Second-Order Know-How Strategies, 2018 Lafayette College

#### Second-Order Know-How Strategies, Pavel Naumov, Jia Tao

*Faculty Research and Reports*

The fact that a coalition has a strategy does not mean that the coalition knows what the strategy is. If the coalition knows the strategy, then such a strategy is called a know-how strategy of the coalition. The paper proposes the notion of a second-order know-how strategy for the case when one coalition knows what the strategy of another coalition is. The main technical result is a sound and complete logical system describing the interplay between the distributed knowledge modality and the second-order coalition know-how modality.

Coincidence Of Bargaining Solutions And Rationalizability In Epistemic Games, 2018 The Graduate Center, City University of New York

#### Coincidence Of Bargaining Solutions And Rationalizability In Epistemic Games, Todd Stambaugh

*All Dissertations, Theses, and Capstone Projects*

**Chapter 1**: In 1950, John Nash proposed the Bargaining Problem, for which a solution is a function that assigns to each space of possible utility assignments a single point in the space, in some sense representing the ’fair’ deal for the agents involved. Nash provided a solution of his own, and several others have been presented since then, including a notable solution by Ehud Kalai and Meir Smorodinsky. In chapter 1, a complete account is given for the conditions under which the two solutions will coincide for two player bargaining scenarios.

**Chapter 2**: In the same year, Nash presented one ...

The Structure Of Models Of Second-Order Set Theories, 2018 The Graduate Center, City University of New York

#### The Structure Of Models Of Second-Order Set Theories, Kameryn J. Williams

*All Dissertations, Theses, and Capstone Projects*

This dissertation is a contribution to the project of second-order set theory, which has seen a revival in recent years. The approach is to understand second-order set theory by studying the structure of models of second-order set theories. The main results are the following, organized by chapter. First, I investigate the poset of T-realizations of a fixed countable model of ZFC, where T is a reasonable second-order set theory such as GBC or KM, showing that it has a rich structure. In particular, every countable partial order embeds into this structure. Moreover, we can arrange so that these embedding preserve ...

Information Flow Under Budget Constraints, 2018 Lafayette College

#### Information Flow Under Budget Constraints, Pavel Naumov, Jia Tao

*Faculty Research and Reports*

Although first proposed in the database theory as properties of functional dependencies between attributes, Armstrong's axioms capture general principles of information flow by describing properties of dependencies between sets of pieces of information. This article generalizes Armstrong's axioms to a setting in which there is a cost associated with information. The proposed logical system captures general principles of dependencies between pieces of information constrained by a given budget.

Armstrong's Axioms And Navigation Strategies, 2018 Vassar College

#### Armstrong's Axioms And Navigation Strategies, Kaya Deuser, Pavel Naumov

*Faculty Research and Reports*

The paper investigates navigability with imperfect information. It shows that the properties of navigability with perfect recall are exactly those captured by Armstrong's axioms from database theory. If the assumption of perfect recall is omitted, then Armstrong's transitivity axiom is not valid, but it can be replaced by a weaker principle. The main technical results are soundness and completeness theorems for the logical systems describing properties of navigability with and without perfect recall.

Strategic Coalitions With Perfect Recall, 2018 Lafayette College

#### Strategic Coalitions With Perfect Recall, Pavel Naumov, Jia Tao

*Faculty Research and Reports*

The paper proposes a bimodal logic that describes an interplay between distributed knowledge modality and coalition know-how modality. Unlike other similar systems, the one proposed here assumes perfect recall by all agents. Perfect recall is captured in the system by a single axiom. The main technical results are the soundness and the completeness theorems for the proposed logical system.

What Makes A Theory Of Infinitesimals Useful? A View By Klein And Fraenkel, 2018 Bar-Ilan University

#### What Makes A Theory Of Infinitesimals Useful? A View By Klein And Fraenkel, Vladimir Kanovei, Karin Katz, Mikhail Katz, Thomas Mormann

*Journal of Humanistic Mathematics*

Felix Klein and Abraham Fraenkel each formulated a criterion for a theory of infinitesimals to be successful, in terms of the feasibility of implementation of the Mean Value Theorem. We explore the evolution of the idea over the past century, and the role of Abraham Robinson's framework therein.

Lighthouse Principle For Diffusion In Social Networks, 2018 Vassar College

#### Lighthouse Principle For Diffusion In Social Networks, Sanaz Azimipoor, Pavel Naumov

*Faculty Research and Reports*

The article investigates an influence relation between two sets of agents in a social network. It proposes a logical system that captures propositional properties of this relation valid in all threshold models of social networks with the same structure. The logical system consists of Armstrong axioms for functional dependence and an additional Lighthouse axiom. The main results are soundness, completeness, and decidability theorems for this logical system.

Model-Completions And Model-Companions, 2017 Wesleyan University

#### Model-Completions And Model-Companions, Mostafa Mirabi

*Mostafa Mirabi*

Interstructure Lattices And Types Of Peano Arithmetic, 2017 The Graduate Center, City University of New York

#### Interstructure Lattices And Types Of Peano Arithmetic, Athar Abdul-Quader

*All Dissertations, Theses, and Capstone Projects*

The collection of elementary substructures of a model of PA forms a lattice, and is referred to as the substructure lattice of the model. In this thesis, we study substructure and interstructure lattices of models of PA. We apply techniques used in studying these lattices to other problems in the model theory of PA.

In Chapter 2, we study a problem that had its origin in Simpson, who used arithmetic forcing to show that every countable model of PA has an expansion to PA^{∗} that is pointwise definable. Enayat later showed that there are 2^{ℵ0} models with the ...

The Common Invariant Subspace Problem And Tarski’S Theorem, 2017 Nicolaus Copernicus University of Toruń

#### The Common Invariant Subspace Problem And Tarski’S Theorem, Grzegorz Pastuszak

*Electronic Journal of Linear Algebra*

This article presents a computable criterion for the existence of a common invariant subspace of $n\times n$ complex matrices $A_{1}, \dots ,A_{s}$ of a fixed dimension $1\leq d\leq n$. The approach taken in the paper is model-theoretic. Namely, the criterion is based on a constructive proof of the renowned Tarski's theorem on quantifier elimination in the theory $\ACF$ of algebraically closed fields. This means that for an arbitrary formula $\varphi$ of the language of fields, a quantifier-free formula $\varphi'$ such that $\varphi\lra\varphi'$ in $\ACF$ is given explicitly. The construction of $\varphi'$ is ...

The Feferman-Vaught Theorem, 2017 Wesleyan University

#### The Feferman-Vaught Theorem, Mostafa Mirabi

*Mostafa Mirabi*

Joint Laver Diamonds And Grounded Forcing Axioms, 2017 The Graduate Center, City University of New York

#### Joint Laver Diamonds And Grounded Forcing Axioms, Miha Habič

*All Dissertations, Theses, and Capstone Projects*

In chapter 1 a notion of independence for diamonds and Laver diamonds is investigated. A sequence of Laver diamonds for κ is *joint* if for any sequence of targets there is a single elementary embedding *j* with critical point κ such that each Laver diamond guesses its respective target via *j*. In the case of measurable cardinals (with similar results holding for (partially) supercompact cardinals) I show that a single Laver diamond for κ yields a joint sequence of length κ, and I give strict separation results for all larger lengths of joint sequences. Even though the principles get strictly ...

Choice Of Choice: Paradoxical Results Surrounding Of The Axiom Of Choice, 2017 Union College - Schenectady, NY

#### Choice Of Choice: Paradoxical Results Surrounding Of The Axiom Of Choice, Connor Hurley

*Honors Theses*

When people think of mathematics they think "right or wrong," "empirically correct" or "empirically incorrect." Formalized logically valid arguments are one important step to achieving this definitive answer; however, what about the underlying assumptions to the argument? In the early 20th century, mathematicians set out to formalize these assumptions, which in mathematics are known as axioms. The most common of these axiomatic systems was the Zermelo-Fraenkel axioms. The standard axioms in this system were accepted by mathematicians as obvious, and deemed by some to be sufficiently powerful to prove all the intuitive theorems already known to mathematicians. However, this system ...

On Tarski's Axiomatic Foundations Of The Calculus Of Relations, 2017 Hungarian Academy of Sciences

#### On Tarski's Axiomatic Foundations Of The Calculus Of Relations, Hajnal Andréka, Steven Givant, Peter Jipsen, István Németi

*Mathematics, Physics, and Computer Science Faculty Articles and Research*

It is shown that Tarski’s set of ten axioms for the calculus of relations is independent in the sense that no axiom can be derived from the remaining axioms. It is also shown that by modifying one of Tarski’s axioms slightly, and in fact by replacing the right-hand distributive law for relative multiplication with its left-hand version, we arrive at an equivalent set of axioms which is redundant in the sense that one of the axioms, namely the second involution law, is derivable from the other axioms. The set of remaining axioms is independent. Finally, it is shown ...

Relation Algebras, Idempotent Semirings And Generalized Bunched Implication Algebras, 2017 Chapman University

#### Relation Algebras, Idempotent Semirings And Generalized Bunched Implication Algebras, Peter Jipsen

*Mathematics, Physics, and Computer Science Faculty Articles and Research*

This paper investigates connections between algebraic structures that are common in theoretical computer science and algebraic logic. Idempotent semirings are the basis of Kleene algebras, relation algebras, residuated lattices and bunched implication algebras. Extending a result of Chajda and Länger, we show that involutive residuated lattices are determined by a pair of dually isomorphic idempotent semirings on the same set, and this result also applies to relation algebras. Generalized bunched implication algebras (GBI-algebras for short) are residuated lattices expanded with a Heyting implication. We construct bounded cyclic involutive GBI-algebras from so-called weakening relations, and prove that the class of weakening ...

Does Logic Help Us Beat Monty Hall?, 2017 Cedarville University

#### Does Logic Help Us Beat Monty Hall?, Adam J. Hammett, Nathan A. Harold, Tucker R. Rhodes

*The Research and Scholarship Symposium*

The classical Monty Hall problem entails that a hypothetical game show contestant be presented three doors and told that behind one door is a car and behind the other two are far less appealing prizes, like goats. The contestant then picks a door, and the host (Monty) is to open a different door which contains one of the bad prizes. At this point in the game, the contestant is given the option of keeping the door she chose or changing her selection to the remaining door (since one has already been opened by Monty), after which Monty opens the chosen ...

From Pythagoreans And Weierstrassians To True Infinitesimal Calculus, 2017 Bar-Ilan University

#### From Pythagoreans And Weierstrassians To True Infinitesimal Calculus, Mikhail Katz, Luie Polev

*Journal of Humanistic Mathematics*

In teaching infinitesimal calculus we sought to present basic concepts like continuity and convergence by comparing and contrasting various definitions, rather than presenting “the definition” to the students as a monolithic absolute. We hope that our experiences could be useful to other instructors wishing to follow this method of instruction. A poll run at the conclusion of the course indicates that students tend to favor infinitesimal definitions over epsilon-delta ones.

The Proscriptive Principle And Logics Of Analytic Implication, 2017 The Graduate Center, City University of New York

#### The Proscriptive Principle And Logics Of Analytic Implication, Thomas M. Ferguson

*All Dissertations, Theses, and Capstone Projects*

The analogy between inference and mereological containment goes at least back to Aristotle, whose discussion in the *Prior Analytics* motivates the validity of the syllogism by way of talk of parts and wholes. On this picture, the application of syllogistic is merely the *analysis* of concepts, a term that presupposes—through the root ἀνά + λύω —a mereological background.

In the 1930s, such considerations led William T. Parry to attempt to codify this notion of logical containment in his system of analytic implication AI. Parry’s original system AI was later expanded to the system PAI. The hallmark of Parry’s ...

Together We Know How To Achieve: An Epistemic Logic Of Know-How, 2017 Lafayette College

#### Together We Know How To Achieve: An Epistemic Logic Of Know-How, Pavel Naumov, Jia Tao

*Faculty Research and Reports*

The existence of a coalition strategy to achieve a goal does not necessarily mean that the coalition has enough information to know how to follow the strategy. Neither does it mean that the coalition knows that such a strategy exists. The paper studies an interplay between the distributed knowledge, coalition strategies, and coalition "know-how" strategies. The main technical result is a sound and complete trimodal logical system that describes the properties of this interplay.