Recently a number of works in meta-ontology have used a variant of J.H. Harris's collapse argument in the philosophy of logic as an argument against Eli Hirsch's quantifier variance. There have been several responses to the argument in the literature, but none of them have identified the central failing of the argument, viz., the argument has two readings: one on which it is sound but doesn't refute quantifier variance and another on which it is unsound. The central (...) lesson I draw is that arguments against quantifier variance must pay strict attention to issues of translation and interpretation. The paper also has a substantial appendix in which I prove the equivalence of plural mereological nihilism and standard first-order atomistic mereology; results of this kind are often appealed to in the literature on quantifier variance but without many details on the nature or proof of the result. (shrink)

Quantifier variance holds that different languages can have unrestricted quantifier expressions that differ in meaning, where an expression is a “quantifier” just in case it plays the right inferential role. Several critics argued that J.H. Harris’s “collapse” argument refutes variance by showing that identity of inferential role is incompatible with meaning variance. This standard, syntactic collapse argument has generated several responses. More recently, Cian Dorr proved semantic collapse theorems to generate a semantic collapse (...) argument against variance. The argument is significantly different from standard collapse, so it requires a new response. Here I clarify and analyze the semantic collapse argument, and explain how variantists can and should respond to it. The paper also includes an appendix showing the difficulties of positing identity variance without quantifier variance. The argument in the appendix has yet to appear in print, but is familiar to specialists. (shrink)

The aim of the paper is to bring clarity regarding the doctrine of quantifier variance (due to Eli Hirsch), and two prominent arguments against this doctrine, the collapse argument and the Eklund-Hawthorne argument. Different versions of the doctrine of quantifier variance are distinguished, and it is shown that the effectiveness of the arguments against it depends on what version of the doctrine is at issue. The metaontological significance of the different versions of the doctrine are also assessed. (...) Roughly, quantifier variance concerns there being different possible existential quantifier meanings, and often the doctrine involves a claim to the effect there is no unique “best” quantifier meaning. Much of the discussion in the paper concerns what it is to be an existential quantifier meaning in the sense at issue. (shrink)

The thesis of quantifier variance is consistent and cannot be refuted via a collapse argument.

Quantifier variantists accept multiple alternative ontological languages in which quantifiers obey the usual inference rules despite having different meanings. But collapse arguments seem to show that these quantifiers would be provably equivalent to one another. Cian Dorr has pushed this discussion forward by formulating the collapse argument in terms of an algebra of meanings that are common amongst the languages. I attempt to show that quantifier variantists can respond. But an important distinction between types of (...) class='Hi'>quantifier variance emerges, between those in which quantifier meanings draw on a single objective backbone of “portions of reality,” and those (such as the type that is arguably associated with neo-Fregeanism) in which they do not. (shrink)

Henkin quantifiers have been introduced in Henkin (1961). Walkoe (1970) studied basic model-theoretical properties of an extension $L_{*}^{1}$ (H) of ordinary first-order languages in which every sentence is a first-order sentence prefixed with a Henkin quantifier. In this paper we consider a generalization of Walkoe's languages: we close $L_{*}^{1}$ (H) with respect to Boolean operations, and obtain the language L¹(H). At the next level, we consider an extension $L_{*}^{2}$ (H) of L¹(H) in which every sentence is an L¹(H)-sentence prefixed (...) with a Henkin quantifier. We repeat this construction to infinity. Using the (un)definability of truth - in - N for these languages, we show that this hierarchy does not collapse. In addition, we compare some of the present results to the ones obtained by Kripke (1975), McGee (1991), and Hintikka (1996). (shrink)

Quantifier variance faces a number of difficulties. In this paper we first formulate the view as holding that the meanings of the quantifiers may vary, and that languages using different quantifiers may be charitably translated into each other. We then object to the view on the basis of four claims: (i) quantifiers cannot vary their meaning extensionally by changing the domain of quantification; (ii) quantifiers cannot vary their meaning intensionally without collapsing into logical pluralism; (iii) quantifier variance is (...) not an ontological doctrine; (iv) quantifier variance is not compatible with charitable translation and as such is internally inconsistent. In light of these troubles, we recommend the dissolution of quantifier variance and suggest that the view be laid to rest. (shrink)

Eli Hirsch recently suggested the metaontological doctrine of so-called "quantifier variance", according to which ontological disputes—e.g. concerning the question whether arbitrary, possibly scattered, mereological fusions exist, in the sense that these are recognised as objects proper in our ontology—can be defused as insubstantial. His proposal is that the meaning of the quanti er `there exists' varies in such debates: according to one opponent in this dispute, some existential statement claiming the existence of, e.g., a scattered object is true, according (...) to the other it is not. This paper argues that Hirsch's proposal leads into inconsistency. (shrink)

Quantifier variance entails that ‘there exists’ has a variety of meanings. Determining what makes all these meanings quantifier meanings is a problem associated with this view. A reasonable candidate suggested by Hirsch is the set of formal rules governing quantification. However, the collapse argument presents a notorious objection to the viability of the candidate: there cannot be more than one quantifier obeying the same rules up to logical equivalence. It is proposed that a quantifier variantist (...) who intends to retain Hirsch’s proposed solution can admit a many-sorted language in order to block the collapse argument. Within the many-sorted language, a Kaplanian thesis about the rules governing sorted quantifiers saves quantifier variance from meaning variance. (shrink)

One of the main criticisms of the theory of collections of indiscernible objects is that once we quantify over one of them, we are quantifying over all of them since they cannot be discerned from one another. In this way, we would call the collapse of quantifiers: ‘There exists one x such as P’ would entail ‘All x are P’. In this paper we argue that there are situations (quantum theory is the sample case) where we do refer to (...) a certain quantum entity, saying that it has a certain property, even without committing all other indistinguishable entities with the considered property. Mathematically, within the realm of the theory of quasi-sets $$\mathfrak {Q}$$, we can give sense to this claim. We show that the above-mentioned ‘collapse of quantifiers’ depends on the interpretation of the quantifiers and on the mathematical background where they are ranging. In this way, we hope to strengthen the idea that quantification over indiscernibles, in particular in the quantum domain, does not conform with quantification in the standard sense of classical logic. (shrink)

The Hilbert program was actually a specific approach for proving consistency, a kind of constructive model theory. Quantifiers were supposed to be replaced by ε-terms. εxA(x) was supposed to denote a witness to ∃xA(x), or something arbitrary if there is none. The Hilbertians claimed that in any proof in a number-theoretic system S, each ε-term can be replaced by a numeral, making each line provable and true. This implies that S must not only be consistent, but also 1-consistent. Here we (...) show that if the result is supposed to be provable within S, a statement about all Pi-0-2 statements that subsumes itself within its own scope must be provable, yielding a contradiction. The result resembles Gödel's but arises naturally out of the Hilbert program itself. (shrink)

There has been an upsurge of interest lately in developing Wigner’s hypothesis that conscious observation causes collapse by exploring dynamical collapse models in which some purportedly quantifiable aspect of consciousness resist superposition. Kremnizer–Ranchin, Chalmers–McQueen and Okon–Sebastián have explored the idea that collapse may be associated with a numerical measure of consciousness. More recently, Chalmers–McQueen have argued that any single measure is inadequate because it will allow superpositions of distinct states of equal consciousness measure to persist. They suggest (...) a satisfactory model needs to associate collapse with a set of measures quantifying aspects of consciousness, such as the “Q-shapes” defined by Tononi et al. in their “integrated information theory” of consciousness. I argue here that Chalmers–McQueen’s argument against associating a single measure with collapse requires a precise symmetry between brain states associated with different experiences and thus does not apply to the only case where we have strong intuitions, namely human observers. In defence of Chalmers–McQueen’s stance, it might be argued that idealized artificial information processing networks could display such symmetries. However, I argue that the most natural form of any theory that postulates a map from network states to mind states is one that assigns identical mind states to isomorphic network states. This suggests that, if such a map exists, no familiar components of mind states, such as viewing different colours, or experiencing pleasure or pain, are likely to be related by symmetries. (shrink)

We shall try to defend two non-standard views that run counter to two well-entrenched familiar views. The standard views are the universal and existential quantifiers of first-order logic are not modal operators, and the quantifiers are extensional. If that is correct then the counterclaims create genuine problems for some traditional philosophical doctrines.

This paper raises the question under what circumstances a plurality forms a set. My main point is that not always all things form sets. A provocative way of presenting my position is that, as a result of my approach, there are more pluralities than sets. Another way of presenting the same thesis claims that there are ways of talking about objects that do not always collapse into sets. My argument is related to expressive powers of formal languages. Assuming classical (...) logic, I show that if all plurality form a set and the quantifiers are absolutely general, then one gets a trivial theory. So, by reductio, one has to abandon one of the premiss. Then, I argue against the collapse of the pluralities into sets. What I am advocating is that the thesis of collapse limits important applications of the plural logic in model theory, when it is assumed that the quantifiers are absolutely general. (shrink)

Reinhardt’s conjecture, a formalization of the statement that a truthful knowing machine can know its own truthfulness and mechanicalness, was proved by Carlson using sophisticated structural results about the ordinals and transfinite induction just beyond the first epsilon number. We prove a weaker version of the conjecture, by elementary methods and transfinite induction up to a smaller ordinal.

A combinatorial criterium is given when a monadic quantifier is expressible by means of universe-independent monadic quantifiers of width n. It is proved that the corresponding hierarchy does not collapse. As an application, it is shown that the second resumption (or vectorization) of the Hartig quantifier is not definable by monadic quantifiers. The techniques rely on Ramsey theory.

There are logics where necessity is defined by means of a given identity connective: \ is a tautology). On the other hand, in many standard modal logics the concept of propositional identity \ can be defined by strict equivalence \}\). All these approaches to modality involve a principle that we call the Collapse Axiom : “There is only one necessary proposition.” In this paper, we consider a notion of PI which relies on the identity axioms of Suszko’s non-Fregean logic (...) SCI. Then S3 proves to be the smallest Lewis modal system where PI can be defined as SE. We extend S3 to a non-Fregean logic with propositional quantifiers such that necessity and PI are integrated as non-interdefinable concepts. CA is not valid and PI refines SE. Models are expansions of SCI-models. We show that SCI-models are Boolean prealgebras, and vice-versa. This associates non-Fregean logic with research on Hyperintensional Semantics. PI equals SE iff models are Boolean algebras and CA holds. A representation result establishes a connection to Fine’s approach to propositional quantifiers and shows that our theories are conservative extensions of S3–S5, respectively. If we exclude the Barcan formula and a related axiom, then the resulting systems are still complete w.r.t. a simpler denotational semantics. (shrink)

When restricted to proving $\Sigma _{i}^{q}$ formulas, the quantified propositional proof system $G_{i}^{\ast}$ is closely related to the $\Sigma _{i}^{b}$ theorems of Buss's theory $S_{2}^{i}$ . Namely, $G_{i}^{\ast}$ has polynomial-size proofs of the translations of theorems of $S_{2}^{i}$ , and $S_{2}^{i}$ proves that $G_{i}^{\ast}$ is sound. However, little is known about $G_{i}^{\ast}$ when proving more complex formulas. In this paper, we prove a witnessing theorem for $G_{i}^{\ast}$ similar in style to the KPT witnessing theorem for $T_{2}^{i}$ . This witnessing theorem (...) is then used to show that $S_{2}^{i}$ proves $G_{i}^{\ast}$ is sound with respect to $\Sigma _{i+1}^{q}$ formulas. Note that unless the polynomial-time hierarchy collapses $S_{2}^{i}$ is the weakest theory in the s₂ hierarchy for which this is true. The witnessing theorem is also used to show that $G_{i}^{\ast}$ is p-equivalent to a quantified version of extended-Frege for prenex formulas. This is followed by a proof that Gi p-simulates $G_{i+1}^{\ast}$ with respect to all quantified propositional formulas. We finish by proving that S₂ can be axiomatized by $S_{2}^{1}$ plus axioms stating that the cut-free version of $G_{0}^{\ast}$ is sound. All together this shows that connection between $G_{i}^{\ast}$ and $S_{2}^{i}$ does not extend to more complex formulas. (shrink)

We prove convergence laws for logics of the form equation image, where equation image is a properly chosen collection of generalized quantifiers, on very sparse finite random structures. We also study probabilistic collapsing of the logics equation image, where equation image is a collection of generalized quantifiers and k ∈ ℕ+, under arbitrary probability measures of finite structures.

We consider the problem of obtaining logical characterisations of oracle complexity classes. In particular, we consider the complexity classes LOGSPACENP and PTIMENP. For these classes, characterisations are known in terms of NP computable Lindström quantifiers which hold on ordered structures. We show that these characterisations are unlikely to extend to arbitrary structures, since this would imply the collapse of certain exponential complexity hierarchies. We also observe, however, that PTIMENP can be characterised in terms of Lindström quantifers , though it (...) remains open whether this can be done for LOGSPACENP. (shrink)

In this paper we develop a novel propositional semantics based on the framework of branching time. The basic idea is to replace the moment-history pairs employed as parameters of truth in the standard Ockhamist semantics by pairs consisting of a moment and a consistent, downward closed set of so-called transitions. Whereas histories represent complete possible courses of events, sets of transitions can represent incomplete parts thereof as well. Each transition captures one of the alternative immediate future possibilities open at a (...) branching point. The transition semantics exploits the structural resources a branching time structure has to offer and provides a fine-grained picture of the interrelation of modality and time. In addition to temporal and modal operators, a so-called stability operator becomes interpretable as a universal quantifier over the possible future extensions of a given transition set. The stability operator allows us to specify how and how far time has to unfold for the truth value of a sentence at a moment to become settled and enables a perspicuous treatment of future contingents. We show that the semantics developed along those lines generalizes and extends extant approaches: both Peirceanism and Ockhamism can be viewed as limiting cases of the transition approach that build on restricted resources only, and on both accounts, stability collapses into truth. (shrink)

Considering the effect of the local topology structure of an edge on cascading failures, we investigate the cascading reaction behaviors on scale-free networks with respect to small edge-based initial attacks. Our aim is to explore the relationship between some parameters and universal robustness characteristics against cascading failures on scale-free networks. We find by the theoretical analysis that the Baraba'si-Albert (BA) scale-free networks can reach the strongest robustness level against cascading failures when α + β = 1, where the robustness is (...) quantified by a transition from normal state to collapse. And the network robustness has a positive correlation with the average degree. We furthermore confirm by the numerical simulations these results. (shrink)

An appraisal of the current status of the modalities and of quine's arguments against them. The author accepts "quine's thesis," that one cannot quantify into referentially opaque contexts, And argues that nobody has succeeded in making sense of such quantification. However, It is shown that modal constructions, Being constructions on general terms and sentences, Can be referentially transparent and extensionally opaque and that consequently the collapse of modal distinctions warned against by quine in "word and object" can be avoided. (...) This combination of referential transparency and extensional opacity is just what quine means by essentialism, And the author therefore agrees with quine that quantified modal logic commits one to essentialism. (shrink)

Data involving epistemic modals suggest that some classically valid argument forms, such as _reductio_, are invalid in natural language reasoning as they lead to modal collapses. We adduce further data showing that the classical argument forms governing the existential quantifier are similarly defective, as they lead to a _de re–de dicto_ collapse. We observe a similar problem for disjunction. But if the classical argument forms for negation, disjunction and existential quantification are invalid, what are the correct forms that (...) govern the use of these items? Our diagnosis is that epistemic modals interfere with hypothetical reasoning. We present a modal first-order logic and model theory that characterizes hypothetical reasoning with epistemic modals in a principled manner. One upshot is a sound and complete natural deduction system for reasoning with epistemic modals in first-order logic. (shrink)

Inspired by the text entitled The Collapse of the Fact/Value Dichotomy and Other Essays (2004) of Hilary Putnam, the volume focuses on the theory and practice of knowledge, but one can legitimately extend it to other fields, most especially in aesthetics. Certain observable features in the fields of aesthetics, practice and artistic creation show that old evaluation criteria may now be obsolete. This is because upon further consideration, the definition of value remains opaque : should the artwork be judged (...) according to its moral value, its market value, or its formal value ? To side with or against the concept of value in art and aesthetics does not preclude a certain number of differences concerning the very nature of what is meant by value. If traditionally the ‘fact’ was the work, taken in all its tangibility, currently the materiality of the object no longer seems to play such a major role. Art is increasingly populated by so-called ‘immaterial’ or ‘ephemeral’ works and is therefore rarely, or badly, quantifiable according to old aesthetic and economic evaluation criteria. The progressively pragmatic contextualization of works within the social space puts forward a new definition of aesthetic value, no longer eternal and ideal, but rather anchored in the sensible and the politico-economic issues of a culturally specific situation. The current changes too support the idea that the question of ‘value’ and its confrontation with the concept of ‘fact’ is urgent. What role do artistic practices and aesthetic theories play ? A role of emancipation, of liberation, escape, or transformation ? Or on the contrary, could art become another means to subject individuals to the status quo ? As can easily be noticed, the question of maintaining or rejecting the dichotomy of facts and values is at the heart of the most pressing issues. (shrink)

Following David Lewis, Ted Sider has famously argued that unrestricted first-order quantification cannot be vague. His argument was intended as a type of reductio: its strategy was to show that the mere hypothesis of unrestricted quantifier vagueness collapses into the claim that unrestricted quantification is precise. However, this short article considers two natural reconstructions of the argument, and shows that each can be resisted. The theme will be that each reconstruction of the argument involves assumptions which advocates of vague (...) quantification have independent reason to reject. (shrink)

This paper is motivated by the search for a natural and deductively powerful extension of classical set theory. A theory of properties U is developed, based on a system of relevant logic related to RQ. In U the set {a, b, c,...} is identified with the property [x: x=a x=b x=c...]. The universe of all sets V, is identified with the property of being a hereditary set. The main result is that relevant implication collapses to material implication for sentences with (...) quantifiers restricted to V. This demonstrates the naturalness of the system. However, an aparent lack of deductive power leads to the conclusion that the best extension of classical set theory is to be found in intensional theories with the unrestricted comprehension schema based on weak relevant logics. The author has obtained similar collapses of to for these systems. (shrink)

We consider several natural fragments of the alternating-time temporal logics ATL* and ATL with restrictions on the nesting between temporal operators and strategic quantifiers. We develop optimal decision procedures for satisfiability in these fragments, showing that they have much lower complexities than the full languages. In particular, we prove that the satisfiability problem for state formulae in the full `strategically flat' fragment of ATL* is PSPACE-complete, whereas the satisfiability problems in the flat fragments of ATL and ATL$^{+}$ are $\Sigma^P_3$-complete. We (...) note that the nesting hierarchies for fragments of ATL* collapse in terms of expressiveness above nesting depth 1, hence our results cover all such fragments with lower complexities. (shrink)

In his recent article, ``Self-Consciousness'’, George Bealer has set outa novel and interesting argument against functionalism in the philosophyof mind. I shall attempt to show, however, that Bealer's argument cannotbe sustained.In arguing for this conclusion, I shall be defending three main theses.The first is connected with the problem of defining theoreticalpredicates that occur in theories where the following two features arepresent: first, the theoretical predicate in question occurswithin both extensional and non-extensional contexts; secondly, thetheory in question asserts that the relevant (...) theoretical states enterinto causal relations. What I shall argue is that a Ramsey-styleapproach to the definition of such theoretical terms requires twodistinct quantifiers: one which ranges over concepts, and theother which ranges over properties in the world.My second thesis is a corollary: since the theories on whichBealer is focusing have both of the features just mentioned, and sincethe method that he employs to define theoretical terms in his argumentagainst functionalism does not involve both quantifiers that range overproperties and quantifiers that range over concepts, that method isunsound.My final thesis is that when a sound method is used, Bealer's argumentagainst functionalism no longer goes through.The structure of my discussion is as follows. I begin by setting out twoarguments – the one, a condensed version of Bealer's argument, andthe other, an argument that parallels Bealer's argument very closely.The parallel argument leads to a conclusion, however, that, rather thanbeing merely somewhat surprising, seems very implausible indeed. Forwhat the second argument establishes, if sound, is that there can betheoretical terms that apply to objects by virtue of their first-orderphysical properties, but whose meaning cannot be defined via aRamsey-style approach.Having set out the two parallel arguments, I then go on to focus uponthe second, to determine what is wrong with it. My diagnosis will bethat the problem with the argument arises from the fact that it involvesdefining a theoretical term that occurs both inside and outside ofopaque contexts, for the method employed fails to take into account thefact that the types of entities that are involved in the relevanttruthmakers are different when a sentence occurs within an extensionalcontext from those involved when a sentence occurs within anon-extensional context.I then go on to discuss how one should define a theoretical term thatoccurs within such theories, and I argue that in such a case one needstwo quantifiers, ranging over different types of entities – on theone hand, over properties and relations, and the other, over concepts. Ithen show that, when such an approach is followed, the argument inquestion collapses.I then turn to Bealer's argument against functionalism, and I show,first, that precisely the same method of defining theoretical terms canbe applied there, and, secondly, that, when this is done, it turns outthat that argument is also unsound.Next, I consider two responses that Bealer might make to my argument,and I argue that those responses would not succeed.Finally, I conclude by asking exactly where the problem lies in the caseof Bealer's argument. My answer will be that it is not simply the factthat one is dealing with a theoretical term that occurs in bothextensional and non-extensional contexts. It is rather the combinationof that feature together with the fact that the theory in questionasserts that the relevant type of theoretical state enters into causalrelations. For the first of these features means that the Ramseysentence for the theory must involve quantification over concepts, whilethe presence of the second feature means that the Ramsey sentence mustinvolve quantification over properties in the world, and so no attemptto offer a Ramsey-style account of the meaning of the relevanttheoretical term can succeed unless one employs both quantification overconcepts and quantification over properties. Bealer, however, in hisargument against functionalism, uses a method of defining theoreticalterms that does not involve both types of quantification, and it isprecisely because of this that his argument does not in the end succeed. (shrink)

This document collects natural derivation systems for logics described in Priest, An Introduction to Non-Classical Logic [4]. It provides an alternative or supplement to the semantic tableaux of his text. Except that some chapters are collapsed, there are sections for each chapter in Priest, with an additional, final section on quantified modal logic. In each case, (i) the language is briefly described and key semantic definitions stated, (ii) the derivation system is presented with a few examples given, and (iii) soundness (...) and completeness are proved. There should be enough detail to make the parts accessible to students would work through parallel sections of Priest. (shrink)

It is and has been widely assumed, e.g., in Hintikka and Hintikka, that the logical theory available from Wittgenstein’s Tractatus Logico-Philosophicus affords a foundation for the conventional logic represented in standard formulations of classical propositional, first-order predicate, and perhaps higher-order formal systems. The present article is a detailed attempt at a mathematical demonstration, or as much demonstration as the sources will allow, that this assumption is false by contemporary lights and according to a preferred account of argument validity. When Wittgenstein’s (...) description of the forms of propositions or Sätze in the 5-numbered remarks and Remark 6 is given a close reconstruction, one sees that no Tractarian proposition is logically equivalent to a simple universally or existentially quantified formula of first-order predicate logic. Therefore, although Wittgenstein employs the sign ∀—or, in his notation—occasionally in explanations and illustrations, e.g., 4.0411 and 5.1311, when it comes to logic, the sign ∀ should receive in Tractatus a semantical treatment that is nonstandard. En route to that result, we show that the hierarchy of variables–and, hence, of propositions–defined at 5.501 incorporates the expressive power of finitary classical propositional logic. Also, when constructed over a first-order language for arithmetic, formulae corresponding to the specification of 5.501 pick out all and only arithmetic sets of numbers. Consequently, Wittgenstein’s hierarchy of iterated N-propositions–as described in vide Remark 6–does not collapse: at any level k, one finds propositions at k + 1 or above that are not logically equivalent to any proposition formed at k or below. (shrink)

In a well-known passage in the last chapter of Frege: Philosophy of Mathematics Michael Dummett suggests that Frege’s major “mistake”—the key to the collapse of the project of Grundgesetze—consisted in “his supposing there to be a totality containing the extension of every concept defined over it; more generally [the mistake] lay in his not having the glimmering of a suspicion of the existence of indefinitely extensible concepts” (Dummett [1991, 317]). Now, claims of the form, Frege fell into paradox because……. (...) are notoriously difficult to assess even when what replaces the dots is relatively straightforward. Offerings have included, for instance, that — (A) Unrestricted quantification: Frege fell into paradox because he allowed himself to quantify over a single, all-inclusive domain of objects (Russell, Dummett). (shrink)

A Kaufmann model is an \(\omega _1\) -like, recursively saturated, rather classless model of \({{\mathsf {P}}}{{\mathsf {A}}}\) (or \({{\mathsf {Z}}}{{\mathsf {F}}} \) ). Such models were constructed by Kaufmann under the combinatorial principle \(\diamondsuit _{\omega _1}\) and Shelah showed they exist in \(\mathsf {ZFC}\) by an absoluteness argument. Kaufmann models are an important witness to the incompactness of \(\omega _1\) similar to Aronszajn trees. In this paper we look at some set theoretic issues related to this motivated by the seemingly (...) naïve question of whether such a model can be “killed” by forcing without collapsing \(\omega _1\). We show that the answer to this question is independent of \(\mathsf {ZFC}\) and closely related to similar questions about Aronszajn trees. As an application of these methods we also show that it is independent of \(\mathsf {ZFC}\) whether or not Kaufmann models can be axiomatized in the logic \(L_{\omega _1, \omega } (Q)\) where _Q_ is the quantifier “there exists uncountably many”. (shrink)

By a result of L.D. Beklemishev, the hierarchy of nested applications of the $$\Sigma _1$$ -collection rule over any $$\Pi _2$$ -axiomatizable base theory extending Elementary Arithmetic collapses to its first level. We prove that this result cannot in general be extended to base theories of arbitrary quantifier complexity. In fact, given any recursively enumerable set of true $$\Pi _2$$ -sentences, S, we construct a sound $$(\Sigma _2 \! \vee \! \Pi _2)$$ -axiomatized theory T extending S such that (...) the hierarchy of nested applications of the $$\Sigma _1$$ -collection rule over T is proper. Our construction uses some results on subrecursive degree theory obtained by L. Kristiansen. (shrink)

In this article, we discuss a simple argument that modal metaphysics is misconceived, and responses to it. Unlike Quine's, this argument begins with the simple observation that there are different candidate interpretations of the predicate ‘could have been the case’. This is analogous to the observation that there are different candidate interpretations of the predicate ‘is a member of’. The argument then infers that the search for metaphysical necessities is misguided in much the way the ‘set-theoretic pluralist’ claims that the (...) search for the true axioms of set theory is. We show that the obvious responses to this argument fail. However, a new response has emerged that purports to prove, from higher-order logical principles, that metaphysical possibility is the broadest kind of possibility applying to propositions, and is to that extent special. We distill two lines of reasoning from the literature, and argue that their import depends on premises that any ‘modal pluralist’ should deny. Both presuppose that there is a unique typed hierarchy, which is what the modal pluralist, in the context of higher-order logic, should disavow. In other words, both presupposes that there is a unique candidate for what higher-order claims could mean. We consider the worry that, in a higher-order setting, modal pluralism faces an insuperable problem of articulation, collapses into modal monism, is vulnerable to the Russell-Myhill paradox, or even contravenes the truism that there is a unique actual world, and argue that these worries are misplaced. We also sketch the bearing of the resulting ‘Higher-Order Pluralism’ on the theory of content. One theme of the discussion is that, if Higher-Order Pluralism is correct, then there is no fixed metatheory from which to characterize higher-order reality. (shrink)

This work examines a number of arguments to the effect that quantification requires identity of the objects that are quantified over; the arguments concern the domain of quantification, the range of quantifiers, the collapse of the existential and the universal quantifiers, and the intelligibility of quantification. The central role of identity in quantification is identified in each case. Also considered is quantification in non-classical contexts, and it is argued that even in logics and set theories that supposedly do not (...) demand identity for quantification, identity is still presupposed. Along the way, some recent challenges to this overall approach are considered. (shrink)

ABSTRACT There has been recent interest in the idea that, when metaphysicians disagree over the truth of (say) ‘There are numbers’ or ‘Chairs exist’, their dispute is merely verbal. This idea has been taken to motivate quantifier variance, the view that the meanings of quantifier expressions vary across different ontological languages, and that each of these meanings is of equal metaphysical merit. I argue that quantifier variance cannot be upheld in light of natural language theorists’ analyses of (...) quantifier expressions. The idea that metaphysicians are engaged in verbal disputes can be maintained only through alternative strategies that have nothing to do with quantifier expressions. (shrink)

Føllesdal (1966 and 1969) claims that modal distinctions collapse, that is that p and Necessarily p are equivalent, in any system of modal logic that incorporates a standard theory of definite descriptions, like the one proposed by Carnap in M&N. I argue that his argument fails.

This work presents a model-theoretic approach to the study of the amalgamation property for varieties of semilinear commutative residuated lattices. It is well-known that if a first-order theory T enjoys quantifier elimination in some language L, the class of models of the set of its universal consequences \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\rm T_\forall}$$\end{document} has the amalgamation property. Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\rm Th}(\mathbb{K})}$$\end{document} be the theory of an elementary (...) subclass \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{K}}$$\end{document} of the linearly ordered members of a variety \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{V}}$$\end{document} of semilinear commutative residuated lattices. We show that whenever \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\rm Th}(\mathbb{K})}$$\end{document} has elimination of quantifiers, and every linearly ordered structure in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{V}}$$\end{document} is a model of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\rm Th}_\forall(\mathbb{K})}$$\end{document}, then \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{V}}$$\end{document} has the amalgamation property. We exploit this fact to provide a purely model-theoretic proof of amalgamation for particular varieties of semilinear commutative residuated lattices. (shrink)

There is a good deal of disagreement about composition. There is firstorder disagreement: there are radically different answers to the special composition question—the question of under what circumstances the xs compose a y. There is second-order disagreement: there are different answers to the question of whether first-order disagreement is real or merely semantic. Virtually all disputants with respect to both the first-and second-order issues agree that the answer or answers to the special composition question will take the form of a (...) necessary truth or truths even though, as I will argue, such answers do not appear to be good candidates to be necessary truths. This paper provides an analysis of the concept of as it pertains to concrete objects, that fulfils two functions. First, it explicates the sense in which claims about composition are contingent and the sense in which they are necessary, and second, it provides a way of understanding when first-order disputes are substantial and when they are merely semantic. (shrink)

The paper argues that the view that the particular quantifier is ‘existentially loaded’ is a relatively new one historically and that it has become entrenched in modern philosophical logic for less than happy reasons.

It is widely agreed that Hume's description of human inductive reasoning is inadequate. But many philosophers think that this inadequacy in no way affects the force of Hume's argument for the unjustifiability of inductive reasoning. I argue that this constellation of opinions contains a serious tension, given that Hume was not merely pointing out that induction is fallible. I then explore a recent diagnosis of where Hume's sceptical argument goes wrong, due to Elliott Sober. Sober argues that Hume committed a (...) quantifier-shift fallacy, i.e. inferred a statement of ?? form from one of ?? form. The implications of this diagnosis for the traditional problem of induction are briefly examined. (shrink)