Semantics II/Formal Semantics Spring 2009 Take-home test #1: Answers I. Since students applied rules/applications in different ways, I will not suggest my own caslculations. II. 1. ‘met’ is a 1st order 2-place predicate whose type is <e, <e,t>> 2. ‘(is a) color’ is a 2nd order 1-place predicate, whose type is <<e,t>, t> 3. ‘John’ is an individual, whose type is <e> 4. ‘a small elephant’ is a 1st order 1-place predicate, whose type is <e,t> 5. ‘a city in Florida’ is a 1st order 1-place predicate, whose type is <e,t> 6. ‘Lily does not live in Gainesville’ is a sentence, i.e. a truth-value, whose type is <t> 7. ‘John lives in Gainesville but Mary lives in Orlando’ is a sentence, i.e. a truth-value, whose type is <t> 8. ‘It is true that’ is a sentential operator, whose type is <t,t> 9. ‘fast’ is a predicate modifier whose type is <<e,t>, <e,t>> 10. ‘the’ is a determiner, whose type is <<e,t>,e> (if we adopt H & K’s theory) III. 1. α(β) ∈ D<e,t> δ(β) ∈ D<t> 2. β is a (saturated) truth-value, rather than function, so it cannot apply to anything. IV. 1. F 2. T 3. T 4. F 5. F V. 1. F 2. T 3. T 4. T 5. F VI. Modification is a type of syntactic construction in which a head is accompanied by an element typically not required by it (syntactically speaking). The accompanying element is a modifier. The syntactic category of the modifier can be an AP, a PP or a relative clause. Semantically speaking, the modifier adds some information, restricting the set of the head. Our textbook (H & K) discusses four kinds of modifiers, three of which are the following. Intersective modifiers denote sets that may intersect with the set of the head noun. E.g., the adjective Floridian denotes the set of Floridian things; when it intersects with the noun farmer it creates a new set, which consists of members that are both Floridian and farmers. Thus, the NP (a) Floridian farmer denotes farmers that are Floridian. Similarly, the PP in Florida denotes the set of all individuals that are in Florida. When it intersects with the set of farmer we get a set of all farmers in Florida. Thus, the NP (a) farmer in Florida denotes farmers in Florida, i.e. a set of individuals who are both farmers and in Florida. Subsective modifiers can only apply to sets, subsecting them into sub-sets. I.e. they do not form a set of themselves. Most modifiers in language seem to be subsective: large, wide, narrow, small, short, good (usually they have to do with size or degree). It does not make sense, e.g. to talk about all large things and then intersect that set with the set of cities. Doing that would mean to have large cities, large countries, large lakes, and large classes in the same set. We cannot define large by itself but only with respect to some set. Thus, large city is a city whose size is beyond the size of an average city, which is probably much smaller than a large country. Context dependent modifiers are interpreted with respect to a context. The example given in our text book is of an elephant, call it Jumbo, which may be big or even bigger than an average elephant, but when in an imaginative scenario where it has to fight with an army of monsters like King Kong, we may say something like “Jumbo doesn’t have a chance; he’s only a small elephant”. In this case, “small” is not used as a subsective modifier. VII. Shönfinkelization, or currying, is a technique developed by M. Shönfinkel, adopted by the logician H. B. Curry, which is a reduction of n-place functions to one-place functions. Examples: The predicate LOVE takes two arguments to be saturated. There are two ways to apply Shönfinkelization and reduce LOVE to a one-place predicate. The way we adopted is in accord with its syntax. We apply the verb to the internal argument (expressed by the NP object) and yield the function LOVE-x, which is a one-place function. The predicate GIVE is a three-place predicate. We reduce it in two steps. First, we apply it to one of the internal arguments (the direct object) and yield GIVE-x, which is a two- place function. Then we apply GIVE-x to the other internal argument and yield GIVE-x- (to)y, which is now a one-place function. .