Internal Semantics and Algebraic Logic
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TL;DR
This chapter discusses internal semantics and algebraic logic, the study of the way in which language relates to its domains of discourse, and the role of function in this study.
Abstract
This chapter discusses internal semantics and algebraic logic. Semantics is the study of the way in which language relates to its domains of discourse, and hence no full, adequate, semantical analysis can confine itself to the study of language by itself. When a given language is interpreted with respect to a given domain, the sentences of the language fall into two classes: the true and the false. When one begins to analyze the relation of a language to an external domain by mathematical means, the most basic and natural tool is that of a function. For example, there is the function that assigns, to each name-symbol of the language, the object of the domain that it denotes and to each n-placed predicate symbol its meaning as an n-ary relation over the domain. Starting from such a function, each sentence comes to take on a definite truth-value.