and is it inductive or deductive?
Deductive And Inductive Paragraph Organization
It is now generally held that the core idea of Bayesian logicism is fatally flawed—that syntactic logical structure cannot be the sole determiner of the degree to which premises inductively support conclusions. A crucial facet of the problem faced by Bayesian logicism involves how the logic is supposed to apply to scientific contexts where the conclusion sentence is some hypothesis or theory, and the premises are evidence claims. The difficulty is that in any probabilistic logic that satisfies the usual axioms for probabilities, the inductive support for a hypothesis must depend in part on its prior probability. This prior probability representshow plausible the hypothesis is supposed to be based on considerationsother than the observational and experimental evidence (e.g., perhapsdue to relevant plausibility arguments). A Bayesian logicist must tellus how to assign values to these pre-evidential priorprobabilities of hypotheses, for each of the hypotheses ortheories under consideration. Furthermore, this kind of Bayesianlogicist must determine these prior probability values in away that relies only on the syntactic logical structure of thesehypotheses, perhaps based on some measure of their syntacticsimplicities. There are severe technical problems with getting thisidea to work. Moreover, various kinds of examples seem to show thatsuch an approach must assign intuitively quite unreasonable priorprobabilities to hypotheses in specific cases (see the footnote citednear the end of section 3.2 for details). Furthermore, for this ideato apply to the evidential support of real scientific theories,scientists would have to formalize theories in a way that makes theirrelevant syntactic structures apparent, and then evaluate theoriessolely on that syntactic basis (together with their syntacticrelationships to evidence statements). Are we to evaluate alternativetheories of gravitation (and alternative quantum theories) this way?This seems an extremely doubtful approach to the evaluation of realscientific theories and hypotheses. Thus, it seems that logicalstructure alone cannot suffice for the inductive evaluation ofscientific hypotheses. (This issue will be treated in more detail inSection 3, after we first see how probabilistic logics employ Bayes'theorem to represent the evidential support for hypotheses as afunction of prior probabilities together withtheir evidential likelihoods.)
Inductive and deductive reasoning
Some inductive logicians have tried to follow the deductive paradigm very closely by attempting to specify inductive support probabilities in terms of the syntactic structures of premise and conclusion sentences. In deductive logic the syntactic structure of the sentences involved completely determines whether premises logically entail a conclusion. So these logicians attempted to specify inductive support probabilities solely in terms of the syntactic structure of premise and conclusion sentences. In such a system each sentence confers a syntactically specified degree of support on each of the other sentences of the language. The inductive probabilities in such a system are logical in the sense that they depend on syntactic structure alone. This kind of conception was articulated to some extent by John Maynard Keynes in his Treatise on Probability (1921). Rudolf Carnap pursued this idea with greater rigor in his Logical Foundations of Probability (1950) and in several subsequent works (e.g., Carnap 1952). (For details of Carnap's approach see the section on in the entry on , in this Encyclopedia.)