Aspects of Consequence

Abstract

This dissertation consists of three papers on issues concerning logical consequence. My first chapter addresses Tarski's invariance account of logicality. I argue that his account is set up in such a way as to exclude indefinite expressions from counting as logical. After arguing that some indefinite expressions such as David Hilbert's epsilon operator are logical, I show how to amend his framework to allow indefinite expressions a chance at logicality and that doing this sheds light on a dispute in the philosophy of mathematics concerning the logical status of abstraction principles.

My second chapter demonstrates that moderate inferentialism has some problematic consequences. Moderate inferentialism generates meanings for logical expressions in terms of classes of models over which the inference rules for these expressions preserve validity. James Garson has shown that we can generate <italic>categorical</italic> meanings for the conditional, conjunction, and negation in this way. My negative result shows that this does not generalize to the full set of standard logical constants. After proving this, lessons are drawn for both intuitionistic and classical versions of moderate inferentialism.

The third chapter investigates the Stoic account of themata---argument-to-argument rules which are used to test arguments for validity in light of their rejection of redundant arguments. I introduce the basics of the Stoic notion of validity, argue for a straightforward and literal interpretation of the rejection of redundant arguments, and show how prior reconstructions violate this constraint. I then give a <italic>partial</italic> reconstruction for their proof system, and discuss some problems in understanding the Stoic notion of validity. I draw no firm conclusions about the Stoics's notion of validity, but suggest some reasons to think that comparisons with modern conceptions of validity are misleading.