A Two Dimensional Tense-modal Sortal Logic

We consider a formal language whose logical syntax involves both modal and tense propositional operators, as well as sortal quantifiers, sortal identities and (second order) quantifiers over sortals. We construct an intensional semantics for the language and characterize a formal logical system which we prove to be sound and complete with respect to the semantics. Conceptualism is the philosophical background of the semantic system.     

A Temporal Logic for Sortals

With the past and future tense propositional operators in its syntax, a formal logical system for sortal quantifiers, sortal identity and (second order) quantification over sortal concepts is formulated. A completeness proof for the system is constructed and its absolute consistency proved. The completeness proof is given relative to a notion of logical validity provided by an intensional semantic system, which assumes an approach to sortals from a modern form of conceptualism.     

Nominalizing Quantifiers

Quantified expressions in natural language generally are taken to act like quantifiers in logic, which either range over entities that need to satisfy or not satisfy the predicate in order for the sentence to be true or otherwise are substitutional quantifiers. I will argue that there is a philosophically rather important class of quantified expressions in English that act quite differently, a class that includes something, nothing, and several things. In addition to expressing quantification, such expressions act like nominalizations, introducing a new domain of objects that would not have been present in the semantic structure of the sentence otherwise. The entities those expressions introduce are of just the same sort as those that certain ordinary nominalizations refer to (such as John's wisdom or John's belief that S), namely they are tropes or entities related to tropes. Analysing certain quantifiers as nominalizing quantifiers will shed a new light on philosophical issues such as the status of properties and the nature of propositional attitudes.     

A Complete and Consistent Formal System for Sortals

A formal logical system for sortal quantifiers, sortal identity and (second order) quantification over sortal concepts is formulated. The absolute consistency of the system is proved. A completeness proof for the system is also constructed. This proof is relative to a concept of logical validity provided by a semantics, which assumes as its philosophical background an approach to sortals from a modern form of conceptualism.     

Semantic representation of Kinship systems

Four representational systems are examined with respect to their adequacy for representing reasoning processes within kinship systems; the systems are associative, semantic feature, logical predicate, and an “algebraic” one based on set mappings. The associative and semantic feature systems are shown to be inadequate, and the logical predicate system is shown to be somewhat clumsy and unintuitive. The algebraic is proposed as the correct representation of kinship terms. It is suggested that certain other conceptual domains can also be best represented by an algebraic system.     

Natural language semantics in biproduct dagger categories

Biproduct dagger categories serve as models for natural language. In particular, the biproduct dagger category of finite dimensional vector spaces over the field of real numbers accommodates both the extensional models of predicate calculus and the intensional models of quantum logic. The morphisms representing the extensional meanings of a grammatical string are translated to morphisms representing the intensional meanings such that truth is preserved. Pregroup grammars serve as the tool that transforms a grammatical string into a morphism. The chosen linguistic examples concern negation, relative noun phrases, comprehension and quantifiers.     

Function and Argument in Begriffsschrift

It is well known that the formal system developed by Frege in Begriffsschrift is based upon the distinction between function and argument—as opposed to the traditional distinction between subject and predicate. Almost all of the modern commentaries on Frege's work suggest a semantic interpretation of this distinction, and identify it with the ontological structure of function and object, upon which Grundgesetze is based. Those commentaries agree that the system proposed by Frege in Begriffsschrift has some gaps, but it is taken as an essentially correct formal system for second-order logic: the first one in the history of logic. However, there is strong textual evidence that such an interpretation should be rejected. This evidence shows that the nature of the distinction between function and argument is stated by Frege in a significantly different way: it applies only to expressions and not to entities. The formal system based on this distinction is tremendously flexible and is suitable for making explicit the logical structure of contents as well as of deductive chains. We put forward a new reconstruction of the function-argument scheme and the quantification theory in Begriffsschrift. After that, we discuss the usual semantic interpretation of Begriffsschrift and show its inconsistencies with a rigorous reading of the text.     

Existence,Negation, and Abstraction in the Neoplatonic Hierarchy 1

The paper is a study of the logic of existence, negation, and order in the Neoplatonic tradition. The central idea is that Neoplatonists assume a logic in which the existence predicate is a comparative adjective and in which monadic predicates function as scalar adjectives that nest the background order. Various scalar predicate negations are then identifiable with various Neoplatonic negations, including a privative negation appropriate for the lower orders of reality and a hyper-negation appropriate for the higher. Reversion to the One can then be explained as the logical inference of hyper-negations from mundane knowledge. Part I develops the relevant linguistic and logical theory, and Part II defends Wolfson and the scalar interpretation against the more traditional Aristotelian understanding of Whittaker and others of reversion as intensional abstraction     

Hobnobbing with the Nonexistent

Recent discussions of Geach sentences by Braun and Salmon are reprised. It is shown that the intractability of providing semantics for Geach sentences (using standard logical tools) is due to the assumption that quantifiers are ontologically committing. Representing the content of these statements is easy using neutral quantifiers. An important concern is consistent identity conditions for nonreferring terms. It may be thought that Meinongian-object approaches handle this better than Azzouni's no-objects-in-any-sense-at-all approach. This is shown to be false. How our truth-inducing practices determine truth values for empty-term statements is indicated—in particular, it is argued that our identification and distinguishing practices with “fictional entities” are parasitic on our practices with entities we take to be real. Examples from Austin—white dots on the horizon and specks—are among the test cases for the views developed here.     

Quantification in English is Inherently Sortal

Within Linguistics the semantic analysis of natural languages (English, Swahili, for example) has drawn extensively on semantical concepts first formulated and studied within classical logic, principally first order logic. Nowhere has this contribution been more substantive than in the domain of quantification and variable binding. As studies of these notions in natural language have developed they have taken on a life of their own, resulting in refinements and generalizations of the classical quantifiers as well as the discovery of new types of quantification which exceed the expressive capacity of the classical quantifiers. We refer the reader to Keenan and Westerståhl (1997) for an overview of results in this area. Here, we focus on one property of quantification in natural language—its inherently sortal nature—which distinguishes it from quantification in classical logic.     

Eternalism and Propositional Multitasking: in defence of the Operator Argument

It is widely held that propositions perform a plethora of theoretical roles. They are believed to be the semantic values of sentences in contexts, the objects of attitudes, the contents of illocutionary acts, the referents of ??that??-clauses, and the primary bearers of truth. This assumption is often combined with the claim that propositions have their truth-values eternally. Following Kaplan??s and Lewis??s Operator Argument, I argue that the compositional semantic values of sentences do not correspond to eternal propositions. Therefore, we cannot hold on to both assumptions at the same time: either we regard the non-eternal entities that realize the compositional role of propositions as fulfilling the remaining theoretical roles, or we abandon the assumption that there is a unique realizer. The Operator Argument has recently come under attack, mainly for its intensional assumptions. However, rejecting these assumptions is not a sufficient defense of eternal propositions as compositional semantic values of sentences. Firstly, we can give a generalized version of the Operator Argument that seems independent of the contested assumptions. Secondly, the extensional alternative to the intensional framework does not allow us to retain eternal propositions as unique semantic values either.     

Quine on intensional entities: Modality and quantification,truth and satisfaction

In this paper, I reconstruct Quine?s arguments against quantified modal logic, from the early 1940?s to the early 1960?s. Quine?s concerns were not technical. Quine was looking for a coherent interpretation of quantified-in English modal sentences. I argue that Quine?s main thesis is that the intended objectual interpretation of the quantifiers is incompatible with any semantic reading of the modal operators, for example as expressing analytic necessity, unless the entities in the domain of quantification are intensions, i.e. definitional entities. The difficulty is that it makes no sense to say of an ordinary object that it bears a property necessarily or contingently when the necessity or contingency in question is analytic. However, starting in 1960, Quine claims that quantified-in modal sentences can be coherently interpreted only as essentialist predications. When we say about an object that it necessarily F?s, we can only coherently mean that it essentially F?s. In the paper, I argue that adequately qualified the thesis is plausible. Two important qualifications are needed. The first is the assumption that satisfaction is an irreducibly predicative notion, making any explication of satisfaction in terms of truth inadequate. The second is the ontological rejection of purely semantic, i.e. merely definitional, entities. With these qualifications in place, Quine?s rejection of the combination of objectual quantifiers and semantic modalities can be upheld. In this way, we vindicate a qualified version of Quine?s conjecture that quantified modal logic is committed to essentialism.     

The Logicality of Language: A new take on Triviality, “Ungrammaticality”, and Logical Form*

Recent work in formal semantics suggests that the language system includes not only a structure building device, as standardly assumed, but also a natural deductive system which can determine when expressions have trivial truth‐conditions (e.g., are logically true/false) and mark them as unacceptable. This hypothesis, called the ‘logicality of language’, accounts for many acceptability patterns, including systematic restrictions on the distribution of quantifiers. To deal with apparent counter‐examples consisting of acceptable tautologies and contradictions, the logicality of language is often paired with an additional assumption according to which logical forms are radically underspecified: i.e., the language system can see functional terms but is ‘blind’ to open class terms to the extent that different tokens of the same term are treated as if independent. This conception of logical form has profound implications: it suggests an extreme version of the modularity of language, and can only be paired with non‐classical—indeed quite exotic—kinds of deductive systems. The aim of this paper is to show that we can pair the logicality of language with a different and ultimately more traditional account of logical form. This framework accounts for the basic acceptability patterns which motivated the logicality of language, can explain why some tautologies and contradictions are acceptable, and makes better predictions in key cases. As a result, we can pursue versions of the logicality of language in frameworks compatible with the view that the language system is not radically modular vis‐á‐vis its open class terms and employs a deductive system that is basically classical.     

Realization, Micro-Realization, and Coincidence

Let thin properties be properties shared by coincident entities, e.g., a person and her body, and thick properties ones that are not shared. Thick properties entail sortal properties, e.g., being a person, and the associated persistence conditions. On the first account of realization defined here, the realized property and its realizers will belong to the same individual. This restricts the physical realizers of mental properties, which are thick, to thick physical properties. We also need a sense in which mental properties can be realized in thin physical properties shared by a person and her body. Defining this in turn requires defining a sense in which the instantiations of sortal properties and of thick properties are realized in micro-structural states of affairs. A fourth notion of realization is needed to allow for the possibility of coincident entities that share a sortal property, e.g., coincident persons.     

The Ambiguity of Quantifiers

In the tradition of substructural logics, it has been claimed for a long time that conjunction and inclusive disjunction are ambiguous:we should, in fact, distinguish between ‘lattice’ connectives (also called additive or extensional) and ‘group’ connectives (also called multiplicative or intensional). We argue that an analogous ambiguity affects the quantifiers. Moreover, we show how such a perspective could yield solutions for two well-known logical puzzles: McGee’s counterexample to modus ponens and the lottery paradox.     

On the Logic of Nonmonotonic Conditionals and Conditional Probabilities: Predicate Logic

In a previous paper I described a range of nonmonotonic conditionals that behave like conditional probability functions at various levels of probabilistic support. These conditionals were defined as semantic relations on an object language for sentential logic. In this paper I extend the most prominent family of these conditionals to a language for predicate logic. My approach to quantifiers is closely related to Hartry Field's probabilistic semantics. Along the way I will show how Field's semantics differs from a substitutional interpretation of quantifiers in crucial ways, and show that Field's approach is closely related to the usual objectual semantics. One of Field's quantifier rules, however, must be significantly modified to be adapted to nonmonotonic conditional semantics. And this modification suggests, in turn, an alternative quantifier rule for probabilistic semantics.     

Processing of Numerical and Proportional Quantifiers

Quantifier expressions like “many” and “at least” are part of a rich repository of words in language representing magnitude information. The role of numerical processing in comprehending quantifiers was studied in a semantic truth value judgment task, asking adults to quickly verify sentences about visual displays using numerical (at least seven, at least thirteen, at most seven, at most thirteen) or proportional (many, few) quantifiers. The visual displays were composed of systematically varied proportions of yellow and blue circles. The results demonstrated that numerical estimation and numerical reference information are fundamental in encoding the meaning of quantifiers in terms of response times and acceptability judgments. However, a difference emerges in the comparison strategies when a fixed external reference numerosity (seven or thirteen) is used for numerical quantifiers, whereas an internal numerical criterion is invoked for proportional quantifiers. Moreover, for both quantifier types, quantifier semantics and its polarity (positive vs. negative) biased the response direction (accept/reject). Overall, our results indicate that quantifier comprehension involves core numerical and lexical semantic properties, demonstrating integrated processing of language and numbers.     

Partial Semantics for Quantified Modal Logic

When it comes to Kripke-style semantics for quantified modal logic, there’s a choice to be made concerning the interpretation of the quantifiers. The simple approach is to let quantifiers range over all possible objects, not just objects existing in the world of evaluation, and use a special predicate to make claims about existence (an existence predicate). This is the constant domain approach. The more complicated approach is to assign a domain of objects to each world. This is the varying domain approach. Assuming that all terms denote, the semantics of predication on the constant domain approach is obvious: either the denoted object has the denoted property in the world of evaluation, or it hasn’t. On the varying domain approach, there’s a third possibility: the object in question doesn’t exist. Terms may denote objects not included in the domain of the world of evaluation. The question is whether an atomic formula then should be evaluated as true or false, or if its truth value should be undefined. This question, however, cannot be answered in isolation. The consequences of one’s choice depends on the interpretation of molecular formulas. Should the negation of a formula whose truth value is undefined also be undefined? What about conjunction, universal quantification and necessitation? The main contribution of this paper is to identify two partial semantics for logical operators, a weak and a strong one, which uniquely satisfy a list of reasonable constraints (Theorem 2.1). I also show that, provided that the point of using varying domains is to be able to make certain true claims about existence without using any existence predicate, this result yields two possible partial semantics for quantified modal logic with varying domains.     

Predication in Conceptual Realism

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.