a closer look at manifest consequence
Fine (2007) argues that Frege’s puzzle and its relatives demonstrate a need for a basic reorientation of the field of semantics. According to this reorientation, the domain of semantic facts would be closed not under the classical consequence relation but only under a stronger relation Fine calls “manifest consequence.” I examine Fine’s informally sketched analyses of manifest consequence, showing that each can be amended to determine a class of strong consequence relations. A best candidate relation emerges from each of the two classes, and I prove that the two candidates extensionally coincide. The resulting consequence relation is of independent interest, for it might be held to constitute a cogent standard of reasoning that proceeds under a deficient grasp on the identity of objects.
logic in the tractatus
I present a reconstruction of the logical system of the Tractatus, which differs from classical logic in two ways. It includes an account of Wittgenstein’s “formseries” device, which suffices to express some effectively generated countably infinite disjunctions. And its attendant notion of structure is relativized to the fixed underlying universe of what is named.
There follow three results. First, the class of concepts definable in the system is closed under finitary induction. Second, if the universe of objects is countably infinite, then the property of being a tautology is \Pi^1_1 -complete. But third, it is only granted the assumption of countability that the class of tautologies is \Sigma_1-definable in set theory.
Wittgenstein famously urges that logical relationships must show themselves in the structure of signs. He also urges that the size of the universe cannot be prejudged. The results of this paper indicate that there is no single way in which logical relationships could be held to make themselves manifest in signs, which does not prejudge the number of objects.
spinoza's proof of the existence of the human mind
According to Descartes, the existence of one’s own mind is recognized in any of one’s thoughts. Spinoza, however, begins only with an axiom that man thinks, which does not imply that there is such a thing as a man. Instead, recognizing the existence of a human mind requires identifying a single thing as an object of emotions like love and desire. So for Spinoza, recognition of one’s own existence, like existence itself, is a struggle for understanding.
facts as figures
Russell (1913/1984), Williamson (1985), Fine (2000), and Dorr (2004) reject the distinction of a relation from its converse. The basis of this rejection is a principle, call it the Uniqueness of Unifiers (UU), to the effect that there's at most one relation in whose exemplification any given relational state consists. Now, say that a proper contraction of a dyadic relation R is a monadic property F such that for something to have F is just for it to bear R to itself. I argue that principle UU also undermines the distinction between a relation and its contraction. Since a relation and its contraction could not be distinct yet cannot be identical, therefore UU implies that nothing bears a relation to itself. So Russell et al must either abandon UU and the objection to the concept of converse, or embrace a more radical understanding of relations than they've acknowledged.
logic in the tractatus ii
In 'Logic in the Tractatus I', I argued that the conception of logic in the Tractatus might be coherent only if the number of objects is countable. In this paper, I evaluate the converse.