1 Introduction. 2 Set-Theory Formalisms. Formal Semantics -W2: Limitations of a Set-Theoretic Model SWU LI713 Meagan Louie August 2015

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1 Formal Semantics -W2: Limitations of a Set-Theoretic Model SWU LI713 Meagan Louie August Introduction Recall from last week: The Semantic System 1. The Model/Ontology 2. Lexical Entries 3. Compositional Rules Elements in the ontology model real-world (or not) referents. These elements are organized into domains: 1. D e, Individuals = {a, b, c, d,...} 2. D t, Truth Values = {1, 0} Lexical Entries are how we use words to refer to elements in the ontology, or mathematical objects based on those elements (eg., sets, pairs, sets of pairs, sets of sets, etc.) 1. Names refer directly to individuals in the ontology, eg., Harry Potter= a, Draco Malfoy = d 2. Common Nouns refer to sets of individuals, eg., girl = {b, f, g}, animal={e, h} Compositional Rules tell us how to interpret syntactic rules of combination: 1. Predication: S DP VP. S =1 iff, DP VP 2. Modification: NP Adj N. NP = Adj N 3. Select: DP D NP. DP = the singleton/maximal member of NP 4. Saturate: VP V DP. VP = {x: x, DP V } These guys,. are interpretation brackets, so read DP as the interpretation of DP - i.e., some individual in the ontology 2 Set-Theory Formalisms The toy model I set up last week is based on set theory In general, it s useful to be comfortable with set-theoretic formalisms 2.1 Set Theory Basics (Partee et al. (1990), Chapter 1) A set is an abstract collection of distinct objects;. these objects are called members of that set A, B, C...for sets, a, b, c...for for members of sets 3. Adjectives refer to sets of individuals, eg., red-haired = {c, g}, blond = {d, f} 4. Verbs refer to intransitive verbs: sets of individuals eg., flies = {h,e,...} transitive verbs: sets of ordered pairs of individuals eg., love = { g,a, i,e, a,q, c,q, d, q, g, q... } set membership: x Y Sets are also objects, so they can be members of other sets Three ways to specify a set: 1. List: If the set is finite, you can just list all of its members. eg., love = { g,a, i,e, a,q, c,q, d, q, g, q } (x Y)

2 2. Property/Predicate: {x: x book & x red } 1 Why do we need Set Theory? eg., love ={ x,y : loves(x,y) } 3. Recursive Rules: (a) 4 E (b) If x E, then x +2 E (c) Nothing else belongs to E Sets are defined by their members. sets with identical members are the same set (State finite members) (Give recursive rules) Why do we need all of this math? Why can t we just describe what words mean using plain-speaking non-technical terminology? The goal of natural-language semantics is to account for. what words mean, and. how their meanings combine into new meanings So if you use natural language words to describe meaning... cardinality: A subset: A B proper subset: A B Intersection: A B Union: A B ( ) ( ) Difference: A - B Complement: A The empty set: (Note: A, for every A) Cartesian Product A B = { x, y : x A and y B}...you re assuming that someone can explain what those natural language words mean!...and if that person uses natural language to explain what those natural language words mean...etc. Mathematical languages defined independently of natural languages i.e., a way to objectively describe meaning and avoid a methodological infinite regress 2.2 Set-Theoretic Homework Exercise from Partee et al. (1990) Given the following sets: A = {a, b, c, 2, 3, 4} B = {a,b} C = {c,2} D = {b,c} E = {a, b, {c}} F = G = {{a,b}, {c,2}} Predicate Logic Other logical systems are commonly used to represent various kinds of meaning in natural language. Eg., propositional logic and predicate logic. The meaning of lexical entries will often be represented using notation from these formal systems: (1) Are the following true or false? a. c A d. {c} E b. c F e. {c} C c. c E f. B A g. {D} G h. {{c}} E i. F E. love(x,y) Predicate-Logic Notation Although these systems are really useful, I m not focusing them in this course because (i) they are tools that linguistics uses, not properly linguistics, and (more importantly 1 YOu can also use the notation, in place of the : notation, eg., {x x book & x red }. (ii) unlike formal semantics, there are a lot of resources to learn available online, eg.,

3 3 Towards A More Complex Model The goal of last week s toy model was to (hopefully) convince you 1. Truth-conditions are a way to formalize what (at least one aspect of) NL meaning is, and 2. A semantic system consisting of an ontology, lexical entries and compositional rules can model these truth-conditions in a compositional way Treating slowly, with a knife, in the bathroom, at midnight as arguments is problematic from a learnability perspective! These phrases seem to behave like modifiers with respect to entailment patterns: Definition: Entailment A sentence S 1 entails a sentence S 2 ( S 1 = S 2 ),. iff in every context in which S 1 is true,.. S 2 is also true. But the simple set-theoretic model is quite limited in terms of the range of meanings it can express... So far we can model situations about individuals and the relations between them But meaning in language is more nuanced and complex than this! In order to accommodate the meaning of tense, aspect, modal marking, etc, we need a more nuanced and complex system We ll be Adding elements/domains to our ontology Using functions as opposed to sets to formalize lexical entries. (But we ll have fewer compositional rules) 3.1 Adding to our Ontology Lexical Entries so far:. eat = { x, y : eats(x,y)} The individuals, x, y, are the verb s individual arguments Davidson (1967): Famously argued that verbs must also refer to an event argument, using the sentence (2): (2) Jones buttered the toast slowly with a knife in the bathroom at midnight. (3) a. It s a [big, red, sparkly balloon] NP b. {x: balloon(x) & big(x) & red(x) & sparkly(x) }. = {x: balloon(x)} {x: big(x)} {x: red(x)} {x: sparkly(x)} (4) a. It s a [red, sparkly balloon] NP b. {x: balloon(x) & red(x) & sparkly(x) } (3a) = (4a) - as expected given our intersection rule! (5) a. Jones buttered the toast slowly with a knife in the bathroom b. Jones buttered the toast slowly with a knife c. Jones buttered the toast slowly d. Jones buttered the toast Observation: (5a) = (5b) = (5c) = (5d) Q:...but if they are modifiers, what are they semantically modifying?. Not the agent (butter-er) or theme (butter-ee)! Davidson: They re modifying a buttering event argument. Idea: A verb s lexical entry would then look something like this:

4 (6) butter = { e, x, y : butter(e,x,y) }. (Read butter(e,x,y) as e is an event of x buttering y ) 3.2 Modifying Lexical Entries Problem: It s not clear how to incorporate something like aspect into the set-theoretic model we ve set up 1. D e, Individuals = {a, b, c, d,...} type e (9) VP 2 2. D t, Truth Values = {1, 0} type t 3. E, Events = {e 1, e 2, e 3, e 4,... } type l This morning =t prog VP 1 VP = {e: make(e, m, b)} Events can be used to capture the meaning of aspects: (7) Context: I noticed I was running late this morning and left my breakfast on the counter half-made in order to get to campus on time NP Meagan V make breakfast a. This morning, I made simple breakfast False! b. This morning, I was making prog breakfast True! (8) a. Progressive = the runtime of e overlaps with t 2 b. Simple/Perfective = the runtime of e is contained within t. (Let t = This morning) 1. D e, Individuals = {a, b, c, d,...} type e 2. D t, Truth Values = {1, 0} type t 3. E, Events = {e 1, e 2, e 3, e 4,... } type l 4. I, Times = {t 1, t 2, t 3, t 4,... } type i 2 There is A LOT more to the meaning of progressive/imperfective morphology crosslinguistically than this analysis, but this is a good start. This is based on Kratzer (1998) s adaptation of Klein (1994) s IMPF vs PFV. Attempt: Let prog = {e: e t}. The set of events that overlap with some time, t. And this combines via modification with the VP But this leaves us with VP 1 denoting a set of events...how can this combine with the reference time ( in this case, with t, this morning)? We need VP 1 to combine with the t that this morning reers to, and this has to somehow yield a truth-value... This would require some new kind of compositional rule... But we don t want to create new kinds of compositional rules for each new kind phenomenon we want to incorporate into our system! 3.3 The Solution: Lexical Entries as Functions We already know that language learners need to memorize lexical entries The lexicon is where the main meaning components of our system should be stored! (Not as a bunch of different compositional rules that need to be memorized) By formalizing lexical entries as functions, we can do this and avoid a proliferation of compositional rules

5 4 Primer on Functions 4.1 What s a Function? Recall: Sets are unordered collections pairs and other n-tuples are ordered This is why they are useful for representing relations like eat:. eat(x,y) Order matters, because if the relation eat holds between the pair Harry, chocolate frog......this does not entail that the relation eat holds between the pair chocolate frog, Harry! Definition: Relation A relation R is a mapping from a domain A to a range B c a b d e g f A linguistic phrase that formerly denoted a set. now denotes the characteristic function of that set. 1. Names refer directly to individuals in the ontology eg., Harry Potter= a, Draco Malfoy = d 2. Common Nouns refer to e, t functions from sets of individuals to truth-values 3. Adjectives refer to e, t functions from sets of individuals to truth-values 4. Intransitive verbs refer to e, t functions from sets of individuals to truth-values If we follow this pattern: Transitive verbs should refer to functions from sets of ordered pairs of individuals to truth-values...but this doesn t reflect syntax - individual arguments aren t merged in pairs VP, t A B DP V, e, t Definition: Function subject, e V DP A relation R from A to B is a function iff it both:. 1. Each element in A is paired with just one element in Y. 2. The domain of R is equal to A. 4.2 Lexical Entries as Functions The Characteristic Function of Sets Each set, X, has a characteristic function This function takes an object, a. and yields true (1) if a X object,e We want a transitive verb to take a single argument And we want the resulting V to yield something like a intransitive verb (i.e., something that maps from individuals to truth-values) 5. Transitive verbs refer to e, e, t functions from sets of individuals... to functions from sets of individuals to truth-values

6 And what about determiners? Before: They apply to a set of individuals (eg., an NP), and select an individual from that set Now: They apply to a function, f, from individuals to truth-values (eg., an NP), and yield an individual that satisfies f. 6. Determiners refer to e, t, e functions from sets of individuals to individuals Next Week: Alonzo Church s Typed Lambda Calculus! Function Application, Modification Optional Reading: Partee et al. (1990): pp Heim & Kratzer (1998): Ch. 2 Lexical entries formalized this way will allow us to collapse the compositional rules predication, select and saturate,. into one compositional rule: function application We ll formalize this next week What sort of Notation should we use for functions? Pair-Notation? We can represent a function from individuals to truth-values as a set of ordered pairs: References Davidson, Donald The logical form of action sentences Heim, Irene & Angelika Kratzer Semantics in generative grammar Blackwell Textbooks in Linguistics. Blackwell Oxford. Klein, Wolfgang Time in language. Psychology Press. Kratzer, Angelika Scope or Pseudoscope? Are there Wide-scope Indefinites? In Susan Rothstein (ed.), Events and grammar, Kluwer Academic Publishers. Partee, Barbara, Alice Ter Meulen & Robert Wall Mathematical Methods in Linguistics, vol. 30 Studies in Linguistics and Philosophy. Kluwer Academic Publishers.. hungry = { a, 0, b, 0, c, 1, d, 0, e, 1, f, 1,...} But this is very unwieldy! It also fails to represent the fact that there is a non-arbitrary relationship between the argument and value...