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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.
The first part of the chapter surveys some of the main ways in which the Sorites Paradox has figured in arguments in practical philosophy in recent decades, with special attention to arguments where the paradox is used as a basis for criticism. Not coincidentally, the relevant arguments all involve the transitivity of value in some way. The second part of the chapter is more probative, focusing on two main themes. First, it further addresses the relationship between the Sorites Paradox and the main arguments discussed in the first part, by elucidating in what sense they rely on (something like) tolerance principles. Second, it briefly discusses the prospect of rejecting the respective principles, aiming to show that we can do so for some of the arguments but not for others. The reason is that in the latter cases the principles do not function as independent premises in the reasoning but, rather, follow from certain fundamental features of the relevant scenarios. It is also argued that not even adopting what is arguably the most radical way to block the Sorites Paradox – that of weakening the consequence relation – suffices to invalidate these arguments.
This essay argues that the logical significance of most natural language expressions is indefinitely elastic. This, it is argued, undermines the idea that the meaning of a word is an item for which it stands, and puts pressure on the methods of conceptual analysis and theoretical elucidation that require context-invariant stable application conditions. Furthermore, it is argued that the insistence that such semantic content is needed which—impervious to local pragmatic concerns—remains stable and available for reasoning, gets things back to front. For in order to determine the correct inflection of any given use of an expression, its inferential relations—in the context of an utterance—must already be discerned. The lack of contextually independent content, however, presents no mystery about language use. For the ability to understand what is said is explained not only by shared practices and common interests, but also by the capacity for interlocutors to ask questions and explain what they mean.