I first show that most authors who developed Plural Quantification Logic (PQL) argued it could capture various features of natural language better than can other logic systems. I then show that it fails to do so: it radically departs from natural language in two of its essential features; namely, in distinguishing plural from singular quantification and in its use of an relation. Next, I sketch a different approach that is more adequate than PQL for capturing plural aspects of natural language (...) semantics and logic. I conclude with a criticism of the claim that PQL should replace natural language for specific philosophical or scientific purposes. (shrink)

Inspired by the grammar of natural language, the paper presents a variant of first-order logic, in which quantifiers are not sentential operators, but are used as subnectors . A quantified term formed by a subnector is an argument of a predicate. The logic is defined by means of a meaning-conferring natural-deduction proof-system, according to the proof-theoretic semantics program. The harmony of the I/E-rules is shown. The paper then presents a translation, called the Frege translation, from the defined logic to standard (...) first-order logic, and shows that the proof-theoretic meanings of both logics coincide. The paper criticizes Frege’s original regimentation of quantified sentences of natural language, and argues for advantages of the proposed variant. (shrink)

The article concerns the treatment of the so-called denoting phrases, of the forms ?every A?, ?any A?, ?an A? and ?some A?, in Russell's Principles of Mathematics. An initially attractive interpretation of what Russell's theory was has been proposed by P.T. Geach, in his Reference and Generality (1962). A different interpretation has been proposed by P. Dau (Notre Dame Journal, 1986). The article argues that neither of these is correct, because both credit Russell with a more thought-out theory than he (...) actually had. The conclusion is mainly negative: at this date Russell has no coherent theory of these phrases. An appendix notes that his understanding of the quantifiers in predicate logic is also, at this date, not entirely secure. (shrink)

The formalisation of Natural Language arguments in a formal language close to it in syntax has been a central aim of Moss’s Natural Logic. I examine how the Quantified Argument Calculus (Quarc) can handle the inferences Moss has considered. I show that they can be incorporated in existing versions of Quarc or in straightforward extensions of it, all within sound and complete systems. Moreover, Quarc is closer in some respects to Natural Language than are Moss’s systems – for instance, is (...) does not use negative nouns. The process also sheds light on formal properties and presuppositions of some inferences it formalises. Directions for future work are outlined. (shrink)

The Modal Predicate Calculus gives rise to issues surrounding the Barcan formulas, their converses, and necessary existence. I examine these issues by means of the Quantified Argument Calculus, a recently developed, powerful formal logic system. Quarc is closer in syntax and logical properties to Natural Language than is the Predicate Calculus, a fact that lends additional interest to this examination, as Quarc might offer a better representation of our modal concepts. The validity of the Barcan formulas and their converses is (...) shown by Quarc to be a result of the specific incorporation of quantification in the Predicate Calculus, and not as reflecting a feature of the interaction of quantification and modality more generally. Necessary existence is shown to follow from the identification, in the Predicate Calculus on its canonical interpretation, of particular quantification, ascription of existence and the ‘there is’ construction, three constructions which are distinguished in both Quarc and Natural Language. The issues surrounding the Barcan formulas, their converses and necessary existence are thus shown to be an artefact of a specific logic system, not an essential feature of our relevant modal concepts or of formal logic. (shrink)

In a series of publications I have claimed that by contrast to standard formal languages, quantifiers in natural language combine with a general term to form a quantified argument, in which the general term's role is to determine the domain or plurality over which the quantifier ranges. In a recent paper Zoltán Gendler Szabó tried to provide a counterexample to this analysis and derived from it various conclusions concerning quantification in natural language, claiming it is often ‘bare’. I show that (...) Szabó's example fails, and that even if it were successful his conclusions would not be supported by it. (shrink)

Frege analyzed the grammatical subject-term 'S' in quantified subject-predicate sentences, 'q S are P', as being logically predicative. This is in contrast to Aristotelian Logic, according to which it is a logical subject-term, like the proper name 'a' in 'a is P' – albeit a plural one, designating many particulars. I show that Frege's arguments for his analysis are unsound, and explain how he was misled to his position by the mathematical concept of function. If common nouns in this grammatical (...) subject position are indeed logical subject-terms, this should require a thorough reevaluation of the adequacy of Frege's Predicate Calculus as a tool for the analysis of the logic and semantics of natural language. (shrink)

I address what I call the Number Issue, which is raised by our ordinary talk and beliefs about certain social groups and institutions, and I take the Hallé orchestra as my example. The Number Issue is that of whether the Hallé is one individual or several individuals. I observe that if one holds that it is one individual, one faces an accusation of metaphysical extravagance. The bulk of the paper examines the difficulty of reconciling the view that the Hallé is (...) several individuals with two prima facie plausible theses about the manner of its persistence through time. The paper is structured around some remarks made by Peter Simons about groups, and the Hallé in particular, in his 'Parts'. (shrink)

It is natural to think that social groups are concrete material particulars, but this view faces an important objection. Suppose the chess club and nature club have the same members. Intuitively, these are different clubs even though they have a common material basis. Some philosophers take these intuitions to show that the materialist view must be abandoned. I propose an alternative explanation. Social groups are concrete material particulars, but there is a psychological explanation of nonidentity intuitions. Social groups appear coincident (...) but nonidentical because they are perceived to be governed by conflicting social norms. (shrink)

The goal of this paper is to present a new reconstruction of Aristotle's assertoric logic as he develops it in Prior Analytics, A1-7. This reconstruction will be much closer to Aristotle's original...

Philosophers disagree whether composition as identity entails mereological universalism. Bricker :264–294, 2016) has recently considered an argument which concludes that composition as identity supports universalism. The key step in this argument is the thesis that any objects are identical to some object, which Bricker justifies with the principle of the universality of identity. I will spell out this principle in more detail and argue that it has an unexpected consequence. If the universality of identity holds, then composition as identity not (...) only leads us to universalism, but also leads to the view that there are no mereological atoms. (shrink)

The present article employs a model-theoretic semantics to interpret a fragment of the language of the Quantified Argument Calculus (Quarc), a recently introduced logical system whose main aim is capturing the structure of natural language sentences in a closer way than does the language of classical logic. The main contribution is an axiomatization for the set of formulas that are valid in all standard interpretations within the employed semantics.

Composition as Identity is the view that an object is identical to its parts taken collectively. I elaborate and defend a theory based on this idea: composition is a kind of identity. Since this claim is best presented within a plural logic, I develop a formal system of plural logic. The principles of this system differ from the standard views on plural logic because one of my central claims is that identity is a relation which comes in a variety of (...) forms and only one of them obeys substitution unrestrictedly. I justify this departure from orthodoxy by showing some problems which result from attempts to avoid inconsistencies within plural logic by means of postulating other non-singular terms besides plural terms. Thereby, some of the main criticisms raised against Composition as Identity can be addressed. Further, I argue that the way objects are arranged is relevant with respect to the question which object they compose, i.e. to which object they are identical to. This helps to meet a second group of arguments against Composition as Identity. These arguments aim to show that identifying composite objects on the basis of the identity of their parts entails, contrary to our common sense view, that rearranging the parts of a composite object does not leave us with a different object. Moreover, it allows us to carve out the intensional aspects of Composition as Identity and to defend mereological universalism, the claim that any objects compose some object. Much of the pressure put on the latter view can be avoided by distinguishing the question whether some objects compose an object from the question what object they compose. Eventually, I conclude that Composition as Identity is a coherent and plausible position, as long as we take identity to be a more complex relation than commonly assumed. (shrink)

Can there be relational universals? If so, how can they be exemplified? A monadic universal is by definition capable of having a scattered spatiotemporal localization of its different exemplifications, but the problem of relational universals is that one single exemplification seems to have to be scattered in the many places where the relata are. The paper argues that it is possible to bite this bullet, and to accept a hitherto un-discussed kind of exemplification relation called ‘scattered exemplification’. It has no (...) immediate symbolic counterpart in any Indo-European natural language or in any so far constructed logical language. In order to remedy this, a notion called ‘many-place copula’ is introduced, too. (shrink)

I show that the contemporary dominant analysis of natural language quantifiers that are one-place determiners by means of binary generalized quantifiers has failed to explain why they are, according to it, conservative. I then present an alternative, Geachean analysis, according to which common nouns in the grammatical subject position are plural logical subject-terms, and show how it does explain that fact and other features of natural language quantification.

Ordinary English contains different forms of quantification over objects. In addition to the usual singular quantification, as in 'There is an apple on the table', there is plural quantification, as in 'There are some apples on the table'. Ever since Frege, formal logic has favored the two singular quantifiers ∀x and ∃x over their plural counterparts ∀xx and ∃xx (to be read as for any things xx and there are some things xx). But in recent decades it has been argued (...) that we have good reason to admit among our primitive logical notions also the plural quantifiers ∀xx and ∃xx. More controversially, it has been argued that the resulting formal system with plural as well as singular quantification qualifies as ‘pure logic’; in particular, that it is universally applicable, ontologically innocent, and perfectly well understood. In addition to being interesting in its own right, this thesis will, if correct, make plural quantification available as an innocent but extremely powerful tool in metaphysics, philosophy of mathematics, and philosophical logic. For instance, George Boolos has used plural quantification to interpret monadic second-order logic and has argued on this basis that monadic second-order logic qualifies as “pure logic.” Plural quantification has also been used in attempts to defend logicist ideas, to account for set theory, and to eliminate ontological commitments to mathematical objects and complex objects. (shrink)