One-Variable Fragments of First-Order Many-Valued Logics

In this thesis we study one-variable fragments of first-order logics. Such a one-variable fragment consists of those first-order formulas that contain only unary predicates and a single variable. These fragments can be viewed from a modal perspective by replacing the universal and existential quantifier with a box and diamond modality, respectively, and the unary predicates with corresponding propositional variables. Under this correspondence, the one-variable fragment of first-order classical logic famously corresponds to the modal logic S5. This thesis explores some such correspondences between first-order and modal logics. Firstly, we study first-order intuitionistic logics based on linear intuitionistic Kripke frames. We show that their one-variable fragments correspond to particular modal Gödel logics, defined over many-valued S5-Kripke frames. For a large class of these logics, we prove the validity problem to be decidable, even co-NP-complete. Secondly, we investigate the one-variable fragment of first-order Abelian logic, i.e., the first-order logic based on the ordered additive group of the reals. We provide two completeness results with respect to Hilbert-style axiomatizations: one for the one-variable fragment, and one for the one-variable fragment that does not contain any lattice connectives. Both these fragments are proved to be decidable. Finally, we launch a much broader algebraic investigation into one-variable fragments. We turn to the setting of first-order substructural logics (with the rule of exchange). Inspired by work on, among others, monadic Boolean algebras and monadic Heyting algebras, we define monadic commutative pointed residuated lattices as a first (algebraic) investigation into one-variable fragments of this large class of first-order logics. We prove a number of properties for these newly defined algebras, including a characterization in terms of relatively complete subalgebras as well as a characterization of their congruences

An Objection to Naturalism and Atheism from Logic

I proffer a success argument for classical logical consequence. I articulate in what sense that notion of consequence should be regarded as the privileged notion for metaphysical inquiry aimed at uncovering the fundamental nature of the world. Classical logic breeds necessitism. I use necessitism to produce problems for both ontological naturalism and atheism

“Almost Identical with Itself” : A Search for a Logic of Fuzzy Identity

This thesis grows out of a fascination with the vagueness of natural language, its manifestation in the ancient Sorites paradox, and the way in which the paradox is dealt with in fuzzy logic. It is an attempt to resolve the tension between two versions of the paradox, and the related problem of whether identity can be fuzzy. If it can be fuzzy, then the most popular argument against vague objects is mistaken, which would be great news for those who hold that there can be vagueness in the world independently of our representation or knowledge of it. The standard Sorites is made up of conditionals about an ordinary predicate (e.g. “heap”) by the rule of modus ponens. It is typically solved in fuzzy logic by interpreting the predicate as a fuzzy relation and showing that the argument fails as a result. There is another, less known version of the paradox, based on the identity predicate and the rule of substitutivity of identicals. The strong analogy between the two versions suggests that their solutions might be analogical as well, which would make identity just as vague as any relation. Yet the idea of vague identity has traditionally been rejected on both formal and philosophical grounds. Even Nicholas J. J. Smith, who is known for his positive attitude toward fuzzy relations in general, denies that identity could be fuzzy. The opposite position is taken by Graham Priest, who argues for a fuzzy interpretation of identity as a similarity relation. Following Priest, I aim to show that there is a perfectly sensible logic of fuzzy identity and that a fuzzy theoretician of vagueness therefore cannot rule out fuzzy identity on logical grounds alone. I compare two fuzzy solutions to the identity Sorites: Priest’s solution, based on the notion of local validity, and B. Jack Copeland’s solution, based on the failure of contraction in sequent calculus. I provide a synthesis of the two solutions, suggesting that Priest’s local validity counts as a genuine kind of validity even if he might not think so himself. The substitutivity of identicals is not locally valid in Priest’s logic, however; his solution only applies to a special case with the rule of transitivity. Applying L. Valverde’s representation theorem and other mathematical results, I lay the foundation for a stronger logic where the substitutivity rule is locally valid and the two Sorites merge into one paradox with one solution. Finally, I defend fuzzy identity against Gareth Evans’ argument that vague identity leads to contradiction, and Smith’s argument that vague identity is not really identity. The former relies on a fallacious application of the substitutivity rule; to the latter, my principal response is to question Smith’s understanding of identity and argue for a broader one. I conclude that not only is fuzzy identity logically possible, but it also has potential applicability in metaphysics and elsewhere

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

Redefining logical constants as inference markers

There is currently no universally accepted general de nition of logical constanthood. With a view to addressing this issue, we follow a pragmatist ratio- nale, according to which, some notion can be identi ed as a logical constant by considering the way in which it is used in our everyday reasoning practices, and argue that a logical constant has to be seen as encoding some kind of dynamic meaning, which marks the presence of an inferential transition among proposi- tional contents. We then put forth a characterisation of logical constants that takes into account their syntactic, semantic and pragmatic roles. What follows from our proposal is that logical constanthood can be best understood as a func- tional property that is satis ed only by certain uses of the relevant notions.peer-reviewe

Epistemic Modality and Hyperintensionality in 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. I also develop a novel topic-sensitive truthmaker semantics for dynamic epistemic logic, and develop a novel dynamic epistemic two-dimensional hyperintensional semantics. 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. Chapters \textbf{8-12} provide cases demonstrating how the two-dimensional intensions of epistemic two-dimensional semantics solve the access problem in the epistemology of mathematics. Chapter \textbf{8} examines the interaction between topic-sensitive epistemic two-dimensional truthmaker semantics, the axioms of epistemic set theory, large cardinal axioms, the Epistemic Church-Turing Thesis, the modal axioms governing the modal profile of $\Omega$-logic, Orey sentences such as the Generalized Continuum Hypothesis, and absolute decidability. Chapter \textbf{9} examines the modal profile of $\Omega$-logic in set theory. Chapter \textbf{10} examines the modal commitments of abstractionism, in particular necessitism, and epistemic modality and the epistemology of abstraction. Chapter \textbf{11} avails of modal coalgebras 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. Chapter \textbf{12} provides a modal logic for rational intuition and provides a 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 multi-hyperintensional, topic-sensitive epistemic two-dimensional truthmaker semantics developed in chapters \textbf{2} and \textbf{4} is applied in chapters \textbf{7}, \textbf{8}, \textbf{10}, \textbf{11}, \textbf{12}, and \textbf{14}.} *Please know that the 5 axiom was meant rather than the B axiom in ch. 10