In this paper we study a notion of HL-extension for a structure in a finite relational language $\mathcal {L}$. We give a description of all finite minimal HL-extensions of a given finite $\mathcal {L}$ -structure. In addition, we study a group-theoretic property considered by Herwig–Lascar and show that it is closed under taking free products. We also introduce notions of coherent extensions and ultraextensive $\mathcal {L}$ -structures and show that every countable $\mathcal {L}$ -structure can be extended to (...) a countable ultraextensive structure. Finally, it follows from our results that the automorphism group of any countable ultraextensive $\mathcal {L}$ -structure has a dense locally finite subgroup. (shrink)

We provide a canonical construction of conformal covers for finite hypergraphs and present two immediate applications to the finite model theory of relational structures. In the setting of relational structures, conformal covers serve to construct guarded bisimilar companion structures that avoid all incidental Gaifman cliques-thus serving as a partial analogue in finite model theory for the usually infinite guarded unravellings. In hypergraph theoretic terms, we show that every finite hypergraph admits a bisimilar cover by a finite conformal hypergraph. In (...) terms of relational structures, we show that every finite relational structure admits a guarded bisimilar cover by a finite structure whose Gaifman cliques are guarded. One of our applications answers an open question about a clique constrained strengthening of the extension property for partial automorphisms (EPPA) of Hrushovski, Herwig and Lascar. A second application provides an alternative proof of the finite model property (FMP) for the clique guarded fragment of first-order logic CGF, by reducing (finite) satisfiability in CGF to (finite) satisfiability in the guarded fragment, GF. (shrink)

The logics considered here are the propositional Linear Logic and propositional Intuitionistic Linear Logic extended by a knotted structural rule: γ, xn → y / γ, xm → y. It is proved that the class of algebraic models for such a logic has the finite embeddability property, meaning that every finite partial subalgebra of an algebra in the class can be embedded into a finite full algebra in the class. It follows that each such logic has the finite (...) model property with respect to its algebraic semantics and hence that the logic is decidable. (shrink)

A partial ordering $\mathbb{P}$ is $n$-Ramsey if, for every coloring of $n$-element chains from $\mathbb{P}$ in finitely many colors, $\mathbb{P}$ has a homogeneous subordering isomorphic to $\mathbb{P}$. In their paper on Ramsey properties of the complete binary tree, Chubb, Hirst, and McNicholl ask about Ramsey properties of other partial orderings. They also ask whether there is some Ramsey property for pairs equivalent to $\mathit{ACA}_{0}$ over $\mathit{RCA}_{0}$. A characterization theorem for finite-level partial orderings with Ramsey properties has (...) been proven by the second author. We show, over $\mathit{RCA}_{0}$, that one direction of the equivalence given by this theorem is equivalent to $\mathit{ACA}_{0}$, and the other is provable in $\mathit{ATR}_{0}$. We answer Chubb, Hirst, and McNicholl’s second question by showing that there is a primitive recursive partial ordering $\mathbb{P}$ such that, over $\mathit{RCA}_{0}$, “$\mathbb{P}$ is 2-Ramsey” is equivalent to $\mathit{ACA}_{0}$. (shrink)

We study those relational structures S with the property that each partition of S contains a block isomorphic to S. We show that the Fraïsse limits of parametric classes K. have property ; over a binary language, every countable structure in K satisfying along with a condition on 1-extensions must be isomorphic to this limit.

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

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)

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

We show, relative to the base theory RCA₀: A nontrivial tree satisfies Ramsey's Theorem only if it is biembeddable with the complete binary tree. There is a class of partial orderings for which Ramsey's Theorem for pairs is equivalent to ACA₀. Ramsey's Theorem for singletons for the complete binary tree is stronger than $B\sum_{2}^{0}$ , hence stronger than Ramsey's Theorem for singletons for ω. These results lead to extensions of results, or answers to questions, of Chubb, Hirst, and McNicholl (...) [3]. (shrink)

We prove the following version of Hechler's classical theorem: For each partially ordered set (Q, ≤) with the property that every countable subset of Q has a strict upper bound in Q, there is a ccc forcing notion such that in the generic extension for each tall analytic P-ideal J (coded in the ground model) a cofinal subset of (J, ⊆*) is order isomorphic to (Q, ≤).

We give an axiomatic characterization for complete elementary extensions, that is, elementary extensions of the first-order structure consisting of all finitary relations and functions on the underlying set. Such axiom systems have been studied using various types of primitive notions . Our system uses the notion of partial functions as primitive. Properties of nonstandard extensions are derived from five axioms in a rather algebraic way, without the use of metamathematical notions such as formulas or satisfaction. For example, when applied (...) to the real number system, it provides a complete framework for developing nonstandard analysis based on hyperreals without having to construct them and without any use of logic. This has possible pedagogical and expository applications as presented in, e.g., [5], [6]. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim. (shrink)

A game for testing the equivalence of Kripke models with respect to finitary and infinitary intuitionistic predicate logic is introduced and applied to discuss a concept of categoricity for intuitionistic theories.

Logics in which a relation R is semantically incomplete in a particular universe E, i.e. the union of the extension of R with its anti-extension does not exhaust the whole universe E, have been studied quite extensively in the last years. (Cf. van Benthem (1985), Blamey (1986), and Langholm (1988), for partial predicate logic; Muskens (1996), for the applications of partial predicates to formal semantics, and Doherty (1996) for applications to modal logic.) This is not so (...) with semantically incomplete generalized quantifiers which constitute the subject of the present paper. The only systematic study of these quantifiers from a purely logical point of view, is, to the best of my knowledge, that by van Eijck (1995). We shall take here a different approach than that of van Eijck and mention some of the abstract properties of the resulting logic. Finally we shall prove that the two approaches are interdefinable. (shrink)

We study partiality in propositional logics containing formulas with either undefined or over-defined truth-values. Undefined values are created by adding a four-place connective W termed transjunction to complete models which, together with the usual Boolean connectives is shown to be functionally complete for all partial functions. Transjunction is seen to be motivated from a game-theoretic perspective, emerging from a two-stage extensive form semantic game of imperfect information between two players. This game-theoretic approach yields an interpretation where partiality is generated (...) as a property of non-determinacy of games. Over-defined values are produced by adding a weak, contradictory negation or, alternatively, by relaxing the assumption that games are strictly competitive. In general, particular forms of extensive imperfect information games give rise to a generalised propositional logic where various forms of informational dependencies and independencies of connectives can be studied. (shrink)

We present a systematic study of join-extensions and join-completions of partially ordered algebras, which naturally leads to a refined and simplified treatment of fundamental results and constructions in the theory of ordered structures ranging from properties of the Dedekind–MacNeille completion to the proof of the finite embeddability property for a number of varieties of lattice-ordered algebras.

This paper continues the investigation, started in Lávička and Noguera : 521–551, 2017), of infinitary propositional logics from the perspective of their algebraic completeness and filter extension properties in abstract algebraic logic. If follows from the Lindenbaum Lemma used in standard proofs of algebraic completeness that, in every finitary logic, intersection-prime theories form a basis of the closure system of all theories. In this article we consider the open problem of whether these properties can be transferred to lattices of (...) filters over arbitrary algebras of the logic. We show that in general the answer is negative, obtaining a richer hierarchy of pairwise different classes of infinitary logics that we separate with natural examples. As by-products we obtain a characterization of subdirect representation for arbitrary logics, develop a fruitful new notion of natural expansion, and contribute to the understanding of semilinear logics. (shrink)

Kleene [5] mentions two ways of extending the constructive ordinals. The first is by relativizing the setOof notations for the constructive ordinals, using fundamental sequences which are partial recursive inO. In this way we obtain the setOOwhich provides notations for the ordinals less than ω1O. Continuing the process, the sequenceO,OO,, … and the corresponding ordinalsare obtained. A second possibility is to define higher number classes in which partial recursive functions are used at limit ordinals to provide an “accessibility” (...) mapping from a previously defined number class. The relationship between the ordinals obtained by the two methods of extension has been an open problem. Methods developed in this article are used to show that the two ways of extending the constructive ordinals are equivalent, provided the sets of notations for the higher number classes satisfy certain natural conditions. Equivalence is obtained, not only with respect to ordinals, but also with respect to the forms of the sets of notations for the higher number classes. Specifically, the fundamental fact that the sets of notations for the constructive ordinals are complete Π11sets generalizes to suitably defined higher number classes. As an application we prove that the ordinals of the Addison and Kleene [1] constructive third number class are exactly the ordinals less than ω1Oand the setof notations for their third number class is recursively isomorphic toOO. (shrink)

In this paper, we develop the system LZF of set theory with the unrestricted comprehension in full linear logic and show that LZF is a conservative extension of ZF– i.e., the Zermelo-Fraenkel set theory without the axiom of regularity. We formulate LZF as a sequent calculus with abstraction terms and prove the partial cut-elimination theorem for it. The cut-elimination result ensures the subterm property for those formulas which contain only terms corresponding to sets in ZF–. This implies (...) that LZF is a conservative extension of ZF– and therefore the former is consistent relative to the latter. (shrink)

Firstly I characterize Simple Partial Logic (SPL) as the generalization and extension of a certain two-valued logic. Based on the characterization I present two definitions of validity in SPL. Finally I show that given my characterization these two definitions are more appropriate than other definitions that have been prevalent, since both have some desirable semantic properties that the others lack.

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

In this paper we define the relation t of elementary extension of topological models in the language L t and show a Back and Forth criterion for t. We introduce some new operations on partial homeomorphisms preserving Back and Forth properties. Some properties of t are proved by the Back and Forth technique.

We study Basic algebra, the algebraic structure associated with basic propositional calculus, and some of its natural extensions. Among other things, we prove the amalgamation property for the class of Basic algebras, faithful Basic algebras and linear faithful Basic algebras. We also show that a faithful theory has the interpolation property if and only if its correspondence class of algebras has the amalgamation property.

In the logic of theory change, the standard model is AGM, proposed by Alchourrón et al. (J Symb Log 50:510–530, 1985 ). This paper focuses on the extension of AGM that accounts for contractions of a theory by a set of sentences instead of only by a single sentence. Hansson (Theoria 55:114–132, 1989 ), Fuhrmann and Hansson (J Logic Lang Inf 3:39–74, 1994 ) generalized Partial Meet Contraction to the case of contractions by (possibly non-singleton) sets of sentences. (...) In this paper we present the possible worlds semantics for partial meet multiple contractions. (shrink)

A rather promising value theory for environmental philosophers combines the well-known fitting attitude (FA) account of value with the rather less well-known account of value as richness. If the value of an entity is proportional to its degree of richness (which has been cashed out in terms of unified complexity and organic unity), then since natural entities, such as species or ecosystems, exhibit varying degrees of richness quite independently of what we happen to feel about them, they also possess differing (...) degrees of mind-independent and subject-independent value. In particular, their value is not dependent on the desires or preferences of humans. The fitting attitudes account of value, at least as it is standardly developed, demands isomorphic evaluative responses on the part of all valuers. In particular, it entails that all valuers should have isomorphic preferences. But this seems absurd. I consider three different strategies with which the fitting attitude theorist can deflect this challenge. The first makes use of an account of non-standard value relations in terms of permissible preference orderings. The second appeals to value appearances and the associated notions of value distance and value perspective. The third involves an account of the ultimate bearers of value as properties, rather than as propositions or states of affairs. These strategies are not all mutually incompatible. While it isn’t possible to combine the first and second strategies, it is possible to combine the first and third strategies, and also to combine the second and the third. (shrink)

A semantics may be compositional and yet partial, in the sense that not all well-formed expressions are assigned meanings by it. Examples come from both natural and formal languages. When can such a semantics be extended to a total one, preserving compositionality? This sort of extension problem was formulated by Hodges, and solved there in a particular case, in which the total extension respects a precise version of the fregean dictum that the meaning of an expression is (...) the contribution it makes to the meanings of complex phrases of which it is a part. Hodges' result presupposes the so-called Husserl property, which says roughly that synonymous expressions must have the same category. Here I solve a different version of the compositional extension problem, corresponding to another type of linguistic situation in which we only have a partial semantics, and without assuming the Husserl property. I also briefly compare Hodges' framework for grammars in terms of partial algebras with more familiar ones, going back to Montague, which use many-sorted algebras instead. (shrink)

We prove a generalization of Maehara’s lemma to show that the extensions of classical and intuitionistic first-order logic with a special type of geometric axioms, called singular geometric axioms, have Craig’s interpolation property. As a corollary, we obtain a direct proof of interpolation for (classical and intuitionistic) first-order logic with identity, as well as interpolation for several mathematical theories, including the theory of equivalence relations, (strict) partial and linear orders, and various intuitionistic order theories such as apartness and (...) positive partial and linear orders. (shrink)

A theory of higher-order vagueness for partially-defined, context-sensitive predicates like is blue is offered. According to the theory, the predicate is determinately blue means roughly is an object o such that the claim that o is blue is a necessary consequence of the rules of the language plus the underlying non-linguistic facts in the world. Because the question of which rules count as rules of the language is itself vague, the predicate is determinately blue is both vague and partial (...) in a sense analogous (though not completely identical) to the sense in which is blue is vague and partial. However, higher-order vagueness stops there. Although the anti-extension of is determinately blue differs from that of is blue, further iterations of determinately have no semantic effect. The paper closes with a discussion of the need to posit sharp and precise lines dividing the range of potential application of a vague predicate into semantically distinct categories. It is argued that this is less of a problem for the present theory than for many others, because the categories into which the range is divided by the present theory -- like the category of counting as blue because the rules of the language offer speakers no discretion, as opposed to the category of counting as blue because speakers have invoked an absolute minimum of discretion in adjusting the contextually sensitive boundaries of the predicate -- are not those that the linguistic competence of ordinary speakers equips them to make highly accurate and reliable judgments about. (shrink)

The main result proves Lyndon’s and Craig’s interpolation properties for the logic of strict implication ${\textsf{F}}$, with a purely syntactical method. A cut-free G3-style sequent calculus $ {\textsf{GF}} $ and its single-succedent variant $ \textsf{GF}_{\textsf{s}} $ are introduced. $ {\textsf{GF}} $ can be extended to a G3-variant of the sequent calculus GBPC3 for Visser’s basic logic. Also a simple syntactic proof of known embedding result of $ {\textsf{F}} $ into $ {\textsf{K}} $ is provided. An extension of $ {\textsf{F}} (...) $, namely $ \textsf{FD}, $ is considered as well. (shrink)

In this paper, we study the notion of arborescent ordered sets, a generalizationof the notion of tree-property for cardinals. This notion was already studied previously in the case of directed sets. Our main result gives a geometric condition for an order to be ℵ0-arborescent.

The aim of the paper is to develop the notion of partial probability distributions as being more realistic models of belief systems than the standard accounts. We formulate the theory of partial probability functions independently of any classical semantic notions. We use the partial probability distributions to develop a formal semantics for partial propositional calculi, with extensions to predicate logic and higher order languages. We give a proof theory for the partial logics and obtain soundness (...) and completeness results. (shrink)

We obtain in this paper a representation of the formulae of extensions ofL by generalized quantifiers through functors between categories of first-order structures and partial isomorphisms. The main tool in the proofs is the back-and-forth technique. As a corollary we obtain the Caicedo's version of Fraïssés theorem characterizing elementary equivalence for such languages. We also discuss informally some geometrical interpretations of our results.

The following two assertions are equivalent for an o-minimal expansion of an ordered group $$\mathcal M=(M,<,+,0,\ldots )$$. There exists a definable bijection between a bounded interval and an unbounded interval. Any definable continuous function $$f:A \rightarrow M$$ defined on a definable closed subset of $$M^n$$ has a definable continuous extension $$F:M^n \rightarrow M$$.

We offer a new outlook on the vexed question of the reference of natural-kind terms. Since Kripke and Putnam, there is a widespread assumption that natural-kind terms function just like proper names: they designate their referents directly and they are rigid designators: their reference is unchanged even in worlds in which the referent lacks some or all the properties associated with it in the actual world, and which are useful to us in identifying that referent. There have, however, been heated (...) debates about what should be taken as a natural-kind term. Some challenge the very existence of a separate category of natural-kind terms ; some their being directly referential ; some raise the possibility that direct reference extends to terms beyond those usually assumed to fall under the category, e.g. to ‘polemical’ terms. When these debates turn on the question of what natural kinds are, they take on a strong epistemological or metaphysical dimension. We think the issues can be clarified within the limits of the philosophy of language: by looking into what ranges of general terms are perceived by speakers as rigid designators of natural kinds. The first step to take is to ground the various kinds of semantic externalism in distinct brands of semantic deference. This we define as speakers’ being disposed to use words in line with the norms of their linguistic community and as consenting to being corrected when it is manifest that their use and understanding of a word does not match common practice. When those dispositions are present, speakers defer semantically to something beyond themselves. Here, our focus is on spotting the words for which speakers would defer not to the current usage of the word in the linguistic community, nor to the current experts of the field to which the word pertains, but ultimately to the very nature of the referent of the term. When speakers’ deference conforms to that pattern, we argue, that is evidence that indexical externalism provides the right metasemantic account of how the meaning of the word is determined. In other words, one can say that the word is treated like a natural-kind term. But how can patterns of deference be measured? In an ongoing survey, participants are confronted with conditions that may prompt them to revise certain classificatory statements, e.g. An emu is a bird. Each condition makes salient one of the targets we have identified for deference: the community usage, the experts, the ‘world as it is’. In the condition that seeks to tap into the latter kind of deference, participants are presented with a scenario in which future scientific discoveries result in excluding from the extension of a term certain members currently thought to fall under that extension, e.g. discoveries that require excluding certain species now thought to be birds from the Aves class. The scenario is such that it is clear that the ‘discoveries’ bring us closer to the actual essence, if there is one, of birds. Our reasoning is that, if participants significantly modify their statements in the light of that scenario, they can be taken to ‘defer’ to the nature of the referent, thus vindicating indexical externalism. We test if words not normally assumed to be natural-kind terms – for instance summer, contract or rape – exhibit patterns of deference similar to those for bird. If so, then there’s a case for an extension of indexical externalism beyond the usual set of terms. What would be shown in this way is that speakers have something like realist intuitions with respect to words whose meaning is usually taken to be purely conventional or polemical. We are at the pre-test stage for the survey. We cannot yet report on our results. We should, however, have initial results by March, which, we believe, will enrich the observations made by previous empirical studies. References: Braisby, N., Franks, B. & Hampton, J. 1996. Essentialism, word use, and concepts. Cognition 59, 247-74./Genone, J. & Lombrozo, T. 2012. Concept possession, experimental semantics, and hybrid theories of reference. Philosophical Psychology 25, 717-742./Häggqvist, S. & Wikforss, Å. 2015. Experimental semantics: the case of natural kind terms, in J. Haukioja, Experimental Philosophy of Language, London, Bloomsbury./Jylkkä, J., Railo, H. & Haukioja, J. 2009. Psychological essentialism and semantic externalism: Evidence for externalism in lay speakers’ language use. Philosophical Psychology 22, 37-60./Mallon, R., Machery, E., Nichols, S. & Stich, S. 2009. Against arguments from reference, Philosophy and Phenomenological Research 79, 332-356./ Marconi, D. 1997. Language, Speech, and Communication. Lexical Competence. Cambridge : MIT Press./Moravcsik, J. 2016. Meaning, Creativity, and the Partial Inscrutability of the Human Mind, 2nd ed. Stanford: CSLI./Schroeter, L. & Schroeter, F. 2014. Normative concepts: a connectedness model, Philosophers’ Imprint 14, 1-26./ Wikforss, Å. 2010. Are natural kind terms special?, in H. Beebee and N. Sabbarton-Leary, The Semantics and Metaphysics of Natural Kinds, New York, Routledge, 64-83. (shrink)

This paper proves that the disjunction property, the numerical existence property, Church’s rule, and several other metamathematical properties hold true for Constructive Zermelo-Fraenkel Set Theory, CZF, and also for the theory CZF augmented by the Regular Extension Axiom.As regards the proof technique, it features a self-validating semantics for CZF that combines realizability for extensional set theory and truth.

We prove that Basic Arithmetic, BA, has the de Jongh property, i.e., for any propositional formula A built up of atoms p1,..., pn, BPC⊢\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\vdash}$$\end{document}A if and only if for all arithmetical sentences B1,..., Bn, BA⊢\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\vdash}$$\end{document}A. The technique used in our proof can easily be applied to some known extensions of BA.

John Locke's labor theory of property is one of the seminal ideas of political philosophy and served to establish its author's reputation as one of the leading social and political thinkers of all time. Through it Locke addressed many of his most pressing concerns, and earned a reputation as an outstanding spokesman for political individualism - a reputation that lingers widely despite some partial challenges that have been raised in recent years. In this major new study Matthew Kramer (...) offers an extensive critique of the labor theory and investigates the consequences of its downfall. With incisive analyses of the merits and failings of many aspects of Locke's political thought, Kramer advances a powerful challenge to Locke's image as an individualist. Employing a rigorously philosophical methodology, but remaining aware of the insights generated by historical approaches to Locke, Kramer concludes that Locke's political vision was in fact profoundly communitarian. (shrink)

In this paper, we show that the finite model property fails for certain non‐integral semilinear substructural logics including Metcalfe and Montagna's uninorm logic and involutive uninorm logic, and a suitable extension of Metcalfe, Olivetti and Gabbay's pseudo‐uninorm logic. Algebraically, the results show that certain classes of bounded residuated lattices that are generated as varieties by their linearly ordered members are not generated as varieties by their finite members.

Objects partially resemble when they are alike in some way but not entirely alike. Partial resemblance, then, involves similarity in a respect. It has been observed that talk of “respects” appears to be thinly‐veiled talk of properties. So some theorists take similarity in a respect to require property realism. I will go a step further and argue that similarity in intrinsic respects (partial intrinsic resemblance) requires properties to be immanent in objects. For a property to be (...) immanent in an object is roughly for it to be wholly present in that object. (So understood, immanence does not imply repeatability, and thus tropes count as immanent.) If two objects are intrinsically similar in one way but not another, there must be a difference between the two ways. And if the similarity is between the objects as they are in themselves, then this difference, too, must be within the objects. Partial intrinsic resemblance thus requires internal structure. Without it, there can be no distinctions between ways in which things are intrinsically alike and ways in which they aren't. I will argue, further, that distinctions between properties that transcend objects fail to account for the possibility of partial intrinsic resemblance. (shrink)

We prove the following theorem: For a partially ordered set Q such that every countable subset of Q has a strict upper bound, there is a forcing notion satisfying the countable chain condition such that, in the forcing extension, there is a basis of the null ideal of the real line which is order-isomorphic to Q with respect to set-inclusion. This is a variation of Hechler’s classical result in the theory of forcing. The corresponding theorem for the meager ideal (...) was established by Bartoszyński and Kada. (shrink)

ABSTRACT:The extension of human rights obligations to corporations raises questions about whose rights and which rights corporations are responsible for. This paper gives a partial answer by asking what legal rights corporations would need to have to fulfil various sorts of human rights obligations. We should compare the chances of human rights fulfilment (and violations) that are likely to result from assigning human rights obligations to corporations with the chances of human rights fulfilment (and violations) that are likely (...) to result from giving corporations the legal rights needed to undertake those human rights obligations. Corporations should respect basic human rights of all people. Non-complicity in human rights violations requires that corporations have the right to political freedom of speech. To actively protect people from human rights violations, corporations need the right to hire armed security personnel; such obligations should be limited to protecting corporate property and narrowly defined stakeholders. Obligations to spend corporate resources on human rights fulfilment are confined to contributing to specific projects. Corporations have no obligation to ensure a society in which human rights are fulfilled. This principle helps us understand why corporate obligations are substantially different from those of governments. (shrink)

The present article, which is a revised version of part of [Hu1], deals with various relations between models which might serve as exact formulations for the vague concept "similar" or "almost isomorphic". One natural class of such formulations is equivalence in a given logic. Another way to express similarity is by potential isomorphism, i.e., isomorphism in some extension of the set-theoretic universe. The class of extensions may be restricted to give different notions of potential isomorphism. A third method is (...) to study the winning strategies for an Ehrenfeucht-Fraisse-game played between the two models, and the properties of the resulting equivalence and nonequivalence trees. The basic question studied here is whether one such notion of similarity implies another. Some implications and counterexamples listed in this part are previously known or trivial, but all are mentioned for completeness' sake. Only models of cardinality ℵ 1 are considered. Some results are therefore connected with the Continuum Hypothesis. (shrink)

A modified subformula property for the modal logic KD with the additionalaxiom □ ◊ ⊃ □ ◊ A ∨ □ ◊B is shown. A new modification of the notion of subformula is proposed for this purpose. This modification forms a natural extension of our former one on which modified subformula property for the modal logics K5, K5D and S4.2 has been shown. The finite model property as well as decidability for the logic follows from this.