chapter13(pred) - CHAPTER 13 PREDICATE LANGUAGES 1...

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Unformatted text preview: CHAPTER 13 PREDICATE LANGUAGES 1 Predicate Languages Propositional Languages are also called Zero Order Languages, as opposed to Predicate Languages that are called First Order Languages. The same applies to the use of terms Propositional and Predicate Logic; they are often called zero Order and First Order Logics and we will use both terms equally. We will work with several different predicate languages, depending on what ap- plications we have in mind. All of those languages have some common features, and we begin with these. Propositional connectives We define the set of propositional connectives CON in the same way as in the case of the propositional languages. It means that we assume the following. 1. The set of connectives is non-empty and finite, i.e. < cardCON < . 2. We consider only the connectives with one or two arguments. Quantifiers We adopt two quantifiers; (for all, the universal quantifier) and (there exists, the existential quantifier), i.e. we have the following set of quantifiers Q = { , } . In a case of the classical logic and the logics that extend it, it is possible to adopt only one quantifier and to define the other in terms of it and propositional connectives. It is impossible in a case of some non-classical logics, for example the intuitionistic logic. But even in the case of classical logic two quantifiers express better the common intuition, so we assume that we have two of them. Parenthesis. As in the propositional case, we adopt the signs ( and ) for our parenthesis., i.e. we define a set PAR as PAR = { ( , ) } . 1 Variables We assume that we always have a countably infinite set V AR of variables, i.e. we assume that cardV AR = . We denote variables by x,y,z,... , with indices, if necessary, what we often express by writing V AR = { x 1 ,x 2 ,.... } . The set of propositional connectives CON defines a propositional part of the predicate logic language. What really differ one predicate language from the other is the choice of additional symbols to the symbols described above. These are called predicate symbols, function symbols, and constant symbols. I.e. a particular predicate language is determined by specifying the following sets of symbols. Predicate symbols Predicate symbols represent relations. We assume that we have an non empty, finite or countably infinite set P of predicate, or relation symbols. I.e. we assume that < card P . We denote predicate symbols by P,Q,R,... , with indices, if necessary. Each predicate symbol P P has a positive integer # P assigned to it; if # P = n then say P is called an n-ary (n - place) predicate (relation) symbol. Function symbols We assume that we have a finite (may be empty) or count- ably infinite set F of function symbols. I.e. we assume that card F ....
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This note was uploaded on 02/12/2011 for the course CSE 541 taught by Professor Bachmair,l during the Spring '08 term at SUNY Stony Brook.

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chapter13(pred) - CHAPTER 13 PREDICATE LANGUAGES 1...

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