Some Logical Properties of Natural Language Quantifiers

Abstract: We discuss and illustrate (some of) the following logical properties of quantification in natural language (NL):

- Natural classes of NL Determiners: intersective (INT; generalized existential), co-intersective (CO-INT; generalized universal), proportional; "presuppositional"
- The fundamental role of the (co-)intersective classes:
- the boolean closure of INT union CO-INT is the set of conservative (CONS) quantifiers;
- CONS is a universal property of NL quantifiers (with possibly two exceptions): CONS (Keenan) + EXTENSIONS (Van Benthem) = Domain Independence
- a quantifier is sortally reducible iff it is intersective or co-intersective (so restricting the range of "the variable" is an essential feature of NL quantification, given the existence of non-((CO)-INT quantifiers).

- Non-Fregean Quantifiers:
- n>1-ary quantifiers that are not reducible to iterated application of unary quantifiers.
- A new entailment paradigm for reducible quantifiers

- Characterizing syntactic classes of NPs:
- Those licensing nagative polarity items are the monotone decreasing NPs
- Those occurring in Existential There Ss are those built from CONS(2) Dets
- DPs occurring in the post of- position in partitives are those denoting principal filters

Reading: E.L. Keenan "Some Properties of Natural Language Quantifiers" in Linguistics and Philosophy vol 25: Nos 5-6. pp. 627-654

'One is a lonely number':

some recent trends in the logic of communication

Logic is not just about single agents describing fixed situations by monologues. Every speech act changes information states of interacting groups of people. This calls for dynamic-epistemic logics keeping track of such changes, due to Plaza, Gerbrandy, Baltag, van Ditmarsch, and others, such as Fagin-Halpern-Moses- Vardi, or Parikh. We explain how update logics work, and what broader issues arise. In particular, how do statements produce common knowledge (if/when they do), and how can we plan discourse in public settings without informing everyone about everything (since ignorance is the basis of civilised social life)? Such issues have clear analogies with analysis/synthesis of programs in computer science, and I conclude with new examples of what Parikh has called 'social software'.

References:

J. van Benthem, 2001, 'Logics for Information Update', Proceedings TARK VIII, Morgan Kaufmann, Los Altos, 51-88.

J. van Benthem, 2002, 'One is a lonely number: the logic of communication', to appear in P. Koepke et al., eds., "Logic Colloquium & Colloquium Logicum, Muenster 2002".

Kant's Philosophy of Mathematics in Perspective

Kant's approach to the philosophy of mathematics is virtually unique in the history of philosophy in that he assigns the capacity for a priori knowledge in this science to the faculty of sensibility rather than the intellect. (Kant takes the intellect to be the source of a priori knowledge in pure logic but sensibility to be the source of pure mathematics.) In this way Kant arrives at the quite unusual idea of a pure or a priori faculty of sensibility, whose structure is given by the "pure intuitions" of space and time. I explore the variety of factors motivating and sustaining Kant's unique conception, including his views on the relationship between pure mathematics and logic, the roles of construction and calculation in geometry and arithmetic, and the relationship between pure mathematics and sense perception (empirical intuition). In this way we see how pure and applied mathematics are related, for Kant, including especially a particular application in mathematical physics.

On the thesis that human languages are infinite

Introductory textbooks on formal linguistics lay considerable stress on the thesis that human languages are infinite collections of expressions. The arguments they give, however, are uniformly unsound. This paper dissects the logic of the arguments and exposes their failings, their hidden assumptions, and their links (or lack of links) to issues that are really about other things, like the abilities of speakers or the form in which grammars should be cast. It is argued that we do not have to accept that human languages are infinite collections, yet we also do not have to accept the (even more indefensible) position that they are finite. Despite appearances, there is no contradiction here.

A formalization of an approach to modality suggested by C. Peacocke

C. Peacocke suggested in Being Known, Oxford : Clarendon Press, 1999, a "principle based" approach to modality where the roles of actual world and other possible world are drastically different. The talk describes one of possible formalizations of this approach allowing to make precise some of the suggestions.

A User-Friendly Decision Procedure for the Probability Calculus, with Some Applications to Bayesian Philosophy of Science

I will present a simple, Mathematica-based decision procedure for a rather broad class of arguments in the probability calculus. Some background on the decision procedure will be presented, and then some applications of the procedure to Bayesian philosophy of science will be presented. The relevant background about Bayesianism will be provided as well. The procedure in question has already been used to solve dozens of problems in contemporary Bayesian confirmation theory (these applications were reported in the author's dissertation, and in several recent publications of the author as well).

On Gödel's philosophical development

Abstract: It is by now well known that Gödel first advocated the philosophy of Leibniz and then, since 1959, that of Husserl. This raises three questions:

- How is this turn to Husserl to be interpreted? Is it a complete dismissal of the Leibnizian philosophy, or a different way to achieve similar goals, or could the relation be even closer than that?
- Why did Gödel turn specifically to the later Husserl's transcendental idealism?
- Is there any detectable influence from Husserl on Gödel's writings?

Last modified: Fri Apr 4 13:47:13 PST 2003