1. a. Briefly explain the following terms using appropriate examples: (6 marks) i. Entailments ii. Reference (6 marks) iii. Homonym (6 marks) iv. Speech Act (6 marks) b.How is Frege's distinction between sense (Sinn)and denotation (Bedeutung) supposed to explain how an identity sentence can be both true and informative (non-trivial)? (6 Marks) 2. a. What is a function? Explain with reference to the types of function. (5 Marks) b. Give the analysis of the following sentences using tree diagrams and lambda Calculi. (5 Marks) (i.) Mary cried. (5 Marks) (ii.) Andrew dried her tears (iii.) Andrew gave Mary some flowers. (5 Marks) 3. a. Translate the following into propositional logic , analyzing in as much detail as possible, and giving a key for each one. Explain any problems that you have encountered in the process. (i.) Erica is going to the party unless it rains. (4 Marks) (ii.) Frank wants a bicycle and a game for Christmas and he will get both. (4 Marks) b. Define the following lexical semantic relations, giving an example of each: (i.) gradable antonyms (4 Marks) (ii.)hyponyms (4 Marks) (iii.) hypernyms (5 Marks)

1. a. i. Entailments: Entailments refer to the logical consequences or implications of a statement or proposition. For example, if the statement "All birds have feathers" is true, then the entailment "All penguins have feathers" is also true, as penguins are a type of bird.<br /><br />ii. Reference: Reference refers to the relationship between a word or phrase and the object or concept it represents in the real world. For example, the word "dog" refers to a specific type of animal.<br /><br />iii. Homonym: Homonyms are words that have the same spelling and pronunciation but different meanings. For example, "bank" can refer to a financial institution or the side of a river.<br /><br />iv. Speech Act: A speech act is an action performed by a speaker through their utterance, which has an effect on the listener. For example, when someone says "I promise to help you," they are performing the speech act of promising.<br /><br />b. Frege's distinction between sense (Sinn) and denotation (Bedeutung) explains how an identity sentence can be both true and informative by distinguishing between the meaning (sense) and the reference (denotation) of a term. For example, in the sentence "The morning star is the evening star," the sense of "morning star" and "evening star" refers to the same object, but their denotations are different. This allows the sentence to be informative while still being true.<br /><br />2. a. A function is a mathematical relation between a set of inputs and a set of outputs, where each input is associated with exactly one output. There are different types of functions, such as linear functions, quadratic functions, and exponential functions.<br /><br />b. i. Mary cried.<br />Tree diagram:<br />```<br />S<br />├── NP<br />│ └── N<br />├── VP<br />│ └── V<br />└── AdvP<br /> └── Adv<br />```<br />Lambda Calculi:<br />```<br />λx.λy.λz.((x y) (z y) (x y z))<br />```<br />ii. Andrew dried her tears.<br />Tree diagram:<br />```<br />S<br />├── NP<br />│ └── N<br />├── VP<br />│ ├── V<br />│ └── NP<br />│ └── N<br />└── AdvP<br /> └── Adv<br />```<br />Lambda Calculi:<br />```<br />λx.λy.λz.((x y) (z y) (x y z))<br />```<br />iii. Andrew gave Mary some flowers.<br />Tree diagram:<br />```<br />S<br />├── NP<br />│ └── N<br />├── VP<br />│ ├── V<br />│ └── NP<br />│ └── N<br />└── AdvP<br /> └── Adv<br />```<br />Lambda Calculi:<br />```<br />λx.λy.λz.((x y) (z y) (x y z))<br />```<br />3. a. i. Erica is going to the party unless it rains.<br />Propositional logic: (E ∧ ¬R) → P<br />Key: E = Erica is going to the party, R = it rains, P = Erica is going to the party<br />ii. Frank wants a bicycle and a game for Christmas and he will get both.<br />Propositional logic: (W ∧ G) → B<br />Key: W = Frank wants a bicycle, G = Frank wants a game, B = Frank will get both<br /><br />b. i. Gradable antonyms: Gradable antonyms are words that can be ordered in terms of degree or intensity. For example, "hot" and "cold" are gradable antonyms, as "hot" can be seen as a higher degree of temperature than "cold."<br /><br />ii. Hyponyms: Hyponyms are words that are more specific than a given word or concept. For example, "car" is a hypernym of "sedan," as a sedan is a specific type of car.<br /><br />iii. Hypernyms: Hypernyms are words that are more general than a given word or concept. For example, "animal" is a hypernym of "dog," as a dog is a specific type of animal.