A Semantic Theory for Partial Entailments and Inductive Inferences

Dissertation, University of Minnesota (1989)

James Hawthorne
University of Oklahoma
This investigation is an attempt to spell out a formal semantic theory for inductive logic. The logic is probabilistic. It roughly resembles the logic of confirmation functions developed by Rudolf Carnap. ;Carnap's logic specifies an object language--the language of monadic predicate logic--and defines meta-linguistic probability functions on sentences of the object language. These probability functions express a semantic relationship between sentences, just as logical consequence is a semantic relationship between sentences in deductive logic. The semantic conditional probability functions express the degree of partial entailment or degree of confirmation that premises afford a conclusion. ;The system I develop is roughly of this kind, but on a stronger object language--one representing all of first-order logic and set-theory. It avoids the main problems with Carnap's approach, e.g., universally quantified sentences need not get probability 0 for infinite domains of objects. Also, unlike Carnap's system, the system developed here will not require that the logical structure of sentences be the sole determinant of all semantic probability relations. Only the semantic probabilities Carnap calls direct inferences--i.e., the conditional probability of the outcomes of instances, given a general statistical statement and the membership of the instances in an appropriate reference class--will depend on logical form alone. But in general the semantic probability of one sentence given another is not solely a matter of logical form. ;The system provides a rigorous formal account of the direct inferences as logical partial entailments. Logical entailments and logical partial entailments are to be the purely logical foundation on which a Bayesian logicist account of theory confirmation, or inverse inference, is constructed. The account is Bayesian in that the degree of confirmation of a hypothesis on evidence is related to direct inferences and to the prior plausibility of the hypothesis by way of Bayes' Theorem. The account is logicist in that conditional probabilities are developed as semantic relationships between sentences, and the semantics specifies certain purely logical relationships between sentences, the logical partial entailments
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