Grounding,Quantifiers, and Paradoxes

Journal of Philosophical Logic - The notion of grounding is usually conceived as an objective and explanatory relation. It connects two relata if one—the ground—determines or explains...     

Quantifiers,Knowledge, and Counterfactuals

Many of the motivations in favor of contextualism about knowledge apply also to a contextualist approach to counterfactuals. I motivate and articulate such an approach, in terms of the context‐sensitive ‘all cases’, in the spirit of David Lewis’s contextualist view about knowledge. The resulting view explains intuitive data, resolves a puzzle parallel to the skeptical paradox, and renders safety and sensitivity, construed as counterfactuals, necessary conditions on knowledge.     

Truth,Partial Logic and Infinitary Proof Systems

In this paper we apply proof theoretic methods used for classical systems in order to obtain upper bounds for systems in partial logic. We focus on a truth predicate interpreted in a Kripke style way via strong Kleene; whereas the aim is to connect harmoniously the partial version of Kripke–Feferman with its intended semantics. The method we apply is based on infinitary proof systems containing an \(\omega \)-rule.     

Infinitary Action Logic: Complexity, Models and Grammars

Action logic of Pratt [21] can be presented as Full Lambek Calculus FL [14, 17] enriched with Kleene star *; it is equivalent to the equational theory of residuated Kleene algebras (lattices). Some results on axiom systems, complexity and models of this logic were obtained in [4, 3, 18]. Here we prove a stronger form of *-elimination for the logic of *-continuous action lattices and the –completeness of the equational theories of action lattices of subsets of a finite monoid and action lattices of binary relations on a finite universe. We also discuss possible applications in linguistics. Presented by Jacek Malinowski     

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.     

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.     

Proof Paradoxes,Agency, and Stereotyping

Many have attempted to justify various courts’ position that bare or naked statistical evidence is not sufficient for findings of liability. I provide a particular explanation by examining a different, but related, issue about when and why stereotyping is wrong. One natural explanation of wrongness of stereotyping appeals to agency. However, this has been scrutinised. In this paper, I argue that we should broaden our understanding of when and how our agency can be undermined. In particular, I argue that when we take seriously that our agency is exercised in the social world, we can see that stereotyping can and does undermine our agency by fixing the social meaning of our choices and actions as well as by reducing the quality and the kinds of choices that are available to us. Although this improves the agency-based explanation, it must be noted that undermining agency is not an overriding reason against stereotyping. Much depends on the balance of reasons that take into account moral stakes involved in a case of stereotyping. This results in a messier picture of when and why stereotyping is wrong, but I argue that this is a feature, not a bug. I end by applying this agency-based explanation to cases that have motivated the so-called Proof Paradoxes.     

Logical, Semantic and Cultural Paradoxes

The property common to three kinds of paradoxes (logical, semantic, and cultural) is the underlying presence of an exclusive disjunction: even when it is put to a check by the paradox, it is still invoked at the level of implicit discourse. Hence the argumentative strength of paradoxical propositions is derived. Logical paradoxes (insolubilia) always involve two contradictory, mutually exclusive, truths. One truth is always perceived to the detriment of the other, in accordance with a succession which is endlessly repetitive. A check is put on the principle of the excluded middle by the logical paradoxes, because self-reference leads to an endlessly repeating circle, out of which no resolution is conceivable. Logical paradoxes are to be compared with the `objective ambiguity' prevalent in oracles (Gallet, 1990). Semantic paradoxes are contextually-determined occurrences, whose resolution at the metalinguistic level is made possible by the discovery of a middle term. They express a wilful ambiguity, in which the interlocutor is invited to take an active part in the construction of sense, since what must be found is the unexpected sense thanks to which A and not-A can be asserted simultaneously. Cultural paradoxes play about doxa (`common sense') and openly challenge common opinion because of their character as inopinata (`unexpected'). My aim is to show that even cultural paradoxes hide sometimes a flaw of argumentation similar to logical or semantic paradox; they too imply an exclusive disjunction leading to the disappearance of the middle terms. Finally, basing myself on the theory of topoi (Anscombre and Ducrot, 1983), a tentative resolution of the cultural paradoxes will be suggested.     

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).     

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.     

Fluids,Molecules and Paradoxes of Infinity

Several paradoxes of infinity have recently featured in this journal involving gases distributed in a denumerable infinite series of compartments. I shall demonstrate in this paper that:

a) None of these new paradoxes applies where the gases comply with both Boyle’s law and Avogadro’s law. As several of these new paradoxes expressly require compliance with Boyle’s law, it is unclear, in principle, as to whether there is a plausible model of gas that is able to uphold them all.

b) Notwithstanding a), any of the above paradoxes (and their variations) can be reinstated by acknowledging (contrary to what is widely assumed in the literature) that there are two distinct, non-equivalent concepts of ideal gas. Indeed, the various infinity puzzles actually enable a distinction to be made between the two concepts (which is a particularly elegant way of doing so).

     

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.     

Minimalism and Paradoxes

This paper argues against minimalism about truth. It does so by way of acomparison of the theory of truth with the theory of sets, and considerationof where paradoxes may arise in each. The paper proceeds by asking twoseemingly unrelated questions. First, what is the theory of truth about?Answering this question shows that minimalism bears important similaritiesto naive set theory. Second, why is there no strengthened version ofRussell's paradox, as there is a strengthened Liar paradox? Answering thisquestion shows that like naive set theory, minimalism is unable to makeadequate progress in resolving the paradoxes, and must be replaced by adrastically different sort of theory. Such a theory, it is shown, must befundamentally non-minimalist.     

Antonyms and Paradoxes

Adjectives can be gradable or non-gradable and this aspect of their meaning is responsible for their different distribution and also for their classification into two different classes of antonyms. Non-gradable antonyms are called contradictories: they are neither true nor false together and exclude any middle term; gradable antonyms are called contraries: they are not simultaneously true, but may be simultaneously false. While with contraries a negative disjunction (neque...neque) can define an intermediate level, with contradictories it simply means that either term of the disjunction is excluded. There are however some Latin examples, such as neque vivus neque mortuus (`neither alive nor dead'), where the negation of a contradictory pair is used to convey a third, intermediate value. This third possibility is precisely what gives place to a paradox. Such an intermediate level can be defined also by terms like semivivus, semianimis (`half-dead'). Following Ducrot's theory on argumentation, such terms represent an argumentative attenuation, not with respect to life, rather with respect to death. With contradictories, in fact, the use of semi-, like the use of negation, gives the assertion of the opposite term as a result.     

Paradoxes: Their Roots,Range and Resolution

Book Information Paradoxes: Their Roots, Range and Resolution. Paradoxes: Their Roots, Range and Resolution Nicholas Rescher, Chicago and La Salle : Open Court, 2001, xxiii + 293, US$24.95 (paper). By Nicholas Rescher. Open Court. Chicago and La Salle. Pp. xxiii + 293. US$24.95 (paper:).,