Online research in philosophy

Every recursively enumerable extension of arithmetic which obeys the disjunction property obeys the numerical existence property [Fr, 1]. The requirement of recursive enumerability is essential. For extensions of intuitionistic second order arithmetic by means of sentences (in its language) with no existential set quantifiers, the numerical existence property implies the set existence property. The restriction on existential set quantifiers is essential. The numerical existence property cannot be eliminated, but in the case of finite extensions of HAS, can be replaced by a weaker form of it. As a consequence, the set existence property for intuitionistic second order arithmetic can be proved within itself.

We characterize the generalized quantifiers Q which satisfy the scheme $QxQy\phi \leftrightarrow QyQx\phi$ , the so-called self-commuting quantifiers, or quantifiers with the Fubini property.

We say that a first order formula ϕ distinguishes a structure M over a vocabulary L from another structure M′ over the same vocabulary if ϕ is true on M but false on M′. A formula ϕ defines an L-structure M if ϕ distinguishes M from any other non-isomorphic L-structure M′. A formula ϕ identifies an n-element L-structure M if ϕ distinguishes M from any other non-isomorphic n-element L-structure M′. We prove that every n-element structure M is identifiable by a formula with quantifier rank less than (1 − 1/2k)n + k2 − k + 4 and at most one quantifier alternation, where k is the maximum relation arity of M. Moreover, if the automorphism group of M contains no transposition of two elements, the same result holds for definability rather than identification. The Bernays-Schönfinkel class consists of prenex formulas in which the existential quantifiers all precede the universal quantifiers. We prove that every n-element structure M is identifiable by a formula in the Bernays-Schönfinkel class with less than $(1-\frac{1}{2k})n+k^{2}-k+4$ quantifiers. If in this class of identifying formulas we restrict the number of universal quantifiers to k, then less than $n-\sqrt{n}+k^{2}+k$ quantifiers suffice to identify M and, as long as we keep the number of universal quantifiers bounded by a constant, at total $n-O(\sqrt{n})$ quantifiers are necessary.

Machine generated contents note: 1. What this book is about and how to use it; 2. Generalized quantifiers and their elements: operators and their scopes; 3. Generalized quantifiers in non-nominal domains; 4. Some empirically significant properties of quantifiers and determiners; 5. Potential challenges for generalized quantifiers; 6. Scope is not uniform and not a primitive; 7. Existential scope versus distributive scope; 8. Distributivity and scope; 9. Bare numeral indefinites; 10. Modified numerals; 11. Clause-internal scopal diversity; 12. Towards a compositional semantics of quantifier words.

Summary The article argues thatceteris paribus clauses have to be separated from another type of clauses called closure clauses. The former are associated with laws and theories, the latter with test situations of a particular kind. It is also argued that closure clauses, but notceteris paribus clauses, make Popper's falsifiability principle untenable. In that way, it also resolves the quarrel between Popper and Lakatos aboutceteris paribus clauses and falsifiability by saying that both are partly wrong and partly right.

The interpretation of if -clauses in the scope of ordinary quantifiers has provoked semanticists into extraordinary measures, such as abandoning compositionality or claiming that if has no meaning. We argue that if -clauses have a normal conditional meaning, even in the scope of ordinary quantifiers, and that the trick is to have the right semantics for conditionals.