Chang's modal operators in algebraic logic

Chang algebras as algebraic models for Chang's modal logics [1] are defined. The main result of the paper is a representation theorem for these algebras.     

Formal systems for modal operators on locales

In the paper [8], the first author developped a topos- theoretic approach to reference and modality. (See also [5]). This approach leads naturally to modal operators on locales (or spaces without points). The aim of this paper is to develop the theory of such modal operators in the context of the theory of locales, to axiomatize the propositional modal logics arising in this context and to study completeness and decidability of the resulting systems.     

Nominalizing Quantifiers

Quantified expressions in natural language generally are taken to act like quantifiers in logic, which either range over entities that need to satisfy or not satisfy the predicate in order for the sentence to be true or otherwise are substitutional quantifiers. I will argue that there is a philosophically rather important class of quantified expressions in English that act quite differently, a class that includes something, nothing, and several things. In addition to expressing quantification, such expressions act like nominalizations, introducing a new domain of objects that would not have been present in the semantic structure of the sentence otherwise. The entities those expressions introduce are of just the same sort as those that certain ordinary nominalizations refer to (such as John's wisdom or John's belief that S), namely they are tropes or entities related to tropes. Analysing certain quantifiers as nominalizing quantifiers will shed a new light on philosophical issues such as the status of properties and the nature of propositional attitudes.     

Set theory as modal logic

A logical systemBM ⁺ is proposed, which, is a prepositional calculus enlarged with prepositional quantifiers and with two modal signs, and These modalities are submitted to a finite number of axioms. is the usual sign of necessity, corresponds to transmutation of a property (to be white) into the abstract property (to be the whiteness). An imbedding of the usual theory of classesM intoBM ⁺ is constructed, such that a formulaA is provable inM if and only if(A) is provable inBM ⁺. There is also an inverse imbedding with an analogous property.     

Quantifiers and 'If'-Clauses

Stephen Barker ( The Philosophical Quarterly , 47 (1997), pp. 195–211) has presented a new argument for a pure material implication analysis of indicative conditionals. His argument relies crucially on the assumption that general indicatives such as 'Every girl, if she gets a chance, bungee-jumps' are correctly analysed as having the formal structure (for all x)(if x gets a chance, x bungee-jumps). This paper argues that an approach first proposed by David Lewis must be pursued: the 'if'-clause in these sentences restricts the quantifier. Only the Lewis-style analysis can deal with sentences involving non-universal quantifiers such as 'Most letters are answered if they are shorter than 5 pages'. I show that Barker's reasons for rejecting the restrictor analysis are not cogent and that the restrictor analysis connects widely with recent work in natural language semantics.     

Quantifiers in ontology

This paper is a reaction to G. Küng's and J. T. Canty's Substitutional Quantification and Leniewskian quantifiers'Theoria 36 (1970), 165–182. I reject their arguments that quantifiers in Ontology cannot be referentially interpreted but I grant that there is what can be called objectual — referential interpretation of quantifiers and that because of the unrestricted quantification in Ontology the quantifiers in Ontology should not be given a so-called objectual-referential interpretation. I explain why I am in agreement with Küng and Canty's recommendation that Ontology's quantifiers not be substitutionally interpreted even if Leniewski intended them to be so interpreted. A notion of an interpretation which is referential but yet which does not interpret as an assertor of existence of objects in a domain is developed. It is then shown that a first order version of Ontology is satisfied by those special kind of referential interpretations which read as Something as epposed to Something existing.Allatum est die 1 Junii 1976     

Quality and Quantifiers

I examine three ‘anti-object’ metaphysical views: nihilism (there are no objects at all), generalism (reality is ultimately qualitative), and anti-quantificationalism (quantification over objects does not perspicuously represent the world). After setting aside nihilism, I argue that generalists should be anti-quantificationalists. Along the way, I attempt to articulate what a ‘metaphysically perspicuous’ language might even be.     

The Ambiguity of Quantifiers

In the tradition of substructural logics, it has been claimed for a long time that conjunction and inclusive disjunction are ambiguous:we should, in fact, distinguish between ‘lattice’ connectives (also called additive or extensional) and ‘group’ connectives (also called multiplicative or intensional). We argue that an analogous ambiguity affects the quantifiers. Moreover, we show how such a perspective could yield solutions for two well-known logical puzzles: McGee’s counterexample to modus ponens and the lottery paradox.     

How to Precisify Quantifiers

I here argue that Ted Sider's indeterminacy argument against vagueness in quantifiers fails. Sider claims that vagueness entails precisifications, but holds that precisifications of quantifiers cannot be coherently described: they will either deliver the wrong logical form to quantified sentences, or involve a presupposition that contradicts the claim that the quantifier is vague. Assuming (as does Sider) that the “connectedness” of objects can be precisely defined, I present a counter-example to Sider's contention, consisting of a partial, implicit definition of the existential quantifier that in effect sets a given degree of connectedness among the putative parts of an object as a condition upon there being something (in the sense in question) with those parts. I then argue that such an implicit definition, taken together with an “auxiliary logic” (e.g., introduction and elimination rules), proves to function as a precisification in just the same way as paradigmatic precisifications of, e.g., “red”. I also argue that with a quantifier that is stipulated as maximally tolerant as to what mereological sums there are, precisifications can be given in the form of truth-conditions of quantified sentences, rather than by implicit definition.     

Quantifiers and Congruence Closure

We prove some results about the limitations of the expressive power of quantifiers on finite structures. We define the concept of a bounded quantifier and prove that every relativizing quantifier which is bounded is already first-order definable (Theorem 3.8). We weaken the concept of congruence closed (see [6]) to weakly congruence closed by restricting to congruence relations where all classes have the same size. Adapting the concept of a thin quantifier (Caicedo [1]) to the framework of finite structures, we define the concept of a meager quantifier. We show that no proper extension of first-order logic by means of meager quantifiers is weakly congruence closed (Theorem 4.9). We prove the failure of the full congruence closure property for logics which extend first-order logic by means of meager quantifiers, arbitrary monadic quantifiers, and the Härtig quantifier (Theorem 6.1).     

Socratic Proofs for Quantifiers

First-order logic is formalized by means of tools taken from the logic of questions. A calculus of questions which is a counterpart of the Pure Calculus of Quantifiers is presented. A direct proof of completeness of the calculus is given. *Research for this paper was supported by The Foundation for Polish Science (both authors), and indirectly (in the case of the first author) by a bilateral exchange project funded by the Ministry of the Flemish Community (project BIL 01/80) and the State Committee for Scientific Research, Poland.     

Voting by Eliminating Quantifiers

Mathematical theory of voting and social choice has attracted much attention. In the general setting one can view social choice as a method of aggregating individual, often conflicting preferences and making a choice that is the best compromise. How preferences are expressed and what is the “best compromise” varies and heavily depends on a particular situation. The method we propose in this paper depends on expressing individual preferences of voters and specifying properties of the resulting ranking by means of first-order formulas. Then, as a technical tool, we use methods of second-order quantifier elimination to analyze and compute results of voting. We show how to specify voting, how to compute resulting rankings and how to verify voting protocols.     

Quantifiers and epistemic contextualism

I defend a neo-Lewisean form of contextualism about knowledge attributions. Understanding the context-sensitivity of knowledge attributions in terms of the context-sensitivity of universal quantifiers provides an appealing approach to knowledge. Among the virtues of this approach are solutions to the skeptical paradox and the Gettier problem. I respond to influential objections to Lewis’s account.     

Modal Ontology and Generalized Quantifiers

Timothy Williamson has argued that in the debate on modal ontology, the familiar distinction between actualism and possibilism should be replaced by a distinction between positions he calls contingentism and necessitism. He has also argued in favor of necessitism, using results on quantified modal logic with plurally interpreted second-order quantifiers showing that necessitists can draw distinctions contingentists cannot draw. Some of these results are similar to well-known results on the relative expressivity of quantified modal logics with so-called inner and outer quantifiers. The present paper deals with these issues in the context of quantified modal logics with generalized quantifiers. Its main aim is to establish two results for such a logic: Firstly, contingentists can draw the distinctions necessitists can draw if and only if the logic with inner quantifiers is at least as expressive as the logic with outer quantifiers, and necessitists can draw the distinctions contingentists can draw if and only if the logic with outer quantifiers is at least as expressive as the logic with inner quantifiers. Secondly, the former two items are the case if and only if all of the generalized quantifiers are first-order definable, and the latter two items are the case if and only if first-order logic with these generalized quantifiers relativizes.     

Symmetric Propositions and Logical Quantifiers

Symmetric propositions over domain and signature are characterized following Zermelo, and a correlation of such propositions with logical type- quantifiers over is described. Boolean algebras of symmetric propositions over and Σ are shown to be isomorphic to algebras of logical type- quantifiers over . This last result may provide empirical support for Tarski’s claim that logical terms over fixed domain are all and only those invariant under domain permutations.     

Quantifiers Defined by Parametric Extensions

This paper develops a metaphysically flexible theory of quantification broad enough to incorporate many distinct theories of objects. Quite different, mutually incompatible conceptions of the nature of objects and of reference find representation within it. Some conceptions yield classical first-order logic; some yield weaker logics. Yet others yield notions of validity that are proper extensions of classical logic.