Home  - Mathematical_Logic - Logic With Extra Quantifiers And Operators
 Images Newsgroups
 Page 1     1-57 of 57    1

1. First-order Logic - Wikipedia, The Free Encyclopedia
First order Logic with extra quantifiers has new quantifiers Qx, , Logic they include all the quantifiers and Logical operators of first order Logic
http://en.wikipedia.org/wiki/First-order_logic
var wgNotice = ""; var wgNoticeLocal = ""; var wgNoticeLang = "en"; var wgNoticeProject = "wikipedia";
##### First-order logic
Jump to: navigation search First-order logic (FOL) is a formal deductive system used by mathematicians, philosophers, linguists, and computer scientists. It goes by many names, including: first-order predicate calculus FOPC the lower predicate calculus the language of first-order logic or predicate logic . Unlike natural languages such as English, FOL uses a wholly unambiguous formal language interpreted by mathematical structures. FOL is a system of deduction extending propositional logic by allowing quantification over individuals of a given domain (universe) of discourse. For example, it can be stated in FOL "Every individual has the property P". While propositional logic deals with simple declarative propositions, first-order logic additionally covers predicates and quantification. Take for example the following sentences: "Socrates is a man", "Plato is a man". In propositional logic these will be two unrelated propositions, denoted for example by

2. List KWIC DDC22 510 And MSC+ZDM E-N Lexical Connection
Logic to commutative algebra applications of 13L05 Logic to group theory applications of 20A15 Logic with extra quantifiers and operators 03C80
http://www.math.unipd.it/~biblio/kwic/msc-cdd/dml2_11_33.htm
 linear integral equations # systems of linear integral equations # systems of nonsingular linear integral equations # systems of singular linear logic and other substructural logics linear logic, Lambek calculus, BCK and BCI logics) # substructural logics (including relevance, entailment, linear mappings, matrices, determinants, theory of equation) # linear algebra. multilinear algebra. (vector spaces, linear models # generalized linear operators linear operators # equations and inequalities involving linear operators # equations with linear operators # general theory of linear operators # groups and semigroups of linear operators # special classes of linear operators (operator and matrix valued functions, etc., including analytic and meromorphic ones) # functions whose values are linear operators as elements of algebraic systems # individual linear operators) # linear relations (multivalued linear operators, their generalizations and applications # groups and semigroups of linear operators, with operator unknowns # equations involving

3. 03Cxx
03C10 Quantifier elimination, model completeness and related topics 03C80 Logic with extra quantifiers and operators See also 03B42, 03B44, 03B45,
http://www.ams.org/msc/03Cxx.html

 4. MSC 2000 : CC = Operators 03C80 Logic with extra quantifiers and operators See also 03B42, 03B44, 13N10 Rings of differential operators and their modules See also 16S32, 32C38http://portail.mathdoc.fr/cgi-bin/msc2000.py?L=fr&T=Q&C=msc2000&CC=Operators

5. Mhb03.htm
03C70, Logic on admissible sets. 03C75, Other infinitary Logic. 03C80, Logic with extra quantifiers and operators See also 03B42, 03B44, 03B45, 03B48
http://www.mi.imati.cnr.it/~alberto/mhb03.htm

6. Sachgebiete Der AMS-Klassifikation: 00-09
topics 03XX Mathematical Logic and foundations 03-00 General reference Other infinitary Logic 03C80 Logic with extra quantifiers and operators,
http://www.math.fu-berlin.de/litrech/Class/ams-00-09.html
##### nach 90-99 Weiter nach 10-19 Suche in allen Klassifikationen
01-XX 03-XX 04-XX 05-XX 06-XX 08-XX
##### nach 90-99 Weiter nach 10-19 Suche in allen Klassifikationen

 7. HeiDOK 03C75 Other infinitary Logic ( 0 Dok. ) 03C80 Logic with extra quantifiers and operators ( 0 Dok. ) 03C85 Second and higher-order model theory ( 0 Dok.http://archiv.ub.uni-heidelberg.de/volltextserver/msc_ebene3.php?zahl=03C&anzahl

8. PlanetMath: Generalized Quantifier
Generalized quantifiers are an abstract way of defining quantifiers. and foundations Model theory Logic with extra quantifiers and operators)
http://planetmath.org/encyclopedia/GeneralizedQuantifier.html
 (more info) Math for the people, by the people. Encyclopedia Requests Forums Docs ... RSS Login create new user name: pass: forget your password? Main Menu sections EncyclopÂ¦dia Papers Books Expositions meta Requests Orphanage Unclass'd Unproven ... Classification talkback Polls Forums Feedback Bug Reports downloads Snapshots PM Book information News Docs Wiki ChangeLog ... About generalized quantifier (Definition) Generalized quantifiers are an abstract way of defining quantifiers The underlying principle is that formulas quantified by a generalized quantifier are true if the set of elements satisfying those formulas belong in some relation associated with the quantifier. Every generalized quantifier has an arity , which is the number of formulas it takes as arguments , and a type , which for an -ary quantifier is a tuple of length . The tuple represents the number of quantified variables for each argument. The most common quantifiers are those of type , including and . If is a quantifier of type is the universe of a model, and is the relation associated with in that model, then

 9. JSTOR Probability Quantifiers And Operators. There are also analogous biprobability Logics with integral operators instead of probability quantifiers. Chapter 7 presents a Logic that has bothhttp://links.jstor.org/sici?sici=0022-4812(199809)63:3<1191:PQAO>2.0.CO;2-K

10. Teaching Freshman Logic With MIZAR-MSE
The power of the equational argument comes not so much from the equational reasoning, as from the treatment of quantifiers as operators, and the ability to
http://www.cs.ualberta.ca/~hoover/dimacs-teaching-logic/paper.html
##### Teaching freshman logic with MIZAR-MSE
H. James Hoover and Piotr Rudnicki
Department of Computing Science
University of Alberta
Edmonton, Alberta, Canada T6J 2H1
[hoover,piotr]@cs.ualberta.ca
http://www.ualberta.ca/[,] June 25, 1996
##### Introduction
We would like to share our experience of using a proof-checker in teaching introductory logic to first year science students who plan to enroll into computing science. For several years, we have been teaching an introduction to predicate logic as a part of a course which also covers: (a) elementary material of discrete mathematics (sets, relations, functions, and induction), and (b) reasoning about iterative programming constructs using variants and invariants. The presentation of the material from these three areas is spread over the thirteen week course. The logic component of the course stresses the practical skills of deductive reasoning in predicate calculus. We use the MIZAR-MSE proof-checker to check students' assignments-typically about 50 small proofs for the course. The syntax of the MIZAR-MSE input language is a notation of natural deduction derived from the style of Gentzen, Jaskowski and Fitch. We have chosen a logical system which is an extension of a natural deduction system for the following two reasons:
• Jaskowski's goal (see [ ] reprinted in [ ]) was to identify the methods used by mathematicians in their proofs, ``to put those methods under the form of structural rules and to analyze their relation to the theory of deduction.'' Inspection shows that he has achieved his goal: indeed the proof structures of his logical system are frequently used by proof authors. It is worthwhile noting here that Jaskowski did not use the term

11. Tree Structure Of LoLaLi Concept Hierarchy Updated On 2004624
490 boolean operators . . . . SbC 211 alethic Logic g . .. 227 Logic with extra quantifiers . . . . . SbC 457 modal model theory (7) + .
http://remote.science.uva.nl/~caterina/LoLaLi/soft/ch-data/tree.txt

12. Ockham Algebras With Additional Operators -- Figallo Et Al. 12 (6): 447 -- Logic
Logic Journal of IGPL 2004 12(6)447459; doi10.1093/jigpal/12.6.447 Priestley spaces, quantifiers, interior operators, congruence relations
http://jigpal.oxfordjournals.org/cgi/content/abstract/12/6/447
@import "/resource/css/hw.css"; @import "/resource/css/igpl.css"; Skip Navigation Oxford Journals Logic Journal of IGPL 2004 12(6):447-459; doi:10.1093/jigpal/12.6.447
Oxford University Press

##### Ockham Algebras with Additional Operators
Aldo V. Figallo Paolo Landini and Alicia Zilliani Here we initiate an investigation of the equational classes of Ockham algebras endowed with a quantifier (or OQ-algebras) and of monadic distributive lattioes endowed with a dual endomorphism (or MOL-algebras). These varieties are natural generalizations of the Q-distributive lattices introduced by R. Cignoli and the monadic the De Morgan algebras considered by A. Petrovich

13. Review Miodrag Raskovic, Radosav Dordevic, Probability
Review Miodrag Raskovic, Radosav Dordevic, Probability quantifiers and operators. H. Jerome Keisler. Source J. Symbolic Logic Volume 63, Issue 3 (1998),
http://projecteuclid.org/handle/euclid.jsl/1183745594
Log in RSS Title Author(s) Abstract Subject Keyword All Fields FullText more options
• Home Browse Search ... next
##### Review: Miodrag Raskovic, Radosav Dordevic, Probability Quantifiers and Operators
H. Jerome Keisler Source: J. Symbolic Logic Volume 63, Issue 3 (1998), 1191-1193.
##### Journal of Symbolic Logic

 14. 1. Statements And Logical Operators Negations of statements involving the quantifiers all or some are tricky. . not boring even though Logic is a boring subject in Logical form.http://people.hofstra.edu/Stefan_waner/realworld/logic/logic1.html

 15. Storage Operators And Multiplicative Quantifiers In Many-valued Logics To every manyvalued Logic L we associate a Logic LS obtained from L by the operator allows one to obtain a multiplicative universal quantifier whichhttp://portal.acm.org/citation.cfm?id=1094367.1094375

16. Semantics And Logical Form
More complex propositions can be constructed using logical operators. (NOT (LOVES1 SUE1 JACK1)) . and extra quantifiers. HOWMANY HOW-MUCH
http://www.cse.unsw.edu.au/~billw/cs9414/notes/nlp/logicalform/semanticslogfm-20
##### Semantics and Logical Form
Reference: Chapter 8 of Allen Aim: To describe a language for representing logical forms - that is, intermediate representations on the way to transforming a parse tree into the final meaning representation. Logical forms must be able to encode possible ambiguities of meaning of a particular parse of a sentence. Keywords: co-agent compositional semantics exists experiencer ... victim Plan:
• Definition of compositional semantics
• Word senses and ambiguity
• Logical form language - terms, predicates, propositions, logical operators, quantifiers, predicate operators, modal operators.
• Ambiguity in logical forms
• Verbs and states in logical forms - thematics roles
• Logical forms for speech acts and for embedded sentences
##### Semantics
• Syntax concerns structure; semantics concerns "meaning".
• Semantics is often assumed to be compositional : the meaning of a phrase like three green boxes is constructed from the meanings of the words: three, green, and boxes
• Compositional semantics is violated in idioms like kick the bucket which can mean "die", and

17. Storage Operators And Multiplicative Quantifiers In Many-valued Logics -- Montag
To every manyvalued Logic L we associate a Logic LS obtained from L by the adding of storage operators, multiplicative quantifiers, algebraic semantics
http://logcom.oxfordjournals.org/cgi/content/abstract/14/2/299
@import "/resource/css/hw.css"; @import "/resource/css/logcom.css"; Skip Navigation Oxford Journals Journal of Logic and Computation 2004 14(2):299-322; doi:10.1093/logcom/14.2.299
Oxford University Press

ISI Web of Science (1)
Request Permissions Google Scholar Articles by Montagna, F. Search for Related Content
##### Storage Operators and Multiplicative Quantifiers in Many-valued Logics
Franco Montagna University of Siena, Department of Mathematics, Via del Capitano 15, 53100 Siena, Italy. E-mail: To every many-valued logic L we associate a logic LS obtained from L by the adding of a storage operator which has some analogies with Girard's exponential !. The algebraic counterpart of *

18. A Short Bibliography On Spatial Logics For Concurrency
Adds to the ambient Logic of CG00 two operators to deal with restricted names, the revelation operator and the freshness quantifier, which is defined here
http://ctp.di.fct.unl.pt/SLMC/bibpage.html
 A short bibliography on Spatial Logics for Concurrency This list is not claimed to me complete; papers are listed in backwards chronological order. We only mention here works related to concurrency, but there are other applications of spatial logics, in particular to semi-structured data (studied by Cardelli, Gardner, Ghelli, DalZilio, Lugiez, Meyssonnier). Separation and Bunched logics are also closely related, see links on Peter O'Hearn's web site for works related to the semantics, proof theory and tools for various fragments of separation logic. We are also aware of some ongoing work of Ghelli, Gardner and Calcagon on tools for static spatial logics. Suggestions, corrections, and additions are very welcome. Electronic version of the papers can, in general, be found in the respective author's web site. ExpressivitÃ© des logiques d'espaces Ãtienne Lozes. PhD Thesis (in French). Gives a complete survey on the expressiveness of spatial logics, namely what properties can be specified, up to what point can the logic distinguish between elements of the model, what are the minimal operations in the logic that provide this expressiveness and what is the complexity of the associated algorithms. An Observational Model for Spatial Logics Emilio Tuosto and Hugo Torres Vieira.

19. AKRI : Artificial Intelligence : From Logic To Fuzzy Logic
Two quantifiers are also added to the list of logical operators. an extra function (skolem function) to remove an existential quantifier from the scope
http://akri.org/ai/flogic.htm
@import url(../css/sophist.css); Skip Navigation Home Search Site Index ... Artificial Intelligence John L. Gordon
##### Abstract:
Fuzzy logic enables a computer to make decisions which care more in line with the sort of decisions which a human would make. Computer logic is rigorous and deterministic and relates to finite states and numbering systems. Computer logic marks distinct boundaries between any states. For instance, given various weather conditions to process such as, stormy, rainy, cloudy, sunny, ordinary logic would assign one of these values to any weather condition being observed. People however would recognise all sorts of shades in between theses states such as dull or drizzle etc. This is exactly what fuzzy logic can do. What is more impressive is that fuzzy logic offers a way of processing these decisions so that a final result is still correct. This text provides an overview of the general subject of logic, taking a multi-perspective view. Fuzzy logic is considered following a background description of other logics. Concluding statements present fuzzy logic as an element in the set of logic types.
##### The Nature of Logic
What you know about logic will often depend on what subjects you have studied. If you have not studied logic you may still be familiar with the philosophy of the Vulcan race from the TV show 'Star Trek'. This popularised view of logic is nevertheless a valuable one. It implies that Vulcans makes decisions in a strictly deductive way, ignoring emotions and having the ability to place events into well defined categories. The deductive power of the Vulcans, and in particular, Mr Spock, produces verifiable decisions based on available evidence or recognises that a decision cannot be made due to lack of information.

Yet every operator in Logic is a specialization of some word or phrase in natural to represent the quantifiers and operators of firstorder Logic.
http://doi.ieeecomputersociety.org/10.1109/MIS.2007.29

21. Good Math Has Moved To
operators in first order predicate Logic are and, written ; OR, For FOPL, we need to add inference rules for the new things quantifiers and variables
http://goodmath.blogspot.com/2006/03/calculus-no-not-that-calculus_29.html
ScienceBlogs : Calculus - no, not that calculus! @import url("http://www.blogger.com/css/blog_controls.css"); @import url("http://www.blogger.com/dyn-css/authorization.css?targetBlogID=23588438"); var BL_backlinkURL = "http://www.blogger.com/dyn-js/backlink_count.js";var BL_blogId = "23588438";
##### Calculus - no, not that calculus!
This post is about more logic. It's also about a kind of calculus.
When you hear the word calculus, you probably think of derivatives and integrals, differential equations, and stuff like that. Well, that's a calculus: differential calculus. But it's not the only calculus. In fact, a calculus is really any system of rules for manipulating symbolic expressions, which has a formal mathematical or logical basis for the meanings of its rules.
That's really a fancy way of saying that a calculus is a bunch of mechanical rules with some special properties.
When I talked about propositional logic, the rules for working with propositions were really simple. In basic propositional logic, there's a finite set of propositions, and so there aren't a lot of deep things you can infer. Overall, it's really pretty trivial. When you get into more complicated logics, like first order predicate logic (FOPL), then things can get really interesting. In FOPL, there are variables, and you can find statements where you can create infinite loops using inference rules to try to prove some statement.
In logics like FOPL, the inference rules form a general symbolic/mechanic reasoning system. You can manipulate the rules and derive inferences

22. Alur/Henzinger/Kupferman: Alternating-time Temporal Logic
Temporal Logic comes in two varieties lineartime temporal Logic assumes admit arbitrary nesting of selective path quantifiers and temporal operators,
http://www.eecs.berkeley.edu/~tah/Publications/alternating-time_temporal_logic.h
##### Alternating-time Temporal Logic
Rajeev Alur, Thomas A. Henzinger , and Orna Kupferman Temporal logic comes in two varieties: linear-time temporal logic assumes implicit universal quantification over all paths that are generated by system moves; branching-time temporal logic allows explicit existential and universal quantification over all paths. We introduce a third, more general variety of temporal logic: alternating-time temporal logic offers selective quantification over those paths that are possible outcomes of games, such as the game in which the system and the environment alternate moves. While linear-time and branching-time logics are natural specification languages for closed systems, alternating-time logics are natural specification languages for open systems. For example, by preceding the temporal operator "eventually" with a selective path quantifier, we can specify that in the game between the system and the environment, the system has a strategy to reach a certain state. The problems of receptiveness, realizability, and controllability can be formulated as model-checking problems for alternating-time formulas. Depending on whether or not we admit arbitrary nesting of selective path quantifiers and temporal operators, we obtain the two alternating-time temporal logics ATL and ATL*. ATL and ATL* are interpreted over concurrent game structures Journal of the ACM 49:672-713, 2002. Preliminary versions appeared in the

23. Propositions For The Common Logic Interchange Format
The lexicon must include identity, the usual boolean operators and quantifiers, and a distinguished constant ist for McCarthy s true in a context
http://www.ihmc.us/users/phayes/CLIF.html
##### Syntax
(foo bar (baz a b)) baz (baz a b) baz a b a b (ist ?con (baz a b)) (baz a b) proposition ?con ist ist ist ist global vocabulary, i.e., one that contains only denoting names, as I'm still not convinced Pat's treatment of nondenoting names quite works as stated. I'll present the semantics simply by modifying the HTML in Pat's SCL document.
##### Interpretations
The semantics of CLIF is defined conventionally in terms of a satisfaction relation between CLIF text and structures called interpretations . Since CLIF core syntax contains a number of 'syntactic sugar' constructions, we give the semantic conditions for a basic subset called the CLIF kernel, and then translate the remaining syntactic constructions into the kernel. A vocabulary is a set of names. The vocabulary of an CLIF text is the set of names occurring in the text,. The vocabulary of an CLIF module is the union of the vocabularies of all text in the module. I ,P I I comprises the objects of I and P I the propositions of I. U

24. CLC:Meetings
with a extra talk by Joachim Lambek 14001500 . Differently from Logic, natural language quantifiers remain in situ, taking semantic scope around an
http://www.let.uu.nl/~ctl/workshops/
##### Workshops on Computational Linguistics and Logic
This page contains the programmes of meetings, workshops, conferences etc. organized from September 2000 onwards by the group of Computational Linguistics working on Categorial Grammar at the UiL OTS. The group is coordinated by Michael Moortgat and composed of Raffaella Bernardi Christophe Costa Florencio Herman Hendriks Paola Monachesi ... Willemijn Vermaat . For further information about the activities carried out within the group, please feel free of contact any of its members.
##### Past:

 25. Oxford University Press: The Many Worlds Of Logic: Paul Herrick Sentences with Overlapping quantifiers. What Are You Talking About? The Universe of Discourse. Dean Martin, Universal Love, and a Summary of Logichttp://www.oup.com/us/catalog/he/subject/Philosophy/Logic/IntroductiontoLogic/?v

 26. [0712.1345] Sequential Operators In Computability Logic The main groups of operators on which CL has been focused so far are the parallel, and disjunction, sequential quantifiers, and sequential recurrences.http://export.arxiv.org/abs/0712.1345

27. Modal Logic
BarcanMarcus has urged an unconventional reading of the quantifiers to avoid of modal Logic without using modal operators by constructing an ordinary
http://www-formal.stanford.edu/jmc/mcchay69/node22.html
Next: Logic of Knowledge Up: DISCUSSION OF LITERATURE Previous: DISCUSSION OF LITERATURE
##### Modal Logic
It is difficult to give a concise definition of modal logic. It was originally invented by Lewis (1918) in an attempt to avoid the `paradoxes' of implication (a false proposition implies any proposition). The idea was to distinguish two sorts of truth: necessary truth and mere contingent truth. A contingently true proposition is one which, though true, could be false. This is formalized by introducing the modal operator (read `necessarily') which forms propositions from propositions. Then p 's being a necessary truth is expressed by 's being true. More recently, modal logic has become a much-used tool for analyzing the logic of such various propositional operators as belief, knowledge and tense. There are very many possible axiomatizations of the logic of none of which seem more intuitively plausible than many others. A full account of the main classical systems is given by Feys (1965), who also includes an excellent bibliography. We shall give here an axiomatization of a fairly simple modal logic, the system M of Feys - von Wright. One adds to any full axiomatization of propositional calculus the following:

28. First-Order Predicate Logic
A short description of what predicate Logic is about. these are really just a bit of extra structure necessary to permit the study of quantifiers.
http://rbjones.com/rbjpub/logic/log019.htm
##### predicates in natural languagesquantifiers in natural languagespredicate logics
semi-formal
and formal descriptions of a first-order predicate logic.
informal
semi-formal and formal descriptions of propositional logic.
##### Predicates in Natural Languages
A predicate is a feature of language which you can use to make a statement about something, e.g. to attribute a property to that thing. If you say "Peter is tall", then you have applied to Peter the predicate "is tall". We also might say that you have predicated tallness of Peter or attributed tallness to Peter. A predicate may be thought of as a kind of function which applies to individuals (which would not usually themselves be propositions) and yields a proposition. They are therefore sometimes known as propositional function s Analysing the predicate structure of sentences permits us to make use of the internal structure of atomic sentences, and to understand the structure of arguments which cannot be accounted for by propositional logic alone.

29. Mathematical Background
The most commonly used operators in propositional Logic correspond to the . The order of quantifiers in predicate Logic makes a crucial difference,
http://www.jfsowa.com/logic/math.htm
##### Mathematical Background
by John F. Sowa This web page is a revised and extended version of Appendix A from the book Conceptual Structures by John F. Sowa. It presents a brief summary of the following topics for students and general readers of that book and related books such as Knowledge Representation and books on logic, linguistics, and computer science.
• Sets, Bags, and Sequences
• Functions
• Lambda Calculus
• Graphs ...
• References Note: Special symbols in this file that are outside the Latin-1 character set (ISO 8859-1) are represented by a .gif image for each character. The alt tag for each image gives the name of the character. Students who are just learning the symbols can move the mouse to any symbol to get a brief reminder of its name.
##### 1. Sets, Bags, and Sequences
Elementary or "naive" set theory is used to define basic mathematical structures. A set is an arbitrary collection of elements, which may be real or imaginary, physical or abstract. In mathematics, sets are usually composed of abstract things like numbers and points, but one can also talk about sets of apples, oranges, people, or canaries. In computer science, sets are composed of bits, bytes, pointers, and blocks of storage. In many applications, the elements are never defined, but are left as abstractions that could be represented in many different ways in the human brain, on a piece of paper, or in computer storage. Curly braces are used to enclose a set specification. For small, finite sets, the specification of a set can be an exhaustive list of all its elements:
• 30. SCAN: Computing Correspondences
SCAN terminates and returns a secondorder predicate Logic formula with a parallel Henkin quantifier. For example the translation of the modal Logic
http://www.mpi-inf.mpg.de/departments/d2/software/SCAN/corr.html
##### Computing Correspondences
Let us use the modal logic example to illustrate how SCAN can be used for computing the corresponding frame properties. Suppose we are given the Hilbert axiom and the standard possible worlds semantics of the modal operator:
iff for all v : if r(w,v) then
is the satisfiability relation.) This semantics, together with the usual possible worlds semantics of the ordinary propositional connectives can be taken as a rewrite rules for translating the Hilbert axiom into predicate logic. For the above axiom we get:
The outer quantifier all p comes because Hilbert axioms implicitly assume universal quantification over all formulae. The quantifier all w comes because Hilbert axioms are required to hold in all worlds. is the translation of []p where p(v) means that p is true in world v . This is now a second-order predicate logic formula. Since we want to apply SCAN, we negate it first:
and give it as input to SCAN. The clause form is
-r(w,v) v p(v)
-p(w)
where w is a Skolem constant and v is a variable. There is only one resolvent possible:

31. Common Temporal Logic Constructs For CTL And LTL
Only the following six path Logic quantifier combinations can occur in CTL The F and G quantifiers are trivial forms of the until operator (McMillan
http://www.cl.cam.ac.uk/~djg11/pubs/temporal.html
##### Revision Note: Common Temporal Logic Constructs
A number of different basis sets for temporal logic are possible and a number of restrictions on the allowable uses of negation and quantification lead to various classes of temporal logic. Here are some notes I blogged. A Linear Temporal Logic formula describes a pattern for a sequence of events. Any actual sequence of events may match or not match that pattern. A Branching Temporal Logic formula is more expressive, since it may contain quantifiers that range over a number of possible sequences.
##### Primitives
The following primitives are assumed:
• true The true operator is always true and takes one cycle.
• p Boolean predicate p is true if it is satisfied by the environment.
• X->Y The successor/chop operator holds if X transits to Y.
##### Derived Operators
o circle / next state oF = skip;F diamond / eventually weak next []F square / always []F upto upto F upto G F U G until F U G = F W G unless / weak until F W G atnext atnext F atnext G leadsto leadsto
##### Saftey and Liveness Theorem
A safety property is an assertion that a particular state will not be reached.
• 32. Guide To Logic, Quantifiers II
GUIDE TO Logic Negation of Statements with Two Existential quantifiers . Then Ex Ey P(x, y) means At least one computer operator knows how to use
http://www.jgsee.kmutt.ac.th/exell/Logic/Logic22.htm
GUIDE TO LOGIC
##### Contents
20. Two Variables
21. Symbols for Incomplete Statements with Two Variables

22. Quantifiers with One Variable

23. Two Universal Quantifiers
...
30. Negation of Statements with Two Existential Quantifiers
##### 20. Two Variables
An incomplete statement may have two variables. In such a sentence one variable stands for an individual in one set, and the other variable stands for an individual in another set. For example, suppose we have a number of men and women in a group. Let x stand for a man in the group, and let y stand for a woman in the group. Then x is older than y is an incomplete statement about the men and women in the group. When we replace one of the variables by an individual, we obtain an incomplete statement containing the other variable. For example, if the men are John, Peter and George, and the women are Anne, Mary and Susan, then x is older than Anne
John is older than y
are incomplete statements with one variable replaced and the other variable remaining. When we replace both variables by individuals we obtain a statement. For example

33. ScienceDirect - Journal Of Applied Logic : On Modal μ-calculus With Explic
We also provide the Logic extended with the bisimulation quantifier with a complete . operator with a set of cover operators, one for each natural n.
 Athens/Institution Login Not Registered? User Name: Password: Remember me on this computer Forgotten password? Home Browse My Settings ... Help Quick Search Title, abstract, keywords Author e.g. j s smith Journal/book title Volume Issue Page Journal of Applied Logic Volume 4, Issue 3 , September 2006, Pages 256-278 Methods for Modalities 3 (M4M-3) Abstract Full Text + Links PDF (212 K) Related Articles in ScienceDirect A resolution (minimal model) of the PROP for bialgebras Journal of Pure and Applied Algebra A resolution (minimal model) of the PROP for bialgebras Journal of Pure and Applied Algebra Volume 205, Issue 2 May 2006 Pages 341-374 Martin Markl Abstract This paper is concerned with a minimal resolution of the PROP for bialgebras (Hopf algebras without unit, counit and antipode). We prove a theorem about the form of this resolution (Theorem 15) and give, in Section 5, a lot of explicit formulas for the differential. Abstract Full Text + Links PDF (399 K) An axiomatization of bisimulation quantifiers via the [... ... Theoretical Computer Science An axiomatization of bisimulation quantifiers via the calculus Theoretical Computer Science Volume 338, Issues 1-3

34. Shadow Â» Blog Archive Â» John McArthy On Modal Logic
calculi, that is, when we attempt to introduce quantifiers. This was of modal Logic directly. To do this we give every predicate an extra
http://www.garyfeng.com/wordpress/2004/11/30/john-mcarthy-on-modal-logic/
Mike Terry Modal logic
##### John McArthy on Modal logic
Tags: Original URL John McArthy, creater of Lisp, in an extension of his book on philosophical problems in AI, touched on modal logic The idea is that modal calculi describe several possible
worlds at once, instead of just one. Statements are not assigned a
single truth-value, but rather a spectum of truth-values, one in each
possible world. Now, a statement is necessary when it is true in
all
different modal logics (and even then not all of them) one has to be
a bit more subtle, and have a binary relation on the set of possible
in a world when it is true in all alternatives to that world. Now it
turns out that many common axioms of modal propositional logics
correspond directly to conditions of alternativeness. Thus for
instance in the system M above, Ax . 1 corresponds to the reflexiveness of the alternativeness relation; corresponds to its transitivity. If we make the alternativeness relation into an equivalence relation, then this is just like not

35. Correspondences Between The Primitives Of The Natural Semantic Metalanguage And
Formal language operators (e.g. simple/extended quantifiers, Logic operators) they are directly provided via syntactic sugar by highlevel general-purpose
http://www.webkb.org/kb/nsm.html
##### Correspondences between the primitives of the Natural Semantic Metalanguage and concepts in the ontology or languages of WebKB-2
From the viewpoint of the knowledge representation task and languages, the primitives of the Natural Semantic Metalanguage (NSM) may be categorized as follow.
• Formal language operators (e.g. simple/extended quantifiers, logic operators): they are directly provided via syntactic sugar by high-level general-purpose formal languages (e.g. FCG and FE ) or are provided by "language ontologies" (i.e. ontologies about components of formal languages) that are used in low-level languages (i.e. those with minimal syntactic sugar). Some low-level languages, such as KIF, are expressive enough to define some of these operators (e.g. extended quantifiers can be defined using sets and basic quantifiers
Context-dependent shortcuts (e.g. indexicals such as "this", "you", "here") and straightforward shortcuts (e.g. "when" as an interrogative over a concept of time).
General-purpose content-related categories (e.g. for time), i.e. those that belong to general-purpose "content ontologies" (as opposed to "language ontologies"). Only these categories have been represented in the ontology of WebKB-2 (i.e. no category has been introduced to represent the NSM-related language operators and shortcuts).

• Language operators
• Syntactic shortcuts
• General-purpose content-related categories
##### 1. Language operators
• 36. NON-FREGEAN LOGIC AND ONTOLOGY OF SITUATIONS - T.E.MIECZYSLAW OMYLA - Athenaeum
According to Suszko the author of non-Fregean Logic - the variables running over letters (for example quantifiers, description operator and so on).
http://evans-experientialism.freewebspace.com/omyla.htm
One of the Largest and Most Visited Sources of Philosophical Texts on the Internet.
Evans Experientialism Evans Experientialism
SEARCH THE WHOLE SITE? SEARCH CLICK THE SEARCH BUTTON
The Athenaeum Library
The Nominalist Library Athenaeum Reading Room
##### T.E.Mieczyslaw Omyla
(Symposium on Logic and Its Applications, Torun 21 IX 1987 r.)
Let L be any language. According to Tractatus Logico-Philosophicus by L. Wittgenstein sentences of any language present situations. Thus, for every language L there is associated set of situations UL given by the sentences of the language. The situattions are assigned according to non-Fregean semantics of sentences, the principles of which were described in [4] and [5]. Recently many authors discuss the problems: What are situations?, What is formal representation of situation? What is the role of the notion of situation in the theory of meaning? These problems shall not be discussed in my paper. What I would like to do is just to introduce the notion of ontology ofsituations understood as a set of formulas having three properties:
1. The ontology of situations is a theory in a so called first order language, i. e. language containing only one kind of variables.

Abstract The core of modal Logic is elegantly simple classical propositional Logic with one extra operator, corresponding to a welldefined fragment of
http://hylo.loria.fr/content/Hylo02/abstracts.html
 Abstracts M. Fitting. AddOns Abstract: C. Areces, P. Blackburn, M. Marx and U. Sattler. Welcome to the Workshop Abstract: To be provided. C. Areces and C. Lutz. Concrete Domains and Nominals United Abstract: While the complexity of concept satisfiability in both ALCO , the basic description logic ALC enriched with nominals, and ALC(D) , the extension of ALC with concrete domains, is known to be PSpace-complete, in this article we show that the combination ALCo(D) of these two logics can have a NExpTime-hard concept satisfiability problem (depending on the concrete domain D used). The proof is by a reduction of a NExpTime-complete variant of the domino problem to ALCO(D) -concept satisfiability. Postscript PDF Natural Deduction for First-Order Hybrid Logic Abstract: This is a compainion to a previous paper where a natural deduction system for propositional hybrid logic is given. In the present paper we generalize the system to the first-order case. Our natural deduction system for first-order hybrid logic can be extended with additional inference rules corresponding to conditions on the accessibility relaitons and the quantifier domains expressed by so-called geometric theories. We prove soundness and completeness and we prove a normalisation theorem. Postscript PDF B. Heinemann.

38. Neural Network World
5 IvÃ¡nek J. (Czech Republic) Using fuzzy Logic operators for construction of data mining quantifiers, 403410. Relations between two Boolean attributes
http://www.cs.cas.cz/nnw/contents2004/number5.shtml
##### Contents of Volume 14 (2004)
[1] Editorial, 365. [2] Berka P., LaÂ¹ V., SvÃ¡tek V. (Czech Republic): NEST: re-engineering the compositional approach to rule-based inference, 367-380. BlaÂ»Ã¡k J., PopelÃ­nskÃ½ L. (Czech Republic): Mining first-order maximal frequent patterns, 381-390.
The frequent patterns discovery is one of the most important data mining tasks. We introduce RAP, the first system for finding first-order maximal frequent patterns. We describe search strategies and methods of pruning the search space. RAP which generates long patterns much faster than other systems has been used for feature construction for propositional as well as multi-relational data. We prove that a partial search for maximal frequent patterns as new features is competitive with other approaches and results in classification accuracy increase.
[4] HorvÃ¡th T., KrajÃ¨i S. (Slovakia): Integration of two fuzzy data mining methods, 391-402. [5] IvÃ¡nek J. (Czech Republic): Using fuzzy logic operators for construction of data mining quantifiers, 403-410.
Relations between two Boolean attributes derived from data can be quantified by truth functions defined on four-fold tables corresponding to pairs of the attributes. Several classes of such quantifiers (implicational, double implicational, equivalence ones) with truth values in the unit interval were investigated in the frame of the theory of data mining methods. In the fuzzy logic theory, there are well-defined classes of fuzzy operators, namely t-norms representing various types of evaluations of fuzzy conjunction (and t-conorms representing fuzzy disjunction), and operators of fuzzy implications.

39. III
The idea was to secure (the consistency of) classical Logic and classical mathematics such that we can define the quantifiers in terms of this operator.
http://www.hf.uio.no/ifikk/filosofi/njpl/vol1no2/howlogic/node4.html
Next: References Up: On How Logic Became Previous: II
##### III
If arguments (or ``urgings'') from Skolem and GÃ¶del did not play a major role in the AFOL development, what did? Of course, there were several causes of the AFOL development. A (most certainly incomplete) list of probable causes will be presented below. What I will do in the remainder of this essay is to add some pieces to the answer to the question of the AFOL development. First I will sketch an account of how the analysis of quantification in the 1920s might have helped cause the AFOL development, and below I will present the (even more tentative) suggestion that there might be a connection between Tarski's model-theoretic analyses of the notions of logical truth and logical consequence (and, quite generally, the emergence of model theory as a mathematical discipline) and the emergence of first-order logic as the de facto standard in logic.
Because of the paradoxes that had been discovered (e.g. Russell's and other paradoxes) and to some extent because of the intuitionistic challenge, several logicians in the 1920s felt induced to embrace (Hilbertian) finitism. The idea was to secure (the consistency of) classical logic and classical mathematics by ``finitary'', and hence epistemologically innocuous, methods. Even logicians not directly belonging to Hilbert's school, like Thoralf Skolem, were clearly influenced by this development. The quantifiers constituted a major obstacle for any finitistic analysis of logic: they brought in the possibly

40. Formal Logic/Predicate Logic/Formal Syntax - Wikibooks, Collection Of Open-conte
In predicate Logic it is no longer true that all molecular formulae have a main connective. Some main operators are now quantifiers rather than sentential
http://en.wikibooks.org/wiki/Formal_Logic/Predicate_Logic/Formal_Syntax
var wgNotice = ""; var wgNoticeLocal = ""; var wgNoticeLang = "en"; var wgNoticeProject = "wikibooks";
##### From Wikibooks, the open-content textbooks collection
Formal Logic Predicate Logic Jump to: navigation search Ã¢ÂÂ The Predicate Language Ã¢ÂÂ Predicate Logic ... Free and Bound Variables Ã¢ÂÂ
##### Contents
• Formal Syntax
##### edit Formal Syntax
In The Predicate Language , we informally described our sentential language. Here we give its formal syntax or grammar. We will call our languange . This is an expansion of the sentential language and will include as a subset.
##### edit Vocabulary
• Variables: Lower case letters 'n'Ã¢ÂÂ'z' with a natural number subscript. Thus the variables are: Operation letters: Lower case letters 'a'Ã¢ÂÂ'm' with (1) a natural number superscript and (2) a natural number subscript.
A constant symbol is a zero-place operation letter. This piece of terminology is not completely standard.
• Predicate letters: Upper case letters 'A'Ã¢ÂÂ'Z' with (1) a natural number superscript and (2) a natural number subscript.
A sentence letter is a zero-place predicate letter.

41. Mathematical Logic Research Of Victor Porton: 21 Century Math Method, Operator T
My Math Logic Research 21 Century Math Method, Algebraic Theory of Formulas It does not use quantifiers and set theory at all making mathematics much
http://www.mathematics21.org/math-logic.html
##### My Math Logic Research: 21 Century Math Method, Algebraic Theory of Formulas
Announces of my publications and the current state of the research Journal of post-Axiomatic Mathematics and Logic - I started this online journal.

42. [FOM] Modal Logic With Scope-modifying Operators
In modal Logic it is natural to distinguish between two types of dependency relation between states that interpret modal operators.
http://cs.nyu.edu/pipermail/fom/2005-December/009506.html
##### [FOM] Modal logic with scope-modifying operators
tulenhei@mappi.helsinki.fi tulenhei at mappi.helsinki.fi
Thu Dec 29 18:26:04 EST 2005 aatu.koskensilta at xortec.fi Indeed it does, if for no other reason than the underlying idea being allowing the scope of a modal operator to be non-contiguous. There are independence friendly modal propositional logics, studied by Tero Tulenheimo in his dissertation. More information about the FOM mailing list

 43. Publikationer FrÃ¥n Uppsala Universitet : 823 - Term-modal Logic And Quantifier- Thalmann, Lars Termmodal Logic and quantifier-free dynamic assignment Logic. A mainfeature of our Logics is the use of modal operators indexed byhttp://publications.uu.se/abstract.xsql?lang=sv&dbid=823

44. TECHNICAL REPORTS
Keywords Group decision making, fuzzy Logic, linguistic preferences, fuzzy majority, fuzzy linguistic quantifiers, nondominance degree.
http://decsai.ugr.es/difuso/tre.html
##### Technical Reports (1994)
• #DECSAI-94102, F. Herrera, E. Herrera-Viedma, J. L. Verdegay A Linguistic Decision Process in Group Decision Making (14 pages) February, 1994. Keywords : Group decision making, fuzzy logic, linguistic preferences, fuzzy majority, fuzzy linguistic quantifiers, nondominance degree. (Abstract) (Full paper)
• #DECSAI-94103, F. Herrera, E. Herrera-Viedma, J. L. Verdegay A Sequential Selection Process in Group Decision Making with a Linguistic Assessment Approach (17 pages) February, 1994. Keywords : Group decision making, linguistic labels, linguistic preferences, fuzzy majority, fuzzy linguistic quantifiers, linguistic nondominance degree, linguistic dominance degree, strict dominance degree. (Abstract) (Full paper)
• #DECSAI-94107, J. M. Medina, O. Pons, M. A. Vila GEFRED. A Generalized Model of Fuzzy Relational Databases , (23 pages) July, 1994. Keywords : Fuzzy Relational Database, Database, Fuzzy Sets, Relational Model. (Abstract) (Full paper)
• #DECSAI-94108

 45. Mathematics And Social Sciences - MathÃ©matiques & Sciences Humaines Indeed, a simple quantifier is an operator which applies to a predicate by building of functions in Analysis; the expression of quantification in LogicÂhttp://www.ehess.fr/revue-msh/recherche_gb.php?theme=216

46. Ian Hodkinson: Monodic Fragments Of First-order Temporal Logic
The onevariable fragment of linear first-order temporal Logic even with sole temporal operator Box is EXPSPACE-complete (this solves an open problem of
http://www.doc.ic.ac.uk/~imh/frames_website/monodic.html
##### Decidable fragments of first-order temporal logics
Ian Hodkinson, Frank Wolter, and Michael Zakharyaschev
Ann. Pure. Appl. Logic 106 (2000) 85-134.
##### Monodic packed fragment with equality is decidable
Ian Hodkinson
Studia Logica 72 (2002) 185-197. This paper proves decidability of satisfiability of sentences of the monodic packed fragment of first-order temporal logic with equality and connectives Until and Since, in models with various flows of time and domains of arbitrary cardinality. It also proves decidability over models with finite domains, over flows of time including the real order.
##### Monodic fragments of first-order temporal logics: 20002001 A.D.
I Hodkinson, F Wolter, M Zakharyaschev
In R. Nieuwenhuis and A. Voronkov, editors, Logic for Programming, Artificial Intelligence and Reasoning, number 2250 of LNAI, Springer, 2001, pages 1-23. The aim of this paper is to summarize and analyze some results obtained in 20002001 about decidable and undecidable fragments of various first-order temporal logics, give some applications in the field of knowledge representation and reasoning, and attract the attention of the `temporal community' to a number of interesting open problems.
##### Decidable and undecidable fragments of first-order branching temporal logics
I Hodkinson, F Wolter, M Zakharyaschev

47. Self Join Elimination Through Union - US Patent 7107255
The apparatus of claim 8, wherein the Logic for determining (a) and Logic for simplifying (b) are performed for each quantifier in the query,
http://www.patentstorm.us/patents/7107255-claims.html
United States Patent 7107255
##### Self join elimination through union
US Patent Issued on September 12, 2006
##### Application
No. 09887759 filed on 2001-06-21
##### Current US Class
Access augmentation or optimizing Query formulation, input preparation, or translation Query augmenting or refining (e.g., inexact access)
##### US Patent References
Method for optimizing processing of join queries by determining optimal processing order and assigning optimal join methods to each of the join operations
Issued on: September 6, 1994
Inventor: Iyer, et al.
System for optimizing correlated SQL queries in a relational database using magic decorrelation
Issued on: August 20, 1996
Inventor: Leung, et al.
Exploitation of uniqueness properties using a 1-tuple condition for the optimization of SQL queries
Issued on: March 25, 1997

48. Logical Forms: An Introduction To Philosophical Logic. - Book Reviews | Mind | F
Mark Sainsbury s Logical Forms, as a handbook to Philosophical Logic, is a secondlevel quantifier-like operator and Arthur Prior s challenge to the
http://findarticles.com/p/articles/mi_m2346/is_n405_v102/ai_13634993
@import url(/css/us/pub_page_article.css); @import url(/css/us/template_503.css); @import url(/css/us/tabs_503.css); @import url(/css/us/fa_bnet.css); @import url(http://i.bnet.com/css/fa.css); BNET Research Center Find 10 Million Articles BNET.com Advanced Search Find in free and premium articles free articles only premium articles only this publication Arts Autos Business Health News Reference Sports Technology
##### Explore Publications in:
all Arts Autos Business ... Technology
Content provided in partnership with
FIND IN free and premium articles free articles only premium articles only this publication Arts Autos Business Health News Reference Sports Technology Advanced
Search
##### Logical Forms: An Introduction to Philosophical Logic. - book reviews
Mind Jan, 1993 by Maria J. Frapolli Mark Sainsbury's Logical Forms, as a handbook to Philosophical Logic, is obviously intended for beginners. It can also be used to gain some familiarity with the languages of propositional, first-order and modal logics. It is very clearly written with a large number of exercises, an extensive glossary (in which the most important terms in every chapter are explained) and a set of extremely useful bibliographical notes.
##### Most Popular Articles in Reference

49. Peter Suber, "Translation Tips"
A predicate Logic expression is in prenex normal form if (1) all its quantifiers are clustered at the left, (2) no quantifier is negated, (3) the scope of
http://www.earlham.edu/~peters/courses/log/transtip.htm
 Translation Tips Peter Suber Philosophy Department Earlham College In this hand-out I treat the notation of truth-functional propositional logic and first-order predicate logic as a language, and give guidance on translating from English into this foreign language. In general, "logical" issues, such as methods for making use of the expressions once translated, are omitted here. References to Irving Copi's Symbolic Logic are to the fifth edition, Macmillan, 1979. Truth-Functional Propositional Logic Bivalence . While there are 3-valued and many-valued logics, remember that our logic is 2-valued (or bivalent). Therefore, "She was not unhappy" must be translated as if it were synonymous with "She was happy." If you dislike this restriction, then you dislike bivalence and will have a reason to use a 3-valued or many-valued logic. Exclusive disjunction . Remember that " " in our notation expresses inclusive disjunction: "p q" means that either p is true or q is true or both . The exclusive disjunction of p and q asserts that either p is true or q is true but not both . The natural, but long-winded, way to express exclusive disjunction, then, is "(p

50. Joint-committee Mailing List Archive: RuleML's Horn Logic Program Semantics: Dra
Let R be a Horn Logic program, consisting of rules R_1, , R_k. We define an operator T_R which takes as input any subset V of HB, and which generates as
http://www.daml.org/listarchive/joint-committee/1346.html
##### RuleML's Horn logic program semantics: draft (inline'd and attached)
From: Benjamin Grosof ( bgrosof@MIT.EDU
Date:
• Next message: Benjamin Grosof: "revised version: some suggestions on roadmap for Rules (inline'd and attached too)" Hi folks, I got more ambitious than promised at the end of the last JC telecon, and have drafted a self-contained writeup on the standard (least fixed point / least Herbrand model) semantics of Horn logic programs. That semantics is normative for the Horn logic programs sublanguage of RuleML. Benjamin P.S. (I intend to post this on my webpage, along with other related RuleML stuff, in the next few weeks when I do a major webpage update.) % Draft of a self-contained writeup on the standard % (least fixed point / least Herbrand model) semantics of Horn logic programs. % That semantics is normative for the Horn logic programs sublanguage of RuleML. % The corresponding current RuleML DTD is % http://www.ruleml.org/dtd/0.8/ruleml-urhornlog-monolith.dtd
•  51. 0 Top The TOP Concept In The Hierarchy. 1 Adverbial Modification 253 negation 254 quantifier A form of operator introduced by Frege. It indicates what was, in traditional Logic, called the quantity of a statement,http://staff.science.uva.nl/~caterina/LoLaLi/soft/ch-data/gloss.txt

52. Descriptive Complexity
A transitive closure operator added to secondorder Logic captures this. Second-Order quantifier blocks iterated t(n) steps times describe the boolean
http://www.cs.umass.edu/~immerman/descriptive_complexity.html
##### Descriptive Complexity
Computational complexity was originally defined in terms of the natural entities of time and space, and the term complexity was used to denote the time or space used in the computation. Rather than checking whether an input satisfies a property S, a more natural question might be, what is the complexity of expressing the property S? These two issues checking and expressing are closely related. It is startling how closely tied they are when the latter refers to expressing the property in first-order logic of finite and ordered structures. In 1974 Fagin gave a characterization of nondeterministic polynomial time as the set of properties expressible in second-order existential logic. Extending this theorem, our research has related first-order expressibility to computational complexity. Some of the results arising from this approach include characterizing polynomial time as the set of properties expressible in first-order logic plus a least fixed-point operator, and showing that parallel time on a Parallel Random Access Machine is linearly related to first-order inductive depth. This research has settled a major, long standing question in complexity theory by proving the following result: For all s(n) greater than or equal to log n, nondeterministic space s(n) is closed under complementation. See Neil Immerman's Recent Publications , for available on-line publications on descriptive complexity, and, Descriptive Complexity Survey for the slides of a recent survey talk on descriptive complexity.

53. Description Operator (logic) --Â Britannica Online Encyclopedia
description operator (Logic). EncyclopÃ¦dia Britannica Related Articles (ix) is analogous to a quantifier in that, when prefixed to a wff a, it binds.
http://www.britannica.com/eb/topic-158941/description-operator
Already a member? LOGIN EncyclopÃ¦dia Britannica - the Online Encyclopedia Home Blog Advocacy Board ... Free Trial Britannica Online Content Related to
this Topic Shopping
Revised, updated, and still unrivaled.
2008 Britannica Ultimate DVD/CD-ROM

The world's premier software reference source.
Great Books of the Western World

The greatest written works in one magnificent collection. Visit Britannica Store
##### description operator (logic)
A selection of articles discussing this topic.
##### definite descriptions in LPC
...then stands for the single value of a that makes a i x ), known as a description operator, can be thought of as forming a name of an individual out of a proposition form. ( i x ) is analogous to a quantifier in that, when prefixed to a wff a , it binds... No results were returned. Please consider rephrasing your query. For additional help, please review Search Tips Search Britannica for description operator About Us Legal Notices ... Test Prep Other Britannica sites: Australia France India Korea ... Encyclopedia

 Page 1     1-57 of 57    1