The ontology of intentionality I: the dependence ontological account of order: mediate and immediate moments and pieces of dependent and independent objects

This is the first of three essays which use Edmund Husserl's dependence ontology to formulate a non-Diodorean and non-Kantian temporal semantics for two-valued, first-order predicate modal languages suitable for expressing ontologies of experience (like physics and cognitive science). This essay's primary desideratum is to formulate an adequate dependence-ontological account of order. To do so it uses primitive (proper) part and (weak) foundation relations to formulate seven axioms and 28 definitions as a basis for Husserl's dependence ontological theory of relating moments. The essay distinguishes between dependence v. independence, pieces v. moments, mediate v. immediate pieces and moments, maximal v. non-maximal pieces, founded v. unfounded qualities, integrative v. disintegrative dependence, and defines the concepts of the completion of an object, the adumbrational equivalence relation of objects, moments of unity which unify objects, and relating moments which relate objects. The eight theorems [CUT90]-[CUT97] show that relating moments of unity provide an adequate account of order in terms of primitive (proper) part and (weak) foundation relations.     

Two-Valued Logics of Intentionality: Temporality,Truth, Modality,and Identity

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.     

How to Lewis a Kripke–Hintikka

It has been argued that a combination of game-theoretic semantics and independence-friendly (IF) languages can provide a novel approach to the conceptual foundations of mathematics and the sciences. I introduce and motivate an IF first-order modal language endowed with a game-theoretic semantics of perfect information. The resulting interpretive independence-friendly logic (IIF) allows to formulate some basic model-theoretic notions that are inexpressible in the ordinary quantified modal logic. Moreover, I argue that some key concepts of Kripke’s new theory of reference are adequately modeled within IIF. Finally, I compare the logic IIF to David Lewis counterpart theory, drawing some morals concerning the interrelation between metaphysical and semantic issues in possible-world semantics.     

Neighbourhood Semantics for Quantified Relevant Logics

The Mares-Goldblatt semantics for quantified relevant logics have been developed for first-order extensions of R, and a range of other relevant logics and modal extensions thereof. All such work has taken place in the the ternary relation semantic framework, most famously developed by Sylvan (née Routley) and Meyer. In this paper, the Mares-Goldblatt technique for the interpretation of quantifiers is adapted to the more general neighbourhood semantic framework, developed by Sylvan, Meyer, and, more recently, Goble. This more algebraic semantics allows one to characterise a still wider range of logics, and provides the grist for some new results. To showcase this, we show, using some non-augmented models, that some quantified relevant logics are not conservatively extended by connectives the addition of which do conservatively extend the associated propositional logics, namely fusion and the dual implication. We close by proposing some further uses to which the neighbourhood Mares-Goldblatt semantics may be put.

     

The Logic and Meaning of Plurals. Part II

In this sequel to “The logic and meaning of plurals. Part I”, I continue to present an account of logic and language that acknowledges limitations of singular constructions of natural languages and recognizes plural constructions as their peers. To this end, I present a non-reductive account of plural constructions that results from the conception of plurals as devices for talking about the many. In this paper, I give an informal semantics of plurals, formulate a formal characterization of truth for the regimented languages that results from augmenting elementary languages with refinements of basic plural constructions of natural languages, and account for the logic of plural constructions by characterizing the logic of those regimented languages.

Propositional attitudes and formal ontology

This paper develops — within an axiomatic theory of properties, relations, and propositions which accords them well-defined existence and identity conditions — a sententialist-functionalist account of belief as a symbolically mediated relation to a special kind of propositional entity, theproxy-encoding abstract proposition. It is then shown how, in terms of this account, the truth conditions of English belief reports may be captured in a formally precise and empirically adequate way that accords genuinely semantic status to familiar opacity data.I am deeply indebted to Edward Zalta for many helpful comments and suggestions.     

First-order indefinite and uniform neighbourhood semantics

The main purpose of this paper is to define and study a particular variety of Montague-Scott neighborhood semantics for modal propositional logic. We call this variety the first-order neighborhood semantics because it consists of the neighborhood frames whose neighborhood operations are, in a certain sense, first-order definable. The paper consists of two parts. In Part I we begin by presenting a family of modal systems. We recall the Montague-Scott semantics and apply it to some of our systems that have hitherto be uncharacterized. Then, we define the notion of a first-order indefinite semantics, along with the more specific notion of a first-order uniform semantics, the latter containing as special cases the possible world semantics of Kripke. In Part II we prove consistency and completeness for a broad range of the systems considered, with respect to the first-order indefinite semantics, and for a selected list of systems, with respect to the first-order uniform semantics. The completeness proofs are algebraic in character and make essential use of the finite model property. A by-product of our investigations is a result relating provability in S-systems and provability in T-systems, which generalizes a known theorem relating provability in the systems S 2° and C 2.The author would like to thank Prof. Nuel D. Belnap of the University of Pittsburg for many indispensable contributions to earlier versions of this work. The author also thanks the referee for several helpful comments and corrections.     

Inexact Knowledge with Introspection

Standard Kripke models are inadequate to model situations of inexact knowledge with introspection, since positive and negative introspection force the relation of epistemic indiscernibility to be transitive and euclidean. Correlatively, Williamson’s margin for error semantics for inexact knowledge invalidates axioms 4 and 5. We present a new semantics for modal logic which is shown to be complete for K45, without constraining the accessibility relation to be transitive or euclidean. The semantics corresponds to a system of modular knowledge, in which iterated modalities and simple modalities are not on a par. We show how the semantics helps to solve Williamson’s luminosity paradox, and argue that it corresponds to an integrated model of perceptual and introspective knowledge that is psychologically more plausible than the one defended by Williamson. We formulate a generalized version of the semantics, called token semantics, in which modalities are iteration-sensitive up to degree n and insensitive beyond n. The multi-agent version of the semantics yields a resource-sensitive logic with implications for the representation of common knowledge in situations of bounded rationality.     

Associative Substitutional Semantics and Quantified Modal Logic

The paper presents an alternative substitutional semantics for first-order modal logic which, in contrast to traditional substitutional (or truth-value) semantics, allows for a fine-grained explanation of the semantical behavior of the terms from which atomic formulae are composed. In contrast to denotational semantics, which is inherently reference-guided, this semantics supports a non-referential conception of modal truth and does not give rise to the problems which pertain to the philosophical interpretation of objectual domains (concerning, e.g., possibilia or trans-world identity). The paper also proposes the notion of modality de nomine as an alternative to the denotational notion of modality de re.     

Framework for formal ontology

The discussions which follow rest on a distinction, first expounded by Husserl, between formal logic and formal ontology. The former concerns itself with (formal) meaning-structures; the latter with formal structures amongst objects and their parts. The paper attempts to show how, when formal ontological considerations are brought into play, contemporary extensionalist theories of part and whole, and above all the mereology of Leniewski, can be generalised to embrace not only relations between concrete objects and object-pieces, but also relations between what we shall call dependent parts or moments. A two-dimensional formal language is canvassed for the resultant ontological theory, a language which owes more to the tradition of Euler, Boole and Venn than to the quantifier-centred languages which have predominated amongst analytic philosophers since the time of Frege and Russell. Analytic philosophical arguments against moments, and against the entire project of a formal ontology, are considered and rejected. The paper concludes with a brief account of some applications of the theory presented.     

Towards a semantics for the artifactual theory of fiction and beyond

In her book Fiction and Metaphysics (1999) Amie Thomasson, influenced by the work of Roman Ingarden, develops a phenomenological approach to fictional entities in order to explain how non-fictional entities can be referred to intrafictionally and transfictionally, for example in the context of literary interpretation. As our starting point we take Thomasson’s realist theory of literary fictional objects, according to which such objects actually exist, albeit as abstract and artifactual entities. Thomasson’s approach relies heavily on the notion of ontological dependence, but its precise semantics has not yet been developed. Moreover, the modal approach to the notion of ontological dependence underlying the Artifactual Theory has recently been contested by several scholars. The main aims of this paper are (i) to develop a semantic approach to the notion of ontological dependence in the context of the Artifactual Theory of fiction, and in so doing bridge a number of philosophical and logical gaps; (ii) to generalize Thomasson’s categorial theory of ontological dependence by reconstructing ontological categories of entities purely in terms of different structures of ontological dependence, rather than in terms of the basic kinds of entities the categorical entities depend on.     

Weyl’s Appropriation of Husserl’s and Poincar“s Thought

This article locates Weyl's philosophy of mathematics and its relationship to his philosophy of science within the epistemological and ontological framework of Husserl's phenomenology as expressed in the Logical Investigations and Ideas. This interpretation permits a unified reading of Weyl's scattered philosophical comments in The Continuum and Space-Time-Matter. But the article also indicates that Weyl employed Poincaré's predicativist concerns to modify Husserl's semantics and trim Husserl's ontology. Using Poincaré's razor to shave Husserl's beard leads to limitations on the least upper bound theorem in the foundations of analysis and Dirichlet's principle in the foundations of physics. Finally, the article opens the possibility of reading Weyl as a systematic thinker, that he follows Husserl's so-called transcendental turn in the Ideas. This permits an even more unified reading of Weyl's scattered philosophical comments.     

On the ontology of branching quantifiers

Conclusion Still, some may still want to say it. If so, my replies may gain nothing better than a stalemate against such persistence, though I can hope that earlier revelations will discourage others from persisting. But two replies are possible. Both come down, one circuitously, to an issue with us from the beginning: whether the language of the right side of (10) is suspect. For if (10) is to support instances for (6) which are about objects, that clause must itself be about objects. (These would be ones assigned by variants of I it mentions to constants it mentions.) Yet Barwise and I would call it implicitly about functions. Ironically, the discussion surrounding (10), hoping to settle that issue, could only do so if the issue were already settled, revealing decisively the ontology of (10). If irritating, this is also inevitable, given the Tarskian spirit of the ensivioned semantics for QLB.A second reply begins by noting that a QLB semantics which implies (10) cannot simply be assumed. Even a QL semantics augmented to imply (11) is not trivial to frame, as I will let readers confirm. On the QLB project, Barwise comments thus:It is not possible to explain the meaning of an essential use of branching quantification ... inductively, by treating one quantifier at a time in a first-order fashion. Some use of higher-type abstract objects is essential. (BQE75)But believers in a modest ontology for QLB can claim a non sequitur here. For (10), as they read it, succeeds without mentioning abstract objects, though not in the one quantifier at a time way which Barwise rightly finds impossible. This is not yet to say the same about a QLB semantics implying (10). But that too can be said, as it happens, with as good a conscience as with (10). A suitable treatment can begin by somehow linearizing non QL sentences like (6). Still assuming prefixes whose rows each consist, speaking loosely, of n universal quantifiers followed by a single existential, we could simply line up these rows in any order, with unique deconcatenatability being assured. From then on, it gets both tedious and complex. A full syntax is essential, and some surprising categories arose in mine. I will spare readers all details, except to say that one can indeed treat one quantifier at a time, if not in first-order fashion: schematic rules for treating n quantifiers at once can be eschewed. Unsurprisingly, heavy use is made of the depending only on idiom seen in (10). Nor does it surprise me that this semantics can succeed, with that idiom available. Roughly, if open talk about functions works in a semantics for QLF, what I read as implicit talk about them should work in a semantics for QLB.So even from a bare sketch, we can see that this new semantics settles nothing. It just leads back to the stalemate. Barwise and I will read crucial clauses as talking implicitly about functions, but this general charge against an idiom is equally deniable wherever the idiom occurs: in a reading for (6), say, or in the metalanguage can hardly repress a motherhood slogan: better dead than obscurely read. But that just denies the denial, unhelpfully. A better comment is the reminder that the claim stays alive only in a form which no one has ever imagined for it. The QLB quantifiers that cannot be shown not to range over objects are not the items anyone would ever have pointed to in illustrating the unkillable claim.

---

     

Line-based affine reasoning in Euclidean plane

We consider the binary relations of parallelism and convergence between lines in a 2-dimensional affine space. Associating with parallelism and convergence the binary predicates P and C and the modal connectives [P] and [C], we consider a first-order theory based on these predicates and a modal logic based on these modal connectives. We investigate the axiomatization/completeness and the decidability/complexity of this first-order theory and this modal logic.     

Prescribed spatial prepositions influence how we think about time

Prepositions combine with nouns flexibly when describing concrete locative relations (e.g. at/on/in the school) but are rigidly prescribed when paired with abstract concepts (e.g. at risk; on Wednesday; in trouble). In the former case they do linguistic work based on their discrete semantic qualities, and in the latter they appear to serve a primarily grammatical function. We used the abstract concept of time as a test case to see if specific grammatically prescribed prepositions retain semantic content. Using ambiguous questions designed to interrogate one’s meaningful representation of temporal relations, we found that the semantics of prescribed prepositions modulate how we think about time. Although prescribed preposition use is unlikely to be based on a core representational organization shared between space and time, results demonstrate that the semantics of particular locative prepositions do constrain how we think about paired temporal concepts.     

A program for the semantics of science

Concluding remarks Our program is ambitious, as is any attempt to match life (in our case real science) with virtue (e.g., exactness). We want our semantics to be not only simia mathematicae but also ancilla scientiae: built more geometrico and at the same time relevant, nay useful, to live science. The goal of exactness may sound arrogant but is actually modest, for the more we rigorize the more we are forced to leave out of consideration, at least for the time being. As to the service intention: we should try to be of some help to science because the latter faces semantic problems but has no tools of its own for solving them. If it had such tools scientists would not engage in spirited polemics over matters of sense and reference, as they often do. Witness the debates on whether the relativistic and quantum theories are concerned with sentient observers, whether population genetics refers to populations taken as wholes, whether psychology is actually concerned with the brain, and whether the sense of a theory is excreted by its mathematical formalism or is determined by the way the theory is tested.A semantics of science should help settle these and similar issues. Moreover it should give sound advice as to how to formulate scientific theories so as to avoid such imprecisions and ambiguities as may give rise to debates of the kind. Constructing such a semantics, both exact and relevant to science, should be more rewarding than either manufacturing neat but irrelevant theories or pursuing erratic polemics on meaning and meaning changes.

---

     

Actualism,ontological commitment,and possible world semantics

Actualism is the doctrine that the only things there are, that have being in any sense, are the things that actually exist. In particular, actualism eschews possibilism, the doctrine that there are merely possible objects. It is widely held that one cannot both be an actualist and at the same time take possible world semantics seriously — that is, take it as the basis for a genuine theory of truth for modal languages, or look to it for insight into the modal structure of reality. For possible world semantics, it is supposed, commits one to possibilism. In this paper I take issue with this view. To the contrary, I argue that one can take possible world semantics seriously and yet remain in full compliance with actualist scruples.     

Squares in Fork Arrow Logic

In this paper we show that the class of fork squares has a complete orthodox axiomatization in fork arrow logic (FAL). This result may be seen as an orthodox counterpart of Venema's non-orthodox axiomatization for the class of squares in arrow logic. FAL is the modal logic of fork algebras (FAs) just as arrow logic is the modal logic of relation algebras (RAs). FAs extend RAs by a binary fork operator and are axiomatized by adding three equations to RAs equational axiomatization. A proper FA is an algebra of relations where the fork is induced by an injective operation coding pair formation. In contrast to RAs, FAs are representable by proper ones and their equational theory has the expressive power of full first-order logic. A square semantics (the set of arrows is U×U for some set U) for arrow logic was defined by Y. Venema. Due to the negative results about the finite axiomatizability of representable RAs, Venema provided a non-orthodox finite axiomatization for arrow logic by adding a new rule governing the applications of a difference operator. We address here the question of extending the type of relational structures to define orthodox axiomatizations for the class of squares. Given the connections between this problem and the finitization problem addressed by I. Németi, we suspect that this cannot be done by using only logical operations. The modal version of the FA equations provides an orthodox axiomatization for FAL which is complete in view of the representability of FAs. Here we review this result and carry it further to prove that this orthodox axiomatization for FAL also axiomatizes the class of fork squares.