We have two polynomial time results for the uniform word problem for a quasivariety Q: The uniform word problem for Q can be solved in polynomial time iff one can find a certain congruence on finite partial algebras in polynomial time. Let Q* be the relational class determined by Q. If any universal Horn class between the universal closure S and the weak embedding closure S̄ of Q* is finitely axiomatizable then the uniform (...) word problem for Q is solvable in polynomial time. This covers Skolem's 1920 solution to the uniform word problem for lattices and Evans' 1953 applications of the weak embeddability property for finite partial V algebras. (shrink)

The set of all words in the alphabet {l, r} forms the full binary tree T. If x ∈ T then xl and xr are the left and the right successors of x respectively. We consider the monadic second-order language of the full binary tree with the two successor relations. This language allows quantification over elements of T and over arbitrary subsets of T. We prove that there is no monadic second-order formula φ * (X, y) such that for every (...) nonempty subset X of T there is a unique y ∈ X that satisfies φ * (X, y) in T. (shrink)

For a finite von Neumann algebra factor M, the projections form a modular ortholattice L(M). We show that the equational theory of L(M) coincides with that of some resp. all L(ℂ n × n ) and is decidable. In contrast, the uniform word problem for the variety generated by all L(ℂ n × n ) is shown to be undecidable.

In lieu of an abstract, here is a brief excerpt of the content:Hume and the Lockean Background: Induction and the Uniformity Principle David Owen Introduction What has come to be called Hume's problem of induction is special in many ways. It is arguably his most important and influential argument, especially when seen in its overall context of the more general argument about causaUty. It has come to be one of the great "standard problems" ofphilosophyandyetis,by most accounts, almost unique in (...) having no ancient precursor. Interestingly, the argument is frequently presented in terms that the author never used. Itis difficult for us not to think of the problem in terms of the contrast between deductive vs. inductive reasoning (or logic, or whatever). But this is a contrast nowhere to be found in Hume: he talks of demonstrative vs. probable (or moral) reasoning. Hume uses the word "induction" only twice in the Treatise1 and in neither case is it clear that the word is used toreportapieceofprobablereasoning.2One ofthefewoccurrences of"deduce" or "deducing" occurs in the second Enquiry, where he talks about "following the experimental method, and deducing general maximsfrom acomparisonofparticularinstances,"3hardlyaparadigm of what we would call deductive reasoning.4 Many modern studies of Hume that make a real attempt to understand what Hume meant by demonstration and probability, in the context of his argument about induction, tend towards the ahistorical,6 treating Hume's writings as contemporary texts. But failure to take into account some of the background concerning the nature of demonstrative and probable reason makes it unlikely, I think, that we can understand Hume's argument. Part ofthe problem is the failure to identify Hume's targets in that famous negative argument about induction (vague reference is sometimes made to the "rationalists") and part of the problem is identifying the role the uniformity principle plays in Hume's famous argumentand thelightitsheds on the conception ofprobablereasoning Hume was attacking. In this paper I want to argue that a careful look at Locke solves some ofthese problems: he has a well developed view ofdemonstrative and probable reasoning, his confidence in conformity as a ground of probability is a Ukely target of Hume's attack and his Volume XVIII Number 2 179 DAVID OWEN accountshedsmuchlightonwhatrole the principleofuniformitymight play in Hume's account.6 Of course, at a certain level of philosophical abstraction it is irrelevant whether we have understood what Hume "really meant." Theproblemofinductionhas emergedas a central onein epistemology, philosophical logic, and philosophy of science. It would be an interesting question in philosophical historiography to trace the development ofthe "problem" from Hume's time to our own but that is not my task here. I trust one needs no justification for taking an historical look at a famous problem rather than tackling the problem head-on in one's own terms. But if one is needed, here is mine: sometimes, by looking at a problem from a different, historical perspective, one sees a different problem or afamiliar problem in a new light. One's philosophical horizons are broadened. Furthermore, one may make an interesting historical discovery, or at least draw some interesting historical connections. I take it as uncontroversial that Locke, following Descartes, rejected a formal, logical account of demonstrative reasoning. Instead ofusing a formal, syllogistic account ofhow two propositions can entail a third, Locke concentrated on the relation two ideas bear one to another.7 Thus a demonstration, for Locke, getsits force, notfrom what we would call a deductive relation between one set of propositions (.the premises) and anotherproposition (the conclusion), butrather from the relation one idea bears to another, either directly, or via a chain of intermediate ideas. It is true that Locke frequently talks of inferring one proposition from another, but this is always cashed out in terms of relations ofideas via a chain ofintermediate ideas.8 It is also true that Locke occasionally uses the terms "deduce" and "deduction." But these are never used in our sense that has to do with formal deductive validity. Rather, they are simply used loosely in the informal sense of "argue" or "infer," and "argument" or "inference."9 In the rest of this paper, all uses ofthe word"deduction" andits cognates shouldbe taken as having the modern sense, unless flagged as having "Locke... (shrink)

The essays that make up my dissertation share a methodological approach that aims to explore the philosophical implications of linguists' accounts of ordinary language use. In particular, all of them focus on epistemic natural language and the implications that linguists' accounts of such language has for epistemology. The first essay focuses on the debate about the norms that govern assertion and shows the ways in which research on natural language evidentiality has direct bearing. This essay uses existing cross-linguistic data about (...) assertions in Quechua and Cheyenne to argue that assertions and the norms that govern them are more complex than allowed for in extant views. What makes Quechua and Cheyenne important is that they allow speakers to assert sentences the content of which they do not believe, or even believe to be false, as long as the sentence contains the right evidential marker. This is a problem for the current theories as they all take belief in the proposition being expressed as a minimal requirement for a speaker to felicitously assert. Given the data, I argue that we should see evidential markers as modifying the norm that is in place governing felicity of the current content. I go on to present three implementations of a context-sensitive norm and argue that only the ones that are evidence-based or completely contextual can properly capture the entire set of linguistic data. In the second essay I argue that epistemic uses of 'should' can be modelling using the standard Kratzerian modal canon. In Kratzer's system, modals induce quantification over some partially ordered, restricted class of worlds. The relevant partially ordered, restricted class of worlds is generally fixed by two ingredients: a modal base and an ordering source. Modelling epistemic uses of 'should' requires us to rethink and expand what has traditionally been thought of as making up an epistemic modal base. Traditionally it has been thought that epistemic modal bases just include information about probabilities but this thought needs to be updated in order to bring epistemic 'should' into the fold. I argue that there are good theoretical reasons for having as uniform a semantics as possible then show that the context-sensitive semantic model that I develop meets all empirical demands. I end the essay by arguing that this updated way of thinking about epistemic modals bases has implications for epistemology. The final essay outlines the broad type of methodological approach mentioned above that guides the research throughout the dissertation. In it, I argue that the results of linguistics' theorizing about semantics, especially within the epistemic domain, ought to be seen as an accurate guide to reality. Looking to the way that language evolves over time and what it takes for information to become encoded into language is what drives this result. Buttressing the evolutionary explanation with results from the Condorcet Jury Theorem gives us more reason to believe that language is an accurate, albeit defeasible, guide to reality. If these results hold, they not only have broad implications for how we ought to be conducting our epistemological theorizing but our philosophical theorizing in general. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Journal of the History of Philosophy 38.1 (2000) 103-110 [Access article in PDF] Hutcheson's Moral Sense and the Problem of Innateness Daniel Carey National University of Ireland Francis Hutcheson's philosophy arguably represented a delicate, and at times precarious, synthesis of positions laid out by John Locke and the third Earl of Shaftesbury. From Shaftesbury, whose influence he acknowledged explicitly in the title page of the first edition of (...) the Inquiry into the Original of our Ideas of Beauty and Virtue (1725), Hutcheson gained an appreciation of the dignity of human nature, reflected in sociable instincts, benevolent impulses, and the moral sense. From Locke he adopted the bias of empiricism, placing moral philosophy on an observational footing, ostensibly free from a priori assumptions.1 The challenge posed by integrating the work of these two mentors arose from their deep incompatibility, evident in Shaftesbury's declared opposition to Locke's mode of philosophizing and rejection of his disagreeable representation of human nature. Nowhere was Hutcheson's combination of Locke and Shaftesbury more difficult to sustain than over the issue of innateness. Locke was intent on jettisoning innate ideas, but Shaftesbury remained keen to rescue them.Of course there are different versions and definitions of innateness at stake here: on one account, it entailed the possession of particular, specifiable ideas or conceptions, somehow inscribed in the soul which might or might not require the trigger of experience to emerge (for example, ideas of God, good and, evil); on another, a predisposition to hold certain beliefs and make certain judgments, provided that 'normal' conditions of maturity and rationality have been met; finally, a faculty, that is, an inborn structure enabling the subject to experience the world in a moral way, and to register ethical responses to it, whatever they may be.2 All of these orientations on innateness impinge on Hutcheson's approach to the problem. On the one hand, he conceded Locke's critique as consistent with modern philosophizing, but he restricted its force to innate ideas or propositions, leaving the moral sense conveniently exempt from the [End Page 103] objection. On the other hand, Hutcheson was determined to reintroduce the content of innate ideas if not the name, assigning the moral sense a familiar, normative role in underwriting perceptions and judgments.The intricacy of the issue has led critics to arrive at contradictory assessments of Hutcheson's outlook on innateness. Stephen Buckle and Knud Haakonssen take Hutcheson at his word and argue that he abandoned innate ideas altogether.3 David Fate Norton agrees that Hutcheson found Locke's critique persuasive, at least against neo-Platonic theories of ethics (such as Cudworth's), but elsewhere Norton maintains that Hutcheson sets up the moral sense as an "innate and independent" moral faculty.4 Charles Taylor, meanwhile, stresses the dispositional inclination of Hutcheson's argument: "Goodness and generosity are natural to us. He might have been tempted in an earlier generation to say 'innate,' but he has internalized enough of the Lockean psychology to shy away from this. So he claims that these sentiments are ones we naturally grow to have, just as we develop to our normal stature and shape."5 Given the disparate of conclusions drawn by historians of philosophy, the issue clearly merits reinvestigation. Differing lines of interpretation have emerged precisely because Hutcheson frequently changed his tack. His shifting strategy appears both in the revisions he made to later editions of the Inquiry, and in the development of his work over the course of his career. Closer attention to these considerations offers an insight into the predicament of moral philosophy in the early decades of the eighteenth century.In the critique of innate practical ideas launched in the first book of the Essay concerning Human Understanding (1690), Locke took a sceptical view of Stoic assertions about human uniformity and the existence of "common notions" indicating moral consensus around the world.6 Evidence drawn from contemporary travelers suggested the absence of any moral agreement of this kind; Locke based his conclusion on an investigation of human practice influenced by techniques... (shrink)

Let M be a smooth, compact manifold of dimension n ≥ 5 and sectional curvature | K | ≤ 1. Let Met (M) = Riem(M)/Diff(M) be the space of Riemannian metrics on M modulo isometries. Nabutovsky and Weinberger studied the connected components of sublevel sets (and local minima) for certain functions on Met (M) such as the diameter. They showed that for every Turing machine T e , e ∈ ω, there is a sequence (uniformly effective in e) of homology (...) n-spheres {P k e } k ∈ ω which are also hypersurfaces, such that P k e is diffeomorphic to the standard n-sphere S n (denoted P k e ≈ diff S n ) iff T e halts on input k, and in this case the connected sum N k e =M ♯ P k e ≈ diff M , so N k e ∈ Met(M), and N k e is associated with a local minimum of the diameter function on Met(M) whose depth is roughly equal to the settling time σ e (k) of T e on inputs y i } ∈ ω of c.e. sets so that for all i the settling time of the associated Turing machine for A i dominates that for A i + 1 , even when the latter is composed with an arbitrary computable function. From this, Nabutovsky and Weinberger showed that the basins exhibit a "fractal" like behavior with extremely big basins, and very much smaller basins coming off them, and so on. This reveals what Nabutovsky and Weinberger describe in their paper on fractals as "the astonishing richness of the space of Riemannian metrics on a smooth manifold, up to reparametrization." From the point of view of logic and computability, the Nabutovsky-Weinberger results are especially interesting because: (1) they use c.e. sets to prove structural complexity of the geometry and topology, not merely undecidability results as in the word problem for groups, Hilbert's Tenth Problem, or most other applications; (2) they use nontrivial information about c.e. sets, the Soare sequence {A i } i ∈ ω above, not merely G öodel's c.e. noncomputable set K of the 1930's; and (3) without using computability theory there is no known proof that local minima exist even for simple manifolds like the torus T 5 (see §). (shrink)

A number of theories have been developed to characterize ALogTime (or uniform NC 1, or just NC 1), the class of languages accepted by alternating logtime Turing machines, in the same way that Buss’s theory ${{\bf S}^{1}_{2}}$ characterizes polytime functions. Among these, ALV′ (by Clote) is particularly interesting because it is developed based on Barrington’s theorem that the word problem for the permutation group S 5 is complete for ALogTime. On the other hand, ALV (by Clote), T (...) 0 NC 0 (by Clote and Takeuti) as well as Arai’s theory ${{\bf AID}+\Sigma_0^B{-}{\bf CA}}$ and its two-sorted version VNC 1 (by Cook and Morioka) are based on the circuit characterization of ALogTime. While the last three theories have been known to be equivalent, their relationship to ALV′ has been an open problem. Here we show that ALV′ is indeed equivalent to the other theories. (shrink)

In this paper, we will prove the inevitable non-uniformity of two constructions from combinatorial group theory related to the word problem for finitely generated groups and the Higman—Neumann—Neumann Embedding Theorem.

This note addresses the issue as to which ceers can be realized by word problems of computably enumerable (or, simply, c.e.) structures (such as c.e. semigroups, groups, and rings), where being realized means to fall in the same reducibility degree (under the notion of reducibility for equivalence relations usually called “computable reducibility”), or in the same isomorphism type (with the isomorphism induced by a computable function), or in the same strong isomorphism type (with the isomorphism induced by a computable (...) permutation of the natural numbers). We observe, e.g., that every ceer is isomorphic to the word problem of some c.e. semigroup, but (answering a question of Gao and Gerdes) not every ceer is in the same reducibility degree of the word problem of some finitely presented semigroup, nor is it in the same reducibility degree of some non‐periodic semigroup. We also show that the ceer provided by provable equivalence of Peano Arithmetic is in the same strong isomorphism type as the word problem of some non‐commutative and non‐Boolean c.e. ring. (shrink)

Taking inspiration from Searle’s ‘The Logic of Fictional Discourse’, this essay presents an argument against different versions of the so-called ‘speech act theory of fiction’. In particular, it argues that a Uniformity Argument may be constructed, which is additional to the Determination Argument commonly attributed to Searle, and which does not rely on his presumably controversial Determination Principle. This Uniformity Argument is equally powerful against the ‘Dedicated Speech Act’ theories that Searle originally targeted, and the more recent, Grice-inspired versions of (...) speech-act approaches to fictional discourse. (shrink)

By theword problemfor some class of algebraic structures we mean the problem of determining, given a finite setEof equations between words and an additional equationx=y, whetherx=ymust hold in all structures satisfying each member ofE. In 1947 Post [P] showed the word problem for semigroups to be undecidable. This result was strengthened in 1950 by Turing, who showed the word problem to be undecidable forcancellation semigroups,i.e. semigroups satisfying thecancellation propertyNovikov [N] eventually showed the word (...) class='Hi'>problem for groups to be undecidable.In 1966 Gurevich [G] showed the word problem to be undecidable forfinitesemigroups. However, this result on finite structures has not been extended to cancellation semigroups or groups; indeed it is easy to see that a finite cancellation semigroup is a group, so both questions are the same. We do not here settle the word problem for finite groups, but we do show that the word problem is undecidable for finite semigroups with zero satisfying an approximation to the cancellation property. (shrink)

Every science has its technical vocabulary, consisting in part of terms coined for explicit purposes and in part of words borrowed from ordinary discourse and used with greater or lesser degrees of precision. Words of the latter sort pose curious problems, some of them familiar to those historians of science concerned with, for example, what Galileo meant by forza and Newton by attraction. Indeed, analogous problems face any historian seeking to understand the older meanings of terms still in use today.

Mathematics word problems provide students with an opportunity to apply what they are learning in their mathematics classes to the world around them. However, students often neglect their knowledge of the world and provide nonsensical responses (e.g., they may answer that a school needs 12.5 buses for a field trip). This study examined if the question design of word problems affects students' mindset in ways that affect subsequent sense‐making. The hypothesis was that rewriting standard word problems to (...) introduce inherent uncertainty about the result would be beneficial to student performance and sense‐making because it requires students to reason explicitly about the context described in the problem. Middle school students (N = 229) were randomly assigned to one of three conditions. In the standard textbook condition, students solved a set of six word problems taken from current textbooks. In the modified yes/no condition, students solved the same six problems rewritten so the solution helped answer a “yes” or “no” question. In the disfluency control condition, students solved the standard problems each rewritten in a variety of fonts to make them look unusual. After solving the six problems in their assigned condition, all students solved the same three “problematic” problems designed to assess sense‐making. Contrary to predictions, results showed that students in the modified yes/no condition solved the fewest problems correctly in their assigned condition problem set. However, consistent with predictions, they subsequently demonstrated more sense‐making on the three problematic problems. Results suggest that standard textbook word problems may be able to be rewritten in ways that mitigate a “senseless” mindset. (shrink)

Three experiments explored the effects of word problem cueing on debiasing versions of the bat-and-ball problem. In the experimental condition order, participants solved a simpler isomorphic version of the problem prior to solving a standard version that, critically, had the same item-and-dollar amounts. Conversely, in the control condition order, participants solved the standard version prior to solving the isomorph. Across the first 2 experiments, participants cued with the isomorph were more likely to correctly solve the standard (...) version of the problem. In the third experiment, this cueing effect was not replicated, but importantly cued participants were more likely to correctly solve the standard version of the problem with other item-and-dollar amounts, indicating near transfer. Far transfer was evident in faster response times to other problems in the Cognitive Reflection Test following practice. Along with providing converging evidence for the cueing account, these results are consistent with recent attentional debiasing findings. (shrink)

The uniform halting problem (UH) can be stated as follows.Give a decision procedure which for any given Turing machine (TM) will decide whether or not it has an immortal instantaneous description (ID).An ID is called immortal if it has no terminal successor. As it is generally the case in the literature (see e.g. Minsky [3, p. 118]) we assume that in an ID the tape must be blank except for some finite numbers of squares. If we remove this (...) restriction the UH becomes the immortality problem (IP). The UH should not be confused with the initialised uniform halting problem (whether or not a TM has an immortal ID when started in a specified state) which can easily be shown to be undecidable (see e.g. Minsky [3, p. 151]). (shrink)

Two experiments were conducted to investigate children’s interpretations of standard arithmetic word problems and the factors that influence their interpretations. In Experiment 1, children were required to solve a series of problems and then to draw and select pictures that represented the problems’ structures. Solution performance was found to vary systematically with the nature of the representations drawn and chosen. The crucial determinant of solution success was the interpretation a child assigned to certain phrases used in the problems. In (...) Experiment 2, solution and drawing accuracy were found to be significantly improved by rewording problems to avoid ambiguous linguistic forms. Together, these results imply that (a) word-problem solution errors are caused by misinterpretations of certain verbal expressions commonly used in problem texts, and (b) these misinterpretations are the result of missing or inadequate mappings of these verbal expressions to partwhole knowledge. (shrink)

The uniform halting problem (UH) can be stated as follows:Give a decision procedure which for any given Turing machine (TM) will decide whether or not it has an immortal instantaneous description (ID).An ID is called immortal if it has no terminal successor. As it is generally the case in the literature (see e.g. Minsky [4, p. 118]) we assume that in an ID the tape must be blank except for some finite number of squares. If we remove this (...) restriction the UH becomes the immortality problem (IP). (shrink)

Word problems are notoriously difficult to solve. We suggest that much of the difficulty children experience with word problems can be attributed to difficulty in comprehending abstract or ambiguous language. We tested this hypothesis by (1) requiring children to recall problems either before or after solving them, (2) requiring them to generate f'mal questions to incomplete word problems, and (3) modeling performance pattems using a computer simulation. Solution performance was found to be systematically related to recall and (...) question generation performance. Correct solutions were associated with accurate recall of the problem structure and with appropriate question generation. Solution "errors" were found to be correct solutions to miscomprehended problems. Word problems that contained abstract or ambiguous language tended to be miscomprehended more often than those using simpler language, and there was a great deal of systematicity in the way these problems were miscomprehended. Solution error pattems were successfully simulated by manipulating a computer model’s language comprehension strategies, as opposed to its knowledge of logical set relations. o was.. (shrink)

This paper presents a uniform general account of regress problems in the form of a pentalemma—i.e., a set of five mutually inconsistent claims. Specific regress problems can be analyzed as instances of such a general schema, and this Regress Pentalemma Schema can be employed to generate deductively valid arguments from the truth of a subset of four claims to the falsity of the fifth. Thus, a uniform account of the nature of regress problems allows for an improved understanding (...) of specific regress objections or arguments, and, correspondingly, of the general logical geography of the debate about infinite regresses. This uniform approach is illustrated by a treatment of the classical epistemological problem of justification, but it encompasses a whole variety of cases including explanation and ontological grounding. Furthermore, this general account is compared and contrasted with the existing literature discussing argument schemata for regress objections, particularly with the work of Jan Willem Wieland. It is shown how such other schemata can be incorporated and superseded by the general Regress Pentalemma Schema. (shrink)

A pseudo-natural algorithm for the word problem of a finitely presented group is an algorithm which not only tells us whether or not a word w equals 1 in the group but also gives a derivation of 1 from w when w equals 1. In [13], [14] Madlener and Otto show that, if we measure complexity of a primitive recursive algorithm by its level in the Grzegorczyk hierarchy, there are groups in which a pseudo-natural algorithm is arbitrarily (...) more complicated than an algorithm which simply solves the word problem. In a given group the lowest degree of complexity that can be realised by a pseudo-natural algorithm is essentially the derivational complexity of that group. Thus the result separates the derivational complexity of the word problem of a finitely presented group from its intrinsic complexity. The proof given in [13] involves the construction of a finitely presented group G from a Turing machine T such that the intrinsic complexity of the word problem for G reflects the complexity of the halting problem of T, while the derivational complexity of the word problem for G reflects the runtime complexity of T. The proof of one of the crucial lemmas in [13] is only sketched, and part of the purpose of this paper is to give the full details of this proof. We will also obtain a variant of their proof, using modular machines rather than Turing machines. As for several other results, this simplifies the proofs considerably. MSC: 03D40, 20F10. (shrink)

The practice of mathematical word problem is ubiquitous and thought to impact academic achievement. However, the underlying neural mechanisms are still poorly understood. In this study, we investigate how lexical consistency of word problem description is modulated in adults' brain responses during word problem solution. Using functional magnetic resonance imaging methods, we examined compare word problems that included relational statements, such as “A dumpling costs 9 dollars. A wonton is 2 dollars less than (...) a dumpling. How much does a wonton cost?” and manipulated lexical consistency and problem operation. We found a consistency by operation interaction in the widespread fronto-insular-parietal activations, including the anterior insula, dorsoanterior cingulate cortex, middle frontal gyrus, and intraparietal sulcus, such that inconsistent problems engaged stronger activations than consistent problems for addition, whereas the consistency effect was inverse for subtraction. Critically, these results were more salient in the less successful problem solvers than their more successful peers. Our study is the first to demonstrate that lexical consistency effects on arithmetic neural networks are modulated during reading word problem that required distinct arithmetic operations. More broadly, our study has strong potentials to add linkage between neuroscience and education by remediating deficits and enhance instruction design in the school curriculum. (shrink)

ABSTRACTIn a bilingual educational setting, even when mathematical word problems are presented in one’s first language, students may still perform poorly if cognitive constraints such as working me...

Understanding arithmetic word problems involves a complex interaction of text comprehension and mathematical processes. This article presents a computer simulation designed to capture the working memory demands required in “bottomup” comprehension of arithmetic word problems. The simulation's sentence‐level parser and text integration component reflect the importance of processing the problem from its original natural language presentation. Children's probability of solution was analyzed in exploratory regression analyses as a function of the simulation's sentence‐level and text integration processes. Working (...) memory variables measuring the combined effects of concepts to remember and text integration inferences account for a significant proportion of variance in children's solution probabilities across the first four grade levels (K‐3). Consistent with previous results from others, which highlighted the significance of small changes in problem wording, the simulation offers a process‐oriented perspective as to why natural language presentation constrains the comprehension of mathematical relationships. (shrink)

This book surveys research in quantification starting with the foundational work in the 1970s. It paints a vivid picture of generalized quantifiers and Boolean semantics. It explains how the discovery of diverse scope behavior in the 1990s transformed the view of quantification, and how the study of the internal composition of quantifiers has become central in recent years. It presents different approaches to the same problems, and links modern logic and formal semantics to advances in generative syntax. A unique feature (...) of the book is that it systematically brings cross-linguistic data to bear on the theoretical issues, discussing French, German, Dutch, Hungarian, Russian, Japanese, Telugu (Dravidian), and Shupamem (Grassfield Bantu), and pointing to formal semantic literature involving quantification in around thirty languages. -- -/- 1. What this book is about and how to use it; 2. Generalized quantifiers and their elements: operators and their scopes; 3. Generalized quantifiers in non-nominal domains; 4. Some empirically significant properties of quantifiers and determiners; 5. Potential challenges for generalized quantifiers; 6. Scope is not uniform and not a primitive; 7. Existential scope versus distributive scope; 8. Distributivity and scope; 9. Bare numeral indefinites; 10. Modified numerals; 11. Clause-internal scopal diversity; 12. Towards a compositional semantics of quantifier words. (shrink)

Saul Kripke, in his celebrated book Wittgenstein on Rules and Private Language (1982), offers a novel reading of Ludwig Wittgenstein’s main remarks in his later works, especially in Philosophical Investigations (1953) and, to some extent, in Remarks on the Foundations of Mathematics (1956). Kripke presents Wittgenstein as proposing a skeptical argument against a certain conception of meaning and linguistic understanding, as well as a skeptical solution to such a problem. Many philosophers have called this interpretation of Wittgenstein Kripke’s Wittgenstein (...) or Kripkenstein because, as Kripke himself emphasizes, it is “Wittgenstein’s argument as it struck Kripke, as it presented a problem for him” (Kripke 1982, 5) and “probably many of my formulations and re-castings of the argument are done in a way Wittgenstein would not himself approve” (Kripke 1982, 5). Such an interpretation has been the subject of tremendous discussions since its publication, and this has formed a huge literature on the topic of meaning skepticism in general and Wittgenstein’s later view in particular. -/- According to the skeptical argument that Kripke extracts from Wittgenstein’s later remarks on meaning and rule-following, there is no fact about a speaker’s behavioral, mental or social life that can metaphysically determine, or constitute, what she means by her words and also fix a determinate connection between those meanings and the correctness of her use of these words. Such a skeptical conclusion has a disastrous consequence for the classical realist view of meaning: if we insist on the idea that meaning is essentially a factual matter, we face the bizarre conclusion that there is thereby “no such thing as meaning anything by any word” (Kripke 1982, 55). -/- According to the skeptical solution that Kripke attributes to Wittgenstein, such a radical conclusion is intolerable because we certainly do very often mean certain things by our words. The skeptical solution begins by rejecting the view that results in such a paradoxical conclusion, that is, the classical realist conception of meaning. The skeptical solution offers then a new picture of the practice of meaning-attribution, according to which we can legitimately assert that a speaker means something specific by her words if we, as members of a speech-community, can observe, in enough cases, that her use agrees with ours. We can judge, for instance, that she means by “green” what we mean by this word, namely, green, if we observe that her use of “green” agrees with our way of using it. Attributing meanings to others’ words, therefore, brings in the notion of a speech-community, whose members are uniform in their responses. As a result, there can be no private language. -/- This article begins by introducing Kripke’s Wittgenstein’s skeptical problem presented in Chapter 2 of Kripke’s book. It then explicates Kripke’s Wittgenstein’s skeptical solution to the skeptical problem, which is offered in Chapter 3 of the book. The article ends by reviewing some of the most important responses to the skeptical problem and the skeptical solution. (shrink)