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Summary
Summary All logic can be based on very few general principles.
Procrustes of Attika, around 800 BC
This book has three distinct, but strongly connected, directions: First, we base several types of human reasoning on a small number of basic semantical notions. This is a back and forth procedure, as we take simple common-sense notions like "size" or "distance" to clarify human reasoning and, in turn vary these notions to obtain different forms of reasoning. This is a rather philosophical enterprise. Second, we establish a number of representation theorems for such reasoning, showing how properties of the basic semantical notions are reflected by the logical properties. This is a logical, or even algebraic, enterprise. In more abstract terms, the basic semantical notions result in coherence conditions in corresponding model choice functions, which carry over (often almost one-to-one) to logical properties. Third, we sketch a unified treatment of several such types of reasoning in a uniform modal framework, with several layers of abstraction which, we hope, gives some of the richness and variability of human reasoning. This is almost an engineering effort, and intended as an approach to a flexible system of argumentation. In this part, we will indicate problems and solutions in rough outline only.
XV
xvi
SUMMA R Y
These three parts are interrelated. The second part formalizes the first, it establishes a 1-1 correspondence between semantics and reasoning based on it, or demonstrates the impossibility to do so. Without the ideas in the first part (at least in rough outline), there would not have been anything in the second part. The first and the third parts are related: We not only obtain a principled way of reasoning and arguing, but we can also refine our arguments by going down to the basic notions themselves. Finally, the second part gives the necessary technical rigour to construct a sound system of argumentation. Conversely, the third part gave motivation for some of the results in the second part. In the second and main part (completeness and incompleteness constructions), we have three main directions of thrust. First, we want to show that, for preferential and related structures, a basic (and provably most general) idea can be used and modified to give a number of results for semantics with various strengths. Second, we insist on stressing the role of domain closure conditions, which seem to have been mostly neglected so far. In particular, we will show that some results depend on such seemingly innocent closure conditions. But, we will also show how to avoid such nice closure conditions, and what is the price to pay for it (more complicated conditions elsewhere). Third, we will show that several, at first sight very simple, semantics do not allow a finite characterization, or even no characterization at all by usual means. Again, the third and the second point are related, as finite representation can be possible, provided the domain is sufficiently rich. We will usually decompose representation problems into two steps: as we work with model sets, the basic semantical notions, like a relation between models, give rise to functions on model sets. Thus separating the algebraic from the logical problem brings them better to light, makes their solution easier, and we can re-use results for the (usually more difficult) algebraic part in various logical contexts. Finally, reformulating the logics as "generalized modal logics", expressing the core concepts in the object language has many advantages, like increased flexibility and expressivity, and also results in increased "quality" of the logic, permitting, e.g. contraposition.
SUMMARY
xvii
The reader will probably have guessed that the quotation attributed to Procrustes is a hoax. Procrustes was not so much concerned with a systematization of logic (by perhaps unorthodox and slightly violent means), but rather with a standardization of people (also by unorthodox and slightly violent means).
Procrustes of Attika, around 800 BC
This book has three distinct, but strongly connected, directions: First, we base several types of human reasoning on a small number of basic semantical notions. This is a back and forth procedure, as we take simple common-sense notions like "size" or "distance" to clarify human reasoning and, in turn vary these notions to obtain different forms of reasoning. This is a rather philosophical enterprise. Second, we establish a number of representation theorems for such reasoning, showing how properties of the basic semantical notions are reflected by the logical properties. This is a logical, or even algebraic, enterprise. In more abstract terms, the basic semantical notions result in coherence conditions in corresponding model choice functions, which carry over (often almost one-to-one) to logical properties. Third, we sketch a unified treatment of several such types of reasoning in a uniform modal framework, with several layers of abstraction which, we hope, gives some of the richness and variability of human reasoning. This is almost an engineering effort, and intended as an approach to a flexible system of argumentation. In this part, we will indicate problems and solutions in rough outline only.
XV
xvi
SUMMA R Y
These three parts are interrelated. The second part formalizes the first, it establishes a 1-1 correspondence between semantics and reasoning based on it, or demonstrates the impossibility to do so. Without the ideas in the first part (at least in rough outline), there would not have been anything in the second part. The first and the third parts are related: We not only obtain a principled way of reasoning and arguing, but we can also refine our arguments by going down to the basic notions themselves. Finally, the second part gives the necessary technical rigour to construct a sound system of argumentation. Conversely, the third part gave motivation for some of the results in the second part. In the second and main part (completeness and incompleteness constructions), we have three main directions of thrust. First, we want to show that, for preferential and related structures, a basic (and provably most general) idea can be used and modified to give a number of results for semantics with various strengths. Second, we insist on stressing the role of domain closure conditions, which seem to have been mostly neglected so far. In particular, we will show that some results depend on such seemingly innocent closure conditions. But, we will also show how to avoid such nice closure conditions, and what is the price to pay for it (more complicated conditions elsewhere). Third, we will show that several, at first sight very simple, semantics do not allow a finite characterization, or even no characterization at all by usual means. Again, the third and the second point are related, as finite representation can be possible, provided the domain is sufficiently rich. We will usually decompose representation problems into two steps: as we work with model sets, the basic semantical notions, like a relation between models, give rise to functions on model sets. Thus separating the algebraic from the logical problem brings them better to light, makes their solution easier, and we can re-use results for the (usually more difficult) algebraic part in various logical contexts. Finally, reformulating the logics as "generalized modal logics", expressing the core concepts in the object language has many advantages, like increased flexibility and expressivity, and also results in increased "quality" of the logic, permitting, e.g. contraposition.
SUMMARY
xvii
The reader will probably have guessed that the quotation attributed to Procrustes is a hoax. Procrustes was not so much concerned with a systematization of logic (by perhaps unorthodox and slightly violent means), but rather with a standardization of people (also by unorthodox and slightly violent means).