The author you quote seems to be oversimplifying, but it is possible to understand some work of the logical positivists as an attempt at a purely syntactic approach to expressing the relation between evidence and hypothesis. Rudolf Carnap, in particular, attempted to set out a formal logic of induction in which inductive probabilities can be derived from evidence statements in a probabilistic but deductive fashion. This approach rests heavily on principles such as the use of Bayes' theorem as an updating rule. Similar approaches have been used by Edwin Jaynes and others in deploying purely formal considerations such as maximum entropy. Another related approach was taken by I J Good in his advocacy of the principle of the weight of evidence, defined as W(H:E) = log( P(E|H) / P(E|¬H) ).
Purely formal approaches like this run into a number of common objections. They assume that it is possible to identify evidence statements that are theory-neutral and language-neutral. They run into difficulties in determining objective and scale-invariant priors. They fail to account for the fact that some properties are projectible from known instances to unknown ones, while others are not, so the degree of support of some evidence for a hypothesis depends on the choice and meaning of the predicate terms. They struggle to give an adequate account of how hypotheses explain observations, rather than merely predicting them. Other positivists, such as Carl Hempel, placed more emphasis on the concept of explanation.
Rudolf Carnap, The Logical Foundations of Probability (1950).
Edwin Jaynes, Probability Theory: The Logic of Science (2003).
I J Good "Weight of Evidence: A Brief Survey" in Bernardo, DeGroot, Lindley, Smith. Bayesian Statistics 2, pp 249-270 (1985).
Carl Hempel, Aspects of Scientific Explanation (1965).
See also, the Stanford Encyclopedia entry on Inductive Logic.