Due to Gödel’s incompleteness results, the categoricity of a sufficiently rich mathematical theory and the semantic completeness of its underlying logic are two mutually exclusive ideals. For first- and second-order logics we obtain one of them with the cost of losing the other. In addition, in both these logics the rules of deduction for their quantifiers are non-categorical. In this paper I examine two recent arguments—Warren [43] and Murzi and Topey [30]—for the idea that the natural deduction rules for the first-order universal quantifier are categorical, i.e., they uniquely determine its semantic intended meaning. Both of them make use of McGee’s open-endedness requirement and the second one uses in addition Garson’s [19] local models for defining the validity of these rules. I argue that the success of both these arguments is relative to their semantic or infinitary assumptions, which could be easily discharged if the introduction rule for the universal quantifier is taken to be an infinitary rule, i.e., non-compact. Consequently, I reconsider the use of the $\omega $-rule and I show that the addition of the $\omega $-rule to the standard formalizations of first-order logic is categorical. In addition, I argue that the open-endedness requirement does not make the first-order Peano Arithmetic categorical and I advance an argument for its categoricity based on the inferential conservativity requirement.

This paper reconstructs and defends Kant’s argument for the transcendental status of reason’s principles of the systematic unity of nature in the Appendix to the Transcendental Dialectic. On the present account, these principles neither contain mere methodological recommendations for conducting scientific inquiry nor do they have the normative force of categorical imperatives—two extant interpretations of Kant’s discussion of reason in the Appendix. Instead, they are regulative yet transcendental principles restricted to theoretical cognition. The principles of the systematic unity of nature count as transcendental in virtue of their role as conditions of the inferential articulation of empirical concepts.

The dissertation introduces new sequent-calculi for free first- and second-order logic, and a hyper-sequent calculus for modal logics K, D, T, B, S4, and S5; to attain the calculi for the stronger modal logics, only external structural rules need to be added to the calculus for K, while operational and internal structural rules remain the same. Completeness and cut-elimination are proved for all calculi presented.

Philosophically, the dissertation develops an inferentialist, or proof-theoretic, theory of meaning. It takes as a starting point that the sense of a sentence is determined by the rules governing its use. In particular, there are two features of the use of a sentence that jointly determine its sense, the conditions under which it is coherent to assert that sentence and the conditions under which it is coherent to deny that sentence. The dissertation develops a theory of quantification as marking coherent ways a language can be expanded and modality as the means by which we can reflect on the norms governing the assertion and denial conditions of our language. If the view of quantification that is argued for is correct, then there is no tension between second-order quantification and nominalism. In particular, the ontological commitments one can incur through the use of a quantifier depend wholly on the ontological commitments one can incur through the use of atomic sentences. The dissertation concludes by applying the developed theory of meaning to the metaphysical issue of necessitism and contingentism. Two objections to a logic of contingentism are raised and addressed. The resulting logic is shown to meet all the requirement that the dissertation lays out for a theory of meaning for quantifiers and modal operators.

Abstract prepared by Andrew Parisi

E-mail: [email protected]

URL: https://opencommons.uconn.edu/dissertations/1480/

Inferentialism is a theory in the philosophy of language which claims that the meanings of expressions are constituted by inferential roles or relations. Instead of a traditional model-theoretic semantics, it naturally lends itself to a proof-theoretic semantics, where meaning is understood in terms of inference rules with a proof system. Most work in proof-theoretic semantics has focused on logical constants, with comparatively little work on the semantics of non-logical vocabulary. Drawing on Robert Brandom’s notion of material inference and Greg Restall’s bilateralist interpretation of the multiple conclusion sequent calculus, I present a proof-theoretic semantics for atomic sentences and their constituent names and predicates. The resulting system has several interesting features: (1) the rules are harmonious and stable; (2) the rules create a structure analogous to familiar model-theoretic semantics; and (3) the semantics is compositional, in that the rules for atomic sentences are determined by those for their constituent names and predicates.

Explaining genuine moral disagreement is a challenge for metaethical theories. For expressivists, this challenge comes from the plausibility of agents making seemingly univocal claims while expressing incongruent conative attitudes. I argue that metaethical inferentialism – a deflationary cousin to expressivism, which locates meaning in the inferential import of our moral assertions rather than the attitudes they express – offers a unique solution to this problem. Because inferentialism doesn’t locate the source of moral disagreements in a clash between attitudes, but instead in conflicts between the inferential import of ethical assertions, the traditional problem for expressivism can be avoided. After considering two forms of inferentialism that lead to revenge versions of the problem, I conclude by recommending that we understand the semantics of moral disagreements pragmatically: the source of univocity does not come from moral or semantic facts waiting to be described, but instead from the needs that ethical and semantic discourses answer – a solution to the problems of what we are to do and how we are to talk about it.

The dominant route to nondescriptivist views of normative and evaluative language is through the expressivist idea that normative terms have distinctive expressive roles in conveying our attitudes. This paper explores an alternative route based on two ideas. First, a core normative term ‘ought’ is a modal operator; and second, modal operators play a distinctive nonrepresentational role in generating meanings for the statements in which they figure. I argue that this provides for an attractive alternative to expressivist forms of nondescriptivism about normative language. In the final section of the paper, I explore ways it might be extended to evaluative language.