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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 From Wikipedia, the free encyclopedia 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

Home MathSciNet Journals Books ... Contact Us 201 Charles Street Providence, RI 02904 USA Phone: 401-455-4000 or 800-321-4AMS Or email us at ams@ams.org Open Positions Model theory 03C05 Equational classes, universal algebra [See also 03C07 Basic properties of first-order languages and structures 03C10 Quantifier elimination, model completeness and related topics 03C13 Finite structures [See also 03C15 Denumerable structures 03C20 Ultraproducts and related constructions 03C25 Model-theoretic forcing 03C30 Other model constructions 03C35 Categoricity and completeness of theories 03C40 Interpolation, preservation, definability 03C45 Classification theory, stability and related concepts 03C50 Models with special properties (saturated, rigid, etc.) 03C52 Properties of classes of models 03C55 Set-theoretic model theory 03C57 Effective and recursion-theoretic model theory [See also 03C60 Model-theoretic algebra [See also 03C62 Models of arithmetic and set theory [See also 03C64 Model theory of ordered structures; o-minimality 03C65 Models of other mathematical theories 03C68 Other classical first-order model theory 03C70 Logic on admissible sets 03C75 Other infinitary logic 03C80 Logic with extra quantifiers and operators [See also 03C85 Second- and higher-order model theory 03C90 Nonclassical models (Boolean-valued, sheaf, etc.)

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, 32C38 http://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

03-XX Mathematical logic and foundations General reference works (handbooks, dictionaries, bibliographies, etc.) Instructional exposition (textbooks, tutorial papers, etc.) Research exposition (monographs, survey articles) Explicit machine computation and programs (not the theory of computation or programming) Proceedings, conferences, collections, etc. General logic Classical propositional logic Classical first-order logic Higher-order logic and type theory Subsystems of classical logic (including intuitionistic logic) Abstract deductive systems Decidability of theories and sets of sentences [See also Foundations of classical theories (including reverse mathematics) [See also Mechanization of proofs and logical operations [See also Combinatory logic and lambda-calculus [See also Logic of knowledge and belief Temporal logic ; for temporal logic, see ; for provability logic, see also Probability and inductive logic [See also Many-valued logic Fuzzy logic; logic of vagueness [See also Logics admitting inconsistency (paraconsistent logics, discussive logics, etc.)

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

Sachgebiete der AMS-Klassifikation: 00-09 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 Encyclop¦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 both http://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

Tree structure of LoLaLi Concept Hierarchy Updated on 2004:6:24, 13:16 In each line the following information is shown (in order from left to right, [OPT] indicates information that can be missing): Type of relation with the parent concept (see below for the legend) [OPT] Id of the node Name of the node Number of children, in parenthesis [OPT] + if the concept is repeated somehwere [OPT] (see file path.txt for the list of repeated nodes) LEGEND: SbC Subclass Par Part-of Not Notion Res Mathematical results His historical view Ins Instance Uns Unspecified top (4) g . 87 computer science (4) g . . 191 logic (1) (31) + g . . . Par 53 automated reasoning (25) + . . . . 35 belief revision . . . . . 76 update . . . . 67 nonmonotonic reasoning . . . . 63 mathematical induction . . . . 71 rewrite system (3) . . . . . 350 termination . . . . . 348 confluence . . . . . 349 critical pair . . . . 70 resolution (7) + . . . . . 339 purity principle . . . . . 342 simplification . . . . . 337 demodulation . . . . . 338 ordering . . . . . 340 removal of tautologies . . . . . 341 resolution refinement (4) . . . . . . 345 lock resolution . . . . . . 344 hyper resolution . . . . . . 347 theory resolution . . . . . . 346 set of support . . . . . 343 subsumption . . . . 68 paramodulation . . . . Not 72 skolemisation . . . . 65 model checking . . . . 55 clause 55 (2) . . . . . 80 horn clause g . . . . . 79 Gentzen clause . . . . 74 uncertainty . . . . 75 unification + . . . . 57 connection graph procedure . . . . 64 metatheory . . . . 61 literal . . . . 58 connection matrix . . . . 81 clause 81 . . . . . SbC 82 relative clause . . . . 69 reason extraction . . . . 59 deduction (7) + . . . . . Not 109 inconsistency . . . . . 106 consequence g . . . . . SbC 494 labelled deductive system . . . . . 111 rule-based deduction . . . . . Not 108 entailment + . . . . . 110 natural deduction (2) + g . . . . . . Not 482 hypothetical reasoning + . . . . . . Not 483 normalization . . . . . Not 107 consistency + . . . . Res 60 Herbrand's theorem . . . . 56 completion . . . . . 86 Knuth Bendix completion . . . . 73 theorem prover (3) . . . . . 427 Bliksem g . . . . . 428 Boyer-Moore theorem prover . . . . . 429 SPASS g . . . . 66 narrowing . . . . 62 logic programming g . . . . 54 answer extraction . . . . 247 nonmonotonic logic + g . . . . . 248 default inference . . . Par 198 proof theory (22) g . . . . SbC 503 sequent calculus . . . . . Not 484 structural rules . . . . 289 interpretation . . . . 282 constructive analysis . . . . 295 recursive ordinal . . . . 287 Goedel numbering . . . . 288 higher-order arithmetic . . . . 281 complexity of proofs . . . . 294 recursive analysis . . . . Res 292 normal form theorem . . . . 297 second-order arithmetic . . . . SbC 110 natural deduction (2) + g . . . . . Not 482 hypothetical reasoning + . . . . . Not 483 normalization . . . . 290 intuitionistic mathematics . . . . 286 functionals in proof theory . . . . 298 structure of proofs g . . . . 283 constructive system . . . . 291 metamathematics . . . . 59 deduction (7) + . . . . . Not 109 inconsistency . . . . . 106 consequence g . . . . . SbC 494 labelled deductive system . . . . . 111 rule-based deduction . . . . . Not 108 entailment + . . . . . 110 natural deduction (2) + g . . . . . . Not 482 hypothetical reasoning + . . . . . . Not 483 normalization . . . . . Not 107 consistency + . . . . 296 relative consistency . . . . Not 284 cut elimination theorem g . . . . 293 ordinal notation . . . . 285 first-order arithmetic . . . . SbC 485 proof nets . . . SbC 475 first order logic (4) g . . . . 254 quantifier (5) + g . . . . . Not 516 bound variable + g . . . . . His 514 Frege on quantification + g . . . . . Not 517 free variable + g . . . . . His 513 Aristotle on quantification + . . . . . Not 301 scope . . . . . . 351 scoping algorithm . . . . Par 476 first order language g . . . . . Not 477 fragment (3) g . . . . . . SbC 479 finite-variable fragment g . . . . . . SbC 480 guarded fragment g . . . . . . SbC 478 modal fragment g . . . . . . . Not 470 standard translation + g . . . . 511 SPASS g . . . . Par 515 quantification (4) + . . . . . Not 516 bound variable + g . . . . . His 514 Frege on quantification + g . . . . . Not 517 free variable + g . . . . . His 513 Aristotle on quantification + . . . 193 computability theory . . . SbC 167 temporal logic (2) + g . . . 435 type theory (2) + . . . . 433 type . . . . . 434 type shifting . . . . Not 23 polymorphism + g . . . 495 substructural logic . . . SbC 200 relevance logic + . . . . 108 entailment + . . . Res 180 Lindstroem's theorem + . . . SbC 481 linear logic . . . 526 variable g . . . . SbC 517 free variable + g . . . Res 179 Goedel's 1st incompleteness theorem (1931) + g . . . SbC 125 feature logic + . . . . 75 unification + . . . 197 model theory (29) . . . . 237 set-theoretic model theory . . . . 11 universal algebra + . . . . 225 infinitary logic . . . . 217 admissible set . . . . 234 recursion-theoretic model theory . . . . 239 ultraproduct . . . . 227 logic with extra quantifiers . . . . SbC 457 modal model theory (7) + . . . . . SbC 215 Kripke semantics + g . . . . . . Not 489 accessability relation + . . . . . Not 461 generated submodel g . . . . . 462 model (4) + . . . . . . SbC 464 finite model g . . . . . . SbC 466 image finite model . . . . . . . Res 467 Hennessy-Milner theorem g . . . . . . Par 463 valuation g . . . . . . SbC 465 tree model g . . . . . Not 459 disjoint union of models g . . . . . 455 homomorphism (2) + g . . . . . . SbC 456 bounded homomorphism g . . . . . . SbC 468 bounded morphism . . . . . Not 469 expressive power g . . . . . . Not 470 standard translation + g . . . . . Not 460 bisimulation g . . . . 219 completeness of theories . . . . 235 saturation . . . . 222 equational class . . . . 238 stability . . . . 233 quantifier elimination . . . . 221 denumerable structure . . . . 228 model-theoretic algebra . . . . 236 second-order model theory . . . . 230 model of arithmetic . . . . 218 categoricity g . . . . 220 definability . . . . 226 interpolation . . . . SbC 454 first order model theory . . . . . 455 homomorphism (2) + g . . . . . . SbC 456 bounded homomorphism g . . . . . . SbC 468 bounded morphism . . . . 231 nonclassical model (2) . . . . . 246 sheaf model . . . . . 245 boolean valued . . . . 201 set theory (24) + g . . . . . 398 set-theoretic definability . . . . . Not 391 iota operator . . . . . 384 determinacy . . . . . 387 fuzzy relation . . . . . Not 385 filter . . . . . 389 generalized continuum hypothesis . . . . . 386 function (3) g . . . . . . 482 hypothetical reasoning + . . . . . . 509 functional application . . . . . . 508 functional composition . . . . . Not 394 ordinal definability . . . . . Not 107 consistency + . . . . . 397 set algebra . . . . . 399 Suslin scheme . . . . . SbC 383 descriptive set theory g . . . . . 388 fuzzy set g . . . . . 378 borel classification g . . . . . SbC 380 combinatorial set theory . . . . . Not 390 independence . . . . . 381 constructibility . . . . . 396 relation g . . . . . 377 axiom of choice g . . . . . 392 large cardinal . . . . . Not 395 ordinal number . . . . . 393 Martin's axiom . . . . . 382 continuum hypothesis g . . . . . Not 379 cardinal number . . . . 232 preservation . . . . 216 abstract model theory + . . . . . 254 quantifier (5) + g . . . . . . Not 516 bound variable + g . . . . . . His 514 Frege on quantification + g . . . . . . Not 517 free variable + g . . . . . . His 513 Aristotle on quantification + . . . . . . Not 301 scope . . . . . . . 351 scoping algorithm . . . . 229 model-theoretic forcing . . . . 224 higher-order model theory . . . . Par 493 correspondence theory . . . . 223 finite structure . . . Res 182 Loewenheim-Skolem-Tarski theorem + . . . Not 83 completeness (2) + g . . . . SbC 84 axiomatic completeness . . . . SbC 85 functional completeness + . . . SbC 156 modal logic (13) + g . . . . Ins 512 S4 . . . . 488 modes . . . . 486 frame (2) . . . . . SbC 487 frame constraints . . . . Par 210 modal operator (2) + g . . . . . SbC 472 diamond g . . . . . SbC 473 box g . . . . SbC 213 doxastic logic g . . . . Not 489 accessability relation + . . . . Par 471 modal language (2) g . . . . . Par 210 modal operator (2) + g . . . . . . SbC 472 diamond g . . . . . . SbC 473 box g . . . . . 490 boolean operators . . . . SbC 211 alethic logic g . . . . SbC 212 deontic logic (3) g . . . . . SbC 521 standard deontic logic g . . . . . SbC 523 two-sorted deontic logic g . . . . . SbC 522 dyadic deontic logic g . . . . Par 215 Kripke semantics + g . . . . . Not 489 accessability relation + . . . . Par 457 modal model theory (7) + . . . . . SbC 215 Kripke semantics + g . . . . . . Not 489 accessability relation + . . . . . Not 461 generated submodel g . . . . . 462 model (4) + . . . . . . SbC 464 finite model g . . . . . . SbC 466 image finite model . . . . . . . Res 467 Hennessy-Milner theorem g . . . . . . Par 463 valuation g . . . . . . SbC 465 tree model g . . . . . Not 459 disjoint union of models g . . . . . 455 homomorphism (2) + g . . . . . . SbC 456 bounded homomorphism g . . . . . . SbC 468 bounded morphism . . . . . Not 469 expressive power g . . . . . . Not 470 standard translation + g . . . . . Not 460 bisimulation g . . . . SbC 214 epistemic logic g . . . . Not 462 model (4) + . . . . . SbC 464 finite model g . . . . . SbC 466 image finite model . . . . . . Res 467 Hennessy-Milner theorem g . . . . . Par 463 valuation g . . . . . SbC 465 tree model g . . . 194 computational logic (2) . . . Not 183 operator (4) + g . . . . 254 quantifier (5) + g . . . . . Not 516 bound variable + g . . . . . His 514 Frege on quantification + g . . . . . Not 517 free variable + g . . . . . His 513 Aristotle on quantification + . . . . . Not 301 scope . . . . . . 351 scoping algorithm . . . . SbC 210 modal operator (2) + g . . . . . SbC 472 diamond g . . . . . SbC 473 box g . . . . 518 truth-funcional operator (2) g . . . . . SbC 252 iff g . . . . . SbC 253 negation . . . . Not 525 arity g . . . SbC 192 combinatory logic g . . . Par 199 recursive function theory . . . 361 formal semantics (10) + g . . . . 365 property theory . . . . 240 Montague grammar (4) . . . . . 243 sense 243 (4) g . . . . . . 203 meaning relation (5) . . . . . . . 205 hyponymy g . . . . . . . 204 antonymy g . . . . . . . 207 synonymy g . . . . . . . . 149 intensional isomorphism + . . . . . . . 206 paraphrase g . . . . . . . 108 entailment + . . . . . . 375 metaphor g . . . . . . 376 metonymy g . . . . . . 374 literal meaning . . . . . 244 sense 244 g . . . . . 241 meaning postulate . . . . . 242 ptq g . . . . . . 300 quantifying in . . . . 254 quantifier (5) + g . . . . . Not 516 bound variable + g . . . . . His 514 Frege on quantification + g . . . . . Not 517 free variable + g . . . . . His 513 Aristotle on quantification + . . . . . Not 301 scope . . . . . . 351 scoping algorithm . . . . 353 truth (4) + . . . . . 431 truth definition g . . . . . 432 truth value . . . . . 372 truth function + g . . . . . 430 truth condition . . . . 362 dynamic semantics . . . . 363 lexical semantics . . . . 366 situation semantics (2) g . . . . . 402 partiality . . . . . 400 situation . . . . . . 401 scene . . . . Not 507 compositionality . . . . 364 natural logic + . . . . Par 515 quantification (4) + . . . . . Not 516 bound variable + g . . . . . His 514 Frege on quantification + g . . . . . Not 517 free variable + g . . . . . His 513 Aristotle on quantification + . . . SbC 168 lambda calculus (4) g . . . . 170 application . . . . 172 lambda operator . . . . 169 abstraction . . . . 171 conversion . . . 38 knowledge representation (20) + g . . . . 152 frame (1) . . . . 104 database + g . . . . . 105 query g . . . . 165 situation calculus . . . . 167 temporal logic (2) + g . . . . 166 temporal logic (1) g . . . . 93 concept formation . . . . . 90 concept + . . . . . . 91 individual concept . . . . 154 logical omniscience . . . . 162 rule-based representation . . . . 157 predicate logic + g . . . . 159 procedural representation . . . . 161 representation language . . . . 156 modal logic (13) + g . . . . . Ins 512 S4 . . . . . 488 modes . . . . . 486 frame (2) . . . . . . SbC 487 frame constraints . . . . . Par 210 modal operator (2) + g . . . . . . SbC 472 diamond g . . . . . . SbC 473 box g . . . . . SbC 213 doxastic logic g . . . . . Not 489 accessability relation + . . . . . Par 471 modal language (2) g . . . . . . Par 210 modal operator (2) + g . . . . . . . SbC 472 diamond g . . . . . . . SbC 473 box g . . . . . . 490 boolean operators . . . . . SbC 211 alethic logic g . . . . . SbC 212 deontic logic (3) g . . . . . . SbC 521 standard deontic logic g . . . . . . SbC 523 two-sorted deontic logic g . . . . . . SbC 522 dyadic deontic logic g . . . . . Par 215 Kripke semantics + g . . . . . . Not 489 accessability relation + . . . . . Par 457 modal model theory (7) + . . . . . . SbC 215 Kripke semantics + g . . . . . . . Not 489 accessability relation + . . . . . . Not 461 generated submodel g . . . . . . 462 model (4) + . . . . . . . SbC 464 finite model g . . . . . . . SbC 466 image finite model . . . . . . . . Res 467 Hennessy-Milner theorem g . . . . . . . Par 463 valuation g . . . . . . . SbC 465 tree model g . . . . . . Not 459 disjoint union of models g . . . . . . 455 homomorphism (2) + g . . . . . . . SbC 456 bounded homomorphism g . . . . . . . SbC 468 bounded morphism . . . . . . Not 469 expressive power g . . . . . . . Not 470 standard translation + g . . . . . . Not 460 bisimulation g . . . . . SbC 214 epistemic logic g . . . . . Not 462 model (4) + . . . . . . SbC 464 finite model g . . . . . . SbC 466 image finite model . . . . . . . Res 467 Hennessy-Milner theorem g . . . . . . Par 463 valuation g . . . . . . SbC 465 tree model g . . . . 97 context (2) . . . . . 99 context dependence . . . . . 98 context change . . . . 160 relation system . . . . 153 frame problem g . . . . 92 concept analysis . . . . . 90 concept + . . . . . . 91 individual concept . . . . 163 script . . . . 145 idea g . . . . . 90 concept + . . . . . . 91 individual concept . . . . 164 semantic network g . . . . 247 nonmonotonic logic + g . . . . . 248 default inference . . . Par 367 semantics 367 (8) g . . . . 371 truth conditional semantics . . . . 373 truth table . . . . SbC 215 Kripke semantics + g . . . . . Not 489 accessability relation + . . . . 85 functional completeness + . . . . 370 satisfaction . . . . 369 material implication g . . . . 368 assignment . . . . Not 372 truth function + g . . . Par 201 set theory (24) + g . . . . 398 set-theoretic definability . . . . Not 391 iota operator . . . . 384 determinacy . . . . 387 fuzzy relation . . . . Not 385 filter . . . . 389 generalized continuum hypothesis . . . . 386 function (3) g . . . . . 482 hypothetical reasoning + . . . . . 509 functional application . . . . . 508 functional composition . . . . Not 394 ordinal definability . . . . Not 107 consistency + . . . . 397 set algebra . . . . 399 Suslin scheme . . . . SbC 383 descriptive set theory g . . . . 388 fuzzy set g . . . . 378 borel classification g . . . . SbC 380 combinatorial set theory . . . . Not 390 independence . . . . 381 constructibility . . . . 396 relation g . . . . 377 axiom of choice g . . . . 392 large cardinal . . . . Not 395 ordinal number . . . . 393 Martin's axiom . . . . 382 continuum hypothesis g . . . . Not 379 cardinal number . . . Par 216 abstract model theory + . . . . 254 quantifier (5) + g . . . . . Not 516 bound variable + g . . . . . His 514 Frege on quantification + g . . . . . Not 517 free variable + g . . . . . His 513 Aristotle on quantification + . . . . . Not 301 scope . . . . . . 351 scoping algorithm . . . 178 compactness + . . . His 177 aristotelean logic (2) + g . . . . Par 39 syllogism g . . . . Par 513 Aristotle on quantification + . . . Par 196 foundations of theories . . . 195 constraint programming . . Not 88 software (2) . . . 104 database + g . . . . 105 query g . . . 275 programming language (3) . . . . 190 semantics 190 (8) + g . . . . . 356 denotational semantics . . . . . 119 domain theory g . . . . . . 120 domain . . . . . 360 program analysis . . . . . 359 process model . . . . . SbC 215 Kripke semantics + g . . . . . . Not 489 accessability relation + . . . . . 357 operational semantics . . . . . 358 partial evaluation . . . . . 355 algebraic semantics . . . . 276 syntax 276 . . . . 277 prolog g . . . . . 70 resolution (7) + . . . . . . 339 purity principle . . . . . . 342 simplification . . . . . . 337 demodulation . . . . . . 338 ordering . . . . . . 340 removal of tautologies . . . . . . 341 resolution refinement (4) . . . . . . . 345 lock resolution . . . . . . . 344 hyper resolution . . . . . . . 347 theory resolution . . . . . . . 346 set of support . . . . . . 343 subsumption . . Par 34 artificial intelligence (5) g . . . Par 38 knowledge representation (20) + g . . . . 152 frame (1) . . . . 104 database + g . . . . . 105 query g . . . . 165 situation calculus . . . . 167 temporal logic (2) + g . . . . 166 temporal logic (1) g . . . . 93 concept formation . . . . . 90 concept + . . . . . . 91 individual concept . . . . 154 logical omniscience . . . . 162 rule-based representation . . . . 157 predicate logic + g . . . . 159 procedural representation . . . . 161 representation language . . . . 156 modal logic (13) + g . . . . . Ins 512 S4 . . . . . 488 modes . . . . . 486 frame (2) . . . . . . SbC 487 frame constraints . . . . . Par 210 modal operator (2) + g . . . . . . SbC 472 diamond g . . . . . . SbC 473 box g . . . . . SbC 213 doxastic logic g . . . . . Not 489 accessability relation + . . . . . Par 471 modal language (2) g . . . . . . Par 210 modal operator (2) + g . . . . . . . SbC 472 diamond g . . . . . . . SbC 473 box g . . . . . . 490 boolean operators . . . . . SbC 211 alethic logic g . . . . . SbC 212 deontic logic (3) g . . . . . . SbC 521 standard deontic logic g . . . . . . SbC 523 two-sorted deontic logic g . . . . . . SbC 522 dyadic deontic logic g . . . . . Par 215 Kripke semantics + g . . . . . . Not 489 accessability relation + . . . . . Par 457 modal model theory (7) + . . . . . . SbC 215 Kripke semantics + g . . . . . . . Not 489 accessability relation + . . . . . . Not 461 generated submodel g . . . . . . 462 model (4) + . . . . . . . SbC 464 finite model g . . . . . . . SbC 466 image finite model . . . . . . . . Res 467 Hennessy-Milner theorem g . . . . . . . Par 463 valuation g . . . . . . . SbC 465 tree model g . . . . . . Not 459 disjoint union of models g . . . . . . 455 homomorphism (2) + g . . . . . . . SbC 456 bounded homomorphism g . . . . . . . SbC 468 bounded morphism . . . . . . Not 469 expressive power g . . . . . . . Not 470 standard translation + g . . . . . . Not 460 bisimulation g . . . . . SbC 214 epistemic logic g . . . . . Not 462 model (4) + . . . . . . SbC 464 finite model g . . . . . . SbC 466 image finite model . . . . . . . Res 467 Hennessy-Milner theorem g . . . . . . Par 463 valuation g . . . . . . SbC 465 tree model g . . . . 97 context (2) . . . . . 99 context dependence . . . . . 98 context change . . . . 160 relation system . . . . 153 frame problem g . . . . 92 concept analysis . . . . . 90 concept + . . . . . . 91 individual concept . . . . 163 script . . . . 145 idea g . . . . . 90 concept + . . . . . . 91 individual concept . . . . 164 semantic network g . . . . 247 nonmonotonic logic + g . . . . . 248 default inference . . . 191 logic (1) (31) + g . . . . Par 53 automated reasoning (25) + . . . . . 35 belief revision . . . . . . 76 update . . . . . 67 nonmonotonic reasoning . . . . . 63 mathematical induction . . . . . 71 rewrite system (3) . . . . . . 350 termination . . . . . . 348 confluence . . . . . . 349 critical pair . . . . . 70 resolution (7) + . . . . . . 339 purity principle . . . . . . 342 simplification . . . . . . 337 demodulation . . . . . . 338 ordering . . . . . . 340 removal of tautologies . . . . . . 341 resolution refinement (4) . . . . . . . 345 lock resolution . . . . . . . 344 hyper resolution . . . . . . . 347 theory resolution . . . . . . . 346 set of support . . . . . . 343 subsumption . . . . . 68 paramodulation . . . . . Not 72 skolemisation . . . . . 65 model checking . . . . . 55 clause 55 (2) . . . . . . 80 horn clause g . . . . . . 79 Gentzen clause . . . . . 74 uncertainty . . . . . 75 unification + . . . . . 57 connection graph procedure . . . . . 64 metatheory . . . . . 61 literal . . . . . 58 connection matrix . . . . . 81 clause 81 . . . . . . SbC 82 relative clause . . . . . 69 reason extraction . . . . . 59 deduction (7) + . . . . . . Not 109 inconsistency . . . . . . 106 consequence g . . . . . . SbC 494 labelled deductive system . . . . . . 111 rule-based deduction . . . . . . Not 108 entailment + . . . . . . 110 natural deduction (2) + g . . . . . . . Not 482 hypothetical reasoning + . . . . . . . Not 483 normalization . . . . . . Not 107 consistency + . . . . . Res 60 Herbrand's theorem . . . . . 56 completion . . . . . . 86 Knuth Bendix completion . . . . . 73 theorem prover (3) . . . . . . 427 Bliksem g . . . . . . 428 Boyer-Moore theorem prover . . . . . . 429 SPASS g . . . . . 66 narrowing . . . . . 62 logic programming g . . . . . 54 answer extraction . . . . . 247 nonmonotonic logic + g . . . . . . 248 default inference . . . . Par 198 proof theory (22) g . . . . . SbC 503 sequent calculus . . . . . . Not 484 structural rules . . . . . 289 interpretation . . . . . 282 constructive analysis . . . . . 295 recursive ordinal . . . . . 287 Goedel numbering . . . . . 288 higher-order arithmetic . . . . . 281 complexity of proofs . . . . . 294 recursive analysis . . . . . Res 292 normal form theorem . . . . . 297 second-order arithmetic . . . . . SbC 110 natural deduction (2) + g . . . . . . Not 482 hypothetical reasoning + . . . . . . Not 483 normalization . . . . . 290 intuitionistic mathematics . . . . . 286 functionals in proof theory . . . . . 298 structure of proofs g . . . . . 283 constructive system . . . . . 291 metamathematics . . . . . 59 deduction (7) + . . . . . . Not 109 inconsistency . . . . . . 106 consequence g . . . . . . SbC 494 labelled deductive system . . . . . . 111 rule-based deduction . . . . . . Not 108 entailment + . . . . . . 110 natural deduction (2) + g . . . . . . . Not 482 hypothetical reasoning + . . . . . . . Not 483 normalization . . . . . . Not 107 consistency + . . . . . 296 relative consistency . . . . . Not 284 cut elimination theorem g . . . . . 293 ordinal notation . . . . . 285 first-order arithmetic . . . . . SbC 485 proof nets . . . . SbC 475 first order logic (4) g . . . . . 254 quantifier (5) + g . . . . . . Not 516 bound variable + g . . . . . . His 514 Frege on quantification + g . . . . . . Not 517 free variable + g . . . . . . His 513 Aristotle on quantification + . . . . . . Not 301 scope . . . . . . . 351 scoping algorithm . . . . . Par 476 first order language g . . . . . . Not 477 fragment (3) g . . . . . . . SbC 479 finite-variable fragment g . . . . . . . SbC 480 guarded fragment g . . . . . . . SbC 478 modal fragment g . . . . . . . . Not 470 standard translation + g . . . . . 511 SPASS g . . . . . Par 515 quantification (4) + . . . . . . Not 516 bound variable + g . . . . . . His 514 Frege on quantification + g . . . . . . Not 517 free variable + g . . . . . . His 513 Aristotle on quantification + . . . . 193 computability theory . . . . SbC 167 temporal logic (2) + g . . . . 435 type theory (2) + . . . . . 433 type . . . . . . 434 type shifting . . . . . Not 23 polymorphism + g . . . . 495 substructural logic . . . . SbC 200 relevance logic + . . . . . 108 entailment + . . . . Res 180 Lindstroem's theorem + . . . . SbC 481 linear logic . . . . 526 variable g . . . . . SbC 517 free variable + g . . . . Res 179 Goedel's 1st incompleteness theorem (1931) + g . . . . SbC 125 feature logic + . . . . . 75 unification + . . . . 197 model theory (29) . . . . . 237 set-theoretic model theory . . . . . 11 universal algebra + . . . . . 225 infinitary logic . . . . . 217 admissible set . . . . . 234 recursion-theoretic model theory . . . . . 239 ultraproduct . . . . . 227 logic with extra quantifiers . . . . . SbC 457 modal model theory (7) + . . . . . . SbC 215 Kripke semantics + g . . . . . . . Not 489 accessability relation + . . . . . . Not 461 generated submodel g . . . . . . 462 model (4) + . . . . . . . SbC 464 finite model g . . . . . . . SbC 466 image finite model . . . . . . . . Res 467 Hennessy-Milner theorem g . . . . . . . Par 463 valuation g . . . . . . . SbC 465 tree model g . . . . . . Not 459 disjoint union of models g . . . . . . 455 homomorphism (2) + g . . . . . . . SbC 456 bounded homomorphism g . . . . . . . SbC 468 bounded morphism . . . . . . Not 469 expressive power g . . . . . . . Not 470 standard translation + g . . . . . . Not 460 bisimulation g . . . . . 219 completeness of theories . . . . . 235 saturation . . . . . 222 equational class . . . . . 238 stability . . . . . 233 quantifier elimination . . . . . 221 denumerable structure . . . . . 228 model-theoretic algebra . . . . . 236 second-order model theory . . . . . 230 model of arithmetic . . . . . 218 categoricity g . . . . . 220 definability . . . . . 226 interpolation . . . . . SbC 454 first order model theory . . . . . . 455 homomorphism (2) + g . . . . . . . SbC 456 bounded homomorphism g . . . . . . . SbC 468 bounded morphism . . . . . 231 nonclassical model (2) . . . . . . 246 sheaf model . . . . . . 245 boolean valued . . . . . 201 set theory (24) + g . . . . . . 398 set-theoretic definability . . . . . . Not 391 iota operator . . . . . . 384 determinacy . . . . . . 387 fuzzy relation . . . . . . Not 385 filter . . . . . . 389 generalized continuum hypothesis . . . . . . 386 function (3) g . . . . . . . 482 hypothetical reasoning + . . . . . . . 509 functional application . . . . . . . 508 functional composition . . . . . . Not 394 ordinal definability . . . . . . Not 107 consistency + . . . . . . 397 set algebra . . . . . . 399 Suslin scheme . . . . . . SbC 383 descriptive set theory g . . . . . . 388 fuzzy set g . . . . . . 378 borel classification g . . . . . . SbC 380 combinatorial set theory . . . . . . Not 390 independence . . . . . . 381 constructibility . . . . . . 396 relation g . . . . . . 377 axiom of choice g . . . . . . 392 large cardinal . . . . . . Not 395 ordinal number . . . . . . 393 Martin's axiom . . . . . . 382 continuum hypothesis g . . . . . . Not 379 cardinal number . . . . . 232 preservation . . . . . 216 abstract model theory + . . . . . . 254 quantifier (5) + g . . . . . . . Not 516 bound variable + g . . . . . . . His 514 Frege on quantification + g . . . . . . . Not 517 free variable + g . . . . . . . His 513 Aristotle on quantification + . . . . . . . Not 301 scope . . . . . . . . 351 scoping algorithm . . . . . 229 model-theoretic forcing . . . . . 224 higher-order model theory . . . . . Par 493 correspondence theory . . . . . 223 finite structure . . . . Res 182 Loewenheim-Skolem-Tarski theorem + . . . . Not 83 completeness (2) + g . . . . . SbC 84 axiomatic completeness . . . . . SbC 85 functional completeness + . . . . SbC 156 modal logic (13) + g . . . . . Ins 512 S4 . . . . . 488 modes . . . . . 486 frame (2) . . . . . . SbC 487 frame constraints . . . . . Par 210 modal operator (2) + g . . . . . . SbC 472 diamond g . . . . . . SbC 473 box g . . . . . SbC 213 doxastic logic g . . . . . Not 489 accessability relation + . . . . . Par 471 modal language (2) g . . . . . . Par 210 modal operator (2) + g . . . . . . . SbC 472 diamond g . . . . . . . SbC 473 box g . . . . . . 490 boolean operators . . . . . SbC 211 alethic logic g . . . . . SbC 212 deontic logic (3) g . . . . . . SbC 521 standard deontic logic g . . . . . . SbC 523 two-sorted deontic logic g . . . . . . SbC 522 dyadic deontic logic g . . . . . Par 215 Kripke semantics + g . . . . . . Not 489 accessability relation + . . . . . Par 457 modal model theory (7) + . . . . . . SbC 215 Kripke semantics + g . . . . . . . Not 489 accessability relation + . . . . . . Not 461 generated submodel g . . . . . . 462 model (4) + . . . . . . . SbC 464 finite model g . . . . . . . SbC 466 image finite model . . . . . . . . Res 467 Hennessy-Milner theorem g . . . . . . . Par 463 valuation g . . . . . . . SbC 465 tree model g . . . . . . Not 459 disjoint union of models g . . . . . . 455 homomorphism (2) + g . . . . . . . SbC 456 bounded homomorphism g . . . . . . . SbC 468 bounded morphism . . . . . . Not 469 expressive power g . . . . . . . Not 470 standard translation + g . . . . . . Not 460 bisimulation g . . . . . SbC 214 epistemic logic g . . . . . Not 462 model (4) + . . . . . . SbC 464 finite model g . . . . . . SbC 466 image finite model . . . . . . . Res 467 Hennessy-Milner theorem g . . . . . . Par 463 valuation g . . . . . . SbC 465 tree model g . . . . 194 computational logic (2) . . . . Not 183 operator (4) + g . . . . . 254 quantifier (5) + g . . . . . . Not 516 bound variable + g . . . . . . His 514 Frege on quantification + g . . . . . . Not 517 free variable + g . . . . . . His 513 Aristotle on quantification + . . . . . . Not 301 scope . . . . . . . 351 scoping algorithm . . . . . SbC 210 modal operator (2) + g . . . . . . SbC 472 diamond g . . . . . . SbC 473 box g . . . . . 518 truth-funcional operator (2) g . . . . . . SbC 252 iff g . . . . . . SbC 253 negation . . . . . Not 525 arity g . . . . SbC 192 combinatory logic g . . . . Par 199 recursive function theory . . . . 361 formal semantics (10) + g . . . . . 365 property theory . . . . . 240 Montague grammar (4) . . . . . . 243 sense 243 (4) g . . . . . . . 203 meaning relation (5) . . . . . . . . 205 hyponymy g . . . . . . . . 204 antonymy g . . . . . . . . 207 synonymy g . . . . . . . . . 149 intensional isomorphism + . . . . . . . . 206 paraphrase g . . . . . . . . 108 entailment + . . . . . . . 375 metaphor g . . . . . . . 376 metonymy g . . . . . . . 374 literal meaning . . . . . . 244 sense 244 g . . . . . . 241 meaning postulate . . . . . . 242 ptq g . . . . . . . 300 quantifying in . . . . . 254 quantifier (5) + g . . . . . . Not 516 bound variable + g . . . . . . His 514 Frege on quantification + g . . . . . . Not 517 free variable + g . . . . . . His 513 Aristotle on quantification + . . . . . . Not 301 scope . . . . . . . 351 scoping algorithm . . . . . 353 truth (4) + . . . . . . 431 truth definition g . . . . . . 432 truth value . . . . . . 372 truth function + g . . . . . . 430 truth condition . . . . . 362 dynamic semantics . . . . . 363 lexical semantics . . . . . 366 situation semantics (2) g . . . . . . 402 partiality . . . . . . 400 situation . . . . . . . 401 scene . . . . . Not 507 compositionality . . . . . 364 natural logic + . . . . . Par 515 quantification (4) + . . . . . . Not 516 bound variable + g . . . . . . His 514 Frege on quantification + g . . . . . . Not 517 free variable + g . . . . . . His 513 Aristotle on quantification + . . . . SbC 168 lambda calculus (4) g . . . . . 170 application . . . . . 172 lambda operator . . . . . 169 abstraction . . . . . 171 conversion . . . . 38 knowledge representation (20) + g . . . . . 152 frame (1) . . . . . 104 database + g . . . . . . 105 query g . . . . . 165 situation calculus . . . . . 167 temporal logic (2) + g . . . . . 166 temporal logic (1) g . . . . . 93 concept formation . . . . . . 90 concept + . . . . . . . 91 individual concept . . . . . 154 logical omniscience . . . . . 162 rule-based representation . . . . . 157 predicate logic + g . . . . . 159 procedural representation . . . . . 161 representation language . . . . . 156 modal logic (13) + g . . . . . . Ins 512 S4 . . . . . . 488 modes . . . . . . 486 frame (2) . . . . . . . SbC 487 frame constraints . . . . . . Par 210 modal operator (2) + g . . . . . . . SbC 472 diamond g . . . . . . . SbC 473 box g . . . . . . SbC 213 doxastic logic g . . . . . . Not 489 accessability relation + . . . . . . Par 471 modal language (2) g . . . . . . . Par 210 modal operator (2) + g . . . . . . . . SbC 472 diamond g . . . . . . . . SbC 473 box g . . . . . . . 490 boolean operators . . . . . . SbC 211 alethic logic g . . . . . . SbC 212 deontic logic (3) g . . . . . . . SbC 521 standard deontic logic g . . . . . . . SbC 523 two-sorted deontic logic g . . . . . . . SbC 522 dyadic deontic logic g . . . . . . Par 215 Kripke semantics + g . . . . . . . Not 489 accessability relation + . . . . . . Par 457 modal model theory (7) + . . . . . . . SbC 215 Kripke semantics + g . . . . . . . . Not 489 accessability relation + . . . . . . . Not 461 generated submodel g . . . . . . . 462 model (4) + . . . . . . . . SbC 464 finite model g . . . . . . . . SbC 466 image finite model . . . . . . . . . Res 467 Hennessy-Milner theorem g . . . . . . . . Par 463 valuation g . . . . . . . . SbC 465 tree model g . . . . . . . Not 459 disjoint union of models g . . . . . . . 455 homomorphism (2) + g . . . . . . . . SbC 456 bounded homomorphism g . . . . . . . . SbC 468 bounded morphism . . . . . . . Not 469 expressive power g . . . . . . . . Not 470 standard translation + g . . . . . . . Not 460 bisimulation g . . . . . . SbC 214 epistemic logic g . . . . . . Not 462 model (4) + . . . . . . . SbC 464 finite model g . . . . . . . SbC 466 image finite model . . . . . . . . Res 467 Hennessy-Milner theorem g . . . . . . . Par 463 valuation g . . . . . . . SbC 465 tree model g . . . . . 97 context (2) . . . . . . 99 context dependence . . . . . . 98 context change . . . . . 160 relation system . . . . . 153 frame problem g . . . . . 92 concept analysis . . . . . . 90 concept + . . . . . . . 91 individual concept . . . . . 163 script . . . . . 145 idea g . . . . . . 90 concept + . . . . . . . 91 individual concept . . . . . 164 semantic network g . . . . . 247 nonmonotonic logic + g . . . . . . 248 default inference . . . . Par 367 semantics 367 (8) g . . . . . 371 truth conditional semantics . . . . . 373 truth table . . . . . SbC 215 Kripke semantics + g . . . . . . Not 489 accessability relation + . . . . . 85 functional completeness + . . . . . 370 satisfaction . . . . . 369 material implication g . . . . . 368 assignment . . . . . Not 372 truth function + g . . . . Par 201 set theory (24) + g . . . . . 398 set-theoretic definability . . . . . Not 391 iota operator . . . . . 384 determinacy . . . . . 387 fuzzy relation . . . . . Not 385 filter . . . . . 389 generalized continuum hypothesis . . . . . 386 function (3) g . . . . . . 482 hypothetical reasoning + . . . . . . 509 functional application . . . . . . 508 functional composition . . . . . Not 394 ordinal definability . . . . . Not 107 consistency + . . . . . 397 set algebra . . . . . 399 Suslin scheme . . . . . SbC 383 descriptive set theory g . . . . . 388 fuzzy set g . . . . . 378 borel classification g . . . . . SbC 380 combinatorial set theory . . . . . Not 390 independence . . . . . 381 constructibility . . . . . 396 relation g . . . . . 377 axiom of choice g . . . . . 392 large cardinal . . . . . Not 395 ordinal number . . . . . 393 Martin's axiom . . . . . 382 continuum hypothesis g . . . . . Not 379 cardinal number . . . . Par 216 abstract model theory + . . . . . 254 quantifier (5) + g . . . . . . Not 516 bound variable + g . . . . . . His 514 Frege on quantification + g . . . . . . Not 517 free variable + g . . . . . . His 513 Aristotle on quantification + . . . . . . Not 301 scope . . . . . . . 351 scoping algorithm . . . . 178 compactness + . . . . His 177 aristotelean logic (2) + g . . . . . Par 39 syllogism g . . . . . Par 513 Aristotle on quantification + . . . . Par 196 foundations of theories . . . . 195 constraint programming . . . 40 planning . . . Not 36 classification . . . Not 37 heuristic g . . Par 89 theory of computation (4) g . . . Par 127 formal language theory (10) g . . . . 128 categorial grammar + . . . . . SbC 528 combinatorial categorial grammar . . . . 131 context free language g . . . . 130 Chomsky hierarchy g . . . . 134 phrase structure grammar . . . . 129 category . . . . 135 recursive language + g . . . . 137 unrestricted language g . . . . 136 regular language . . . . 132 context sensitive language g . . . . 133 feature constraint . . . Par 302 recursion theory (31) g . . . . 306 complexity of computation . . . . 330 undecidability . . . . 328 theory of numerations . . . . 309 effectively presented structure . . . . 314 isol . . . . 307 decidability (2) g . . . . . 474 tree model property g . . . . . 504 subformula property . . . . 322 recursively enumerable degree . . . . 331 word problem . . . . 327 subrecursive hierarchy . . . . 315 post system . . . . 324 recursively enumerable set . . . . 320 recursive function . . . . 318 recursive axiomatizability . . . . 329 thue system . . . . 325 reducibility . . . . 304 automaton . . . . 310 formal grammar . . . . 326 set recursion theory . . . . 303 abstract recursion theory . . . . 323 recursively enumerable language . . . . 305 axiomatic recursion theory . . . . 135 recursive language + g . . . . 313 inductive definability . . . . 316 recursion theory on admissible sets . . . . Not 52 Turing machine + . . . . 308 degrees of sets of sentences . . . . 319 recursive equivalence type . . . . 312 higher type recursion theory . . . . 317 recursion theory on ordinals . . . . 321 recursive relation . . . . 311 hierarchy . . . Par 185 computational logic (1) (8) g . . . . 190 semantics 190 (8) + g . . . . . 356 denotational semantics . . . . . 119 domain theory g . . . . . . 120 domain . . . . . 360 program analysis . . . . . 359 process model . . . . . SbC 215 Kripke semantics + g . . . . . . Not 489 accessability relation + . . . . . 357 operational semantics . . . . . 358 partial evaluation . . . . . 355 algebraic semantics . . . . 189 reasoning about programs . . . . 53 automated reasoning (25) + . . . . . 35 belief revision . . . . . . 76 update . . . . . 67 nonmonotonic reasoning . . . . . 63 mathematical induction . . . . . 71 rewrite system (3) . . . . . . 350 termination . . . . . . 348 confluence . . . . . . 349 critical pair . . . . . 70 resolution (7) + . . . . . . 339 purity principle . . . . . . 342 simplification . . . . . . 337 demodulation . . . . . . 338 ordering . . . . . . 340 removal of tautologies . . . . . . 341 resolution refinement (4) . . . . . . . 345 lock resolution . . . . . . . 344 hyper resolution . . . . . . . 347 theory resolution . . . . . . . 346 set of support . . . . . . 343 subsumption . . . . . 68 paramodulation . . . . . Not 72 skolemisation . . . . . 65 model checking . . . . . 55 clause 55 (2) . . . . . . 80 horn clause g . . . . . . 79 Gentzen clause . . . . . 74 uncertainty . . . . . 75 unification + . . . . . 57 connection graph procedure . . . . . 64 metatheory . . . . . 61 literal . . . . . 58 connection matrix . . . . . 81 clause 81 . . . . . . SbC 82 relative clause . . . . . 69 reason extraction . . . . . 59 deduction (7) + . . . . . . Not 109 inconsistency . . . . . . 106 consequence g . . . . . . SbC 494 labelled deductive system . . . . . . 111 rule-based deduction . . . . . . Not 108 entailment + . . . . . . 110 natural deduction (2) + g . . . . . . . Not 482 hypothetical reasoning + . . . . . . . Not 483 normalization . . . . . . Not 107 consistency + . . . . . Res 60 Herbrand's theorem . . . . . 56 completion . . . . . . 86 Knuth Bendix completion . . . . . 73 theorem prover (3) . . . . . . 427 Bliksem g . . . . . . 428 Boyer-Moore theorem prover . . . . . . 429 SPASS g . . . . . 66 narrowing . . . . . 62 logic programming g . . . . . 54 answer extraction . . . . . 247 nonmonotonic logic + g . . . . . . 248 default inference . . . . Not 83 completeness (2) + g . . . . . SbC 84 axiomatic completeness . . . . . SbC 85 functional completeness + . . . . 188 program verification (4) . . . . . 274 mechanical verification . . . . . 269 invariant + . . . . . 273 logic of programs . . . . . 43 assertion (2) + . . . . . . 45 imperative assertion . . . . . . 44 declarative assertion . . . . 435 type theory (2) + . . . . . 433 type . . . . . . 434 type shifting . . . . . Not 23 polymorphism + g . . . . 186 program construct (5) . . . . . 265 functional construct . . . . . 267 program scheme . . . . . 266 object oriented construct . . . . . 264 control primitive . . . . . 268 type structure . . . . 187 program specification (5) . . . . . 271 pre-condition . . . . . 269 invariant + . . . . . 272 specification technique . . . . . 43 assertion (2) + . . . . . . 45 imperative assertion . . . . . . 44 declarative assertion . . . . . 270 post-condition . . . Par 48 automata theory (4) . . . . Not 52 Turing machine + . . . . 50 linear bounded automaton . . . . 49 finite state machine g . . . . 51 push down automaton . 173 linguistics (13) g . . Par 446 descriptive linguistics g . . . 142 grammar (5) g . . . . Not 519 derivation g . . . . 452 grammatical constituent g . . . . . 121 ellipsis g . . . . . . 122 antecedent of ellipsis . . . . 444 linguistic unit (3) g . . . . . SbC 440 word (5) g . . . . . . 28 anaphor (2) g . . . . . . . 30 antecedent of an anaphor . . . . . . . 29 anaphora resolution . . . . . . 278 pronoun (2) g . . . . . . . 280 pronoun resolution . . . . . . . 279 demonstrative g . . . . . . 138 function word (2) g . . . . . . . SbC 139 determiner g . . . . . . . SbC 441 modifier g . . . . . . . . 445 adjective (4) g . . . . . . . . . 4 predicative position . . . . . . . . . 1 adverbial modification g . . . . . . . . . 3 intersective adjective . . . . . . . . . 2 graded adjective . . . . . . 442 content word g . . . . . . 425 term (2) g . . . . . . . 426 singular term g . . . . . . . 260 plural term (2) g . . . . . . . . 261 collective reading . . . . . . . . 262 distributive reading . . . . . SbC 500 quantified phrases + . . . . . SbC 115 discourse (3) g . . . . . . 116 discourse particle . . . . . . 118 discourse representation theory g . . . . . . 117 discourse referent . . . . 144 syntax 144 (2) g . . . . . 453 logical syntax g . . . . . . 12 algebraic logic (10) + . . . . . . . 6 boolean algebra + . . . . . . . . SbC 7 boolean algebra with operators . . . . . . . 17 post algebra . . . . . . . 15 Lukasiewicz algebra . . . . . . . 14 cylindric algebra g . . . . . . . 8 lattice + g . . . . . . . 18 quantum logic . . . . . . . 10 relation algebra + . . . . . . . 13 categorical logic . . . . . . . 16 polyadic algebra . . . . . . . 19 topos . . . . . 423 syntactic category (3) g . . . . . . 447 part of speech g . . . . . . SbC 249 noun (2) g . . . . . . . SbC 251 proper name . . . . . . . SbC 250 mass noun g . . . . . . SbC 438 verb g . . . . . . . SbC 439 perception verb . . . . 143 sentence g . . 443 linguistic geography g . . Not 502 discontinuity . . Par 361 formal semantics (10) + g . . . 365 property theory . . . 240 Montague grammar (4) . . . . 243 sense 243 (4) g . . . . . 203 meaning relation (5) . . . . . . 205 hyponymy g . . . . . . 204 antonymy g . . . . . . 207 synonymy g . . . . . . . 149 intensional isomorphism + . . . . . . 206 paraphrase g . . . . . . 108 entailment + . . . . . 375 metaphor g . . . . . 376 metonymy g . . . . . 374 literal meaning . . . . 244 sense 244 g . . . . 241 meaning postulate . . . . 242 ptq g . . . . . 300 quantifying in . . . 254 quantifier (5) + g . . . . Not 516 bound variable + g . . . . His 514 Frege on quantification + g . . . . Not 517 free variable + g . . . . His 513 Aristotle on quantification + . . . . Not 301 scope . . . . . 351 scoping algorithm . . . 353 truth (4) + . . . . 431 truth definition g . . . . 432 truth value . . . . 372 truth function + g . . . . 430 truth condition . . . 362 dynamic semantics . . . 363 lexical semantics . . . 366 situation semantics (2) g . . . . 402 partiality . . . . 400 situation . . . . . 401 scene . . . Not 507 compositionality . . . 364 natural logic + . . . Par 515 quantification (4) + . . . . Not 516 bound variable + g . . . . His 514 Frege on quantification + g . . . . Not 517 free variable + g . . . . His 513 Aristotle on quantification + . . Not 20 ambiguity (7) g . . . SbC 27 syntactic ambiguity . . . SbC 25 semantic ambiguity + g . . . SbC 22 lexical ambiguity g . . . SbC 21 derivational ambiguity . . . SbC 24 pragmatic ambiguity . . . SbC 26 structural ambiguity . . . 23 polymorphism + g . . 510 frameworks (7) . . . 535 LFG . . . 128 categorial grammar + . . . . SbC 528 combinatorial categorial grammar . . . 530 TAG . . . 532 DRT . . . 529 GB . . . 534 HPSG . . . 531 dynamic syntax . . 506 linguistic phenomena . . Not 174 language acquisition g . . Par 450 pragmatics (2) g . . . 403 speech act (5) g . . . . 408 statement (2) g . . . . . 112 description (2) g . . . . . . SbC 114 indefinite description . . . . . . SbC 113 definite description . . . . . 409 indicative statement . . . . 405 indirect speech act . . . . 406 performative . . . . 407 performative hypothesis . . . . 404 illocutionary force . . . 100 conversational maxim (3) g . . . . 103 implicature + g . . . . 102 cooperative principle . . . . 101 conversational implicature g . . 499 syntax and semantic interface + . . Par 175 semantics 175 (16) g . . . 25 semantic ambiguity + g . . . Not 123 extension g . . . . 124 extensionality g . . . 334 referent g . . . Not 332 reference (2) g . . . . 333 identity puzzle . . . . 335 referential term . . . . . SbC 336 anchor . . . Not 263 presupposition g . . . . 103 implicature + g . . . Not 146 indexicality . . . . 147 indexical expression g . . . Par 41 aspect . . . . 42 aspectual classification . . . SbC 361 formal semantics (10) + g . . . . 365 property theory . . . . 240 Montague grammar (4) . . . . . 243 sense 243 (4) g . . . . . . 203 meaning relation (5) . . . . . . . 205 hyponymy g . . . . . . . 204 antonymy g . . . . . . . 207 synonymy g . . . . . . . . 149 intensional isomorphism + . . . . . . . 206 paraphrase g . . . . . . . 108 entailment + . . . . . . 375 metaphor g . . . . . . 376 metonymy g . . . . . . 374 literal meaning . . . . . 244 sense 244 g . . . . . 241 meaning postulate . . . . . 242 ptq g . . . . . . 300 quantifying in . . . . 254 quantifier (5) + g . . . . . Not 516 bound variable + g . . . . . His 514 Frege on quantification + g . . . . . Not 517 free variable + g . . . . . His 513 Aristotle on quantification + . . . . . Not 301 scope . . . . . . 351 scoping algorithm . . . . 353 truth (4) + . . . . . 431 truth definition g . . . . . 432 truth value . . . . . 372 truth function + g . . . . . 430 truth condition . . . . 362 dynamic semantics . . . . 363 lexical semantics . . . . 366 situation semantics (2) g . . . . . 402 partiality . . . . . 400 situation . . . . . . 401 scene . . . . Not 507 compositionality . . . . 364 natural logic + . . . . Par 515 quantification (4) + . . . . . Not 516 bound variable + g . . . . . His 514 Frege on quantification + g . . . . . Not 517 free variable + g . . . . . His 513 Aristotle on quantification + . . . Not 501 coordination . . . Not 353 truth (4) + . . . . 431 truth definition g . . . . 432 truth value . . . . 372 truth function + g . . . . 430 truth condition . . . Not 354 underspecification (2) . . . . 437 quasi-logical form . . . . 436 monotonic semantics . . . 499 syntax and semantic interface + . . . Par 46 attitude . . . . SbC 47 propositional attitude . . . . . Not 299 belief . . . Not 500 quantified phrases + . . . Not 148 intension (3) g . . . . 149 intensional isomorphism + . . . . 151 intensionality . . . . 150 intensional verb . . . 31 animal (3) g . . . . SbC 33 unicorn . . . . SbC 32 donkey . . . . SbC 352 rabbit . . Par 496 syntax 496 (2) g . . . Par 498 word order . . . Par 497 movement . . Par 140 language generation . . . 141 reversibility . 202 mathematics (5) g . . Not 527 algebra 2 g . . 191 logic (1) (31) + g . . . Par 53 automated reasoning (25) + . . . . 35 belief revision . . . . . 76 update . . . . 67 nonmonotonic reasoning . . . . 63 mathematical induction . . . . 71 rewrite system (3) . . . . . 350 termination . . . . . 348 confluence . . . . . 349 critical pair . . . . 70 resolution (7) + . . . . . 339 purity principle . . . . . 342 simplification . . . . . 337 demodulation . . . . . 338 ordering . . . . . 340 removal of tautologies . . . . . 341 resolution refinement (4) . . . . . . 345 lock resolution . . . . . . 344 hyper resolution . . . . . . 347 theory resolution . . . . . . 346 set of support . . . . . 343 subsumption . . . . 68 paramodulation . . . . Not 72 skolemisation . . . . 65 model checking . . . . 55 clause 55 (2) . . . . . 80 horn clause g . . . . . 79 Gentzen clause . . . . 74 uncertainty . . . . 75 unification + . . . . 57 connection graph procedure . . . . 64 metatheory . . . . 61 literal . . . . 58 connection matrix . . . . 81 clause 81 . . . . . SbC 82 relative clause . . . . 69 reason extraction . . . . 59 deduction (7) + . . . . . Not 109 inconsistency . . . . . 106 consequence g . . . . . SbC 494 labelled deductive system . . . . . 111 rule-based deduction . . . . . Not 108 entailment + . . . . . 110 natural deduction (2) + g . . . . . . Not 482 hypothetical reasoning + . . . . . . Not 483 normalization . . . . . Not 107 consistency + . . . . Res 60 Herbrand's theorem . . . . 56 completion . . . . . 86 Knuth Bendix completion . . . . 73 theorem prover (3) . . . . . 427 Bliksem g . . . . . 428 Boyer-Moore theorem prover . . . . . 429 SPASS g . . . . 66 narrowing . . . . 62 logic programming g . . . . 54 answer extraction . . . . 247 nonmonotonic logic + g . . . . . 248 default inference . . . Par 198 proof theory (22) g . . . . SbC 503 sequent calculus . . . . . Not 484 structural rules . . . . 289 interpretation . . . . 282 constructive analysis . . . . 295 recursive ordinal . . . . 287 Goedel numbering . . . . 288 higher-order arithmetic . . . . 281 complexity of proofs . . . . 294 recursive analysis . . . . Res 292 normal form theorem . . . . 297 second-order arithmetic . . . . SbC 110 natural deduction (2) + g . . . . . Not 482 hypothetical reasoning + . . . . . Not 483 normalization . . . . 290 intuitionistic mathematics . . . . 286 functionals in proof theory . . . . 298 structure of proofs g . . . . 283 constructive system . . . . 291 metamathematics . . . . 59 deduction (7) + . . . . . Not 109 inconsistency . . . . . 106 consequence g . . . . . SbC 494 labelled deductive system . . . . . 111 rule-based deduction . . . . . Not 108 entailment + . . . . . 110 natural deduction (2) + g . . . . . . Not 482 hypothetical reasoning + . . . . . . Not 483 normalization . . . . . Not 107 consistency + . . . . 296 relative consistency . . . . Not 284 cut elimination theorem g . . . . 293 ordinal notation . . . . 285 first-order arithmetic . . . . SbC 485 proof nets . . . SbC 475 first order logic (4) g . . . . 254 quantifier (5) + g . . . . . Not 516 bound variable + g . . . . . His 514 Frege on quantification + g . . . . . Not 517 free variable + g . . . . . His 513 Aristotle on quantification + . . . . . Not 301 scope . . . . . . 351 scoping algorithm . . . . Par 476 first order language g . . . . . Not 477 fragment (3) g . . . . . . SbC 479 finite-variable fragment g . . . . . . SbC 480 guarded fragment g . . . . . . SbC 478 modal fragment g . . . . . . . Not 470 standard translation + g . . . . 511 SPASS g . . . . Par 515 quantification (4) + . . . . . Not 516 bound variable + g . . . . . His 514 Frege on quantification + g . . . . . Not 517 free variable + g . . . . . His 513 Aristotle on quantification + . . . 193 computability theory . . . SbC 167 temporal logic (2) + g . . . 435 type theory (2) + . . . . 433 type . . . . . 434 type shifting . . . . Not 23 polymorphism + g . . . 495 substructural logic . . . SbC 200 relevance logic + . . . . 108 entailment + . . . Res 180 Lindstroem's theorem + . . . SbC 481 linear logic . . . 526 variable g . . . . SbC 517 free variable + g . . . Res 179 Goedel's 1st incompleteness theorem (1931) + g . . . SbC 125 feature logic + . . . . 75 unification + . . . 197 model theory (29) . . . . 237 set-theoretic model theory . . . . 11 universal algebra + . . . . 225 infinitary logic . . . . 217 admissible set . . . . 234 recursion-theoretic model theory . . . . 239 ultraproduct . . . . 227 logic with extra quantifiers . . . . SbC 457 modal model theory (7) + . . . . . SbC 215 Kripke semantics + g . . . . . . Not 489 accessability relation + . . . . . Not 461 generated submodel g . . . . . 462 model (4) + . . . . . . SbC 464 finite model g . . . . . . SbC 466 image finite model . . . . . . . Res 467 Hennessy-Milner theorem g . . . . . . Par 463 valuation g . . . . . . SbC 465 tree model g . . . . . Not 459 disjoint union of models g . . . . . 455 homomorphism (2) + g . . . . . . SbC 456 bounded homomorphism g . . . . . . SbC 468 bounded morphism . . . . . Not 469 expressive power g . . . . . . Not 470 standard translation + g . . . . . Not 460 bisimulation g . . . . 219 completeness of theories . . . . 235 saturation . . . . 222 equational class . . . . 238 stability . . . . 233 quantifier elimination . . . . 221 denumerable structure . . . . 228 model-theoretic algebra . . . . 236 second-order model theory . . . . 230 model of arithmetic . . . . 218 categoricity g . . . . 220 definability . . . . 226 interpolation . . . . SbC 454 first order model theory . . . . . 455 homomorphism (2) + g . . . . . . SbC 456 bounded homomorphism g . . . . . . SbC 468 bounded morphism . . . . 231 nonclassical model (2) . . . . . 246 sheaf model . . . . . 245 boolean valued . . . . 201 set theory (24) + g . . . . . 398 set-theoretic definability . . . . . Not 391 iota operator . . . . . 384 determinacy . . . . . 387 fuzzy relation . . . . . Not 385 filter . . . . . 389 generalized continuum hypothesis . . . . . 386 function (3) g . . . . . . 482 hypothetical reasoning + . . . . . . 509 functional application . . . . . . 508 functional composition . . . . . Not 394 ordinal definability . . . . . Not 107 consistency + . . . . . 397 set algebra . . . . . 399 Suslin scheme . . . . . SbC 383 descriptive set theory g . . . . . 388 fuzzy set g . . . . . 378 borel classification g . . . . . SbC 380 combinatorial set theory . . . . . Not 390 independence . . . . . 381 constructibility . . . . . 396 relation g . . . . . 377 axiom of choice g . . . . . 392 large cardinal . . . . . Not 395 ordinal number . . . . . 393 Martin's axiom . . . . . 382 continuum hypothesis g . . . . . Not 379 cardinal number . . . . 232 preservation . . . . 216 abstract model theory + . . . . . 254 quantifier (5) + g . . . . . . Not 516 bound variable + g . . . . . . His 514 Frege on quantification + g . . . . . . Not 517 free variable + g . . . . . . His 513 Aristotle on quantification + . . . . . . Not 301 scope . . . . . . . 351 scoping algorithm . . . . 229 model-theoretic forcing . . . . 224 higher-order model theory . . . . Par 493 correspondence theory . . . . 223 finite structure . . . Res 182 Loewenheim-Skolem-Tarski theorem + . . . Not 83 completeness (2) + g . . . . SbC 84 axiomatic completeness . . . . SbC 85 functional completeness + . . . SbC 156 modal logic (13) + g . . . . Ins 512 S4 . . . . 488 modes . . . . 486 frame (2) . . . . . SbC 487 frame constraints . . . . Par 210 modal operator (2) + g . . . . . SbC 472 diamond g . . . . . SbC 473 box g . . . . SbC 213 doxastic logic g . . . . Not 489 accessability relation + . . . . Par 471 modal language (2) g . . . . . Par 210 modal operator (2) + g . . . . . . SbC 472 diamond g . . . . . . SbC 473 box g . . . . . 490 boolean operators . . . . SbC 211 alethic logic g . . . . SbC 212 deontic logic (3) g . . . . . SbC 521 standard deontic logic g . . . . . SbC 523 two-sorted deontic logic g . . . . . SbC 522 dyadic deontic logic g . . . . Par 215 Kripke semantics + g . . . . . Not 489 accessability relation + . . . . Par 457 modal model theory (7) + . . . . . SbC 215 Kripke semantics + g . . . . . . Not 489 accessability relation + . . . . . Not 461 generated submodel g . . . . . 462 model (4) + . . . . . . SbC 464 finite model g . . . . . . SbC 466 image finite model . . . . . . . Res 467 Hennessy-Milner theorem g . . . . . . Par 463 valuation g . . . . . . SbC 465 tree model g . . . . . Not 459 disjoint union of models g . . . . . 455 homomorphism (2) + g . . . . . . SbC 456 bounded homomorphism g . . . . . . SbC 468 bounded morphism . . . . . Not 469 expressive power g . . . . . . Not 470 standard translation + g . . . . . Not 460 bisimulation g . . . . SbC 214 epistemic logic g . . . . Not 462 model (4) + . . . . . SbC 464 finite model g . . . . . SbC 466 image finite model . . . . . . Res 467 Hennessy-Milner theorem g . . . . . Par 463 valuation g . . . . . SbC 465 tree model g . . . 194 computational logic (2) . . . Not 183 operator (4) + g . . . . 254 quantifier (5) + g . . . . . Not 516 bound variable + g . . . . . His 514 Frege on quantification + g . . . . . Not 517 free variable + g . . . . . His 513 Aristotle on quantification + . . . . . Not 301 scope . . . . . . 351 scoping algorithm . . . . SbC 210 modal operator (2) + g . . . . . SbC 472 diamond g . . . . . SbC 473 box g . . . . 518 truth-funcional operator (2) g . . . . . SbC 252 iff g . . . . . SbC 253 negation . . . . Not 525 arity g . . . SbC 192 combinatory logic g . . . Par 199 recursive function theory . . . 361 formal semantics (10) + g . . . . 365 property theory . . . . 240 Montague grammar (4) . . . . . 243 sense 243 (4) g . . . . . . 203 meaning relation (5) . . . . . . . 205 hyponymy g . . . . . . . 204 antonymy g . . . . . . . 207 synonymy g . . . . . . . . 149 intensional isomorphism + . . . . . . . 206 paraphrase g . . . . . . . 108 entailment + . . . . . . 375 metaphor g . . . . . . 376 metonymy g . . . . . . 374 literal meaning . . . . . 244 sense 244 g . . . . . 241 meaning postulate . . . . . 242 ptq g . . . . . . 300 quantifying in . . . . 254 quantifier (5) + g . . . . . Not 516 bound variable + g . . . . . His 514 Frege on quantification + g . . . . . Not 517 free variable + g . . . . . His 513 Aristotle on quantification + . . . . . Not 301 scope . . . . . . 351 scoping algorithm . . . . 353 truth (4) + . . . . . 431 truth definition g . . . . . 432 truth value . . . . . 372 truth function + g . . . . . 430 truth condition . . . . 362 dynamic semantics . . . . 363 lexical semantics . . . . 366 situation semantics (2) g . . . . . 402 partiality . . . . . 400 situation . . . . . . 401 scene . . . . Not 507 compositionality . . . . 364 natural logic + . . . . Par 515 quantification (4) + . . . . . Not 516 bound variable + g . . . . . His 514 Frege on quantification + g . . . . . Not 517 free variable + g . . . . . His 513 Aristotle on quantification + . . . SbC 168 lambda calculus (4) g . . . . 170 application . . . . 172 lambda operator . . . . 169 abstraction . . . . 171 conversion . . . 38 knowledge representation (20) + g . . . . 152 frame (1) . . . . 104 database + g . . . . . 105 query g . . . . 165 situation calculus . . . . 167 temporal logic (2) + g . . . . 166 temporal logic (1) g . . . . 93 concept formation . . . . . 90 concept + . . . . . . 91 individual concept . . . . 154 logical omniscience . . . . 162 rule-based representation . . . . 157 predicate logic + g . . . . 159 procedural representation . . . . 161 representation language . . . . 156 modal logic (13) + g . . . . . Ins 512 S4 . . . . . 488 modes . . . . . 486 frame (2) . . . . . . SbC 487 frame constraints . . . . . Par 210 modal operator (2) + g . . . . . . SbC 472 diamond g . . . . . . SbC 473 box g . . . . . SbC 213 doxastic logic g . . . . . Not 489 accessability relation + . . . . . Par 471 modal language (2) g . . . . . . Par 210 modal operator (2) + g . . . . . . . SbC 472 diamond g . . . . . . . SbC 473 box g . . . . . . 490 boolean operators . . . . . SbC 211 alethic logic g . . . . . SbC 212 deontic logic (3) g . . . . . . SbC 521 standard deontic logic g . . . . . . SbC 523 two-sorted deontic logic g . . . . . . SbC 522 dyadic deontic logic g . . . . . Par 215 Kripke semantics + g . . . . . . Not 489 accessability relation + . . . . . Par 457 modal model theory (7) + . . . . . . SbC 215 Kripke semantics + g . . . . . . . Not 489 accessability relation + . . . . . . Not 461 generated submodel g . . . . . . 462 model (4) + . . . . . . . SbC 464 finite model g . . . . . . . SbC 466 image finite model . . . . . . . . Res 467 Hennessy-Milner theorem g . . . . . . . Par 463 valuation g . . . . . . . SbC 465 tree model g . . . . . . Not 459 disjoint union of models g . . . . . . 455 homomorphism (2) + g . . . . . . . SbC 456 bounded homomorphism g . . . . . . . SbC 468 bounded morphism . . . . . . Not 469 expressive power g . . . . . . . Not 470 standard translation + g . . . . . . Not 460 bisimulation g . . . . . SbC 214 epistemic logic g . . . . . Not 462 model (4) + . . . . . . SbC 464 finite model g . . . . . . SbC 466 image finite model . . . . . . . Res 467 Hennessy-Milner theorem g . . . . . . Par 463 valuation g . . . . . . SbC 465 tree model g . . . . 97 context (2) . . . . . 99 context dependence . . . . . 98 context change . . . . 160 relation system . . . . 153 frame problem g . . . . 92 concept analysis . . . . . 90 concept + . . . . . . 91 individual concept . . . . 163 script . . . . 145 idea g . . . . . 90 concept + . . . . . . 91 individual concept . . . . 164 semantic network g . . . . 247 nonmonotonic logic + g . . . . . 248 default inference . . . Par 367 semantics 367 (8) g . . . . 371 truth conditional semantics . . . . 373 truth table . . . . SbC 215 Kripke semantics + g . . . . . Not 489 accessability relation + . . . . 85 functional completeness + . . . . 370 satisfaction . . . . 369 material implication g . . . . 368 assignment . . . . Not 372 truth function + g . . . Par 201 set theory (24) + g . . . . 398 set-theoretic definability . . . . Not 391 iota operator . . . . 384 determinacy . . . . 387 fuzzy relation . . . . Not 385 filter . . . . 389 generalized continuum hypothesis . . . . 386 function (3) g . . . . . 482 hypothetical reasoning + . . . . . 509 functional application . . . . . 508 functional composition . . . . Not 394 ordinal definability . . . . Not 107 consistency + . . . . 397 set algebra . . . . 399 Suslin scheme . . . . SbC 383 descriptive set theory g . . . . 388 fuzzy set g . . . . 378 borel classification g . . . . SbC 380 combinatorial set theory . . . . Not 390 independence . . . . 381 constructibility . . . . 396 relation g . . . . 377 axiom of choice g . . . . 392 large cardinal . . . . Not 395 ordinal number . . . . 393 Martin's axiom . . . . 382 continuum hypothesis g . . . . Not 379 cardinal number . . . Par 216 abstract model theory + . . . . 254 quantifier (5) + g . . . . . Not 516 bound variable + g . . . . . His 514 Frege on quantification + g . . . . . Not 517 free variable + g . . . . . His 513 Aristotle on quantification + . . . . . Not 301 scope . . . . . . 351 scoping algorithm . . . 178 compactness + . . . His 177 aristotelean logic (2) + g . . . . Par 39 syllogism g . . . . Par 513 Aristotle on quantification + . . . Par 196 foundations of theories . . . 195 constraint programming . . 424 system g . . Par 5 algebra 1 (8) g . . . 8 lattice + g . . . SbC 6 boolean algebra + . . . . SbC 7 boolean algebra with operators . . . 11 universal algebra + . . . 77 category theory + g . . . . 78 bottom . . . SbC 9 Lindenbaum algebra . . . 10 relation algebra + . . . 12 algebraic logic (10) + . . . . 6 boolean algebra + . . . . . SbC 7 boolean algebra with operators . . . . 17 post algebra . . . . 15 Lukasiewicz algebra . . . . 14 cylindric algebra g . . . . 8 lattice + g . . . . 18 quantum logic . . . . 10 relation algebra + . . . . 13 categorical logic . . . . 16 polyadic algebra . . . . 19 topos . . . Par 491 algebraic principles . . . . SbC 492 residuation . . 176 mathematical logic (12) g . . . Res 180 Lindstroem's theorem + . . . 77 category theory + g . . . . 78 bottom . . . 53 automated reasoning (25) + . . . . 35 belief revision . . . . . 76 update . . . . 67 nonmonotonic reasoning . . . . 63 mathematical induction . . . . 71 rewrite system (3) . . . . . 350 termination . . . . . 348 confluence . . . . . 349 critical pair . . . . 70 resolution (7) + . . . . . 339 purity principle . . . . . 342 simplification . . . . . 337 demodulation . . . . . 338 ordering . . . . . 340 removal of tautologies . . . . . 341 resolution refinement (4) . . . . . . 345 lock resolution . . . . . . 344 hyper resolution . . . . . . 347 theory resolution . . . . . . 346 set of support . . . . . 343 subsumption . . . . 68 paramodulation . . . . Not 72 skolemisation . . . . 65 model checking . . . . 55 clause 55 (2) . . . . . 80 horn clause g . . . . . 79 Gentzen clause . . . . 74 uncertainty . . . . 75 unification + . . . . 57 connection graph procedure . . . . 64 metatheory . . . . 61 literal . . . . 58 connection matrix . . . . 81 clause 81 . . . . . SbC 82 relative clause . . . . 69 reason extraction . . . . 59 deduction (7) + . . . . . Not 109 inconsistency . . . . . 106 consequence g . . . . . SbC 494 labelled deductive system . . . . . 111 rule-based deduction . . . . . Not 108 entailment + . . . . . 110 natural deduction (2) + g . . . . . . Not 482 hypothetical reasoning + . . . . . . Not 483 normalization . . . . . Not 107 consistency + . . . . Res 60 Herbrand's theorem . . . . 56 completion . . . . . 86 Knuth Bendix completion . . . . 73 theorem prover (3) . . . . . 427 Bliksem g . . . . . 428 Boyer-Moore theorem prover . . . . . 429 SPASS g . . . . 66 narrowing . . . . 62 logic programming g . . . . 54 answer extraction . . . . 247 nonmonotonic logic + g . . . . . 248 default inference . . . Res 182 Loewenheim-Skolem-Tarski theorem + . . . 181 logical constants . . . Not 83 completeness (2) + g . . . . SbC 84 axiomatic completeness . . . . SbC 85 functional completeness + . . . Res 179 Goedel's 1st incompleteness theorem (1931) + g . . . Not 183 operator (4) + g . . . . 254 quantifier (5) + g . . . . . Not 516 bound variable + g . . . . . His 514 Frege on quantification + g . . . . . Not 517 free variable + g . . . . . His 513 Aristotle on quantification + . . . . . Not 301 scope . . . . . . 351 scoping algorithm . . . . SbC 210 modal operator (2) + g . . . . . SbC 472 diamond g . . . . . SbC 473 box g . . . . 518 truth-funcional operator (2) g . . . . . SbC 252 iff g . . . . . SbC 253 negation . . . . Not 525 arity g . . . Not 178 compactness + . . . Res 520 Goedel's 2nd incompleteness theorem (1931) g . . . 435 type theory (2) + . . . . 433 type . . . . . 434 type shifting . . . . Not 23 polymorphism + g . . . 184 symbolic logic (18) g . . . . SbC 412 dynamic logic . . . . 420 partial logic . . . . SbC 413 fuzzy logic g . . . . 200 relevance logic + . . . . . 108 entailment + . . . . SbC 419 paraconsistent logic . . . . 416 intermediate logic . . . . 125 feature logic + . . . . . 75 unification + . . . . 157 predicate logic + g . . . . 364 natural logic + . . . . SbC 422 propositional logic g . . . . SbC 410 boolean logic g . . . . SbC 156 modal logic (13) + g . . . . . Ins 512 S4 . . . . . 488 modes . . . . . 486 frame (2) . . . . . . SbC 487 frame constraints . . . . . Par 210 modal operator (2) + g . . . . . . SbC 472 diamond g . . . . . . SbC 473 box g . . . . . SbC 213 doxastic logic g . . . . . Not 489 accessability relation + . . . . . Par 471 modal language (2) g . . . . . . Par 210 modal operator (2) + g . . . . . . . SbC 472 diamond g . . . . . . . SbC 473 box g . . . . . . 490 boolean operators . . . . . SbC 211 alethic logic g . . . . . SbC 212 deontic logic (3) g . . . . . . SbC 521 standard deontic logic g . . . . . . SbC 523 two-sorted deontic logic g . . . . . . SbC 522 dyadic deontic logic g . . . . . Par 215 Kripke semantics + g . . . . . . Not 489 accessability relation + . . . . . Par 457 modal model theory (7) + . . . . . . SbC 215 Kripke semantics + g . . . . . . . Not 489 accessability relation + . . . . . . Not 461 generated submodel g . . . . . . 462 model (4) + . . . . . . . SbC 464 finite model g . . . . . . . SbC 466 image finite model . . . . . . . . Res 467 Hennessy-Milner theorem g . . . . . . . Par 463 valuation g . . . . . . . SbC 465 tree model g . . . . . . Not 459 disjoint union of models g . . . . . . 455 homomorphism (2) + g . . . . . . . SbC 456 bounded homomorphism g . . . . . . . SbC 468 bounded morphism . . . . . . Not 469 expressive power g . . . . . . . Not 470 standard translation + g . . . . . . Not 460 bisimulation g . . . . . SbC 214 epistemic logic g . . . . . Not 462 model (4) + . . . . . . SbC 464 finite model g . . . . . . SbC 466 image finite model . . . . . . . Res 467 Hennessy-Milner theorem g . . . . . . Par 463 valuation g . . . . . . SbC 465 tree model g . . . . SbC 418 many-valued logic g . . . . SbC 417 intuitionistic logic g . . . . SbC 421 probability logic . . . . 411 conditional logic . . . . SbC 414 higher-order logic . . . . 415 inductive logic . 258 philosophy (3) g . . Par 524 philosophy of language g . . Par 259 logic 259 (2) g . . . His 177 aristotelean logic (2) + g . . . . Par 39 syllogism g . . . . Par 513 Aristotle on quantification + . . . 449 proposition (2) g . . . . 448 contradiction g . . . . . 255 paradox (2) g . . . . . . 256 liar paradox g . . . . . . 257 semantic paradox . . . . 94 conditional statement (2) . . . . . 95 antecedent . . . . . 96 counterfactual g . . Par 208 metaphysics g . . . 209 common sense world g

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 Contact Us My Basket ... Volume 12, Number 6 Pp. 447-459 Logic Journal of IGPL 2004 12(6):447-459; doi:10.1093/jigpal/12.6.447 Oxford University Press This Article Full Text (PDF) Alert me when this article is cited Alert me if a correction is posted Services Email this article to a friend Similar articles in this journal Alert me to new issues of the journal Add to My Personal Archive ... Request Permissions Google Scholar Articles by Figallo, A. V. Articles by Zilliani, A. Search for Related Content 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. Reviewed Items: Miodrag Raskovic, Radosav Dordevic, Probability Quantifiers and Operators. Full-text: Remote access If you are a member of the ASL, log in to Euclid for access. Full-text is available via JSTOR, for JSTOR subscribers. Go to this article in JSTOR. Links and Identifiers Permanent link to this document: http://projecteuclid.org/euclid.jsl/1183745594 JSTOR: links.jstor.org back to Table of Contents previous next Journal of Symbolic Logic Current Issue All Issues Search this Journal Editorial Board ... Log in RSS Cornell University Library

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 which http://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 Contact Us My Basket ... Volume 14, Number 2 Pp. 299-322 Journal of Logic and Computation 2004 14(2):299-322; doi:10.1093/logcom/14.2.299 Oxford University Press This Article Full Text (PDF) Alert me when this article is cited Alert me if a correction is posted Services Email this article to a friend Similar articles in this journal Similar articles in ISI Web of Science Alert me to new issues of the journal ... Download to citation manager Search for citing articles in: ISI Web of Science (1) Request Permissions Google Scholar Articles by Montagna, F. Search for Related Content Original Article 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. Expressivité des logiques d'espaces Étienne 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.

20. Fads And Fallacies About Logic 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

var sc_project=2763585; var sc_invisible=0; var sc_partition=28; var sc_security="3c678009"; var addtoLayout=0; var addtoMethod=0; var AddURL = escape(document.location.href); // this is the page's URL var AddTitle = escape(document.title); // this is the page title Advanced Search CS Search Google Search Home Digital Library Podcasts Site Map ... Table of Contents Abstract March/April 2007 (Vol. 22, No. 2) pp. 84-87 Fads and Fallacies about Logic John F. Sowa , VivoMind Intelligence Inc. Full Article Text: DOI Bookmark: http://doi.ieeecomputersociety.org/10.1109/MIS.2007.29 Send link to a friend Abstract Misunderstandings about the nature of logic and its relationship to language, reasoning, and computation have plagued AI from the earliest days. The author analyzes six of the major issues and their implications for design decisions. The goal is to support more readable user interfaces and better integrated tools for software design and development. About References Back to Top References [1] J. Limber, "The Genesis of Complex Sentences,"

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"; Good Math has moved to ScienceBlogs Wednesday, March 29, 2006 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

Propositions for the Common Logic Interchange Format 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. CLIF semantics 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. Meetings: Colloquia Computational Linguistic Colloquia in 2002 Fridays, 2002 Computational Linguistic Colloquia in 2001 Fridays, 2001 Amsterdam-Utrecht Workshops Special events Past: Grammars and Learning Friday, November 14th, 2003 Linguistic Corpora and Logic Based Grammar Formalisms November 27th and 28th, 2002 Logical Tools for Linguistics Tuesday, June 18th, 2002 Games in Logic, Language and Computation 6 Thursday, June 20th, 2002 with a Extra talk by Joachim Lambek Location: Centrum-Gebouw Zuid, Room F119, De Uithof FGTrento, The 7th conference on Formal Grammar

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 Logic http://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

First-Order Predicate Logic predicates in natural languages quantifiers in natural languages predicate logics see also: 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

SCAN 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. Common Linear Temporal Logic (LTL) Operators 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 Quantifiers II 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. http://linkinghub.elsevier.com/retrieve/pii/S1570868305000455

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 » Blog Archive » 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/

Shadow 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). Table of contents 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 Academy Library The Athenaeum Library The Nominalist Library Athenaeum Reading Room Non-Fregean Logic and Ontology of Situations 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.

37. Hybrid Logic's Home Page 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 Ivánek 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

Neural Network world Contents of Volume 14 (2004) [1] Editorial, 365. [2] Berka P., La¹ V., Svátek V. (Czech Republic): NEST: re-engineering the compositional approach to rule-based inference, 367-380. Bla»ák J., Popelínský 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] Horváth T., Krajèi S. (Slovakia): Integration of two fuzzy data mining methods, 391-402. [5] Ivánek 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 Gödel 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"; Formal Logic/Predicate Logic/Formal Syntax From Wikibooks, the open-content textbooks collection Formal Logic Predicate Logic Jump to: navigation search ← The Predicate Language ↑ Predicate Logic ... Free and Bound Variables → Contents Formal Syntax Vocabulary Expressions Formation rules ... 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'–'z' with a natural number subscript. Thus the variables are: Operation letters: Lower case letters 'a'–'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'–'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

google_ad_client = "pub-9523722979947731"; google_alternate_ad_url = "http://www.mathematics21.org/ads/728x15.html"; google_ad_width = 728; google_ad_height = 15; google_ad_format = "728x15_0ads_al"; google_ad_channel ="3391171102"; google_ad_client = "pub-9523722979947731"; google_ad_width = 728; google_ad_height = 90; google_ad_format = "728x90_as"; google_ad_channel ="3391171102"; google_alternate_ad_url = "http://www.mathematics21.org/ads/728x90"; google_ad_client = "pub-9523722979947731"; google_ad_width = 160; google_ad_height = 600; google_ad_format = "160x600_as"; google_ad_channel ="3391171102"; google_alternate_ad_url = "http://www.mathematics21.org/ads/160x600"; My homepage My math page My math news Donate for the research 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. Axiomatic/Algebraic Theory of Formulas - new theory describing mathematical formulas (and any objects which have constituent parts) with axiomatic method, a new foundation for math logic, that is new foundation of the foundation of mathematics. I advise to add this new theory to the University program . See also my informal article Different Viewpoints on Axiomatic Theory of Formulas 21 Century Math Method (preliminary draft) - new mathematical method replacing axiomatic method with a simple definitional formal system.

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 Previous message: [FOM] Towards a Real Finitism? Messages sorted by: [ date ] [ thread ] [ subject ] [ author ] ... 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. Previous message: [FOM] Towards a Real Finitism? Messages sorted by: [ date ] [ thread ] [ subject ] [ author ] ... More information about the FOM mailing list

43. Publikationer Från 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 by http://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 TR-1994 TR-1995 TR-1996 TR-1997 ... TR-1999 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 - Mathématiques & 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 Logic 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

Monodic fragments of predicate temporal logic Go to home page Related Papers 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

Home Browse by Inventor Browse by Date Links ... Contact Us United States Patent 7107255 Self join elimination through union US Patent Issued on September 12, 2006 Inventor(s) Gerald George Kiernan Jayavel Shanmugasundaram Assignee International Business Machines Corporation 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) Examiners Primary: Greta Robinson Assistant: Jacques Veillard Attorney, Agent or Firm 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 print share ... link 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 The importance of ...

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 Truth-Functional Propositional Logic ... Polyadic 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)" Previous message: Benjamin Grosof: "my notes from JC telecon 5/6/03 on roadmap for rules (inline'd and attached)" Messages sorted by: [ date ] [ thread ] [ subject ] [ author ] ... [ attachment ] Mail actions: [ respond to this message ] [ mail a new topic ] 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) -- Britannica Online Encyclopedia description operator (Logic). Encyclopædia 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 Encyclopædia 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