Logical Semantics - Assignment Example

Semantics Question Truth seeks to find logical forms of ments. If an argument or a sentence is in logical form then it is valid when all arguments of the same logical form are valid and true respectively. It is important to stress that logical form is not the same as grammatical form (Allwood et al., 19). If the preposition of an argument is true the conclusion is supposed to be true as well (18). The worked examples in 1 (a) and (b) seek to calculate truth values. a) (P^Q) If P is true (T) and Q is false (F), and ^ is the logical conjunction ‘and’ is conditional for, if’ then, P Q → Q (P^Q) →Q T F → F (F) Then the truth value is false. b) (P→Q) ^ (P ^R) If P is true (T), Q is false (F) and R is true (T), and → is conditional for, if’ then, and ^ is the logical conjunction “and’ then: P Q (P→Q) T F (F) P R (P ^R) T T (T) Question 2 Key to symbols &= conjunction __> = conditional implication E= Existential quantifier A= universal quantifier a) John read any book Mary bought J(x)= John reads book M(x)=Mary buys book E(x)=[J(x)__> M(x)] b) Every man who is bald is thin B(x)= Man is bald T(x)= man is thin A(x)= [ B(x)&T(x)] c) Dyfed and Lillian love everyone D(x) = Dyfed love everyone L(x) = Lillian love everyone A(x) = [ D(x) & L (x)] d) Ninnian is loved by Myfanwy and Olwen M(x) = Myfanwy loves Ninnian O(x) = Olwen loves Ninnian E(x) = [ M(x) & O (x)] Question 3 Analysis construction a) Kim saw Fred The syntactic structure of the formula will be: Saw(kim, fred) Kim is put in the first argument in relation of seeing. The first argument is reserved for semantic representation of the noun phrase to be combined with verb phrase to form a sentence. When using lambda expression, the value of x can be used to show the missing variable. For instance ƛx.x.@kim is used to represent proper name of Kim in the noun phrase. Lambda calculus is a tool for controlling processes of making substitutions. When in the noun phrase, it can be uses as a function. Appropriate lambda expressions are applied in lexical levels. The verb saw, can be expressed using the expression ƛw.ƛz(w@ƛx.saw(z.x)) The figure shows that representation of left child node is applied to representation of right child node in every level of the tree. Most of the work is done before the combinations in constructing appropriate lambda expressions (Bekiroğlu 15). b) Peter is ill The formulae will be Peter is ill The analysis construction using Lambda in line with Partee (3) will be: TR(peter is) = λPEx[peter (x) → λP(λx)[p x]] λP(λx)[p x]]= means ‘is’ Ex is used because Peter is a person who is in existence and not a universal quantifier. TR( ill) = ill R(peter is ill) = λPEx[Peter(x) → λP(λx)[p x]]= Ex[Peter(x) → ill(x)] Question 4 Overview: Carlson A, Unified Analysis of the English Bare Plural The study deals with bare plural constructions. Bare plurals are Noun Phrases (NP) of English that do not show determiner before the head noun such as ‘ineffective arguments’. Noun phrases have been a conundrum for philosophers and grammarians because of the possibilities of many interpretations (413). Carlson mentions two distinct uses of NP i.e. , a) there is a split between generic and existential uses of NP and b) semantic plural of NP is determined by singular indefinite article. His thesis is that the uses of bare plural (øNP) are facets of syntactically and semantically unified phenomena, and the different interpretations can be attributed in a predictable manner (414). A base rule introducing articles is: Art ͢ [+-definite] The + definite is the, which occurs after singulars and plurals while – definite is the ‘a’ deleted before plurals by the transformational rule. Therefore, if ø serves as a plural counterpart of indefinite article a, then the two will share all the relevant semantic properties, except for those explainable by the presence or absence of plurality (415). For instance, consider the sentences 1aband 2ab in which E is existential quantifier, Sl singular and Pl, plural: (1) a. A dog chased Mavin down the street b. Dogs chased Mavin down the street (2) a. (ESl )(Dog (x) & x chased M. down the street) b. (EPl )(Dog (x) & x chased M. down the street) Indefinite plurals show ambiguity compared to their singular forms. Carlson identifies sentences derived from ø as opacity form while sentences containing indefinite singular as the transparency form (417). But some cases exist where the plural has narrower scope than the singular (421). For instance in sentences such as: (a) A dog was everywhere and dogs were everywhere (b) Max discovered a rabbit in his yard for two hours and Max discovered rabbits in his yard for two hours (c) It was a tomato that everyone ate and it was tomatoes that everyone ate The second sentence treats time adverbials as a universal quantification. Adverbials are strange with achievement verbs unless the subject or the direct object is a bare plural. The third sentence shows an example of differentiated scope arising from cleft sentences. The first part of the third sentence implies that each person ate the same tomato. Differentiated scope sets apart not only ø from a but also from other quantifiers and determinants. øi s the determiner or quantifier among the indefinites that fails to show ambiguities or opaque-transparent distinctions. The ø indefinite plural is thus not semantically parallel to the singular form a. To describe the anaphoric process, Carlson uses two sentences (425): (a) Kelly is seeking a unicorn (b) Kelly is seeking a unicorn, and Milllie is seeking it too. In the first sentence there is ambiguity between transparent and opaque readings. In the second sentence however, the ambiguity disappears even though the first sentence is in the second sentence. This arises because it is apparent that Kelly and Millie are seeking the same unicorn. Carlson concludes that the indefinite plural (ø) cannot be the plural of a in any semantically relevant way. The bare plural is shown to have opaque readings in opaque contexts and does not participate in quantifier scope ambiguities, but take the narrowest scope. Indefinite plural use of bare plural is not to be distinguished from generic uses. Indefinite plural can serve as an antecedent for generic use (a), and generic may serve as antecedent for existential use (b) as exemplified below: (a) My brother thinks snakes are nasty creatures but that hasn’t stopped me from keeping them as pets. (b) I’ve had snakes my whole life as pets my whole life, but my brother thinks that they are nasty creatures. The generic (universal) in the example serves as an antecedent for indefinite plural (an existential), and vice versa. It is unexplainable how this occurs if ø were at least two ways ambiguous. According to Carlson, the complicated factors that were presented by the described example call for a unified analysis. The case for pronominalization is poorly misunderstood as exemplified in the sentence, Max killed a few rabbits, but Hiram killed them in abundance. The pronoun them does not expect the reader to imagine that some of the rabbits killed by Max resurrected so that Hiram could kill them. However, Carlson identifies the strongest point of unified analysis of bare plural arises from the observation that contextual factors that result to generic and indefinite plural interpretation are independently motivated, and are needed elsewhere in grammar (435). For sentences to have natural reading, the noun phrases ought to refer to kinds/kind, for example, No reptiles are indigenous to Philippines. These too have existential or indefinite plural definitions, e.g., My dog has been known to attack leopards. Generic and indefinite plural are not to be differentiated syntactically or symmetrically. The concluding suggestion by Carlson is that bare plurals ought to be treated as denoting kinds or things. Similarity between bare plurals and proper names was also noted. The singular generic with “a, the, and ø” is interesting, but difficult, and it was not looked at by Carlson (455). Works Cited Allwood, Jens, Lars-Gunnar Andersson, and Osten Dahl, eds. Logic in linguistics. Cambridge University Press, 1977. Bekiroğlu. Natural Language Processing of Basic Arithmetic Operations. Computer Science, Roskilde University, Denmark May 2005 Carlson, Greg N. "A unified analysis of the English bare plural." Linguistics and philosophy 1.3 (1977): 413-457. Gruber, Thomas R. "A translation approach to portable ontology specifications."Knowledge acquisition 5.2 (1993): 199-220. Partee, B. Lambda abstraction, NP semantics, and a Fragment of English. Formal Semantics, Lecture 2. MGU, February 22, 2005 Read More