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### Page 557 of American Mathematical Society. Bulletin of the American Mathematical Society Vol. 59, Issue 6 [page]

1953 American Mathematical Society. Bulletin of the American Mathematical Society
Theodore Hailperin: Quantification theory for empty individual domains. A set of axioms for quantification formulas valid in all individual domains, includ- ing the empty domain, is presented.  ...  The axioms are so phrased that by the addition of one further axiom the usual quantification theory for formulas valid in all nonempty domains is obtained. A simple proof of completeness is given.  ...

### Against the iterative conception of set

Edward Ferrier
2018 Philosophical Studies
Perhaps the strongest reason for this is the discovery and development of consistent, axiomatic set theories, in which quantification is absolute. 2 These set theories are philosophically motivated by  ...  (If there are no urelements, one begins by forming the empty collection, ∅.) One then proceeds to form new collections out of these collections and the urelements (if there are any).  ...  Because empty properties exist, though no things instantiate them, we cannot always render existential plural quantification as quantification over properties.  ...

### Page 277 of Mathematical Reviews Vol. 15, Issue 4 [page]

1954 Mathematical Reviews
Quantification theory and empty individual-domains. J. Symbolic Logic 18, 197-200 (1953).  ...  Mostowski [same J. 16, 107-111 (1951); these Rev. 13, 3] examines those formulas of quantification theory which are valid on all domains, including the empty domain and gives a complete set of axioms for  ...

### The Syntax of the World

W. D. Hart
1996 Crítica. Revista Hispanoamericana de Filosofía
Given a domain D, pure set theory yields from the relations on D and the members of D as individuals, the totality of propositions and facts on D. This is a theorem of pure set theory.  ...  . , n k in a non-empty domain D are d 1 , . . . , d k , and that the extension of F in D is a set S of ordered k-tuples of members of D.  ...

### Skolem fragments

Daniel Bonevac
1984 Notre Dame Journal of Formal Logic
Provided that we employ an inclusive logic and regard vacuous quantification as ill-formed, the Skolem fragment of a theory will not in general be equivalent to the theory itself.  ...  I have argued in [1] that we can characterize ontological commitment only by taking empty domains into account.  ...

### Plural quantifiers: a modal interpretation

Rafal Urbaniak
2013 Synthese
On this view claims like 'some logicians admire only each other' involve ineliminable quantification over subsets of a salient domain.  ...  The main motivation for this view is that plural quantification has to be given some sort of semantics, and among the two main candidates-substitutional and set-theoretic-only the latter can provide the  ...  and the source are credited.  ...

### Page 49 of American Mathematical Society. Bulletin of the American Mathematical Society Vol. 59, Issue 1 [page]

1953 American Mathematical Society. Bulletin of the American Mathematical Society
The theory is applied to obtain reduction-undecidability-definability for the elementary theory of a single binary dis- joint relation (domain and co-domain disjoint), of a single symmetric reflexive rela  ...  A decision method for the elementary theory of a single equivalence is known, also for monadic quantification theory with identity, i.e., the elementary theory of a single equivalence R and any number  ...

### Incomplete Descriptions and the Underdetermination Problem

Andrei Moldovan
2015 Research in Language
Proponents of other theories, such as the Fregean, which does not treat descriptions as quantifiers, might benefit from this account of domain restriction.  ...  Relative to such uses, Wettstein (1981) and others have argued that the Russellian theory faces a problem of underdetermination of semantic content.  ...  The underdetermination problem Consider again sentence (2) ('Every bottle is empty.'). What is the contextually determined property that restricts the domain of quantification?  ...

### The Price of Universality

Gabriel Uzquiano
2006 Philosophical Studies
For we could treat some mereological sums as individuals. 13 When the null set is treated as an individual, this tells us that the parts of a set are all and only its non-empty subsets. absent specific  ...  the field of the part-whole relation. • The framework of (4) -(6) according to which the axioms of applied set theory govern the membership relation and all there is is either an individual or a set. 12  ...  Not only are proper classes not sets, they are not individuals either. And they lie outside of the domain of ZFCU, which is exclusively concerned with individuals and sets.  ...

### 'Neo-Logicist' Logic is not Epistemically Innocent

STEWART SHAPIRO, ALAN WEIR
2000 Philosophia Mathematica
of individual constants, into two classes, the inner domain of 'real' individuals, this domain being the range of the individual quantifiers, and an outer domain of "dummy" or 'virtual' individuals, which  ...  If d is the sole individual, there are two subsets of domain D: ∅ and {d} and so only one non-empty subset falling in the range of the predicate variables.  ...

### "Neo-Logicist" Logic is not Epistemically Innocent [chapter]

Stewart Shapiro, Alan Weir
2007 The Western Ontario Series in Philosophy of Science
of individual constants, into two classes, the inner domain of 'real' individuals, this domain being the range of the individual quantifiers, and an outer domain of "dummy" or 'virtual' individuals, which  ...  If d is the sole individual, there are two subsets of domain D: ∅ and {d} and so only one non-empty subset falling in the range of the predicate variables.  ...

### A formulation of the simple theory of types (for Isabelle) [chapter]

Lawrence C. Paulson
1990 Lecture Notes in Computer Science
Simple type theory is formulated for use with the generic theorem prover Isabelle. This requires explicit type inference rules. There are function, product, and subset types, which may be empty.  ...  Higher-order logic is obtained through reflection between formulae and terms of type bool . Recursive types and functions can be formally constructed. Isabelle proof procedures are described.  ...  First-order logic allows quantification over individuals; second-order logic allows quantification over properties of individuals; third-order logic allows quantification over properties of properties  ...

### A Semantic Annotation Scheme for Quantification

Harry Bunt
2019 Proceedings of the 13th International Conference on Computational Semantics - Long Papers
of generalised quantifiers, neo-Davidsonian semantics, and DRT, covers a wide range of aspects of quantification.  ...  annotation scheme called 'QuantML' which was proposed last December to the International Organisation for Standardisation (ISO) as a starting point for developing a standard for interoperable rooted in the theory  ...  Individuation and distributivity are distinct aspects of quantification; elements from a domain with count/parts individuation can for example participate collectively in a quantification, as in the report  ...

### The Mereological Foundation of Megethology

Massimiliano Carrara, Enrico Martino
2015 Journal of Philosophical Logic
a reconstruction of classical set theory grounded on mereology and plural quantification (see also (Lewis, 1991) ). 1 Mereology is the theory of parthood relation.  ...  Acccording to Boolos, second order quantification differs from first order quantification only in that it refers to individuals plurally, while the latter refers to individuals singularly.  ...  In this way an a-relation can be understood as a fusion of atoms, i.e. as a single individual, and quantification over relations is reduced to first order quantification.  ...

### Collapse, Plurals and Sets

Eduardo Alejandro Barrio
2014 Principia: An International Journal of Epistemology
Assuming classical logic, I show that if all plurality form a set and the quantifiers are absolutely general, then one gets a trivial theory. So, by reductio, one has to abandon one of the premiss.  ...  What I am advocating is that the thesis of collapse limits important applications of the plural logic in model theory, when it is assumed that the quantifiers are absolutely general.  ...  I am also heavily indebted to Agustín Rayo for detailed and helpful comments. His suggestions allowed me to significantly improve my paper.  ...
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