Speaker: Harold Hodes (Cornell)
Title: Why Ramify?
Abstract: Are there any good reasons to prefer a ramified type-assignment system to a simple type-assignment system? I will address this question from the perspective of a reconstruction of Russell’s views between 1903 and 1919. The first reason that I’ll consider is the purported existence of paradoxes that arise for a simple type-assignment system, beginning with the so-called “Russell’s propositional paradox”. These arguments depend on certain converse-compositional principles. Russell’s “syntactic picture” of propositions may seem to recommend such principles. But when we take account of Russell’s doctrine that a propositional function is not a constituent of its values, these principles become too implausible to make these arguments troubling. The second that I’ll consider is conditional on a substitutional interpretation of quantification over types other than that of individuals (call them “higher types”). There is evidence that Russell was tacitly inclined towards this interpretation. And as a reason for ramification it stands up well. A strong construal of this interpretation opens a way to make sense of Russell’s puzzling simultaneous repudiation of propositions and his willingness to quantify over them and over propositional functions. But that route runs into trouble with Russell’s commitment to the finitude of human understanding. It seems that there is no coherent Russellian philosophy that can stand behind Russell’s approach to logic.
Speaker: Donald Martin (UCLA)
Title: Theorems about a concept proved from axioms about a stronger concept: What do they show?
Abstract: I will talk about the use of strong mathematical concepts to prove theorems about weaker ones.
One of the many classes of examples is that of proofs of first order arithmetical statements in second order Peano arithmetic. By second order Peano arithmetic (or just second order arithmetic), I mean the standard two-sorted first order theory with a comprehension schema. The axioms of this theory are derived from the concept of the natural numbers and sets of natural numbers. We can see that these axioms have to be true of any structure instantiating this concept. In second order arithmetic one can prove many sentences of first order arithmetic that are unprovable in first order (Peano) arithmetic. The axioms of first order arithmetic are derived from the concept of the natural numbers. Without consideration of second order objects, we can see that these axioms have to be true of any structure instantiating that concept. There are proper extensions of first order arithmetic whose axioms also have this property, but there are first order arithmetical consequences of the second order axioms that we have no idea how to prove from axioms with the property.
The talk will be about the question of whether, when, and in what ways such a use of a strong concept to prove a theorem about a weaker one shows that the theorem has to be true of any instantiation of the weaker concept.
Speaker: Gila Sher (UC San Diego)
Title: The Foundational Problem of Logic
Abstract: The construction of a systematic philosophical foundation for logic is a notoriously difficult problem. In Part 1, I suggest that the problem is largely methodological, having to do with the common philosophical conception of “providing a foundation”. I offer an alternative to the common methodology - "foundational holism" - which combines a strong foundational requirement (veridical justification) with the use of non-traditional, holistic tools to achieve this result. In Part 2 I delineate an outline of a foundation for logic, employing this methodology. The foundational inquiry begins with the question whether logic requires a veridical justification, one involving the world and not just the mind. My answer is positive. Logic, the investigation suggests, is grounded in the formal aspect of reality. The outline explains the nature of this aspect and the way it both constrains and enables logic. It then proceeds to explain logic’s role in our overall system of knowledge, the relation between logic and mathematics, the normativity of logic, the characteristic traits of logic, and error and revision in logic.