For various reasons several authors have enriched classical first order syntax by adding a predicate abstraction operator. “Conservatives” have done so without disturbing the syntax of the formal quantifiers but “revisionists” have argued that predicate abstraction motivates the universal quantifier’s re-classification from an expression that combines with a variable to yield a sentence from a sentence, to an expression that combines with a one-place predicate to yield a sentence. My main aim is to advance the (...) cause of predicate abstraction while cautioning against revisionism. In so doing, however, I shall pursue a secondary aim by conveying mixed blessings to those who hold the view that in the logical sense of “existence” some existing object is such as to exist contingently. Advocates of this view must concede Williamson’s recent contention that the domain of unrestricted objectual quantification could not have been narrower than it is actually, but predicate abstraction affords them some hope of accommodating this concession. (shrink)

Natural language semanticists have often found it useful to assume that all expressions denote sets of values. The approach is most prominent in the study of questions and prosodic focus, but also common in work on indefinites, disjunction, negative polarity, and scalar implicature. However, the most popular compositional implementation of this idea is known to face technical obstacles in the presence of object-language binding constructs, including, chiefly, lambda abstraction. The problem has been well-described on several occasions in the literature, (...) and in fact several solutions have been explored. This paper seeks to formalize the challenge of defining an indeterminate semantics for binding operators, and to formally establish the intuition that the challenge is in fact insurmountable. The primary benefit to this exercise is that it offers an abstract characterization of what it means to lift an operation from one semantic space to another, a notion which may be applied to domains having nothing to do with sets of alternatives. (shrink)

Frege's theorem says that second-order Peano arithmetic is interpretable in Hume's Principle and full impredicative comprehension. Hume's Principle is one example of an abstraction principle, while another paradigmatic example is Basic Law V from Frege's Grundgesetze. In this paper we study the strength of abstraction principles in the presence of predicative restrictions on the comprehension schema, and in particular we study a predicative Fregean theory which contains all the abstraction principles whose underlying equivalence relations can be proven (...) to be equivalence relations in a weak background second-order logic. We show that this predicative Fregean theory interprets second-order Peano arithmetic. (shrink)

Abstractionist programs in the philosophy of mathematics have focused on abstraction principles, taken as implicit definitions of the objects in the range of their operators. In second-order logic (SOL) with predicative comprehension, such principles are consistent but also (individually) mathematically weak. This paper, inspired by the work of Boolos (Proceedings of the Aristotelian Society 87, 137–151, 1986) and Zalta (Abstract Objects, vol. 160 of Synthese Library, 1983), examines explicit definitions of abstract objects. These axioms state that there is a (...) unique abstract encoding all concepts satisfying a given formula $$\phi (F)$$, with F a concept variable. Such a system is inconsistent in full SOL. It can be made consistent with several intricate tweaks, as Zalta has shown. Our approach in this article is simpler: we use a novel method to establish consistency in a restrictive version of predicative SOL. The resulting system, RPEAO, interprets first-order PA in extensional contexts, and has a natural extension delivering a peculiar interpretation of PA $$^2$$. (shrink)

Abstract.Given a class C of t-norm BL-algebras, one may wonder which is the complexity of the set Taut(C∀) of predicate formulas which are valid in any algebra in C. We first characterize the classes C for which Taut(C∀) is recursively axiomatizable, and we show that this is the case iff C only consists of the Gödel algebra on [0,1]. We then prove that in all cases except from a finite number Taut(C∀) is not even arithmetical. Finally we consider (...) class='Hi'>predicate monadic logics TautM(C∀) of classes C of t-norm BL-algebras, and we prove that (possibly with finitely many exceptions) they are undecidable. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Self-Predication and the "Third Man" Argument ROGER A. SHINER 1.1. IN COMMPm'mO on the 'Third Man' Argument (TMA), Proclus z produces the following line of thought. He argues that. if the relation of resemblance between Form and particular were symmetrical, the argument in question would be valid; the relation is not, however, symmetrical. Where a Form and particular are both alike, have the quality of likeness, the likeness of (...) that which is derived is not the same as the likeness of that from which it is derived, just because of the former's derivatory nature. Consequently, ~k tt~v ala01]x~ bttotoi3c0at ~re[vot~ (sc. xo~ ~Semv), abx~ 8~ xo6~ot~ oSr~xt, 'the sensible particulars resemble the Forms, but they are not in the same relation of resemblance to the particulars' (ibid., 913.29-30). That is to say, the bare relation of similarity may be symmetrical, but the relation of a similar copy to a similar original is not; the copy is 'a copy of' the original, but the original/s not 'a copy of' the copy. This way of defending the Theory of Forms against the TMA I shall for purposes of identification call the 'Ontological Defence'. It lays stress on the ontological difference between Form and particular in order to avoid the fatal implication that the relation between them is symmetrical. If we use the language of predication, the point is as follows: the TMA is valid, if the Forms have a given term predicated of them severally in exactly the same sense as it is predicated of their respective dependent particulars. This, however, is not the case, for the quality in virtue of which a given term is predicated of a given Form differs in ontological status from the quality in virtue of which that term is predicated of that Form's dependent particulars. Thus we see that underlying the Ontological x I am grateful for helpful comments on an earlier version of this paper to Professor Gilbert Ryle, Professor John F. Malcolm and two anonymous readers for this Journal. Since this paper went to press, the following relevant articles have appeared: G. Vlastos, "'Self Predication' in Plato's Later Period," Philosophical Review, LXXVIII (1969), 74-78, and "Plato's 'Third Man' Argument (Parm. 132al-b2): Text and Logic," Philosophical Quarterly, XIX (1969), 289-301; K. W. Rankin, "The Duplicity of Plato's Third Man," Mind, LXXVIII (1969), 178-197; T. O. Smith, "'The Theory of Forms, Relations and Infinite Regress," Dialogue, VIII (1969), 116-123. 2 Proelus, In Platonis Parmenidem, ed. V. Cousin, Paris, 1864; 912.27 ft. [3711 372 HISTORY OF PHILOSOPHY Defence is the following assumption--the fact that the Forms are self-predicating only causes problems for the Theory of Forms, if a given Form F and its corresponding particulars, a set of f things, have 'F' predicated of them univocally, are F in precisely the same sense. My thesis in this paper is that this latter assumption is false, it is, rather, quite sufficient for the Self-Predication Assumption (SPA) to be vicious, if 'F' is predicated of both Form and particular not equivocally. This is not the same as 'univocally', and furthermore it is not possible that Plato could have thought of the predication of a term of both Form and particular as equivocal, the only alternative left to him to escape the TMA. The SPA, therefore, is required for the Theory of Forms to make sense, and leads to an ineradicable defect in that Theory. 1.2. One minor terminological point must be noted before we proceed. There are two versions of the TMA in the Parmenides; the first runs from 131eS-132b2, and the second from 132c11-133a7. The second of these explicitly employs the language of copy-resemblance, 7mpdS~tytm ('paradigm'), 6lxoio)tta ('likeness'), ~tKao0~vat ('resemble') as an explication of ~t~0e~tq ('participation'): the first employs the lan~.,uage of cttx6 ~ xai3xa.., eIvat ('that by which things are...'), which, as Plato has already indicated (Phaedo 100d6-8), is deliberatety noncommittal as to the relation between Forms and particulars. I shall refer to the first argument as TMA(1) and the second... (shrink)

Theories of relations between language and conceptual development benefit from empirical evidence for concepts available in infancy, but such evidence is comparatively scarce. Here, we examine early representations of specific concepts, namely, sets of dynamic events corresponding either to predicates involving two variables with a reversible, asymmetric relation between them (such as the set of all events that correspond to a linguistic phrase like “a dog is pushing a car,”) or to comparatively simpler, one-variable predicates (such as the set of (...) events corresponding to a phrase like “a dog is jumping.”). We develop a non-linguistic, anticipatory eye-tracking task that can be administered to both infants and adults, and we use this task to gather evidence for the formation and use of such one-and two-place-predicate classes (which we refer to as event sortals) in 12–24-mo-old infants, and in adults with and without concurrent verbal prose shadowing. Using visually similar stimuli for both the simpler (one-place) and the more complex (reversible, asymmetric, two-place) concepts, we find that infants only show evidence for forming and generalizing one-place event sortals, and, while adults succeed with both kinds in the absence of verbal shadowing, shadowing hampers their ability to form and use the asymmetric two-place event sortals. In a subsequent experiment with adults, we find that if the shadowing material is grammatically impoverished, adults now succeed in forming and using both one- and two-place event sortals. We discuss implications of these results for theories of concept acquisition, and the role of language in this process. (shrink)

Abstract: In this brief commentary, I focus on two issues, first on Moltmann’s proposed Davidsonian event semantics for transitive verb attitude predicates, and second on the import of what she calls ‘the underspecification of content’ for the proper semantic interpretation of that-clauses. With respect to the first of these issues, I question the empirical justification of her proposed semantics, suggesting that she needs a syntactic rationale for her semantics. With respect to the second issue, I question whether, as she claims, (...) the that-clauses in transitive verb attitude predicates specify (or even partially specify) the truth or satisfaction conditions for the attributed attitudes. (shrink)

Conceptual realism begins with a conceptualist theory of the nexus of predication in our speech and mental acts, a theory that explains the unity of those acts in terms of their referential and predicable aspects. This theory also contains as an integral part an intensional realism based on predicate nominalization and a reflexive abstraction in which the intensional contents of our concepts are “object”-ified, and by which an analysis of predication with intensional verbs can be given. Through a (...) second nominalization of the common names that are part of conceptual realism’s theory of reference (via quantifier phrases), the theory also accounts for both plural reference and predication and mass noun reference and predication. Finally, a separate nexus of predication based on natural kinds and the natural properties and relations nomologically related to those natural kinds, is also an integral part of the framework of conceptual realism. (shrink)

This paper presents a formalization of first-order arithmetic characterizing the natural numbers as abstracta of the equinumerosity relation. The formalization turns on the interaction of a nonstandard cardinality quantifier with an abstraction operator assigning objects to predicates. The project draws its philosophical motivation from a nonreductionist conception of logicism, a deflationary view of abstraction, and an approach to formal arithmetic that emphasizes the cardinal properties of the natural numbers over the structural ones.

Those who believe that ordinary things have temporal as well as spatial parts must give an account of the truth conditions of temporally modified predications of the form ‘a is F at t ’ in terms of temporal parts. I will argue that the friend of temporal parts is committed to an account of temporal predication that is incompatible with the classical principle of predicate abstraction.

We argue that no attempt of reducing meaning to a systematic set of rules, according to which the role of linguistic expressions is to be normatively defined, can be abstracted from an irreducibly decisional compound. By comparing Lorenzen ’s project of building an Ortho-language and Brandom ’s inferentialist take on meaning, we distinguish two ways of acknowledging this fact, while claiming that Lorenzen ’s take is more genuinely constructive, insofar as choices be thought of as genuine features of constructions. It (...) brings into a new perspective the relation between dialogical constructivism and Brouwer ’s intuitionism. Finally we bring up a philosophical argument for the claim that interaction rules should be indexed on players and on their choices, when providing deontic bases to semantics. (shrink)

ABSTRACT Leśniewskian Ontology (LO) is a system in which the basic subject-predicate formula takes the form of a b and express one-argument predication, e.g. John is a student. In LO’s language, there is no many-argument form of predication given that would allow for the structural expression of, for example, the sentence John is Anne’s son. In this article, a simple and natural extension of LO is suggested to encompass many-argument predication. The system thus obtained corresponds to polyadic second-order logic.

This experimental study provides further support for a theory of meaning first put forward by Bar-Hillel and Carnap in 1953 and foreshadowed by Asimov in 1951. The theory is the Popperian notion that the meaningfulness of a proposition is its a priori falsity. We tested this theory in the first part of this paper by translating to logical form a long, tightly written, published text and computed the meaningfulness of each proposition using the a priori falsity measure. We then selected (...) the top propositions—by a priori falsity—and strung them together to form ad hoc abstracts and compared these abstracts with the published summary. The results were startling: translation to logical form, followed by application of the Asimovian idea and Bar-Hillel/Carnap mathematics as elaborated into an AI/NLP proposal in Fulda (1986, 1988), produced excellent abstracts, thereby providing a proof-of-concept that merely by knowing the logical form of long text passages, one can produce reasonable abstracts of them—without actually understanding the text. We here report on a second experiment analyzing, in the exact same manner, the correspondence that followed the published text of the first experiment. While the results of this confirming experiment are less startling, they nevertheless provide additional confidence in the promise of the technique. In other words, were the results of these two experiments to generalize, that would show that logical form captures much more semantics than has heretofore been considered likely. Far from (as is commonly supposed) being merely the syntactical rewrite of text into formal notation, translation to logical form, even when undertaken with almost no knowledge about the particular predicates, individual constants, or other objects referred to in that form, might capture the core of the meaning in some important sense. -/- Note: There is a key difference in the /style/ of the texts analyzed in I and II in this long paper, divided into two only for practical reasons—space; the first text analyzed is written in a tight mathematical style, while the second is written in the discursive style of philosophy. -/- Finally, the unusual (for me) length of this two-part paper is justified by its being data-driven. (shrink)

ABSTRACT Perry contends that an utterance of ‘It is raining’ must be assigned a location before being truth assessed. The location is famously argued to be an unarticulated constituent of the proposition an utterance of expresses. My paper examines this view from a pluri-propositionalist perspective. The sentence contains an impersonal pronoun, ‘it’ and the impersonal verb ‘to rain. I suggest that the utterance of semantically determines ‘to rain’, which is an event, and that that event is instantiated at a time (...) indicated by the tense at a location It is assumed that all event are located in space and time. (shrink)

Abstract The paper is concerned with the status of vague predicates. It is argued that they are for the most part ?classifiers?, which are covertly comparatives and name not monadic properties but relations. The Sorites Paradox, it is claimed, is thus defused and a verdict theory of vague predicates is presented. Our practice in using vague words is described and it is contended that in our use of these predicates we always have a permanent possibility of independent demarcation. Wittgenstein's picture (...) of the wall and the swamp is deployed to avoid the Transition Problem and it is argued against Fregeans that we need vague language for the advancement of knowledge and understanding. (shrink)

I want to motivate an account of what it is for an object to have a property, which may as well be called a deflationary view about properties. Such a view follows from a conception of predication I ground in the work of Donald Davidson, some of which remains unpublished. I claim that if we take seriously Davidson’s account of predication, by maintaining that sentences are the primary linguistic unit, we can define properties in terms of predicates. The aim of (...) this paper is twofold. First, I argue that this account is present in Davidson’s systematic treatment of the problem of predication. Second, I claim that this account is serviceable and economical, as it can accommodate a wide scope of properties and abstract objects without appealing to entities such as truthmakers or joints in nature. (shrink)

abstract: Aristotle claims that cross-sense comparisons—that is, comparisons with respect to homonymous predicates—are impossible. At the same time, he uses such comparisons in arguments that are fundamental to his philosophical project, such as when he claims that happiness is better than instrumental goods. In this paper, I discuss how this tension arises, and I explain why the cross-sense comparisons Aristotle uses are nevertheless possible. Using evidence from the Protrepticus, I claim Aristotle distinguishes two kinds of comparisons: comparisons of degrees of (...) a quality, and comparisons of the priority with which a quality applies. Only the latter admits of some cross-sense comparisons. (shrink)

We suggest a new framework for the Weyl-Feferman predicativist program by constructing a formal predicative set theory P ZF which resembles ZF , and is suitable for mechanization. The basic idea is that the predicatively acceptable instances of the comprehension schema are those which determine the collections they define in an absolute way, independent of the extension of the “surrounding universe”. The language of P ZF is type-free, and it reflects real mathematical practice in making an extensive use of statically (...) defined abstract set terms. Another important feature of P ZF is that its underlying logic is ancestral logic (i.e. the extension of first-order logic with a transitive closure operation). (shrink)

This essay introduces the concept of deictic abstraction , taking as a point of departure Husserl’s prototypical but insufficient description of the act of ideation in which a shade of color comes to givenness as an ideal object, i.e., a non-individual or abstract object, on the basis of a perceived individual object. This concept comprises not only color-ideation and ideations of universalities of the sensuous sphere , but all acts founded in perceptions in which ideal objects are directly referred (...) to by means of demonstrative expressions . This allows various types of deictic ideations or abstractions, corresponding to the degree of universality and the kind of ideal object concerned, to be brought to light through the analysis of particular cases of everyday acts of deictic abstraction, including deictic abstractions of predicative universals of the lowest level of universality ; deictic abstractions of non-predicative universals, especially of universal ways of doing something ; and deictic abstractions of individual styles of doing something . Deictic abstractions not only turn out to be constitutive for the formation of perceptual concepts of the lower and lowest levels, but also prove to be foundational for all kinds of learning in which schemata of actions are acquired from a model. (shrink)

The Quantified argument calculus (Quarc) has received a lot of attention recently as an interesting system of quantified logic which eschews the use of variables and unrestricted quantification, but nonetheless achieves results similar to the Predicate calculus (PC) by employing quantifiers applied directly to predicates instead. Despite this noted similarity, the issue of the relationship between Quarc and PC has so far not been definitively resolved. We address this question in the present paper, and then expand upon that result. (...) Utilizing recent developments in structural proof theory, we develop a G3-style sequent calculus for Quarc and briefly demonstrate its structural properties. We put these properties to use immediately to construct direct proofs of the meta-theoretical properties of the system. We then incorporate an abstract (and, as we shall see, logical) predicate into the system in a way that preserves all the structural properties. This allows us to identify a system of Quarc which is deductively equivalent to PC, and also yields a constructive method of demonstrating the Craig interpolation theorem (which speaks in favor of the aforementioned predicate being logical). We further generalize this extension to develop a bivalent system of Quarc with defining clauses that still maintains all the desirable properties of a good proof system. (shrink)

Non-reductive physicalists have long held that the special sciences offer explanations of some phenomena that are objectively superior to physical explanations. This explanatory “autonomy” has largely been based on the multiple realizability argument. Recently, in the face of the local reduction and disjunctive property responses to multiple realizability, some defenders of non-reductive physicalism have suggested that autonomy can be grounded merely in human cognitive limitations. In this paper, I argue that this is mistaken. By distinguishing between two kinds of (...) class='Hi'>abstraction I show that the greater explanatory relevance of some special science predicates (to certain explananda) is both non-anthropocentric and not solely based on considerations of multiple realizability. This shows that the explanatory autonomy of the special sciences is safe from the local reduction and disjunctive property strategies, even if they are successful responses to the multiple realizability argument. (shrink)

Feminists, critical race scholars, and other social‐justice theorists sometimes object to “abstraction” in liberal normative theory. Arguing that oppression affects individual agents in powerful yet subtle ways, they contend that allegedly abstract theories often reinforce oppressive power structures. Here I critically examine and ultimately reject Onora O'Neill's “abstraction without idealization” as a solution to this problem. Because O'Neill defines abstraction as simply the “bracketing of certain predicates,” her methodology fails to guide decisions about what to bracket and (...) what to include in the theory. Moreover, it may not be possible to abstract without also relying on some particular ideals. While abstraction is unavoidable, I conclude that it must be employed with greater attention to sociopolitical hierarchies and by using ideals that do not collude with structures of oppression. I discuss the work of Susan Babbitt and Elizabeth Anderson as examples of how nonoppressive ideals might be incorporated into normative theory. (shrink)

By the term nominalization I mean any process which transforms a predicate or predicate phrase into a noun or noun phrase, e.g. feminine is transformed into feminity. I call these derivative nouns abstract singular terms. Our aim is to provide a model-theoretic interpretation for a formal language which admits the occurrence of such abstract singular terms.

Abstraction principles and grounding can be combined in a natural way Modality: metaphysics, logic, and epistemology, Oxford University Press, Oxford, pp 109–136, 2010; Schwartzkopff in Grazer philosophische studien 82:353–373, 2011). However, some ground-theoretic abstraction principles entail that there are circles of partial ground :775–801, 2017). I call this problem auto-abstraction. In this paper I sketch a solution. Sections 1 and 2 are introductory. In Sect. 3 I start comparing different solutions to the problem. In Sect. 4 I (...) contend that the thesis that the right-hand side of an abstraction principle is prior to its left-hand side motivates an independence constraint, and that this constraint leads to predicative restrictions on the acceptable instances of ground-theoretic abstraction principles. In Sect. 5 I argue that auto-abstraction is acceptable unless the left-hand side is essentially grounded by the right-hand side. In Sect. 6 I highlight several parallelisms between auto-abstraction and the puzzles of ground. I finally compare my solution with the strategies listed in Sect. 3. (shrink)

In an attempt to accommodate natural language phenomena involving nominalization and self-application, various researchers in formal semantics have proposed abandoning the hierarchical type system which Montague inherited from Russell, in favour of more flexible type regimes. We briefly review the main extant proposals, and then develop a new approach, based semantically on Aczel's notion of Frege structure, which implements a version ofsubsumption polymorphism. Nominalization is achieved by virtue of the fact that the types of predicative and propositional complements are contained (...) in the type of individuals. Russell's paradox is avoided by placing a type-constraint on lambda-abstraction, rather than by restricting comprehension. (shrink)

Kripke’s notion of groundedness plays a central role in many responses to the semantic paradoxes. Can the notion of groundedness be brought to bear on the paradoxes that arise in connection with abstraction principles? We explore a version of grounded abstraction whereby term models are built up in a ‘grounded’ manner. The results are mixed. Our method solves a problem concerning circularity and yields a ‘grounded’ model for the predicative theory based on Frege’s Basic Law V. However, the (...) method is poorly behaved unless the background second-order logic is predicative. (shrink)

Abstract The notion of necessary new terms (predicates) is proposed. It is shown that necessary new predicates in first?order logic must be directly, recursively defined. I present a first?order inductive learning algorithm that introduces new necessary predicates to model scientific revolution in which a new language is adopted. I demonstrate that my learning system can learn a genetic theory with theoretical terms which, after being induced by my system, can be interpreted as either types of genetic properties (dominant or recessive) (...) or genes, depending on the representation of the hypotheses of the same theoretical terms. (shrink)

Distinct from nominal metaphors, predicate metaphors entail metaphorical abstraction from concrete verbs, which generally involve more action and stronger motor simulation than nouns. It remains unclear whether and how the concrete, embodied aspects of verbs are connected with abstract, disembodied thinking in the brains of L2 learners. Since English predicate metaphors are unfamiliar to Chinese L2 learners, the study of embodiment effect on English predicate metaphor processing may provide new evidence for embodied cognition and categorization models (...) that remain controversial, and offer practical insights into L2 metaphor processing and pedagogy. Hence, we aim to investigate whether the embodiment of verbs, via the activation of sensorimotor information, influences two groups of L2 learners during their comprehension of conventional and novel predicate metaphors. The results show a significant effect of embodiment: a stronger facilitation for novel predicate metaphors in both higher-level and lower-level groups, and a weaker facilitation for conventional predicate metaphors in the lower-level group. The findings demonstrate preliminary evidence for a graded effect of embodiment on predicate metaphors processing, modulated by L2 proficiency and metaphor novelty. The study supports a hybrid view of embodied cognition and reveals that sensorimotor aspects of verbs may be the intermediate entity involved in the indirect categorization. (shrink)

Recent work on mood choice considers fine-grained semantic differences among desire predicates (notably, ‘want’ and ‘hope’) and their consequences for the distribution of indicative and subjunctive complement clauses. In that vein, this paper takes a close look at ‘intend’. I show that cross-linguistically, ‘intend’ accepts nonfinite and subjunctive complements and rejects indicative complements. This fact poses difficulties for recent approaches to mood choice. Toward a solution, a broad aim of this paper is to argue that—while ‘intend’ is loosely in the (...) family of desire predicates—it differs from ‘want’ and ‘hope’ in that it has a causative component, and this is relevant to its mood choice behavior, given that causative predicates also systematically reject indicative complements. More concretely, my analysis has three ingredients: (i) following related proposals in philosophy, intention reports have causally self-referential content; (ii) encoding causal self-reference requires abstraction over the complement clause’s eventuality argument; and (iii) nonfinite and subjunctive clauses enable such abstraction but indicative clauses do not. Aside from causative predicates, independent support for the proposal comes from the syntax of belief-/intention-hybrid attitude predicates like ‘decide’ and ‘convince’, anankastic conditional antecedents, aspectual predicates, and memory and perception reports. Synthesizing this result with that of previous literature, the emergent generalization is that subjunctive mood occurs in attitude reports that involve either comparison or eventuality abstraction. Toward a unified theory of mood choice, I suggest that both comparison and eventuality abstraction represent departures from the clausal semantics of unembedded assertions and consequently that subjunctive mood signals such a departure. (shrink)

Mandarin Chinese, along with Japanese, Yorùbá, Mòoré, and Samoan, has been argued to lack ‘degree abstraction’, a configuration at LF involving lambda abstraction over a degree variable. These languages are claimed to have a negative setting for a hypothesized ‘Degree Abstraction Parameter’. Recent work, however, has argued for degree abstraction in Japanese and Yorùbá, and degree abstraction has been detected in a number of additional languages. Could it in fact be universal? Here, we focus on (...) the case of Mandarin, and argue that Mandarin has degree abstraction too. We offer three arguments in favor of degree abstraction in Mandarin, based on attributive comparatives, comparatives with embedded predicates, and scope interactions with modals. We also rebut prior arguments for the lack of degree abstraction in Mandarin, considering degree questions, measure phrases, and negative island effects. Taken together, these results show that degree abstraction is not a parameter along which Mandarin and English vary, and suggest rather that degree abstraction may be universally available. (shrink)

Theabstract variable binding calculus (VB-calculus) provides a formal frame-work encompassing such diverse variable-binding phenomena as lambda abstraction, Riemann integration, existential and universal quantification (in both classical and nonclassical logic), and various notions of generalized quantification that have been studied in abstract model theory. All axioms of the VB-calculus are in the form of equations, but like the lambda calculus it is not a true equational theory since substitution of terms for variables is restricted. A similar problem with the standard (...) formalism of the first-order predicate logic led to the development of the theory of cylindric and polyadic Boolean algebras. We take the same course here and introduce the variety of polyadic VB-algebras as a pure equational form of the VB-calculus. In one of the main results of the paper we show that every locally finite polyadic VB-algebra of infinite dimension is isomorphic to a functional polyadic VB-algebra that is obtained from a model of the VB-calculus by a natural coordinatization process. This theorem is a generalization of the functional representation theorem for polyadic Boolean algebras given by P. Halmos. As an application of this theorem we present a strong completeness theorem for the VB-calculus. More precisely, we prove that, for every VB-theory T that is obtained by adjoining new equations to the axioms of the VB-calculus, there exists a model D such that T s=t iff D s=t. This result specializes to a completeness theorem for a number of familiar systems that can be formalized as VB-calculi. For example, the lambda calculus, the classical first-order predicate calculus, the theory of the generalized quantifierexists uncountably many and a fragment of Riemann integration. (shrink)

The paper describes a decidable class of verification problems expressed in first order timed logic. To specify programs we useState Machines. It is known that Abstract State Machines and first order timed logic are two very powerful formalisms apt to represent verification problems for timed distributed systems. However, the general verification problem represented in this way is undecidable. Prior, some decidable classes of verification problems were described in semantical properties that are in their turn undecidable. The decidable class of the (...) present paper is described in syntactical terms. Though it admits no functions, only predicates, it is of practical interest and we give an example illustrating possible applications. (shrink)

The purpose of this paper is to outline an alternative approach to introductory logic courses. Traditional logic courses usually focus on the method of natural deduction or introduce predicate calculus as a system. These approaches complicate the process of learning different techniques for dealing with categorical and hypothetical syllogisms such as alternate notations or alternate forms of analyzing syllogisms. The author's approach takes up observations made by Dijkstrata and assimilates them into a reasoning process based on modified notations. The (...) author's model adopts a notation that addresses the essentials of a problem while remaining easily manipulated to serve other analytic frameworks. The author also discusses the pedagogical benefits of incorporating the model into introductory logic classes for topics ranging from syllogisms to predicate calculus. Since this method emphasizes the development of a clear and manipulable notation, students can worry less about issues of translation, can spend more energy solving problems in the terms in which they are expressed, and are better able to think in abstract terms. (shrink)

The logical formalization of sentences containing adverbial modification, e.g., “John walks slowly”, remains demanding. According to Bennett (1988, 2002), as facts are exemplifications of properties, they are abstract and not basic items in the universe, whereas events constitute a special subcategory of facts. The names of facts are complete, but the names of events are incomplete. This paper aims to provide an alternative logical formalization of facts and events, reflecting Bennett’s metaphysical and semantic insights better than his formalization based on (...) Davidson (1967a, 1985) and Parsons (1990). After discussing Bennett’s theory, I will develop my logical formalization based on Clark’s (1970) predicate modifiers theories. Using the abstraction operator will allow different modifiers to have the same scope and simultaneously guarantee the unicity of the fact or event. There is no quantification over facts or events. The resulting formalization will have advantages in coping with verb dependent and nonstandard adverbs and allowing for easier conjunctive formalization. The semantic insight that the names of facts are complete describers, and the names of events are not will manifest in the distinct formalization of events. Multigrade predicates will enable further logical entailments. As a result, we get a formalization fitting into Bennett’s theory. (shrink)

Abstract:Bipolar or related disorders (BoRD) present unique practical and existential problems for people who live with them. All agents experience changes in the things they care about over time. However people living with BoRD face drastic shifts in what seems valuable to them, which upset their longitudinal values (if, indeed, any stable longitudinal values are available in the first place). Navigating these evaluative high seas presents agents living with BoRD with a distinctive existential question, not shared by those on calmer (...) waters. I draw out two contrasting ways in which someone living with BoRD might seek to support their self-determination in these circumstances, by crafting appropriate self care and support regimes. The first strategy involves managing one’s affective episodes so that they do not interrupt one’s plans and long-term agency over time. The second involves a regimen that allows one the greatest degree of freedom in adapting to changes in one’s experiences of value. What distinguishes these sorts of self-determination is the scope at which they are predicated. Although both sorts of self-determination allow an individual to rule themselves, they alter the overall shape of one’s autonomy in quite different ways. (shrink)

abstract: In the Categories, Aristotle relies on the truth of claims like ‘Socrates is an individual’ and ‘human is a species,’ but it is not clear how terms like ‘species’ and ‘individual’ fit into the framework of the Categories. Do these terms introduce substances or accidents? When we truly apply them to a subject, is the predication we express essential or accidental? These questions puzzled ancient commentators on the Categories but have largely been neglected in modern scholarship. My central contention (...) is that these terms correspond to entities outside of the categories. An upshot of my account is that sentences like ‘human is a species’ and ‘Socrates is an individual’ express neither essential nor accidental predications. (shrink)

In "Abstracts from Logical Form I/II," it was stated in the abstract that it remained necessary to put the pilot experiments into a "comprehensive theory." It is suggested here that the comprehensive theory is nothing other than classical logic modestly extended to include higher-order predicates, functions, and epistemic predicates, as well as a quantitative quantifier to deal with cases other than "all" (taken literally) or "some" in the sense of at least one. It is further suggested that up to a (...) point the theory should generalize across languages, and that logical form is very powerful. (shrink)

An important lesson that philosophy can learn from the Turing Test and computer science more generally concerns the careful use of the method of Levels of Abstraction (LoA). In this paper, the method is first briefly summarised. The constituents of the method are “observables”, collected together and moderated by predicates restraining their “behaviour”. The resulting collection of sets of observables is called a “gradient of abstractions” and it formalises the minimum consistency conditions that the chosen abstractions must satisfy. Two (...) useful kinds of gradient of abstraction – disjoint and nested – are identified. It is then argued that in any discrete (as distinct from analogue) domain of discourse, a complex phenomenon may be explicated in terms of simple approximations organised together in a gradient of abstractions. Thus, the method replaces, for discrete disciplines, the differential and integral calculus, which form the basis for understanding the complex analogue phenomena of science and engineering. The result formalises an approach that is rather common in computer science but has hitherto found little application in philosophy. So the philosophical value of the method is demonstrated by showing how making the LoA of discourse explicit can be fruitful for phenomenological and conceptual analysis. To this end, the method is applied to the Turing Test, the concept of agenthood, the definition of emergence, the notion of artificial life, quantum observation and decidable observation. It is hoped that this treatment will promote the use of the method in certain areas of the humanities and especially in philosophy. (shrink)

Persian possesses a number of stative complex predicates with _dâshtan_ ‘to have’ that express certain kinds of mental state. I propose that these _possessive experiencer complex predicates_ be given a formal semantic treatment involving possession of a portion of an abstract quality by an individual, as in the analysis of property concept lexemes due to Francez and Koontz-Garboden (Language 91(3):533–563, 2015 ; Natural Language and Linguistic Theory 34:93–106, 2016 ; Semantics and morphosyntactic variation: Qualities and the grammar of property concepts, (...) Oxford University Press, 2017 ). Augmented with an analysis of prepositional phrases introducing the target of the mental state and an approach to gradability in terms of measure functions (Wellwood in Measuring predicates, PhD dissertation, University of Maryland, College Park, 2014 ), the analysis explains various properties of possessive experiencer complex predicates, including the behavior of target phrases, the ability of the non-verbal element to be modified by a range of adjectives, the direct participation of the non-verbal element in comparative constructions, and the ability of degree expressions to modify both the non-verbal element and the VP containing the complex predicate. Theoretically, the analysis ties transitive mental state expressions to the grammar and semantics of property concept sentences, which are expressed via possessive morphosyntax cross-linguistically, and connects with syntactic proposals that independently argue for a universal underlyingly possessive morphosyntax for mental state predicates (Noonan in Case and syntactic geometry, PhD dissertation, McGill University, 1992 ; Hale and Keyser in Prolegomenon to a theory of argument structure, MIT Press, 2002 ). The work here also motivates modifications to Francez and Koontz-Garboden’s original proposal, and opens new questions in the original empirical domain of the analysis of possessive predicating strategies for the expression of property concept sentences. (shrink)

Finitely algebraizable deductive systems were introduced by Blok and Pigozzi to capture the essential properties of those deductive systems that are very tightly connected to quasivarieties of universal algebras. They include the equivalential logics of Czelakowski. Based on Blok and Pigozzi’s work, Herrmann defined algebraizable deductive systems. These are the equivalential deductive systems that are also truth-equational, in the sense that the truth predicate of the class of their reduced matrix models is explicitly definable by some set of unary (...) equations. Raftery undertook the task of characterizing the property of truth-equationality for arbitrary deductive systems. In this paper, following Raftery, we extend the notion of truth-equationality for logics formalized as $\pi$-institutions and abstract several of the results that hold for deductive systems in this more general categorical context. (shrink)

Peirce’s beta graphs are roughly equivalent to our first-order predicate logic. However, Bellucci and Pietarinen have recently argued that the beta graphs are not well-equipped to handle asymmetric relative terms. I survey four proposed solutions to the problem and find them all wanting. I offer a fifth solution according to which Peirce’s beta graphs function at three different levels of abstractness from natural language. I diagnose the problem of asymmetric relative terms as arising when we transition from the first (...) to the second level of abstractness. Making grammatical information encoded in natural language explicit at the first level of abstractness and interpreting graphs at the second level of abstractness as shorn of grammatical information resolves the problem. The solution is both well-motivated by Peirce’s own commitments and increases the expressiveness of the beta graphs. (shrink)

In the opening chapter of Subject and Predicate in Logic and Grammar, [1] Professor Strawson develops an explanation of the subjectpredicate distinction on the basis of a supposedly more fundamental distinction or contrast between, on the one hand, spatio-temporal particulars and, on the other, general concepts applicable to such particulars. At a basic level, he argues, these contrasted items occupy a central position in our thought about the world. They form the constituents of a fundamental type of judgment about (...) the worldcomprising judgments to the effect that a certain spatio-temporal particular (pair, triple, etc., of such particulars) exemplifies a certain general concept. The basic class of subject-predicate sentences-the class for which the explanation is initially presented-comprises, Strawson claims, those sentences which are apt for the expression of judgments of this fundamental type. With respect to sentences of this basic class, a subject is, in effect, defined to be an expression which serves to specify the spatio-temporal particular (or one of the spatiotemporal particulars) in question in whatever judgment the sentence expresses, and a predicate is defined to be an expression which serves to specify the general concept in question. This explanation is subsequently generalized to cover subject-predicate sentences outside the basic class, such as sentences expressing judgments about abstract entities. I shall not be concerned with this part of Strawson's explanation. One advantage Strawson claims for this explanation is that it brings out that, and how, the subject-predicate distinction reflects certain fundamental features of our thought about the world. Another, not unconnected, advantage he claims for it is that it enables us to explain certain 'purely formal or syntactical' differences between subjects and predicates-such differences as that a predicate has always a fixed number of argument-places, whereas there is nothing analogous to this with singular terms (subject-expressions), and that predicates take the form of verb-phrases, subjects the form of noun phrases. Such purely formal or syntactical differences, he claims, themselves stand in need of explanation, and so cannot provide the materials for an adequate explanation of the subject-predicate distinction. Perhaps Strawson does not explicitly assert that no purely formal or syntactical characterization of the subject-predicate distinction could constitute an adequate explanation of it; but he leaves us in little doubt that this is his view of the matter. (shrink)

By language we understand a lower predicate calculus with identity and (perhaps) relation and function symbols. It is convenient to allow for more than one sort of variable. Now each individual constant (if there are any) is of a specified sort, the formal expressions R(t1, … tn), f(t1,…, tn) are well formed only if the terms t1, …, tn are of specified sorts determined by the relation symbol R and the function symbol f, and the term f(t1, …, tn) (...) (if well formed) is of a sort determined by f. We admit s = t as well formed, no matter what the sorts of s and t. (shrink)

Recent studies in semantics of modal and superintuitionistic predicate logics provided many examples of incompleteness, especially for Kripke semantics. So there is a problem: to find an appropriate possible- world semantics which is equivalent to Kripke semantics at the propositional level and which is strong enough to prove general completeness results. The present paper introduces a new semantics of Kripke metaframes' generalizing some earlier notions. The main innovation is in considering "n"-tuples of individuals as abstract "n"-dimensional vectors', together with (...) some transformations of these vectors. Soundness of the semantics is proved to be equivalent to some non- logical properties of metaframes; and thus we describe the maximal semantics of Kripke- type. (shrink)

A proposition containing an adjectival predicate has customarily been described as one which predicates some quality of its subject; thus "William is white" is said to attribute whiteness to William. The concrete adjectival form in such a situation was sometimes said (e. g. by Boethius) to be derived from the corresponding abstract (as "white" from whiteness, "just" from justice, and so on), thus enabling the subject in question to be "denominated" from the abstract by means of the concrete. The (...) quality is then said to be predicated of its subject in a denominative or paronymous fashion. Involved here also is the shaky assumption that adjectives may indeed be distinguished from substantives on the basis of the former's correlation with available abstract forms which the latter lack, but this need not concern us here. (shrink)