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2000, The Journal of Symbolic Logic
Russell’s type theory has been the standard property theory for years, relying on rigid type distinctions at the grammatical level to circumvent the paradoxes of predication. In recent years it has been convincingly argued by Bealer, Cochiarella, Turner and others that many linguistic and ontological data are best accounted for by using a type-free property theory. In the spirit of exploring alternatives and “to have as many opportunities as possible for theory comparison”, this paper presents another type-free property theory, to be called P*, intended for applications in Montague-style natural language semantics and formal ontology. The theory is philosophically grounded on Gupta’s and Belnap’s revision theory of definitions and its basic idea is viewing predication (exemplification) as a ‘circular concept’ that can be captured by circular definitions. The paper has the following fourteen sections: (1) Introduction; (2) Formal type-free property theory; (3) Applying RTD [revision theory of definitions] to exemplification; (4) The system P*; (5) Some features of P*; (6) A comparison with Turner’s system; (7) Entailment and P*; (8) P* and natural language semantics; (9) Noun phrases; (10) The need for type-freedom in semantics; (11) Arithmetic in P*; (12) Arithmetic in PN*; (13) Complexity of P*; (14) Conclusion, and acknowledgments.
2016 •
Modern Type Theories (MTTs) provide us with a new framework for formal semantics with attractive advantages as compared to Montague Grammar. First, MTTs have rich type structures that can be employed effectively to capture various linguistic features that have proved difficult in the Montagovian setting. Second, MTTs are prooftheoretically specified and can hence be usefully implemented in proof assistants such as Coq, where the MTT-semantics has been implemented for computer-assisted reasoning. These two respects may be characterised as saying that the MTT-semantics is both modeltheoretic and proof-theoretic. They offer unique features unavailable in traditional logical systems that have proved very useful in formal semantics. We shall introduce MTTs and how they can be used for formal semantics. The lectures will be informal and explanatory. They will be rigorous but contain a lot of examples, to illustrate the use of MTTs, on the one hand, and to compare the MTT-semantics with Mo...
2017 •
Type theories, from the early days of Montague Semantics (Montague 1974) to the recent work of using rich or modern type theories, have a long history of being employed as foundational languages of natural language semantics. In this introductory chapter, we will describe and discuss the development of type theories as foundational languages of mathematics, as well as their applications as foundational languages for formal semantics. In the end, a brief description of each chapter in the volume will follow.
Lecture Notes in Computer Science
Formal Semantics in Modern Type Theories: Is It Model-Theoretic, Proof-Theoretic, or Both?2014 •
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On the Interpretation of Common Nouns: Types Versus Predicates2017 •
In type theoretical approaches to formal semantics of natural languages, a lexical item in a sentence is assigned a type and using a number of composition rules and based on the syntactic structure of the sentence, the types compose to converge to a single type called t, the semantic type for sentences, together with a statement specifying the conditions under which the sentence is true. For example consider the example (1-a) below (1) a. Every linguist offended Fred. b. Fred offended every linguist. The syntactic structure of the sentence, its type assignments and the compositions are as follows(we omit the obvious truth conditions since it does not concern our discussion of type shifting): (2) t DP < et, t > < et, < et, t >> every < e, t > linguist < e, t > V < e, et > offended NP e N e Fred So the above type system correctly predicts that the example (1-a) is semantically interpretable. Consider the word every which has been assigned the type <et , ett>. We say the type <et , ett> is a locally correct type for the word every since this type assignment gives the expected result for a context, namely example (1-a). We call a type assignment, a globally correct assignment if the type assignment converges in all contexts containing the word. Whether there exist globally correct type assignments for lexical entries or not is a challenging question. There are at least four possibilities to be considered: (1) (Hypothesis A) Lexical items have globally correct type assignments. If the type assignment for a lexical item does not converge in a given interpretable context, then either the syntactic structure of the context needs to be changed or a new lexical entry needs to be defined. (2) (Hypothesis B) Some lexical items do not have globally correct type assignments but their locally correct assignments are systematically related.
2012 •
0. A universal, or almost universal distinction, in syntax: Sentence and NP.........................................................1 1. A possibly universal foundation for natural language semantics: types e and t..................................................1 2. Thought experiments: “Monocategoric”? and just one basic semantic type?....................................................3 3. Ingredients for a possible one-basic-type semantics..........................................................................................3 3.1. Neo-Davidsonian semantics of event sentences..........................................................................................3 3.2. Kamp-Heim semantics for indefinite NPs..................................................................................................3 3.3. Open formulas are “almost ” type-neutral....................................................................................................4 3.4. Exploit the similarit...
1991 •
This paper presents a type-free property-theoretic system in the spirit of a framework proposed by Menzel and then supplements it with a theory of truth and exemplification. The notions of a truth-relevantly complex (simple) sentence and of a truth-relevant subsentence are introduced and then used in order to motivate the proposed theory. Finally, it is shown how the theory avoids Russell's paradox and similar problems. Some potential applications to the foundations of mathematics and to natural language semantics are sketched in the introduction.
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