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)

We study what the existence of a classical embedding between computable structures implies about the existence of computable embeddings. In particular, we consider the effect of fixing and varying the computable presentations of the computable structures.

We study the notion of weak amalgamation in the context of diagonal conjugacy classes. Generalizing results of Kechris and Rosendal, we prove that for every countable structure M, Polish group G of permutations of M, and \, G has a comeager n-diagonal conjugacy class iff the family of all n-tuples of G-extendable bijections between finitely generated substructures of M, has the joint embedding property and the weak amalgamation property. We characterize limits of weak Fraïssé (...) classes that are not homogenizable. Finally, we investigate 1- and 2-diagonal conjugacy classes in groups of ball-preserving bijections of certain ordered ultrametric spaces. (shrink)

We present several examples of hereditary classes of finite structures satisfying the joint embedding property and the weak amalgamation property, but failing the cofinal amalgamation property. These include a continuum‐sized family of classes of finite undirected graphs, as well as an example due to Pouzet with countably categorical generic limit.

Downey and Lempp 1215–1240) have shown that the contiguous computably enumerable degrees, i.e. the c.e. Turing degrees containing only one c.e. weak truth-table degree, can be characterized by a local distributivity property. Here we extend their result by showing that a c.e. degree a is noncontiguous if and only if there is an embedding of the nonmodular 5-element lattice N5 into the c.e. degrees which maps the top to the degree a. In particular, this shows that local (...) nondistributivity coincides with local nonmodularity in the computably enumerable degrees. (shrink)

Journal of Mathematical Logic, Volume 22, Issue 02, August 2022. Motivated by results of Bagaria, Magidor and Väänänen, we study characterizations of large cardinal properties through reflection principles for classes of structures. More specifically, we aim to characterize notions from the lower end of the large cardinal hierarchy through the principle [math] introduced by Bagaria and Väänänen. Our results isolate a narrow interval in the large cardinal hierarchy that is bounded from below by total indescribability and from above by subtleness, (...) and contains all large cardinals that can be characterized through the validity of the principle [math] for all classes of structures defined by formulas in a fixed level of the Lévy hierarchy. Moreover, it turns out that no property that can be characterized through this principle can provably imply strong inaccessibility. The proofs of these results rely heavily on the notion of shrewd cardinals, introduced by Rathjen in a proof-theoretic context, and embedding characterizations of these cardinals that resembles Magidor’s classical characterization of supercompactness. In addition, we show that several important weak large cardinal properties, like weak inaccessibility, weak Mahloness or weak [math]-indescribability, can be canonically characterized through localized versions of the principle [math]. Finally, the techniques developed in the proofs of these characterizations also allow us to show that Hamkin’s weakly compact embedding property is equivalent to Lévy’s notion of weak [math]-indescribability. (shrink)

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)

Emotive-factive predicates, such as surprise or be happy, are a source of empirical and theoretical puzzles in the literature on embedded questions. Although they embed wh-questions, they seem not to embed whether-questions. They have complex interactions with negative polarity items such as any or even, and they have been argued to preferentially give rise to weakly exhaustive readings with embedded questions. We offer an empirical overview of the situation in three experiments collecting acceptability judgments, monotonicity judgments, and truth-value judgments. The (...) results straightforwardly confirm the special selectional properties of emotive-factive predicates. More interestingly, they reveal the existence of strongly exhaustive readings for surprise. The results also suggest that the special properties of emotive-factives cannot be solely explained by their monotonicity profiles, because they were not found to differ from the profiles of other responsive predicates. (shrink)

This is a defence of the authenticity of Plato’s Epistula vii against the recent onslaught by Frede and Burnyeat (2015). It focusses on what Ep. vii has to say about writing and the embedded philosophical Digression and evaluates this in the context of other mainly late dialogues. In the Cratylus, Socrates ends with resignation regarding the potential of language study as a source of truth. This is also the case in Ep. vii, where the four means of knowledge (names, definitions, (...) images (diagrams) and knowledge/insight/true opinion) do not offer the essence of reality due to the weakness of language, in this case owing to a diagram with contrary properties. Consequently, name and definition are impermanent and conventional. Contra Burnyeat, definition is not impossible but useful in the acquisition of knowledge. Only after dialectical examination comes a flash of insight into reality. So, reality (truth) must, as in the Cratylus, be studied directly. Significantly for Plato (as opposed to Socrates), the epistemology is illustrated by a mathematical case. Hence it is relevant to look at mathematical procedure in middle and late dialogues. Moreover, the possible role of division is considered. Finally, the critique of writing justified by the Digression is placed in the context of the general aversion to the written word. The epistemology of the Digression is shown to be a fair and competent synopsis of the later Plato’s epistemology. Hence, there seems to be no reason thus far to doubt that Ep. vii is a genuine work of Plato. (shrink)

Some algebraic structures of the set of all effects are investigated and summarized in the notion of a(weak) orthoalgebra. It is shown that these structures can be embedded in a natural way in lattices, via the so-calledMacNeille completion. These structures serve as a model ofparaconsistent quantum logic, orthologic, andorthomodular quantum logic.

One of the numerous characterizations of a Ramsey cardinal κ involves the existence of certain types of elementary embeddings for transitive sets of size κ satisfying a large fragment of ZFC. We introduce new large cardinal axioms generalizing the Ramsey elementary embeddings characterization and show that they form a natural hierarchy between weakly compact cardinals and measurable cardinals. These new axioms serve to further our knowledge about the elementary embedding properties of smaller large cardinals, in particular those still consistent (...) with V = L. (shrink)

A general method for constructing a new class of topological Ramsey spaces is presented. Members of such spaces are infinite sequences of products of Fraïssé classes of finite relational structures satisfying the Ramsey property. The Product Ramsey Theorem of Sokič is extended to equivalence relations for finite products of structures from Fraïssé classes of finite relational structures satisfying the Ramsey property and the Order-Prescribed Free Amalgamation Property. This is essential to proving Ramsey-classification theorems for equivalence relations on (...) fronts, generalizing the Pudlák–Rödl Theorem to this class of topological Ramsey spaces. To each topological Ramsey space in this framework corresponds an associated ultrafilter satisfying some weak partition property. By using the correct Fraïssé classes, we construct topological Ramsey spaces which are dense in the partial orders of Baumgartner and Taylor generating p-points which are k-arrow but not \-arrow, and in a partial order of Blass producing a diamond shape in the Rudin-Keisler structure of p-points. Any space in our framework in which blocks are products of n many structures produces ultrafilters with initial Tukey structure exactly the Boolean algebra \\). If the number of Fraïssé classes on each block grows without bound, then the Tukey types of the p-points below the space’s associated ultrafilter have the structure exactly \. In contrast, the set of isomorphism types of any product of finitely many Fraïssé classes of finite relational structures satisfying the Ramsey property and the OPFAP, partially ordered by embedding, is realized as the initial Rudin-Keisler structure of some p-point generated by a space constructed from our template. (shrink)

LetMbe a countably infinite ω-categorical structure. Consider Aut(M) as a complete metric space by definingd(g, h) = Ω{2−n:g(xn) ≠h(xn) org−1(xn) ≠h−1(xn)} where {xn:n∈ ω} is an enumeration ofMAn automorphism α ∈ Aut(M) is generic if its conjugacy class is comeagre. J. Truss has shown in [11] that if the set P of all finite partial isomorphisms contains a co-final subset P1closed under conjugacy and having the amalgamation property and the joint embedding property then there is a generic (...) automorphism. In the present paper we give a weaker condition of this kind which is equivalent to the existence of generic automorphisms. Really we give more: a characterization of the existence of generic expansions (defined in an appropriate way) of an ω-categorical structure. We also show that Truss' condition guarantees the existence of a countable structure consisting of automorphisms ofMwhich can be considered as an atomic model of some theory naturally associated toM. We do it in a general context of weak models for second-order quantifiers.The author thanks Ludomir Newelski for pointing out a mistake in the first version of Theorem 1.2 and for interesting discussions. Also, the author is grateful to the referee for very helpful remarks. (shrink)

We study the embedding property in the category of sorted profinite groups. We introduce a notion of the sorted embedding property (SEP), analogous to the embedding property for profinite groups. We show that any sorted profinite group has a universal SEP-cover. Our proof gives an alternative proof for the existence of a universal embedding cover of a profinite group. Also our proof works for any full subcategory of the sorted profinite groups, which is (...) closed under taking finite quotients, fibre products, and inverse limits. We also show that any sorted profinite group having SEP has a sorted complete system whose theory is $\omega $ -categorical and $\omega $ -stable under the assumption that the set of sorts is countable. (shrink)

Within Bishop's constructive mathematics we provide conditions that ensure weak continuity properties of mappings between metric and normed spaces.

We investigate the following weak Ramsey property of a cardinal κ: If χ is coloring of nodes of the tree κ <ω by countably many colors, call a tree ${T \subseteq \kappa^{ < \omega}}$ χ-homogeneous if the number of colors on each level of T is finite. Write ${\kappa \rightsquigarrow (\lambda)^{ < \omega}_{\omega}}$ to denote that for any such coloring there is a χ-homogeneous λ-branching tree of height ω. We prove, e.g., that if ${\kappa < \mathfrak{p}}$ or ${\kappa (...) > \mathfrak{d}}$ is regular, then ${{\kappa \rightsquigarrow (\kappa)^{ < \omega}_{\omega}}}$ and that ${\mathfrak{b}}$ ${(\mathfrak{b})^{ < \omega}_{\omega}}$ and ${\mathfrak{d}}$ ${(\mathfrak{d})^{ < \omega}_{\omega}}$ . The arrow is applied to prove a generalization of a theorem of Hurewicz: A Čech-analytic space is σ-locally compact iff it does not contain a closed homeomorphic copy of irrationals. (shrink)

We study the two model-theoretic concepts of weak saturation and weak amalgamation property in the context of accessible categories. We relate these two concepts providing sufficient conditions for existence and uniqueness of weakly saturated objects of an accessible category ${\cal K}$. We discuss the implications of this fact in classical model theory.

The models of normal open induction are those discretely ordered rings, integrally closed in their fraction field whose nonnegative part satisfy Peano's induction axioms for open formulas in the language of ordered semirings.It is known that neither open induction nor the usually studied stronger fragments of arithmetic , have the joint embedding property.We prove that normal models of open induction have the joint embedding property.

According to Thomas Metzinger, many human cognitive processes in the waking state are spontaneous and are deprived of the experience of epistemic agency. He considers mind wandering as a paradigm example of our recurring loss of epistemic agency. I will enrich this view by extending the scope of the concept of epistemic agency to include cases of depressive rumination and creative cognition, which are additional types of spontaneous cognition. Like mind wandering, they are characterized by unique phenomenal and functional properties (...) that give rise to varying degrees of epistemic agency. The main claim of this paper will be that the experience of being an epistemic agent within a certain time frame is a relational phenomenon that emerges from the organism’s capacity to interact with its cognitive niche. To explore this relation, I develop a new framework that integrates phenomenological considerations on epistemic agency with a functional account of the reciprocal coupling of the embodied organism with its cognitive niche. This account rests upon dynamical accounts of strong embodied and embedded cognition and recent work on cognitive niche construction. Importantly, epistemic agency and organism-niche coupling are gradual phenomena ranging from weak to strong realizations. The emerging framework will be employed to analyze mind wandering, depressive rumination, and creative cognition as well as their commonalities and differences. Mind wandering and depressive rumination are cases of weak epistemic agency and organism-niche coupling. However, there are also important phenomenological, functional, and neuronal differences. In contrast, creative cognition is a case of strong epistemic agency and organism-niche coupling. By providing a phenomenological and functional analysis of these distinct types of spontaneous cognition, we can gain a better understanding of the importance of organism-niche interaction for the realization of epistemic agency. (shrink)

We construct a nonlow2 r.e. degree d such that every positive extension of embeddings property that holds below every low2 degree holds below d. Indeed, we can also guarantee the converse so that there is a low r.e. degree c such that that the extension of embeddings properties true below c are exactly the ones true belowd.Moreover, we can also guarantee that no b ≤ d is the base of a nonsplitting pair.

THE ART OF CAUSAL CONJECTURE Glenn Shafer Table of Contents Chapter 1. Introduction........................................................................................ ...........1 1.1. Probability Trees..........................................................................................3 1.2. Many Observers, Many Stances, Many Natures..........................................8 1.3. Causal Relations as Relations in Nature’s Tree...........................................9 1.4. Evidence............................................................................................ ...........13 1.5. Measuring the Average Effect of a Cause....................................................17 1.6. Causal Diagrams..........................................................................................20 1.7. Humean Events............................................................................................23 1.8. Three Levels of Causal Language................................................................27 1.9. An Outline of the Book................................................................................27 Chapter 2. Event Trees............................................................................................... .....31 2.1. Situations and Events...................................................................................32 2.2. The Ordering of Situations and Moivrean Events.......................................35 2.3. Cuts................................................................................................ ..............39 2.4. Humean Events............................................................................................43 2.5. (...) Moivrean Variables......................................................................................49 2.6. Humean Variables........................................................................................53 2.7. Event Trees for Stochastic Processes...........................................................54 2.8. Timing in Event Trees.................................................................................56 2.9. Intersecting Event Trees...............................................................................60 2.10. Notes on the Literature...............................................................................61 Chapter 3. Probability Trees...........................................................................................63 3.1. Some Types of Probability Trees.................................................................64 ii 3.2. Axioms for the Probabilities of Moivrean Events.......................................68 3.3. Zero Probabilities....................................................................................... ..70 3.4. A Sample-Space Analysis of the Event-Tree Axioms.................................72 3.5. Probabilities and Expected Values for Variables.........................................74 3.6. Martingales......................................................................................... .........79 3.7. The Expectation of a Variable in a Cut........................................................83 3.8. Conditional Expected Value and Conditional Expectation.........................87 Chapter 4. The Meaning of Probability...........................................................................91 4.1. The Interpretation of Expected Value..........................................................92 4.2. The Interpretation of Expectation................................................................95 4.3. The Long Run..............................................................................................98 4.4. Changes in Belief.........................................................................................101 4.5. The Empirical Validation of Probability......................................................106 4.6. The Diversity of Uses of Probability...........................................................108 4.7. Notes on the Literature.................................................................................110 Chapter 5. Independent Events.......................................................................................113 5.1. Independence........................................................................................ .......114 5.2. Weak Independence.....................................................................................118 5.3. The Principle of the Common Cause...........................................................121 5.4. Conditional Independence............................................................................128 5.5. Notes on the Literature.................................................................................133 Chapter 6. Events Tracking Events.................................................................................135 6.1. Tracking............................................................................................ ...........137 6.2. Tracking and Conditional Independence.....................................................142 6.3. Stochastic Subsequence...............................................................................143 6.4. Singular Diagrams for Stochastic Subsequence...........................................147 6.5. Conjunctive and Interactive Forks...............................................................149 Chapter 7. Events as Signs of Events..............................................................................153 iii 7.1. Sign................................................................................................ ..............154 7.2. Weak Sign................................................................................................ ....159 7.3. The Ethics of Causal Talk............................................................................160 7.4. Screening Off...............................................................................................16 2 Chapter 8. Independent Variables...................................................................................167 8.1. Unconditional Independence........................................................................170 8.2. Conditional Independence............................................................................175 8.3. Independence for Partitions.........................................................................177 8.4. Independence for Families of Variables......................................................182 8.5. Individual Properties of the Independence Relations...................................186 Chapter 9. Variables Tracking Variables........................................................................189 9.1. Tracking and Conditional Independence: A Summary...............................190 9.2. Strong Tracking............................................................................................ 192 9.3. Strong Tracking and Conditional Independence..........................................198 9.4. Stochastic Subsequence...............................................................................201 9.5. Functional Dependence................................................................................203 9.6. Tracking in Mean.........................................................................................204 9.7. Linear Tracking............................................................................................ 207 9.8. Tracking by Partitions..................................................................................210 9.9. Tracking by Families of Variables...............................................................212 Chapter 10. Variables as Signs of Variables...................................................................215 10.1. Sign................................................................................................ ............219 10.2. Linear Sign................................................................................................ .222 10.3. Scored Sign................................................................................................ 225 10.4. Families of Variables.................................................................................227 Chapter 11. AnTheory of Event Trees.............................................................229 11.1. Event Trees as Sets of Sets........................................................................230 11.2. Event Trees as Partially Ordered Sets........................................................232 iv 11.3. Regular Event Trees...................................................................................240 11.4. The Resolution of Moivrean Variables......................................................244 11.5. Humean Events and Variables...................................................................246 Chapter 12. Martingale Trees..........................................................................................247 12.1. Examples of Decision Trees......................................................................249 12.2. The Meaning of Probability in a Decision Tree.........................................253 12.3. Martingales......................................................................................... .......257 12.4. The Structure of Martingale Trees.............................................................261 12.5. Probability and Causality...........................................................................265 12.6. Lower and Upper Probability.....................................................................269 12.7. The Law of Large Numbers.......................................................................272 12.8. Notes on the Literature...............................................................................274 Chapter 13. Refining............................................................................................ ...........275 13.1. Examples of Refinement............................................................................277 13.2. A Constructive Definition of Finite Refinement.......................................281 13.3. Axioms for Refinement..............................................................................282 13.4. Lifting Moivrean Events and Variables.....................................................288 13.5. Refining Martingale Trees.........................................................................288 13.6. Grounding........................................................................................... .......294 Chapter 14. Principles of Causal Conjecture..................................................................299 14.1. The Diversity of Causal Explanation.........................................................302 14.2. The Mean Effect of the Happening of a Moivrean Event..........................305 14.3. The Effect of a Humean Variable..............................................................311 14.4. Attribution and Generality.........................................................................316 14.5. The Statistical Measurement of the Effect of a Cause...............................319 14.6. Measurement by Experiment.....................................................................320 14.7. Using Our Knowledge of How Things Work............................................322 v 14.8. Sampling Error...........................................................................................329 14.9. The Sampling Frame..................................................................................329 14.10. Notes on the Literature.............................................................................330 Chapter 15. Causal Models.............................................................................................3 31 15.1. The Causal Interpretation of Statistical Prediction....................................333 15.2. Generalizing to a Family of Exogenous Variables....................................337 15.3 Some Joint Causal Diagrams......................................................................339 15.4. Causal Path Diagrams................................................................................342 15.5. Causal Relevance Diagrams.......................................................................346 15.6. The Meaning of Latent Variables..............................................................352 15.7. Notes on the Literature...............................................................................357 Chapter 16. Representing Probability Trees...................................................................359 16.1. Three Graphical Representations...............................................................361 16.2. Skeletal Simplifications.............................................................................368 16.3. Martingale Trees in Type Theory...............................................................371 Appendix A. Huygens’s Probability Trees.....................................................................379 Huygens’s Manuscript in Translation..................................................................380 Appendix B. Some Elements of Graph Theory..............................................................385 B1. Undirected Graphs........................................................................................385 B2. Directed Graphs............................................................................................38 6 Appendix C. Some Elements of Order Theory...............................................................393 C1. Partial and Quasi Orderings.........................................................................393 C2. Singular and Joint Diagrams for Binary Relations.......................................394 C3. Lattices............................................................................................ .............395 C4. The Lattice of Partitions of a Set..................................................................396 Appendix D. The Sample-Space Framework for Probability.........................................399 D1. Probability Measures....................................................................................399 D2. Variables........................................................................................... ...........400 vi D3. Families of Variables...................................................................................401 D4. Expected Value............................................................................................402 D5. The Law of Large Numbers.........................................................................405 D6. Conditional Probability................................................................................406 D7. Conditional Expected Value........................................................................407 Appendix E. Prediction in Probability Spaces................................................................409 E1. Conditional Distribution...............................................................................411 E2. Regression on a Single Variable...................................................................412 E3. Regression on a Partition or a Family of Variables......................................415 E4. Linear Regression on a Single Variable.......................................................418 E5. Linear Regression on a Family of Variables................................................422 Appendix F. Sample-Space Concepts of Independence.................................................425 F1. Overview............................................................................................ ...........426 F2. Independence Proper.....................................................................................432 F3. Unpredictability in Mean..............................................................................434 F4. Simple Uncorrelatedness..............................................................................437 F5. Mixed Uncorrelatedness...............................................................................438 F6. Partial Uncorrelatedness...............................................................................440 F7. Independence for Partitions..........................................................................442 F8. Independence for Families of Variables.......................................................445 F9. The Basic Role of Uncorrelatedness.............................................................448 F10. Dawid’s Axioms.........................................................................................449 Appendix G. Prediction Diagrams..................................................................................453 G1. Path Diagrams............................................................................................ ..454 G2. Generalized Path Diagrams..........................................................................462 G3. Relevance Diagrams.....................................................................................466 G4. Bubbled Relevance Diagrams......................................................................475 Appendix H. Abstract Stochastic Processes...................................................................479 vii H1. Probability Conditionals and Probability Distributions...............................477 H2. Abstract Stochastic Processes......................................................................479 H3. Embedding Variables and Processes in a Sample Space.............................480 References.......................................................................................... ..............................491. (shrink)

We formulate and prove a combinatorial property assuming AD + V = L(R). As a consequence, we show that every regular κ which is either a Suslin cardinal or the successor of a Suslin cardinal is δ 2 1 -supercompact. In particular, all the projective ordinals δ 1 n are δ 2 1 -supercompact.

Expressivism promises an illuminating account of the nature of normative judgment. But worries about the details of expressivist semantics have led many to doubt whether expressivism's putative advantages can be secured. Drawing on insights from linguistic semantics and decision theory, I develop a novel framework for implementing an expressivist semantics that I call ordering expressivism. I argue that by systematically interpreting the orderings that figure in analyses of normative terms in terms of the basic practical attitude of conditional weak (...) preference, the expressivist can explain the semantic properties of normative sentences in terms of the logical properties of that attitude. Expressivism's problems with capturing the logical relations among normative sentences can be reduced to the familiar, more tractable problem of explaining certain coherence constraints on preferences. Particular attention is given to the interpretation of wide-scope negation. The proposed solution is also extended to other types of embedded contexts—most notably, disjunctions. (shrink)

Moral encroachment holds that the epistemic justification of a belief can be affected by moral factors. If the belief might wrong a person or group more evidence is required to justify the belief. Moral encroachment thereby opposes evidentialism, and kindred views, which holds that epistemic justification is determined solely by factors pertaining to evidence and truth. In this essay I explain how beliefs such as ‘that woman is probably an administrative assistant’—based on the evidence that most women employees at the (...) firm are administrative assistants—motivate moral encroachment. I then describe weaknesses of moral encroachment. Finally I explain how we can countenance the moral properties of such beliefs without endorsing moral encroachment, and I argue that the moral status of such beliefs cannot be evaluated independently from the understanding in which they are embedded. (shrink)

An algebraization of the notion of topology has been proposed more than 70 years ago in a classical paper by McKinsey and Tarski, leading to an area of research still active today, with connections to algebra, geometry, logic and many applications, in particular, to modal logics. In McKinsey and Tarski’s setting the model theoretical notion of homomorphism does not correspond to the notion of continuity. We notice that the two notions correspond if instead we consider a preorder relation \( \sqsubseteq (...) \) defined by \(a \sqsubseteq b\) if _a_ is contained in the topological closure of _b_, for _a_, _b_ subsets of some topological space. A _specialization poset_ is a partially ordered set endowed with a further coarser preorder relation \( \sqsubseteq \). We show that every specialization poset can be embedded in the specialization poset naturally associated to some topological space, where the order relation corresponds to set-theoretical inclusion. Specialization semilattices are defined in an analogous way and the corresponding embedding theorem is proved. Specialization semilattices have the amalgamation property. Some basic topological facts and notions are recovered in this apparently very weak setting. The interest of these structures arises from the fact that they also occur in many rather disparate contexts, even far removed from topology. (shrink)

We define what it means for a function on ω1 to be a collapsing function for λ and show that if there exists a collapsing function for +, then there is no precipitous ideal on ω1. We show that a collapsing function for ω2 can be added by forcing. We define what it means to be a weakly ω1-Erdös cardinal and show that in L[E], there is a collapsing function for λ iff λ is less than the least weakly ω1-Erdös (...) cardinal. As a corollary to our results and a theorem of Neeman, the existence of a Woodin limit of Woodin cardinals does not imply the existence of precipitous ideals on ω1. We also show that the following statements hold in L[E]. The least cardinal λ with the Chang property ↠ is equal to the least ω1-Erdös cardinal. In particular, if j is a generic elementary embedding that arises from non-stationary tower forcing up to a Woodin cardinal, then the minimum possible value of j is the least ω1-Erdös cardinal. (shrink)

A normalizable natural deduction formulation, with subformula property, of the implicative fragment of classical logic is presented. A traditional notion of normal deduction is adapted and the corresponding weak normalization theorem is proved. An embedding of the classical logic into the intuitionistic logic, restricted on propositional implicational language, is described as well. We believe that this multiple-conclusion approach places the classical logic in the same plane with the intuitionistic logic, from the proof-theoretical viewpoint.

Any two models of arithmetic can be jointly embedded in a third with any prescribed isomorphic submodels as intersection and any prescribed relative ordering of the skies above the intersection. Corollaries include some known and some new theorems about ultrafilters on the natural numbers, for example that every ultrafilter with the "4 to 3" weak Ramsey partition property is a P-point. We also give examples showing that ultrafilters with the "5 to 4" partition property need not be (...) P-points and that the main theorem cannot be improved to allow a prescribed ordering of lower skies. (shrink)

A number of counterexamples have recently been leveled against Alvin Plantinga's Proper Functionalism, counterexamples aimed at showing that Plantinga's theory fads to provide sufficient conditions for warrant — that elusive epistemic property which together with true belief yields knowledge Among these counterexamples, Laurence Bonjour s is perhaps the most formidable and, if successful, shows that Proper Functionalism is simply too weak to serve as an acceptable theory of warrant In this paper, I argue that, contrary to initial appearances, (...) BonJour's counterexample is not successful More exactly, I argue that, once it is recognized that a defeasibility constraint is deeply embedded within Plantinga's proper function condition for warrant — a constraint which says, in effect, that a belief B is warranted for an agent S only of S does not possess any defeaters against B — BonJour's counterexample to Proper Functionalism can be handled quite straightforwardly. (shrink)

Classically, any structure for a signature ${\rm{\Sigma }}$ may be completed to a model of a desired regular theory ${T}}$ by means of the chase construction or small object argument. Moreover, this exhibits ${\rm{Mod}}\left$ as weakly reflective in ${\rm{Str}}\left$.We investigate this in the constructive setting. The basic construction is unproblematic; however, it is no longer a weak reflection. Indeed, we show that various reflectivity principles for models of regular theories are equivalent to choice principles in the ambient set theory. (...) However, the embedding of a structure into its chase-completion still satisfies a conservativity property, which suffices for applications such as the completeness of regular logic with respect to Tarski models.Unlike most constructive developments of predicate logic, we do not assume that equality between symbols in the signature is decidable. While in this setting, we also give a version of one classical lemma which is trivial over discrete signatures but more interesting here: the abstraction of constants in a proof to variables. (shrink)

This thesis divides naturally into two parts, each concerned with the extent to which the theory of $L$ can be changed by forcing.The first part focuses primarily on applying generic-absoluteness principles to how that definable sets of reals enjoy regularity properties. The work in Part I is joint with Itay Neeman and is adapted from our paper Happy and mad families in $L$, JSL, 2018. The project was motivated by questions about mad families, maximal families of infinite subsets of $\omega (...) $ of which any two have only finitely many members in common. We begin, in the spirit of Mathias, by establishing a strong Ramsey property for sets of reals in the Solovay model, giving a new proof of Törnquist’s theorem that there are no infinite mad families in the Solovay model.In Chapter 3 we stray from the main line of inquiry to briefly study a game-theoretic characterization of filters with the Baire Property.Neeman and Zapletal showed, assuming roughly the existence of a proper class of Woodin cardinals, that the boldface theory of $L$ cannot be changed by proper forcing. They call their result the Embedding Theorem, because they conclude that in fact there is an elementary embedding from the $L$ of the ground model to that of the proper forcing extension. With a view toward analyzing mad families under $\mathsf {AD}^+$ and in $L$ under large-cardinal hypotheses, in Chapter 4 we establish triangular versions of the Embedding Theorem. These are enough for us to use Mathias’s methods to show that there are no infinite mad families in $L$ under large cardinals and that $\mathsf {AD}^+$ implies that there are no infinite mad families. These are again corollaries of theorems about strong Ramsey properties under large-cardinal assumptions and $\mathsf {AD}^+$, respectively. Our first theorem improves the large-cardinal assumption under which Todorcevic established the nonexistence of infinite mad families in $L$. Part I concludes with Chapter 5, a short list of open questions.In the second part of the thesis, we undertake a finer analysis of the Embedding Theorem and its consistency strength. Schindler found that the the Embedding Theorem is consistent relative to much weaker assumptions than the existence of Woodin cardinals. He defined remarkable cardinals, which can exist even in L, and showed that the Embedding Theorem is equiconsistent with the existence of a remarkable cardinal. His theorem resembles a theorem of Harrington–Shelah and Kunen from the 1980s: the absoluteness of the theory of $L$ to ccc forcing extensions is equiconsistent with a weakly compact cardinal. Joint with Itay Neeman, we improve Schindler’s theorem by showing that absoluteness for $\sigma $ -closed $\ast $ ccc posets—instead of the larger class of proper posets—implies the remarkability of $\aleph _1^V$ in L. This requires a fundamental change in the proof, since Schindler’s lower-bound argument uses Jensen’s reshaping forcing, which, though proper, need not be $\sigma $ -closed $\ast $ ccc in that context. Our proof bears more resemblance to that of Harrington–Shelah than to Schindler’s.The proof of Theorem 6.2 splits naturally into two arguments. In Chapter 7 we extend the Harrington–Shelah method of coding reals into a specializing function to allow for trees with uncountable levels that may not belong to L. This culminates in Theorem 7.4, which asserts that if there are $X\subseteq \omega _1$ and a tree $T\subseteq \omega _1$ of height $\omega _1$ such that X is codable along T, then $L$ -absoluteness for ccc posets must fail.We complete the argument in Chapter 8, where we show that if in any $\sigma $ -closed extension of V there is no $X\subseteq \omega _1$ codable along a tree T, then $\aleph _1^V$ must be remarkable in L.In Chapter 9 we review Schindler’s proof of generic absoluteness from a remarkable cardinal to show that the argument gives a level-by-level upper bound: a strongly $\lambda ^+$ -remarkable cardinal is enough to get $L$ -absoluteness for $\lambda $ -linked proper posets.Chapter 10 is devoted to partially reversing the level-by-level upper bound of Chapter 9. Adapting the methods of Neeman, Hierarchies of forcing axioms II, we are able to show that $L$ -absoluteness for $\left |\mathbf {R}\right |\cdot \left |\lambda \right |$ -linked posets implies that the interval $[\aleph _1^V,\lambda ]$ is $\Sigma ^2_1$ -remarkable in L.prepared by Zach Norwood.E-mail: [email protected]. (shrink)

Social Epistemology, as formulated by Steve Fuller, is based on the suggestion that rational knowledge policy must be held accountable to ‘brute facts’ about the nature of our human cognitive pursuits, whatever these may be. One difficulty for Fuller concerns the conception of the social which underwrites social epistemology. I argue that social epistemology conflates the social with human psychological properties that are available for public scrutiny and, accordingly, that social epistemology is best viewed as a brand of psychologism. Though (...) Fuller's proposal signifies an important step in the ongoing attempt by scholars to eradicate the last traces of Descartes' epistemological device of a disembodiedres cogitans, I conclude that his conception of the social is too weak to serve as the basis for a socially-embedded discipline in anything but name only. (shrink)

We investigate in this work the meaning of weak values through the prism of property ascription in quantum systems. Indeed, the weak measurements framework contains only ingredients of the standard quantum formalism, and as such weak measurements are from a technical point of view uncontroversial. However attempting to describe properties of quantum systems through weak values—the output of weak measurements—goes beyond the usual interpretation of quantum mechanics, that relies on eigenvalues. We first recall the (...) usual form of property ascription, based on the eigenstate-eigenvalue link and the existence of “elements of reality”. We then describe against this backdrop the different meanings that have been given to weak values. We finally argue that weak values can be related to a specific form of property ascription, weaker than the eigenvalues case but still relevant to a partial description of a quantum system. (shrink)

Let M=\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal{M}}=}$$\end{document} be a weakly o-minimal expansion of a dense linear order without endpoints. Some tame properties of sets and functions definable in M\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal{M}}}$$\end{document} which hold in o-minimal structures, are examined. One of them is the intermediate value property, say IVP. It is shown that strongly continuous definable functions in M\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal{M}}}$$\end{document} satisfy an (...) extended version of IVP. After introducing a weak version of definable connectedness in M\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal{M}}}$$\end{document}, we prove that strong cells in M\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal{M}}}$$\end{document} are weakly definably connected, so every set definable in M\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal{M}}}$$\end{document} is a finite union of its weakly definably connected components, provided that M\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal{M}}}$$\end{document} has the strong cell decomposition property. Then, we consider a local continuity property for definable functions in M\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal{M}}}$$\end{document} and conclude some results on cell decomposition regarding that property. Finally, we extend the notion of having no dense graph which was examined for definable functions in and related to uniform finiteness, definable completeness, and others. We show that every weakly o-minimal structure M\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal{M}}}$$\end{document} having cell decomposition, satisfies NDG, i.e. every definable function in M\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal{M}}}$$\end{document} has no dense graph. (shrink)

We introduce the anti-rectangle refining property for forcing notions and investigate fragments of Martin’s axiom for ℵ1 dense sets related to the anti-rectangle refining property, which is close to some fragment of Martin’s axiom for ℵ1 dense sets related to the rectangle refining property, and prove that they are really weaker fragments.

A weak constructive sequential compactness property of metric spaces is introduced. It is proved that for complete, totally bounded metric spaces this property is equivalent to Brouwer's fan theorem for detachable bars. Our results form a part of constructive reverse mathematics.

As it is well known, Relevance Logic R plus the axiom mingle (R-Mingle) does not have the variable-sharing property (vsp). The aim of this paper is to improve this result by defining a weak logic with the axiom mingle and not included in minimal logic BM lacking the vsp.

An algebraic type of structure is shown forth which is such that if it is a characteristic matrix for a logic, then that logic satisfies Meyer's weak variable sharing property. As a corollary, it is shown that RM and all its odd-valued extensions \(\mathbf{RM}_{2n\mathord{-}1}\) satisfy the weak variable sharing property. It is also shown that a proof to the effect that the "fuzzy" version of the relevant logic R satisfies the property is incorrect.

The main result of this paper is Theorem 3.1 which is a criterion for weak o-minimality of a linearly ordered structure in terms of realizations of 1-types. Here we also prove some other properties of weakly o-minimal structures. In particular, we characterize all weakly o-minimal linear orderings in the signature $\{ . Moreover, we present a criterion for density of isolated types of a weakly o-minimal theory. Lastly, at the end of the paper we present some remarks on the (...) Exchange Principle for algebraic closure in a weakly o-minimal structure. (shrink)

The paper concerns the uncertainty in current propositions for the regulation of tissue donation. It focuses mainly on two statements issued in Germany. The scope of the paper is to give a systematic approach to ethical problems coming up in this field. Both statements try to maintain the idea of positive autonomy in regard to tissue donation, but their attempt eventually is forced to fail. Different procedures are proposed that most often are not practicable (because a truly “informed” consent is (...) impossible, or because optional models tend to overwhelm donors). Blanket consent is also proposed, but this form of consent cannot be seen as an expression of self-determination. Under the pretence of autonomy, donors are left alone with the task to control scientific research and to have their personal and property rights respected. Following this rather weak position of autonomy, in one statement there is a clear tendency to place the intrinsic value of research over the autonomy of the donor. In order to avoid this conclusion, autonomy has to be more than individual decision making. It has to be embedded in social and institutional rules which support and protect individual choice. If the benefits of research are supposed to lead to a common good and to satisfy public interests, then research has to be controlled through public institutions. Autonomy does not exclude institutional support, as institutional support is the only way to take the autonomy of donors seriously. (shrink)

We investigate, within the framework of Zermelo-Fraenkel set theory ZF, the interrelations between weak forms of the Axiom of Choice AC restricted to sets of reals.