Intensional Logic - Its Place Inside Logic

Its Place Inside Logic

Logic is the study of proof and deduction as manifested in language (abstracting from any underlying psychological or biological processes). Logic is not a closed, completed science, and presumably, it will never stop developing: the logical analysis can penetrate into varying depths of the language (sentences regarded as atomic, or splitting them to predicates applied to individual terms, or even revealing such fine logical structures like modal, temporal, dynamic, epistemic ones).

In order to achieve its special goal, logic was forced to develop its own formal tools, most notably its own grammar, detached from simply making direct use of the underlying natural language. Functors belong to the most important categories in logical grammar (alongside with basic categories like sentence and individual name): a functor can be regarded as an "incomplete" expression with argument places to fill in. If we fill them in with appropriate subexpressions, then the resulting entirely completed expression can be regarded as a result, an output. Thus, a functor acts like a function sign, taking on input expressions, resulting in a new, output expression.

Semantics links expressions of language to the outside world. Also logical semantics has developed its own structure. Semantic values can be attributed to expressions in basic categories: the reference of an individual name (the "designated" object named by that) is called its extension; and as for sentences, their truth value is called also extension.

As for functors, some of them are simpler than others: extension can be attributed to them in a simple way. In case of a so-called extensional functor we can in a sense abstract from the "material" part of its inputs and output, and regard the functor as a function turning directly the extension of its input(s) into the extension of its output. Of course, it is assumed that we can do so at all: the extension of input expression(s) determines the extension of the resulting expression. Functors for which this assumption does not hold are called intensional.

Natural languages abound with intensional functors, this can be illustrated by intensional statements. Extensional logic cannot reach inside such fine logical structures of the language, it stops at a coarser level. The attempts for such deep logical analysis have a long past: authors as early as Aristotle had already studied modal syllogisms. Gottlob Frege developed a kind of two dimensional semantics: for resolving questions like those of intensional statements, he has introduced a distinction between two semantic values: sentences (and individual terms) have both an extension and an intension. These semantic values can be interpreted, transferred also for functors (except for intensional functors, they have only intension).

As mentioned, motivations for settling problems that belong today to intensional logic have a long past. As for attempts of formalizations. the development of calculi often preceded the finding of their corresponding formal semantics. Intensional logic is not alone in that: also Gottlob Frege accompanied his (extensional) calculus with detailed explanations of the semantical motivations, but the formal foundation of its semantics appeared only in the 20th century. Thus sometimes similar patterns repeated themselves for the history of development of intensional logic like earlier for that of extensional logic.

There are some intensional logic systems that claim to fully analyze the common language:

• Transparent Intensional Logic
• Modal logic