CAS LX 502

8a. Formal semantics10.1-10.4

Truth and meaning

• The basis of formal semantics: knowing the meaning of a sentence is knowing under what conditions it is true.• Formal semantics, a.k.a. truth conditional semantics,

a.k.a. model-theoretic semantics, related to Montague Grammar.

• Using logical language as a metalanguage to describe meaning precisely.

Models and predicate logic

• One of the basic pieces of formal semantics is the model of the world, which has individuals, with properties, that can be referred to in the logical metalanguage.

• Predicate logic is an extension of the propositional logic that we were examining before. Propositional logic is about combining propositions and determining truth values; predicate logic is about properties of individuals (which can in the end still lead to the determination of truth values).

Translation• What we are going to be doing is translating from

natural language (e.g., English) into the propositional logic metalanguage. Parts of this we already know how to do (from propositional logic):

Connective Syntax English

p it is not the case that p

p q p and q

p q p and/or q

e p e q p or q, but not both

p q if p, then q

p q p if and only if q

Predicates• Predicate logic adds the notion of predicate, something

that is true of certain individuals, false of certain others.• For example, A, the property of being asleep, or S, the property

of being a smoker.

• If A is true of an individual, say Mary, who we will refer to as m, then the following is true:• A(m). ‘Mary is asleep.’S(m). ‘Mary doesn’t smoke.’

• For individuals in our model, we can use these individual constants to refer to them. We refer to John with j, Mary with m, and so forth.

Relations

• Intransitive verbs and adjectives can generally be modeled with one-place predicates like A and S above.

• Other verbs seem to denote relations (two-place predicates) among individuals. Transitive verbs, for example, relate two individuals. If R is the resemble relation, the following is true if John resembles Mary.• R(j, m). ‘John resembles Mary.’

Three-place relations

• We can also find three-place relations in natural language, such as introduce or give.• I(j, m, b) ‘John introduces Mary to Bill.’• G(j, m, c) ‘John gave Mary the cake.’

• This is essentially the same notion as -roles discussed in previous weeks. Different predicates have different numbers of arguments, which play different roles in the truth conditions.

Quantification

• Predicate logic is particularly useful when we want to translate quantificational statements, such as Every student is happy or A student wrote a paper.

• Every student is happy says that for each individual (say, in the model) that is a student, the predicate H (‘is happy’) is true of that individual.

• A student wrote a paper says that there is at least one individual in the model that has the property S (‘is a student’) and that the predicate W (‘wrote a paper’) is true of.

Existential and Universal quantifiers

• There are two special operators in predicate logic that allow this kind of statement to be made. ‘for each’ ‘there exists’

universal existential

• Each of these quantifiers binds a variable over individuals. Its scope is notated by parentheses: x ( S(x) H(x) ) ‘Every student is happy.’ x ( S(x) W(x) ) ‘A student wrote a paper.’

Composite quantifiers

• No student is happy.• There are two ways we can think about this, both

of which are equivalent with respect to the models in which they are true:

x ( S(x) H(x) )‘Every individual x is such that if x is a student, x is not happy.’

x ( S(x) W(x) )‘It is not the case that there is an individual x such that x is a student and x is happy.’

The syntax of predicate logic

• In order for a formula to be well-formed, it must obey the rules of formula formation, the syntax of predicate logic.

• The symbols: Predicates (A, B, C, …)Individual constants (a, b, c, …)Individual variables (x, y, z, …)Connectives (, , , e, , )Quantifiers (, )

The syntax of predicate logic

• The rules:• Individual constants and variables are terms.

• If A is an n-place predicate and t1, …, tn are terms, then A(t1, …, tn) is a formula.

• If is a formula, then is a formula.• If and are formulas, then ( ), ( ), ( e ),

( ), ( ) are all formulae.• If is a formula and x is a variable, then x and x

are formulae.• (The outer parentheses can be omitted)

Scope ambiguities

• In natural language, the scope of quantifiers and certain operators is not unambiguously specified in the surface form—but in predicate logic it is.• Pat is not tall and happy.T(p) H(p) ( T(p) H(p) )

• All doors will not open.x(D(x) O(x)) x(D(x) O(x))

• Someone admires everyone.x y A(x, y) y x A(x, y)

Syllogisms

• Recall that we saw reasoning patterns such as modus tollens:• If Pat is hungry, then Pat is irritable.

Pat is not irritable.Pat is not hungry.

• p = Pat is hungry, q = Pat is irritable.• p q

q p

Syllogisms requiring predicate logic

• We get the same feeling of logical conclusion from something like this:• All men are mortal.

Socrates is a man.Socrates is mortal.

• But yet we were not able to represent that with propositional logic.• p

qr

Syllogisms requiring predicate logic

• With predicate logic, we can see how this follows.

• All men are mortal x(H(x) M(x)) Socrates is a man H(s)Socrates is mortal M(s)

The components of a formal semantics

• Semantic interpretation• Rules for interpreting the symbols of predicate logic.

• Domain (model)• Identifies (linguistically) relevant individuals,

properties, and relations.

• Denotation assignment function• A procedure that matches the symbols for nouns, verbs,

etc., with items in the model that they denote.

Functions

• A function is something that, given one or more arguments, returns a value.

• In elementary mathematics, we can write a doubling function like this:• f(x) = 2x• This takes any number, which we will call x,

and returns a value (a number) that is twice x.

Denotation function

• The denotation function takes symbols or formulae and returns whatever in the model the symbols/formulae denotes.• For individual constants, the denotation function

returns the individual in the model.

• Suppose the individuals in our model are Pat, Tracy, and Chris. Then, we might have a denotation function like:• F(t) = Tracy

F(c) = ChrisF(p) = Pat

Predicate denotations

• The denotation of predicates (like T or H) is trickier.

• In our model, certain individuals have the property denoted by T, certain individuals have the property denoted by H.

• Those individuals that have property T have that property in common.

• We can in fact think of the property T as being that property common to those individuals.

Predicate denotations

• Another way to think about this is as sets of individuals. A predicate denotes a set of individuals that share the property (such that no individual that shares the property is not in the set, and all individuals in the set share the property).

• Whatever H is, Pat and Tracy share that property, while Chris does not.• F(H) = {Pat, Tracy}.

Truth

• So, F(t) = Tracy, and F(H) = {Pat, Tracy}.• Then, what about the whole sentence Pat is

happy?• H(p)• Is it true? How do we know?

• F(A(a)) is true when A(a) is true—that is, when F(a) is in the set denoted by A, namely F(A).

[ ] vs. F

• The function F represents the situation.• In our model, there are individuals.• But the relation among the individuals and the

properties the individuals will vary from situation to situation. So, we have a different denotation function for each situation.

• When we evaluate a certain bit of a formula, we write it like this: [ ].• For example, [p] or [H(p)].

[ ] vs. F

• [ ] is a function too, it is the evaluation function. We use it to evaluate the denotation of formulae or subparts of formulae.

• [ ] makes use of the denotation function F to perform some of its duties.• For example, [p] = F(p) = Pat.

[ ] vs. F

• However, F and [ ] are different. We have some general rules we can state about meaning that transcend which particular denotation function we are using (i.e. which particular situation we are in).

• For example, if p is an individual constant and F is the denotation function we are evaluating with, then: [p] = F(p).

Complete model

• In general, we are going to be evaluating formulae in the context of a complete model, which is both a specification of the individuals, and a function F specifying the properties of and relations among the individuals.

• A complete model M has two parts, U (the set of individuals) and F (the denotation function for individuals, properties, relations).

• M = <U, F>

[ ] with respect to a model

• M = <U, F>• We evaluate formulae (and parts thereof) in

the context of a model.

• [ H(p) ]<U,F> or [ H(p) ]M

• This will be true or false, depending on the particular model we’re talking about.

General rules about [ ]

• We can state some general rules of interpretation with respect to [ ] now.

• Where M=<U,F>, a is an individual constant, A is a predicate,• [a]M = F(a)• [A]M = F(A)• [A(a)]M = true iff [a]M [A]M, otherwise false.

• These are true regardless of the model, regardless of the F.

Relations

• Just as we can think of F(A) as being a set of individuals (A being defined as a property that the members of F(A) have exclusively and in common), we can think of the denotation of a relation F(R) as being a set of pairings. Ordered pairings.

• R(a, b) = true, R(b, c) = true, R(c, a) = true.

• F(R) = {<Pat, Tracy>, <Tracy, Chris>, <Chris, Pat>}

Relations

• So, if R is a relation (a two-place predicate), M=<U,F> is a model, and a and b are individual constants (or, more generally, terms),

• [R(a,b)]M = true iff <[a]M, [b]M> [R]M

• <F(a), F(b)> F(R)