Page 92 - DCAP310_INTRODUCTION_TO_ARTIFICIAL_INTELLIGENCE_AND_EXPERT_SYSTEMS

Introduction to Artificial Intelligence & Expert Systems

Notes theories. Results of Kurt Gödel, Gerhard Gentzen, and others provided partial resolution to the
program, and clarified the issues involved in proving consistency. Work in set theory showed
that almost all ordinary mathematics can be formalized in terms of sets, although there are
some theorems that cannot be proven in common axiom systems for set theory. Contemporary
work in the foundations of mathematics often focuses on establishing which parts of mathematics
can be formalized in particular formal systems (as in reverse mathematics) rather than trying to
find theories in which all of mathematics can be developed.

6.1 Concept of Formalized Symbolic Logics

Mathematical logic emerged in the mid-19th century as a subfield of mathematics independent
of the traditional study of logic. Before this emergence, logic was studied with rhetoric, through
the syllogism, and with philosophy. The first half of the 20th century saw an explosion of
fundamental results, accompanied by vigorous debate over the foundations of mathematics.
Theories of logic were developed in many cultures in history, including China, India, Greece
and the Islamic world. In 18th century Europe, attempts to treat the operations of formal logic in
a symbolic or algebraic way had been made by philosophical mathematicians including Leibniz
and Lambert, but their labors remained isolated and little known. In the middle of the 19th
century, George Boole and then Augustus De Morgan presented systematic mathematical
treatments of logic. Their work, building on work by algebraists such as George Peacock,
extended the traditional Aristotelian doctrine of logic into a sufficient framework for the study
of foundations of mathematics. Charles Sanders Peirce built upon the work of Boole to develop
a logical system for relations and quantifiers, which he published in several papers from 1870 to
1885. Gottlob Frege presented an independent development of logic with quantifiers in his Begri
ffsschrift, published in 1879, a work generally considered as marking a turning point in the
history of logic. Frege’s work remained obscure, however, until Bertrand Russell began to
promote it near the turn of the century. The two-dimensional notation Frege developed was
never widely adopted and is unused in contemporary texts. From 1890 to 1905, Ernst Schröder
published Vorlesungen über die Algebra der Logik in three volumes. This work summarized and
extended the work of Boole, De Morgan, and Peirce, and was a comprehensive reference to
symbolic logic as it was understood at the end of the 19th century.

6.1.1 Syntax Logic

In logic, syntax is anything having to do with formal languages or formal systems without
regard to any interpretation or meaning given to them. Syntax is concerned with the rules used
for constructing, or transforming the symbols and words of a language, as contrasted with the
semantics of a language which is concerned with its meaning. The symbols, formulas, systems,
theorems, proofs, and interpretations expressed in formal languages are syntactic entities whose
properties may be studied without regard to any meaning they may be given, and, in fact, need
not be given any. Syntax is usually associated with the rules (or grammar) governing the
composition of texts in a formal language that constitute the well-formed formulas of a formal
system.

Did u know? In computer science, the term “syntax” refers to the rules governing the
composition of meaningful texts in a formal language, such as a programming language,
that is, those texts for which it makes sense to define the semantics or meaning, or otherwise
provide an interpretation.

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