RésuméAristote n'a pas développé une théorie de la quantification du prédicat, mais une étude récente de Hasnawi a montré qu'Ibn Sīnā a consacré à celle-ci une étude rigoureuse. Assumant la structure aristotélicienne sujet-prédicat, Ibn Sīnā qualifie les propositions qui comportent un prédicat quantifié, de propositions déviantes. Une conséquence de cette approche avicennienne est que la seconde quantification est absorbée par le prédicat. La distinction claire ainsi opérée entre un sujet quantifié, qui pose le domaine de la quantification, et une partie (...) prédicative, qui construit une proposition sur ce domaine, correspond structurellement à la distinction, faite dans la théorie constructive des types, entre le type des ensembles et le type des propositions.Aristote n'a pas non plus combiné son analyse logique de la quantification avec sa théorie ontologique des relations ou de l’égalité. Ibn Sīnā, en revanche, a utilisé les syllogismes qui nécessitent une logique de l’égalité et il a examiné des cas où la quantification se combine, via l’égalité, avec des termes singuliers. En outre, ces réflexions sont essentielles pour sa théorie des nombres qui est fondée sur l'interaction entre l’un et le multiple. Lorsqu'on combine la théorie métaphysique de l’égalité telle qu'on la trouve chez Ibn Sīnā avec son travail sur la quantification du prédicat, il en résulte tout naturellement une logique de l’égalité. En effet, l'interaction entre la quantification du prédicat et l’égalité peut être appliquée aux exemples de syllogismes mentionnés par Ibn Sīnā dans lesquels ces notions interviennent. En utilisant les instruments formels fournis par la théorie constructive des types de Martin-Löf, cet article établit des liens entre la métaphysique d'Ibn Sīnā et son travail logique, liens qui ont été discutés en relation avec d'autres sujets par Thom et Street. Ibn Sīnā n'a pas développé une logique de l'identité, mais il a développé les moyens conceptuels de le faire. (shrink)

In this paper we propose a way to deal with natural language inference by implementing Modern Type Theoretical Semantics in the proof assistant Coq. The paper is a first attempt to deal with NLI and natural language reasoning in general by using the proof assistant technology. Valid NLIs are treated as theorems and as such the adequacy of our account is tested by trying to prove them. We use Luo’s Modern Type Theory with coercive subtyping as the formal language into (...) which we translate natural language semantics, and we further implement these semantics in the Coq proof assistant. It is shown that the use of a MTT with an adequate subtyping mechanism can give us a number of promising results as regards NLI. Specifically, it is shown that a number of inference cases, i.e. quantifiers, adjectives, conjoined noun phrases and temporal reference among other things can be successfully dealt with. It is then shown, that even though Coq is an interactive and not an automated theorem prover, automation of all of the test examples is possible by introducing user-defined automated tactics. Lastly, the paper offers a number of innovative approaches to NL phenomena like adjectives, collective predication, comparatives and factive verbs among other things, contributing in this respect to the theoretical study of formal semantics using MTTs. (shrink)

The analysis of atomic sentences and their subatomic components poses a special problem for proof-theoretic approaches to natural language semantics, as it is far from clear how their semantics could be explained by means of proofs rather than denotations. The paper develops a proof-theoretic semantics for a fragment of English within a type-theoretical formalism that combines subatomic systems for natural deduction [20] with constructive (or Martin-Löf) type theory [8, 9] by stating rules for the formation, introduction, elimination and equality of (...) atomic propositions understood as types (or sets) of subatomic proof-objects. The formalism is extended with dependent types to admit an interpretation of non-atomic sentences. The paper concludes with applications to natural language including internally nested proper names, anaphoric pronouns, simple identity sentences, and intensional transitive verbs. (shrink)

In formal semantics based on modern type theories, some sentences may be interpreted as judgements and some as logical propositions. When interpreting composite sentences, one may want to turn a judgemental interpretation or an ill-typed semantic interpretation into a proposition in order to obtain an intended semantics. For instance, an incorrect judgement $$a:A$$ may be turned into its propositional form $$\textsc {is}(A,a)$$ and an ill-typed application p(a) into $$\textsc {do}(p,a)$$, so that the propositional forms can take part in logical compositions (...) that interpret composite sentences, especially those that involve negations and conditionals.In this paper, we propose an operator not that facilitates such a transformation. Introducing not axiomatically, with five axiomatic laws to govern its behaviour, we shall use it to define $$\textsc {is}$$ and $$\textsc {do}$$ and give examples to illustrate its use in semantic interpretation. The introduction of not into type theories is logically consistent – this is justified by showing that not can be defined by means of the heterogeneous equality $$\textrm{JMeq}$$ so that all of the axiomatic laws for not become provable. Therefore, since the extension with $$\textrm{JMeq}$$ preserves logical consistency, so does the extension with not. We shall also study conditions under which $$\textsc {is}$$ and $$\textsc {do}$$ operators can be used safely without the risk of over-generation. (shrink)

This article is about Avicenna’s account of syllogisms comprising opposite premises. We examine the applications and the truth conditions of these syllogisms. Finally, we discuss the relation between these syllogisms and the principle of non-contradiction.

We present Property Theory with Curry Typing (PTCT), an intensional first-order logic for natural language semantics. PTCT permits fine-grained specifications of meaning. It also supports polymorphic types and separation types. We develop an intensional number theory within PTCT in order to represent proportional generalized quantifiers like “most.” We use the type system and our treatment of generalized quantifiers in natural language to construct a type-theoretic approach to pronominal anaphora that avoids some of the difficulties that undermine previous type-theoretic analyses of (...) this phenomenon. (shrink)

In this paper we present a study of adjectival/adverbial modification using modern type theories, i.e. type theories within the tradition of Martin-Löf. We present an account of various issues concerning adjectival/adverbial modification and argue that MTTs can be used as an adequate language for interpreting NL semantics. MTTs are not only expressive enough to deal with a range of modification phenomena, but are furthermore well-suited to perform reasoning tasks that can be easily implemented given their proof-theoretic nature. In MTT-semantics, common (...) nouns are interpreted as types rather than predicates. Therefore, in order to capture the semantics of adjectives adequately, one needs to meet the challenge of modeling CNs modified by adjectives as types. To explicate that this can be done successfully, we first look at the mainstream classification of adjectives, i.e. intersective, subsective and non-subsective adjectives. There, we show that the rich type structure available in MTTs, along with a suitable subtyping framework, offers an adequate mechanism to model these cases. In particular, this modelling naturally takes care of the characterising inferences associated with each class of adjectives. Then, more advanced issues on adjectival modification are discussed: degree adjectives, comparatives and multidimensional adjectives. There, it is shown that the use of indexed types can be usefully applied in order to deal with these cases. In the same vein, the issue of adverbial modification is discussed. We study two general typings for sentence and VP adverbs respectively. It is shown that the rich type structure in MTTs further provides useful organisational mechanisms in giving formal semantics for adverbs. In particular, we discuss the use of \-types to capture the veridicality/non-veridicality distinction and further discuss cases of intensional adverbs using the type theoretic notion of context. We also look at manner, subject and speech act adverbials and propose solutions using MTTs. Finally, we show that the current proof technology can help mechanically check the associated inferences. A number of our proposals concerning adjectival and adverbial modification have been formalised in the proof assistant Coq and many of the associated inference patterns are checked to be correctly captured. (shrink)