Does Semantic Relationism Solve Frege’s Puzzle?

In a series of recent works, Kit Fine (The Journal of Philosophy, 100(12), 605–631, 2003, 2007) has sketched a novel solution to Frege’s puzzle. Radically departing from previous solutions, Fine argues that Frege’s puzzle forces us to reject compositionality. In this paper we first provide an explicit formalization of the relational semantics for first-order logic suggested, but only briefly sketched, by Fine. We then show why the relational semantics alone is technically inadequate, forcing Fine to enrich the syntax with a coordination schema. Given this enrichment, we argue, that that the semantics is compositional. We then examine the deep consequences of this result for Fine’s proposed solution to Frege’s puzzle. We argue that Fine has mis-diagnosed his own solution–his attempted solution does not deny compositionality. The correct characterization of Fine’s solution fits him more comfortably among familiar solutions to the puzzle.     

Information closure and the sceptical objection

In this article, I define and then defend the principle of information closure (pic) against a sceptical objection similar to the one discussed by Dretske in relation to the principle of epistemic closure. If I am successful, given that pic is equivalent to the axiom of distribution and that the latter is one of the conditions that discriminate between normal and non-normal modal logics, a main result of such a defence is that one potentially good reason to look for a formalization of the logic of “ $S$ is informed that $p$ ” among the non-normal modal logics, which reject the axiom, is also removed. This is not to argue that the logic of “ $S$ is informed that $p$ ” should be a normal modal logic, but that it could still be insofar as the objection that it could not be, based on the sceptical objection against pic, has been removed. In other word, I shall argue that the sceptical objection against pic fails, so such an objection provides no ground to abandon the normal modal logic B (also known as KTB) as a formalization of “ $S$ is informed that $p$ ”, which remains plausible insofar as this specific obstacle is concerned.     

The Modal Logic of Gödel Sentences

The modal logic of Gödel sentences, termed as GS, is introduced to analyze the logical properties of ‘true but unprovable’ sentences in formal arithmetic. The logic GS is, in a sense, dual to Grzegorczyk’s Logic, where modality can be interpreted as ‘true and provable’. As we show, GS and Grzegorczyk’s Logic are, in fact, mutually embeddable. We prove Kripke completeness and arithmetical completeness for GS. GS is also an extended system of the logic of ‘Essence and Accident’ proposed by Marcos (Bull Sect Log 34(1):43–56, 2005). We also clarify the relationships between GS and the provability logic GL and between GS and Intuitionistic Propositional Logic.     

A Defense of Second-Order Logic

Second-order logic has a number of attractive features, in particular the strong expressive resources it offers, and the possibility of articulating categorical mathematical theories (such as arithmetic and analysis). But it also has its costs. Five major charges have been launched against second-order logic: (1) It is not axiomatizable; as opposed to first-order logic, it is inherently incomplete. (2) It also has several semantics, and there is no criterion to choose between them (Putnam, J Symbol Logic 45:464–482, 1980). Therefore, it is not clear how this logic should be interpreted. (3) Second-order logic also has strong ontological commitments: (a) it is ontologically committed to classes (Resnik, J Phil 85:75–87, 1988), and (b) according to Quine (Philosophy of logic, Prentice-Hall: Englewood Cliffs, 1970), it is nothing more than “set theory in sheep’s clothing”. (4) It is also not better than its first-order counterpart, in the following sense: if first-order logic does not characterize adequately mathematical systems, given the existence of non-isomorphic first-order interpretations, second-order logic does not characterize them either, given the existence of different interpretations of second-order theories (Melia, Analysis 55:127–134, 1995). (5) Finally, as opposed to what is claimed by defenders of second-order logic [such as Shapiro (J Symbol Logic 50:714–742, 1985)], this logic does not solve the problem of referential access to mathematical objects (Azzouni, Metaphysical myths, mathematical practice: the logic and epistemology of the exact sciences, Cambridge University Press, Cambridge, 1994). In this paper, I argue that the second-order theorist can solve each of these difficulties. As a result, second-order logic provides the benefits of a rich framework without the associated costs.     

Strong and Weak Readings in the Domain of Worlds: A Negative Polar Modal and Children’s Scope Assignment

This study investigates children’s interpretation of sentences with two logical operators: Dutch universal modal hoeven and negation (niet). In adult Dutch, hoeven is an NPI that necessarily scopes under negation, giving rise to a not > necessary reading. The findings from a hidden-object task with 5- and 6-year-old children showed that children’s performance is suggestive of an interpretation of sentences with hoeft niet in which the modal scopes over negation (necessary > not). This is in line with the Semantic Subset Principle that dictates that children should opt for the strongest possible reading in case of potential scope ambiguities. The full pattern of results, however, seems to be determined, in addition, by a particular strategy children use when facing uncertainty called Premature Closure.     

IF Modal Logic and Classical Negation

The present paper provides novel results on the model theory of Independence friendly modal logic. We concentrate on its particularly well-behaved fragment that was introduced in Tulenheimo and Sevenster (Advances in Modal Logic, 2006). Here we refer to this fragment as ‘Simple IF modal logic’ (IFML _s ). A model-theoretic criterion is presented which serves to tell when a formula of IFML _s is not equivalent to any formula of basic modal logic (ML). We generalize the notion of bisimulation familiar from ML; the resulting asymmetric simulation concept is used to prove that IFML _s is not closed under complementation. In fact we obtain a much stronger result: the only IFML _s formulas admitting their classical negation to be expressed in IFML _s itself are those whose truth-condition is in fact expressible in ML.     

On axiomatizations of public announcement logic

In the literature, different axiomatizations of Public Announcement Logic (PAL) have been proposed. Most of these axiomatizations share a “core set” of the so-called “reduction axioms”. In this paper, by designing non-standard Kripke semantics for the language of PAL, we show that the proof system based on this core set of axioms does not completely axiomatize PAL  without additional axioms and rules. In fact, many of the intuitive axioms and rules we took for granted could not be derived from the core set. Moreover, we also propose and advocate an alternative yet meaningful axiomatization of PAL  without the reduction axioms. The completeness is proved directly by a detour method using the canonical model where announcements are treated as merely labels for modalities as in normal modal logics. This new axiomatization and its completeness proof may sharpen our understanding of PAL  and can be adapted to other dynamic epistemic logics.     

Fer-ho Anaphora in Catalan: Semantic and Discourse Properties

This paper considers the anaphoric status of the pro-form fer-ho (do it) in Catalan [This paper contains some ideas included in Busquets (2005)]. I discuss some anaphoric properties of fer-ho as deep anaphora. I also compare these properties to those of other types of anaphora, like VPE and pseudogapping (pg). I show that its interpretation is strongly constrained by information and discourse structure.     

Jade,Imperial Identity,and Sumptuary Reform in J<Emphasis Type="SmallCaps">ia</Emphasis> Yi’s <Emphasis Type="Italic">Xin Shu</Emphasis>

The founding of the Han 漢 dynasty by a man of common birth, Liu Bang 劉邦, precipitated a new awareness that class boundaries had become more fluid than in prior generations. New fashions threatened the established social order as wealthy individuals pretended to status that they had not yet achieved. To respond to these concerns, Jia Yi 賈誼 (200–168 BCE) proposed a new sumptuary code regulating a range of luxury goods from apparel to accessories to ritual wares. This sumptuary system was designed to bolster the authority of the imperial throne and to distinguish members of the Han government and the imperial family from other elites. This essay will present an overview of Jia Yi’s sumptuary code and consider its relationship to earlier theories of ritual reform. It will argue that his system combined Lord Shang’s 商 theory of circumstantial advantage (shi 勢) with a new scheme of imperial publicity. The final section of this study assesses the implementation of Jia Yi’s recommendations by using jade as a case study. Based on recent archaeological evidence, I argue that Jia Yi’s ideas about jade in the context of his sumptuary guidelines have a contemporaneity with the revitalization of jade craft in the early Western Han.     

On the Costs of Nonclassical Logic

Solutions to semantic paradoxes often involve restrictions of classical logic for semantic vocabulary. In the paper we investigate the costs of these restrictions in a model case. In particular, we fix two systems of truth capturing the same conception of truth: (a variant) of the system KF of Feferman (The Journal of Symbolic Logic, 56, 1–49, 1991) formulated in classical logic, and (a variant of) the system PKF of Halbach and Horsten (The Journal of Symbolic Logic, 71, 677–712, 2006), formulated in basic De Morgan logic. The classical system is known to be much stronger than the nonclassical one. We assess the reasons for this asymmetry by showing that the truth theoretic principles of PKF cannot be blamed: PKF with induction restricted to non-semantic vocabulary coincides in fact with what the restricted version of KF proves true.     

John Venn's Hypothetical Infinite Frequentism and Logic

The goal of this paper is to provide a detailed reading of John Venn's Logic of Chance (1866) as a work of logic or, more specifically, as a specific portion of the general system of so-called ‘material’ logic developed in his Principles of Empirical or Inductive Logic (1889) and to discuss it against the background of his Boolean-inspired views on the connection between logic and mathematics. It is by means of this situating of Venn 1866 Venn, J. 1866. The Logic of Chance. An Essay on the Foundations and Province of the Theory of Probability. With Especial Reference to Its Application to Moral and Social Science, London: Macmillan. [Google Scholar] [The Logic of Chance. An Essay on the Foundations and Province of the Theory of Probability. With Especial Reference to Its Application to Moral and Social Science, London: Macmillan] within the entirety of his oeuvre that it becomes both possible to revisit and necessary to re-articulate its place in the history of the frequency interpretation of probability. For it is clear that if Venn's approach to logic not only allowed him to establish its foundations on the basis of a process of idealization and define it as consisting of so-called hypothetical infinite series, but also brought him to do so in strictly non-mathematical terms, he was able to anticipate much of the content (and problems) of his twentieth-century fellow frequentists.     

Wu Song’s Killing of His Sister-in-law: An Ethical Analysis

The Water Margin is a great Chinese classical novel; Wu Song’s 武松 killing of his sister-in-law, Pan Jinlian 潘金蓮, is one of the most popular episodes of the novel. It depicts Wu as the hero and defender of traditional values, and Pan as the adulterous woman. In contemporary discussion, there has been a dearth of ethical analyses regarding Wu’s killing of Pan. How should we judge the moral status of his action? Does the killing signify Wu Song’s ethical achievement or his ethical failure? What does the killing tell us about Wu’s character or his virtues? Does our appraisal of Wu’s action square with our modern belief regarding the treatment of women? I will examine these questions in the article.     

Natural Deduction for the Sheffer Stroke and Peirce’s Arrow (and any Other Truth-Functional Connective)

Methods available for the axiomatization of arbitrary finite-valued logics can be applied to obtain sound and complete intelim rules for all truth-functional connectives of classical logic including the Sheffer stroke (nand) and Peirce’s arrow (nor). The restriction to a single conclusion in standard systems of natural deduction requires the introduction of additional rules to make the resulting systems complete; these rules are nevertheless still simple and correspond straightforwardly to the classical absurdity rule. Omitting these rules results in systems for intuitionistic versions of the connectives in question.     

An Inconsistency-Adaptive Deontic Logic for Normative Conflicts

We present the inconsistency-adaptive deontic logic DP ^r , a nonmonotonic logic for dealing with conflicts between normative statements. On the one hand, this logic does not lead to explosion in view of normative conflicts such as O A?∧?O ～A, O A?∧?P ～A or even O A?∧?～O A. On the other hand, DP ^r still verifies all intuitively reliable inferences valid in Standard Deontic Logic (SDL). DP ^r interprets a given premise set ‘as normally as possible’ with respect to SDL. Whereas some SDL-rules are verified unconditionally by DP ^r , others are verified conditionally. The latter are applicable unless they rely on formulas that turn out to behave inconsistently in view of the premises. This dynamic process is mirrored by the proof theory of DP ^r .     

Probability logic and combining evidence

In Hailperin 1996 Hailperin, T. 1996. Sentential Probability Logic, Bethlehem and London: Lehigh University Press and Associated University Presses.  [Google Scholar], in addition to its formal development of Probability Logic, there are many sections devoted to historical origins, illustrative examples, and discussion of related work by other authors. Here selected portions of its formal treatment are summarized and then used as a basis for a probability logic treatment of combining evidence.     

Pure Modal Logic of Names and Tableau Systems

By a pure modal logic of names (PMLN) we mean a quantifier-free formulation of such a logic which includes not only traditional categorical, but also modal categorical sentences with modalities de re and which is an extension of Propositional Logic. For categorical sentences we use two interpretations: a “natural” one; and Johnson and Thomason’s interpretation, which is suitable for some reconstructions of Aristotelian modal syllogistic (Johnson in Notre Dame J Form Logic 30(2):271–284, 1989; Thomason in J Philos Logic 22(2):111–128, 1993 and J Philos Logic 26:129–141, 1997. In both cases we use Johnson-like models (1989). We also analyze different kinds of versions of PMLN, for both general and singular names. We present complete tableau systems for the different versions of PMLN. These systems enable us to present some decidability methods. It yields “strong decidability” in the following sense: for every inference starting with a finite set of premises (resp. every syllogism, every formula) we can specify a finite number of steps to check whether it is logically valid. This method gives the upper bound of the cardinality of models needed for the examination of the validity of a given inference (resp. syllogism, formula).     

Steps Toward a Computational Metaphysics

In this paper, the authors describe their initial investigations in computational metaphysics. Our method is to implement axiomatic metaphysics in an automated reasoning system. In this paper, we describe what we have discovered when the theory of abstract objects is implemented in prover9 (a first-order automated reasoning system which is the successor to otter). After reviewing the second-order, axiomatic theory of abstract objects, we show (1) how to represent a fragment of that theory in prover9’s first-order syntax, and (2) how prover9 then finds proofs of interesting theorems of metaphysics, such as that every possible world is maximal. We conclude the paper by discussing some issues for further research.     

At the threshold of knowledge

We explore consequences of the view that to know a proposition your rational credence in the proposition must exceed a certain threshold. In other words, to know something you must have evidence that makes rational a high credence in it. We relate such a threshold view to Dorr et al.’s (Philosophical Studies 170(2):277–287, 2014) argument against the principle they call fair coins: “If you know a coin won’t land tails, then you know it won’t be flipped.” They argue for rejecting fair coins because it leads to a pervasive skepticism about knowledge of the future. We argue that the threshold view of evidence and knowledge gives independent grounds to reject fair coins.