Haim Gaifman Reasoning with Limited Resources and Assigning Probabilities to Arithmetical Statements

Reasoning with Limited Resources and Assigning Probabilities to Arithmetical Statements

There are three sections in this paper. The first is a philosophical discussion of the general problem of reasoning under limited deductive capacity. The second sketches a rigorous way of assigning probabilities to statements in pure arithmetic; motivated by the preceding discussion, it can nonetheless be read separately. The third is a philosophical discussion that highlights the shifting cont...

Isomorphism Types of Trees1

Inductive Coherence

While probability theory is normally applied to external environments, there has been some recent interest in probabilistic modeling of the outputs of computations that are too expensive to run. Since mathematical logic is a powerful tool for reasoning about computer programs, we consider this problem from the perspective of integrating probability and logic. Recent work on assigning probabilit...

Unifying Probability and Logic for Learning

1 Uncertain knowledge can be modeled by using graded probabilities rather than binary truth-values, but so far a completely satisfactory integration of logic and probability has been lacking. In particular the inability of confirming universal hypotheses has plagued most if not all systems so far. We address this problem head on. The main technical problem to be discussed is the following: Give...

What Gödel ’ s Incompleteness Result Does and Does Not Show

In a recent paper S. McCall adds another link to a chain of attempts to enlist Gödel’s incompleteness result as an argument for the thesis that human reasoning cannot be construed as being carried out by a computer. McCall’s paper is undermined by a technical oversight. My concern however is not with the technical point. The argument from Gödel’s result to the no-computer thesis can be made wit...