Thursday, December 15, 2005
Communications on lattice of theories
Communication from John Sowa to ONTAC working group
I don't claim that Wittgenstein envisioned a lattice of theories, and given the vehemence of his rejection of Waismann's attempt to summarize his views [1], I wouldn't hold out much hope for his approval of my version. But I would claim that many of Wittgenstein's games -- including all those that involve declarative language and queries and many that involve commands -- could be mapped to theories of the lattice:
1. Translate Wittgenstein's rules to the axioms (constraints) of a theory T.
2. Every declarative sentence in that game would be compatible with T -- i.e., consistent with T and if not a theorem or axiom of T then an extension of T to a more specialized theory.
3. Every question in that game would be a query that could be answered by methods similar to Prolog goals or SQL queries.
4. Every command in that game could be interpreted as a request to start with theory T in the lattice and move by some sequence of theory revision operators to another theory T'.
Many of Wittgenstein's remarks illustrate issues that are still open research questions, but I believe that the subject matter of the various domains could be characterized by theories in the lattice.
[1] The issues related to Waismann and Wittgenstein are described at http://plato.stanford.edu/entries/wittgenstein-atomism/ .