LIN3021 Formal Semantics Lecture 4

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LIN3021 Formal Semantics Lecture 4. Albert Gatt. In this lecture. Compositionality in Natural Langauge revisited: The role of types The type d lambda calculus. Part 1. Compositionality for Natural Language: the role of types and functions. Montague’s programme.
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LIN3021 Formal SemanticsLecture 4Albert GattIn this lecture
  • Compositionality in Natural Langauge revisited:
  • The role of types
  • The typedlambda calculus
  • Part 1Compositionality for Natural Language: the role of types and functionsMontague’s programme
  • As we saw last week, Montague attempted to define a programme where:
  • There is a precise set of syntactic rules for languages such as English
  • There is a corresponding set of semantic rules.
  • E.g. We can determine the truth value of a sentence by applying the semantic rules corresponding to the syntactic rules that gave rise to it.
  • Note the parallel to what we were doing for logic.
  • Last week, we did some of this informally. Let’s see how we can make these ideas more precise.
  • Predicates revisited
  • We said that simple predicates like sleeps can be viewed like this:
  • If we plug in something of the right kind, saturating the predicate, we get something that returns true or false (i.e. a proposition).
  • Predicates as functions
  • Given a model, the predicate can be thought of as a set:
  • E.g. [[sleep]]M,g= {paul, mary} could be a fragment of our model
  • (Technically, the set is the extension of the predicate)
  • It can also be thought of as a function which, given something of the right kind, returns:
  • TRUE if that thing is in the extension of the predicate,
  • FALSE otherwise.
  • It’s useful to think of this as a function:
  • from entities in U (the set of individuals in the model)
  • to truth values.
  • Predicates and types
  • What we use to saturate a predicate must be of the right kind.
  • Our consists in part of a set of individuals (paul, mary). Our language also has:
  • one-place predicates (sleep, eat, book...)
  • two-place predicates (read, love...)
  • ... (other stuff we might need)
  • So how do we specify that something is of the right kind?
  • Predicates and types
  • Recall one of our interpretation rules from last week:
  • If a node x has two daughters y and z, and [[y]] is an individual and [[z]] is a property, then saturate the meaning of z with the meaning of y and assign the resulting meaning to x;
  • This makes no direct reference to syntax, but it makes reference to the semantic type associated with (the meanings of) syntactic nodes.
  • Many contemporary semantic theories are type-driven in this sense.
  • Types remove the necessity of mixing syntactic categories with semantic ones.
  • Sentences/propositions and types
  • Last week we also suggested that propositions can be thought of as functions too:
  • Basically, a proposition denotes either:
  • The set of worlds in which it’s true; or
  • A function from worlds to truth values
  • Let’s restrict our attention to one world for now (our world, for example). This obviates the need to refer to several worlds. Viewed in this way, a proposition’s meaning is just a truth value.
  • So it’s a different sort of semantic object from a predicate.
  • Predicates are “incomplete”: they need to be saturated before they become propositions.
  • Propositions are “complete” in the sense that they denote truth values.
  • TPropositionFTypes
  • A type system is a set of categories which are semantically motivated.
  • For our purposes today, this has two uses:
  • It allows us to specify restrictions on the kinds of things that can combine in certain ways, thus avoiding semantically anomalous combinations (but without need to mix syntactic and semantic categories).
  • It allows us to distinguish between qualitatively different kinds of linguistic objects (such as predicates, names, and propositions), which denote different kinds of things.
  • What types do we need?
  • Suppose we distinguish between semantic objects depending on what sort of thing they denote:
  • Names (paul) denote entities
  • Predicates (sleep)are unsaturated propositions, functions from entities to truth values
  • Propositions (Paul sleeps)denote truth values (ignoring the complication of possible worlds, for now).
  • A simple type system
  • To ensure that predicates get the right type of objects to saturate them, we could start by assuming that:
  • Type eis the type of entities in our model – this is the type of things like Mary and Paul
  • Type t is the type of truth values (i.e. of the values TRUE/1 and FALSE/0) – this is the type of things which can be true or false, i.e. Propositions (Paul sleeps)
  • What about predicates, which are functions from things of type (1) to things of type (2)?
  • Let’s think of predicates as having a complex type <e,t>
  • This is just shorthand for “a function from entities to truth values”
  • In terms of our model
  • Our model consists of a domain U and an interpretation function. If we assume:
  • U is the domain of entities (type e)
  • T is the set of truth values (i.e. {0,1}; type t)
  • Then anything of type <e,t> is a function from things in U (of type e) to things in T (of type t)
  • U(entities)Type eT (truth values)Type tType <e,t>A non-linguistic example
  • In maths, any numerical operation can be thought of as a function.
  • Let f be a function such that f(x) = x2.
  • In other words, this function is just the process of taking a number x and squaring it.
  • f(2) = 22 = 4
  • Note that this is a function from numbers to numbers.
  • Suppose we call the set of numbers N, and say that anything in N is of type e.
  • Then, f is a function from things of type e to things of type e, that is, <e,e>
  • F(x) = x2Another non-linguistic example
  • We know that we can define sets in terms of their characteristic functions:
  • {x | x > 7} (the set of numbers greater than 7)
  • Can be thought of as a function:
  • The characteristic function of the set just returns true if an element is in the set, false otherwise.
  • Once again, we define U as the domain of numbers, and assign these the type e.
  • The above function takes something of type e(here, a number) and returns a truth value, which is of type t. So the type of the function is <e,t>
  • Defining the type system recursively
  • In general, we could summarise what we’ve done using a small set of recursive rules:
  • e is a type
  • t is a type
  • If a and b are types, then <a,b> (the function from things of type a to things of type b) is also a type.
  • Nothing else is a type
  • Note that (3) is recursive: given any two types (even complex ones), we can define a new complex type. We’ll see why this is useful soon.
  • Consequences
  • In putting together the name pauland the predicate sleeps, we get a proposition.
  • pauldenotes an entity, so it’s of type e
  • sleeps denotes a predicate, so it’s of type <e,t>
  • e[[paul]]: type e[[sleep]]: type <e,t>Putting something of type e into something of type <e,t> gives us back a truth value (of type t).The combination e + <e,t> is something that can be interpreted as a proposition, of type t.Extending this to n-place predicates
  • So what about predicates like kill?
  • Extensionally, this is a set of pairs (the killers and the killees) in our model.
  • What’s the type of this predicate?
  • (Think: what does this predicate need in order to become a proposition and return a truth value?)
  • kill(x,y)N-place predicates
  • Let’s take it piece by piece:
  • Mary killed John
  • We can think of kill as a function that returns a truth value given two entities of type e.
  • The syntax is helpful here:
  • We have:
  • An individual (the subject), combining with a predicate which itself needs to combine with another individual (kill + John)
  • SMarykillJohnN-place predicates
  • In order to be saturated, the transitive verb needs to:
  • Combine with an individual (type e).
  • This leaves an empty slot for a second individual.
  • Once it combines with the second individual, it becomes a proposition.
  • tkillMary<e,<e,t>>eJohneNB: note that we’re working from the inside out (starting with kill + John)Syntactic combination as function application
  • Recall one of our interpretation rules from last week:
  • If a node x has two daughters y and z, and [[y]] is an individual and [[z]] is a property, then saturate the meaning of z with the meaning of y and assign the resulting meaning to x;
  • We can rework this rule as something like the following:
  • If a node x has two daughters y and z and [[y]] is of type e and [[z]] is of type <e,t>, then:
  • [[x]] = the result of applying the function [[z]] to the argument [[y]]
  • [[x]] = [[z]]([[y]])
  • Thus:
  • [[Paul sleeps]] is the result of applying the function [[sleep]] to the argument [[Paul]]
  • [[sleeps]]([[paul]])
  • [[Mary killed John]] is the result of applying the function [[kill]]:
  • first to [[John]]
  • then to [[Mary]]
  • [[kill]](john)(mary)
  • Things to remember
  • Our recursive rule can construct complex types from simpler types:
  • Given type a and type b, we construct type <a,b>
  • This gave us <e,t> for predicates
  • Things that combine with individuals (e) to yield propositions (t)
  • For 2-place predicates, we take the types e and <e,t> and create a new complex type <e,<e,t>>
  • Things that need an individual to yield a predicate that requires another individual for saturation.
  • Our original interpretation rules for predicate saturation can be reinterpreted as function application.
  • Part 2 The lambda calculusNaming a function
  • Given what we’ve said so far, a verb like sleep can be viewed as a function which applies to an individual to return a truth value.
  • In other words, predication is a form of function application.
  • The problem is that compositional interpretation will now require lots of functions (every predicate is a function of some kind).
  • Naming a function
  • The usual way we define functions is contextually:
  • Let f be a function from things in A to things in B
  • (Correspondingly: Let f be a function from things of type a to things of type b)
  • E.g. Let f be the function from numbers to numbers, f(x) = x2
  • Thus, f(2) = 4
  • An alternative is to use a system that would allow us to name functions on the fly:
  • The lambda notation is a way of naming the function, which can now be applied to something of the right type:
  • Naming a function: 1-place predicates
  • A one-place predicate can be represented as a function of this kind using lambda notation:
  • This notation represents the function corresponding to sleep, which indicates that it requires some x to saturate it (and make it return a truth value). E.g.:
  • tepaul<e,t>sleepNaming a function: 2-place predicates
  • We can do the same with 2-place predicates
  • Observe that the order of variables corresponds to the order in which we apply the function, first applying kill to the object, and then applying the result to the subject.
  • tMarykill<e,<e,t>>eeJohnSyntax of lambda abstraction
  • If u is a variable of type a and ß is an expression of type b, then is an expression of type <a,b>
  • In other words, we have a way of expressing complex types as functions.
  • Semantics of lambda abstraction
  • Given u of type a, ß of type b, then is a function f from the set of things of type a to the set of things of type b, such that:
  • For any object k of type a, where g’ is just like g except that g’(u) = k.
  • (In other words, applying the function to the object returns the meaning of the original expression with the variable substituted for the object.)
  • Example
  • This is that function fsuch that, for any number n, f(n) = n2.
  • So, we find the value of the function, for the number 2 by:
  • Assuming a variable assignment g’ whereby x = 2
  • Applying the body of the function to the value of x to give us the value 4.
  • Note that the lambda expression names the function. What we have just done is apply the function to get its value.
  • The lambda calculus allows us to distinguish the function itself from the set of values it returns.
  • Example 2
  • This is the function f such that, for some x of type e (say, the individual Paul), the function returns true just in case it is true in the model that that individual sleeps.
  • To find the value for the individual Paul, we:
  • Assume an assignment function g’ where g’(x) = Paul
  • Apply the function sleep to Paul
  • Check the truth value
  • Lambda conversion
  • The process we’ve just seen entails:
  • Setting the value of a variable
  • Applying the function to the value
  • Checking the result.
  • This process is called lambda conversion
  • Another example
  • Suppose we set:
  • x= paul
  • y = mary
  • We apply this as follows:
  • We can then check the truth value of this proposition
  • Notice that effectively, we have two function applications here.
  • An exercise
  • Express these predicates as lambda functions:
  • smile (1-place)
  • thin (1-place)
  • eat (2-place)
  • give (3-place)
  • An exercise
  • What is their semantic type?
  • smile (1-place)
  • <e,t>
  • thin (1-place)
  • <e,t>
  • eat (2-place)
  • <e,<e,t>>
  • give (3-place)
  • <e,<e,<e,t>>
  • Predicates vs modifiers
  • What is the semantic difference of the use of tall in:
  • The woman is tall
  • Mary is a tall woman
  • In the first case, tall is being predicated of an individual.
  • Once saturated (i.e. once we apply the function), we get a truth value.
  • In the second case, tall has a different use.
  • Here, we first combine tall and woman, and then the resultng complex predicate can be applied to Mary to check if she’s indeed a tall woman.
  • Can you think of the difference in terms of types?
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