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# Topic: First-order logic

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###### In the News (Mon 17 Jun 19)

 Second-order logic - Wikipedia, the free encyclopedia In mathematical logic, second-order logic is an extension of first-order logic, which itself is an extension of propositional logic. Why second-order logic is not reducible to first-order logic In a second order logic that permits quantifying over functions, it is possible to write formal sentences which mean "the domain is finite" or "the domain is of countable cardinality." To say that the domain is finite, use the sentence which says that every injective function on the domain is surjective. en.wikipedia.org /wiki/Second-order_logic   (1791 words)

 First-order logic - Wikipedia, the free encyclopedia Most of these logics are in some sense extensions of first order logic: they include all the quantifiers and logical operators of first order logic with the same meanings. The predicate calculus is an extension of the propositional calculus that defines which statements of first order logic are provable. An exotic equivalence is the exact equivalence between first order logic with an ordered pair construction and a natural system of relation algebra with the projections of an ordered pair as special relations, investigated by Tarski and Givant. en.wikipedia.org /wiki/First-order_predicate_calculus   (2903 words)

 PlanetMath: second order logic In particular, there are some first order logics with additional quantifiers whose strength is comparable to that of second order logic. Second order logic refers to logics with two (or three) types where one type consists of the objects of interest and the second is either sets of those objects or functions on those objects (or both, in the three type case). This is version 5 of second order logic, born on 2002-08-28, modified 2006-03-04. planetmath.org /encyclopedia/SecondOrderLogic.html   (281 words)

 PlanetMath: first order logic This is version 4 of first order logic, born on 2002-08-28, modified 2003-12-02. A logic is first order if it has exactly one type. classical first order logic, FO Cross-references: languages, iff, semantics, quantifiers, term, type, order, logic planetmath.org /encyclopedia/FirstOrderLogic.html   (98 words)

 First order logic compared with intuitionist and modal logic First order logic is defined as that logic which considers only boolean propositions - that is propositions which must be either true or false. First order logic is of course not the only formal system for the representation of thought. First order logic compared with intuitionist and modal logic www.geocities.com /engleerica/firstorder.htm   (1889 words)

 Leivant. Higher Order Logic. In first order, we have Fraisse's Theorem which states that properties of a model are first order definable iff "it can be recognized by a computation with a finite number of alternations between existential (nondeterministic) and universal (co-nondeterministic) guesses" [compare this to "finitely isomorphic" iff elementary equivalence for finite symbol sets]. Quine concludes that second order logic is a mathematical theory instead of a logic. It follows that full second order logic cannot be interpreted in weak second order logic, and that f-validity of a second order formula is reducible to truth in N of a second order formula. andrew.cmu.edu /~cebrown/notes/leivant.html   (4896 words)

 First-order logic - Wikipedia, the free encyclopedia First order logic in which no atomic sentence lies in the scope of more than three quantifiers, has the same expressive power as the relation algebra of Tarski and Givant (1987). Most of these logics are in some sense extensions of first order logic: they include all the quantifiers and logical operators of first order logic with the same meanings. Unlike the propositional logic, first-order logic is undecidable, provided that the language has at least one predicate of valence at least 2. en.wikipedia.org /wiki/First-order_logic#An_important_type_of_well_formed_formulas:_clauses   (3132 words)

 Leivant. Higher Order Logic. Finite order logic (omega-order logic, or type theory) [Church 1940] is introduced in the form of a relational variant [Schutte 1960] which is essentially Andrews F-omega. It follows that full second order logic cannot be interpreted in weak second order logic, and that f-validity of a second order formula is reducible to truth in N of a second order formula. In first order, we have Fraisse's Theorem which states that properties of a model are first order definable iff "it can be recognized by a computation with a finite number of alternations between existential (nondeterministic) and universal (co-nondeterministic) guesses" [compare this to "finitely isomorphic" iff elementary equivalence for finite symbol sets]. www.andrew.cmu.edu /~cebrown/notes/leivant.html   (4896 words)

 First-Order Logic First-order logic can be distinguished from propositional logic in two ways: this introduction of variables and relations and functions which hold for them, and the ability to quantify over these variables. This section discusses the syntax and semantics of first-order logic. In this chapter, Russell and Norvig introduce first-order logic, one of the most common and widely studied representation languages. www.eecs.umich.edu /~jestelle/quals/rn/chapter7_2.html   (1070 words)

 Higher-order logic - Wikipedia, the free encyclopedia Another way in which higher-order logic differs from first-order logic is in the constructions allowed in the underlying type theory. One of these is the type of variables appearing in quantifications; in first-order logic, roughly speaking, it is forbidden to quantify over predicates. In mathematics, higher-order logic is distinguished from first-order logic in a number of ways. en.wikipedia.org /wiki/Higher-order_logic   (231 words)

 First-order logic - Wikipedia, the free encyclopedia First-order logic is mathematical logic that is distinguished from higher-order logic in that it does not allow quantification over properties. Unlike the propositional calculus, first-order logic is undecidable. (logical or), → (logical conditional), ↔ (logical biconditional). en.wikipedia.org /wiki/First-order_logic   (884 words)

 First-Order Predicate Logic First-order logic permits reasoning about the propositional connectives (as in propositional logic) and also about quantification ("all" or "some"). Though predicates are one of the features which distinguish first-order logic from propositional logic, these are really just a bit of extra structure necessary to permit the study of quantifiers. Unlike propositional logics, in which specific propositional operators are identified and treated, predicate logic uses arbitrary names for predicates and relations which have no specific meaning (until an attempt may be made to apply the logic). www.rbjones.com /rbjpub/logic/log019.htm   (521 words)

 Classical Logic First, recall that "and" is an analogue of the English connective "and". If the first symbol in θ is a negation sign "¬", then was θ produced by clause (2), and not by any other clause (since the other clauses produce formulas that begin with either a quantifier or a left parenthesis). Thus, the first symbol in θ must be either a predicate letter, a term, a unary marker, or a left parenthesis. plato.stanford.edu /entries/logic-classical   (11934 words)

 First Order Predicate Logic First order logic is very well understood, and has a sound mathematical foundation. One such artificial language is the first order predicate calculus, also known as FOL, first order logic. Logic was first studied in the 4th century B.C. its invention is attributed to the Greek philosopher Aristotle. www.ryerson.ca /~dgrimsha/courses/cps721/FOLIntro.html   (271 words)

 The undecidability of first order logic A first order logic is given by a set of function symbols and a set of predicate symbols. Thus, in order to decide whether or not M accepts w, it suffices to check whether or not the formula above is a theorem of first order logic. The completeness theorem for first order logic says that a formula is provable from the laws of first order logic (not given here) if and only if it is true in under all possible interpretations, i.e. kilby.stanford.edu /~rvg/154/handouts/fol.html   (607 words)

 first-order logic - a Whatis.com definition - see also: first-order predicate calculus A sentence in first-order logic is written in the form Px or P(x), where P is the predicate and x is the subject, represented as a variable. Rob van Glabbeek proves that first-order logic is undecidable (it is subject to the Incompleteness Theorem). The Incompleteness Theorem, proven in 1930, demonstrates that first-order logic is in general undecidable. whatis.techtarget.com /definition/0,,sid9_gci835674,00.html   (354 words)

 First-order Model Theory The elements of C are the ordered pairs (a,b) where a is an element of A and b is an element of B. As a first step, one easily sees from the upward and downward Loewenheim-Skolem theorems that if T is κ-categorical for some κ at least as large as the number of formulas in the language of T, then T must be a complete theory. First add to L a supply of new individual constants to serve as names for all the elements of A. plato.stanford.edu /entries/modeltheory-fo   (6179 words)

 II Moore says that ``The question of which logic was appropriate for set theory--first-order logic, second-order logic, or an infinitary logic--culminated in a vigorous exchange between Zermelo and Gödel around 1930''. He proved first-order logic complete and he contributed to the development of von Neumann-Bernays-Gödel (NBG) set theory, which, unlike ZF, is finitely axiomatizable in first-order logic. Maybe it seems strange that a logician could be confused concerning what holds within a logic of one order and what holds within a logic of another order; but Gregory Moore (1988) actually argues forcefully that both Fraenkel and von Neumann in the 1920s were confused on the very same issue. www.hf.uio.no /filosofi/njpl/vol1no2/howlogic/node3.html   (2404 words)

 Citebase - The monadic second-order logic of graphs XVI : Canonical graph
decompositions The modular decomposition of countable graphs:Constructions in Monadic Second-Order Logic. http://www.labri.fr/Perso/ courcell/Textes/BCOumSubmitted(2004).pdfVertex-minors, monadic second-order logic and a conjecture by Seese. The monadic second-order logic of graphs X:Linear orderings. citebase.eprints.org /cgi-bin/citations?id=oai:arXiv.org:cs/0510066   (469 words)

 Decision Problems For Second Order Linear Logic LJ2 is second order intuitionistic propositional logic, shown undecidable by Loeb [JSL 41 (1976) 705-718] and Gabbay. Decision Problems For Second Order Linear Logic Patrick Lincoln, Andre Scedrov, and Natarajan Shankar Relating to the study of polymorphic languages based on linear logic, we have been studying fragments of second order linear logic. Subject: Decision Problems For Second Order Linear Logic www.cis.upenn.edu /~bcpierce/types/archives/1994/msg00133.html   (271 words)

 Second order logic Second order logic can be given so called "weak" or "Henkin" semantics, which makes it virtually the same as first order logic, but then the quantifiers do not range any more over all subsets of the universe. Ordinary first order predicate logic is not able take quantify over subsets of the universe even though it seems often a natural thing to do. Second order logic permits existential and universal quantification over subsets (and over relations and functions) of the universe. www.math.helsinki.fi /~logic/opetus/ext_elem_logic/second_order.html   (134 words)

 fomletter8.txt Logic includes the study of the form of sentences; the maneuver involved in second-order logic (quantifying over predicates) is certainly of logical interest and calls for logical analysis. Second-order logic falls within the scope of logic(1); the question of the admissibility of second-order logic as a fundamental system of reasoning is part of the analysis of predication, which is a logical(1) issue. It appears obvious that the question of the meaningfulness of second-order quantifiers falls under this definition; second-order logic is part of the subject matter of logic (though one might belong to a school of logic which regards second-order logic as inadmissible: "thou shalt not quantify over predicates"). math.boisestate.edu /~holmes/holmes/fomletter8.txt   (594 words)

 --The Plan By existential second order logic we refer to either expressions in first order logic with some vocabulary V, which we denote f, or expression of the form \$Pf, where to f’s vocabulary we add the additional relational symbol P which takes some fixed number of arguments. Horn existential second-order logic has the interesting property that if all graphs with n nodes are equally likely, then the probability that a random graph with n nodes will satisfy a given existential second order Horn expression approaches either 1 or 0 as n goes to infinity. Although the full proof was not given, the idea behind the characterization of NP using existential second order logic has been described. www.cs.brown.edu /courses/gs019/papers/logic.html   (3381 words)

 Peter Suber, "Glossary of First-Order Logic" A property possessed by all the wffs in a set is logically hereditary iff the accepted rules of inference pass it on (transmit it) to all the conclusions derivable from that set by those rules. A wff A of propositional logic created from a wff B of predicate logic by (1) removing the quantifiers from B, and (2) replacing each predicate symbol (and its arguments) in B with a propositional symbol. In propositional logic, an interpretation is just such a function; in predicate logic, it is some set (the domain) together with such a function defined for members of that domain. www.earlham.edu /~peters/courses/logsys/glossary.htm   (9715 words)

 Logic, Higher-order Of course, first-order logic is very strong and it is possible to encode such a statement into it. This interpretation of higher-order logic as denoting truth in a standard model is often used by those studying the mathematical properties of integers and structures that can be built from them (Shapiro, 1985). Since logical connectives within substitutions are possible in higher-order logic, as this example shows, atomic formula unification does not suggest enough substitution terms. www.lix.polytechnique.fr /Labo/Dale.Miller/papers/AIencyclopedia   (1791 words)

 HOL 4 Kananaskis 3 It is the latest version of the HOL automated proof system for higher order logic: a programming environment in which theorems can be proved and proof tools implemented. HOL 4 is particularly suitable as a platform for implementing combinations of deduction, execution and property checking. HOL 4 is an open source project with a BSD-style licence that allows its free use in commercial products. hol.sourceforge.net   (214 words)

 FOL RuleML: The First-Order Logic Web Language This paper describes First-Order Logic RuleML (FOL RuleML), which is planned to be the FOL sublanguage of RuleML 0.9, the rule component of SWRL FOL, and an FOL content language for SWSI. In the spirit of the logic in 'FOL RuleML', the conjoined clauses of a rulebase are connected by an explicit 'And'. 2004-09-27: Clarified in the "In the spirit of the logic..." paragraph of the Introduction. www.ruleml.org /fol   (4133 words)

 VFH - First-Order Logic Revisited The volume is the proceedings from the conference FOL75 - 75 Years of First-Order Logic held at Humboldt University, Berlin, Germany, September 18 - 21, 2003 on the occasion of the anniversary of the publication of Hilbert's and Ackermann's Grundzüge der theoretischen Logik. with Plural Reference / Lanzet, R and Ben-Yami, H. On generalizing the logic-approach to space-time towards general relativity: first steps / Madarasz, J., Nemeti, I, and Töke, C. Constructive Predicate Logic and Constructive Modal Logic. Although the celebrated book marks a most important step in the development of logic, the volume in hand proves the actuality of the question "Which logic is the right logic?" www.akira.ruc.dk /~vincent/firstorder.HTM   (302 words)

 Amazon.com: First-Order Logic: Books: Raymond M. Smullyan Second, don't use this as your first exposure to first-order logic (note the title doesnt say "Introduction to...")- although logically self-contained, it requires some experience to appreciate what a neat little book this is. For example, the very first section is a wiz-bang treatment of trees (not the usual graph-theoretic ones), defined in the abstract/axiomatic fashion. Smullyan has divorced logic from its roots: logics are simply recursively-defined sets of sentences and mappings, and that is that. www.amazon.com /exec/obidos/tg/detail/-/0486683702?v=glance   (1690 words)

 Peter Suber, "Glossary of First-Order Logic" A property possessed by all the wffs in a set is logically hereditary iff the accepted rules of inference pass it on (transmit it) to all the conclusions derivable from that set by those rules. A wff A of propositional logic created from a wff B of predicate logic by (1) removing the quantifiers from B, and (2) replacing each predicate symbol (and its arguments) in B with a propositional symbol. The branch of logic dealing with propositions in which subject and predicate are separately signified, reasoning whose validity depends on this level of articulation, and systems containing such propositions and reasoning. www.earlham.edu /~peters/courses/logsys/glossary.htm   (9715 words)

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