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### Arithmetic, First-Order Logic, and Counting Quantifiers [article]

Nicole Schweikardt
2002 arXiv   pre-print
This paper gives a thorough overview of what is known about first-order logic with counting quantifiers and with arithmetic predicates.  ...  Applying this theorem, we obtain an easy proof of Ruhl's result that reachability (and similarly, connectivity) in finite graphs is not expressible in first-order logic with unary counting quantifiers  ...  Syntax and Semantics First-order logic with unary counting quantifiers, FOunC, is the extension of first-order logic obtained by adding unary counting quantifiers of the form ∃ =x y.  ...

### Arithmetic, first-order logic, and counting quantifiers

Nicole Schweikardt
2005 ACM Transactions on Computational Logic
This paper gives a thorough overview of what is known about first-order logic with counting quantifiers and with arithmetic predicates.  ...  For other related work that deals with first-order logic, counting, and/or arithmetic from different points of view see, e.g., the articles [Lee 2003; Llima 1998; Krynicki and Zdanowski 2003; Mostowski  ...  FIRST-ORDER LOGIC WITH COUNTING QUANTIFIERS In this section we fix the syntax and semantics of first-order logic with counting quantifiers, and we summarize some important properties of this logic.  ...

### Page 4089 of Mathematical Reviews Vol. , Issue 98G [page]

1998 Mathematical Reviews
Summary: “We study the expressive power in the finite setting of the logic Fixed-Point + Counting, the extension of first-order logic which is obtained by adding both the fixed-point constructor and the  ...  Summary: “A combinatorial criterion for two finite structures to agree on all sentences with bounded quantifier rank in first-order logic with any set of unary generalized quantifiers is given.  ...

### First-order logic [chapter]

Peter J. Cameron
This paper gives a thorough overview of what is known about first-order logic with counting quantifiers and with arithmetic predicates.  ...  For other related work that deals with first-order logic, counting, and/or arithmetic from different points of view see, e.g., the articles [Lee 2003; Llima 1998; Krynicki and Zdanowski 2003; Mostowski  ...  FIRST-ORDER LOGIC WITH COUNTING QUANTIFIERS In this section we fix the syntax and semantics of first-order logic with counting quantifiers, and we summarize some important properties of this logic.  ...

### The Nature and Purpose of Numbers

G. Aldo Antonelli
2010 Journal of Philosophy
Each is implemented through a specific device, i.e., respectively, an extra-logical operator representing numerical abstraction and a non-standard (but still first-order) cardinality quantifier.  ...  Along the way, we investigate the logical status of arithmetic, the function of abstraction principles, and the respective merits of various strategies for reducing arithmetical notions to those of a theory  ...  A unary quantifier is first-order if and only if it represents a collection of subsets of D (and similarly, a binary quantifier is first-order if and only if it expresses a relation between subsets of  ...

### Extensions of MSO and the monadic counting hierarchy

Juha Kontinen, Hannu Niemistö
2011 Information and Computation
We show that over unary vocabularies the logic MSO(R), where MSO is monadic second-order logic and R is the first-order Rescher quantifier, can be characterized by Presburger arithmetic, whereas the logic  ...  In this paper we study the expressive power of the extension of first-order logic by the unary second-order majority quantifier Most 1 .  ...  Acknowledgments The authors thank Lauri Hella for valuable comments and suggestions.  ...

### Modulo quantifiers over functional vocabularies extending addition [article]

A. Baskar, A. V. Sreejith, R. S. Thinniyam
2021 arXiv   pre-print
We show that first order logic (FO) and first order logic extended with modulo counting quantifiers (FOMOD) over purely functional vocabularies which extend addition, satisfy the Crane beach property (  ...  Showing the existence of regular languages not definable in FOMOD[<, +, *] is equivalent to the separation of the circuit complexity classes ACC0 and NC1 .  ...  In this paper, we identify a sufficient condition for functional vocabularies over first order logic and first order logic extended with modulo quantifiers, to satisfy the Crane Beach property.  ...

### Page 5204 of Mathematical Reviews Vol. , Issue 2002G [page]

2002 Mathematical Reviews
The present paper considers logics formed by extending first- order logic with groupoidal quantifiers.  ...  The paper establishes that extending first-order logic with arithmetic by all groupoidal quantifiers yields a logic that exactly characterizes LOGCFL, which is known to be equivalent to the complexity  ...

### Building a Calculus of Data Structures [chapter]

Viktor Kuncak, Ruzica Piskac, Philippe Suter, Thomas Wies
2010 Lecture Notes in Computer Science
, multiset algebra with Presburger arithmetic, the Bernays-Schönfinkel-Ramsey class of first-order logic, and the logic of algebraic data types with the set content function.  ...  As an instance of this approach, we describe a decidable logic whose formulas are propositional combinations of formulas in: weak monadic second-order logic of two successors, two-variable logic with counting  ...  , multiset algebra with Presburger arithmetic, the Bernays-Schönfinkel-Ramsey class of first-order logic, and the logic of algebraic data types with the set content function.  ...

### Induction and indefinite extensibility: the Gödel sentence is true, but did someone change the subject?

S Shapiro
1998 Mind
If basic arithmetic propositions are logically true-as the logicist contends-then there is tension between this conservation requirement and the ontological commitments of arithmetic.  ...  and elimination rules of the logical terms that occur in S.  ...  rules for negation, identity, and the first-order existential quantifier.  ...

### One-Variable Logic Meets Presburger Arithmetic [article]

Bartosz Bednarczyk
2019 arXiv   pre-print
We consider the one-variable fragment of first-order logic extended with Presburger constraints.  ...  The logic is designed in such a way that it subsumes the previously-known fragments extended with counting, modulo counting or cardinality comparison and combines their expressive powers.  ...  The author would also like to thank the two anonymous reviewers as well as Witold Charatonik, Emanuel Kieroński and Antti Kuusisto for their careful proofreading and for pointing out numerous grammatical  ...

### Combining Theories with Shared Set Operations [chapter]

Thomas Wies, Ruzica Piskac, Viktor Kuncak
2009 Lecture Notes in Computer Science
Algebra with Presburger Arithmetic (with quantifiers over sets and integers), 2) weak monadic second-order logic over trees (with monadic second-order quantifiers), 3) two-variable logic with counting  ...  quantifiers (ranging over elements), 4) the Bernays-Schönfinkel-Ramsey class of first-order logic with equality (with ∃ * ∀ * quantifier prefix), and 5) the quantifier-free logic of multisets with cardinality  ...  Conclusion We have presented a combination result for logics that share operations on sets. This result yields an expressive decidable logic that is useful for software verification.  ...

### Expressing versus Proving: Relating Forms of Complexity in Logic

A. Kolokolova
2010 Journal of Logic and Computation
of arithmetic.  ...  Lastly, we will touch upon the topic of formalizing complexity theory using logic, and the meta-question of complexity of logical reasoning about complexity-theoretic statements.  ...  The results on connections between finite model theory and bounded arithmetic mentioned here come from my joint work with Stephen Cook (which resulted in my PhD thesis).  ...

### Logic and Complexity: Independence results and the complexity of propositional calculus [chapter]

Pavel Pudlák
1995 Proceedings of the International Congress of Mathematicians
I want to thank Sam Buss, Jan Krajicek, Russell Irnpagliazzo, and Jiff Sgall for their comments on a draft of this paper.  ...  Bounded arithmetic We turn now to some first order theories that are important in this field. Let us first recall the classical theory Peano arithmetic used to formalize number theory.  ...  Suppose we use not only the usual quantifiers V and 3, but also a counting quantifier, saying that the number of x 's satisfying ip is divisible by p.  ...

### Number Determiners, Numbers, and Arithmetic

T. Hofweber
2005 Philosophical Review
The latter would have two hundred first-order quantifiers in its logical form.  ...  These quantifiers are part of the first-order predicate calculus, and this calculus, in turn, is part of logic, and thus unproblematic.  ...  In these cases, it is plausible that there is a plural quantifier involved, as in the sentence 'There are numbers between 100 and 200.' 28 Carnap's discussion can be found in his famous 1956 essay.  ...
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