Universal Logic

Structuralist Logic
Arnold Koslow

The Graduate Center

The City University of New York

The first meeting will be devoted to an elementary introduction to the structuralist conception of logic that has its historical roots in the work of P. Hertz and G. Gentzen. We shall develop the notion of an implication relation that generalizes the notion of implication and the corresponding notion of an implication structure will become the central concept of logical structuralism. Given this generality we will show how to define the logical operators for arbitrary implication structures. We shall explore both the extent and the particular features of the variety of implication relations and the variety of those logical systems that these implication structures provide.
The second meeting will define the concept of truth-value assignments to the members of arbitrary implication structures, even when sometimes those members may not have truth-values. We shall prove some completeness theorems which justify the idea that these are genuine truth-value assignments, and we shall explore the problem of determining when the logical operators are extensional and when extensionality fails. We shall also introduce the notion of a modal operator on an implication structure using two simple conditions on implication relations, and prove certain results that show when such modals exist on a structure, and when they do not. It will also be shown that all modal operators thus defined, are non-extensional.
In the third meeting we shall show how the structuralist approach enables one to define an accessibility relation between Tarskian theories of an implication structure and obtain all the Kripkean systematizations of familiar modal systems without the use of possible world semantics. Finally given the variety of logical structures that structuralism generates, we shall consider how one should understand this pluralism and how one should respond to those views which maintain that there is only one correct logic that cannot be revised.

References:

A. Koslow, A Structuralist Theory of Logic, A. Koslow, Cambridge University Press, 1992 (Paperback)

A. Koslow, "The Implicational Nature of Logic", European Review of Philosophy (The Nature of Logic), volume 4, pp.111-155, ed. Achille C. Varzi, CSLI Publications, Stanford 1999.

J.C.Beall and G. Restall, Logical Pluralism, Oxford University Press Paperback, 2006,

G. Priest, Doubt truth to be a Liar, Oxford University Press, 2006, Chapter 10. Logic and Revisability, and Chapter 12. Logical Pluralism.

The first meeting will be devoted to an elementary introduction to the structuralist conception of logic that has its historical roots in the work of P. Hertz and G. Gentzen. We shall develop the notion of an implication relation that generalizes the notion of implication and the corresponding notion of an implication structure will become the central concept of logical structuralism. Given this generality we will show how to define the logical operators for arbitrary implication structures. We shall explore both the extent and the particular features of the variety of implication relations and the variety of those logical systems that these implication structures provide.
The second meeting will define the concept of truth-value assignments to the members of arbitrary implication structures, even when sometimes those members may not have truth-values. We shall prove some completeness theorems which justify the idea that these are genuine truth-value assignments, and we shall explore the problem of determining when the logical operators are extensional and when extensionality fails. We shall also introduce the notion of a modal operator on an implication structure using two simple conditions on implication relations, and prove certain results that show when such modals exist on a structure, and when they do not. It will also be shown that all modal operators thus defined, are non-extensional.
In the third meeting we shall show how the structuralist approach enables one to define an accessibility relation between Tarskian theories of an implication structure and obtain all the Kripkean systematizations of familiar modal systems without the use of possible world semantics. Finally given the variety of logical structures that structuralism generates, we shall consider how one should understand this pluralism and how one should respond to those views which maintain that there is only one correct logic that cannot be revised.