J. R. Schoenfield, Mathematical Logic, Addison-Wesley, Boston, Massachusetts, 1967.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....in mid 2001. i ii Contents 1 Introduction 1 2 Mappings 5 3 Theory Declarations 11 4 Prettyprinting Theory Instances 19 5 Comparison with Other Systems 21 6 Future Work 25 7 Conclusion 27 Bibliography 29 iii iv Introduction Theory interpretations have a long history in first order logic [Sho67,End72,Mon76] They are used to show that the language of a given source theory S can be interpreted within a target theory T such that the corresponding interpretation of axioms of S become theorems of T . This demonstrates the consistency of S relative to T , and also the decidability of S ....

Joseph R. Shoenfield. Mathematical Logic. Addison-Wesley, Reading, MA, 1967. 30

....try to show the effectiveness of combination techniques by choosing one of the most general and adapting it so that it can be incorporated in the CLP scheme with few modification of the scheme itself. 1. 1 Notation and Conventions We adhere rather closely to the notation and definitions given in [15] for what concerns mathematical logic in general and [9] for what concerns constraint logic programming in particular. We report here the most notable notational conventions followed. Other notation which may appear in the sequel follows the common conventions of the two fields. In this paper, we ....

Joseph. R. Shoenfield. Mathematical Logic. Addison-Wesley, Reading, MA, 1967.

....procedure used in [26] and provide a novel proof of Nelson and Oppen s original results. In addition, we provide a characterization of the consistency of the union of first order theories. 1.1. NOTATION AND CONVENTIONS In general, we will adhere to the notation and definitions given in [22]. The most notable notational conventions followed are given below with the understanding that other notations that may appear in the paper follow the common conventions of the field. The letters v, l, z denote logical variables, first order formulas, and a value assignment, or valuation, to ....

....as 7 is consistent by assumption. By construction of 7. the reduct of )4 to i is a model of 7, for i 1, 2. Obviously, both reducts have the same cardinality. Let )4 and )42 be models of T and T2, respectively, and assume they have the same cardinality. By the Craig Robinson Theorem (see [22]) 7. is inconsistent iff there is a sentence o, whose non logical symbols are in ( 2, such that T o and T2 o. If such o exists, we have that fi4 o and J2 =o. 1) 6Nelson Oppen call such formulas simple formulas. Now, as and T2 are signature disjoint, can only be an equational formula. It is a ....

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Joseph. R. Shoenfield. Mathematical Logic. Addison-Wesley, Reading, MA, 1967.

....domains, such as, for instance, the domains of finite trees, real numbers, lists, strings, partial orders, and so on. An earlier, less detailed version of our extension appeared in [TH96a] 1. 1 Notation and Conventions We adhere rather closely to the notation and definitions given in [Sho67] for what concerns mathematical logic in general, and [JM94] for what concerns constraint logic programming in particular. We report here the most notable notational conventions followed. Further notation, which may appear in the sequel, follows the common conventions of the two fields. The ....

Joseph. R. Shoenfield. Mathematical Logic. Addison-Wesley. Reading, MA, 1967.

....in the provability of N sat S when S is valid and the specifications for the processes in N are precise. STL specifications can involve elements of the data domain from which messages are drawn, sequences of such elements, and lengths of sequences. Since number theory itself is incomplete [S67], a valid assertion involving sequence lengths might not be provable in any system. When designing a programming logic, one actually aims for relative completeness [Co78] Assuming that one can prove any valid statement of predicate logic, number theory, and the data domain of the network being ....

J.R. Schoenfield. Mathematical Logic. Add/sonWesley, Reading, Mass., 1967.

....be proved, since there are no earlier laws from which they can be proved. Hence we have certain rst laws, called axioms, which we accept without proof; the remaining laws, called theorems, are proved from the axioms. Axioms, theorems and certain concepts of derivation build up an axiom system (see [13] or [7] for a good introduction to mathematical logic) The study of axioms and theorems as sentences, sequences of glyphs, is called the syntactical study of axiom systems; the study of the meaning of these sentences is called the semantic study of axiom systems. There are two subelds of ....

....One of these has been developed by Brouwer, which considers the proofs as the semantic of mathematics, not the syntax. i.e. a proof of A oe B is considered as an instruction how to construct a proof of B from a proof of A. We will not give detailed exposition of classical logic, for this see [13], 1] 7] or [10] Since we are interested in the relations between proofs and properties of the proven object, we will follow Gentzen like proof theory, so : 3.2 What is Gentzen like Proof Theory The proof theory according to Gentzen is based on the sequent calculus, Gentzen s formulation ....

Joseph R. Shoeneld. Mathematical Logic. Addison-Wesley, 1967.

....contains two symbols: integer and real , and which contain standard arithmetic predicates and function symbols such as 0, 1, Gamma, Delta, both for integers and for reals. We will assume that this theory contains both the standard first order theory of integers (Peano arithmetic [3,11,48]) and a standard first order theory of real numbers [5,10,49,51] One of the possibilities is to consider, as the theory T , axiomatic set theory (e.g. ZF) together with explicit definitions of integers, real numbers, and standard operations and predicates in terms of set theory. Once a theory ....

Schoenfield, J. R. (1967) Mathematical Logic, Addison-Wesley.

....the axiom system of set theory (subject to some reasonable restrictions, e.g. that no contradiction should be derivable and that it should be possible to decide about a statement whether it is an axiom) still there would remain unsolvable problems. For more detail on topics in logic, see e.g. [14]. The second form of the question of undecidability is when we are concerned with a family of problems and are looking for an algorithm that decides each of them. Church in 1936 formulated a family of problems for which he could prove that it is not decidable by any algorithm (see reference in ....

.... x: Theorem 3.3.5. The theory N is minimally adequate. Thus, there is a minimally adequate consistent theory of arithmetic with a finite system of axioms. This fact implies the following theorem of Church, showing that the problem of logical provability is algorithmically undecidable (see e.g. [14]) Theorem 3.3.6 (Undecidability of Predicate Calculus) The set P of all sentences that can be proven without any axioms, is undecidable. Proof. Let N be a finite system of axioms of a minimally adequate consistent theory of arithmetic, and let N be the sentence obtained by taking the ....

Joseph R. Schoenfield, Mathematical logic, Addison-Wesley, New York, 1967.

.... the formula valid, whereas the universally quantified variables have to be left untouched (cf. 3] Within this logical framework it becomes obvious that considering the universally quantified variables as distinct new constants essentially is an application of the Theorem on Constants (cf. [23]) This fundamental result, roughly speaking, states that universally quantified variables in logical formulae may be equivalently replaced by distinct new constants. In our context we do not only have the equivalence between (1) and 9y : Phi(s 1 ) Phi(t 1 ) Phi(s n ) Phi(t n ) ....

Joseph R. Shoenfield. Mathematical Logic. Addison-Wesley, 1973.

....that interpretation. But it is this type of informality, interpretation, and reasoning about the underlying model that leads to the claim that false is not provable, by de nition. A proof of G odel s second incompleteness theorem can be found in textbooks on mathematical logic (e.g. 20] [25], 15] and [16] A study of formal provability via modal logic is conducted by Boolos [0] where on page xvii, Boolos makes the point that although for every theorem S of a suciently strong formal system F, there is also a theorem of F to the e ect that S is a theorem of F, for no non theorem S ....

....is that calculational proofs are described by Dijkstra as formal calculations (e.g. on page 7 of [12] However, even a cursory look at the EWDs reveals that Dijkstra s modus operandi is for the most part informal. In a formal system the notion of proof must be recursive (cf. page 132 of [25]) By recursive we mean, as in the theory of computability, that there is a Turing machine that can decide whether or not its input is a proof. This requirement embodies the nitistic requirements of Hilbert and allows us to calculate : to see who is right , a partial ful llment of ....

Joseph Robert Shoeneld. Mathematical Logic. Addison Wesley, Reading, Ma., 1967.

....contains two symbols: integer and real , and which contain standard arithmetic predicates and function symbols such as 0, 1, Gamma, Delta, both for integers and for reals. We will assume that this theory contains both the standard first order theory of integers (Peano arithmetic [1, 8, 16]) and a standard first order theory of real numbers [3, 7, 17, 18] One of the possibilities is to consider, as the theory T , axiomatic set theory (e.g. ZF) together with explicit definitions of integers, real numbers, and standard operations and predicates in terms of set theory. Once a ....

J.R. Schoenfield. Mathematical logic. AddisonWesley, 1967.

....Let us first define what a condition can look like. In mathematical logic, formal expressions which describe conditions are called formulas, so we want to define the notion of a formula. We will give brief definitions here; readers who are interested in technical details can look, e.g. in [3, 4, 9]. Definition 1. By a fuzzy logic, we mean a finite set of constants from the interval [0; 1] and and a finite set of operations on the interval [0; 1] i.e. function from [0; 1] k [0; 1] The operations from this set will be called logical operations. For example, constants may include 0 ....

J. R. Schoenfield. Mathematical logic. Addison-Wesley, 1967.

....variables that run over arbitrary sets) So, the natural language to use is multisorted first order logic. For the convenience of the readers who may not be well familiar with this notion, let us give sketchy definitions here; readers who are interested in technical details can look, e.g. in [1, 9, 19]. Definition 1. ffl Let a finite set A be fixed. This set will be called an alphabet, and elements of this set will be called symbols. We assume that this set does not contain symbols ( 8, 9, and symbols with subscripts. ffl By a multi sorted first order language, we mean the ....

....is deducible from the theory T (and denote it by T F ) if this formula F is true in every model of the theory T . Comment. In the following text, we will assume that a theory T is fixed. We will assume that this theory contains both the standard first order theory of integers (Peano arithmetic [1, 9, 19]) and a standard first order theory of real numbers [21, 20, 3, 8] As we have already mentioned, one of the possibilities is to consider, as the theory T , axiomatic set theory (e.g. ZF) together with explicit definitions of integers, real numbers, and standard operations and predicates in ....

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J. R. Schoenfield. Mathematical logic. Addison-Wesley, 1967.

....Let us first define what a condition can look like. In mathematical logic, formal expressions which describe conditions are called formulas, so we want to define the notion of a formula. We will give sketchy definitions here; readers who are interested in technical details can look, e.g. in [3, 4, 9]. Let us fix a set of constants (e.g. 0 and 1) and a set of operations on the interval [0; 1] this set can include an and operation (t norm) e , an or operation e , a fuzzy negation e : hedge operations, etc. Some of these operations are binary (like t norm and t conorm) some are ....

J. R. Schoenfield. Mathematical logic. Addison-Wesley, 1967.

....ffl Finally, a formula is any expression that can be obtained from elementary formulas by using propositional connectives (such as , implies) etc. and quantifiers 8x and 9x over all real numbers. Comments. ffl For general notions of logic, see, e.g. Barwise, 1977; Enderton, 1972; Schoenfield, 1967). ffl In our definition, we started with two real numbers 0 and 1. It is known that if we start with arbitrary algebraic real numbers, we end up with the same class of first order formulas. Indeed, if we know that r is the only real number on the interval [a; b] for which P (r) 0 for a given ....

Schoenfield, J. R. (1967), Mathematical logic, Addison-Wesley.

....mid 2001. i ii Contents 1 Introduction 1 2 Mappings 5 3 Theory Declarations 11 4 Prettyprinting Theory Instances 19 5 Comparison with Other Systems 21 6 Future Work 25 7Conclusion 27 Bibliography 29 iii Chapter 1 Introduction Theory interpretations have a long history in first order logic [Sho67,End72,Mon76] They are used to show that the language of a given source theory # can be interpreted within a target theory # such that the corresponding interpretation of axioms of # become theorems of # . This demonstrates the consistency of # relative to # , and also the decidability of # ....

Joseph R. Shoenfield. Mathematical Logic. Addison-Wesley, Reading, MA, 1967. 30

....contains two symbols: integer and real , and which contain standard arithmetic predicates and function symbols such as 0, 1, Gamma, Delta, both for integers and for reals. We will assume that this theory contains both the standard first order theory of integers (Peano arithmetic [4, 13, 56]) and a standard first order theory of real numbers [7, 12, 57, 60] One of the possibilities is to consider, as the theory T , axiomatic set theory (e.g. ZF) together with explicit definitions of integers, real numbers, and standard operations and predicates in terms of set theory. Once a ....

Schoenfield, J.R.: Mathematical logic,Addison-Wesley, 1967.

....be substituted for a variable, which is a syntactic object. Taken literally, this is nonsense. However, there is sound intuition behind this nonsense, as proved by the fact that, if one handles this with care (see e.g. 11, 13] the desired result is achieved anyway. Curiously, Shoenfield [25] tries to avoid the difficulty by assuming that there is a constant in the language for every domain element. As any reasonable language would have a countable set of names, axiomatization of a domain such as the reals seems to be ruled out. 7 Domain means universe of discourse in logic ....

Joseph R. Shoenfield. Mathematical Logic. AddisonWesley, 1967.

....and Reiter identified an important group of theories that are consistent with CWA: 4 Theorem 1.2 If T is a consistent Horn theory, then CWA(T) is consistent. 2 Horn theory is defined as any theory in clausal form with at most one positive literal in each clause for all clauses of the theory [Shoenfield, 1967]. Theorem 1.3 Let T be a theory and suppose that CWA(T) is consistent. If A is a ground positive sentence, then CWA(T ) A iff T A CWA(T ) A iff T 6 A: 2 A ground positive sentence is a conjunction of disjunctions of atomic sentences containing no variables (a variable free conjunctive ....

....propagation computations. Without any change in the NLBN, any repetition of the query with the same network segment and inputs will result in the same belief state. The NLBN therefore converges with a stable belief state. 3 2. 3 Representing Relations This section shows how logical relations [Shoenfield, 1967] and other relations are represented by different combinations of directed links in a NLBN. 2.3.1 Relations Represented by Combinative links The four valued logical relations modeled by the combinative links of a NLBN are direct extensions of Kleene s strong three valued logic [Kleene, 1964] If ....

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Joseph R. Shoenfield. Mathematical Logic. Addison-Wesley, Readings, Massachusetts, 1967.

....propagation computations. Without any change in the NLBN, any repetition of the query with the same network segment and inputs will result in the same belief state. The NLBN therefore converges with a stable belief state. 3 4 Representing Relations This section shows how logical relations [ 33 ] and other relations are represented by different combinations of directed links in a NLBN. 4.1 Relations Represented by Combinative links The four valued logical relations modeled by the combinative links of a NLBN are direct extensions of Kleene s strong three valued logic [ 16 ] If truth ....

....as equivalent in a NLBN. The definition for negation is the same as Belnap s four valued scheme AE [ 13; 3 ] Definition 7 (Logical Relations) The basic logical connectives for a NLBN are AND, OR and NOT: f; g and all logical relations are expressed in Conjunctive Normal Form (CNF) 2; 33 ] k i=1 ( l j=1 ff ij ) where k, l are the numbers of conjuncts and disjuncts in the expression respectively, and ff ij is a proposition p or the negated form of the proposition :p. 2 Material Implication ( A material implication logical relation, for example, a b, can be ....

Joseph R. Shoenfield. Mathematical Logic. Addison-Wesley, Readings, Massachusetts, 1967.

.... is schematized as follows: 8X1 : 8Xn8Y (f n (X1 ; Xn) Y A(X1 ; Xn ; Y ) 8 a1 : 8 an8 b(f n (a1 ; an) b A(a1 ; an ; b) 32 These proofs are obtained by slightly modifying standard results of first order logic as presented in [Sup57] and [Sch67]. Regarding the strictness transformations, see Farmer s Eliminability Theorem for his partial first order logic [Far95] and his remark in [Far90] p. 1289. Regarding the strictness transformation of conditional sentences, also see [Sch65] Regarding the closure transformation of an equational ....

Joseph Schoenfield. Mathematical Logic. Addison-Wesley, Reading, MA, 1967.

....2 , t 1 6= t 2 , and t 1 t 2 . ffl Finally, a formula is any expression that can be obtained from elementary formulas by using propositional connectives (such as , implies) etc. and quantifiers 8x and 9x over all real numbers. Comments. ffl For general notions of logic, see, e.g. [2, 5, 10]. ffl In our definition, we started with two real numbers 0 and 1. It is known that if we start with arbitrary algebraic real numbers, we end up with the same class of first order formulas. Indeed, if we know that r is the only real number on the interval [a; b] for which P (r) 0 for a given ....

J. R. Schoenfield. Mathematical logic. Addison-Wesley, 1967.

....rb) goes to B(a; b) when A(ra; rb) goes to A(a; b) 4 GENERALIZATION OF CATEGORY 1 PROOFS 49 4. 3 Theorem on Constants Argument Although part of the natural development of logic, the Theorem on Constants (TOC) has been traditionally attributed to Shoen eld, and its proof is on page 33 of his book [Sho67]. One version, taken from [FNP99] can be stated as follows. Suppose is a set of formulas and (x) is a formula with a free variable x, and the constant d does not appear in any formula in . Further, suppose [x=d] where the notation [x=d] signi es the substitution of d for x in (x) Then we ....

....regions converge to the square, which has a total area of 1. However, this proof contains another piece of inference that must be accounted for: the observation that the shaded area is always one third of the total area. We will represent this using an invariant. An invariant, as de ned in [Sho67], is something (in this case, a speci c aspect of the diagram) that does not change after each permutation, transformation, or construction step. Another good de nition of invariant comes from [Wei] and follows: A quantity which remains unchanged under certain classes of transformations. ....

Joseph R. Shoeneld. Mathematical Logic. Addison-Wesley, 1967. REFERENCES 73

....names. Thus, qualified names occurring in a theory indicate exactly where this discipline has been violated. Another facet of the Cogito methodology is that if specification modules are validated then the theory derived from an importing module, like Cylinder, will be a conservative extension[Sch67] of the theory derived from imported modules like Circle. In brief, a theory T conservatively extends a theory T 0 if every theorem of T is a theorem of T 0 and every T theorem provable in T 0 is provable in T . In addition, schemas (and all the operations of the schema calculus) are ....

J. R. Schoenfield. Mathematical Logic. Addison-Wesley Publishing Company, 1967.

....in showing that the sequence is always periodic and has a period of 9 or, in other words, that the sequences x 1 ; x 2 ; and x 10 ; x 11 ; are always identical. 2 All results described in this paper hold for real and rational numbers since the two structures are elementary equivalent [9]. Our implementation uses infinite precision numbers to guarantee numerical stability. We also give experimental results on finite precision numbers. 1 invalidate : X1, X2 ] 6= X10, X11 ] sequence( X11,X10,X9,X8,X7,X6,X5,X4,X3,X2,X1] sequence( X2,X1] sequence( X11,X10,X9 Xs] ....

J.R. Schoenfield. Mathematical Logic. Addison-Wesley, 1967.

.... system is considerably richer than the system of arities, it is correspondingly better able to provide a natural representation of syntax (see, for example, the encoding of higher order logic given below) In this section we consider the representation of the abstract syntax of firstorder logic [49] and higher order logic [6] For the sake of specificity, we assume in each case that the language of individuals is that of arithmetic. It will be clear that the method applies to any signature of first order or higher order logic. We shall treat the first order logic example in some detail in ....

Schoenfield, J. R. Mathematical Logic. Addison-Wesley, 1967.

....of our temporal language and may have exceeded the capabilities of the original axiomatization. and (4) 8xy:xy yx. Similarly, the class of temporal structures that are totally ordered fields are characterized by the first order axioms of a totally ordered field (see, e.g. Shoenfield [15] However, there does not exist any axiomatization that characterizes the integer or the real temporal structures. Let BTK T be a BTK system in which any member of T is admissible as the temporal domain T , and let AX T be an axiomatization which characterizes T . Let AX FO be any complete ....

Joseph R. Shoenfield. Mathematical Logic. Addison-Wesley, 1967.

....Rather than duplicate it here we present a version with an additional binding operator. Thus the logic we 2 The splitting of the of the : rules into introduction and elimination schemas, for its combination with other connectives, is a known idea also in the context of classical logic, see e.g. [50], p. 66. present here illustrates, with three examples, how one handles binding operators in the LF. Apart from the quantifiers 9 and 8 we include ffl; a version of Hilbert s choice operator. If x is a variable of type i, then x = x is a term of type o. We can bind x by abstraction obtaining ....

Joseph R. Schoenfield. Mathematical Logic. Addison--Wesley, Reading, Massachusetts, 1967.

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J. R. Schoenfield, Mathematical Logic, Addison-Wesley, Boston, Massachusetts, 1967.

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Joseph R. Shoenfield. Mathematical Logic. Addison-Wesley, Reading, MA, 1967.

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J. R. Schoenfield. Mathematical logic. Addison-Wesley, 1967.

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Joseph Shoenfield. Mathematical Logic. AddisonWesley, 1967.

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Joseph R. Shoeneld, Mathematical Logic, Addison-Wesley, Reading, MA, 1967.

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Joseph R. Shoenfield. Mathematical Logic. Addison-Wesley, 1967.

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Joseph R. Shoenfield. Mathematical Logic. Addison-Wesley, 1967.

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Joseph Shoenfield, Mathematical logic, Addison-Wesley, 1967.

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Joseph R. Shoenfield, Mathematical logic, Addison-Wesley, Reading, Massachusetts, 1967.

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Joseph R. Shoeneld, Mathematical Logic, Reading MA 1967

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Joseph Shoenfield, Mathematical logic, Addison-Wesley, 1967.

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Joseph R. Shoeneld. Mathematical Logic. Addison-Wesley, Duke University, 1967.

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Joseph R. Shoenfield, Mathematical Logic, Addison-Wesley, Reading, MA, 1967.

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