0 top The TOP concept in the hierarchy. 1 adverbial modification gloss 10 relation algebra 100 conversational maxim Pronciple proposed by H.P. Griece as a conversational maxim. The principle is further explicated by four Maxims of Conversation: the Quality Maxim ('try to be truthful'), the Qunatity MAxim ('be informative'), the Maxim of Relevance ('be relevant'), and the MAxim of Manner ('be perspicuous'). 101 conversational implicature Conversational implicatures refer to the implications which can be deduced from the form of an utterance, on the basis of certain co-operative principles which govern the efficiency and normal acceptability of conversations, as when the sentence 'There's some chalk on the floor' is taken to mean 'you ought to pick it up' 102 cooperative principle 103 implicature A term derived from the work of the philosopher H. P. Grice (1913-88) and now frequently used in linguistics as part of the study of conversational structure. 104 database An organized body of related information. 105 query A question that may be implemented in code such as SQL or by completing a query table, that is presented to a database which returns an appropriate result. 106 consequence A is the syntactic consequence of a set L of wffs iff A can be derived from L (and the axioms). Notation: L |- A. 107 consistency 108 entailment 109 inconsistency 11 universal algebra 110 natural deduction A set of rules expressing how valid proofs may be constructed in predicate logic. 111 rule-based deduction 112 description A statement that represents something in words. 113 definite description 114 indefinite description 115 discourse An extended verbal expression in speech or writing. 116 discourse particle 117 discourse referent 118 discourse representation theory A semantic theory which seeks to extend model-theoretic semantics to accommodate sequences of sentences, and in particular to accommodate anaphoric dependencies across sentence boundaries. Central to the theory is an intermediate level of semantic representation called a discourse-representation structure (DRS). 119 domain theory Abstract mathematical theory of functions that provides a systematic way of solving recursive equations X = ...X... This is useful in the semantics of programming languages since recursive definitions of structures and procedures are very common in programming languages. Moreover, domain theory enabled a model theoretic interpretation for the untyped lambda calculus. 12 algebraic logic 120 domain 121 ellipsis The omission or suppression of parts of words or sentences. 122 antecedent of ellipsis 123 extension The class of objects that an expression refers to. 124 extensionality Principle of classical logic, which say that propositions with the same truth value, or predicates with the same extension, may be replaced byu each other salva varitate. 125 feature logic 126 feature structure 127 formal language theory Branch of mathematical linguistics that studies mathematical properties of alngauges. The subject has drawn the joint attention of linguists, logicians and computer scientists. There is a close connection with automata theory. 128 categorial grammar 129 category 13 categorical logic 130 Chomsky hierarchy A much studied classification of languages within the formal language theory. In this hierarchy four types of languages are distinguished. The distinction between the types are based on the properties of the phrase structure grammars that generate the languages. 131 context free language A formal language theory. This type of language is generated by grammars that have to meet the following restriction: in every production X => Y, Y may consist of only one non-terminal symbol. 132 context sensitive language A formal language theory. A context sensitive language is defined to be generated by grammars that satisfy the restriction that all productions are of the form AXB => AYB. 133 feature constraint 134 phrase structure grammar 135 recursive language Language (set of strings) for which the question of whether some string belongs to the language is decidible. 136 regular language 137 unrestricted language A language that can be generated by any phrase structure grammar. 138 function word A word that serves a grammatical function but has no identifiable meaning. 139 determiner One of a limited class of noun modifiers that determine the referents of noun phrases. 14 cylindric algebra Algebras that aris in the algebraic study of first order logic. They are extensions of Boolean algebras with cylindriification operations that are itnended to model quantification. 140 language generation 141 reversibility 142 grammar Studies the formation of basic linguistic units. 143 sentence A string of words satisfying the grammatical rules of a language. 144 syntax 144 Studies the rules for forming admissible sentences. 145 idea The content of cognition. 146 indexicality 147 indexical expression A type of expression whose semantic value is in part determined by features of the context of utterance, and hence may vary with that context. Among indexicals are the personal pronouns, such as "I", "you", "he", "she", and "it"; demonstratives, such as "this" and "that"; temporal expressions, such as "now", "today", "yesterday"; and locative expressions, such as "here", there", etc. 148 intension What you must know in order to determine the reference of an expression. 149 intensional isomorphism 15 Lukasiewicz algebra 150 intensional verb 151 intensionality 152 frame (1) 153 frame problem The problem of efficiently maintaining a correct and consistent world model as some aspects of the world are changed by operators (but most aspects remain the same). 154 logical omniscience 156 modal logic Originally, a system of logic whose formal properties resemble certain moral and epistemological concepts. 157 predicate logic The branch of logic dealing with propositions in which subject and predicate are separately signified 159 procedural representation 16 polyadic algebra 160 relation system 161 representation language 162 rule-based representation 163 script 164 semantic network A knowledge representation scheme in which objects are depicted as nodes in the network and the relationships between objects are represented by connecting arcs; the arcs are labeled by type of relationship such as is, belongs to, and so on. 165 situation calculus 166 temporal logic (1) The logical study of time. 167 temporal logic (2) A system of logic whose formal properties resemble certain temporal concepts. 168 lambda calculus A formalized description of functions and the way in which they combine, developed by Alonzo Church and used in the theory of certain high-level programming languages. 169 abstraction 17 post algebra 170 application 171 conversion 172 lambda operator 173 linguistics The scientific study of language. 174 language acquisition In the study of the growth of language in children, a term referring to the process or result of learning a particular aspect of a language, and ultimately the language as a whole. In generative linguistics, the term refers to a model in which the infant is credited with an innate predisposition to acquire linguistic structure. 175 semantics 175 The study of language meaning. 176 mathematical logic The branch of mathematics that analyzes inference. 177 aristotelean logic The syllogistic logic of Aristotle. 178 compactness 179 Goedel's 1st incompleteness theorem (1931) Roughly, any consistent or omega-consistent formal system of arithmetic of "sufficient strength" is incomplete (negation incomplete and omega-incomplete). To be of sufficient strength, the system must (1) have decidable sets of wffs and proofs, and (2) represent every decidable set of natural numbers. 18 quantum logic 180 Lindstroem's theorem 181 logical constants 182 Loewenheim-Skolem-Tarski theorem 183 operator Any symbol that denotes the operation to be performed on other symbols or numbers (the operand). Examples include: "+," "-," "×," "÷," or the Boolean operators "AND," and "OR" which operate on two operands while "sin," "cos," or the Boolean operator "NOT" have only one operand. 184 symbolic logic Any logical system that abstracts the form of statements away from their content in order to establish abstract criteria of consistency and validity. 185 computational logic (1) The branch of computer science that uses logic to study and understand programs. 186 program construct 187 program specification 188 program verification 189 reasoning about programs 19 topos 190 semantics 190 The computational meaning of computer programs. 191 logic (1) A system or calculus or reasoning. 192 combinatory logic A branch of formal logic that deals with formal systems designed for the study of certain basic operations for constructing and manipulating functions as rules, i.e. as rules of calculation expressed by definitions. 193 computability theory 194 computational logic (2) 195 constraint programming 196 foundations of theories 197 model theory 198 proof theory The branch of logic describing procedures for combining logical statements to show, by a series of truth-preserving transformations, that one statement is a consequence of some other statement or group of statements. 199 recursive function theory 2 graded adjective 20 ambiguity Ambiguity results whenever a word or a phrase can have more than one distinct and valid meaning. Ambiguity can usually be eliminated through extra information. 200 relevance logic 201 set theory The branch of pure mathematics that deals with the nature and relations of sets. 202 mathematics A group of related sciences dealing with the logic of quantity and shape and arrangement. 203 meaning relation 204 antonymy The semantic relation that holds between two words that can (in a given context) express opposite meanings. 205 hyponymy The semantic relation of being subordinate or belonging to a lower rank or class. 206 paraphrase The result or process of producing alternative versions of a sentence or text without changing the meaning. One sentence may have several paraphrases, e.g. The dog is eating a bone, A bone is being eaten by the dog, and so on. Most semantic theories would treat all these sentences as having a single semantic representation (though variations in focus and presupposition could differentiate them). 207 synonymy The semantic relation that holds between two words that can (in a given context) express the same meaning. 208 metaphysics The philosophical study of being and knowing. 209 common sense world n philosophy, the doctrine that we perceive the external world directly, that what we perceive is what there is and how things are. Common-sense realism has been held by Scottish mathematician Thomas Reid and English philosopher G E Moore. Although a useful antidote to complex metaphysical theories, common sense can mislead - for instance, common sense tells us that the world is flat. 21 derivational ambiguity 210 modal operator Operator of modality, i.e. expressing necessity or contingency. 211 alethic logic The modal logic of necessity and possibility and contingency. 212 deontic logic The modal logic of obligation and permission. There are three principal types of formal deontic systems. (1) Standard deontic logic, or SDL; (2) Dyadic deontic logic; (3) Two-sorted deontic logic. 213 doxastic logic The modal logic of belief and disbelief. 214 epistemic logic The modal logic of knowledge, where the modality means "knowing that". It is common to use "epistemic" as a blanket term for logics of knowledge and beliefs. 215 Kripke semantics Type of model theory for modal logic developed by Saul Kripke 216 abstract model theory 217 admissible set 218 categoricity The semantic property belonging to a set of sentences, a postulate set, that implicitly defines (completely describes, or characterizes up to isomorphism) the structure of its intended interpretation or standard model. 219 completeness of theories 22 lexical ambiguity A lexical ambiguity occurs when a lexical item (word) is assigned multiple meanings by the language. It includes (a) homonymy: distinct lexical items having the same sound or form but different senses: knight/night, lead (n.)/lead (v.); and (b) polysemy: a single lexical item having multiple senses: lamb (the animal)/lamb (the flesh). The distinction between homonymy and polysemy is problematic. 220 definability 221 denumerable structure 222 equational class 223 finite structure 224 higher-order model theory 225 infinitary logic 226 interpolation 227 logic with extra quantifiers 228 model-theoretic algebra 229 model-theoretic forcing 23 polymorphism The phenomenon that a term can be assigned more than one type. 230 model of arithmetic 231 nonclassical model 232 preservation 233 quantifier elimination 234 recursion-theoretic model theory 235 saturation 236 second-order model theory 237 set-theoretic model theory 238 stability 239 ultraproduct 24 pragmatic ambiguity 240 Montague grammar 241 meaning postulate 242 ptq The Proper Treatment of Quantification (from the homonimous seminal paper by Richard Montague). 243 sense 243 The meaning of a word or expression. 244 sense 244 What you must know in order to determine the reference of an expression. 245 boolean valued 246 sheaf model 247 nonmonotonic logic A form of logic in which conclusions can be drawn based on assumptions of things that are typically true; later information could force such conclusions to be withdrawn. 248 default inference 249 noun A word that can serve as the subject or object of a verb. 25 semantic ambiguity Unclearness by virtue of having more than one meaning. 250 mass noun A noun that does not form plurals. 251 proper name 252 iff A shortened form of if and only if: it indicates that the two sentences so connected are necessary and sufficient conditions for one another. Usually iff is used for equivalence in the metalanguage, rather than as the biconditional in the object language. 253 negation 254 quantifier A form of operator introduced by Frege. It indicates what was, in traditional logic, called the quantity of a statement, namely whether it is universal, as 'All bats are blind', or particular, as 'Some swans are black'. 255 paradox A self-contradiction. 256 liar paradox The notion of a self-referential statement creating a sematic paradox goes back to Epimenides, a Cretan who is supposed to have said: "All Cretans are liars". 257 semantic paradox 258 philosophy The rational investigation of questions about existence and knowledge and ethics. 259 logic 259 The branch of philosophy that analyzes inference. 26 structural ambiguity 260 plural term A term referring to two or more items or units. 261 collective reading 262 distributive reading 263 presupposition An assumption that is taken for granted. 264 control primitive 265 functional construct 266 object oriented construct 267 program scheme 268 type structure 269 invariant 27 syntactic ambiguity 270 post-condition 271 pre-condition 272 specification technique 273 logic of programs 274 mechanical verification 275 programming language 276 syntax 276 277 prolog A programming language originally designed to support natural language processing. 278 pronoun A function word that is used in place of a noun or noun phrase. 279 demonstrative A pronoun that points out an intended referent. 28 anaphor A word (such as a pronoun) used to avoid repetition; the referent of an anaphor is determined by its antecedent. 280 pronoun resolution 281 complexity of proofs 282 constructive analysis 283 constructive system 284 cut elimination theorem A theorem stating that a certain type of inference rule (including a rule that corresponds to modus ponens) is not needed in classical logic. The idea was anticipated by J. Herbrand; the theorem was proved by G. Gentzen and generalized by S. Kleene. 285 first-order arithmetic 286 functionals in proof theory 287 Goedel numbering 288 higher-order arithmetic 289 interpretation 29 anaphora resolution 290 intuitionistic mathematics 291 metamathematics 292 normal form theorem 293 ordinal notation 294 recursive analysis 295 recursive ordinal 296 relative consistency 297 second-order arithmetic 298 structure of proofs A proof in a formal system is a symbolic structure that can be characterized by referring only to the syntax of the system. The nature of the structure is determined by the axioms and/or rules of inference of the system (see axiom; inference, rule of). As such structures, proofs themselves become objects that can be studied by mathematical techniques. 299 belief 3 intersective adjective 30 antecedent of an anaphor 300 quantifying in 301 scope 302 recursion theory The study of problems that, in principle, cannot be solved by either computers or humans. 303 abstract recursion theory 304 automaton 305 axiomatic recursion theory 306 complexity of computation 307 decidability A (modal logic) is decidable if there exists an algorithm for deciding the satisfiability problem for the logic. 308 degrees of sets of sentences 309 effectively presented structure 31 animal A kind of object that is often used to illustrate semantic phenomena. 310 formal grammar 311 hierarchy 312 higher type recursion theory 313 inductive definability 314 isol 315 post system 316 recursion theory on admissible sets 317 recursion theory on ordinals 318 recursive axiomatizability 319 recursive equivalence type 32 donkey 320 recursive function 321 recursive relation 322 recursively enumerable degree 323 recursively enumerable language 324 recursively enumerable set 325 reducibility 326 set recursion theory 327 subrecursive hierarchy 328 theory of numerations 329 thue system 33 unicorn 330 undecidability 331 word problem 332 reference The class of objects that an expression refers to. 333 identity puzzle 334 referent Something referred to; the object of a reference. 335 referential term 336 anchor 337 demodulation 338 ordering 339 purity principle 34 artificial intelligence The branch of computer science that deals with writing computer programs that can solve problems creatively. 340 removal of tautologies 341 resolution refinement 342 simplification 343 subsumption 344 hyper resolution 345 lock resolution 346 set of support 347 theory resolution 348 confluence 349 critical pair 35 belief revision 350 termination 351 scoping algorithm 352 rabbit 353 truth 354 underspecification 355 algebraic semantics 356 denotational semantics 357 operational semantics 358 partial evaluation 359 process model 36 classification 360 program analysis 361 formal semantics The study of language meaning using mathematical and logical formalisms. 362 dynamic semantics 363 lexical semantics 364 natural logic 365 property theory 366 situation semantics An approach to the semantic analysis of languages developed during the 1980s as an alternative to possible-worlds-based model-theoretic semantics. Sentences are analysed as denoting not truth values but situations (sets of facts which consist of a location, a relation and a truth value) and, contrary to model-theoretic semantics, the interpretation of sentences may depend on the context. 367 semantics 367 A formal description of the meaning of the symbols and expressions used in a logic. 368 assignment 369 material implication The truth-functional connective that forms a compound sentence from two given sentences and assigns the value false to it only when its antecedent is true and its consequent false, without consideration of relevance; loosely corresponds to the English "if ... then". 37 heuristic A "heuristic principle" is a principle of thinking or reasoning which is not judged based upon its truth but rather its pragmatic consequences. It is assumed to be true for the purposes of some problem or inquiry. 370 satisfaction 371 truth conditional semantics 372 truth function A truth function is a function that returns one of two values, one of which is interpreted as "true", and the other which is interpreted as "false". Typically either "T" and "F" are used, or "1" and "0", respectively. 373 truth table 374 literal meaning 375 metaphor A figure of speech in which a word or phrase literally denoting one kind of object or idea is applied to another to suggest a likeness or analogy between them. 376 metonymy The substitution of a word referring to an attribute of a thing for the thing itself, e. the 'crown' to refer to the monarch. 377 axiom of choice A controversial axiom asserting that for a non-empty set A of non-empty disjoint sets, there is a set B with exactly one member from each of the disjoint sets comprising A. Sometimes the axiom is written so as to assert that there is a function for choosing the members of the disjoint sets in A that will become the members of B. Cantor's generalized continuum hypothesis implies this axiom. 378 borel classification A systems of classification for sets of reals, due to Emil Borel, developed in the area of "descriptive set theory". 379 cardinal number 38 knowledge representation A technique for structuring human knowledge about a subject or specialty so that it can be coded for input into a computer; examples are rules, frames, semantic networks, propositional logic, and predicate calculus. 380 combinatorial set theory 381 constructibility 382 continuum hypothesis A hypothesis in set theory first proposed by Cantor. The set of all natural numbers N has a cardinal number Aleph_0. The power set of N will therefore have a cardinality of Aleph_0 to teh power of 2, which is denoted by c-the cardinal number of the set of real numbers (the continuum). Cantor's hypothesis is that no infinite cardinal lies between Aleph_0 and c. 383 descriptive set theory Descriptive set theory is the study of definable sets and functions in Polish (complete separable metric) spaces. 384 determinacy 385 filter 386 function A mathematical relation such that each element of one set is associated with at least one element of another set. A more formal definition: let A and B be sets. A function f:A->B is a relation R from A to B such that a) for every element a in A, there exists an element b in B such that the pair (a,b) belongs to the set R; b) if b1 and b2 are elements of B, and the pairs (a, b1) and (a, b2) both belong to R, then b1=b2. 387 fuzzy relation 388 fuzzy set A set in which membership is a matter of degree. In classical set theory, for every set S and thing x, either x is a member of S (1) or x is not (0). In fuzzy set theory, things x can be members of sets S to any degree between 0 and 1, inclusive. 389 generalized continuum hypothesis 39 syllogism Deductive reasoning in which a conclusion is derived from two premises. The conclusion necessarily follows from the premises; so that, if these are true, the conclusion must be true, and the argument amounts to demonstration. A classical example of syllogism is "All men are mortal; Socrate is a man; Socrate is mortal". 390 independence 391 iota operator 392 large cardinal 393 Martin's axiom 394 ordinal definability 395 ordinal number 396 relation A way in which two or more objects are connected, associated, or related, or (at a different level) a polyadic predicate symbolizing such a relation. 397 set algebra 398 set-theoretic definability 399 Suslin scheme 4 predicative position 40 planning 400 situation 401 scene 402 partiality 403 speech act The use of language to perform some act. 404 illocutionary force 405 indirect speech act 406 performative 407 performative hypothesis 408 statement The act of affirming or asserting or stating something. 409 indicative statement 41 aspect 410 boolean logic A form of algebra that employs only two values, TRUE and FALSE. Boolean logic, developed by the 19th-century English mathematician George Boole, is particularly well suited for use with computers because it works so well with the Binary number system. A bit with value 1 corresponds to TRUE; a bit with value 0 corresponds to FALSE. 411 conditional logic 412 dynamic logic 413 fuzzy logic Developed by Lofti Zadeh, fuzzy logic is a system of reasoning which allows for the possibility that propositions may have degrees of truth or falsity, rather than simply possessing simple truth or simple falsity. 414 higher-order logic 415 inductive logic 416 intermediate logic 417 intuitionistic logic Brouwer's foundational theory of mathematics which says that you should not count a proof of (There exists x such that P(x)) valid unless the proof actually gives a method of constructing such an x. Similarly, a proof of (A or B) is valid only if it actually exhibits either a proof of A or a proof of B. 418 many-valued logic A logical system in which the truth-values that a proposition may have are not restricted to two, representing only truth and falsity. 419 paraconsistent logic 42 aspectual classification 420 partial logic 421 probability logic 422 propositional logic A branch of symbolic logic dealing with propositions as units and with their combinations and the connectives that relate them. 423 syntactic category A category of words having the same grammatical properties. 424 system A complex of methods or rules governing behavior. 425 term A word or expression used for some particular thing. 426 singular term A term referring to a single item or unit. 427 Bliksem A theorem prover for first-order logic developed by Hans de Nivelle. 428 Boyer-Moore theorem prover 429 SPASS An automated theorem prover for first-order logic with equality developed at MPI, Saarbruecken. 43 assertion 430 truth condition 431 truth definition A formal characterization of the set of true sentences of a language which is already used meaningfully - a characterization which is derived from an interpretation of the language. 432 truth value 433 type 434 type shifting 435 type theory 436 monotonic semantics 437 quasi-logical form 438 verb A word that serves as the predicate of a sentence. 439 perception verb 44 declarative assertion 440 word A unit of language that native speakers can identify. 441 modifier A content word that qualifies the meaning of a noun or verb. 442 content word A word to which an independent meaning can be assigned. 443 linguistic geography The study of the geographical distribution of linguistic features. 444 linguistic unit One of the natural units into which linguistics messages can be analyzed. 445 adjective A word that expresses an attribute of something. 446 descriptive linguistics An explanation of a person's mastery of their native language. 447 part of speech One of the traditional categories of words intended to reflect their functions in a grammatical context. 448 contradiction A statement that is necessarily false. 449 proposition A statement that affirms or denies something and is either true or false. 45 imperative assertion 450 pragmatics The study of language use. 452 grammatical constituent A word or phrase or clause forming part of a larger construction. 453 logical syntax Family of linguistic theories that use graphs as the carriers of syntactic, semantic, morphological and phonetic information, and that aim to describe the admissible graphs underlying text or utterances. 454 first order model theory 455 homomorphism Structure preserving mapping between modal models; preserves valuations and relations. 456 bounded homomorphism Homomorphism that also satisfies a "back condition" which ensures that some of the structure in the target model is reflected in the source model. 457 modal model theory 459 disjoint union of models Method to construct a new model out of the domains of the old model. 46 attitude 460 bisimulation Relation between two models in which related states have identical atomic and matching transitions. 461 generated submodel Method to construct a new model of an old model, by restricting the domain to the smallest connected component that contains the generating states 462 model 463 valuation Mapping that assigns subsets of the domain of a model to propositional letters 464 finite model Model whose domain is finite 465 tree model Model whose relation or relations form a tree 466 image finite model 467 Hennessy-Milner theorem Image-finite models are bisimilar if and only if they satisfy the same modal formulas 468 bounded morphism 469 expressive power Characterization of the properties of models that can be captured in a formal language 47 propositional attitude 470 standard translation Mapping from modal languages to first-order (or second order) languages; usually defined by capturing the truth definition of the source language in the terms of the target language 471 modal language Defined using a set of proposition letters, propositional connectives, and modal operators. 472 diamond Existential modal operator; evaluated at a state, it seeks a related state that satisfies the argument of the operator. 473 box Unuiversal modal operator; evaluated at a state, it explores all related states to ceck that they satisfy the argument of the operator. 474 tree model property A logic has the tree model property if a formula from the logic is satisfiable only if is satisfiable on a tree model. 475 first order logic The branch of logic dealing with propositions in which subject and predicate are separately signified, reasoning whose validity depends on this level of articulation, and systems containing such propositions and reasoning. 476 first order language Language built from terms and first-order formulas. Terms are built from variables, constants, and function symbols. Atomic formulas are identity statements and/or relational expressions; complex formulas are formed using propositional connectives and quantifiers. 477 fragment A subset of the legal formulas of a formal language; the subset may be defined by syntactic or semantic means. 478 modal fragment Fragment of first-order logic obtained by applying the standard translation to the modal language; an alternative definition is to admit a very restricted form of first-order quantification only. 479 finite-variable fragment Fragment of first-order logicin which only a fixed finite set of variables may be used. 48 automata theory 480 guarded fragment Generalization of the modal fragment; fragment of first-order logic that only allows guarded quantification. 481 linear logic 482 hypothetical reasoning 483 normalization 484 structural rules 485 proof nets 486 frame (2) 487 frame constraints 488 modes 489 accessability relation 49 finite state machine A machine, or a mathematical model of a machine, which can only reach a finite number of states and transitions between these states. It is used in mathematical problem analysis. 490 boolean operators 491 algebraic principles 492 residuation 493 correspondence theory 494 labelled deductive system 495 substructural logic 496 syntax 496 The study of language form. 497 movement 498 word order 499 syntax and semantic interface 5 algebra 1 The mathematics of generalized arithmetical operations 50 linear bounded automaton 500 quantified phrases 501 coordination 502 discontinuity 503 sequent calculus 504 subformula property 506 linguistic phenomena 507 compositionality 508 functional composition 509 functional application 51 push down automaton 510 frameworks 511 SPASS Automated thorem prover for first order logic with equlity 512 S4 513 Aristotle on quantification 514 Frege on quantification Frege treated 'All bats are blind' as equivalent to 'For any object x, if x is a bat, then x is blind'. This is now symbolically represented by (x) (Bx -> Cx), where 'B' abbreviates 'is a bat', 'C' abbreviates 'is blind', and '(x)' is the universal quantifier, which is read 'For any x' or 'For all x'. It is sometimes also written (ForAllx). 515 quantification 516 bound variable In predicate logic, an individual variable at least one of whose occurrences lies within the scope of a quantifier on the same letter. Because other occurrences may be free, a variable may be both free and bound in the same wffs. 517 free variable A variable which is not quantified. 518 truth-funcional operator Operator that can be added to one or more statements to yield a further statement whose truth-value is systematically dependent on the truth-value(s) of the original statement(s). 'And' is thus a two-place truth-functional operator, while 'not' is a unary truth-functional operator. The truth-functional operators are also called sentential operators because, when applied to sentences, they yield another sentence. 519 derivation Derivation is how new words are created by processes such as inflections,trumpet + er = trumpeter, or compounding wind + mill = windmill. It contrasts with grammatical inflections 52 Turing machine 520 Goedel's 2nd incompleteness theorem (1931) The consistency of a system of "sufficient strength" (same as for the first incompleteness theorem) is not provable in the system, unless the system is inconsistent. The second incompleteness theorem is a corollary of the first. 521 standard deontic logic Standard deontic logic, or SDL, results from adding a pair of monadic deontic operators O and P, read as it ought to be that and it is permissible that, respectively, to the classical propositional calculus. 522 dyadic deontic logic Dyadic deontic logic is obtained by adding a pair of dyadic deontic operators O(I) and P(I), to be read as "it ought to be that..., given that..." and "it is permissible that..., given that ...", respectively. 523 two-sorted deontic logic Two-sorted deontic logic, due to Castañeda (Thinking and Doing, 1975), pivotally distinguishes between propositions, the bearers of truth-values, and practitions, the contents of commands, imperatives, requests, and such. 524 philosophy of language The philosophical study of natural language and its workings, particularly of linguistic meaning and the use of language. 525 arity The number of arguments a function or operator takes. It is common to distingish unary, binary and n-ary functions/operators, taking respectively one, two or n arguments. In some languages functions may have variable arity which sometimes means their last or only argument is actually a list of arguments. 526 variable In logic and mathematics, a symbol interpreted so as to be associated with a range of values, a set of entities any one of which may be temporarily assigned as a value of the variable. An occurrence of a variable in a mathematical or logical expression is a free occurrence if assigning a value is necessary in order for the containing expression to acquire a semantic value-a denotation, truth-value, or other meaning. 527 algebra 2 Any formal mathematical system consisting of a set of objects and operations on those objects. Examples are Boolean algebra, numerical algebra, set algebra and matrix algebra. 528 combinatorial categorial grammar 529 GB 53 automated reasoning 530 TAG 531 dynamic syntax 532 DRT 534 HPSG 535 LFG 54 answer extraction 55 clause 55 56 completion 57 connection graph procedure 58 connection matrix 59 deduction 6 boolean algebra 60 Herbrand's theorem 61 literal 62 logic programming A form of programming, as in PROLOG, in which programs take the form of logical statements, usually as Horn clauses, and the solution to a desired problem is found by a backchaining search using the database of facts and the logical axioms. 63 mathematical induction 64 metatheory 65 model checking 66 narrowing 67 nonmonotonic reasoning 68 paramodulation 69 reason extraction 7 boolean algebra with operators 70 resolution 71 rewrite system 72 skolemisation 73 theorem prover 74 uncertainty 75 unification 76 update 77 category theory A mathematical theory that studies the universal properties of structures via their relationships with one another. 78 bottom 79 Gentzen clause 8 lattice A partially ordered set in which all finite subsets have a least upper bound and greatest lower bound. 80 horn clause A logical formula that is a disjunction of literals and has at most one positive literal. Viewed as a rule, a Horn clause is a rule from a conjunction of positive literals to a single positive conclusion literal. 81 clause 81 82 relative clause 83 completeness A formal system S is said to be simply complete if and only if, for every wff A of S, either A or ~ A is a theorem of S. 84 axiomatic completeness 85 functional completeness 86 Knuth Bendix completion 87 computer science The branch of engineering that studies (with the aid of computers) computable processes and structures. 88 software 89 theory of computation Theory of computation. 9 Lindenbaum algebra 90 concept 91 individual concept 92 concept analysis 93 concept formation 94 conditional statement 95 antecedent 96 counterfactual A counterfactual is any conditional statement (if X, then Y) in which the antecedent (X) is known to be false. 97 context 98 context change 99 context dependence