Jon Barwise. An introduction to first-order logic. In Jon Barwise, editor, Handbook of Mathematical Logic, volume 90 of Studies in Logic and the Foundations of Mathematics, chapter A1, pages 5--46. NorthHolland, Amsterdam, Holland, 1978.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... coming from set theory and logic [1, 8, 23, 25] With a remarkably original view of information (which is fully adapted by situation theory) 27, 28] a logic, based not on truth but on information, is being developed [24] This logic will probably be an extension of first order logic [5] rather than being an alternative to it. Individuals, properties, relations, spatio tempo ral locations, and situations are basic constructs of situation theory. The world is viewed as a collection of objects, sets of objects, properties, and relations. Infons ( unit facts) 25] are discrete ....

....in l 2 (l 1 l 2 ) Permitting only coherent situations gives the advantage of distinguishing between logically equivalent statements. For example, the statements Bob is angry and Bob is angry and Bob is shouting or Bob is not shouting are logically equivalent in the classical sense [5]. In situation semantics, these two sentences will not have the same interpretation. A course of events e describing the situation in which Bob is only angry will not contain any sequence about Bob s shouting, i.e. e will be silent on Bob s shouting whereas another courses of events e ....

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J. Barwise. "An Introduction to First-Order Logic," in J. Barwise, editor, Handbook of Mathematical Logic, Amsterdam, Holland: North-Holland, 1977, pp. 5--46.

....in the goal, say a 1 , a n . To emphasize the dependence we write the parameter as b[ a 1 , a n ] Compare with the function application f( a 1 , a n ) parameters resemble Skolem terms, but do not use fictitious functions. Logically, b[ a 1 , a n ] is like a Henkin constant (Barwise [1977], page 30) Variable a is unifiable with term t just if a does not occur in t. This occurs check can be extended to prevent invalid assignments to meta variables. Folderol associates with every parameter the list of meta variables it depends on: in the derivation above, the parameter is b[ a] ....

John Barwise. An introduction to first-order logic. In John Barwise, editor. Handbook of Mathematical Logic. North-Holland, pages 5--46.

....N is the rational set (1, 1) # of N 2 but it is not recognizable (since any two points (n, 0) and (m, 0) with n #= m are not equivalent for #L ) see for example [58, page 16] 3 Some Notions of Logic In this section, we give a short lesson in first order logic. The reader is referred to [3] for more details about first order logic. 3.1 Structures and Formulae In the sequel, we will often meet the structures #N, # and #N, V p #. In first order logic, a structure S = #D, R i ) i#I , f j ) j#J , c k ) k#K # consists of a domain D (some set) a family of relations (R ....

J. Barwise, An introduction to first-order logic, in : Handbook of Mathematical Logic, J. Barwise, Ed., North Holland, Amsterdam (1977) 5--46.

....a formula in first order logic. In order to interpret a first order logic formula, all one does is give a domain (a nonempty set) associate relations on this domain with predicate 26 symbols, associate operations on this domain with function symbols, and map variables to elements of the domain [Bar77]. Collectively, the domain, the relations, and the operations are called a structure; the mapping of variables to elements is called an assignment. In the case of T , we define the structure from the stores # and # # ; the assignment maps r to v, and maps any other free variables of T to their ....

J. Barwise. An introduction to first-order logic. In J. Barwise, editor, The Handbook of Mathematical Logic, Studies in Logic and Foundations of Mathematics, pages 5--46. North Holland, 1977.

....morning. I spent three wonderful years at CRIN, especially the last one. I am grateful to CRIN and to all people I met here for creating me the excellent research opportunities and environment. 2 Preliminaries We suppose some familiarity with basic notions of the first order model theory [CK73, Bar77, Kei77], like signature, formula, satisfaction, etc. We use the standard notation, for example, z = z 1 ; z n is a vector of variables or values, FV (e) the set of free variables in e, N denotes the set of natural numbers, Zthe set of integers, is j N j, i.e. the cardinality of N, 1 is the ....

J. Barwise. An introduction to first-order logic. In J. Barwise, editor, Handbook of Mathematical Logic, Studies in Logic and the Foundations of Mathematics, pages 5--46. North Nolland, 1977.

....tautologies can be traced back to the Henkin s proof of completeness of predicate calculus [5] but it was first explicitly formulated by Smullyan in the form of his Fundamental theorem [8] He calls the idea central to predicate logic. A modern and very readable exposition is by Barwise in [1]. The semantic reduction of predicate logic to tautologies can be expressed as follows: T j= A iff the formula V 1 in A i A is a tautology for some n 0 such that A i 2 T[Eq[Q. Here Eq are the identity and Q the substitution and Henkin axioms. Smullyan in [8] has investigated the syntactic, ....

....(for details see [4, 9] From now on we will refer to the codes of finite sequences (which are numbers) as lists . 3 Semantic reductions In order to establish the terminology we start with a quick review of the standard semantic notions for first order logic. The reader may refer for details to [1]. We then proceed with the investigation of the refined chain of semantic reductions which is done in a detail sufficient to appreciate the symmetries of the figure 1. The chain of semantic reductions in the order discussed below starts in the left bottom corner of the figure and proceeds ....

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J. Barwise. An introduction to first-order logic. In the Handbook of Mathematical Logic, J. Barwise (editor), North-Holland, 1977.

....(adapted from (Barwise 1975) where P is the power operation, and ff and are ordinals. This universe can be depicted as in Figure 3. It should be noticed that the KPU universe is like the ZF universe (excluding the existence of urelements) since it supports the same idea of cumulative hierarchy (Barwise 1977). If M is a structure 10 for L, then an admissible set over M is a model UM of KPU of the form UM = M ; A; 2) where A is a nonempty set of non urelements and 2 is defined in M Theta A. Such a typical admissible set over M can be depicted as in Figure 4. A pure admissible set is an admissible ....

Barwise, J. (1977). An Introduction to First-Order Logic. In Barwise, J. (ed.) Handbook of Mathematical Logic, 5--46, North-Holland, Amsterdam.

....process. The universe of admissible sets over an arbitrary collection M of urelements is defined recursively: ffl VM (0) ffl VM (ff 1) P (M [ VM (ff) ffl VM ( S ff VM (ff) if is a limit ordinal, ffl VM = S ff VM (ff) This universe is depicted in Figure 3, adapted from Barwise (1977). It should be noticed that the KPU universe is like the ZF universe (excluding the existence of urelements) since it supports the same idea of cumulative hierarchy. If M is a structure 8 for L, then an admissible set over M is a model UM of KPU of the form UM = M ; A; 2) where A is a ....

Barwise, J. (1977). An Introduction to First Order Logic. In Barwise, J. (ed.) Handbook of Mathematical Logic, 5--46, North-Holland, Amsterdam.

....W (y) which in any model holds exactly in those states where this program is guaranteed to terminate. Hence Phi [f8yW (y)g would then characterize the Abelian groups where for any element a there is some n such that a n = 1, the class of torsion Abelian groups. It has been shown however ([Bar77]) that this class is not axiomatizable in L . Hence, L is not globally expressive for WHILEL , and thus not globally expressive for WHILE. 2 Excursion on Local Expressiveness After this negative result, one can ask whether we might achieve a weaker positive result in terms of local ....

Jon Barwise. An introduction to first-order logic. In Jon Barwise, editor, Handbook of Mathematical Logic. North-Holland Publishing Company, 1977.

....which has not yet been proved (or refuted) we would not be able to assert P :P . Note how different the CLASS interpretation of P :P is from the BHK interpretation. Of course, P :P is a law of CLASS logic. 47. There are numerous accounts of Godel s completeness theorem. See, for example (Barwise 1977, Chang and Keisler 1973, Enderton 1972, and Mendelson 1987) 48. For a study of the mathematics and philosophy of Cantor, see (Dauben 1979) 49. For an historical account of the controversy about the Axiom of Choice, see (Moore 1982) for a mathematical discussion of the significance of this ....

Barwise, J. (1977). An introduction to first-order logic. In Handbook of mathematical logic (ed. J. Barwise), pp. 5--46. North-Holland, Amsterdam.

....more general notion of a concrete I category. These concrete I categories have objects which are sets with some internal structure given by operations and predicates defined on the elements of the sets or on their finite subsets, i.e. they are weak second order structures in the terminology of [Bar77]. The partial order on objects corresponds to the substructure relation between objects; and morphisms are relations between elements or finite subsets of the carrier sets of objects. We will call these categories information categories, which like the abstract I categories can be complete or ....

K. J. Barwise. An introduction to first order logic. In K. J. Barwise, editor, The Handbook of Mathematical Logic, Studies in Logic and Foundations of Mathematics, pages 5--46. North Holland, 1977.

....for the reals. That is, any sentence in the theory of real closed fields is provable if and only if it is true of the reals. However, this completeness comes at the price of expressiveness. It is well know that there is no first order axiomatization that characterize the reals up to isomorphism (Barwise [ 1977 ] What the theory of real closed fields does is to carve out a characterizable piece of the theory of the reals. In particular, it restricts the language so that only a limited set of the sentences can be written in the theory. It is for this limited set of sentences that the completeness ....

Jon Barwise. An introduction to first-order logic. In Jon Barwise, editor, Handbook of Mathematical Logic, chapter A.1. North-Holland, Amsterdam, 1977.

....of a variable a formula is true. Universally quantified means for all values (denoted 8) and existentially quantified means there exists some value (denoted 9) The order of a logic specifies what entities 8 and 9 may quantify over. First order logic can only quantify over atomic formulas [5], i.e. true, false or p(t 1 ; t n ) with p a predicate and t 1 ; t n are terms. In first order logic, quantifiers always range over all the elements of the domain of discourse. A set L of connectives for functions, relations and constants (i.e. primitive non logical symbols) for ....

J. Barwise. An introduction to first-order logic. In J. Barwise, editor, Handbook of Mathematical Logic, pages 5--46. North Holland, 1977.

.... from mathematical programming, like ampl or gams [31] or a high level programming language from computer science, like chip [18] ilog solver [29] or oz [32] In order to clarify the relationship of the constraint languages underlying ILP and CP(FD) we propose to use first order predicate logic [9], which gives us a standard syntax and a very well understood semantics to compare the two approaches. There exist also higher order notions in finite domain constraint programming, but we do not consider these in the present paper. In first order predicate logic, a language is defined by a ....

J. Barwise. An introduction to first-order logic. In J. Barwise, editor, Handbook of Mathematical Logic. North Holland, 1977.

No context found.

Jon Barwise. An introduction to first-order logic. In Jon Barwise, editor, Handbook of Mathematical Logic, volume 90 of Studies in Logic and the Foundations of Mathematics, chapter A1, pages 5--46. NorthHolland, Amsterdam, Holland, 1978.

No context found.

J. Barwise. An introduction to first-order logic. In J. Barwise, editor, The Handbook of Mathematical Logic, Studies in Logic and Foundations of Mathematics, pages 5--46. North Holland, 1977.

No context found.

J. Barwise, An introduction to first-order logic, in: Barwise [4], pages 5--46.

No context found.

Barwise, J. "An introduction to first-order logic", in Handbook of Mathematical Logic (J. Barwise, Ed.), 5--46, Amsterdam: North-Holland Publishing Company, 1978.

No context found.

Jon Barwise. An introduction to first-order logic. In Jon Barwise, editor, Handbook of Mathematical Logic, chapter A.1, pages 5--46. North-Holland Publishing Company, Amsterdam, The Netherlands, 1977.

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