Let ZF denote Zermelo-Fraenkel set theory (without the axiom of choice), and let $M$ be a countable transitive model of ZF. The method of forcing extends $M$ to another model $M\lbrack G\rbrack$ of ZF (a "generic extension"). If the axiom of choice holds in $M$ it also holds in $M\lbrack G\rbrack$, that is, the axiom of choice is preserved by generic extensions. We show that this is not true for many weak forms of the axiom of choice, (...) and we derive an application to Boolean toposes. (shrink)

We continue [4] and study sets in generic extensions by the Magidor forcing and by the Prikry forcing with non-normal ultrafilters.

Sureson, C., Symmetric submodels of a Cohen generic extension, Annals of Pure and Applied Logic 58 247–261. We study some symmetric submodels of a Cohen generic extension and the satisfaction of several properties ) which strongly violate the axiom of choice.

Let P be a forcing notion and $G\subseteq P$ its generic subset. Suppose that we have in $V[G]$ a $\kappa{-}$ complete ultrafilter1,2W over $\kappa $. Set $U=W\cap V$.

We show that large fragments of MM, e. g. the tree property and stationary reflection, are preserved by strongly -game-closed forcings. PFA can be destroyed by a strongly -game-closed forcing but not by an ω2-closed.

Given a theoryTextending that of dense linear orders without endpoints, in a language ℒ ⊇ {<}, we are interested in extensionsT′ ofTin languages extending ℒ by unary relation symbols that are each interpreted in models ofT′ as sets that are both dense and codense in the underlying sets of the models.There is a canonically “wild” example, namelyT= Th andT′ = Th. Recall thatTis o-minimal, and so every open set definable in any model ofThas only finitely many definably connected components. But (...) it is well known that 〈ℝ, <, +, · ℚ 〉 defines every real Borel set, in particular, every open subset of any finite cartesian power of ℝ and every subset of any finite cartesian power of ℚ. To put this another way, the definable open sets in models ofTare essentially as simple as possible, whileT′ has a model where the definable open sets are as complicated as possible, as is the structure induced on the new predicate.In contrast to the preceding example, if ℝalgis the set of real algebraic numbers andT′ Th, then no model ofT′ defines any open set that is not definable in the underlying model ofT. (shrink)

This package supplies the infrastructure for extending STEX macros with OMDoc metadata. This package is mainly intended for authors of STEX extension packages.

It is shown how certain generic extensions of a fine structural model in the sense of Mitchell and Steel [MiSt] can be reorganized as relativizations of the model to the generic object. This is then applied to the construction of Steel's core model for one Woodin cardinal [St] and its generalizations.

We define a generic Vopěnka cardinal to be an inaccessible cardinal \ such that for every first-order language \ of cardinality less than \ and every set \ of \-structures, if \ and every structure in \ has cardinality less than \, then an elementary embedding between two structures in \ exists in some generic extension of V. We investigate connections between generic Vopěnka cardinals in models of ZFC and the number and complexity of \-Suslin sets of (...) reals in models of ZF. In particular, we show that ZFC + is equiconsistent with ZF + \\) where \ is the pointclass of all \-Suslin sets of reals, and also with ZF + \\) + \\) where \ is the least ordinal that is not a surjective image of the reals. (shrink)

Generic statements are commonly expressed using the bare plural – ‘tigers are striped’ – or the indefinite singular – ‘a tiger is striped’. Notoriously, some generics can be expressed using the bare plural locution, but not the indefinite singular; bare plural generics and indefinite singular generics pattern differently. I explore this phenomenon as it applies to normative generic statements: expressions like boys don’t cry, women are kind and nurturing, children are seen and not heard – that convey something (...) over and above (and even distinct from) the descriptive reading. I argue that we should look to a metalinguistic theory of generics to understand indefinite singular normative generics. That is, a normative indefinite singular generic is a proposal about what to include in the extension of the generic term. This theory explains why it is more often than not that felicitous indefinite singular generics are about social kind terms. I argue that this explains a crucial way in which language shapes the social world. When we say things like (2), we propose definitions and uses for the word ‘boy’. Such proposals and their consequences actually impact the behavior of the groups they pick out, as well as the attitudes of others towards those groups. (shrink)

It is natural to wish to study miniaturisations of Cohen forcing suitable to sets of low arithmetic complexity. We consider extensions of the work of Schaeffer [9] and Jockusch and Posner [6] by looking at genericity notions within the Δ2 sets. Different equivalent characterisations of 1-genericity suggest different ways in which the definition might be generalised. There are two natural ways of casting the notion of 1-genericity: in terms of sets of strings and in terms of density functions, as (...) we will see here. While these definitions coincide at the first level of the difference hierarchy, they turn out to differ at other levels. Furthermore, these differences remain when the remainder of the Δ02 sets are considered. While the string characterization of 1-genericity collapses at the second level of the difference hierarchy to 2-genericity, the density function definition gives a very interesting hierarchy at level w and above. Both of these results point towards the deep similarities exhibited by the n-c.e. degrees for n ≥ 2. (shrink)

K.J. Devlin has extended Jensen's construction of a model ofZFC andCH without Souslin trees to a model without Kurepa trees either. We modify the construction again to obtain a model with these properties, but in addition, without Kurepa trees inccc-generic extensions. We use a partially defined ◊-sequence, given by a fine structure lemma. We also show that the usual collapse ofκ Mahlo toω 2 will give a model without Kurepa trees not only in the model itself, but also (...) inccc-extensions. (shrink)

The project of this paper is to deliver a semantics for a broad subset of bare plural generics about racial kinds, a class which I will dub 'Type C generics.' Examples include 'Blacks are criminal' and 'Muslims are terrorists.' Type C generics have two interesting features. First, they link racial kinds with ​ socially perspectival predicates ​ (SPPs). SPPs lead interpreters to treat the relationship between kinds and predicates in generic constructions as nomic or non-accidental. Moreover, in computing their (...) content, interpreters must make implicit reference to socially privileged​ ​ perspectives which are treated as authoritative about whether a given object fits into the extension of the predicate. Such deference grants these authorities influence over both the conventional meaning of these terms and over the nature of the ​ objects ​ in the social ontology that these terms purport to describe, much the way a baseball umpire is authoritative over the meaning and metaphysics of 'strike'/​ strike ​. Second, terms like 'criminal' and 'terrorist' receive default ​ racialized ​ interpretations in which these terms conventionally token racial or ethnic identities. I show that neither of these features can be explained by Sarah-Jane Leslie's influential 'weak semantics' for generics, and show how my own 'socially perspectival semantics' fares better on both counts. Finally, I give an analysis of 'Blacks are criminal' which explores the semantic mechanisms that underlie default racialized interpretations. (shrink)

This article maintains that an important class of scientific generalizations should be reinterpreted: they have typically been understood as ceteris paribus laws, but are, in fact, generics. Four arguments are presented to support this thesis. One argument is that the interpretation in terms of ceteris paribus laws is a historical accident. The other three arguments draw on similarities between these generalizations and archetypal generics: they come with similar inferential commitments, they share a syntactic form, and the existing theories to make (...) sense of them are alike. Once these generalizations are properly understood as generics, the recent cognitive approach to generics can be extended to the study of the relevant sciences. The last section indicates ways in which this extension is fruitful for the two strands of research that we combine: the philosophy of science literature on generalizations and the semantics literature on generics. (shrink)

I use generic embeddings induced by generic normal measures on that can be forced to exist if κ is an indestructibly weakly compact cardinal. These embeddings can be applied in order to obtain the forcing axioms in forcing extensions. This has consequences in : The Singular Cardinal Hypothesis holds above κ, and κ has a useful Jónsson-like property. This in turn implies that the countable tower works much like it does when κ is a Woodin limit of (...) Woodin cardinals. One consequence is that every set of reals in the Chang model is Lebesgue measurable and has the Baire Property, the Perfect Set Property and the Ramsey Property. So indestructible weak compactness has effects on cardinal arithmetic high up and also on the structure of sets of real numbers, down low, similar to supercompactness. (shrink)

Henle, Mathias, and Woodin proved in [21] that, provided that${\omega }{\rightarrow }({\omega })^{{\omega }}$holds in a modelMof ZF, then forcing with$([{\omega }]^{{\omega }},{\subseteq }^*)$overMadds no new sets of ordinals, thus earning the name a “barren” extension. Moreover, under an additional assumption, they proved that this generic extension preserves all strong partition cardinals. This forcing thus produces a model$M[\mathcal {U}]$, where$\mathcal {U}$is a Ramsey ultrafilter, with many properties of the original modelM. This begged the question of how important the Ramseyness (...) of$\mathcal {U}$is for these results. In this paper, we show that several classes of$\sigma $-closed forcings which generate non-Ramsey ultrafilters have the same properties. Such ultrafilters include Milliken–Taylor ultrafilters, a class of rapid p-points of Laflamme,k-arrow p-points of Baumgartner and Taylor, and extensions to a class of ultrafilters constructed by Dobrinen, Mijares, and Trujillo. Furthermore, the class of Boolean algebras$\mathcal {P}({\omega }^{{\alpha }})/{\mathrm {Fin}}^{\otimes {\alpha }}$,$2\le {\alpha }<{\omega }_1$, forcing non-p-points also produce barren extensions. (shrink)

In this paper, we present a generic format for adaptive vague logics. Logics based on this format are able to (1) identify sentences as vague or non-vague in light of a given set of premises, and to (2) dynamically adjust the possible set of inferences in accordance with these identifications, i.e. sentences that are identified as vague allow only for the application of vague inference rules and sentences that are identified as non-vague also allow for the application of some (...) extra set of classical logic rules. The generic format consists of a set of minimal criteria that must be satisfied by the vague logic in casu in order to be usable as a basis for an adaptive vague logic. The criteria focus on the way in which the logic deals with a special ⊡-operator. Depending on the kind of logic for vagueness that is used as a basis for the adaptive vague logic, this operator can be interpreted as completely true, definitely true, clearly true , etc. It is proven that a wide range of famous logics for vagueness satisfies these criteria when extended with a specific ⊡-operator, e.g. fuzzy basic logic and its well known extensions, cf. [7], super- and subvaluationist logics, cf. [6], [9], and clarity logic, cf. [13]. Also a fuzzy logic is presented that can be used for an adaptive vague logic that can deal with higher-order vagueness. To illustrate the theory, some toy-examples of adaptive vague proofs are provided. (shrink)

A study of smooth classes whose generic structures have simple theory is carried out in a spirit similar to Hrushovski 147; Simplicity and the Lascar group, preprint, 1997) and Baldwin–Shi 1). We attach to a smooth class K0, of finite -structures a canonical inductive theory TNat, in an extension-by-definition of the language . Here TNat and the class of existentially closed models of =T+,EX, play an important role in description of the theory of the K0,-generic. We show that (...) if M is the K0,-generic then MEX. Furthermore, if this class is an elementary class then Th=Th). The investigations by Hrushovski and Pillay , provide a general theory for forking and simplicity for the nonelementary classes, and using these ideas, we show that if K0,, where {,*}, has the joint embedding property and is closed under the Independence Theorem Diagram then EX is simple. Moreover, we study cases where EX is an elementary class. We introduce the notion of semigenericity and show that if a K0,-semigeneric structure exists then EX is an elementary class and therefore the -theory of K0,-generic is near model complete. By this result we are able to give a new proof for a theorem of Baldwin and Shelah 1359). We conclude this paper by giving an example of a generic structure whose first-order theory is simple. (shrink)

It is unknown to what extent medical researchers generalize study findings beyond their samples when their sample size, sample diversity, or knowledge of conditions that support external validity do not warrant it. It is also unknown to what extent medical researchers describe their results with precise quantifications or unquantified generalizations, i.e., generics, that can obscure variations between individuals. We therefore systematically reviewed all prospective studies (n = 533) published in the top four highest ranking medical journals, Lancet, New England Journal (...) of Medicine (NEJM), Journal of the American Medical Association (JAMA), and the British Medical Journal (BMJ), from January 2022 to May 2023. We additionally reviewed all NEJM Journal Watch clinical research summaries (n = 143) published during the same time. Of all research articles reporting prospective studies, 52.5% included generalizations beyond specific national study populations, with the numbers of articles with generics varying significantly between journals (JAMA = 12%; Lancet = 77%) (p < 0.001, V = 0.48). There was no evidence that articles containing broader generalizations or generics were correlated with larger or more nationally diverse samples. Moreover, only 10.2% of articles with generalizations beyond specific national populations reported external validity strengthening factors that could potentially support such extrapolations. There was no evidence that original research articles and NEJM Journal Watch summaries intended for practitioners differed in their use of broad generalizations, including generics. Finally, from the journal with the highest citation impact, articles containing broader conclusions were correlated with more citations. Since there was no evidence that studies with generalizations beyond specific national study populations or with generics were associated with larger, more nationally diverse samples, or with reports of population similarity that may permit extensions of conclusions, our findings suggest that the generalizations in many articles were insufficiently supported. Caution against overly broad generalizations in medical research is warranted. (shrink)

Genericity is the idea that the same program can work at many different data types. Longo, Milstead and Soloviev proposed to capture the inability of generic programs to probe the structure of their instances by the following equational principle: if two generic programs, viewed as terms of type , are equal at any given instance A[T], then they are equal at all instances. They proved that this rule is admissible in a certain extension of System F, but finding (...) a semantically motivated model satisfying this principle remained an open problem.In the present paper, we construct a categorical model of polymorphism, based on game semantics, which contains a large collection of generic types. This model builds on two novel constructions:•A direct interpretation of variable types as games, with a natural notion of substitution of games. This allows moves in games A[T] to be decomposed into the generic part from A, and the part pertaining to the instance T. This leads to a simple and natural notion of generic strategy.•A “relative polymorphic product” which expresses quantification over the type variable Xi in the variable type A with respect to a “universe” which is explicitly given as an additional parameter B. We then solve a recursive equation involving this relative product to obtain a universe in a suitably “absolute” sense.Full Completeness for ML types is proved for this model. (shrink)

Given a countable transitive model of set theory and a partial order contained in it, there is a natural countable Borel equivalence relation on generic filters over the model; two are equivalent if they yield the same generic extension. We examine the complexity of this equivalence relation for various partial orders, focusing on Cohen and random forcing. We prove, among other results, that the former is an increasing union of countably many hyperfinite Borel equivalence relations, and hence is (...) amenable, while the latter is neither amenable nor treeable. (shrink)

We give strengthened versions of the Herwig–Lascar and Hodkinson–Otto extension theorems for partial automorphisms of finite structures. Such strengthenings yield several combinatorial and group-theoretic consequences for homogeneous structures. For instance, we establish a coherent form of the extension property for partial automorphisms for certain Fraïssé classes. We deduce from these results that the isometry group of the rational Urysohn space, the automorphism group of the Fraïssé limit of any Fraïssé class that is the class of all ${\cal F}$-free structures, and (...) the automorphism group of any free homogeneous structure over a finite relational language all contain a dense locally finite subgroup. We also show that any free homogeneous structure admits ample generics. (shrink)

The forcing method is a powerful tool to prove the consistency of set-theoretic assertions relative to the consistency of the axioms of set theory. Laver’s theorem and Bukovský’s theorem assert that set-generic extensions of a given ground model constitute a quite reasonable and sufficiently general class of standard models of set-theory.In Sects. 2 and 3 of this note, we give a proof of Bukovsky’s theorem in a modern setting ). In Sect. 4 we check that the multiverse of (...) set-generic extensions can be treated as a collection of countable transitive models in a conservative extension of ZFC. The last section then deals with the problem of the existence of infinitely-many independent buttons, which arose in the modal-theoretic approach to the set-generic multiverse by Hamkins and Loewe :1793–1817, 2008). (shrink)

ABSTRACT: This paper offers a generic revenge-proof solution to the Sorites paradox that is compatible with several philosophical approaches to vagueness, including epistemicism, supervaluationism, psychological contextualism and intuitionism. The solution is traditional in that it rejects the Sorites conditional and proposes a modally expressed weakened conditional instead. The modalities are defined by the first-order logic QS4M+FIN. (This logic is a modal companion to the intermediate logic QH+KF, which places the solution between intuitionistic and classical logic.) Borderlineness is introduced modally (...) as usual. The solution is innovative in that its modal system brings out the semi-determinability of vagueness. Whether something is borderline and whether a predicate is vague or precise is only semi-determinable: higher-order vagueness is columnar. Finally, the solution is based entirely on two assumptions. (1) It rejects the Sorites conditional. (2) It maintains that if one specifies borderlineness in terms of the ‒suitably interpreted‒ modal logic QS4M+FIN, then one can explain why the Sorites appears paradoxical. From (1)+(2) it results that one can tell neither where exactly in a Sorites series the borderline zone starts and ends nor what its extension is. Accordingly, the solution is also called agnostic. (shrink)

We present a category-theoretical analysis, based on the concept of generic figures, of a diagrammatic system for propositional logic (Peirce’s Existential Graphs α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document}). The straightforward construction of a presheaf category EGα∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {E}}}{{\mathcal {G}}}_{\alpha ^{*}}$$\end{document} of cuts-only Existential Graphs (equivalent to the well-studied category of finite forests) provides a basis for the further construction of the category EGα\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} (...) \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {E}}}{{\mathcal {G}}}_\alpha $$\end{document} which introduces variables in a reconstructedly generic, or label-free, mode. Morphisms in these categories represent syntactical embeddings or, equivalently but dually, extensions. Through the example of Peirce’s system, it is shown how the generic figures approach facilitates the formal investigation of relations between syntax and semantics in such diagrammatic systems. (shrink)

We show that Dependent Choice is a sufficient choice principle for developing the basic theory of proper forcing, and for deriving generic absoluteness for the Chang model in the presence of large cardinals, even with respect to $\mathsf {DC}$ -preserving symmetric submodels of forcing extensions. Hence, $\mathsf {ZF}+\mathsf {DC}$ not only provides the right framework for developing classical analysis, but is also the right base theory over which to safeguard truth in analysis from the independence phenomenon in the (...) presence of large cardinals. We also investigate some basic consequences of the Proper Forcing Axiom in $\mathsf {ZF}$, and formulate a natural question about the generic absoluteness of the Proper Forcing Axiom in $\mathsf {ZF}+\mathsf {DC}$ and $\mathsf {ZFC}$. Our results confirm $\mathsf {ZF} + \mathsf {DC}$ as a natural foundation for a significant portion of “classical mathematics” and provide support to the idea of this theory being also a natural foundation for a large part of set theory. (shrink)

We present a category-theoretical analysis, based on the concept of generic figures, of a diagrammatic system for propositional logic ). The straightforward construction of a presheaf category \ of cuts-only Existential Graphs provides a basis for the further construction of the category \ which introduces variables in a reconstructedly generic, or label-free, mode. Morphisms in these categories represent syntactical embeddings or, equivalently but dually, extensions. Through the example of Peirce’s system, it is shown how the generic (...) figures approach facilitates the formal investigation of relations between syntax and semantics in such diagrammatic systems. (shrink)

In this paper, we present a generic format for adaptive vague logics. Logics based on this format are able to (1) identify sentences as vague or non-vague in light of a given set of premises, and to (2) dynamically adjust the possible set of inferences in accordance with these identifications, i.e. sentences that are identified as vague allow only for the application of vague inference rules and sentences that are identified as non-vague also allow for the application of some (...) extra set of classical logic rules. The generic format consists of a set of minimal criteria that must be satisfied by the vague logic in casu in order to be usable as a basis for an adaptive vague logic. The criteria focus on the way in which the logic deals with a special ⊡-operator. Depending on the kind of logic for vagueness that is used as a basis for the adaptive vague logic, this operator can be interpreted as completely true, definitely true, clearly true, etc. It is proven that a wide range of famous logics for vagueness satisfies these criteria when extended with a specific ⊡-operator, e.g. fuzzy basic logic and its well known extensions, cf. [7], super- and subvaluationist logics, cf. [6], [9], and clarity logic, cf. [13]. Also a fuzzy logic is presented that can be used for an adaptive vague logic that can deal with higher-order vagueness. To illustrate the theory, some toy-examples of adaptive vague proofs are provided. (shrink)

Up to equivalence, a substitution in propositional logic is an endomorphism of its free algebra. On the dual space, this results in a continuous function, and whenever the space carries a natural measure one may ask about the stochastic properties of the action. In classical logic there is a strong dichotomy: while over finitely many propositional variables everything is trivial, the study of the continuous transformations of the Cantor space is the subject of an extensive literature, and is far from (...) being a completed task. In many-valued logic this dichotomy disappears: already in the finite-variable case many interesting phenomena occur, and the present paper aims at displaying some of these. (shrink)

In this paper we use “generic submodels” to prove that each normal extension of G.3 (K4.3W) has the finite model property, by which we establish that each proper normal extension of G.3 is G.3Altn for some n≥0.

An elliptic curve over a supersimple field with exactly one extension of degree 2 has an s-generic point.

We show that if M is a countable transitive model of $\text {ZF}$ and if $a,b$ are reals not in M, then there is a G generic over M such that $b \in L[a,G]$. We then present several applications such as the following: if J is any countable transitive model of $\text {ZFC}$ and $M \not \subseteq J$ is another countable transitive model of $\text {ZFC}$ of the same ordinal height $\alpha $, then there is a forcing extension N (...) of J such that $M \cup N$ is not included in any transitive model of $\text {ZFC}$ of height $\alpha $. Also, assuming $0^{\#}$ exists, letting S be the set of reals generic over L, although S is disjoint from the Turing cone above $0^{\#}$, we have that for any non-constructible real a, $\{ a \oplus s : s \in S \}$ is cofinal in the Turing degrees. (shrink)

The aim of this paper is to develop the theory of groups definable in the p-adic field ${{\mathbb {Q}}_p}$, with “definable f-generics” in the sense of an ambient saturated elementary extension of ${{\mathbb {Q}}_p}$. We call such groups definable f-generic groups.So, by a “definable f-generic” or $dfg$ group we mean a definable group in a saturated model with a global f-generic type which is definable over a small model. In the present context the group is definable over (...) ${{\mathbb {Q}}_p}$, and the small model will be ${{\mathbb {Q}}_p}$ itself. The notion of a $\mathrm {dfg}$ group is dual, or rather opposite to that of an $\operatorname {\mathrm {fsg}}$ group (group with “finitely satisfiable generics”) and is a useful tool to describe the analogue of torsion-free o-minimal groups in the p-adic context.In the current paper our group will be definable over ${{\mathbb {Q}}_p}$ in an ambient saturated elementary extension $\mathbb {K}$ of ${{\mathbb {Q}}_p}$, so as to make sense of the notions of f-generic type, etc. In this paper we will show that every definable f-generic group definable in ${{\mathbb {Q}}_p}$ is virtually isomorphic to a finite index subgroup of a trigonalizable algebraic group over ${{\mathbb {Q}}_p}$. This is analogous to the o-minimal context, where every connected torsion-free group definable in $\mathbb {R}$ is isomorphic to a trigonalizable algebraic group [5, Lemma 3.4]. We will also show that every open definable f-generic subgroup of a definable f-generic group has finite index, and every f-generic type of a definable f-generic group is almost periodic, which gives a positive answer to the problem raised in [28] of whether f-generic types coincide with almost periodic types in the p-adic case. (shrink)

While generic generalisations have been studied by linguists and philosophers for decades, they have only recently become the focus of concentrated interest by cognitive and developmental psychologists, who propose the generics-as-default view. In this paper we focus on the ‘Generic Overgeneralisation’ (GOG) effect proposed by Leslie and colleagues and the native speaker judgments that have been used to support it, and by extension, the generics-as-default view. We take a step back to look at the history of the GOG (...) effect in order to contextualise it. We review existing experimental evidence and discuss four non-mutually exclusive explanations for the GOG effect: ignorance, subkind interpretation, atypical behaviour of all and quantifier domain restriction. We conclude that a closer look at the semantics and pragmatics of generics and universal quantifiers may provide a more nuanced explanation for the pattern of judgment data than that proposed by the generics-as-default view. (shrink)

In this paper, we present a generic format for adaptive vague logics. Logics based on this format are able to (1) identify sentences as vague or non-vague in light of a given set of premises, and to (2) dynamically adjust the possible set of inferences in accordance with these identifications, i.e. sentences that are identified as vague allow only for the application of vague inference rules and sentences that are identified as non-vague also allow for the application of some (...) extra set of classical logic rules. The generic format consists of a set of minimal criteria that must be satisfied by the vague logic in casu in order to be usable as a basis for an adaptive vague logic. The criteria focus on the way in which the logic deals with a special ⊡-operator. Depending on the kind of logic for vagueness that is used as a basis for the adaptive vague logic, this operator can be interpreted as completely true, definitely true, clearly true, etc. It is proven that a wide range of famous logics for vagueness satisfies these criteria when extended with a specific ⊡-operator, e.g. fuzzy basic logic and its well known extensions, cf. [7], super- and subvaluationist logics, cf. [6], [9], and clarity logic, cf. [13]. Also a fuzzy logic is presented that can be used for an adaptive vague logic that can deal with higher-order vagueness. To illustrate the theory, some toy-examples of adaptive vague proofs are provided. (shrink)

The extensions of vague predicates like ‘is bald’, ‘is tall’, and ‘is a heap’ apparently lack sharp boundaries, and this makes such predicates susceptible to soritical reasoning, i.e. reasoning that leads to some version of the notorious sorites paradox. This essay is concerned with a certain kind of theory of vagueness, according to which the symptoms and puzzles of vagueness should be accounted for in terms of a particular species of context sensitivity exhibited by vague expressions. The basic idea (...) is that the extensions of vague predicates vary with certain contextual factors, and that this fact can explain why they appear to lack sharp boundaries. This kind of view is referred to as contextualism about vagueness. A detailed characterisation of contextualism about vagueness is given in chapter two and three. In chapter two, a generic version of contextualism about vagueness is developed, and some alternative forms of context sensitivity are introduced. In chapter three, the specific contextual factors appealed to by different contextualists are discussed. In chapter four, different contextualist diagnoses of the sorites paradox are considered, and found to be problematic in various ways. It is argued that contrary to what some of its proponents have claimed, contextualism about vagueness is not superior to other comparable theories of vagueness when it comes to explaining the appeal of soritical reasoning. In chapter five, a certain version of the sorites paradox, known as the forced march sorites, is discussed. It is argued that “data” about how speakers would behave in the forced march cannot lend any firm support to contextualism about vagueness. In chapter six, some problems concerning the instability of the contextual factors are considered. One problem is that contextualist diagnoses of the sorites which locate a fallacy of equivocation in the reasoning seem to render non-soritical reasoning fallacious as well. A model for treating this problem is suggested, but on closer consideration, it turns out to be problematic. Moreover, this model is of no help in solving the more general problem that even if classical logic remains valid for vague language on some contextualist views, the instability of the extensions of vague predicates makes it difficult to know when a certain piece of reasoning instantiates a valid argument form. Other difficulties arise with respect to speech reports and belief contents. Chapter seven concludes with a summary and some methodological remarks. (shrink)

It is shown that the boolean prime ideal theorem BPIT: every boolean algebra has a prime ideal, does not follow from the order-extension principle OE: every partial ordering can be extended to a linear ordering. The proof uses a Fraenkel-Mostowski model, where the family of atoms is indexed by a countable universal-homogeneous boolean algebra whose boolean partial ordering has a `generic' extension to a linear ordering. To illustrate the technique for proving that the order-extension principle holds in the model (...) we also study Mostowski's ordered model, and give a direct verification of OE there. The key technical point needed to verify OE in each case is the existence of a support structure. (shrink)

In this paper we prove that, in the Cohen extension of a model M of ZFC+CH containing a simplified -morass, η1-orderings without endpoints having cardinality of the continuum, and satisfying specified technical conditions, are order-isomorphic. Furthermore, any order-isomorphism in M between countable subsets of the η1-orderings can be extended to an order-isomorphism between the η1-orderings in the Cohen extension of M. We use the simplified -morass, and commutativity conditions with morass maps on terms in the forcing language, to extend countable (...) partial functions on terms in the forcing language that are forced in all generic extensions to be order-preserving injections. This technique provides for the construction of functions in Cohen extensions adding 2 generic reals for which the only known arguments require transfinite constructions of order type no greater than ω1 in models of ZFC+CH. The specific example presented in this paper is an extension of Tarski’s classic result that in models of ZFC+CH, η1-orderings are order-isomorphic. (shrink)

Some arguments use ‘generic’, or ‘typical’, objects. An explanation for this idea in terms of ‘almost all’ is suggested. The intuition of ‘almost all’ as ‘but for a few exceptions’ is rendered precise by means of ultrafilters. A logical system, with generalized quantifiers for ‘almost all’, is proposed as a basis for generic reasoning. This logic is monotonic, has a simple sound and complete deductive calculus, and is a conservative extension of classical first-order logic, with which it shares (...) several properties. For generic reasoning, generic individuals are introduced and internalized as generic constants, thereby producing conservative extensions where one can reason about generic objects as intended. A many-sorted version of this logic is introduced to handle distinct notions of ‘large’ subsets. Other possible applications for this logic are indicated. (shrink)

We study the preservation under projective ccc forcing extensions of the property of L(ℝ) being a Solovay model. We prove that this property is preserved by every strongly-̰Σ₃¹ absolutely-ccc forcing extension, and that this is essentially the optimal preservation result, i.e., it does not hold for Σ₃¹ absolutely-ccc forcing notions. We extend these results to the higher projective classes of ccc posets, and to the class of all projective ccc posets, using definably-Mahlo cardinals. As a consequence we obtain an (...) exact equiconsistency result for generic absoluteness under projective absolutely-ccc forcing notions. (shrink)

A classic result of Baumgartner-Harrington-Kleinberg [1] implies that assuming CH a stationary subset of ω1 has a CUB subset in a cardinal-perserving generic extension of V, via a forcing of cardinality ω1. Therefore, assuming that $\omega_2^L$ is countable: { $X \in L \mid X \subseteq \omega_1^L$ and X has a CUB subset in a cardinal -preserving extension of L} is constructible, as it equals the set of constructible subsets of $\omega_1^L$ which in L are stationary. Is there a similar (...) such result for subsets of $\ omega_2^L$ ? Building on work of M. Stanley [9], we show that there is not. We shall also consider a number of related problems, examining the extent to which they are "solvable" in the above sense, as well as defining a notion of reduction between them. (shrink)

Given the assertion of a relation between two types, like: “Epidermis has part some Keratinocyte”, we define silent change as any kind of change of the instance-relata of the relation in question that does not change the truth-value of the respective type-level assertion. Such assertions are notoriously difficult to model in OWL 2. To address this problem, we distinguish different modes of type-level relatedness giving rise to this problem and describe a conservative extension to the BFO top-level ontology that allows (...) expressing these modes. (shrink)

Adults design utterances to match listeners' informational needs by making both “generic” adjustments (e.g., mentioning atypical more often than typical information) and “particular” adjustments tailored to their specific interlocutor (e.g., including things that their addressee cannot see). For children, however, relevant evidence is mixed. Three experiments investigated how generic and particular factors affect children's production. In Experiment 1, 4‐ to 5‐year‐old children and adults described typical and atypical instrument events to a silent listener who could either see or (...) not see the events. In later extensions, participants described the same events to either a silent (Experiment 2) or an interactive (Experiment 3) addressee with a specific goal. Both adults and 4‐ to 5‐year‐olds performed generic adjustments but, unlike adults, children made listener‐particular adjustments inconsistently. These and prior findings can be explained by assuming that particular adjustments can be costlier for children to implement compared to generic adjustments. (shrink)