Complete Axiomatization of Probabilistic Threshold Semantics for
Inferences Among Conditionals: the System T

SPEAKER

Gerhard Schurz

ABSTRACT

I discuss different semantical consequence relations for inferences among of uncertain conditionals which are semantically treated as high conditional probabilities .I consider uncertainty-sum-consequence or P-consequence, an extension of P-consequence by default assumptions, and finally threshold-preservation-conse_quence, or T-consequence, which has been introduced by Hawthorne and Makinson (2006). It is the weakest of the three and the only one which guarantees strict probability preservation. In the first part of the talk, I compare there merits of these semantics in applications to cognitive psychology and to practical reasoning. In the second part I turn to the question of their axiomatization. While P-consequence has a nice and well-known axiomatization, the system P, the axiomatization of threshold-semantics is rather difficult. Hawthorne and Makinson's axiomatization O has been shown to by incomplete by Paris and Simmonds (2009), who have found a complete axiomatization by themselves which, however, is infinite and practically unfeasible. I propose a different axiomatization, the system T, which proved to be complete for the two-premise case. Initially I though the proof can be generalized to the n-premise case, but in the light of the results of Paris and Simmonds (2009) I am now skeptical concerning this question.