**Compositional Logical Semantics** Torbjörn Lager Department of Linguistics Stockholm University

**Logical Semantics Example** • John laughed • laughed'(j) • Nobody laughed • x[laughed'(x)] • But this is just translation! What's semantic about that? NLP1 - Torbjörn Lager

**What Is the Name of This Game?** • Truth conditional semantics • Model theoretic semantics • Logical semantics • Formal semantics • Compositional semantics • Syntax-driven semantic analysis • Compositional, logical, truth conditional, model theoretic semantics .... NLP1 - Torbjörn Lager

**We use language to talk about the world** Meaning = Truth conditions Examples: "John whistles" is true iff John whistles "John visslar" is true iff John whistles "Ogul fautu seq" is true iff... Truth Conditional Semantics Natural language The outside world NLP1 - Torbjörn Lager

**We don't know what the world is really like, so let's talk** about a model of the world instead Such a model does (usually) consists of individuals, sets of individuals, functions and relations. i.e the sort of things set theory talks about Truth becomes truth relative to a model Model Theoretic Semantics Natural language Model The outside world NLP1 - Torbjörn Lager

**The Compositionality Principle:** The meaning of the whole is a function of the meaning of the parts and the mode of combining them. The meaning of a complex expression is uniquely determined by the meaning of its constituents and the syntactic construction used to combine them. Compositional Semantics Natural language Model The World NLP1 - Torbjörn Lager

**A simple model M:** Domain: {John, Mary, Paul} Interpretation: Names: "John" refers to John , "Mary" refers to Mary, etc. Verbs: "whistles" refers to {John, Paul} Example "John whistles" is true in M iff the individual in M referred to as "John" is an element in the set of individuals that "whistles" refer to. Truth Conditional, Model Theoretic and Compositional Semantics Combined Richard Montague (1970): "I reject the contention that an important theoretical difference exists between formal and natural languages" NLP1 - Torbjörn Lager

**Account for the meanings of natural language utterances by** translating them into another language. It could be any language, but only if this language has a formal semantics are we done. Translational Semantics Natural language Logical Form Language Model The World NLP1 - Torbjörn Lager

**Have same truth conditions** Grammar and Logical Form • S -> NP VP[S] = [VP]([NP]) • NP -> john[NP] = j • VP -> whistles[VP] = x[whistles'(x)] • [john whistles] = whistles'(j) cf. The principle of compositionality NLP1 - Torbjörn Lager

**Beta Reduction (Lambda Conversion)** • [S] = [VP]([NP]) • [NP] = j • [VP] = x[whistles'(x)] • Beta reduction rule: u() where every occurrence of u in is replaced by • x[whistles'(x)](j) application • whistles'(j) reduction NLP1 - Torbjörn Lager

**Grammar and Logical Form** • S -> NP VP[S] = [NP]([VP]) • NP -> john[NP] = P[P(j)] • VP -> whistles[VP] = x[whistles'(x)] • [john whistles] = whistles'(j) NLP1 - Torbjörn Lager

**From Logical Form to Truth Conditions** • whistles'(j) is true iffthe individual (in the model) denoted by 'j' has the property denoted by 'whistles' • cf. "John whistles" is true iff John whistles NLP1 - Torbjörn Lager

**Beta Reduction (Lambda Conversion)** • [S] = [NP]([VP]) • [NP] = P[P(j)] • [VP] = x[whistles'(x)] • Beta reduction rule: u() where every occurrence of u in is replaced by • P[P(j)](x[whistles'(x)]) application • x[whistles'(x)](j) reduction • whistles'(j) reduction NLP1 - Torbjörn Lager

**A Larger Example** • S -> NP VP [S] = [NP]([VP]) • NP -> DET N [NP] = [DET]([N]) • DET -> every[DET] = Q[P[z[Q(z) P(z)]]] • N -> man [N] = x[man'(x)] • VP -> whistles[VP] = x[whistles'(x)] • [every man whistles} = z[man'(z) whistles'(z)] NLP1 - Torbjörn Lager

**A Larger Example (cont'd)** • [S] = [NP]([VP]) • [NP] = [DET]([N]) • [DET] = Q[P[z[Q(z) P(z)]]] • [N] = x[man'(x)] • [VP] = x[whistles'(x)] • Q[P[z[Q(z) P(z)]]](x[man'(x)]) application • P[z[x[man'(x)](z) P(z)]] reduction • P[z[man'(z) P(z)]] reduction • P[z[man'(z) P(z)]](x[whistles'(x)]) application • z[man'(z) x[whistles'(x)](z)] reduction • z[man'(z) whistles'(z)] reduction NLP1 - Torbjörn Lager

**Examples in Oz** • X = {fun {$ P} {P j} end fun {$ X} whistles(X) end} • % X gets bound to the tuple 'whistles(j)' • DetSem = fun {$ Q} fun {$ P} all(Z impl({Q Z} {P Z})) end end • NSem = fun {$ X} man(X) end • VPSem = fun {$ X} whistles(X) end • NPSem = {DetSem NSem} • SSem = {NPSem VPSem} • % X gets bound to the tuple 'all(Z impl(man(Z) whistles(Z)))' NLP1 - Torbjörn Lager