Katz's 1972 book Semantic Theory was arguably one of the first radically different theories from historically prevalent notions of formal semantics so far. Moreover, this is one place where the relationship of (Katzian) semantics to cognitive psycohology and learning is prevalent, thus allowing means to tie together formal semantics and logical representations to language use and communication.
It begins with a preface containing a somewhat historical overview of how logic and symbolic structures had encompassed linguistic theories until then. He states that the "classical problem of logical form" was what united linguists and logicians --- for example, Aristotle's On Interpretation and several works by Wittgenstein and Strawson.
Aside: several logicians were devoutly against this, arguing that ordinary language "has no logic". Russel, in 1957 said “I may say, to begin with, that I am totally unable to see any validity whatever in any of Mr. Strawson’s arguments. … I agree, however, with Mr. Strawson’s statement that ordinary language has no logic.” . He was not the first to complain about this very apparent illogicality of natural language, and considered the formulaic structure of first order logic a truer picture of logical form than English sentences.
To come back to Katz's document and the role of logic in language, what interested them was to formulate an artificial logical language that was expressive enough to represent natural language enough to understand meaning. Quine, previously, had looked into identifying the logical particles or atoms that could be composed together to create entire expressions.
Quine segregated these atoms into classes, by reasoning that logical particles (e.g., and, or, not, if, then have inferential powers while others (e.g., table, chair, dog, Adam have only referential powers. While this seemed fairly intuitive, Katz argued against this, saying that there is no non-arbitrary way of dividing a vocabulary into these sets. The document contains a series of step-through proofs for example natural language sentences to show that Quine's logical/extra-logical partioning of a vocabulary could not hold for a range of sentences in different contexts.
Having (somewhat) proved that these theories of logic of Quine, Frege, Carnap and others are ill-suited in a rationalist theory of language such a sgenerative grammar, Katz then attempts to construct a rationalist theory of semantics. He proposed that semantics is a crucial component of generative grammar. Semantic interpretations or senses are assigned to syntactic structures of grammar through projection rules, defined by some of the formal properties of transformations. Moreover, the meanings of propositional units or lexical items (and therefore of the structures containing them) can and should be expressed using combinations of semantic markers that are "intended to relect, in their formal structure, the structure of concepts they represent."
Despite how brittle some of these theories are, Katz's work was the only serious and sufficiently thought-out attempt to draw out a theory of meaning within the framework of generative grammar. These ideas, along with several others developed around the same time were incorporated into the standard theory of generative grammar formulated in Chomsky (1965), where deep structures are mapped to semantic interpretations by projection rules, and by syntactic transformations to surface stuctures, and ultimately to phonetic interpretations.