenn.edu/E/E85/E85-1002.pdf 8. Pullum, Geoffrey K; Barbara C. Scholz; Contrasting Applications of Logic in Natural Language Syntactic Description; http://people.ucsc.edu/pullum/Oviedo.pdf 9. Doyle, Patrick; Natural Language; Al Qual Summary; May 14, 1997; http://www.cs.dartmouth.edu/brd/Teaching/AI/Lectures/Summaries/natlang.html 10. Klement, Kevin C.; Gottlob Frege (1848-1925); The Internet Encyclopedia of Philosophy; 2006 http://www.utm.edu/research/iep/f/frege.htm 11. Fitch, W. Tecumseh: The Evolution of Language: A Comparative Review; Biology and Philosophy; Copyright Springer 2005 pages 193-230 http://psy.st-andrews.ac.uk/people/lect/wtsf.shtml 12. Blakemore, Diane. Relevance and Linguistic Meaning : The Semantics and Pragmatics of Discourse Markers. Cambridge, England: Cambridge University Press, 2002. Questia. 24 May 2006 . Read More