Katz and Fodor, in their 1963 paper The Structure of a Semantic Theory argued that a generative grammar should be thought of as a system of rules relating the sentences in a language in their externalised form, to their meanings. To relate the grammar to meaning, they attempted to express meaning in a universal semantic representation, in the same way that sounds can be expressed in a universal phonetic representation.
The Katz-Fodor framework for a semantic component consists of two parts: a lexicon and a set of projection rules. A lexicon contains a list of all the lexical items of the language that can contain semantic readings, syntactic and phonological information associated with each item, thus forming a repository of varied information about the lexical items that the language is composed of.
Aside: multiple senses and readings, e.g., in bachelor have been the center of discussion in a lot of these works, with attempts at formalisms to capture anomaly and analyticity. This work however, assumes that the lexicon has some specification for each reading of the lexical item, and they can therefore all be enumerated and called upon as needed.
The projection rules, on the other hand, are a set of rules that contribute to the strucutre, organisation and semantic representation of a sentence. This is therefore the part of interpretation that is traceable to syntactic structure. The assumption here is that the projection rules are interpretive i.e., they operate on the syntactic structures generated by the base structures and transformations, in order to produce semantic readings.
The approach to projection rules and this framework is therefore as follows: the projection rules describe the contribution of syntactic structure to meaning, and the syntactic structure is generated by recursively applying a sequence of rules i.e., first phrase structure rules and then transformations. Different sentences of different structure or semantic readings are due to differences in sequences of rules and transformations that generate the sentences --- so two sentences with the same lexical items can differ in meaning only if different sequences of rules have been applied. These projection rules can be of Type I or Type II, depending on whether associated with phrase structure or transformations.
To quote: "The projection problem: A full synchronic description of a natural language is a grammatical and semantic characterisation of that language [...] Hence, a semantic theory must be constructed to have whatever properties are demanded by its role in linguistic description. Since, however, the goals of such description are reasonable well understood and since, in comparison to semantics, the nature of grammar has been clearly articulated, we may expect that by studying the contribution that semantics will be required to make [..] constrains on a semantic theory. [..] Since a fluent speaker is able to use and understand any sentnece drawn from the INFINITE set of sentences of his language, and since, at any time, he has only encountered a FINITE set of sentences, it follows that the speaker's knowledge of his language takes the form of rules which project the finite set of sentences he has fortuitously encountered to the infinite set of sentences of the language. A description of the language which adequately represents the speaker's linguistic knowledge must, accordingly, state these rules. The problem of formulating these rules we shall refer to as the projection problem."
There are several works that later examine and test the hypothesis that (semantic) readings are determined solely on the basis of deep structure as porposed by Chomsky. Some attempts show that semantic generalisations actually involve derived structure, and therefore certain aspects of semantic interpretation can be determined from derived structure, giving lots of advantages over the standard approaches to these problems.