Thesis advisor: Andrea Staiti / This study concerns the nature and role of analyticity in the work of Immanuel Kant and Edmund Husserl. Its initial goal is that of clarifying the place of analytic judgment in Kant's critical project. Against the widely held assumption that analytic judgment has no role to play in the critical project, I show that analytic judgment has a precise and genuinely important role to play in the context of Kant's metaphysics. Analytic judgment has the role of clarifying our a priori conceptual repertoire and thus of making possible the synthetic a priori judgments that are properly constitutive of metaphysics. The next goal of the study is that of unifying and defending Kant's various characterizations of analytic judgment. Whereas a number of commentators have suggested that Kant is vague or ambivalent as regards the properties of analytic judgment, I show that we can extract a clear, consistent picture of analytic judgment from his work. The key to seeing this, I argue, is becoming clear on Kant's basic assumptions concerning concepts, logic, and propositional form. Subsequently, I turn to Husserl. Picking up on the fact that for Husserl, too, analyticity has metaphysical, or ontological significance, I spell out his conception of analyticity in detail. I show that analyticity for Husserl embraces two essentially symmetrical domains of law: the a priori laws of objective givenness and the a priori laws of propositional form. I then bring Husserl and Kant together. After showing that Husserl fails to capture the essence of Kant's theory of analytic judgment, and so fails to see exactly where he stands relative to Kant, I argue that what ultimately distinguishes Husserl from Kant is the claim that analytic truth is properly articulated in a purely formal context. I show that this departure from Kant has extremely significant consequences. For example, it enables Husserl to describe whole systems of judgment, such as mathematics or logic, as analytic; and it enables Husserl to defend the possibility of analytic judgments having empirical content. / Thesis (PhD) — Boston College, 2014. / Submitted to: Boston College. Graduate School of Arts and Sciences. / Discipline: Philosophy.

This dissertation examines the Carnap-Quine debate on analyticity with the objective of identifying exactly what is at stake. Close scrutiny of Quine’s criticism of the definitions of analyticity reveals that most of his objections are convincing only if they are considered in relation to the definitions of analyticity in natural language. Carnap, however, defines analyticity in artificial languages. The dissertation also shows that Carnap can meet the objections to his definitions by using a perspective based within his own philosophy. After examining the presumptions of each party to the debate, the dissertation concludes that the disagreement between Carnap and Quine on the notion of analyticity is rooted in their different approaches to empiricism and that there is nothing said by either philosopher which proves that one approach has ultimate advantages over the other. It is thus impossible to identify a winner in the Carnap-Quine debate on analyticity. The process of arriving at this conclusion starts with a discussion in the first three chapters of Carnap’s philosophy followed by a critical and detailed discussion of his syntactical and semantical definitions of analyticity and the advantages and disadvantages of each. Chapter Four examines Quine’s objections to Carnap’s definitions of essential predication and shows that his objections do not undermine Carnap’s definitions of this notion in artificial languages. It also shows how vital providing a proper definition of essential predication in natural language is for Carnap’s philosophy and examines whether or not he is able to do so. Chapter Five analyzes Quine’s objections to Carnap’s definitions of logical truth and demonstrates that Carnap is able to respond to all of them when the discussion is situated within his philosophical system. Again, Quine’s objections to definitions of logical truth are meaningful only if they are considered in relation to natural language, which is not Carnap’s concern. The dissertation concludes by showing that both Carnap and Quine arrived at their conclusions with respect to the nature of logical sentences, based not on the arguments in their debate on analyticity, but on their philosophical considerations regarding the principle of empiricism: for Carnap, logical sentences are out of the realm of knowledge and independent of matters of fact whereas, for Quine, these are as empirical as other sentences. Nothing either says in their debate can convince the other to accept a different viewpoint.

It is well known that very few models of interacting systems particularly those in dimension higher than two, can be solved exactly. The mean-field treatment is the first step in approximate calculations for such models. Although mean-field approximation leads to sufficiently accurate results of the thermodynamic properties of these systems away from critical points, most often it fails miserably close to the critical points. In this thesis, we propose to study direct methods (not based on any mean-field type approximations) for proving the exponential decay of the two point-correlation functions and the analyticity of the pressure (free energy per unit volume) for models of Kac type. The methods are based on the Helffer-Sjöstrand formula for the covariance in terms of Witten's Laplacians.

This dissertation examines the Carnap-Quine debate on analyticity with the objective of identifying exactly what is at stake. Close scrutiny of Quine’s criticism of the definitions of analyticity reveals that most of his objections are convincing only if they are considered in relation to the definitions of analyticity in natural language. Carnap, however, defines analyticity in artificial languages. The dissertation also shows that Carnap can meet the objections to his definitions by using a perspective based within his own philosophy. After examining the presumptions of each party to the debate, the dissertation concludes that the disagreement between Carnap and Quine on the notion of analyticity is rooted in their different approaches to empiricism and that there is nothing said by either philosopher which proves that one approach has ultimate advantages over the other. It is thus impossible to identify a winner in the Carnap-Quine debate on analyticity. The process of arriving at this conclusion starts with a discussion in the first three chapters of Carnap’s philosophy followed by a critical and detailed discussion of his syntactical and semantical definitions of analyticity and the advantages and disadvantages of each. Chapter Four examines Quine’s objections to Carnap’s definitions of essential predication and shows that his objections do not undermine Carnap’s definitions of this notion in artificial languages. It also shows how vital providing a proper definition of essential predication in natural language is for Carnap’s philosophy and examines whether or not he is able to do so. Chapter Five analyzes Quine’s objections to Carnap’s definitions of logical truth and demonstrates that Carnap is able to respond to all of them when the discussion is situated within his philosophical system. Again, Quine’s objections to definitions of logical truth are meaningful only if they are considered in relation to natural language, which is not Carnap’s concern. The dissertation concludes by showing that both Carnap and Quine arrived at their conclusions with respect to the nature of logical sentences, based not on the arguments in their debate on analyticity, but on their philosophical considerations regarding the principle of empiricism: for Carnap, logical sentences are out of the realm of knowledge and independent of matters of fact whereas, for Quine, these are as empirical as other sentences. Nothing either says in their debate can convince the other to accept a different viewpoint.

This dissertation examines the Carnap-Quine debate on analyticity with the objective of identifying exactly what is at stake. Close scrutiny of Quine’s criticism of the definitions of analyticity reveals that most of his objections are convincing only if they are considered in relation to the definitions of analyticity in natural language. Carnap, however, defines analyticity in artificial languages. The dissertation also shows that Carnap can meet the objections to his definitions by using a perspective based within his own philosophy. After examining the presumptions of each party to the debate, the dissertation concludes that the disagreement between Carnap and Quine on the notion of analyticity is rooted in their different approaches to empiricism and that there is nothing said by either philosopher which proves that one approach has ultimate advantages over the other. It is thus impossible to identify a winner in the Carnap-Quine debate on analyticity. The process of arriving at this conclusion starts with a discussion in the first three chapters of Carnap’s philosophy followed by a critical and detailed discussion of his syntactical and semantical definitions of analyticity and the advantages and disadvantages of each. Chapter Four examines Quine’s objections to Carnap’s definitions of essential predication and shows that his objections do not undermine Carnap’s definitions of this notion in artificial languages. It also shows how vital providing a proper definition of essential predication in natural language is for Carnap’s philosophy and examines whether or not he is able to do so. Chapter Five analyzes Quine’s objections to Carnap’s definitions of logical truth and demonstrates that Carnap is able to respond to all of them when the discussion is situated within his philosophical system. Again, Quine’s objections to definitions of logical truth are meaningful only if they are considered in relation to natural language, which is not Carnap’s concern. The dissertation concludes by showing that both Carnap and Quine arrived at their conclusions with respect to the nature of logical sentences, based not on the arguments in their debate on analyticity, but on their philosophical considerations regarding the principle of empiricism: for Carnap, logical sentences are out of the realm of knowledge and independent of matters of fact whereas, for Quine, these are as empirical as other sentences. Nothing either says in their debate can convince the other to accept a different viewpoint.

Consider a family of cubic parabolic polynomials given by for non-zero complex parameters such that for each the polynomial is a parabolic polynomial, that is, the polynomial has a parabolic fixed point and the Julia set of , denoted by , does not contain any critical points of . We also assumed that for each , one finite critical point of the polynomial escapes to the super-attracting fixed point infinity. So, the Julia sets are disconnected. The concern about the family is that the members of this family are generally not even bi-Lipschitz conjugate on their Julia sets. We have proved that the parameter set is open and contains a deleted neighborhood of the origin 0. Our main result is that the Hausdorff dimension function defined by is real analytic. To prove this we have constructed a holomorphic family of holomorphic parabolic graph directed Markov systems whose limit sets coincide with the Julia sets of polynomials up to a countable set, and hence have the same Hausdorff dimension. Then we associate to this holomorphic family of holomorphic parabolic graph directed Markov systems an analytic family, call it , of conformal graph directed Markov systems with infinite number of edges in order to reduce the problem of real analyticity of Hausdorff dimension for the given family of polynomials to prove the corresponding statement for the family .

Doctor of Philosophy(PhD) / Though Quine's argument against the analytic-synthetic distinction is widely disputed, one of the major effects of his argument has been to popularise the belief that there is no sharp distinction between science and philosophy. This thesis begins by distinguishing reductive from holistic empiricism, showing why reductive empiricism is false, refuting the major objections to holistic empiricism and stating the limits on human knowledge it implies. Quine's arguments (and some arguments that have been mistakenly attributed to him) from holism against the analytic-synthetic are considered, and while many of them are found wanting one good argument is presented. Holism does not, however, imply that there is no sharp distinction between science and philosophy, and indeed implies that the distinction between scientific and philosophical disputes is perfectly sharp. The grounds upon which philosophical disputes may be resolved are then sought for and deliniated.

Doctor of Philosophy(PhD) / Though Quine's argument against the analytic-synthetic distinction is widely disputed, one of the major effects of his argument has been to popularise the belief that there is no sharp distinction between science and philosophy. This thesis begins by distinguishing reductive from holistic empiricism, showing why reductive empiricism is false, refuting the major objections to holistic empiricism and stating the limits on human knowledge it implies. Quine's arguments (and some arguments that have been mistakenly attributed to him) from holism against the analytic-synthetic are considered, and while many of them are found wanting one good argument is presented. Holism does not, however, imply that there is no sharp distinction between science and philosophy, and indeed implies that the distinction between scientific and philosophical disputes is perfectly sharp. The grounds upon which philosophical disputes may be resolved are then sought for and deliniated.

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This thesis is an examination of Bertrand Russell's realist "bundle theory of particulars". In Russell's earlier work, the need to explain the unity and individuality of objects compelled him to accept particulars as well as universals as ultimate kinds of reality. Nevertheless, in carrying out his efforts to economize his ontology, he discovered he could not reduce properties to particulars, because there are some relations that resist nominalistic explanation, but particulars could be reduced to bundles of qualities. In this thesis, I show that the realist 'bundle theory' not only reduces the kinds of ultimate reality to one, i.e., to universal qualities, but also serves all of the purposes for which bare particulars were originally required. Specifically. I examine what I take to be the major criticisms leveled against the realist 'bundle theory': the problem of individuation, the problem of necessity, and the problem of analyticity. I defend the strength and consistency of Russell's theory and argue that it can answer to the objections. / Thesis / Doctor of Philosophy (PhD)