We show that a coherent theory of partially ordered connectives can be developed along the same line as partially ordered quantification. We estimate the expressive power of various partially ordered connectives and use methods like Ehrenfeucht games and infinitary logic to get various undefinability results.

Motivated by constraint satisfaction problems, Feder and Vardi (SIAM Journal of Computing, 28, 57–104, 1998) set out to search for fragments of satisfying the dichotomy property: every problem definable in is either in P or else NP-complete. Feder and Vardi considered in this connection two logics, strict NP (or SNP) and monadic, monotone, strict NP without inequalities (or MMSNP). The former consists of formulas of the form , where is a quantifier-free formula in a relational vocabulary; and the latter is (...) the fragment of SNP whose formulas involve only negative occurrences of relation symbols, only monadic second-order quantifiers, and no occurrences of the equality symbol. It remains an open problem whether MMSNP enjoys the dichotomy property. In the present paper, SNP and MMSNP are characterized in terms of partially ordered connectives. More specifically, SNP is characterized using the logic D of partially ordered connectives introduced in Blass and Gurevich (Annals of Pure and Applied Logic, 32, 1–16, 1986), Sandu and Väänänen (Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, 38, 361–372 1992), and MMSNP employing a generalization C of D introduced in the present paper. (shrink)

Connections between partially ordered connectives and Henkin quantifiers are considered. It is proved that the logic with all partially ordered connectives and the logic with all Henkin quantifiers coincide. This implies that the hierarchy of partially ordered connectives is strongly hierarchical and gives several nondefinability results between some of them. It is also deduced that each Henkin quantifier can be defined by a quantifier of the form equation imagewhat is a strengthening (...) of the Walkoe result. MSC: 03C80. (shrink)

Starting from a connected, partially ordered set of events, it is shown that results of the measurement of time are elements of a partially ordered and filtering field, as used in a previous paper. Moreover, some relations between physical formulas and properties of the field are proved. Finally, some open problems and suggestions are pointed out. For the convenience of the reader not acquainted with elementary algebraic methods, proofs are given in detail.

This paper gives a new, proof-theoretic explanation of partial-order reasoning about time in a nonmonotonic theory of action. The explanation relies on the technique of lifting ground proof systems to compute results using variables and unification. The ground theory uses argumentation in modal logic for sound and complete reasoning about specifications whose semantics follows Gelfond and Lifschitz’s language. The proof theory of modal logic A represents inertia by rules that can be instantiated by sequences of time steps or events. Lifting (...) such rules introduces string variables and associates each proof with a set of string equations; these equations are equivalent to a set of partial-order tree-constraints that can be solved efficiently. The defeasible occlusion of inertia likewise imposes partial-order constraints in the lifted system. By deriving an auxiliary partial order representation of action from the underlying logic, not the input formulas or proofs found, this paper strengthens the connection between practical planners and formal theories of action. Moreover, the general correctness of the theory of action justifies partial-order representations not only for forward reasoning from a completely specified start state, but also for explanatory reasoning and for reasoning by cases. (shrink)

This paper uses a partially ordered set of syntactic categories to accommodate optionality and licensing in natural language syntax. A complex but well-studied data set pertaining to the syntax of quantifier scope and negative polarity licensing in Hungarian is used to illustrate the proposal. The presentation is geared towards both linguists and logicians. The paper highlights that the main ideas can be implemented in different grammar formalisms, and discusses in detail an implementation where the partial ordering on categories (...) is given by the derivability relation of a calculus with residuated and Galois-connected unary operators. (shrink)

In this paper we relax the assumption that the logical constants of ordinary first-order logic be linearly ordered. As a consequence, we shall have formulas involving not only partially ordered quantifiers, but also partially ordered connectives. The resulting language, called the language of informational independence will be given an interpretation in terms of games of imperfect information. The II-logic will be seen to have some interesting properties: It is very natural to define in this (...) logic two negations, weak negation as failure to verify a sentence, and strong negation as the existence of a falsifying strategy: One can express in this logic complete problems of finite structures, like the non-connectedness and 3-colorability of finite graphs, the satisfiability problem for Boolean circuits built up from NAND-gates, etc. (shrink)

Why do relationships of friendship and love support partiality, but not relationships of hatred or commitments of racism? Where does partiality end and why? I take the intuitive starting point that important cases of partiality are meaningful. I develop a view whereby meaning is understood in terms of transcending self-limitations in order to connect with things of external value. I then show how this view can be used to distinguish central cases of legitimate partiality from cases of illegitimate partiality and (...) how it puts pressure on the traditional way of thinking about partiality. (shrink)

We study the partial orderings of the form ⟨P,⊂⟩\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\langle \mathbb{P}, \subset\rangle}$$\end{document}, where X\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{X}}$$\end{document} is a binary relational structure with the connectivity components isomorphic to a strongly connected structure Y\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{Y}}$$\end{document} and P\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{P} }$$\end{document} is the set of substructures of X\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} (...) \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {X}}$$\end{document} isomorphic to X\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{X}}$$\end{document}. We show that, for example, for a countable X\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{X}}$$\end{document}, the poset ⟨P,⊂⟩\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\langle \mathbb {P}, \subset\rangle}$$\end{document} is either isomorphic to a finite power of P\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{P} }$$\end{document} or forcing equivalent to a separative atomless σ-closed poset and, consistently, to P/fin. In particular, this holds for each ultrahomogeneous structure X\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{X}}$$\end{document} such that X\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{X}}$$\end{document} or Xc\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{X}^{c}}$$\end{document} is a disconnected structure and in this case Y\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{Y}}$$\end{document} can be replaced by an ultrahomogeneous connected digraph. (shrink)

This paper provides a defence of the account of partial resemblances between properties according to which such resemblances are due to partial identities of constituent properties. It is argued, first of all, that the account is not only required by realists about universals à la Armstrong, but also useful (of course, in an appropriately re-formulated form) for those who prefer a nominalistic ontology for material objects. For this reason, the paper only briefly considers the problem of how to conceive of (...) the structural universals first posited by Armstrong in order to explain partial resemblances, and focuses instead on criticisms that have been levelled against the theory (by Pautz, Eddon, Denkel and Gibb) and that apply regardless of one’s preferred ontological framework. The partial identity account is defended from these objections and, in doing so, a hitherto quite neglected connection—between the debate about partial similarity as partial identity and that concerning ontological finitism versus infinitism—is looked at in some detail. (shrink)

A continuum is here a primitive notion intended to correspond precisely to a path-connected subset of the usual euclidean space. In contrast, however, to the traditional treatment, we treat here continua not as pointsets, but as irreducible entities equipped only with a partial ordering ≤ interpreted as parthood. Our aim is to examine what basic topological and geometric properties of continua can be expressed in the language of ≤, and what principles we need in order to prove elementary facts about (...) them. Surprisingly enough ≤ suffices to formulate the very heart of continuity (=jumpless and gapless transitions) in a general setting. Further, using a few principles about ≤ (together with the axioms of ZFC), we can define points, joins, meets and infinite closeness. Most important, we can develop a dimension theory based on notions like path, circle, line (=one-dimensional continuum), simple line and surface (=two-dimensional continuum), recovering thereby in a rigorous way Poincaré's well-known intuitive idea that dimension expresses the ways in which a continuum can be torn apart. We outline a classification of lines according to the number of circles and branching points they contain. The ordering (C,≤) is a topped and bottomed, atomic, almost dense and complete partial ordering, weaker than a lattice. Continuous transformations from C to C are also defined in a natural way and results about them are proved. (shrink)

In this paper we apply proof theoretic methods used for classical systems in order to obtain upper bounds for systems in partial logic. We focus on a truth predicate interpreted in a Kripke style way via strong Kleene; whereas the aim is to connect harmoniously the partial version of Kripke–Feferman with its intended semantics. The method we apply is based on infinitary proof systems containing an ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega $$\end{document}-rule.

In this paper we apply proof theoretic methods used for classical systems in order to obtain upper bounds for systems in partial logic. We focus on a truth predicate interpreted in a Kripke style way via strong Kleene; whereas the aim is to connect harmoniously the partial version of Kripke–Feferman with its intended semantics. The method we apply is based on infinitary proof systems containing an ω-rule.

Set theory, it has been contended, developed from its beginnings through a progression ofmathematicalmoves, despite being intertwined with pronounced metaphysical attitudes and exaggerated foundational claims that have been held on its behalf. In this paper, the seminal results of set theory are woven together in terms of a unifying mathematical motif, one whose transmutations serve to illuminate the historical development of the subject. The motif is foreshadowed in Cantor's diagonal proof, and emerges in the interstices of the inclusion vs. membership (...) distinction, a distinction only clarified at the turn of this century, remarkable though this may seem. Russell runs with this distinction, but is quickly caught on the horns of his well-known paradox, an early expression of our motif. The motif becomes fully manifest through the study of functionsof the power set of a set into the set in the fundamental work of Zermelo on set theory. His first proof in 1904 of his Well-Ordering Theoremis a central articulation containing much of what would become familiar in the subsequent development of set theory. Afterwards, the motif is cast by Kuratowski as a fixed point theorem, one subsequently abstracted to partial orders by Bourbaki in connection with Zorn's Lemma. Migrating beyond set theory, that generalization becomes cited as the strongest of fixed point theorems useful in computer science.Section 1 describes the emergence of our guiding motif as a line of development from Cantor's diagonal proof to Russell's Paradox, fueled by the clarification of the inclusion vs. membership distinction. Section 2 engages the motif as fully participating in Zermelo's work on the Well-Ordering Theorem and as newly informing on Cantor's basic result that there is no bijection. Then Section 3 describes in connection with Zorn's Lemma the transformation of the motif into an abstract fixed point theorem, one accorded significance in computer science. (shrink)

Preface 1 Partially ordered sets 1.1 Posets introduced 1.2 Partial orders and strict orders 1.3 Maps between posets 1.4 Compounding maps 1.5 Order similarity 1.6 Inclusion posets as typical..

This paper follows Part I of our essay on case-intensional first-order logic (CIFOL; Belnap and Müller (2013)). We introduce a framework of branching histories to take account of indeterminism. Our system BH-CIFOL adds structure to the cases, which in Part I formed just a set: a case in BH-CIFOL is a moment/history pair, specifying both an element of a partial ordering of moments and one of the total courses of events (extending all the way into the future) that that moment (...) is part of. This framework allows us to define the familiar Ockhamist temporal/modal connectives, most notably for past, future, and settledness. The novelty of our framework becomes visible in our discussion of substances in branching histories, i.e., in its first-order part. That discussion shows how the basic idea of tracing an individual thing from case to case via an absolute property is applicable in a branching histories framework. We stress the importance of keeping apart extensionality and moment-definiteness, and give a formal account of how the specification of natural sortals and natural qualities turns out to be a coordination task in BH-CIFOL. We also provide a detailed answer to Lewis’s well-known argument against branching histories, exposing the fallacy in that argument. (shrink)

The totality of extensional 1-ary connectives distinguishable in a logical framework allowing sequents with multiple or empty (alongside singleton) succedents form a lattice under a natural partial ordering relating one connective to another if all the inferential properties of the former are possessed by the latter. Here we give a complete description of that lattice; its Hasse diagram appears as Figure 1 in §2. Simple syntactic descriptions of the lattice elements are provided in §3; §§4 and 5 give some (...) additional remarks on matrix methods and on alternative terminology. Background: The size of this lattice was underestimated in [3]; some missing cases were noted in [4] in the course of correcting an example from [3] purporting to show the non-distributivity of the lattice. All the 'missing cases' (as well as those originally noted) are covered here. (The present discussion is self-contained.). (shrink)

It is a well-known fact that MV-algebras, the algebraic counterpart of Łukasiewicz logic, correspond to a certain type of partial algebras: lattice-ordered effect algebras fulfilling the Riesz decomposition property. The latter are based on a partial, but cancellative addition, and we may construct from them the representing ℓ-groups in a straightforward manner. In this paper, we consider several logics differing from Łukasiewicz logics in that they contain further connectives: the PŁ-, PŁ'-, PŁ'△-, and ŁΠ-logics. For all their algebraic (...) counterparts, we characterise the corresponding type of partial algebras. We moreover consider the representing f-rings. All in all, we get three-fold correspondences: the total algebras - the partial algebras - the representing rings. (shrink)

This paper follows Part I of our essay on case-intensional first-order logic ). We introduce a framework of branching histories to take account of indeterminism. Our system BH-CIFOL adds structure to the cases, which in Part I formed just a set: a case in BH-CIFOL is a moment/history pair, specifying both an element of a partial ordering of moments and one of the total courses of events that that moment is part of. This framework allows us to define the familiar (...) Ockhamist temporal/modal connectives, most notably for past, future, and settledness. The novelty of our framework becomes visible in our discussion of substances in branching histories, i.e., in its first-order part. That discussion shows how the basic idea of tracing an individual thing from case to case via an absolute property is applicable in a branching histories framework. We stress the importance of keeping apart extensionality and moment-definiteness, and give a formal account of how the specification of natural sortals and natural qualities turns out to be a coordination task in BH-CIFOL. We also provide a detailed answer to Lewis’s well-known argument against branching histories, exposing the fallacy in that argument. (shrink)

In this paper we show that first-order languages extended with partially ordered connectives and partially ordered quantifiers define, under a certain interpretation, their own truth-predicate. The interpretation in question is in terms of games of imperfect information. This result is compared with those of Kripke and Feferman.

It is a curious fact that the reverse surface of many of the so-called Amarna letters sent by Levantine rulers to the Egyptian pharaoh are completely blank or only partially inscribed. In this article, I establish the absolute and relative frequency of the phenomenon within this subcorpus of the Amarna letters. Next, I connect it to a particular type of letter sent by the Levantine rulers that I designate a “replies-to- an-order” letter and offer a suggestion as to why (...) the Egyptian administration may have requested that these letters have blank space on the reverse surface. Finally, I consider one possible origin for the phenomenon in other cuneiform corpora and end the article with a few comments on directions for future research. (shrink)

We investigate the partial orderings of the form 〈P(X),⊂〉\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\langle \mathbb{P}(\mathbb{X}), \subset \rangle}$$\end{document}, where X=〈X,ρ〉\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{X} =\langle X, \rho \rangle }$$\end{document} is a countable binary relational structure and P(X)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{P} (\mathbb{X})}$$\end{document} the set of the domains of its isomorphic substructures and show that if the components of X\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{X}}$$\end{document} (...) are maximally embeddable and satisfy an additional condition related to connectivity, then the poset 〈P(X),⊂〉\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\langle \mathbb{P} (\mathbb{X}), \subset \rangle }$$\end{document} is forcing equivalent to a finite power of (P(ω)/ Fin)+, or to the poset (P(ω × ω)/(fin × Fin))+, or to the product (P(Δ)/EDfin)+×((P(ω)/Fin)+)n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${(P(\Delta )/\fancyscript{e}\fancyscript{d}_{\rm fin})^+ \times ((P(\omega )/{\rm Fin})^+)^n}$$\end{document}, for some n∈ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${n \in \omega}$$\end{document}. In particular we obtain forcing equivalents of the posets of copies of countable equivalence relations, disconnected ultrahomogeneous graphs and some partial orderings. (shrink)

In this paper, we introduce a variant of second-order propositional modal logic interpreted on general (or Henkin) frames, \(SOPML^{\mathcal {H}}\), and present a decidable fragment of this logic, \(SOPML^{\mathcal {H}}_{dec}\), that preserves important expressive capabilities of \(SOPML^{\mathcal {H}}\). \(SOPML^{\mathcal {H}}_{dec}\) is defined as a _modal loosely guarded fragment_ of \(SOPML^{\mathcal {H}}\). We demonstrate the expressive power of \(SOPML^{\mathcal {H}}_{dec}\) using examples in which modal operators obtain (a) the epistemic interpretation, (b) the dynamic interpretation. \(SOPML^{\mathcal {H}}_{dec}\) partially satisfies the principle (...) of non-Fregean logic: two different _atomic_ propositions with the same truth value can have different contents. In \(SOPML^{\mathcal {H}}_{dec}\), we also define _relating connectives_ and show that the _weak Boethius’ Thesis_ built using these connectives is a valid formula of \(SOPML^{\mathcal {H}}_{dec}\). (shrink)

I problematize Grounding-based formulations of physicalism. More specifically, I argue, first, that motivations for adopting a Grounding-based formulation of physicalism are unsound; second, that a Grounding-based formulation lacks illuminating content, and that attempts to imbue Grounding with content by taking it to be a strict partial order are unuseful and problematic ; third, that conceptions of Grounding as constitutively connected to metaphysical explanation conflate metaphysics and epistemology, are ultimately either circular or self-undermining, and controversially assume that physical dependence is incompatible (...) with explanatory gaps; fourth, that in order to appropriately distinguish physicalism from strong emergentism, a Grounding-based formulation must introduce one and likely two primitives in addition to Grounding; and fifth, that understanding physical dependence in terms of Grounding gives rise to ‘spandrel’ questions, including, e.g., “What Grounds Grounding?”, which arise only due to the overly abstract nature of Grounding. (shrink)

Do the partial order and ortholattice operations of a quantum logic correspond to the logical implication and connectives of classical logic? Rephrased, How far might a classical understanding of quantum mechanics be, in principle, possible? A celebrated result of Kochen and Specker answers the above question in the negative. However, this answer is just one among various possible ones, not all negative. It is our aim to discuss the above question in terms of mappings of quantum worlds into classical (...) ones, more specifically, in terms of embeddings of quantum logics into classical logics; depending upon the type of restrictions imposed on embeddings, the question may get negative or positive answers. (shrink)

The history of productivity of the κ-chain condition in partial orders, topological spaces, or Boolean algebras is surveyed, and its connection to the set-theoretic notion of a weakly compact cardinal is highlighted. Then, it is proved that for every regular cardinal κ > א1, the principle □ is equivalent to the existence of a certain strong coloring c : [κ]2 → κ for which the family of fibers T is a nonspecial κ-Aronszajn tree. The theorem follows from an analysis of (...) a new characteristic function for walks on ordinals, and implies in particular that if the κ-chain condition is productive for a given regular cardinal κ > א1, then κ is weakly compact in some inner model of ZFC. This provides a partial converse to the fact that if κ is a weakly compact cardinal, then the κ-chain condition is productive. (shrink)

Introduction. The book is a study in Adam Smith's system of ideas; its aim is to reconstruct the peculiar framework that Adam Smith’s work provided for the shaping of a semi-autonomous new discipline, political economy; the approach adopted lies somewhere in-between the history of ideas and the history of economic analysis. My two claims are: i) The Wealth of Nations has a twofold structure, including a `natural history' of opulence and an `imaginary machine' of wealth. The imaginary machine is a (...) kind of Newtonian theory, whose connecting links are principles; provided either by `partial' characteristics of human nature or by analoga of physical mechanisms transferred to the social world; ii) a domain of the economic, understood as a self-standing social sub-system, was discovered first by Adam Smith. His `discovery' of the new continent of the economic was an `unintended result' of a deviation in his voyage to the never-found archipelago of natural jurisprudence. -/- 1. Imaginary machines and invisible chains: natural philosophy and method. The first chapter reconstructs Smith's views on the method in natural philosophy, presented primarily in the History of Astronomy (HA). The peculiar kind of semi-sceptical Newtonianism which permeates the essay is highlighted. Its reconstruction of the history of one natural science is shown to be based on the assumptions of Hume’s epistemology, and to lead to a self-aware deadlock. Smith's dilemma is between an essentialist realism and sceptical instrumentalism; the Cartesian presuppositions he shares with Hume and with the 18th century as a whole make it impossible for him to overcome his dilemma. The following chapters will show how, on the one hand, Smith's skeptical methodology encourages him in the enterprise to `carve off' a new self-contained discipline and how, on the other hand, his epistemological dilemma is reflected in the inner tensions of his moral and political theory as well as in a number of basic oscillations concerning the status of the new discipline. -/- 2. Chessboards and clocks: moral philosophy and method. -/- The second chapter reconstructs Smith's views on the method in the parallel field of moral philosophy, including the theory of moral sentiments and natural jurisprudence. I argue that The Theory of Moral Sentiments, when considered together with with the Lectures on Jurisprudence, where Smith's peculiar version of a `weaker' form of natural law is presented, wins special interest, not only for the history of ethics but even more for the history of political theory and the social sciences. The two most striking features of Smith's work in this area are highlighted. First, his effort at reformulating the `practical science' is a methodologically self-aware attempt at applying the Newtonian method to moral subjects. Secondly, this attempt ends in a stalemate as two distinguished kinds of normative order are introduced: one ultimate order of Reason, ultimately justifiable but inaccessible, and one weaker order of our `natural sentiments', to which we have empirical access, but which is so variable as to lack any ultimate value as a basis for grounding our normative claims. These two parallel conundrums may arguably account for the author's inability to publish during his lifetime both The History of Astronomy and the projected history and theory of law and government. -/- 3. Wheels, dams, and gravitation: the structure of scientific argument in The Wealth of Nations. -/- The third chapter provides the core of the book, dealing with the structure of the argument in WN. I argue that the main presupposition that makes the shift possible from a `natural history' to a `system' approach is the Newtonian contrast of `mathematical' with `physical' explanation; that is, Smith drops any discussion of the "original qualities" of human nature that could account for economic behaviour, while introducing, as `principles' for the system, a set of `hypothetical' statements of `observed' regularities in human behaviour and of `observed' super-individual self-regulating mechanisms. In bringing this presupposition to light, the coexistence of a teleological with a mechanistic approach is clarified; fresh light is shed on the notion of the invisible hand by a comparison of its occurrence in Smith with the occurrence of the same expression (until now overlooked) in the correspondence between Newton and Cotes. Finally, the peculiar semi-prescriptive and semi-descriptive character of political economy are highlighted, and the consistency of Smith's `impure' semi-prescriptive social science, when understood in his own terms, is defended against familiar charges with inconsistency and against even more familiar strained modernizations. -/- 4. Apples, deer, and frivolous trinkets: the construction of the economic. -/- The fourth chapter draws consequences from the third, examining how Smith's achievement in political economy, marking its transition to scientific status, carried a re-description of the phenomena, creating the comparatively independent and unified field of the economic. Smith's achievement is interpreted not as the `discovery' of an autonomous character already possessed by the economy out there, so much as a Gestalt-switch by which our perception of social phenomena is modified making us `see' the partial order of the economy as an isolated system. To sum up, the autonomy of the economic in social reality and the autonomy of the economic in social consciousness are shown to be two sides of one process. -/- 5. Concluding considerations: Political economy and the Enlightenment halved. -/- A few suggestions on the status of economic theory two centuries after The Wealth of Nations in its relationship to ‘practical philosophy’ are illustrated. (shrink)

We study combinatorial principles known as stick and club. Several variants of these principles and cardinal invariants connected to them are also considered. We introduce a new kind of side by-side product of partial orderings which we call pseudo-product. Using such products, we give several generic extensions where some of these principles hold together with ¬CH and Martin's axiom for countable p.o.-sets. An iterative version of the pseudo-product is used under an inaccessible cardinal to show the consistency of the club (...) principle for every stationary subset of limits of ω1 together with ¬CH and Martin's axiom for countable p.o.-sets. (shrink)

I argue the existence of two tensions in Smith's system of ideas: the first is that between the postulate of an invisible noumenal order of the universe and the imaginary principles by means of which we connect the phenomena; the second is a tension between the noumenal order of the world where 'is' and 'ought' converge, and the various partial orders that may be reconstructed in social phenomena that leave room for irrationality and injustice. My first claim is that these (...) tensions are dialectical tensions in a unitary, albeit rhapsodically presented in writing. My second claim is that, the system’s unanswered question in the moral domain is a ‘metaphysical’ and ‘theological’ question, namely the problem of evil; by implication, I contend also that Smith was no secularist, but instead a post-skeptical fideist or agnostic, who took Bayle's question: "why are men wicked and unhappy?" quite seriously. His private ethics of prudence, justice, benevolence and his public ethics of liberty, justice, equality were modest proposals for coping with the problem of social evil in our imperfect world. (shrink)

Following Birkhoff and von Neumann, quantum logic has traditionally been based on the lattice of closed linear subspaces of some Hilbert space, or, more generally, on the lattice of projections in a von Neumann algebra A. Unfortunately, the logical interpretation of these lattices is impaired by their nondistributivity and by various other problems. We show that a possible resolution of these difficulties, suggested by the ideas of Bohr, emerges if instead of single projections one considers elementary propositions to be families (...) of projections indexed by a partially ordered set \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{C}(A)}$$\end{document} of appropriate commutative subalgebras of A. In fact, to achieve both maximal generality and ease of use within topos theory, we assume that A is a so-called Rickart C*-algebra and that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{C}(A)}$$\end{document} consists of all unital commutative Rickart C*-subalgebras of A. Such families of projections form a Heyting algebra in a natural way, so that the associated propositional logic is intuitionistic: distributivity is recovered at the expense of the law of the excluded middle. Subsequently, generalizing an earlier computation for n × n matrices, we prove that the Heyting algebra thus associated to A arises as a basis for the internal Gelfand spectrum (in the sense of Banaschewski–Mulvey) of the “Bohrification” \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\underline A}$$\end{document} of A, which is a commutative Rickart C*-algebra in the topos of functors from \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{C}A}$$\end{document} to the category of sets. We explain the relationship of this construction to partial Boolean algebras and Bruns–Lakser completions. Finally, we establish a connection between probability measures on the lattice of projections on a Hilbert space H and probability valuations on the internal Gelfand spectrum of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\underline A}$$\end{document} for A = B(H). (shrink)

The notion of a bilattice was rst introduced by Ginsburg (see Gin]) as a general framework for a diversity of applications (such as truth maintenance systems, default inferences and others). The notion was further investigated and applied for various purposes by Fitting (see Fi1]- Fi6]). The main idea behind bilattices is to use structures in which there are two (partial) order relations, having di erent interpretations. The two relations should, of course, be connected somehow in order for the mathematical structure (...) to be useful. It is not clear, however, what this connection should be. Ginsberg, for example, has made the connection through an extra operation of negation. Fitting, on the other hand, has investigated connections in the form of conditions on the structure (such as being interlaced { see below). These conditions are independent of the existence, or even the possibility to de ne, operations like Ginsberg's negation.* Fitting de nes, accordingly, notions like \an interlaced bilattice", \a distributive bilattice", \a bilattice with negation" and others. He does not provide, however, any de nition of the notion of bilattice itself (without an extra modi er). I was unable to nd anywhere, in fact, a de nition which will cover all the structures which were called \bilattice" in the literature. (shrink)

In the course of the eighteenth and nineteenth centuries the paradigm of a new science, political economy, was established. It was a science distinct from the Aristotelian sub-disciplines of practical philosophy named oikonomía and politiké, and emphasis on its character of science not unlike the natural sciences – still called ‘natural philosophy’ – mirrored precisely a willingness to stress its autonomy from two other sub-disciplines of practical philosophy, that is, ethics and politics. However, the new science resulted from a transformation (...) of part of traditional practical philosophy, allowing the inclusion of bodies of knowledge accumulated by experts of commerce and public finance. Such bodies of knowledge were unified by the (true or alleged) discovery of regularities, mechanisms, causal connections making for a new partial order within the overall social order. How far this paved the way to a science similar to mathematics rather left a normative discipline as alive as ever was a recurrent question for at least a century, until the marginalist revolution opened the way for a sharp division, leaving ‘economics’ as a science of causes and effects facing ‘economic policy’ as a discourse on ends. (shrink)

Possibly the most fundamental scientific problem is the origin of time and causality. The inherent difficulty is that all scientific theories of origins and evolution consider the existence of time and causality as given. We tackle this problem by starting from the concept of self-organization, which is seen as the spontaneous emergence of order out of primordial chaos. Self-organization can be explained by the selective retention of invariant or consistent variations, implying a breaking of the initial symmetry exhibited by randomness. (...) In the case of time, we start from a random graph connecting primitive “events”. Selection on the basis of consistency eliminates cyclic parts of the graph, so that transitive closure can transform it into a partial order relation of precedence. Causality is assumed to be carried by causal “agents” which undergo a more traditional variation and selection, giving rise to causal laws that are partly contingent, partly necessary. (shrink)

Putnam and Finkelstein have proposed the abandonment of distributivity in the logic of quantum theory. This change results from defining the connectives, not truth-functionally, but in terms of a certain empirical ordering of propositions. Putnam has argued that the use of this ordering ("implication") to govern proofs resolves certain paradoxes. But his resolutions are faulty; and in any case, the paradoxes may be resolved with no changes in logic. There is therefore no reason to regard the partially (...) class='Hi'>ordered set of propositions as a logic--i.e. as embodying a criterion for soundness of proofs. Its role in quantum theory ought to be understood in an entirely different way. (shrink)

The last chapter examines the tense system used by ordinarily competent speakers of English to discuss past, present, and future events, actual and possible events, and various combinations of these. I present a systematic method for translating English sentences containing certain compound verb tenses and embedded tense constructions into a logical language using tense operators. Finally I show how the usual semantics for these operators reflects the truth conditions of the original English sentences. I argue, however, that a tense logical (...) semantics reflecting the use of tense in English will contain a partially ordered tense structure rather than one that is linearly ordered or branching. ;The third chapter discusses logic of 'unless'. I show that many of the previously puzzling features of this word can be understood by considering the tenses of the verbs in the sentences joined by 'unless'. In a number of cases it turns out that 'unless' should not be interpreted as a truth-functional connective at all, but is best understood as an intensional, tense connective. ;In the second chapter I discuss Quine's doctrine of eternal sentences. I take Quine's position to be that all the information important for logic and science presented in tensed English sentences can also be presented by eternal sentences which use timeless verbs and data and clock times. I argue that in so far as logic and science are concerned with an analysis of the reasons for human action, and with the explanation and prediction of specific events, eternal sentences lose indexical information that is necessary to accomplish these tasks. ;The first chapter investigates what may be called the standard view of propositions. On this view propositions are abstract objects that are timelessly true , and serve both as the meanings of sentences and as the objects of propositional attitudes. Although this view was widely held by a previous generation of logicians and is found in many textbooks today, there are serious difficulties in the view. I show that many of these difficulties stem from the claim that propositions are timelessly true . The standard view has been revised in two general ways. The first sort of revision allows propositions to be sometimes true and sometimes false. This position is consistent with the development of tense logic as one of the intensional logics. The second sort of revision replaces the notion of timeless propositions with that of eternal sentences which always have the same truth value. On this view tense logic is not worth serious philosophic attention since its basic notion, that of truth at a time, can be eliminated in favor of eternal truth. ;The first part of the project examines two major sources of criticism of tense logic: the view that logic is concerned with timeless propositions, and Quine's doctrine of eternal sentences. The second part of the project discusses several problems in logical theory that the use of tense logic can clarify or resolve, and that alternative theories are unable to present accurately, much less to elucidate. ;In the past few years there has been increasing interest in the logic of indexicals, including the temporal indexical 'now', and in the relationship between tense and modality. Although modern tense logic has reached a high level of technical sophistication, often specifically in response to this interest, many philosophers remain skeptical about the value of tense logic as a philosophic tool. This thesis addresses that skepticism and argues that it is unjustified. (shrink)

A partial ordering P is said to have the weak Freese-Nation property if there is a mapping tf : P → [P]0 such that, for any a, b ε P, if a b then there exists c ε tf∩tf such that a c b. In this note, we study the WFN and some of its generalizations. Some features of the class of Boolean algebras with the WFN seem to be quite sensitive to additional axioms of set theory: e.g. under CH, (...) every ccc complete Boolean algebra has this property while, under b 2, there exists no complete Boolean algebra with the WFN. (shrink)

The relation between least and diagonal fixed points is a well known and completely studied question for a large class of partially ordered models of the lambda calculus and combinatory logic. Here we consider this question in the context of algebraic recursion theory, whose close connection with combinatory logic recently become apparent. We find a comparatively simple and rather weak general condition which suffices to prove the equality of least fixed points with canonical (corresponding to those produced by (...) the Curry combinator in lambda calculus) diagonal fixed points in a class of partially ordered algebras which covers both combinatory spaces of Skordev and operative spaces of Ivanov. Especially, this yields an essential improvement of the axiomatization of recursion theory via combinatory spaces. (shrink)

In this paper, semi-Post algebras are introduced and investigated. The generalized Post algebras are subcases of semi-Post algebras. The so called primitive Post constants constitute an arbitrary partially ordered set, not necessarily connected as in the case of the generalized Post algebras examined in [3]. By this generalization, semi-Post products can be defined. It is also shown that the class of all semi-Post algebras is closed under these products and that every semi-Post algebra is a semi-Post product of (...) some generalized Post algebras. (shrink)

Given a Polish groupG, let$E(G)$be the right coset equivalence relation$G^{\omega }/c(G)$, where$c(G)$is the group of all convergent sequences inG. The connected component of the identity of a Polish groupGis denoted by$G_0$.Let$G,H$be locally compact abelian Polish groups. If$E(G)\leq _B E(H)$, then there is a continuous homomorphism$S:G_0\rightarrow H_0$such that$\ker (S)$is non-archimedean. The converse is also true whenGis connected and compact.For$n\in {\mathbb {N}}^+$, the partially ordered set$P(\omega )/\mbox {Fin}$can be embedded into Borel equivalence relations between$E({\mathbb {R}}^n)$and$E({\mathbb {T}}^n)$.

Following Henkin's discovery of partially-ordered (branching) quantification (POQ) with standard quantifiers in 1959, philosophers of language have attempted to extend his definition to POQ with generalized quantifiers. In this paper I propose a general definition of POQ with 1-place generalized quantifiers of the simplest kind: namely, predicative, or "cardinality" quantifiers, e.g., "most", "few", "finitely many", "exactly α", where α is any cardinal, etc. The definition is obtained in a series of generalizations, extending the original, Henkin definition first to (...) a general definition of monotone-increasing (M↑) POQ and then to a general definition of generalized POQ, regardless of monotonicity. The extension is based on (i) Barwise's 1979 analysis of the basic case of M↑ POQ and (ii) my 1990 analysis of the basic case of generalized POQ. POQ is a non-compositional Ist-order structure, hence the problem of extending the definition of the basic case to a general definition is not trivial. The paper concludes with a sample of applications to natural and mathematical languages. (shrink)

A complete recursive description of noetherian linear _KL_-algebras is given. _L_-algebras form a quantum structure that occurs in algebraic logic, combinatorial group theory, measure theory, geometry, and in connection with solutions to the Yang-Baxter equation. It is proved that the self-similar closure of a noetherian linear _KL_-algebra is determined by its partially ordered set of primes, and that its elements admit a unique factorization by a decreasing sequence of prime elements.

In this article we characterize a countable ordinal known as the big Veblen number in terms of natural well-partially ordered tree-like structures. To this end, we consider generalized trees where the immediate subtrees are grouped in pairs with address-like objects. Motivated by natural ordering properties, extracted from the standard notations for the big Veblen number, we investigate different choices for embeddability relations on the generalized trees. We observe that for addresses using one finite sequence only, the embeddability coincides (...) with the classical tree-embeddability, but in this article we are interested in more general situations. We prove that the maximal order type of some of these new embeddability relations hit precisely the big Veblen ordinal ϑΩΩ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\vartheta \Omega^{\Omega}}$$\end{document}. Somewhat surprisingly, changing a little bit the well-partially ordered addresses, the maximal order type hits an ordinal which exceeds the big Veblen number by far, namely ϑΩΩΩ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\vartheta \Omega^{\Omega^\Omega}}$$\end{document}. Our results contribute to the research program on classifying properties of natural well-orderings in terms of order-theoretic properties of the functions generating the orderings. (shrink)

Following Birkhoff and von Neumann, quantum logic has traditionally been based on the lattice of closed linear subspaces of some Hubert space, or, more generally, on the lattice of projections in a von Neumann algebra A. Unfortunately, the logical interpretation of these lattices is impaired by their nondistributivity and by various other problems. We show that a possible resolution of these difficulties, suggested by the ideas of Bohr, emerges if instead of single projections one considers elementary propositions to be families (...) of projections indexed by a partially ordered set C(A) of appropriate commutative subalgebras of A. In fact, to achieve both maximal generality and ease of use within topos theory, we assume that A is a so-called Rickart C*-algebra and that C(A) consists of all unital commutative Rickart C*-subalgebras of A. Such families of projections form a Hey ting algebra in a natural way, so that the associated propositional logic is intuitionistic: distributivity is recovered at the expense of the law of the excluded middle. Subsequently, generalizing an earlier computation for n x n matrices, we prove that the Heyting algebra thus associated to A arises as a basis for the internal Gelfand spectrum (in the sense of Banaschewski-Mulvey) of the "Bohrification" A̱ of A, which is a commutative Rickart C*-algebra in the topos of functors from C(A) to the category of sets. We explain the relationship of this construction to partial Boolean algebras and Bruns-Lakser completions. Finally, we establish a connection between probability measures on the lattice of projections on a Hubert space H and probability valuations on the internal Gelfand spectrum of A̱ for A = B(H). (shrink)

The aim of this paper is to shed light on our understanding of large scale properties of infinite strings. We say that one string $\alpha $ has weaker large scale geometry than that of $\beta $ if there is color preserving bi-Lipschitz map from $\alpha $ into $\beta $ with small distortion. This definition allows us to define a partially ordered set of large scale geometries on the classes of all infinite strings. This partial order compares large scale (...) geometries of infinite strings. As such, it presents an algebraic tool for classification of global patterns. We study properties of this partial order. We prove, for instance, that this partial order has a greatest element and also possess infinite chains and antichains. We also investigate the sets of large scale geometries of strings accepted by finite state machines such as Büchi automata. We provide an algorithm that describes large scale geometries of strings accepted by Büchi automata. This connects the work with the complexity theory. We also prove that the quasi-isometry problem is a $\Sigma _2^0$ -complete set, thus providing a bridge with computability theory. Finally, we build algebraic structures that are invariants of large scale geometries. We invoke asymptotic cones, a key concept in geometric group theory, defined via model-theoretic notion of ultra-product. Partly, we study asymptotic cones of algorithmically random strings, thus connecting the topic with algorithmic randomness. (shrink)

Effective inseparability of pairs of sets is an important notion in logic and computer science. We study the effective inseparability of sets which appear as index sets of subsets of an effectively given topological T0-space and discuss its consequences. It is shown that for two disjoint subsets X and Y of the space one can effectively find a witness that the index set of X cannot be separated from the index set of Y by a recursively enumerable set, if X (...) intersects the topological closure of an effectively enumerable subset of Y. As a consequence of a more general parametric inseparability result a theorem of Rice-Shapiro type is obtained. Moreover, under some additional requirements it follows that nonopen subsets have productive index sets. This implies a generalized Rice theorem: Connected spaces have only trivial completely recursive subsets. As application some decision problems in computable analysis and domain theory are studied. It follows that the complement of the halting problem can be reduced to the problem to decide of a number whether it is a computable irrational. The same is true for the problems to decide whether two numbers are equal, whether one is not greater than the other, and whether a number is equal to a given number. In the case of an effectively given continuous complete partial order the complexity of the last problem depends on whether the given element is the smallest element, in which case the complement of the halting problem is reducible to it, whether it is a base element and maximal, then the decision problem is recursively isomorphic to the halting problem, or whether it is none of these. In this case, both the halting problem and its complement are reducible to the problem. The same is true in nontrivial cases for the problems whether an element belongs to the basis, whether two elements of the partial order are equal, or whether one approximates the other. In general, for any nonempty proper subset of the partial order either the halting problem or its complement can be reduced to the membership problem of the subset. (shrink)

Following Henkin’s discovery of partially-ordered (branching) quantification (POQ) with standard quantifiers in 1959, philosophers of language have attempted to extend his definition to POQ with generalized quantifiers. In this paper I propose a general definition of POQ with 1-place generalized quantifiers of the simplest kind: namely, predicative, or “cardinality” quantifiers, e.g., “most”, “few”, “finitely many”, “exactly α ”, where α is any cardinal, etc. The definition is obtained in a series of generalizations, extending the original, Henkin definition first (...) to a general definition of monotone-increasing (M↑) POQ and then to a general definition of generalized POQ, regardless of monotonicity. The extension is based on (i) Barwise’s 1979 analysis of the basic case of M↑ POQ and (ii) my 1990 analysis of the basic case of generalized POQ. POQ is a non-compositional 1st-order structure, hence the problem of extending the definition of the basic case to a general definition is not trivial. The paper concludes with a sample of applications to natural and mathematical languages. (shrink)

Interest surges in a distinctively metaphysical notion of ground. But a Schism has emerged between Orthodoxy’s view of ground as inducing a strict partial order structure on reality and Heresy’s rejection of this view. What’s at stake is the structure of reality (for proponents of ground), or even ground itself (for those who think this Schism casts doubt upon its coherence). I defend Orthodoxy against Heresy.

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)

Medvedev's intermediate logic (MV) can be defined by means of Kripke semantics as the family of Kripke frames given by finite Boolean algebras without units as partially ordered sets. The aim of this paper is to present a proof of the theorem: For every set of connectives such that the-fragment ofMV equals the fragment of intuitionistic logic. The final part of the paper brings the negative solution to the problem set forth by T. Hosoi and H. Ono, (...) namely: is an intermediate logic based on the axiom (abc) (ab)(a c) separable? (shrink)