Epistemic Modality, Mind, and Mathematics

This book concerns the foundations of epistemic modality. I examine the nature of epistemic modality, when the modal operator is interpreted as concerning both apriority and conceivability, as well as states of knowledge and belief. The book demonstrates how epistemic modality relates to the computational theory of mind; metaphysical modality; the types of mathematical modality; to the epistemic status of large cardinal axioms, undecidable propositions, and abstraction principles in the philosophy of mathematics; to the modal profile of rational intuition; and to the types of intention, when the latter is interpreted as a modal mental state. Chapter \textbf{2} argues for a novel type of expressivism based on the duality between the categories of coalgebras and algebras, and argues that the duality permits of the reconciliation between modal cognitivism and modal expressivism. Chapter \textbf{3} provides an abstraction principle for epistemic intensions. Chapter \textbf{4} advances a topic-sensitive two-dimensional truthmaker semantics, and provides three novel interpretations of the framework along with the epistemic and metasemantic. Chapter \textbf{5} applies the fixed points of the modal $\mu$-calculus in order to account for the iteration of epistemic states, by contrast to availing of modal axiom 4 (i.e. the KK principle). Chapter \textbf{6} advances a solution to the Julius Caesar problem based on Fine's "criterial" identity conditions which incorporate conditions on essentiality and grounding. Chapter \textbf{7} provides a ground-theoretic regimentation of the proposals in the metaphysics of consciousness and examines its bearing on the two-dimensional conceivability argument against physicalism. The topic-sensitive epistemic two-dimensional truthmaker semantics developed in chapter \textbf{4} is availed of in order for epistemic states to be a guide to metaphysical states in the hyperintensional setting. Chapter \textbf{8} examines the modal commitments of abstractionism, in particular necessitism, and epistemic modality and the epistemology of abstraction. Chapter \textbf{9} examines the modal profile of $\Omega$-logic in set theory. Chapter \textbf{10} examines the interaction between epistemic two-dimensional truthmaker semantics, epistemic set theory, and absolute decidability. Chapter \textbf{11} avails of modal coalgebraic automata to interpret the defining properties of indefinite extensibility, and avails of epistemic two-dimensional semantics in order to account for the interaction of the interpretational and objective modalities thereof. The hyperintensional, topic-sensitive epistemic two-dimensional truthmaker semantics developed in chapter \textbf{2} is applied in chapters \textbf{7}, \textbf{8}, \textbf{10}, and \textbf{11}. Chapter \textbf{12} provides a modal logic for rational intuition and provides four models of hyperintensional semantics. Chapter \textbf{13} examines modal responses to the alethic paradoxes. Chapter \textbf{14} examines, finally, the modal semantics for the different types of intention and the relation of the latter to evidential decision theory

The mathematicization of nature.

This thesis defends the Quine-Putnam indispensability argument for mathematical realism and introduces a new indispensability argument for a substantial conception of truth. Chapters 1 and 2 formulate the main components of the Quine-Putnam argument, namely that virtually all scientific laws quantify over mathematical entities and thus logically presuppose the existence thereof. Chapter 2 contains a detailed discussion of the logical structure of some scientific theories that incorporate or apply mathematics. Chapter 3 then reconstructs the central assumptions of Quine's argument, concluding (provocatively) that "science entails platonism". Chapter 4 contains a brief discussion of some major theories of truth, including deflationary views (redundancy, disquotation). Chapter 5 introduces a new argument against such deflationary views, based on certain logical properties of truth theories. Chapter 6 contains a further discussion of mathematical truth. In particular, non-standard conceptions of mathematical truth such as "if-thenism" and "hermeneuticism". Chapter 7 introduces the programmes of reconstrual and reconstruction proposed by recent nominalism. Chapters 8 discusses modal nominalism, concluding that modalism is implausible as an interpretation of mathematics (if taken seriously, it suffers from exactly those epistemological problems allegedly suffered by realism). Chapter 9 discusses Field's deflationism, whose central motivating idea is that mathematics is (pace Quine and Putnam) dispensable in applications. This turns on a conservativeness claim which, as Shapiro pointed out in 1983, must be incorrect (using Godel's Theorems). I conclude in Chapter 10 that nominalistic views of mathematics and deflationist views of truth are both inadequate to the overall explanatory needs of science

Pseudo-contractions as Gentle Repairs

Updating a knowledge base to remove an unwanted consequence is a challenging task. Some of the original sentences must be either deleted or weakened in such a way that the sentence to be removed is no longer entailed by the resulting set. On the other hand, it is desirable that the existing knowledge be preserved as much as possible, minimising the loss of information. Several approaches to this problem can be found in the literature. In particular, when the knowledge is represented by an ontology, two different families of frameworks have been developed in the literature in the past decades with numerous ideas in common but with little interaction between the communities: applications of AGM-like Belief Change and justification-based Ontology Repair. In this paper, we investigate the relationship between pseudo-contraction operations and gentle repairs. Both aim to avoid the complete deletion of sentences when replacing them with weaker versions is enough to prevent the entailment of the unwanted formula. We show the correspondence between concepts on both sides and investigate under which conditions they are equivalent. Furthermore, we propose a unified notation for the two approaches, which might contribute to the integration of the two areas

Variables

Variables is a project at the intersection of the philosophies of language and logic. Frege, in the Begriffsschrift, crystalized the modern notion of formal logic through the first fully successful characterization of the behaviour of quantifiers. In Variables, I suggest that the logical tradition we have inherited from Frege is importantly flawed, and that Frege's move from treating quantifiers as noun phrases bearing word-world connection to sentential operators in the guise of second-order predicates leaves us both philosophically and technically wanting

Logický pluralismus v historické perspektivě

Logický pluralismus v historické perspektivě Pluralita logik je pojata jako výzva k hledání hlubšího pochopení povahy a významu logiky. Jsou zváženy dva základní přístupy k vymezování logiky vůči jiným disciplínám, totiž přístup vycházející z teorie modelů a přístup vycházející z teorie důkazů. Zkoumání historie, která vedla k těmto dvěma přístupům, vyzvihuje postavení logiky v Kantově epistemologii jako klíčové pro další vývoj. Analogický vývoj od Kantova pojetí geometrie ke vzniku plurality geometrických teorií vede k holistickému pohledu jak na geometrii, tak na logiku. Jako klíčové se nakonec ale ukáže především správně uchopit pragmatický význam logiky. Vzhledem k problémům vážícím se ke snahám logiku vymezit se jako nejvhodnější pojetí role logiky jeví inferencialismus a logický expresivismus. Tyto přístupy jsou rozvinuty do koncepce, která zdůraňuje, v návaznosti na historickou perspektivu práce, schopnost logiky vyvíjet se. 1Logical pluralism from historical perspective - Abstract The plurality of logics is understood as a challenge to seek a deeper understanding of the na- ture and import of logic. Two basic approaches to demarcation of logic are considered, the model-theoretic and the proof-theoretic one. Investigation of the history which led to these two appraoches identifies the postion of logic in Kant's epistemology as crucial for the devel- opment. An analogical development from Kant's conception of geometry to the plurality of geometric theories leads to a holistic view both of geometry and of logic. It furthermore proves essential to understand the pragmatic import of logic. Given the problems tied to the attempts to demarcate logic, inferentialism and logical expressivism are arrived at as jointly provid- ing the most appropriate account. These approaches are developed into a conception which stresses, in line with the historical perspective of the work, the ability of logic to develop.Ústav filosofie a religionistikyInstitute of Philosophy and Religious StudiesFaculty of ArtsFilozofická fakult