Semântica formal

O propósito do texto é a apresentação de um mecanismo descritivo de estruturas semânticas, desenvolvido em parceria por lógicos, linguistas e especialistas em processamento computacional de línguas naturais, denominado por vezes semântica de modelo teórico, que se serve de uma metalinguagem de natureza lógica para descrever as estruturas linguísticas. Depois de uma introdução em que se discute a natureza dos modelos analíticos na Linguística e de uma seção em que se discutem as características de um sistema formal, em geral, busca-se a caracterização da semântica formal desenvolvida, em boa parte, com base nos trabalhos de Richard Montague. Para finalizar, apresentam-se alguns exemplos, claramente simplificados, dos procedimentos formais de que faz uso a semântica de modelo teórico.

Modal logic, philosophical issues in

In reasoning we often use words such as ‘necessarily’, ‘possibly’, ‘can’, ‘could’, ‘must’ and so on. For example, if we know that an argument is valid, then we know that it is necessarily true that if the premises are true, then the conclusion is true. Modal logic starts with such modal words and the inferences involving them. The exploration of these inferences has led to a variety of formal systems, and their interpretation is now most often built on the concept of a possible world. Standard non-modal logic shows us how to understand logical words such as ‘not’, ‘and’ and ‘or’, which are truth-functional. The modal concepts are not truth-functional: knowing that p is true (and what ‘necessarily’ means) does not automatically enable one to determine whether ‘Necessarily p’ is true. (‘It is necessary that all people have been people’ is true, but ‘It is necessary that no English monarch was born in Montana’ is false, even though the simpler constituents – ‘All people have been people’ and ‘No English monarch was born in Montana’– are both true.) The study of modal logic has helped in the understanding of many other contexts for sentences that are not truth-functional, such as ‘ought’ (‘It ought to be the case that p’) and ‘believes’ (‘Alice believes that p’); and also in the consideration of the interaction between quantifiers and non-truth-functional contexts. In fact, much work in modern semantics has benefited from the extension of modal semantics introduced by Richard Montague in beginning the development of a systematic semantics for natural language. The framework of possible worlds developed for modal logic has been fruitful in the analysis of many concepts. For example, by introducing the concept of relative possibility, Kripke showed how to model a variety of modal systems: a proposition is necessarily true at a possible world w if and only if it is true at every world that is possible relative to w. To achieve a better analysis of statements of ability, Mark Brown adapted the framework by modelling actions with sets of possible outcomes. John has the ability to hit the bull’s-eye reliably if there is some action of John’s such that every possible outcome of that action includes John’s hitting the bull’s-eye. Modal logic and its semantics also raise many puzzles. What makes a modal claim true? How do we tell what is possible and what is necessary? Are there any possible things that do not exist (and what could that mean anyway)? Does the use of modal logic involve a commitment to essentialism? How can an individual exist in many different possible worlds?

Type-logical semantics

Type-logical semantics studies linguistic meaning with the help of the theory of types. The latter originated with Russell as an answer to the paradoxes, but has the additional virtue that it is very close to ordinary language. In fact, type theory is so much more similar to language than predicate logic is, that adopting it as a vehicle of representation can overcome the mismatches between grammatical form and predicate logical form that were observed by Frege and Russell. The grammatical forms of ordinary language sentences consequently may be taken to be much less misleading than logicians in the first half of the twentieth century often thought them to be. This was realized by Richard Montague, who used the theory of types to translate fragments of ordinary language into a logical language. Semantics is commonly divided into lexical semantics, which studies the meaning of words, and compositional semantics, which studies the way in which complex phrases obtain a meaning from their constituents. The strength of type-logical semantics lies with the latter, but type-logical theories can be combined with many competing hypotheses about lexical meaning, provided these hypotheses are expressed using the language of type theory.

Montague, Richard Merett (1930–71)

Richard Montague was a logician, philosopher and mathematician. His mathematical contributions include work in Boolean algebra, model theory, proof theory, recursion theory, axiomatic set theory and higher-order logic. He developed a modal logic in which necessity appears as a predicate of sentences, showing how analogues of the semantic paradoxes relate to this notion. Analogously, he (with David Kaplan) argued that a special case of the surprise examination paradox can also be seen as an epistemic version of semantic paradox. He made important contributions to the problem of formulating the notion of a ‘deterministic’ theory in science.

Intensional logics

Intensional logics are systems that distinguish an expression’s intension (roughly, its sense or meaning) from its extension (reference, denotation). The purpose of bringing intensions into logic is to explain the logical behaviour of so-called intensional expressions. Intensional expressions create contexts which violate a cluster of standard principles of logic, the most notable of which is the law of substitution of identities – the law that from a = b and P(a) it follows that P(b). For example, ‘obviously’ is intensional because the following instance of the law of substitution is invalid (at least on one reading): Scott = the author of Waverley; obviously Scott = Scott; so, obviously Scott = the author of Waverley. By providing an analysis of meaning, intensional logics attempt to explain the logical behaviour of expressions such as ‘obviously’. On the assumption that it is intensions and not extensions which matter in intensional contexts, the failure of substitution and related anomalies can be understood. Alonzo Church pioneered intensional logic, basing it on his theory of types. However, the widespread application of intensional logic to linguistics and philosophy began with the work of Richard Montague, who crafted a number of systems designed to capture the expressive power of natural languages. One important feature of Montague’s work was the application of possible worlds semantics to the analysis of intensional logic. The most difficult problems concerning intensional logic concern the treatment of propositional attitude verbs, such as ‘believes’, ‘desires’ and ‘knows’. Such expressions pose difficulties for the possible worlds treatment, and have thus spawned alternative approaches.

L2 semantics from a formal linguistic perspective

Ever since Aristotle and Plato (The Categories; Cratylus), linguists have considered language to be the pairing of form (sounds or gestures or written strings) and meaning. This is true for all meaningful linguistic units from morphemes, through words, phrases and sentences, to discourse. Generally speaking, semantics is the study of how form and meaning are related. However, semantics is more narrowly construed as excluding those meanings that derive from speaker intensions and psychological states, as well as sociocultural features of the context. Furthermore, the boundary between semantics proper and pragmatics is intensely debated and to some researchers constitutes an empirical question. Formal semantics came into being as a system describing formal languages, that is, the mathematical and logical languages of computing machines as opposed to the natural languages of human beings. However, in the late 1960s the philosopher Richard Montague argued that natural languages such as English could be fruitfully described using the same rigorous rules and correspondences utilized in the description of formal languages. Modern formal semantics was born and is currently prospering as a branch of linguistics.

Type Theory for Natural Language Semantics

Type theory is a regime for classifying objects (including events) into categories called types. It was originally designed in order to overcome problems relating to the foundations of mathematics relating to Russell’s paradox. It has made an immense contribution to the study of logic and computer science and has also played a central role in formal semantics for natural languages since the initial work of Richard Montague building on the typed λ-calculus. More recently, type theories following in the tradition created by Per Martin-Löf have presented an important alternative to Montague’s type theory for semantic analysis. These more modern type theories yield a rich collection of types which take on a role of representing semantic content rather than simply structuring the universe in order to avoid paradoxes.

Generative Grammar

This article presents different types of generative grammar that can be used as models of natural languages focusing on a small subset of all the systems that have been devised. The central idea behind generative grammar may be rendered in the words of Richard Montague: “I reject the contention that an important theoretical difference exists between formal and natural languages” (“Universal Grammar,” Theoria, 36 [1970], 373–398).

Meaning, Modality, and Possible Worlds Semantics

This chapter begins with a discussion of Kripke-style possible worlds semantics. It considers one of the most important applications of possible worlds semantics, the account of counterfactual conditionals given in Robert Stalnaker and David Lewis. It then goes on to examine the work of Richard Montague. Montague specified syntactic rules that generate English, or English-like, structures directly, while pairing each such rule with a truth-theoretic rule interpreting it. This close parallel between syntax and semantics is what makes the languages of classical logic so transparently tractable, and what they were designed to embody. Montague's bold contention is that we do not have to replace natural language natural languages with formal substitutes to achieve such transparency. The same techniques employed to create formal languages can be used to describe natural languages in mathematically revealing ways.