Modern categorical logic as well as the Kripke and topological models of intuitionistic logic suggest that the interpretation of ordinary “propositional” logic should in general be the logic of subsets of a given universe set. Partitions on a set are dual to subsets of a set in the sense of the category-theoretic duality of epimorphisms and monomorphisms—which is reflected in the duality between quotient objects and subobjects throughout algebra. If “propositional” logic is thus seen as (...) the logic of subsets of a universe set, then the question naturally arises of a dual logic of partitions on a universe set. This paper is an introduction to that logic of partitions dual to classical subset logic. The paper goes from basic concepts up through the correctness and completeness theorems for a tableau system of partition logic. (shrink)

We generalize Moss and Parikh's logic of knowledge, effort, and topological reasoning, in two ways. We develop both a multi-agent and a multi-method setting for it. In each of these cases, we prove a corresponding soundness and completeness theorem, and we show that the new logics are decidable. Our methods of proof rely on those for the original system. This might have been expected, since that system is conservatively extended for the given situation. Several technical details are different nevertheless (...) here. (shrink)

Viewing the language of modal logic as a language for describing directed graphs, a natural type of directed graph to study modally is one where the nodes are sets and the edge relation is the subset or superset relation. A well-known example from the literature on intuitionistic logic is the class of Medvedev frames $\langle W,R\rangle$ where $W$ is the set of nonempty subsets of some nonempty finite set $S$, and $xRy$ iff $x\supseteq y$, or more liberally, (...) where $\langle W,R\rangle$ is isomorphic as a directed graph to $\langle \wp(S)\setminus\{\emptyset\},\supseteq\rangle$. Prucnal [32] proved that the modal logic of Medvedev frames is not finitely axiomatizable. Here we continue the study of Medvedev frames with extended modal languages. Our results concern definability. We show that the class of Medvedev frames is definable by a formula in the language of tense logic, i.e., with a converse modality for quantifying over supersets in Medvedev frames, extended with any one of the following standard devices: nominals (for naming nodes), a difference modality (for quantifying over those $y$ such that $x\not= y$), or a complement modality (for quantifying over those $y$ such that $x\not\supseteq y$). It follows that either the logic of Medvedev frames in one of these tense languages is finitely axiomatizable---which would answer the open question of whether Medvedev's [31] "logic of finite problems'' is decidable---or else the minimal logics in these languages extended with our defining formulas are the beginnings of infinite sequences of frame-incomplete logics. (shrink)

Subset Spaces were introduced by L. Moss and R. Parikh in [8]. These spaces model the reasoning about knowledge of changing states.In [2] a kind of subset space called intersection space was considered and the question about the existence of a set of axioms that is complete for the logic of intersection spaces was addressed. In [9] the first author introduced the class of directed spaces and proved that any set of axioms for directed frames also characterizes (...) intersection spaces.We give here a complete axiomatization for directed spaces. We also show that it is not possible to reduce this set of axioms to a finite set. (shrink)

We reformulate a key definition given by Wáng and Ågotnes to provide semantics for public announcements in subset spaces. More precisely, we interpret the precondition for a public announcement of ???? to be the “local truth” of ????, semantically rendered via an interior operator. This is closely related to the notion of ???? being “knowable”. We argue that these revised semantics improve on the original and offer several motivating examples to this effect. A key insight that emerges is the (...) crucial role of topological structure in this setting. Finally, we provide a simple axiomatization of the resulting logic and prove completeness. (shrink)

We extend Moss and Parikh’s modal logic for subset spaces by adding, among other things, state-valued and set-valued functions. This is done with the aid of some basic concepts from hybrid logic. We prove the soundness and completeness of the derived logics with regard to the class of all correspondingly enriched subset spaces, and show that these logics are decidable.

Classical logic is usually interpreted as the logic of propositions. But from Boole's original development up to modern categorical logic, there has always been the alternative interpretation of classical logic as the logic of subsets of any given (nonempty) universe set. Partitions on a universe set are dual to subsets of a universe set in the sense of the reverse-the-arrows category-theoretic duality--which is reflected in the duality between quotient objects and subobjects throughout algebra. Hence the (...) idea arises of a dual logic of partitions. That dual logic is described here. Partition logic is at the same mathematical level as subset logic since models for both are constructed from (partitions on or subsets of) arbitrary unstructured sets with no ordering relations, compatibility or accessibility relations, or topologies on the sets. Just as Boole developed logical finite probability theory as a quantitative treatment of subset logic, applying the analogous mathematical steps to partition logic yields a logical notion of entropy so that information theory can be refounded on partition logic. But the biggest application is that when partition logic and the accompanying logical information theory are "lifted" to complex vector spaces, then the mathematical framework of quantum mechanics is obtained. Partition logic models indefiniteness (i.e., numerical attributes on a set become more definite as the inverse-image partition becomes more refined) while subset logic models the definiteness of classical physics (an entity either definitely has a property or definitely does not). Hence partition logic provides the backstory so the old idea of "objective indefiniteness" in QM can be fleshed out to a full interpretation of quantum mechanics. (shrink)

This paper proposes a formalization of conditional reasoning using Moss and Parikh’s logic of subsets so that a reasoner can express both conditional assertions about beliefs, as well as beliefs about conditional assertions. We present a complete axiomatization of the logic and show that it is decidable. A version of the Ramsey test is found to be compatible with this logic and provides a correspondence between conditionals and belief contraction.

We consider a question of Pereira as to whether the characteristic function of an internally approachable model can lead to free subsets for functions of the model. Pereira isolated the pertinent Approachable Free Subsets Property (AFSP) in his work on the $${\text {pcf}}$$ pcf -conjecture. A recent related property is the Approachable Bounded Subset Property (ABSP) of Ben-Neria and Adolf, and we here directly show it requires modest large cardinals to establish:TheoremIf ABSP holds for an ascending sequence $$ \langle (...) \aleph _{n_{m}} \rangle _{m}$$ ⟨ ℵ n m ⟩ m $$( n_{m} \in \omega )$$ ( n m ∈ ω ) then there is an inner model with measurables $$\kappa < \aleph _{\omega }$$ κ < ℵ ω of arbitrarily large Mitchell order below $$\aleph _{\omega }$$ ℵ ω, that is: $$\sup \left\{ \alpha \mid {\exists }\kappa < \aleph _{\omega } o ( \kappa ) \ge \alpha \right\} = \aleph _{\omega }$$ sup α ∣ ∃ κ < ℵ ω o ( κ ) ≥ α = ℵ ω. A result of Adolf and Ben Neria then shows that this conclusion is in fact the exact consistency strength of ABSP for such an ascending sequence. Their result went via the consistency of the non-existence of continuous tree-like scales; the result of this paper is direct and avoids the use of PCF scales. (shrink)

The logical basis for information theory is the newly developed logic of partitions that is dual to the usual Boolean logic of subsets. The key concept is a "distinction" of a partition, an ordered pair of elements in distinct blocks of the partition. The logical concept of entropy based on partition logic is the normalized counting measure of the set of distinctions of a partition on a finite set--just as the usual logical notion of probability based on (...) the Boolean logic of subsets is the normalized counting measure of the subsets (events). Thus logical entropy is a measure on the set of ordered pairs, and all the compound notions of entropy (join entropy, conditional entropy, and mutual information) arise in the usual way from the measure (e.g., the inclusion-exclusion principle)--just like the corresponding notions of probability. The usual Shannon entropy of a partition is developed by replacing the normalized count of distinctions (dits) by the average number of binary partitions (bits) necessary to make all the distinctions of the partition. (shrink)

Larry Moss and Rohit Parikh used subset semantics to characterize a family of logics for reasoning about knowledge. An important feature of their framework is that subsets always decrease based on the assumption that knowledge always increases. We drop this assumption and modify the semantics to account for logics of knowledge that handle arbitrary changes, that is, changes that do not necessarily result in knowledge increase, such as the update of our knowledge due to an action. We present a (...) system which is complete for subset spaces and prove its decidability. (shrink)

In this paper we will consider two possible definitions of projective subsets of a separable metric space X. A set A subset of or equal to X is Σ11 iff there exists a complete separable metric space Y and Borel set B subset of or equal to X × Y such that A = {x ε X : there existsy ε Y ε B}. Except for the fact that X may not be completely metrizable, this is the classical (...) definition of analytic set and hence has many equivalent definitions, for example, A is Σ11 iff A is relatively analytic in X, i.e., A is the restriction to X of an analytic set in the completion of X. Another definition of projective we denote by ΣX1 or abstract projective subset of X. A set of A subset of or equal to X is ΣX1 iff there exists an n ε ω and a Borel set B subset of or equal to X × Xn such that A = {x ε X:there existsy ε Xn ε B}. These sets ca n be far more pathological. While the family of sets Σ11 is closed under countable intersections and countable unions, there is a consistent example of a separable metric space X where ΣX1 is not closed under countable intersections or countable unions. This takes place in the Cohen real model. Assuming CH, there exists a separable metric space X such that every Σ11 set is Borel in X but there exists a Σ11 set which is not Borel in X2. The space X2 has Borel subsets of arbitrarily large rank while X has bounded Borel rank. This space is a Luzin set and the technique used here is Steel forcing with tagged trees. We give examples of spaces X illustrating the relationship between Σ11 and ΣX1 and give some consistent examples partially answering an abstract projective hierarchy problem of Ulam. (shrink)

A small large cardinal upper bound in V for proving when certain subsets of ω 1 (including the universally Baire subsets) are precisely those constructible from a real is given. In the core model we find an exact equivalence in terms of the length of the mouse order; we show that $\forall B \subseteq \omega_1 \lbrack B$ is universally Baire $\Leftrightarrow B \in L\lbrack r \rbrack$ for some real r] is preserved under set-sized forcing extensions if and only if there (...) are arbitrarily large "admissibly measurable" cardinals. (shrink)

In this thesis we present two logical systems, $\bf MP$ and $\MP$, for the purpose of reasoning about knowledge and effort. These logical systems will be interpreted in a spatial context and therefore, the abstract concepts of knowledge and effort will be defined by concrete mathematical concepts.

The interplay between ultrafilters and unbounded subsets of \ with the order \ of strict eventual domination is studied. Among the tools are special kinds of non-principal ultrafilters on \. These include simple P-points; that is, ultrafilters with a base that is well-ordered with respect to the reverse of the order \ of almost inclusion. It is shown that the cofinality of such a base must be either \, the least cardinality of \-unbounded set, or \, the least cardinality of (...) a \-cofinal set. The small uncountable cardinal \ is introduced. Consequences of \ and of \ are explored; in particular, both imply \. Here \ is the reaping number, and is also the least cardinality of a \-base for a free ultrafilter. Both of these inequalities are shown to occur if there exist simple P-points of different cofinalities and there exist simple \-points and \-points), but this is a long-standing open problem. Six axioms on nonprincipal ultrafilters on \ and the relationships between them are discussed along with various models of set theory in which one or more are known to hold. The strongest of these, Axiom 1, is that for every free ultrafilter \ and for every \-unbounded \-chain C of increasing functions in \, C is also unbounded in the ultraproduct \. The other axioms replace one or both quantifiers with “there exists.” The negation of Axiom 3 in a model provides a family of normal sequentially compact spaces whose product is not countably compact. The question of whether such a family exists in ZFC, even with “normal” weakened to “regular”, is a famous unsolved problem of set-theoretic topology, known as the Scarborough–Stone problem. (shrink)

This paper discusses two combinatorial problems in stability theory. First we prove a partition result for subsets of stable models: for any A and B, we can partition A into |B |<κ(T ) pieces, Ai | i < |B |<κ(T ) , such that for each Ai there is a Bi ⊆ B where |Bi| < κ(T ) and Ai..

We define certain properties of subsets of models of arithmetic related to their codability in end extensions and elementary end extensions. We characterize these properties using some more familiar notions concerning cuts in models of arithmetic.

In this paper, we are interested in parallels to the classical notions of special subsets in defined in the generalized Cantor and Baire spaces (2κ and ). We consider generalizations of the well‐known classes of special subsets, like Lusin sets, strongly null sets, concentrated sets, perfectly meagre sets, σ‐sets, γ‐sets, sets with the Menger, the Rothberger, or the Hurewicz property, but also of some less‐know classes like X‐small sets, meagre additive sets, Ramsey null sets, Marczewski, Silver, Miller, and Laver‐null sets. (...) We notice that many classical theorems regarding these classes can be relatively easy generalized to higher cardinals although sometimes with some additional assumptions. This paper serves as a catalogue of such results along with some other generalizations and open problems. (shrink)

Assuming the existence of a supercompact cardinal, we construct a model of ZFC + ). Answering a question of Uri Abraham [A], [A-S], we prove that adding a real to the world always makes P ℵ 1 - V stationary.

This article enlarges classical syllogistic logic with assertions having to do with comparisons between the sizes of sets. So it concerns a logical system whose sentences are of the following forms: Allxareyand Somexarey, There are at least as manyxasy, and There are morexthany. Herexandyrange over subsets (not elements) of a giveninfiniteset. Moreover,xandymay appear complemented (i.e., as$\bar{x}$and$\bar{y}$), with the natural meaning. We formulate a logic for our language that is based on the classical syllogistic. The main result is a (...) soundness/completeness theorem. There are efficient algorithms for proof search and model construction. (shrink)

We extend some results of Adam Kolany to show that large sets of satisfiable sentences generally contain equally large subsets of mutually consistent sentences. In particular, this is always true for sets of uncountable cofinality, and remains true for sets of denumerable cofinality if we put appropriate bounding conditions on the sentences. The results apply to both the propositional and the predicate calculus. To obtain these results, we use delta sets for regular cardinals, and, for singular cardinals, a generalization of (...) delta sets. All of our results are theorems in ZFC. (shrink)

Logics L F (M) are considered, in which M ("most") is a new first-order quantifier whose interpretation depends on a given filter F of subsets of ω. It is proved that countable compactness and axiomatizability are each equivalent to the assertion that F is not of the form $\{(\bigcap F) \cup X:|\omega - X| with $|\omega - \bigcap F| = \omega$ . Moreover the set of validities of L F (M) and even of L F ω 1 ω (M) depends (...) only on a few basic properties of F. Similar characterizations are given of the class of filters F for which L F (M) has the interpolation or Robinson properties. An omitting types theorem is also proved. These results sharpen the corresponding known theorems on weak models (U, q), where the collection q is allowed to vary. In addition they provide extensions of first-order logic which possess some nice properties, thus escaping from contradicting Lindstrom's Theorem [1969] only because satisfaction is not isomorphism-invariant (as it is tied to the filter F). However, Lindstrom's argument is applied to characterize the invariant sentences as just those of first-order logic. (shrink)

We show that numerous distinctive concepts of constructive mathematics arise automatically from an “antithesis” translation of affine logic into intuitionistic logic via a Chu/Dialectica construction. This includes apartness relations, complemented subsets, anti-subgroups and anti-ideals, strict and non-strict order pairs, cut-valued metrics, and apartness spaces. We also explain the constructive bifurcation of some classical concepts using the choice between multiplicative and additive affine connectives. Affine logic and the antithesis construction thus systematically “constructivize” classical definitions, handling the resulting bookkeeping (...) automatically. (shrink)

We write and for the cardinalities of the set of finite subsets and the set of permutations with finitely many non‐fixed points, respectively, of a set which is of cardinality. In this paper, we investigate relationships between and for an infinite cardinal in the absence of the Axiom of Choice. We give conditions that make and comparable as well as give related consistency results.

Logical information theory is the quantitative version of the logic of partitions just as logical probability theory is the quantitative version of the dual Boolean logic of subsets. The resulting notion of information is about distinctions, differences and distinguishability and is formalized using the distinctions of a partition. All the definitions of simple, joint, conditional and mutual entropy of Shannon information theory are derived by a uniform transformation from the corresponding definitions at the logical level. The purpose of (...) this paper is to give the direct generalization to quantum logical information theory that similarly focuses on the pairs of eigenstates distinguished by an observable, i.e., qudits of an observable. The fundamental theorem for quantum logical entropy and measurement establishes a direct quantitative connection between the increase in quantum logical entropy due to a projective measurement and the eigenstates that are distinguished by the measurement. Both the classical and quantum versions of logical entropy have simple interpretations as “two-draw” probabilities for distinctions. The conclusion is that quantum logical entropy is the simple and natural notion of information for quantum information theory focusing on the distinguishing of quantum states. (shrink)

It is shown that there is no satisfactory first-order characterization of those subsets of ω 2 that have closed unbounded subsets in ω 1 , ω 2 and GCH preserving outer models. These “anticharacterization” results generalize to subsets of successors of uncountable regular cardinals. Similar results are proved for trees of height and cardinality κ + and for partitions of [ κ + ] 2 , when κ is an infinite cardinal.

The term fuzzy logic is used in this paper to describe an imprecise logical system, FL, in which the truth-values are fuzzy subsets of the unit interval with linguistic labels such as true, false, not true, very true, quite true, not very true and not very false, etc. The truth-value set, , of FL is assumed to be generated by a context-free grammar, with a semantic rule providing a means of computing the meaning of each linguistic truth-value in as (...) a fuzzy subset of [0, 1].Since is not closed under the operations of negation, conjunction, disjunction and implication, the result of an operation on truth-values in requires, in general, a linguistic approximation by a truth-value in . As a consequence, the truth tables and the rules of inference in fuzzy logic are (i) inexact and (ii) dependent on the meaning associated with the primary truth-value true as well as the modifiers very, quite, more or less, etc. (shrink)

An outline is given of the proof that the consistency of a κ⁺-Mahlo cardinal implies that of the statement that I[ω₂] does not include any stationary subsets of Cof(ω₁). An additional discussion of the techniques of this proof includes their use to obtain a model with no ω₂-Aronszajn tree and to add an ω₂-Souslin tree with finite conditions.

In this paper we present two families of probability logics (denoted _QLP_ and \(QLP^{ORT}\) ) suitable for reasoning about quantum observations. Assume that \(\alpha \) means “O = a”. The notion of measuring of an observable _O_ can be expressed using formulas of the form \(\square \lozenge \alpha \) which intuitively means “if we measure _O_ we obtain \(\alpha \) ”. In that way, instead of non-distributive structures (i.e., non-distributive lattices), it is possible to relay on classical logic extended (...) with the corresponding modal laws for the modal logic \({\textbf{B}}\). We consider probability formulas of the form \(CS_{z_{1},\rho _{1}; \ldots ; z_{m},\rho _{m}} \square \lozenge \alpha \) related to an observable _O_ and a possible world (vector) _w_: if _a_ is an eigenvalue of _O_, \(w_{1}\),..., \(w_{m}\) form a base of a closed subspace of the considered Hilbert space which corresponds to eigenvalue _a_, and if _w_ is a linear combination of the basis vectors such that \(w=c_{1}\cdot w_{1}+ \cdots + c_{m}\cdot w_{m}\) for some \(c_{i}\in {\mathbb {C}}\), then \(\Vert c_{1}-z_{1}\Vert \le \rho _{1}\),..., \(\Vert c_{m}-z_{m}\Vert \le \rho _{m}\), and the probability of obtaining _a_ while measuring _O_ in the state _w_ is equal to \(\Sigma _{i=1}^{m}\Vert c_{i}\Vert ^{2}\). Formulas are interpreted in reflexive and symmetric Kripke models equipped with probability distributions over families of subsets of possible worlds that are orthocomplemented lattices, while for \(QLP^{ORT}\) also satisfy ortomodularity. We give infinitary axiomatizations, prove the corresponding soundness and strong completeness theorems, and also decidability for _QLP_-logics. (shrink)

According to Frege's principle the denotation of a sentence coincides with its truth-value. The principle is investigated within the context of abstract algebraic logic, and it is shown that taken together with the deduction theorem it characterizes intuitionistic logic in a certain strong sense.A 2nd-order matrix is an algebra together with an algebraic closed set system on its universe. A deductive system is a second-order matrix over the formula algebra of some fixed but arbitrary language. A second-order matrix (...) A is Fregean if, for any subset X of A, the set of all pairs a,b such that X{a} and X{b} have the same closure is a congruence relation on A. Hence a deductive system is Fregean if interderivability is compositional. The logics intermediate between the classical and intuitionistic propositional calculi are the paradigms for Fregean logics. Normal modal logics are non-Fregean while quasi-normal modal logics are generally Fregean.The main results of the paper: Fregean deductive systems that either have the deduction theorem, or are protoalgebraic and have conjunction, are completely characterized. They are essentially the intermediate logics, possibly with additional connectives. All the full matrix models of a protoalgebraic Fregean deductive system are Fregean, and, conversely, the deductive system determined by any class of Fregean second-order matrices is Fregean. The latter result is used to construct an example of a protoalgebraic Fregean deductive system that is not strongly algebraizable. (shrink)

For the computability of subsets of real numbers, several reasonable notions have been suggested in the literature. We compare these notions in a systematic way by relating them to pairs of ‘basic’ ones. They turn out to coincide for full-dimensional convex sets; but on the more general class of regular sets, they reveal rather interesting ‘weaker/stronger’ relations. This is in contrast to single real numbers and vectors where all ‘reasonable’ notions coincide.

We present a canonical form for definable subsets of algebraically closed valued fields by means of decompositions into sets of a simple form, and do the same for definable subsets of real closed valued fields. Both cases involve discs, forming "Swiss cheeses" in the algebraically closed case, and cuts in the real closed case. As a step in the development, we give a proof for the fact that in "most" valued fields F, if f(x),g(x) ∈ F[ x] and v is (...) the valuation map, then the set {x : v(f(x)) ≤ v(g(x))} is a Boolean combination of discs; in fact, it is a finite union of Swiss cheeses. The development also depends on the introduction of "valued trees", which we define formally. (shrink)

For a subset A of a Polish group G, we study the (almost) packing index pack( A) (respectively, Pack( A)) of A, equal to the supremum of cardinalities |S| of subsets $S\subset G$ such that the family of shifts $\{xA\}_{x\in S}$ is (almost) disjoint (in the sense that $|xA\cap yA|<|G|$ for any distinct points $x,y\in S$). Subsets $A\subset G$ with small (almost) packing index are large in a geometric sense. We show that $\pack}(A)\in\mathbb{N}\cup\{\aleph_0,\mathfrak{c}\}$ for any σ-compact (...) class='Hi'>subset A of a Polish group. In each nondiscrete Polish Abelian group G we construct two closed subsets $A,B\subset G$ with $\mathrm{pack}(A)=\mathrm{pack}(B)=\mathfrak{c}$ and \mathrm{Pack}(A\cup B)=1 and then apply this result to show that G contains a nowhere dense Haar null subset $C\subset G$ with pack(C)=Pack(C)=κ for any given cardinal number $\kappa\in[4,\mathfrak{c}]$. (shrink)

Hamkins and Kikuchi (2016, 2017) show that in both set theory and class theory the definable subset ordering of the universe interprets a complete and decidable theory. This paper identifies the minimum subsystem of,, that ensures that the definable subset ordering of the universe interprets a complete theory, and classifies the structures that can be realised as the subset relation in a model of this set theory. Extending and refining Hamkins and Kikuchi's result for class theory, a (...) complete extension,, of the theory of infinite atomic boolean algebras and a minimum subsystem,, of are identified with the property that if is a model of, then is a model of, where M is the underlying set of, is the unary predicate that distinguishes sets from classes and is the definable subset relation. These results are used to show that that decides every 2‐stratified sentence of set theory and decides every 2‐stratified sentence of class theory. (shrink)

We show that if κ is an inaccessible cardinal then P κ λ splits into $\lambda^{ many disjoint stationary subsets. We also show that if P κ λ carries a strongly saturated ideal then the nonstationary ideal cannot be λ + -saturated.

There are two general conceptions on the relationship between probability and logic. In the first, these systems are viewed as complementary—having offsetting strengths and weaknesses—and there exists a fusion of the two that creates a reasoning system that improves upon each. In the second, probability is viewed as an instance of logic, given some sufficiently broad formulation of it, and it is this that should inform the development of more general reasoning systems. These two conceptions are in conflict (...) with each other, where the root issue of contention is the proper abstraction of the concept of logical consequence. In this work, we put forth a proposal on this abstraction based on an extension of the subset relation through the use of projections, which in turn, allows for the formalization of valid inferences to more general settings. Our proposal results in a formalism that encompasses probability and classical logic, and importantly, does so with minimal machinery. This formalism makes assertions about the relationship between these two systems that are explicit, and suggests a path forward in the development of alternatives to them. (shrink)

We introduce the notion of skinniness for subsets of $\mathcal{P}_\kappa \lambda$ and its variants, namely skinnier and skinniest. We show that under some cardinal arithmetical assumptions, precipitousness or $2^\lambda$-saturation of $\mathrm{NS}_{\kappa\lambda}\mid X$, where $\mathrm{NS}_{\kappa\lambda}$ denotes the non-stationary ideal over $\mathcal{P}_\kappa \lambda$, implies the existence of a skinny stationary subset of $X$. We also show that if $\lambda$ is a singular cardinal, then there is no skinnier stationary subset of $\mathcal{P}_\kappa \lambda$. Furthermore, if $\lambda$ is a strong limit singular (...) cardinal, there is no skinny stationary subset of $\mathcal{P}_\kappa \lambda$. Combining these results, we show that if $\lambda$ is a strong limit singular cardinal, then $\mathrm{NS}_{\kappa\lambda}\mid X$ can satisfy neither precipitousness nor $2^\lambda$-saturation for every stationary $X \subseteq \mathcal{P}_\kappa \lambda$. We also indicate that $\diamondsuit_\lambda(E^{\lambda}_{<\kappa})$, where $E^{\lambda}_{<\kappa} \stackrel{\mathrm{def}}{=} \{\alpha < \lambda \mid \mathrm{cf}(\alpha) < \kappa\}$, is equivalent to the existence of a skinnier (or skinniest) stationary subset of $\mathcal{P}_\kappa \lambda$ under some cardinal arithmetical hypotheses. (shrink)

Quine’s most important charge against second-, and more generally, higher-order logic is that it carries massive existential commitments. The force of this charge does not depend upon Quine’s questionable assimilation of second-order logic to set theory. Even if we take second-order variables to range over properties, rather than sets, the charge remains in force, as long as properties are individuated purely extensionally. I argue that if we interpret them as ranging over properties more reasonably construed, in accordance with (...) an abundant or deflationary conception, Quine’s charge can be resisted. This interpretation need not be seen as precluding the use of model-theoretic semantics for second-order languages; but it will preclude the use of the standard semantics, along with the more general Henkin semantics, of which it is a special case. To that extent, the approach I recommend has revisionary implications which some may find unpalatable; it is, however, compatible with the quite different special case in which the second-order variables are taken to range over definable subsets of the first-order domain, and with respect to such a semantics, some important metalogical results obtainable under the standard semantics may still be obtained. In my final section, I discuss the relations between second-order logic, interpreted as I recommend, and a strong version of schematic ancestral logic promoted in recent work by Richard Heck. I argue that while there is an interpretation on which Heck’s logic can be contrasted with second-order logic as standardly interpreted, when it is so interpreted, its differences from the more modest form of second-order logic I advocate are much less substantial, and may be largely presentational. (shrink)

For a regular cardinal κ with κ<κ = κ and κ ≤ λ , we construct generically a subset S of {x ∈ Pκλ : x ∩ κ is a singular ordinal} such that S is stationary in a strong sense but the stationarity of S can be destroyed by a κ+-c. c. forcing ℙ* which does not add any new element of Pκλ . Actually ℙ* can be chosen so that ℙ* is κ-strategically closed. However we show that (...) such ℙ* itself cannot be κ-strategically closed or even <κ-strategically closed if κ is inaccessible. (shrink)

Erdős proved that for every infinite \ there is \ with \, such that all pairs of points from Y have distinct distances, and he gave partial results for general a-ary volume. In this paper, we search for the strongest possible canonization results for a-ary volume, making use of general model-theoretic machinery. The main difficulty is for singular cardinals; to handle this case we prove the following. Suppose T is a stable theory, \ is a finite set of formulas of (...) T, \, and X is an infinite subset of M. Then there is \ with \ and an equivalence relation E on Y with infinitely many classes, each class infinite, such that Y is \\)-indiscernible. We also consider the definable version of these problems, for example we assume \ is perfect and we find some perfect \ with all distances distinct. Finally we show that Erdős’s theorem requires some use of the axiom of choice. (shrink)

We give a proof ofTheorem 1. Let κ be the smallest cardinal such that the free subset property Fr ω (κ,ω 1)holds. Assume κ is singular. Then there is an inner model with ω1 measurable cardinals.