Cross-correlation analysis of logic and language studies

A formal semantics theory - the role of logic

Formal Semantics is the product of the cross-over study of logic and language, is built within the logical framework of natural language semantics. Formal semantics of natural language theory, although the goal is the semantic processing of natural language, but its implementation steps is first constructed natural language syntax (this syntax for the semantic interpretation of syntactic and semantic corresponding, the syntactic concept) is different from the traditional linguistics to understand transformational grammar with another linguistic theory as compared to formal semantics focuses semantic research, but from its own internal division of labor see formal semantics involves syntax, including the syntax and semantics of the two levels of study.

Formal Semantics main theory: Montague Grammar, generalized quantifier theory of discourse representation theory, situation semantics and the type of logic syntax are described below:
Was founded in the 1960s and 1970s by the American logician Montague Montague Grammar (Montague Grammar), natural language as a the same logic language is essentially the same system of symbols, creating a form of natural language semantics research areas of Montague Grammar constructed the PTQ English part of the statement is successfully described the quantitative expression of natural language, semantic features connotation context and propositional attitude sentences. "Over the years, linguists, logicians and computer scientists has been engaged in research on the form of natural language processing system on the English part of the statement. Montague formalized program is an extremely important step in this research direction ....... Richard Montague the grid introduction of the syntax and semantics of two strong level analysis of natural language, he developed a formalized tools to provide the necessary technical background is a deep understanding of the semantics of natural language. "[1] 301 Formal Semantics most significant characteristic is the natural language as a formal methods of modern logic processing object, natural language and logical language there is no substantial difference between the constructed form of natural language system to solve the semantic problem. specific operation means the establishment of the syntax and semantics of the corresponding principle is constructed based on the significance of a combination of the principles of the semantic model. these ideas and technical tools are the cornerstone of formal semantics (Montague, 1974), Montague and Cresswell, Partee et al earliest explicitly proposed and implemented, so that Montague Grammar is the beginning of the study of formal semantics.

In a word, Montague Grammar points emphasized: (1) natural language and logic languages ​​in the deep structure is the same, from the algebraic structure of the operator's point of view, mathematics and logic will enter the natural language research areas in the form of natural language semantics is a branch of mathematics not belong to psychology, (2) the syntax and semantics of the corresponding principle that each syntactic rule corresponds to a semantic rules. syntactic rules are natural language terms to form a phrase phrases eventually forming statement rules (like the formation of a logical system formed formula Rules), corresponding thereto semantic rules is developed syntax expressions in accordance with the forming process and the significance of the combination rule. syntax by a small string of symbols adjoining symbol string, semantic also by the part of the expression of semantic synthetic compound expression semantics. compound expression semantics is part of its semantics letter items. syntactic and semantic correspondence is the significance of the combination principle, (3) the meaning of natural language sentences model On the semantics of so-called truth conditions, natural language terms, phrases phrase meaning all service on the description of the the sentence true value conditions. above thinking is logical concept of the strong impact of the results of the natural language research.

Example, syntactic rules in Montague Grammar: @ noun phrase and β is the verb phrase, F (@, β) = _AT_ β statement corresponding semantic rules: @ the semantics of the "@" semantics of β ‖ β ‖ @ β semantics ‖ @ β ‖ = G ("@" the ‖ β ‖) = "@" (‖ β ‖) @ β statement expression meaning "@ β ‖ is clearly part of its significance "@ ‖ and ‖ β ‖ letter items, its true value:" @ ‖ (‖ β ‖) = 1 if and only if ‖ β ‖ ∈ "@" @ the significance of the noun phrase ‖ @ "and the verb phrase meaning ‖ β ‖ β work in the conditions described in the statement true value.

Thinking: Montague Grammar emphasized natural language and logic languages ​​in common, whether to give enough attention to different points? Especially the differences of the two languages ​​in the initial part of the system they pursue similar layers and the formation of expression and follow the principle of meaningful combinations. then you parted ways, the logic system and focus instead on the logic effective-proof, natural language systems still in syntactic formation mechanism in depth refinement. Secondly, the logic system logical discussion of the reliability and complete such element, natural language statement any similar nature? whether this be from the syntax and semantics of the corresponding angle to discussion a similar soundness and completeness of that nature? Furthermore, the Chinese statement statement is different from the English system, in addition to syntactic morphological and syntactic differences generated semantic interpretation whether the characteristics? most essential difference where?
Generalized quantifier theory GQT (Generalized Quantifier Theory) the significance of the quantitative study of natural language expressions and their semantic similarities. Generalized quantifier theory as a form of semantic theory proposed in 1980s, but its ideological roots can be traced back to 20 the beginning of the century: the founder of modern logic Frege the first to propose a basic idea of ​​generalized quantifier next 50-60 years the Mostowski and Lindstrm, work to deepen the understanding of generalized quantifier, since the 1970s and 1980s, Montague and Barwise quantifier concept extended to the field of natural language, the generalized quantifier theory become important category in the field of formal semantics, Keenan and after that people continue to focus on natural language quantify expression various quantify different types of quantifiers corresponding to natural language expression: the type <1> quantifiers corresponding to natural language noun phrase "every man", "somedog" and logic systems, "" and "", and <1,1> type even << 1,1>, 1> type quantifiers corresponding to natural language qualifier "non-continuous expression of all", the "natural language" five more ... than ... "and the like, and logic systems no corresponding counterpart. visible, GQT depth development is increasingly dependent on the field of natural language [2].

GQT strictly speaking not about the framework of the theory of natural language, it is only concerned about the quantitative significance of natural language showed the one hand, it is the promotion of classical logic quantifiers concept in the field of natural language, on the other hand thought is Montague Grammar natural language extending quantization expression studies. GQT main content: (1) explained based on set theory based on quantitative expression semantics. if natural language quantization sentence is a split into two, then one of the noun phrase is <1> types of quantifiers. quantifier is the letter items, its theory element is the the sentence verb phrase represented a collection of natural language quantified sentence be split into three, the qualifier is <1,1> type quantifiers such quantifiers binary letter entries, the first argument is limited to the collection of the word rhetoric noun represented, the second theory element is the verb phrase collection, (2) since quantify expression showed a collection of various levels of the relationship between, GQT of set theory perspective to discuss quantifier various mathematical properties, such as the reside, quantity, and scalability. GQT further concern quantitative expressions of natural language and its relationship of set theory counterpart that quantify the expression of natural language able to express whether a given collection of all the relationships involved, this is the so-called expressive (3) GQT research involving diverse patterns quantifiers, quantifiers superimposed composite with quantifiers enhance quantifiers such as definable problem, GQT the so-called non-standard quantitative expression concept extends the scope of its research, the expression of non-standard quantitative expression, including adverbs and conjunctions.

Examples Interpretation: English quantify "Every boy runs" one divides into three, limited to the words "Every" semantic ‖ every ‖ <1,1> type quantifier, the semantics of the term "boy" "boy" this quantifier The first argument, the verb phrase "runs" semantic "runs" is the the quantifier second argument. truth conditions of the semantics of the whole English sentence "Every boy runs ‖ this collection:" boy "with" run ‖ A collection of sequence belonging ‖ Every ‖ This collection sequence pairs {<X,Y> D [2]: XY} intuitive to see the English sentence "Every boy runs" is true if and only if the boy " The corresponding collection belonging to "runs" corresponds to a collection, that is, where a "boy" classes are the "runs" class.

Thinking problems: generalized quantifiers as a set-theoretic letter items, not only is the object of the abstract model theory studies, but also from the point of view of the higher-order logic, this direction has emerged in recent years. Secondly, the quantitative expression of the Chinese in addition to different from the syntactic form of English, the existence of a unique semantic definition its semantic nature? expression in general is based on set theory, the concept of quantifiers letter items with natural language quantify unequal, and European and American scholars English some comparison, whether it can be considered to extend this work to the field of Chinese quantify expression?
Discourse representation theory DRT (Discourse Representation Theory) specializes in the care of the relationship as well as the verb of the sentence between nouns and pronouns in time of Contact, dynamic method of the sentence sequence semantic analysis using a gradual increments. DRT past Montague Grammar expand the analysis to a single sentence of natural language sentence sequence, is a dynamic description of the form of semantic theory of natural language meaning its founder Kamp pointed out: "DRT from the Montague Grammar headed model theory of natural language semantics where development. "[3] 253 In addition, the the DRT also syntactic structure analysis of the tree with its semantic model, the addition of a semantic performance as a natural language called intermediate level of discourse representation structure DRS.

In short, DRT has two main points: First, the breakthrough in the form of semantic theory isolated the traditional analysis of a single sentence, the perspective is expanded to the level of the sentence sequence so as to grasp the relationship of nouns and pronouns take care analysis method innovation by static development to a dynamic sequence of English sentences "John walked He whistled" Montague Grammar only the first sentence analysis into the walk (j), the second sentence analysis whistle (x), the two results are independent of each other, do not rely on contact DRT put two sentences as an interdependent whole, the analysis of the first sentence to affect the analysis of the second sentence, the second sentence analysis based on the analysis of the first sentence as the previous sentence DRT's dynamic analysis:

See here, the second sentence of the analysis of the information obtained is add an analysis is based on the analysis of the first sentence of the growth and expansion of the results. Brought forward sentences affect the subsequent sentence, and subsequent sentence can not be separated from the sentence brought before this language express the sense of language requirements embodied here in DRT. analysis also showed a dynamic process: the second shown DRS covering first shown DRS the second DRS directly based on the first DRS growth out by This replaced the first DRS like human growth, adult form substituted profile of juvenile form dynamic mean growth, meaning that the profile addition, the second DRS formula "x = y" finishing touch to reveal the series is designed for the sentences The name and Pronoun relationship.

The thinking problems: DRT same dynamic Montague grammar DMG What kind of relationship? Both the pros and cons of each? DMG is a dynamic form of semantic theory based on the combination of the principles of the DRT flexible attitude taken by the principle of combination explains what? with dynamic semantic theory, DRT with the same strain dynamic predicate logic the DPL and ordinary dynamic logic why there are minor differences in the semantic interpretation DRT semantic model established embedded confirmation items have any value? how technically processing verb phrase even more types of expression anaphora phenomenon? ability to back from the Chinese and English, refers to the phenomenon in syntactically different, digging a deep semantic root cause?
Situation Semantics SS (Situation Semantics) psychological characteristics of propositional attitude sentence statement contextual factors to give full attention to the use of information entry way to describe the semantics of natural language pragmatics phenomena. Situation semantics is generated in Europe and the United States in the 1980s a new theory, the founder of the the American logician Barwise and linguistic philosophers Perry, whose masterpiece is a book published in 1983 Situationand Attitudes. had a internationally famous the publication << linguistics and philosophy >> (Linguistics & Philosophy) album, a collection of many philosophers, logicians, linguists, psychologists and computer artificial intelligence scientists discuss the article about the situation semantics. better University in the United States and Europe also opened the the Situation Semantics specialized courses. Situation Semantics semantic theory throughout the "information" thinking about mathematics and logic, is a unique form of semantic theories, since the 80s of the last century in the field of logic, linguistics, computer science, and philosophy of language appears from the far-reaching impact.

SS, the most notable feature is its "alternative" nature of the a Tarski logic semantics traditional challenges. SS in order to effectively solve the psychological cognitive problems involved in the so-called propositional attitude sentence, advocate epitaxy as a sentence sentences describe scenario rather than the true value. seems logically, the the propositional attitudes word as operator, its argument is a cognitive subject and object clause. the entire propositional attitude sentence true, however, is not dependent clause as part of its object true false, where the combined principle VALID SS that propositional attitude sentence is satisfied, and does not depend on the true value of the object clause, the scenario description based on the object clause and whether the subject has a specific cognitive relationship Secondly, SS is a kinds of semantic theory of the nature of the world body. situational information entry information entry: a relationship, a number of the different role of the individual as well as the time and space units arrangement of elements. relations exist independently, not by the individual collection or individual sequence pairs to define. again, SS that the meaning of a sentence is a statement of the sentence context and the context sentence an association statement situational factors involving the speaker and hearer. SS the Context as a special context the tentacles of SS has been extended to the areas of pragmatics., SS context abstract concept of situational type, which helps to characterize the various conditions of the natural language sentence. situational type the related ideas lead to the concept of information flow, and accordingly produce information flow logic [4]. posted in the free papers Download Center

about the SS alternative nature, its understanding of the need sentence parsing:

a. Bob believes that New York is between Boston and Washington.

b. Bob believes that 1 +2 = 3.

Formula as well as the significance of the object clause of propositional attitude sentence a and b is true, but if Bob only know the geographical knowledge in the United States do not understand arithmetic calculation, a true and b is false. Analysis proceeds in accordance with the usual logic of a and b Functional Principles combinations, this is impossible, so can not adopt the traditional forms semantics processing a and b in the SS, the scenario described in the a and b to the respective object clause is not the same. a The object clauses describe the context of: e '< >, And b the object of clause scenario described as: e '< >, A set up and b does not hold entirely possible.

Ponder the question: With the in-depth study of the formal semantics of natural language, feel that the original concept of logic is indeed a considerable gap between natural language and its own unique perspective of the SS reasonable combination of the principles of natural language failure of the field of cognitive psychology, but its role in other areas of the natural language semantic analysis can not be denied, the combination of a certain scope of principles?, or whether it can be considered to make new understanding of fusion SS concept of combination principle? SS with today's cognitive logic What is the relationship? such as information flow logic based on situational type of thinking whether characterize belief change can make a difference?
Type logical grammar (Type Logical Grammar) also known as the scope of the type of logic. Theoretical principles as thorough sense combination, the type of logic syntax not only the abstract study of the laws governing the operation of the natural language syntactic category, but also through the introduction of a simple type λ-calculus tools to show the syntax and semantics of the parallel interface (interface). stage of development of the type of logical syntax is divided into: classical areas of grammar, Lambek syntactic calculus "[5], type - logical semantics and syntax of logic. classical Categorial Grammar Language string of symbols by small to large drill progressively generate adjoining converted into areas computing Lambek syntactic calculus-based areas, constitute a form of expression law, type theorem areas - logical semantics through syntactic category (term parallel inference, to show the corresponding natural language syntax and semantics, syntax, logic features the the functor category slash operator adjoining areas as binary modal operator, which opened up the field of type logical grammar a multi-modal system direction.

Type logical grammar points: (1) to establish a clearer concept of syntactic and semantic. Syntax is relatively simple Lambek calculus explicitly natural language syntax starting point. Formal semantics semantics is the core type - logical semantics into the semantic field of science is bound, and give direct pairing of syntactic and semantic rules while providing the syntactic category deduce portfolio based and semantic lexical items under the influence of ideas in the modal logic, the syntax of the scope of the system as a multi-modal logic have come into being, the more profound the abstract syntax logic specific laws governing the operation of the syntactic category. (2) the type of logic syntax with particular emphasis on the spirit of deduction and calculation that the syntax is logical, cognition is calculated, analysis is the interpretation of the type of logic syntax not only the continuation of the tradition of of Montague Grammar constructor natural language statement system, extensively absorb the Gentzen Order of the calculus and frame semantics modern logic tools.

Type the logic syntax of natural language analysis is a syntactic category and semantic lexical items parallel inference, we give the sentence "John loves Mary" analysis:

The deduction the top of the chart is "three times the use of the same axiom", corresponding to three entries "John", "loves" and "Mary" is the type of logic semantics assigned the lowest end of the deduction graph shows that the three entry type logic semantics type logic semantics adjoining introduced the sentence, that sentence corresponding logical formula and the true value category. sentences constitute a part of the type logic semantics determines the type logic semantics of the whole sentence.

Thinking: Type logical syntax syntactic calculus formulation: axioms expressed in the formulation of the Order of the presentation and natural deduction, Gentzen. Axiom statements help metalogical discussion Gentzen Order of the representation is conducive to the solution of the decision problem, while the natural deduction expressed stressed that contact with the natural language. addition, each should have a more in-depth discussion on the pros and cons. Secondly, in the scope of the type of multi-modal system of logic, the logical deduction axioms and structure of public correspond exactly to set whether the the transformational grammar schools of the so-called syntactic generated and the concept of syntactic transformation?
In short, the formal semantics theories have in common: the treatment of natural language, establish the given syntactic rules based on strict correspondence with the syntactic semantic computing rules, which follow the logical semantics of the logic of the combination of the principles of (1) the requirements under the strong influence of natural After all, language semantic analysis suitable computing requirements, easy-to-computer information processing.

Second, language study prompted the logical tools of innovation

Clearly, the formal semantics Over the years various theories is logical impact of a product of natural language research, the role of the logic of natural language research is no doubt, this is one aspect of things John template the Intermot wrote: Frege logic language and the relationship between the ratio of the natural language for the relationship between the microscope and the human eye before a tool is more accurate, but as viewed in a very limited range, while the latter may be less accurate, but its functions, in principle, a very wide range of applications With the deepening of the formal semantics of natural language rich comparative paucity of out logic tools, so the interaction between the two in the other direction, the richness of natural language to promote logical changes that gave birth to the logic tools innovation from the 1970s to the present, the natural language research, in turn, act on logic, and give a positive impact on the development of logic.

The so-called natural language propositional attitude sentences containing words such as "know", "believe" the cognitive verb of the sentence, the verb should not be interpreted as its object clause of the true value of On the formal semantics seems yuan the truth function, that can not be decided by "Morningstar is the true value of the Hesperus' Mr Tan believes that the morning star is the evening star" true value. extension of first-order logic can not reveal the significance of the propositional attitude sentence, which prompted logic tools of innovation, to Intensional Logic by epitaxial logic development, to upgrade to a higher order logic by first-order logic, so have the connotation type on a logical tool IL in Montague students and colleagues Gallin seems the connotation type theory also development the another simple logic theory - two-body theory of types (Two-Sorted Type Theory) Gallin noticed connotations type language is not that possible worlds variables so you can not syntactic computing world connotation entity may directly thus λ-conversion operation is restricted variables can not be converted in the connotation Operators in the Scope of lexical items that you want to convert is a certain kind of semantic value is a constant value of the letter entry lexical items to overcome the connotation type the limitations of the theory, Gallin create a type of two-body theory. syntactic front desk to type s of the world may be counted as basic types, syntax language to express what may be the world's lexical items, the concept of possible world semantics behind the scenes "Go" . "What is more, around the propositional attitude sentence reflects the cognitive and psychological characteristics, Situation Semantics challenge traditional logical semantic concepts, and then gave birth to the so-called" situational body logic "[6] of the birth.

Natural language both syntactic structure, semantic level, two levels coexist. Example, say Sentence "U.S. subprime mortgage crisis is spreading, and we both know the syntactic structure of the sentence, the proper name U.S. subprime mortgage crisis "to act as the NP," is spreading VP. we can also understand the semantics of the sentence: NP VP nature referred to within the meaning of the individual has to say, the actual performance of natural language syntax and semantics of the coexistence based type logical grammar formal semantics to expand the analysis of natural language syntax and semantics of the parallel inference (interface), the presentation of this syntactic and semantic parallel logic theory soon affect British logician Gabbay proposed spiked deductive theory LDS to logical proof every step formula with a label (label), formula and the mark parallel inference "[7]. LDS performance in the field of coherent logic rules and its illustration:

Logical proof of the implementation of parallel inference at least three benefits: (1) can distinguish between logical proof object language's features with meta-logical characteristics. Formulas run the law of object language features, corresponding mark of running law of nature belong to the scope of yuan logic of a description of this meta-logic has a stronger explanatory power, (2) from the point of view of the marked run law illustrate the different features of the different logical proof system, (3) can be seen from the conclusions of logical proof mark it depends What is the assumption, that is, What is the source of the conclusions.

In the field of natural language, the sentence is not the smallest units of language, the sequence of a sentence by sentence or sentence group chapter is larger linguistic units. Sentence sequence has a variety of links between sentences, which pronouns nouns take care of an important contact. objects involved in the noun or noun phrase in the first sentence in the subsequent sentences with pronouns to refer to, this is the so-called natural language anaphora phenomenon. formal semantics of discourse representation theory DRT pay special attention to this class of phenomena, and to characterize the dynamic method with the accumulation of incremental information between sentences, nouns and pronouns contact. DRT set up a so-called DRS level performance natural language semantics, the various levels of the DRS is constructed gradually from outside to inside the most within the meaning of discourse in the outer layer of DRS can pass inner DRS, the corresponding DRS semantic interpretation involved embedding confirmation letter term g can be expanded into more discourse referred mapped to the model theory domain to g '. processing inspired new ideas to logical assignment: the semantic value of the formula to determine the set of assignment letter term order <g,g'> This is dynamic predicate logic of Groenendijk and Stokhof DPL practice.

DPL syntactically with classical logic makes no difference, the difference is mainly reflected in the semantic interpretation of section DPL semantic model the M = <D,F> performance: D is a non-empty set of individuals, F is the individual regular items and the interpretation of the predicate letter items, g individual variable assignment letter items, more classic logic formula semantic value on this basis, DPL understand different: interpretations issued items mapped to the formula G × G (G is the assignment letter The collection of items g) up semantic value is assigned to the formula that constitute the set of assignment letter entry sequence, expressed as follows:

In DPL, a formula reference model M and assignment letter item g true if and only if g assigned as input conditions, the formula also assign an output h (Groenendijk & Stokhof, 1991). Above definition shows: The only conjunctive formula and quantified semantic assignment, assign input and output assignments do not have to be identical. DRS discourse within the meaning of the "may be reached between the extension of thinking of the same strain.

The actual state of the natural language logic characteristics are intertwined, such as tense verb or a sentence of time characteristics (tense), body (aspect) and phase (phase). Posture and phase in the logic of the usual time does not correspond to the concept. especially in Chinese, the verb phrase "written" contains both the contents of the past tense, and is also involved in the completion of the body and results phase factors. formal semantics using a hybrid logic language that both the traditional time logic operator, such as operator in the past The P operator F and the future, there body operator Pros and completed body operator Perf whereby Galton founded the temporal logic axioms system, the system in addition to the traditional time logic axioms specialized body of the axioms of arithmetic. different time characteristics in natural language is intertwined the mixed logic language interpreted state and posture of the two types of operator The two types of model will have some connection to this is the basic idea of ​​Gabbay interleaving logic (Fibring Logic) [8].

Interwoven logic to establish on the basis of the so-called interleaving semantics (Fibring Semantics) or intertwined model (Fibring Model). Gabbay through modal logic formula to elaborate intertwined semantics of the concept: make two modal language, its atomic propositions a collection of

Logical and natural language research is a relationship of interaction logic applied to the field of natural language, formal semantics, formal semantics, the characteristics of natural language, in turn, promote the innovation of the logical method should be said , the moment, after all, no more than the level of mathematical logic, can achieve in the future in order to answer the problem. Nevertheless, this mining theme from the richness of natural language logic to broaden the study of logic in the form of natural language and even change the scope of the logical development of ideas trend is cause for concern.

At this point, the situation semantics is the exception, DRT has a special performance posted in the free papers Download Center

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