We describe a logic which is the same as first-order logic except that it allows control over the information that passes down from formulas to subformulas. For example the logic is adequate to express branching quantifiers. We describe a compositional semantics for this logic; in particular this gives a compositional meaning to formulas of the 'information-friendly' language of Hintikka and Sandu. For first-order formulas the semantics reduces to Tarski's semantics for first-order logic. We prove that two formulas have the same (...) interpretation in all structures if and only if replacing an occurrence of one by an occurrence of the other in a sentence never alters the truth-value of the sentence in any structure. (shrink)

Branching quantifiers were first introduced by L. Henkin in his 1959 paper ‘Some Remarks on Infmitely Long Formulas’. By ‘branching quantifiers’ Henkin meant a new, non-linearly structured quantiiier-prefix whose discovery was triggered by the problem of interpreting infinitistic formulas of a certain form} The branching (or partially-ordered) quantifier-prefix is, however, not essentially infinitistic, and the issues it raises have largely been discussed in the literature in the context of finitistic logic, as they will be here. Our discussion transcends, however, the (...) resources of standard lst-order languages and we will consider the new form in the context of 1st-order logic with 1- and 2-place ‘Mostowskian` generalized quantifiers.2.. (shrink)

Quantification is a topic which brings together linguistics, logic, and philosophy. Quantifiers are the essential tools with which, in language or logic, we refer to quantity of things or amount of stuff. In English they include such expressions as no, some, all, both, many. Peters and Westerstahl present the definitive interdisciplinary exploration of how they work - their syntax, semantics, and inferential role.

Connections between partially ordered connectives and Henkin quantifiers are considered. It is proved that the logic with all partially ordered connectives and the logic with all Henkin quantifiers coincide. This implies that the hierarchy of partially ordered connectives is strongly hierarchical and gives several nondefinability results between some of them. It is also deduced that each Henkin quantifier can be defined by a quantifier of the form equation imagewhat is a strengthening of the Walkoe result. MSC: 03C80.

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. (shrink)

We introduce an atomic formula ${\vec{y} \bot_{\vec{x}}\vec{z}}$ intuitively saying that the variables ${\vec{y}}$ are independent from the variables ${\vec{z}}$ if the variables ${\vec{x}}$ are kept constant. We contrast this with dependence logic ${\mathcal{D}}$ based on the atomic formula = ${(\vec{x}, \vec{y})}$ , actually equivalent to ${\vec{y} \bot_{\vec{x}}\vec{y}}$ , saying that the variables ${\vec{y}}$ are totally determined by the variables ${\vec{x}}$ . We show that ${\vec{y} \bot_{\vec{x}}\vec{z}}$ gives rise to a natural logic capable of formalizing basic intuitions about independence and dependence. (...) We show that ${\vec{y} \bot_{\vec{x}}\vec{z}}$ can be used to give partially ordered quantifiers and IF-logic an alternative interpretation without some of the shortcomings related to so called signaling that interpretations using = ${(\vec{x}, \vec{y})}$ have. (shrink)

Next SectionWe discuss the thesis formulated by Hintikka (1973) that certain natural language sentences require non-linear quantification to express their meaning. We investigate sentences with combinations of quantifiers similar to Hintikka's examples and propose a novel alternative reading expressible by linear formulae. This interpretation is based on linguistic and logical observations. We report on our experiments showing that people tend to interpret sentences similar to Hintikka sentence in a way consistent with our interpretation.

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. (shrink)

We discuss the thesis formulated by Hintikka that certain natural language sentences require non-linear quantification to express their meaning. We investigate sentences with combinations of quantifiers similar to Hintikka's examples and propose a novel alternative reading expressible by linear formulae. This interpretation is based on linguistic and logical observations. We report on our experiments showing that people tend to interpret sentences similar to Hintikka sentence in a way consistent with our interpretation.

We study the expressive power of open formulas of dependence logic introduced in Väänänen [Dependence logic (Vol. 70 of London Mathematical Society Student Texts), 2007]. In particular, we answer a question raised by Wilfrid Hodges: how to characterize the sets of teams definable by means of identity only in dependence logic, or equivalently in independence friendly logic.