formal semantics

Formal Semantics

A Brief Introduction

Contents

1. Richard Montague

2. The FS’s Target and Tools

3. Functional Type Theory

Richard Merritt Montague (1930 –1971) was an American mathematician and philosopher.

• There is in my opinion no important theoretical difference between natural languages and the artificial languages of logicians; indeed, I consider it possible to comprehend the syntax and semantics of both kinds of languages within a single natural and mathematically precise theory. On this point I differ from a number of philosophers, but agree, I believe, with Chomsky and his associates. (Montague 1970 )

While some of the earlier work in this tradition was directed at purposesof no concern to the linguist (as when philosophers hoped to expose theircolleagues’ fallacious arguments more convincingly by applying logical analysisto them), there has been increasing interest in logical semantics as a contributionto linguistic theory, more specically, a contribution to a theoryof one subsystem of grammar, the (or one of the) semantic component(s).（ Heim 1982)

. The FS’s target

• FS assumes an intersubjective level of shared information, a conceptualization of the world, the world as we jointly structure it (Landman).

• FS explores the structure of conceptualized world and how it is related to natural language.

Tools

1.Logic languages: predicate logic, type theory, λ -calculation

2.Algebra

3.Discourse Representation theory.

…

Natural language → Logic language →Model →Conceptualized world

Benefits brought by formal instruments

• We are armed with unambiguous instruments which are favored in dealing with natural language where ambiguity is widespread.

• Direct study of a formal system can yield results which can be applied generally to all systems which are models of it. Also, once we show that a certain system is equivalent in its formal structure to another better known system, what we know about the latter may transfer to new insights about the former (Partee).

Logic language

• Predicate logic • John is fat • Fat’ (John’) • Everyone is fat.• ∀x[ man(x) → fat(x)]• Lexicon: term (individual constant), predicate co

nstant, variable, logical constant.• Grammar: All variables in a predicate must be b

ound to turn a predicate into a proposition.

Model• A Model provides a description of conceptualized wo

rld and an interpretation for logic language. • A simplified model for predicate logical language L i

s a triple:• M = <D,F,g> where:• 1. D, the domain of M, is a non-empty set (the

domain of individuals)• 2. F, the interpretation function of M for the no

n-logical constants of L is a function such that:• a. for every c CONL: F(c) D• b. for every P PREDn

L: F(P) pow(Dn)• Here Dn = {<d1,...,dn>: d1,...,dn D}• c. for every S StatementL: F(S) {0,1}• 3. if x VAR then x’ = g(x)

Tarski’, Montague’, Chomsky’, Partee’,…is the teacher of ’

Chomsky is the teacher of Montague=0

Tarski, Montague, Chomsky, Partee,…

< Chomsky , Partee>, <Tarski, Montague>

• A Model is driven by a logic language. We may include time, possible worlds, events, etc. into the model. It is a vital work to build a proper model, the structure of which may be an object of research.

Mapping

• The relationship of negation • “A true statement” 1 1“A true statement”

• “ A false statement”0 0“A false statement” • t t• <t, t> = ¬ • t+ <t, t> =t

Type theory

• Let e and t be two symbols.

• TYPE is the smallest set such that:

• 1. e,t TYPE

• 2. if a,b TYPE then <a,b> TYPE

• In FL, e=individual; t=truth value (you can take it as the meaning of a statement for the moment)

Tarski, Montague, Chomsky, Partee…: e

Chomsky is alive: tAlive =?

Since statement’s type is always t

If in natural language either one from two elements which are grammatically combined is the functor and takes the other as the argument, then Alive =<e, t>

• Alive:• Tarski,• a true statement 1

• Montague, • a false statement 0• Chomsky, • • Partee

• Vt: Tarski taught Montague

• taught=<e,<e,t>>

• taught Montague= <e,t>

• Tarski taught Montague=t

• Adv= <<e,t>, <e,t>>

• Connectives=?

Thank you!Thank you!