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Rautenberg - A concise introduction to mathematical logic

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Rautenberg A concise introduction to mathematical logic
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Traditional logic as a part of philosophy is one of the oldest scientific disciplines and can be traced back to the Stoics and to Aristotle. Mathematical logic, however, is a relatively young discipline and arose from the endeavors of Peano, Frege, and others to create a logistic foundation for mathematics. It steadily developed during the twentieth century into a broad discipline with several sub-areas and numerous applications in mathematics, informatics, linguistics and philosophy.

This book treats the most important material in a concise and streamlined fashion. The third edition is a thorough and expanded revision of the former. Although the book is intended for use as a graduate text, the first three chapters can easily be read by undergraduates interested in mathematical logic. These initial chapters cover the material for an introductory course on mathematical logic, combined with applications of formalization techniques to set theory. Chapter 3 is partly of descriptive nature, providing a view towards algorithmic decision problems, automated theorem proving, non-standard models including non-standard analysis, and related topics.

The remaining chapters contain basic material on logic programming for logicians and computer scientists, model theory, recursion theory, Gdels Incompleteness Theorems, and applications of mathematical logic. Philosophical and foundational problems of mathematics are discussed throughout the text. Each section of the seven chapters ends with exercises some of which of importance for the text itself. There are hints to most of the exercises in a separate file Solution Hints to the Exercises which is not part of the book but is available from the authors website.

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Wolfgang Rautenberg Universitext A Concise Introduction to Mathematical Logic 10.1007/978-1-4419-1221-3_1 Springer Science+Business Media, LLC 2010
1. Propositional Logic
Wolfgang Rautenberg 1
(1)
Fachbereich Mathematik und Informatik, 14195 Berlin, Germany
Wolfgang Rautenberg
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Abstract
Propositional logic, by which we here mean two-valued propositional logic, arises from analyzing connections of given sentences A , B , such as
These connection operations can be approximately described by two-valued logic - photo 1
These connection operations can be approximately described by two-valued logic. There are other connections that have temporal or local features, for instance, first A then B or here A there B , as well as unary modal operators like it is necessarily true that , whose analysis goes beyond the scope of two-valued logic. These operators are the subject of temporal, modal, or other subdisciplines of many-valued or nonclassical logic. Furthermore, the connections that we began with may have a meaning in other versions of logic that two-valued logic only incompletely captures. This pertains in particular to their meaning in natural or everyday language, where meaning may strongly depend on context.
Propositional logic, by which we here mean two-valued propositional logic, arises from analyzing connections of given sentences A , B , such as
These connection operations can be approximately described by two-valued logic - photo 2
These connection operations can be approximately described by two-valued logic. There are other connections that have temporal or local features, for instance, first A then B or here A there B , as well as unary modal operators like it is necessarily true that , whose analysis goes beyond the scope of two-valued logic. These operators are the subject of temporal, modal, or other subdisciplines of many-valued or nonclassical logic. Furthermore, the connections that we began with may have a meaning in other versions of logic that two-valued logic only incompletely captures. This pertains in particular to their meaning in natural or everyday language, where meaning may strongly depend on context.
In two-valued propositional logic such phenomena are set aside. This approach not only considerably simplifies matters, but has the advantage of presenting many concepts, for instance those of consequence, rule induction, or resolution, on a simpler and more perspicuous level. This will in turn save a lot of writing in Chapter when we consider the corresponding concepts in the framework of predicate logic.
We will not consider everything that would make sense in two-valued propositional logic, such as two-valued fragments and problems of definability and interpolation. The reader is referred instead to []. We will concentrate our attention more on propositional calculi. While there exists a multitude of applications of propositional logic, we will not consider technical applications such as the designing of Boolean circuits and problems of optimization. These topics have meanwhile been integrated into computer science. Rather, some useful applications of the propositional compactness theorem are described comprehensively.
1.1 Boolean Functions and Formulas
Two-valued logic is based on two foundational principles: the principle of bivalence , which allows only two truth values, namely true and false , and the principle of extensionality , according to which the truth value of a connected sentence depends only on the truth values of its parts, not on their meaning. Clearly, these principles form only an idealization of the actual relationships.
Questions regarding degrees of truth or the sense-content of sentences are ignored in two-valued logic. Despite this simplification, or indeed because of it, such a method is scientifically successful. One does not even have to know exactly what the truth values true and false actually are. Indeed, in what follows we will identify them with the two symbols 1 and 0. Of course, one could have chosen any other apt symbols such as and or t and f. The advantage here is that all conceivable interpretations of true and false remain open, including those of a purely technical nature, for instance the two states of a gate in a Boolean circuit.
According to the meaning of the word and , the conjunction A and B of sentences A , B , in formalized languages written as A B or A & B , is true if and only if A , B are both true and is false otherwise. So conjunction corresponds to a binary function or operation over the set {0, 1} of truth values, named the -function and denoted by . It is given by its value matrix A concise introduction to mathematical logic - image 3 , where, in general, A concise introduction to mathematical logic - image 4 represents the value matrix or truth table of a binary function with arguments and values in {0, 1}. The delimiters of these small matrices will usually be omitted.
A function f : {0, 1} n { 0, 1} is called an n -ary Boolean function or truth function . Since there are 2 n n -tuples of 0, 1, it is easy to see that the number of n -ary Boolean functions is Picture 5 . We denote their totality by Bn . While B 2 has 24 = 16 members, there are only four unary Boolean functions. One of these is negation , denoted by and defined by 1 = 0 and 0 = 1. B 0 consists just of the constants 0 and 1.
The first column of the table below contains the common binary connections with examples of their instantiation in English. The second column lists some of its traditional symbols, which also denote the corresponding truth function, and the third its truth table. Disjunction is the inclusive or and is to be distinguished from the exclusive disjunction . The latter corresponds to addition modulo 2 and is therefore given the symbol +. In Boolean circuits the functions +, , are often denoted by xor , nor , and nand ; the latter is also known as the Sheffer function . Recall our agreement in the section Notation that the symbols &, , , and will be used only on the metatheoretic level.
A connected sentence and its corresponding truth function need not be denoted by the same symbol; for example, one might take for conjunction and et as the corresponding truth function. But in doing so one would only be creating extra notation, but no new insights. The meaning of a symbol will always be clear from the context: if , are sentences of a formal language, then denotes their conjunction; if a , b are truth values, then a b just denotes a truth value. Occasionally, we may want to refer to the symbols Picture 6 themselves, setting their meaning temporarily aside. Then we talk of the connectives or truth functors Picture 7
Sentences formed using connectives given in the table are said to be logically equivalent if their corresponding truth tables are identical. This is the case, for example, for the sentences A provided B and A or not B , which represent the converse implication , denoted by A B . It does not appear in the table, since it arises by swapping A , B in the implication. This and similar reasons explain why only a few of the sixteen binary Boolean functions require notation. Another example of logical equivalent sentences are if A and B then C , and if B then C provided A.
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