Logic

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A system that is incomplete in the latter sense can nevertheless be complete in the sense that all the relevant logical truths are valid model-theoretical consequences of the system. This kind of completeness, known as descriptive completeness, is also sometimes confusingly called axiomatizability, despite the more common use of this term to refer to the mechanical generation of theorems from axioms and rules of inference. There is a further reason why the formulation of systems of rules of inference does not exhaust the science of logic.

Rule-governed, goal-directed activities are often best understood by means of concepts borrowed from the study of games. For example, one of the most fundamental ideas of game theory is the distinction between the definitory rules of a game and its strategic rules.

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Definitory rules define what is and what is not admissible in a game—for example, how chessmen may be moved on a board, what counts as checking and mating, and so on. But knowledge of the definitory rules of a game does not constitute knowledge of how to play the game. For that purpose, one must also have some grasp of the strategic rules, which tell one how to play the game well—for example, which moves are likely to be better or worse than their alternatives. They are merely permissive. That is, given a set of premises, the rules of inference indicate which conclusions one is permitted to draw, but they do not indicate which of the permitted conclusions one should or should not draw.

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Hence, any exhaustive study of logic—indeed, any useful study of logic—should include a discussion of strategic principles of inference. Unfortunately, few, if any, textbooks deal with this aspect of logic. In most nontrivial cases, however, the strategic rules cannot be mechanically recursively applied. The rules that license such inferences need not be truth-preserving, but many will be ampliative, in the sense that they lead or are likely to lead eventually to new or useful information.

There are many kinds of ampliative reasoning. Inductive logic offers familiar examples. Thus a rule of inductive logic might tell one what inferences may be drawn from observed relative frequencies concerning the next observed individual. In some cases, the truth of the premises will make the conclusion probable, though not necessarily true.

In other cases, although there is no guarantee that the conclusion is probable, application of the rule will lead to true conclusions in the long run if it is applied in accordance with a good reasoning strategy. It can be shown that there is in fact a close connection between optimal strategies of ampliative reasoning and optimal strategies of deductive reasoning.

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For example, the choice of the best question to ask in a given situation is closely related to the choice of the best deductive inference to draw in that situation. This connection throws important light on the nature of logic.

At first sight, it might seem odd to include the study of ampliative reasoning in the theory of logic. Such reasoning might seem to be part of the subject of epistemology rather than of logic. In so far as definitory rules are concerned, ampliative reasoning does in fact differ radically from deductive reasoning. But since the study of the strategies of ampliative reasoning overlaps with the study of the strategies of deductive reasoning, there is a good reason to include both in the theory of logic in a wide sense.

Some recently developed logical theories can be thought of as attempts to make the definitory rules of a logical system imitate the strategic rules of ampliative inference. Cases in point include paraconsistent logics, nonmonotonic logics, default reasoning, and reasoning by circumscription, among other examples.

Most of these logics have been used in computer science , especially in studies of artificial intelligence.

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Further research will be needed to determine whether they have much application in general logical theory or epistemology. The distinction between definitory and strategic rules can be extended from deductive logic to logic in the wide sense.

Often it is not clear whether the rules governing certain types of inference in the wide sense should be construed as definitory rules for step-by-step inferences or as strategic rules for longer sequences of inferences. Furthermore, since both strategic rules and definitory rules can in principle be explicitly formulated for both deductive and ampliative inference, it is possible to compare strategic rules of deduction with different types of ampliative inference.

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Learn More in these related Britannica articles: History of logic , the history of the discipline from its origins among the ancient Greeks to the present time. The work of the English mathematician George Boole and the American Charles Sanders Peirce had contributed to the development of a symbolism adequate to explore all elementary logical deductions. Logic could be conceived as a special study complete in itself only if the logician were allowed to postulate…. They also added to the growing realization of the immense potential of algebraic thinking, freed from its narrow conception….

For Frege and Russell, the propositions of logic and mathematics are pristinely independent of sense experience, depending for their truth only on the structures of the abstract world they describe—a world made accessible to human beings through the…. More About Logic 28 references found in Britannica articles mathematical aspects algebra In algebra: Matrices automata theory In automata theory: Nature and origin of modern automata In automata theory: Automata with random elements binary system In numerals and numeral systems: The binary system foundations In mathematics: The foundations of mathematics fuzzy logic In fuzzy logic philosophical aspects branches of philosophy metaphysics In metaphysics: Metaphysics and other branches of philosophy philosophical schools and doctrines Aristotelianism In Aristotelianism: Early development In Aristotelianism: From the Renaissance to the 18th century Empiricism In empiricism: Substantive empiricism View More.

Articles from Britannica Encyclopedias for elementary and high school students. Help us improve this article!

Contact our editors with your feedback. Introduction Scope and basic concepts Logical notation Logical systems Definitory and strategic inference rules Rules of ampliative reasoning. You may find it helpful to search within the site to see how similar or related subjects are covered. Any text you add should be original, not copied from other sources. At the bottom of the article, feel free to list any sources that support your changes, so that we can fully understand their context.

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