The monoidal t-norm based logic MTL is obtained from Hájek''s Basic Fuzzy logic BL by dropping the divisibility condition for the strong (or monoidal) conjunction. Recently, Jenei and Montgana have shown MTL to be standard complete, i.e. complete with respect to the class of residuated lattices in the real unit interval [0,1] defined by left-continuous t-norms and their residua. Its corresponding algebraic semantics is given by pre-linear residuated lattices. In this paper we address the (...) issue of standard and rational completeness (rational completeness meaning completeness with respect to a class of algebras in the rational unit interval [0,1]) of some important axiomatic extensions of MTL corresponding to well-known parallel extensions of BL. Moreover, we investigate varieties of MTL algebras whose linearly ordered countable algebras embed into algebras whose lattice reduct is the real and/or the rational interval [0,1]. These embedding properties are used to investigate finite strong standard and/or rational completeness of the corresponding logics. (shrink)

This paper aims at being a systematic investigation of different completeness properties of first-order predicate logics with truth-constants based on a large class of left-continuous t-norms . We consider standard semantics over the real unit interval but also we explore alternative semantics based on the rational unit interval and on finite chains. We prove that expansions with truth-constants are conservative and we study their real, rational and finite chain completeness properties. Particularly interesting is the case of (...) considering canonical real and rational semantics provided by the algebras where the truth-constants are interpreted as the numbers they actually name. Finally, we study completeness properties restricted to evaluated formulae of the kind , where φ has no additional truth-constants. (shrink)

In this paper we prove strong completeness of axiomatic extensions of first-order strict core fuzzy logics with the so-called quasi-witnessed axioms with respect to quasi-witnessed models. As a consequence we obtain strong completeness of Product Predicate Logic with respect to quasi-witnessed models, already proven by M.C. Laskowski and S. Malekpour in [19]. Finally we study similar problems for expansions with Δ, define Δ-quasi-witnessed axioms and prove that any axiomatic extension of a first-order strict core fuzzy (...) logic, expanded with Δ, and Δ-quasi-witnessed axioms are complete with respect to Δ-quasi-witnessed models. (shrink)

This paper is a contribution to Mathematical fuzzy logic, in particular to the algebraic study of t-norm based fuzzy logics. In the general framework of propositional core and Δ-core fuzzy logics we consider three properties of completeness with respect to any semantics of linearly ordered algebras. Useful algebraic characterizations of these completeness properties are obtained and their relations are studied. Moreover, we concentrate on five kinds of distinguished semantics for these logics–namely the class of algebras defined over (...) the real unit interval, the rational unit interval, the hyperreals , the strict hyperreals and finite chains, respectively–and we survey the known completeness methods and results for prominent logics. We also obtain new interesting relations between the real, rational and hyperreal semantics, and good characterizations for the completeness with respect to the semantics of finite chains. Finally, all completeness properties and distinguished semantics are also considered for the first-order versions of the logics where a number of new results are proved. (shrink)

Monoidal t-norm based logic $\mathbf {MTL}$ is the weakest t-norm based residuated fuzzy logic, which is a $[0,1]$ -valued propositional logical system having a t-norm and its residuum as truth function for conjunction and implication. Monadic fuzzy predicate logic $\mathbf {mMTL\forall }$ that consists of the formulas with unary predicates and just one object variable, is the monadic fragment of fuzzy predicate logic $\mathbf {MTL\forall }$, which is indeed the predicate (...) version of monoidal t-norm based logic $\mathbf {MTL}$. The main aim of this paper is to give an algebraic proof of the completeness theorem for monadic fuzzy predicate logic $\mathbf {mMTL\forall }$ and some of its axiomatic extensions. Firstly, we survey the axiomatic system of monadic algebras for t-norm based residuated fuzzy logic and amend some of them, thus showing that the relationships for these monadic algebras completely inherit those for corresponding algebras. Subsequently, using the equivalence between monadic fuzzy predicate logic $\mathbf {mMTL\forall }$ and S5-like fuzzy modal logic $\mathbf {S5(MTL)}$, we prove that the variety of monadic MTL-algebras is actually the equivalent algebraic semantics of the logic $\mathbf {mMTL\forall }$, giving an algebraic proof of the completeness theorem for this logic via functional monadic MTL-algebras. Finally, we further obtain the completeness theorem of some axiomatic extensions for the logic $\mathbf {mMTL\forall }$, and thus give a major application, namely, proving the strong completeness theorem for monadic fuzzy predicate logic based on involutive monoidal t-norm logic $\mathbf {mIMTL\forall }$ via functional representation of finitely subdirectly irreducible monadic IMTL-algebras. (shrink)

In this paper we carry out an algebraic investigation of the weak nilpotent minimum logic and its t-norm based axiomatic extensions. We consider the algebraic counterpart of WNM, the variety of WNM-algebras and prove that it is locally finite, so all its subvarieties are generated by finite chains. We give criteria to compare varieties generated by finite families of WNM-chains, in particular varieties generated by standard WNM-chains, or equivalently t-norm based axiomatic extensions of WNM, and (...) we study their standard completeness properties. We also characterize the generic WNM-chains, i. e. those that generate the variety [MATHEMATICAL DOUBLE-STRUCK CAPITAL W]ℕ[MATHEMATICAL DOUBLE-STRUCK CAPITAL M], and we give finite axiomatizations for some t-norm based extensions of WNM. (shrink)

Axiomatizations are presented for fuzzy logics characterized by uninorms continuous on the half-open real unit interval [0,1), generalizing the continuous t-norm based approach of Hájek. Basic uninorm logic BUL is defined and completeness is established with respect to algebras with lattice reduct [0,1] whose monoid operations are uninorms continuous on [0,1). Several extensions of BUL are also introduced. In particular, Cross ratio logic CRL, is shown to be complete with respect to one special uninorm. A Gentzen-style (...) hypersequent calculus is provided for CRL and used to establish co-NP completeness results for these logics. (shrink)

The paper is inspired by and explicitly presupposes the readers’ knowledge of the Kürbis normalization procedure for the Milne tree-like natural deduction system _C_ for classical propositional logic. The novelty of _C_ is that for each conventional connective, it has only _general_ introduction and elimination rules, whose paradigm is the rule of proof by cases. The present paper deals with the Milne–Kürbis troublemaker—adding universal quantifier—caused by extending the normalization procedure to $\mathbf {C^{\exists }_{\forall }}$, the first-order (...) variant of _C_. For both quantifiers, $\mathbf {C^{\exists }_{\forall }}$ has only general quantifier rules, whose paradigm is the indirect rule of existential elimination. We propose the classical first-order tree-like natural deduction system $\mathbf {NC^{\exists }_{\forall }}$ such that it contains all the propositional and half the quantifier rules of $\mathbf {C^{\exists }_{\forall }}$ and differs from it with respect to the other half of the quantifier rules and the notion of a deduction. We show that in the case of $\mathbf {NC^{\exists }_{\forall }}$, whose specifics is based on the Quine treatment of _flagged_ variables in the linear-style natural deduction system as well as on the Aguilera-Baaz treatment of _characteristic_ and _side_ variables for sequent-style calculi, the troublemaker doesn’t occur. In order to handle another specifics of $\mathbf {NC^{\exists }_{\forall }}$ —an auxiliary notion of a _finished_ deduction that makes deductions _nonhereditary_—we show soundness and completeness of $\mathbf {NC^{\exists }_{\forall }}$. While emphasizing the former with the help of the Smirnov technique, we additionally indicate a fixable gap in the Quine soundness argument. At last, the philosophical significance of the main result of this paper is developed in more detail. (shrink)

Abstract.ΠMTL is a schematic extension of the monoidal t-norm based logic (MTL) by the characteristic axioms of product logic. In this paper we prove that ΠMTL satisfies the standard completeness theorem. From the algebraic point of view, we show that the class of ΠMTL-algebras (bounded commutative cancellative residuated l-monoids) in the real unit interval [0,1] generates the variety of all ΠMTL-algebras.

For each continuous t-norm &, a class of fuzzy topological spaces, called &-topological spaces, is introduced. The motivation stems from the idea that to each many-valued logic there may correspond a theory of many-valued topology, in particular, each continuous t-norm may lead to a theory of fuzzy topology. It is shown that for each continuous t-norm &, the subcategory consisting of &-topological spaces is simultaneously reflective and coreflective in the category of fuzzy topological spaces, hence gives (...) rise to an autonomous theory of fuzzy topology. Topologizing a fuzzy pre-ordered set with the fuzzy Scott topology yields a functor from the category of fuzzy pre-ordered sets and maps that preserve suprema of flat ideals to the category of &-topological spaces. It is proved that this functor is a full one if and only if the t-norm & is Archimedean. (shrink)

We study the forcing operators on MTL-algebras, an algebraic notion inspired by the Kripke semantics of the monoidal t -norm based logic . At logical level, they provide the notion of the forcing value of an MTL-formula. We characterize the forcing operators in terms of some MTL-algebras morphisms. From this result we derive the equality of the forcing value and the truth value of an MTL-formula.

A continuoxis t- norm is a continuous map * from [0, 1]² into [0,1] such that is a commutative totally ordered monoid. Since the natural ordering on [0,1] is a complete lattice ordering, each continuous t-norm induces naturally a residuation → and becomes a commutative naturally ordered residuated monoid, also called a hoop. The variety of basic hoops is precisely the variety generated by all algebras, where * is a continuous t-norm. In this paper we investigate the (...) structure of the variety of basic hoops and some of its subvarieties. In particular we provide a complete description of the finite subdirectly irreducible basic hoops, and we show that the variety of basic hoops is generated as a quasivariety by its finite algebras. We extend these results to Hájek's BL-algebras, and we give an alternative proof of the fact that the variety of BL-algebras is generated by all algebras arising from continuous t-norms on [ 0,1] and their residua. The last part of the paper is devoted to the investigation of the subreducts of BL- algebras, of Gödel algebras and of product algebras. (shrink)

Whenever we communicate, we inevitably have to say one thing before another. This means introducing particularly subtle patterns of salience into our language. In this paper, I introduce ‘order-based salience patterns,’ referring to the ordering of syntactic contents where that ordering, pretheoretically, does not appear to be of consequence. For instance, if one is to describe a colourful scarf, it wouldn’t seem to matter if one were to say it is ‘orange and blue’ or ‘blue and orange.’ Despite (...) their apparent triviality, I argue that order-based salience patterns tend to make the content positioned first more salient – in the sense of attention-grabbing – in a way that can have surprising normative implications. Giving relative salience to gender differences over similarities, for instance, can result in the activation of cognitively accessible beliefs about gender differences. Where those beliefs are epistemically and/or ethically flawed, we can critique the salience pattern that led to them, providing an instrumental way of evaluating those patterns. I suggest that order-based salience patterns can also be evaluated on constitutive grounds; talking about gender differences before similarities might constitute a subtle form of bias. Finally, I reflect on how the apparent triviality of order-based salience patterns in language gives them an insidious strength. (shrink)

The thesis that agents should calibrate their beliefs in the face of higher-order evidence—i.e., should adjust their first-order beliefs in response to evidence suggesting that the reasoning underlying those beliefs is faulty—is sometimes thought to be in tension with Bayesian approaches to belief update: in order to obey Bayesian norms, it’s claimed, agents must remain steadfast in the face of higher-order evidence. But I argue that this claim is incorrect. In particular, I motivate a minimal (...) constraint on a reasonable treatment of the evolution of self-locating beliefs over time and show that calibrationism is compatible with any generalized Bayesian approach that respects this constraint. I then use this result to argue that remaining steadfast isn’t the response to higher-order evidence that maximizes expected accuracy. (shrink)

Substructural fuzzy logics are substructural logics that are complete with respect to algebras whose lattice reduct is the real unit interval [0.1]. In this paper, we introduce Uninorm logic UL as Multiplicative additive intuitionistic linear logic MAILL extended with the prelinearity axiom ((A → B) ∧ t) ∨ ((B → A) ∧ t). Axiomatic extensions of UL include known fuzzy logics such as Monoidal t-norm logic MTL and Gödel logic G, and new weakening-free logics. (...) Algebraic semantics for these logics are provided by subvarieties of (representable) pointed bounded commutative residuated lattices. Gentzen systems admitting cut-elimination are given in the framework of hypersequents. Completeness with respect to algebras with lattice reduct [0, 1] is established for UL and several extensions using a two-part strategy. First, completeness is proved for the logic extended with Takeuti and Titani's density rule. A syntactic elimination of the rule is then given using a hypersequent calculus. As an algebraic corollary, it follows that certain varieties of residuated lattices are generated by their members with lattice reduct [0, 1]. (shrink)

Belnap–Dunn’s relevance logic, $\textsf{BD}$, was designed seeking a suitable logical device for dealing with multiple information sources which sometimes may provide inconsistent and/or incomplete pieces of information. $\textsf{BD}$ is a four-valued logic which is both paraconsistent and paracomplete. On the other hand, De and Omori, while investigating what classical negation amounts to in a paracomplete and paraconsistent four-valued setting, proposed the expansion $\textsf{BD2}$ of the four valued Belnap–Dunn logic by a classical negation. In this paper, we introduce (...) a four-valued expansion of BD called ${\textsf{BD}^\copyright }$, obtained by adding an unary connective ${\copyright }\,\$ which is a consistency operator (in the sense of the Logics of Formal Inconsistency, _LFI_s). In addition, this operator is the unique one with the following features: it extends to $\textsf{BD}$ the consistency operator of LFI1, a well-known three-valued _LFI_, still satisfying axiom _ciw_ (which states that any sentence is either consistent or contradictory), and allowing to define an undeterminedness operator (in the sense of Logic of Formal Undeterminedness, _LFU_s). Moreover, ${\textsf{BD}^\copyright }$ is maximal w.r.t. LFI1, and it is proved to be equivalent to BD2, up to signature. After presenting a natural Hilbert-style characterization of ${\textsf{BD}^\copyright }$ obtained by means of twist-structures semantics, we propose a first-order version of ${\textsf{BD}^\copyright }$ called ${\textsf{QBD}^\copyright }$, with semantics based on an appropriate notion of four-valued Tarskian-like structures called $\textbf{4}$ -structures. We show that in ${\textsf{QBD}^\copyright }$, the existential and universal quantifiers are interdefinable in terms of the paracomplete and paraconsistent negation, and not by means of the classical negation. Finally, a Hilbert-style calculus for ${\textsf{QBD}^\copyright }$ is presented, proving the corresponding soundness and completeness theorems. (shrink)

This paper expands upon the work by Wang (Proceedings of TARK, pp. 493–512, 2017) who proposes a new framework based on quantifier-free predicate language extended by a new bundled modality $\exists x\Box$ and axiomatizes the logic over S5 frames. This paper first gives complete axiomatizations of the logics over K, D, T, 4, S4 frames with increasing domains and constant domains, respectively. The systems w.r.t. constant domains feature infinitely many additional rules defined inductively than systems w.r.t. (...) increasing domains. In the second part of the paper, we show that these rules are admissible in the systems without them by providing a general strategy for proving completeness theorems for the logics without these rules over constant domain models (also increasing domain models). (shrink)

The essay introduces a non-Diodorean, non-Kantian temporal modal semantics based on part-whole, rather than class, theory. Formalizing Edmund Husserl’s theory of inner time consciousness, §3 uses his protention and retention concepts to define a relation of self-awareness on intentional events. §4 introduces a syntax and two-valued semantics for modal first-order predicate object-languages, defines semantic assignments for variables and predicates, and truth for formulae in terms of the axiomatic version of Edmund Husserl’s dependence ontology (viz. the Calculus [CU] (...) of Urelements) introduced by The Ontology of Intentionality I & II. It then uses the §3 results to define the modalities of truth, and §5 extends the semantics to identity claims. §6 defines and contrasts synthetic a priori truths to analytic a priori truths, and §7 compares Brentano School noetic semantic and Leibnizian possible-world semantic perspectives on modality. The essay argues that the modal logics it defines semantically are two-valued, first-order versions of the type of language which Husserl viewed as the language of any ontology of experience (i.e. of any science), and conceived as the logic of intentionality. (shrink)

Belnap–Dunn’s relevance logic, $\textsf{BD}$, was designed seeking a suitable logical device for dealing with multiple information sources which sometimes may provide inconsistent and/or incomplete pieces of information. $\textsf{BD}$ is a four-valued logic which is both paraconsistent and paracomplete. On the other hand, De and Omori, while investigating what classical negation amounts to in a paracomplete and paraconsistent four-valued setting, proposed the expansion $\textsf{BD2}$ of the four valued Belnap–Dunn logic by a classical negation. In this paper, we introduce (...) a four-valued expansion of BD called ${\textsf{BD}^\copyright }$, obtained by adding an unary connective ${\copyright }\,\$ which is a consistency operator (in the sense of the Logics of Formal Inconsistency, _LFI_s). In addition, this operator is the unique one with the following features: it extends to $\textsf{BD}$ the consistency operator of LFI1, a well-known three-valued _LFI_, still satisfying axiom _ciw_ (which states that any sentence is either consistent or contradictory), and allowing to define an undeterminedness operator (in the sense of Logic of Formal Undeterminedness, _LFU_s). Moreover, ${\textsf{BD}^\copyright }$ is maximal w.r.t. LFI1, and it is proved to be equivalent to BD2, up to signature. After presenting a natural Hilbert-style characterization of ${\textsf{BD}^\copyright }$ obtained by means of twist-structures semantics, we propose a first-order version of ${\textsf{BD}^\copyright }$ called ${\textsf{QBD}^\copyright }$, with semantics based on an appropriate notion of four-valued Tarskian-like structures called $\textbf{4}$ -structures. We show that in ${\textsf{QBD}^\copyright }$, the existential and universal quantifiers are interdefinable in terms of the paracomplete and paraconsistent negation, and not by means of the classical negation. Finally, a Hilbert-style calculus for ${\textsf{QBD}^\copyright }$ is presented, proving the corresponding soundness and completeness theorems. (shrink)

In this paper we characterize, classify and axiomatize all axiomatic extensions of the IMT3 logic. This logic is the axiomatic extension of the involutive monoidal t-norm logic given by ¬φ3 ∨ φ. For our purpose we study the lattice of all subvarieties of the class IMT3, which is the variety of IMTL-algebras given by the equation ¬(x 3) ∨ x ≈ ⊤, and it is the algebraic counterpart of IMT3 logic. Since every subvariety of (...) IMT3 is generated by their totally ordered members, we study the structure of all IMT3-chains in order to determine the lattice of all subvarieties of IMT3. Given a family of IMT3-chains the number of elements of the largest odd finite subalgebra in the family and the number of elements of the largest even finite subalgebra in the family turns out to be a complete classifier of the variety generated. We obtain a canonical set of generators and a finite equational axiomatization for each subvariety and, for each corresponding logic, a finite set of characteristic matrices and a finite set of axioms. (shrink)

Timothy Williamson’s Modal Logic as Metaphysics is a book-length defense of necessitism about objects—roughly put, the view that, necessarily, any object that exists, exists necessarily. In more formal terms, Williamson argues for the validity of necessitism for objects (NO: ◻︎∀x◻︎∃y(x=y)). NO entails both the (first-order) Barcan formula (BF: ◇∃xΦ → ∃x◇Φ, for any formula Φ) and the (first-order) converse Barcan formula (CBF: ∃x◇Φ → ◇∃xΦ, for any formula Φ). The purpose of this essay is not (...) to assess Williamson’s arguments either for necessitism (although discussion of these arguments will play a central role in the dialectic) or for necessitism’s two famous corollaries. Instead, the focus shall be a general principle governing abstract objects—the abstract of principle (or AOP) —instances of which seems to be at work in some of Williamson’s central arguments for necessitism. The AOP can be straightforwardly formulated and applied within the neo-logicist framework—in fact, arguably the principle is most naturally formulated in neo-logicist terms. -/- After closely examining, and carefully formalizing, the AOP, the remainder of the paper focuses on arguments for necessitism-like claims (the exact meaning of “necessitism-like” will become clearer as the essay progresses) based on the AOP. In particular, we shall focus on the instance of the AOP that applies to the abstract objects governed by the most well-known and most fully studied abstraction principle: Hume’s Principle (HP). It turns out that, although we cannot reconstruct a valid argument for necessitism based on this numerical instance of the AOP, we can obtain valid arguments for weaker, but equally interesting conclusions. In particular, we shall show that, although HP combined with the AOP (and some additional, related assumptions) allows the contents of the domains of possible worlds to vary, the size of those domains must remain constant. The paper concludes by developing and critiquing some related arguments for necessitism based on applying relevant instances of the AOP to abstraction principles governing sets (or extensions), and to a simple objectual abstraction principle. (shrink)

This thesis examines the prospects for mechanical procedures that can identify true, complete, universal, first-order logical theories on the basis of a complete enumeration of true atomic sentences. A sense of identification is defined that is more general than those which are usually studied in the learning theoretic and inductive inference literature. Some identification algorithms based on confirmation relations familiar in the philosophy of science are presented. Each of these algorithms is shown to identify all purely universal (...) theories without function symbols. It is demonstrated that no procedure can solve this universal theory inference problem in the more usual senses of identification. The question of efficiency for theory inference systems is addressed, and some definitions of limiting complexity are examined. It is shown that several aspects of obvious strategies for solving the universal theory inference problem are NP-hard. Finally, some non-worst case heuristic search strategies are examined in light of these NP-completeness results. These strategies are based upon an isomorphism between clausal entailments of a certain class and partition lattices, and are applicable to the improvement of earlier work on language acquisition and logical inductive inference. (shrink)

Each Girard quantale (i.e., commutative quantale with a selected dualizing element) provides a support for a semantics for linear propositional formulas (but not for linear derivations). Several constructions of Girard quantales are known. We give two more constructions, one using an arbitrary partially ordered monoid and one using a partially ordered group (both commutative). In both cases the semantics can be controlled be a relation between pairs of elements of the support and formulas. This gives us a neat way of (...) handling duality. (shrink)

In this slender volume in The Humanist Library series, Nielsen not only argues for the independence of morality from religion, but as well outlines a normative theory as an alternative to religious [[sic]] morality. The basis of religious morality is the belief that God is all good, and thus we should do what he commands. In response to this, Nielsen elaborates Plato’s argument that morality cannot be based upon religious belief. However one understands the claim "God is good," i.e., (...) whether it is analytic or not, understanding "good" is logically presupposed for one to be able to grasp the meaning or truth of the claim. It is more correct to say that religion is based upon morality since the basis of religious belief is that there is a being worthy of worship, and this conclusion must rest upon autonomous moral insight. The remainder of the book is concerned with two themes involving the competing claims of secular and religious morality. The first is that only the religious perspective provides an adequate account of man’s origin and destiny, and thus the perspective from which one can find happiness and fulfillment. In response, Nielsen invokes "all the old as well as some new objections to religion" as the perspective which provides a true account of the nature of things. Moreover, there is no reason to believe that only religious persons find happiness and meaning in life. The second claim of religious morality is that it alone can provide a satisfactory basis for moral theory. Secular moral theory can provide only conventional relativism. In reply, Nielsen outlines and defends a normative theory of the consequentialist variety. The last part of the book is a general defense of consequential ethical theories against Anscombe-like objections which are associated with "religious absolutism" for reasons not specified.—T.K.H. (shrink)

The author presents an ethical theory which, as he admits, has much in common with the theories of M. Cohen, R. Sellars, H. Feigl, C. Lamont, and G. Williams. His first task is to define the scientific world view on which his ethical conclusions will be based. It comprises the following suppositions, logically derived from and justified by scientific practice: there is a real world independent of the knower, natural events are uniform, every event is related to some (...) other events, everything is in constant change, all changes are orderly, causes are efficient not final. Before the good life can be defined, science must define human nature, i.e., man's basic and universal needs, which are: physical health, recognition, love, excitement and novelty, knowledge. The good life must satisfy all these needs, but since this may be done improperly, a condition must be added: that satisfaction must come from a real adjustment to the environment in the long run. Moral rules must be both happiness-producing and life-promoting. This framework allows for diversity in moral codes. Determinism does not conflict with morality, for the only freedom that morality presupposes is the absence of external or irrational causes. An act is free to the extent that it is determined by the agent's self-knowledge. The book ends with a brief discussion of the problems of overpopulation, pollution, and world-scale planning.--T. D. Z. (shrink)

While the Value-Free Ideal (VFI) had many precursors, it became a solidified bulwark of normative claims about scientific reasoning and practice in the mid-twentieth century. Since then, it has played a central role in the philosophy of science, first as a basic presupposition of how science should work, then as a target for critique, and now as a target for replacement. In this paper, we will argue that a narrow focus on the VFI is misguided, because the VFI coalesced (...) in the midst of other important shifts in the relationship between science and society. In particular, the mid-twentieth century saw the acceptance of the “social contract for science,” a tacit agreement between scientists and government officials, and more broadly between science and society. It was built around three core concepts: a distinction between basic and applied science, a conception of scientific freedom that limited social responsibility for scientists, and a justification for public funding of basic science in the form of the linear model. Within the conceptual framework of the social contract for science, it is clearer both (1) why the VFI was adopted, (2) why it is difficult to replace the VFI within the old social contract, and (3) how we need to revise the social contract for science in order to replace the VFI. (shrink)

Epistemic utility theory seeks to establish epistemic norms by combining principles from decision theory and social choice theory with ways of determining the epistemic utility of agents’ attitudes. Recently, Moss, 1053–69, 2011) has applied this strategy to the problem of finding epistemic compromises between disagreeing agents. She shows that the norm “form compromises by maximizing average expected epistemic utility”, when applied to agents who share the same proper epistemic utility function, yields the result that agents must form compromises by (...) splitting the difference between their credence functions. However, this “split the difference” norm is in conflict with conditionalization, since applications of the two norms don’t commute. A common response in the literature seems to be to abandon the procedure of splitting the difference in favor of compromise strategies that avoid non-commutativity. This would also entail abandoning Moss’ norm. I explore whether a different response is feasible. If agents can use epistemic utility-based considerations to agree on an order in which they will apply the two norms, they might be able to avoid diachronic incoherence. I show that this response can’t save Moss’ norm, because the agreements concerning the order of compromising and updating it generates are not stable over time, and hence cannot avoid diachronic incoherence. I also show that a variant of Moss’ norm, which requires that the weights given to each agent’s epistemic utility change in a way that ensures commutativity, cannot be justified on epistemological grounds. (shrink)

Abstract:In this essay, I argue against the bright-line distinction between ideal and nonideal normative political theory, a distinction used to distinguish “stages” of theorizing such that ideal political principles can be deduced and examined before compromises with the flawed political world are made. The distinction took on its familiar form in Rawls and has enjoyed a resurgence of interest in the past few years. I argue that the idea of a categorical distinction — the kind that could allow for a (...) sequencing of stages of theorizing — is misconceived, because wholly “ideal” normative political theory is a conceptual mistake, the equivalent of taking the simplifying models of introductory physics (“frictionless movement in a vacuum”) and trying to develop an ideal theory of aerodynamics. Political organization and justice are about moral friction in the first instance. I examine both logical and epistemological arguments for the position that we need the uniquely idealizing assumptions of ideal theory in order to arrive at, or to know, a genuine theory of justice or political morality; and I find them wanting. Such assumptions as full compliance, consensus, and the publicity principle of universal knowledge about consensus can sometimes be useful, if used carefully and with justification; but they are not categorically different from other idealizing and abstracting assumptions in generating normative theory. What is referred to as “nonideal” theory is all that there is, and it is many kinds of theory, not one — the many ways in which we learn about justice and injustice, and seek to answer questions of practical reason about what ought to be done in our political world. (shrink)

This paper analyses two methods commonly used to understand legal language: deontic logic and the analysis of concepts taken as fundamental for any one or more areas of the law (sometimes called the philosophical foundations of law project). In doing so I introduce what I call the phenomenon of linguistic regress, and I do so in order to show why and how these methods necessarily fail as theories of legal language. I argue, in short, that any form of (...) content-determination of concepts or norms fails as a method for understanding legal language because of the phenomenon of linguistic regress. Given the failure of these two methods, I argue that legal theorists should instead adopt an approach that focuses on explaining the community-based use of commonly-recurring terms and phrases found in legal language - a use, furthermore, that is most commonly exercised by legal experts (of varying degrees of skills), operating in different and interacting contexts of expertise (judicial, legislative and scholarly), and more broadly, as members of a legal community not impervious to influence from other kinds of communities. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:EMBODIED HUMAN AGENTS INHABITING A MATERIAL WORLD? CHARLES T. HUGHES Chapman University Orange, California I. /n;troduction HE CONCEPT of a "logically possible world" has roven useful in the investigation of issues within many ranches of philosophy, including the philosophy of religion.1 Since this paper includes an analysis of one "possible worlds" objection to Christian theism, based upon the problem of evil, it will prove useful to preface my (...) discussion with a paragraph which summarizes some of the ideas important to the logic of possible worlds. Each logically possible world contains a complete set of compossible facts, which includes the past, present, and future of its objects, properties, and relations. Those complete sets of compossible facts, each of which describes the way the actual world could have been, have in common the truth of all logically necessary propositions (for example, 2 +2 = 4, "Circles cannot be squared "), and the falsity of all logically impossible propositions (for example, 2 +6 = 7, " Circles can be squared"), but differ in their truth values with respect to at least some logically contingent propositions (for example, "Elephants exist" is true in one or more possible worlds but false in all others, and "Unicorns exist " is true in one or more possible worlds but false in all others). An impossible world includes an affirmation of the falsity of at least one necessary proposition (for example,,.._,(2 +2 = 4)), and/or affirms the truth of at least one logical1 See Alvin Plantinga, The Nature of Necessity (Oxford: Clarendon Press, 1974), chapters IX and X. 389 390 CHARLES T. HUGHES ly impossible proposition (for example, " Logically necessary propositions are false in all possible worlds ").2 The actual world, which is our world, is that complete set of compossible propositions which describe the way things actually are. The actual world, therefore, differs from other possible worlds in that all of its contingent propositions correspond to the past, present, and future of all of the objects, properties, and relations which actually obtain. There is an on-going debate about the nature and ontological status of possible worlds and their objects.3 The purpose of this paper is to analyze and reject a particular "possible worlds" objection to Christian theism, based upon the problem of evil, which is not frequently discussed in philosophical literature. The problem is this : If God must actualize a possible world in order to provide human agents with the initial stage of existence they require to develop freely as rational, moral, and religious beings, who choose their own destinies, why did God not create them as incorporeal agents within an immaterial world which would, by its very nature, eliminate the physical dimension of evil and suffering which plagues the actual world? For it seems, prima facie, that God could have realized his purposes by actualizing an immaterial world, populated with human-like incorporeal agents, thereby avoiding all of the terrible evil and suffering made possible by physicality. Therefore, the existence of the actual physical world, with all its evil and suffering, is evidence against the existence of God.4 In order to show that the previously stated objection from evil could pose a genuine threat to theism, I will first identify some of my important assumptions in this paper, and then outline particular Christian ideas to determine how the " possible worlds " 2 For an argument that some impossible propositions are not self-contradictory, see Linda Zagzebski, " What If the Impossible Had Been Actual," in Michael D. Beatty, ed., Christian Theism aml the Problems of Philosophy (Notre Dame: University of Notre Dame Press, 1990), pp. 165-183. a Compare, for example, the positions of D. M. Armstrong, A Combinatorial Theory of Possibility (Cambridge: Cambridge University Press, 1989) and D. Lewis, On the Plurality of Worlds (Oxford: Blackwell, 1986). 4 Some interesting discussion of this type of problem can be found in Austin Farrer, Love Almighty and Ills Unlimited (London: Collins, 1962), pp. 69-76. EMBODIED HUMAN AGENTS 391 objection to Christian theism could arise in the first place. After that, I will discuss the nature of the " possible worlds " objection and how, in principle, it might be rebutted by theists who possess a great amount of... (shrink)

In order to modelize the reasoning of an intelligent agent represented by a poset T, H. Rasiowa introduced logic systems called “Approximation Logics”. In these systems a set of constants constitutes a fundamental tool. In this papers, we consider logic systems called L′T without this kind of constants but limited to the case where T is a finite poset. We prove a weak deduction theorem. We introduce also an algebraic semantics using Hey ting algebra with operators. To (...) prove the completeness theorem of the L′T system with respect to the algebraic semantics, we use the method of H. Rasiowa and R. Sikorski for first order logic. In the propositional case, a corollary allows us to assert that it is decidable to know “if a propositional formula is valid”. We study also certain relations between the L′T logic and the intuitionistic and classical logics. (shrink)

Online trading invariably involves dealings between strangers, so it is important for one party to be able to judge objectively the trustworthiness of the other. In such a setting, the decision to trust a user may sensibly be based on that user’s past behaviour. We introduce a specification language based on linear temporal logic for expressing a policy for categorising the behaviour patterns of a user depending on its transaction history. We also present an algorithm for checking (...) whether the transaction history obeys the stated policy. To be useful in a real setting, such a language should allow one to express realistic policies which may involve parameter quantification and quantitative or statistical patterns. We introduce several extensions of linear temporal logic to cater for such needs: a restricted form of universal and existential quantification; arbitrary computable functions and relations in the term language; and a “counting” quantifier for counting how many times a formula holds in the past. We then show that model checking a transaction history against a policy, which we call the history-based transaction monitoring problem, is PSPACE-complete in the size of the policy formula and the length of the history, assuming that the underlying interpreted func-. (shrink)

Let L be a quantum logic, here an orthoalgebra, and let Δ be a convex set of states on L. Then Δ generates a base-normed space, and the dual-order unit-normed space contains a canonically constructed homomorphic copy of L, denoted by eΔ(L). A convex set Δ of states on L is said to be ample provided that every state on L is obtained by restricting an element of the base of the bi-dual order unit-normed space to eΔ(L). (...) For a quantum logic L we show that a convex set of states Δ is ample if and only if Δ is weakly dense in the convex set of all states on L. The notion of ampleness is then discussed in the context of Gleason-type theorems for W* algebras and JBW algebras and also in the context of classical logics. (shrink)

The paper expands upon the work by Wang [4], who proposes a new framework based on quantifier-free predicate language extended by a new modality ∃x□\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\exists x\Box$$\end{document} and axiomatizes the logic over S5 frames. This paper gives the logics over K, D, T, 4, S4 frames with increasing and constant domains. And we provide a general strategy for proving completeness theorems for logics w.r.t. the increasing domain and logics w.r.t. (...) the constant domain respectively. (shrink)

In this paper we analyze some aspects of the question of using methods from model theory to study structures of functional analysis.By a well known result of P. Lindström, one cannot extend the expressive power of first order logic and yet preserve its most outstanding model theoretic characteristics (e.g., compactness and the Löwenheim-Skolem theorem). However, one may consider extending the scope of first order in a different sense, specifically, by expanding the class of structures that (...) are regarded as models (e.g., including Banach algebras or other structures of functional analysis), and ask whether the resulting extensions of first order model theory preserve some of its desirable characteristics.A formal framework for the study of structures based on Banach spaces from the perspective of model theory was first introduced by C. W. Henson in [8] and [6]. Notions of syntax and semantics for these structures were defined, and it was shown that using them one obtains a model theoretic apparatus that satisfies many of the fundamental properties of first order model theory. For instance, one has compactness, Löwenheim-Skolem, and omitting types theorems. Further aspects of the theory, namely, the fundamentals of stability and forking, were first introduced in [10] and [9].The classes of mathematical structures formally encompassed by this framework are normed linear spaces, possibly expanded with additional structure,e.g., operations, real-valued relations, and constants. This notion subsumes wide classes of structures from functional analysis. However, the restriction that the universe of a structure be a normed space is not necessary. (This restriction has a historical, rather than technical origin; specifically, the development of the theory was originally motivated by questions in Banach space geometry.) Analogous techniques can be applied if the universe is a metric space. Now, when the underlying metric topology is discrete, the resulting model theory coincides with first order model theory, so this logic extends first order in the sense described above. Furthermore, without any cost in the mathematical complexity, one can also work in multi-sorted contexts, so, for instance, one sort could be an operator algebra while another is. say, a metric space. (shrink)

We consider a new fragment of first-order logic with two variables. This logic is defined over interval structures. It constitutes unary predicates, a binary predicate and a function symbol. Considering such a fragment of first-order logic is motivated by defining a general framework for event-based interval temporal logics. In this paper, we present a sound, complete and terminating decision procedure for this logic. We show that the logic is decidable, and (...) provide a NEXPTIME complexity bound for satisfiability. This result shows that even a simple decidable fragment of first-order logic has NEXPTIME complexity. (shrink)

Let T be the theory of dense cyclically ordered sets with at least two elements. We determine the classifying space of $\mathsf {Mod}(T)$ to be homotopically equivalent to $\mathbb {CP}^\infty $. In particular, $\pi _2(\lvert \mathsf {Mod}(T)\rvert )=\mathbb {Z}$, which answers a question in our previous work. The computation is based on Connes’ cycle category $\Lambda $.

In this paper, we define a first-order logic CFʹ with strong negation and bounded static quantifiers, which is a variant of Thomason's logic CF. For the logic CFʹ, the usual Kripke formal semantics is defined based on situations, and a sound and complete axiomatic system is established based on the axiomatic systems of constructive logics with strong negation and Thomason's completeness proof techniques. With the use of bounded quantifiers, CFʹ allows the domain of (...) quantification to be empty and allows for nondenoting constants. CFʹ is intended as a fragment of a logic for situation theory. Thus the connection between CFʹ and infon logic is discussed. (shrink)

We describe a decidable class of formulas in a first order timed logic that covers a good amount of properties of real-time distributed systems. Earlier we described a decidable class based on some finiteness properties, and sketched a decidable class in a weaker logic that captures periodicity properties, though without complete proof. The new feature of the decidable class presented here is to be able to treat parametric properties, in particular, properties that concern an arbitrary (...) number of processes as compared to the just mentioned class that could express only properties for a fixed, explicitly given number of processes. As before the properties may use non-trivial arithmetics. We also state another important feature of the class: if a verification formula is not true then our algorithm gives a quantifier-free description of all its counter-models of the complexity involved in the definition of the decidable class and that is quite representative. In conclusion we discuss theoretical and practical issues of the complexity and some open questions. (shrink)

In Hájek et al. (J Symb Logic 65(2):669–682, 2000) the authors introduce the concept of supersound logic, proving that first-order Gödel logic enjoys this property, whilst first-order Łukasiewicz and product logics do not; in Hájek and Shepherdson (Ann Pure Appl Logic 109(1–2):65–69, 2001) this result is improved showing that, among the logics given by continuous t-norms, Gödel logic is the only one that is supersound. In this paper we will generalize the (...) previous results. Two conditions will be presented: the first one implies the supersoundness and the second one non-supersoundness. To develop these results we will use, between the other machineries, the techniques of completions of MTL-chains developed in Labuschagne and van Alten (Proceedings of the ninth international conference on intelligent technologies, 2008) and van Alten (2009). We list some of the main results. The first-order versions of MTL, SMTL, IMTL, WNM, NM, RDP are supersound; the first-order version of an axiomatic extension of BL is supersound if and only it is n-potent (i.e. it proves the formula ${\varphi^{n}\,\to\,\varphi^{n\,{+}\,1}}$ for some ${n\,\in\,\mathbb{N}^+}$ ). Concerning the negative results, we have that the first-order versions of ΠMTL, WCMTL and of each non-n-potent axiomatic extension of BL are not supersound. (shrink)

According to a common definition, the argument of truthful ones is an argument in which the existence of Necessary Being is proved with no presumption of the existence of the possible being. Avicenna proposed different versions of this style of argument and the version in the book of Nejat is one of them. This paper is intended to examine the possibility of proving the logical validity of this version in first-order predicate logic and explain the principles which (...) the argument is based on. In this way, it becomes clear to what extent Avicenna's version is in accordance with the criteria he himself introduces for the argument of truthful ones. In this respect, while providing a clear exposition of the version in the book of Nejat, bugs on some of the statements and proposals to obviate them in terms of replacement arises. Then explaining the concepts and premises which the argument in the book of Nejat is besed on, it is formulated in first-order predicate logic with its full proof. Finally, the proposed model's accordance with the presented criteria for the argument of truthful ones will be shown. (shrink)

European and international regulation of human health research is typified by a morass of interconnecting laws, diverse and divergent ethical frameworks, and national and transnational standards. There is also a tendency for legislators to regulate in silos—that is, in discrete fields of scientific activity without due regard to the need to make new knowledge as generalisable as possible. There are myriad challenges for the stakeholders—researchers and regulators alike—who attempt to navigate these landscapes. This Delphi study was undertaken in order (...) to provide the first interdisciplinary and crosscutting analysis of health research regulation, as it is experienced by such stakeholders in the UK context. As well as reinforcing existing understandings of the regulatory environment, Delphi participants called for greater collaboration, and even co-production, of processes involved in health research regulation. On the basis of this research, we offer insights about how health research regulation can become a matter with which a wider range of stakeholders—including researchers, regulators, publics and research sponsors—can engage. The evidence supports the normative claim that health research regulation should continue to move away from strict, prescriptive rules-based approaches, and towards flexible principle-based regimes that allow researchers, regulators and publics to co-produce regulatory systems serving core principles. By unpacking thorny concepts and practices at the heart of health research regulation—including the public interest and public engagement—our results have the potential to situate and breathe life into them. The results also demonstrate that while proportionality is well-recognised as a crucial element of flexible regulatory systems, more must be done to operationalise this as an ethical assessment of the values and risks at stake at multiple junctures in the research trajectory. This is required if we are to move beyond proportionality as a mere risk-management tool. Compliance culture no longer accurately reflects the needs and expectations of researchers or regulators, nor does it necessarily produce the best research. Embracing uncertainty—both as a human practice and a regulatory objective—may represent the brighter future for health research. (shrink)

This paper defines the form of prior knowledge that is required for sound inferences by analogy and single-instance generalizations, in both logical and probabilistic reasoning. In the logical case, the first order determination rule defined in Davies (1985) is shown to solve both the justification and non-redundancy problems for analogical inference. The statistical analogue of determination that is put forward is termed 'uniformity'. Based on the semantics of determination and uniformity, a third notion of "relevance" is defined, (...) both logically and probabilistically. The statistical relevance of one function in determining another is put forward as a way of defining the value of information: The statistical relevance of a function F to a function G is the absolute value of the change in one's information about the value of G afforded by specifying the value of F. This theory provides normative justifications for conclusions projected by analogy from one case to another, and for generalization from an instance to a rule. The soundness of such conclusions, in either the logical or the probabilistic case, can be identified with the extent to which the corresponding criteria (determination and uniformity) actually hold for the features being related. (shrink)