![]() The clean mathematical semantics of relational database enabled programmers to say “what” they wanted instead of “how” to compute it (as was the case with so called “hierarchical database” then the dominant database style). (The connection was not direct, that is, Codd did not reference Tarski, but it can be traced ). This new approach can be traced to concepts which were created by Tarski. Codd of IBM Research defined a new approach to database systems, i.e. His approach also had important practical implications. Tarski’s approach revolutionized logic by making semantics mathematical. He then defined a relationship in terms of that compound structure (sentences, facts, truth) which he claimed both generated and explained the consequence relation for the language. ![]() Tarski supplemented the set of statements with a way of representing the “facts” the statements were about, and a recursive definition of truth for the statements that defined which statements were true in which sets of facts. In the original application the sentences were statements from number theory. The statements in that language were sentences in first order logic. He began with the language for which the consequence relation was to be modeled. Tarski developed a mathematical model of consequence that consisted of several parts. The original work was done by Alfred Tarski, a Polish mathematician. These properties included soundness (do the proof rules always generate valid conclusions) and completeness (does a set of axioms and proof rules generate all valid conclusions). Having a mathematical model of consequence that was independent of syntactic deductive rules enabled the properties of deductive systems to be proven. Working out a mathematically rigorous way of modeling consequence for sentences of first order logic was a fundamental advance in the field of logic. One could say it is the central concept in logic. Consequence is an aspect of semantics, the practice of modeling the meaning of expressions in languages. More formally, for some language L, for statements P and Q in L, Q is a consequence of P just in case if P is true, then Q must be true. What does that mean? Consequence is the relation that tells you, for any statement x in some language, what other statements in the language you know, if you know x. The original problem was to create an effective technique for modeling the consequence relation for object-oriented data. ![]() ![]() The published version of this work may be found here. My dissertation project developed a framework for understanding why logic works, and applied that framework to create new techniques for building models of logical consequence that are more effective for modeling the meaning of feature structures (an abstraction related to object-oriented expressions). “I think what you have done in your dissertation is quite interesting and makes a real contribution to the program of understanding what we mean by logical consequence.” ![]()
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