Online research in philosophy

We show how sequent calculi for some generalized quantifiers can be obtained by generalizing the Herbrand approach to ordinary first order proof theory. Typical of the Herbrand approach, as compared to plain sequent calculus, is increased control over relations of dependence between variables. In the case of generalized quantifiers, explicit attention to relations of dependence becomes indispensible for setting up proof systems. It is shown that this can be done by turning variables into structured objects, governed by various types of structural rules. These structured variables are interpreted semantically by means of a dependence relation. This relation is an analogue of the accessibility relation in modal logic. We then isolate a class of axioms for generalized quantifiers which correspond to first-order conditions on the dependence relation.

Abstract: I explore Rousseau's account of the problem of dependence by means of an analysis of the distinction he makes between dependence on things and dependence on men. With reference to his Second Discourse, I argue that dependence on things alone exists only in the case of primitive man in the earliest stages of the state of nature, while dependence on men is more properly to be understood as dependence on other human beings as mediated by dependence on things. I go on to argue that in the light of Rousseau's account of dependence and his description in the Second Discourse of a spontaneous dependence and inequality generating process, there is a significant problem with his solution to the problem of dependence on other human beings proposed in the Social Contract. This problem can be understood in terms of the relation of the idea of will to that of necessity, and I suggest that Rousseau was himself aware of it.

The working assumption of this paper is that noncommuting variables are irreducibly interdependent. The logic of such dependence relations is the author's independence-friendly (IF) logic, extended by adding to it sentence-initial contradictory negation ¬ over and above the dual (strong) negation . Then in a Hilbert space turns out to express orthocomplementation. This can be extended to any logical space, which makes it possible to define the dimension of a logical space. The received Birkhoff and von Neumann quantum logic can be interpreted by taking their disjunction to be ¬(A & B). Their logic can thus be mapped into a Boolean structure to which an additional operator has been added.

We study definability of second-order generalized quantifiers. We show that the question whether a second-order generalized quantifier $\sQ_1$ is definable in terms of another quantifier $\sQ_2$, the base logic being monadic second-order logic, reduces to the question if a quantifier $\sQ^{\star}_1$ is definable in $\FO(\sQ^{\star}_2,<,+,\times)$ for certain first-order quantifiers $\sQ^{\star}_1$ and $\sQ^{\star}_2$. We use our characterization to show new definability and non-definability results for second-order generalized quantifiers. In particular, we show that the monadic second-order majority quantifier $\most^1$ is not definable in second-order logic.

and Data The essence of scope in natural language semantics can be characterized as follows: an expression e1 takes scope over an expression e2 iff the interpretation of the former affects the interpretation of the latter. Consider, for example, the sentence in (1) below, which is typical of the cases discussed in this paper in that it involves an indefinite and a universal (or, more generally, a non-existential) quantifier. (1) Everyx student in my class read ay paper about scope. How can we tell whether the indefinite in (1) is in the scope of the universal or not? We can answer this question in two ways. From a dependence-based perspective, Q y is in the scope of Qx if the values of the variable y (possibly) covary with the values of x. From an independence-based perspective, Q y is outside the scope of Qx if y’s value is fixed relative to the values of x. This brings us to the first of our two central questions: should the scopal properties of ordinary, ‘unmarked’ indefinites be characterized in terms of dependence or in terms of independence? The difference between these two conceptualizations is that a dependence-based approach establishes which quantifier(s) Q y is dependent on, while an independence-based approach establishes which quantifier(s) Q y is independent of. Logical semantics has taken both paths to the notion of scope: compare the standard, dependence-based semantics of first-order logic (FOL) – or the dependence-driven Skolemization procedure – with the independence-based semantics of Independence-Friendly Logic (IFL, Hintikka 1973, Sandu 1993, Hodges 1997, Väänänen 2007 among others). Natural language semantics has only taken the dependence-based path.

We take a fresh look at the logics of informational dependence and independence of Hintikka and Sandu and Väänänen, and their compositional semantics due to Hodges. We show how Hodges’ semantics can be seen as a special case of a general construction, which provides a context for a useful completeness theorem with respect to a wider class of models. We shed some new light on each aspect of the logic. We show that the natural propositional logic carried by the semantics is the logic of Bunched Implications due to Pym and O’Hearn, which combines intuitionistic and multiplicative connectives. This introduces several new connectives not previously considered in logics of informational dependence, but which we show play a very natural rôle, most notably intuitionistic implication. As regards the quantifiers, we show that their interpretation in the Hodges semantics is forced, in that they are the image under the general construction of the usual Tarski semantics; this implies that they are adjoints to substitution, and hence uniquely determined. As for the dependence predicate, we show that this is definable from a simpler predicate, of constancy or dependence on nothing. This makes essential use of the intuitionistic implication. The Armstrong axioms for functional dependence are then recovered as a standard set of axioms for intuitionistic implication. We also prove a full abstraction result in the style of Hodges, in which the intuitionistic implication plays a very natural rôle.

We show that for any pair $\phi$ and $\psi$ of contradictory formulas of dependence logic there is a formula $\theta$ of the same logic such that $\phi\equiv\theta$ and $\psi\equiv\neg\theta$. This generalizes a result of Burgess.

Dependence is a common phenomenon, wherever one looks: ecological systems, astronomy, human history, stock markets - but what is the logic of dependence? This book is the first to carry out a systematic logical study of this important concept, giving on the way a precise mathematical treatment of Hintikka’s independence friendly logic. Dependence logic adds the concept of dependence to first order logic. Here the syntax and semantics of dependence logic are studied, dependence logic is given an alternative game theoretic semantics, and results about its complexity are proven. This is a graduate textbook suitable for a special course in logic in mathematics, philosophy and computer science departments, and contains over 200 exercises, many of which have a full solution at the end of the book. It is also accessible to readers, with a basic knowledge of logic, interested in new phenomena in logic.