This paper defends the case against (sparse) disjunctive properties by means of four Armstrongian arguments. The first of these is a logical atomist argument from truthmaking, which is, broadly speaking, ‘Armstrongian’ (Armstrong 1997). This argument is strong – although it stands or falls with the relevant notion of truthmaking, as it were. However, three arguments, which are prima facie independent of truthmaking, can be found explicitly early in Armstrong’s middle period. Two of these early arguments face a serious objection put (...) forward forcefully by Louise Antony (2003) and Alan Penczek (1997), respectively. I consider these objections and argue that they fail. Thus, even if the argument from truthmaking is indecisive, disjunctive properties should be rejected. (shrink)

This paper is a survey of results concerning the disjunction property, Halldén-completeness, and other related properties of intermediate prepositional logics and normal modal logics containing S4.

A partial solution to Ono’s problem P54 is given. Here Ono’s problem P54 is whether Harrop disjunction property is equivalent to disjunction property or not in intermediate predicate logics. As an application of this result it is shown that some intermediate predicate logics satisfy Harrop disjunction property.

A pigeon has been conditioned to peck at red objects and has also been conditioned to peck at triangular objects. The pigeon is now presented with a red triangle and pecks. In virtue of which of the object's properties did the pigeon peck? I argue that the disjunctive property "red or triangular" best answers this question and that this in turn gives us reason to admit such disjunctive properties to our ontology. I also show how the criterion for causal (...) efficacy that I employ, expressed in terms of counterfactual conditionals, works to preclude spurious disjunctive properties from counting as efficacious. (shrink)

With any structural inference rule A/B, we associate the rule $${(A \lor p)/(B \lor p)}$$, providing that formulas A and B do not contain the variable p. We call the latter rule a join-extension ( $${\lor}$$ -extension, for short) of the former. Obviously, for any intermediate logic with disjunction property, a $${\lor}$$ -extension of any admissible rule is also admissible in this logic. We investigate intermediate logics, in which the $${\lor}$$ -extension of each admissible rule is admissible. We (...) prove that any structural finitary consequence operator (for intermediate logic) can be defined by a set of $${\lor}$$ -extended rules if and only if it can be defined through a set of well-connected Heyting algebras of a corresponding quasivariety. As we exemplify, the latter condition is satisfied for a broad class of algebraizable logics. (shrink)

Are there disjunctive properties? This question is important for at least two reasons. First, disjunctive properties are invoked in defense of certain philosophical theories, especially in the philosophy of mind. Second, the question raises the prior issue of what counts as a genuine property, a central concern in the metaphysics of properties. I argue here, on the basis of general considerations in the metaphysics of properties, that there are no disjunctive properties. Specifically, I argue that genuine properties must guarantee (...) similarity-in-a-respect among their instances, and must inhere in their bearers (two notions to be clarified). Disjunctive properties would fail both requirements. I compare the case of disjunctive properties with “mere Cambridge properties,” determinable properties, and functional properties, and show how my conception of properties remains compatible with determinables and functional properties while ruling out disjunctive and mere Cambridge properties. (shrink)

Meinertsen has recently put forward three arguments against disjunctive properties: the arguments from truthmaking, commonality, and causation. In this paper, I argue that all three arguments fail. The argument from truthmaking rests on the problematic notion of different types of truthmakers and is therefore itself problematic. The argument from commonality may hold but only at the cost of losing much of the philosophical significance of its conclusion. The argument from causation essentially collapses into the argument from truthmaking and is therefore (...) problematic for the same reason as the latter is. I conclude that Meinertsen has not made a case against disjunctive properties. (shrink)

Let \ be a class of formulas. We say that a theory T in classical logic has the \-disjunction property if for any \ sentences \ and \, either \ or \ whenever \. First, we characterize the \-disjunction property in terms of the notion of partial conservativity. Secondly, we prove a model theoretic characterization result for \-disjunction property. Thirdly, we investigate relationships between partial disjunction properties and several other properties of theories containing (...) Peano arithmetic. Finally, we investigate unprovability of formalized partial disjunction properties. (shrink)

We show that there is a purely proof-theoretic proof of the Rasiowa–Harrop disjunction property for the full intuitionistic propositional calculus ), via natural deduction, in which commuting conversions are not needed. Such proof is based on a sound and faithful embedding of \ into an atomic polymorphic system. This result strengthens a homologous result for the disjunction property of \ and answers a question then posed by Pierluigi Minari.

In this paper it is shown that there exist a structural complete interme- diate logics with the disjunction property, which was previously conjectured by H. Friedman. The intermedi.

Some of the basic substructural logics are shown by Ono to have the disjunction property (DP) by using cut elimination of sequent calculi for these logics. On the other hand, this syntactic method works only for a limited number of substructural logics. Here we show that Maksimova's criterion on the DP of superintuitionistic logics can be naturally extended to one on the DP of substructural logics over FL. By using this, we show the DP for some of the (...) substructural logics for which syntactic methods don't work well. (shrink)

The existence of disjunctive properties has been put in question by many philosophers. In this article, I shall offer an argument for the existence of such properties. I shall show that they are required in order to ground certain objective and unique resemblances between things. In Section 1, I develop the argument by presenting a toy example involving three entities: a red fish (Bill), a glass and round fish bowl (Bowl), and a red ball (Ball). In Section 2, I deal (...) with two strategies to rebut my argument. Such strategies respectively consist in denying that there is any objective and unique similarity between Bill, Bowl and Ball, and in holding that such a similarity is based on non-disjunctive properties. In Section 3, I examine and reject two further strategies, that consist in holding that grounding objective and unique similarities is not enough in order for something to exist as a property and in pointing to some more general concerns about disjunctive properties. (shrink)

We systematically study several versions of the disjunction and the existence properties in modal arithmetic. First, we newly introduce three classes $\mathrm {B}$, $\Delta (\mathrm {B})$, and $\Sigma (\mathrm {B})$ of formulas of modal arithmetic and study basic properties of them. Then, we prove several implications between the properties. In particular, among other things, we prove that for any consistent recursively enumerable extension T of $\mathbf {PA}(\mathbf {K})$ with $T \nvdash \Box \bot $, the $\Sigma (\mathrm {B})$ -disjunction (...) property, the $\Sigma (\mathrm {B})$ -existence property, and the $\mathrm {B}$ -existence property are pairwise equivalent. Moreover, we introduce the notion of the $\Sigma (\mathrm {B})$ -soundness of theories and prove that for any consistent recursively enumerable extension of $\mathbf {PA}(\mathbf {K4})$, the modal disjunction property is equivalent to the $\Sigma (\mathrm {B})$ -soundness. (shrink)

For intermediate logics, there is obtained in the paper an algebraic equivalent of the disjunction propertyDP. It is proved that the logic of finite binary trees is not maximal among intermediate logics withDP. Introduced is a logicND, which has the only maximal extension withDP, namely, the logicML of finite problems.

Infinitely many intermediate propositional logics with the disjunction property are defined, each logic being characterized both in terms of a finite axiomatization and in terms of a Kripke semantics with the finite model property. The completeness theorems are used to prove that any two logics are constructively incompatible. As a consequence, one deduces that there are infinitely many maximal intermediate propositional logics with the disjunction property.

We study axiomatic extensions of the propositional constructive logic with strong negation having the disjunction property in terms of corresponding to them varieties of Nelson algebras. Any such varietyV is characterized by the property: (PQWC) ifA,B V, thenA×B is a homomorphic image of some well-connected algebra ofV.We prove:• each varietyV of Nelson algebras with PQWC lies in the fibre –1(W) for some varietyW of Heyting algebras having PQWC, • for any varietyW of Heyting algebras with PQWC the (...) least and the greatest varieties in –1(W) have PQWC, • there exist varietiesW of Heyting algebras having PQWC such that –1(W) contains infinitely many varieties (of Nelson algebras) with PQWC. (shrink)

This is the second part of a paper devoted to the study of the maximal intermediate propositional logics with the disjunction property , whose first part has appeared in this journal with the title “A method to single out maximal propositional logics with the disjunction property I”. In the first part we have explained the general results upon which a method to single out maximal constructive logics is based and have illustrated such a method by exhibiting (...) the Kripke semantics of maximal constructive logics extending the logic ST of Scott, for which, in turn, a semantical characterization in terms of Kripke frames has been given. In the present part we complete the illustration of the method of the first part, having in mind some aspects which might be interesting for a classification of the maximal constructive logics, and an application of the heuristic content of the method to detect the nonmaximality of apparently maximal constructive logics. Thus, on the one hand we introduce the logic AST , which is compared with ST and is seen as a logic “alternative” to it, in a sense which will be precisely explained. We provide a Kripke semantics for AST and show that there are maximal constructive logics which neither are extensions of ST nor are extensions of AST. Finally, we give a further application of the results of the first part by exhibiting the Kripke semantics of a maximal constructive logic extending AST. On the other hand, we compare the maximal constructive logics presented in both parts of the paper with a constructive logic introduced by Maksimova , which has been conjectured to be maximal by Chagrov and Zacharyashchev ; from this comparison a disproof of the conjecture arises. (shrink)

We discuss relationships among the existence property, the disjunction property, and their weak variants in the setting of intermediate predicate logics. We deal with the weak and sentential existence properties, and the Z-normality, which is a weak variant of the disjunction property. These weak variants were presented in the author’s previous paper [16]. In the present paper, the Kripke sheaf semantics is used.

This is the first part of a paper concerning intermediate propositional logics with the disjunction property which cannot be properly extended into logics of the same kind, and are therefore called maximal. To deal with these logics, we use a method based on the search of suitable nonstandard logics, which has an heuristic content and has allowed us to discover a wide family of logics, as well as to get their maximality proofs in a uniform way. The present (...) part illustrates infinitely many maximal logics with the disjunction property extending the well-known logic of Scott, and aims to provide a first picture of the method, sufficient for the reader who wish to achieve an overall understanding of it without entering into the further aspects developed in the second part. From this point of view, the latter will not be self-standing, but will be seen as a prosecution and a complement of the former, with the aim that the material presented in the whole paper can be used as a starting point for a classification of the subject. (shrink)

We provide results allowing to state, by the simple inspection of suitable classes of posets , that the corresponding intermediate propositional logics are maximal among the ones which satisfy the disjunction property. Starting from these results, we directly exhibit, without using the axiom of choice, the Kripke frames semantics of 2No maximal intermediate propositional logics with the disjunction property. This improves previous evaluations, giving rise to the same conclusion but made with an essential use of the (...) axiom of choice, of the cardinality of the set of the maximal intermediate propositional logics with the disjunction property. (shrink)

This paper shows that grounded dispositions are necessarily coextensive with disjunctive properties. It responds to several objections against this thesis, and then shows how to construct a disjunctive property necessarily coextensive with an arbitrary grounded disposition.

This paper shows that grounded dispositions are necessarily coextensive with disjunctive properties. It responds to several objections against this thesis, and then shows how to construct a disjunctive property necessarily coextensive with an arbitrary grounded disposition.

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.

Classical first-order logic \ is commonly used to study logical connections between statements, that is sentences that in every context have an associated truth-value. Inquisitive first-order logic \ is a conservative extension of \ which captures not only connections between statements, but also between questions. In this paper we prove the disjunction and existence properties for \ relative to inquisitive disjunction Open image in new window and inquisitive existential quantifier \. Moreover we extend these results to several families (...) of theories, among which the one in the language of \. To this end, we initiate a model-theoretic approach to the study of \. In particular, we develop a toolkit of basic constructions in order to transform and combine models of \. (shrink)

Classical first-order logic \ is commonly used to study logical connections between statements, that is sentences that in every context have an associated truth-value. Inquisitive first-order logic \ is a conservative extension of \ which captures not only connections between statements, but also between questions. In this paper we prove the disjunction and existence properties for \ relative to inquisitive disjunction Open image in new window and inquisitive existential quantifier \. Moreover we extend these results to several families (...) of theories, among which the one in the language of \. To this end, we initiate a model-theoretic approach to the study of \. In particular, we develop a toolkit of basic constructions in order to transform and combine models of \. (shrink)

In a language for arithmetic with a predicate T, intended to mean “ x is the Gödel number of a true sentence”, a set S of axioms and rules of inference has the truth disjunction property if whenever S ⊢ T ∨ T, either S ⊢ T or S ⊢ T. Similarly, S has the truth existence property if whenever S ⊢ ∃χ T ), there is some n such that S ⊢ T ). Continuing previous work, (...) we establish whether these properties hold or fail for a large collection of possible axiomatic systems. (shrink)

This paper considers the computational complexity of the disjunction and existential properties of intuitionistic logic. We prove that the disjunction property holds feasibly for intuitionistic propositional logic; i.e., from a proof of A v B, a proof either of A or of B can be found in polynomial time. For intuitionistic predicate logic, we prove superexponential lower bounds for the disjunction property, namely, there is a superexponential lower bound on the time required, given a proof (...) of A v B, to produce one of A and B which is true. In addition, there is superexponential lower bound on the size of terms which fulfill the existential property of intuitionistic predicate logic. There are superexponential upper bounds for these problems, so the lower bounds are essentially optimal. (shrink)

In a modal system of arithmetic, a theory S has the modal disjunction property if whenever $S \vdash \square\varphi \vee \square\psi$ , either $S \vdash \square\varphi$ or $S \vdash \square\psi. S$ has the modal numerical existence property if whenever $S \vdash \exists x\square\varphi(x)$ , there is some natural number n such that $S \vdash \square\varphi(\mathbf{n})$ . Under certain broadly applicable assumptions, these two properties are equivalent.

Are there any disjunctive properties—features of things such as being either red or round, or Nelson Goodman’s infamous example of being grue? As esoteric as the question may seem at first, central issues about the metaphysics of properties hinge upon its answer, such as whether reductive views about special science properties can handle the phenomenon of multiple realizability. A familiar argument for a negative answer is that disjunctive properties fail to guarantee that their instances are similar in some genuine respect. (...) In this paper, I respond to a novel, sophisticated version of this argument developed in recent work by Paul Audi. Along the way, I develop two new accounts of what it is for a property to be disjunctive—which rely on important recent work on the nature of essence and analysis—and clarify what one is committed to in believing that there are any disjunctive properties at all. (shrink)

Starting from a recent paper by S. Kaufmann, we introduce a notion of conjunction of two conditional events and then we analyze it in the setting of coherence. We give a representation of the conjoined conditional and we show that this new object is a conditional random quantity, whose set of possible values normally contains the probabilities assessed for the two conditional events. We examine some cases of logical dependencies, where the conjunction is a conditional event; moreover, we give the (...) lower and upper bounds on the conjunction. We also examine an apparent paradox concerning stochastic independence which can actually be explained in terms of uncorrelation. We briefly introduce the notions of disjunction and iterated conditioning and we show that the usual probabilistic properties still hold. (shrink)

This paper is a contribution to the general study of consequence relations which contain (definable) connective of “disjunction”. Our work is centered around the “proof by cases property”, we present several of its equivalent definitions, and show some interesting applications, namely in constructing axiomatic systems for intersections of logics and recognizing weakly implicative fuzzy logics among the weakly implicative ones.

I develop a ‘duty-plus’ approach to supererogation based on a simple intuition: if I am required to do x or y, doing x and y is a candidate for, though not necessarily, supererogation. This is an appealing view to take, located midway between two extreme positions, supererogationism and rigorism. I give a precise statement of the view through the notion of disjunctive duties, and discuss the commitments a duty-plus theorist should make, independent from the Kantian context in which this position (...) is often discussed. I also advocate the novel claim that we should take supererogation as a property of sets of actions, rather than single actions. The latter view suffers from problems in cases of concurrent acts and accumulative supererogation. (shrink)

Two different kinds of multiple-conclusion consequence relations taken from Shoesmith and Smiley and Galatos and Tsinakis or Nowak, called here disjunctive and conjunctive, respectively, defined on a formal language, are considered. They are transferred into a bounded lattice and a complete lattice, respectively. The properties of such abstract consequence relations are presented.

I present a problem for theories of explanation, concerning explanations involving disjunctive properties. The problem is particular acute for the explanatory non-fundamentalist, according to whom non-fundamental scientific explanations are sometimes superior to fundamental physical explanations. I criticise solutions to the problem due to Woodward, Strevens and Sober, and Lewis, and then defend a solution inspired by an account of non-fundamental laws recently defended by Callender and Cohen.

In Coecke (2002) we proposed the intuitionistic or disjunctive representation of quantum logic, i.e., a representation of the property lattice of physical systems as a complete Heyting algebra of logical propositions on these properties, where this complete Heyting algebra goes equipped with an additional operation, the operational resolution, which identifies the properties within the logic of propositions. This representation has an important application “towards dynamic quantum logic”, namely in describing the temporal indeterministic propagation of actual properties of physical systems. (...) This paper can as such by conceived as an addendum to “Quantum Logic in Intuitionistic Perspective” that discusses spin-off and thus provides an additional motivation. We derive a quantaloidal semantics for dynamic disjunctive quantum logic and illustrate it for the particular case of a perfect (quantum) measurement. (shrink)

Direct Informational Semantics, according to which [X]s represent X if ‘Xs cause [X]s’ is a law, and Fodorian naturalistic semantics both suffer from double disjunctivitis. I argue that robustness, properly construed, characterizes both represented properties and representing symbols: two or more properties normally regarded as non-disjunctive may each be nomologically connected to a non-disjunctive symbol, and two or more non-disjunctive symbols may each be nomologically connected to a property. This kind of robustness bifurcates the so-called disjunction problem into (...) a Represented-Disjunction Problem, of which Fodor was aware, and a Representer-Disjunction Problem, of which he was on the whole oblivious. Fodor fails to solve these problems: his solution to the former, the Asymmetric Dependence Condition, presupposes a successful solution to the latter, while possible responses that Fodor might make to the latter either beg the former or cannot be met or else flout the Naturalistic Requirement and the Atomistic Requirement. Even setting the Representer-Disjunction Problem aside, the Represented-Disjunction Problem does not get solved, because the robustness involving phonological/orthographic sequences guarantees that nothing can meet the Asymmetrical Dependence Condition. Indeed there is a serious problem of individuating phonological/orthographic tokens and types in a manner that satisfies Fodor’s expectations. This is made manifest by the presence of orthographic tokens embedded in larger tokens. (shrink)

In Coecke (2002) we proposed the intuitionistic or disjunctive representation of quantum logic, i.e., a representation of the property lattice of physical systems as a complete Heyting algebra of logical propositions on these properties, where this complete Heyting algebra goes equipped with an additional operation, the operational resolution, which identifies the properties within the logic of propositions. This representation has an important application towards dynamic quantum logic, namely in describing the temporal indeterministic propagation of actual properties of physical systems. (...) This paper can as such by conceived as an addendum to Quantum Logic in Intuitionistic Perspective that discusses spin-off and thus provides an additional motivation. We derive a quantaloidal semantics for dynamic disjunctive quantum logic and illustrate it for the particular case of a perfect (quantum) measurement. (shrink)

According to the Multiple Realizability Argument, a higher-level property typically has many physical realizers, so it cannot be type-identical to any one of them. This enables the non-reductive physicalist to claim that some higher-level properties are type-distinct from physical properties. The reductive physicalist can counter with the Disjunctive Strategy: nothing prevents us from type-identifying the higher-level property with the disjunction of its realizers. Developing a powers-based ontology of properties, Shoemaker and Wilson present responses to the Disjunctive Strategy, (...) wherein higher-level property instances are token-distinct from their realizers, while instances of disjunctions are not. In this paper, I argue that such responses to the Disjunctive Strategy can be reasonably resisted, either by denying token-distinctness, or by insisting that exhaustively overlapping disjunctive properties are also token-distinct from the realizers. This secures the Disjunctive Strategy as a core component of the argument for reductive physicalism. (shrink)

In this paper, we make a case for the disjunctive view of phenomenal consciousness: consciousness is essentially disjunctive in being either physical or non-physical in the sense that it has both physical and non-physical possible instances. We motivate this view by showing that it undermines two well-known conceivability arguments in philosophy of mind: the zombie argument for anti-physicalism, and the anti-zombie argument for physicalism. By appealing to the disjunctive view, we argue that two hitherto unquestioned premises of these arguments are (...) false. Furthermore, making use of the resources of this view, we formulate distinct forms of both physicalism and anti-physicalism. On these formulations, it is easy to see how physicalists and anti-physicalists can accommodate the modal intuitions of their opponents regarding zombies and anti-zombies. We conclude that these formulations of physicalism and anti-physicalism are superior to their more traditional counterparts. (shrink)