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<?xml version="1.0" standalone="yes"?> <Paper uid="J79-1074"> <Title>SEllANTIC INTERPFETATION I N ROMTASUE GFbAMNAR</Title> <Section position="3" start_page="0" end_page="0" type="metho"> <SectionTitle> I. INTRODUCTION </SectionTitle> <Paragraph position="0"> In Thc plaopt;.r trt:ut~/i~:nl of ~llii~?~t i f icn t ir))t ii: or&i?~tz~vj dngrish (PTQ) , Richard Montague sets up a system which uses model-theoretic semantics to provide meanings for English sentences. Expressions of intensional logic hold a position intermediate between the English syntax and the model.</Paragraph> <Paragraph position="1"> For each syntactic structure of dn English phrase there is a corresponding formula of intknsional logic. The meaning of the English phra-se is taken to b.e the interpretation of the logical formula in the model.</Paragraph> <Paragraph position="2"> In this paper, which is primarily tutdrial, we show by example how a moikel can be explicitly constructed and how a logical formula is interpreted in a model. Our paper provides concrete examples of the semantic model and the definPtion of interpretation, given only formally by Monthgue. It is intended to be helpful to readers of PTQ. The reader of. this paper may need to have a copy of PTQ hn hand.</Paragraph> <Paragraph position="3"> We first review the definition. of intensional model, and begin ta specify a model. Then we examine the way in which meaning postulates constrain the model to be reasonable. In a reasonable model the interpretations of Enqlish words are consistent with their usual meanings. We select a pafticular reasonable model and use it to evaluate some formulas. Problems in building a larger explicit model are illustrated by considering the case of adverbs..</Paragraph> <Paragraph position="4"> We conclude with calculations of the size of a model and a brief discussion of the possible use of computer+ for Montague grammar.</Paragraph> </Section> <Section position="4" start_page="0" end_page="0" type="metho"> <SectionTitle> 11. INTENSIONAL MODELS </SectionTitle> <Paragraph position="0"> An intensional mvdeL (or interprclation) a is a quintuple @ = (A, 1, J, 2, F) such that 1) A, I, ,and J are non-empty sets, the set ofi entities, the set of possible worlds, and the set of moments in time, respectively. 2) - < is a simple (linear) ordering on J.</Paragraph> <Paragraph position="1"> F is the meaning function, which assigns each logical constant an appropriate element from the model. If the constant is oE type a, the value of F ia of type <s,a). Each element of I x J, the qet of co-ordinates, is a single point of rcj'erencc or indax.</Paragraph> <Paragraph position="2"> To siwplify the notation, we use S for this cross-product. A model then becomes a quadruple a = ( A, S, - , F). E.6r a given A and S the- set of possible denotations of type a, D is given by: a,A, 5' - for simple types - for complex types (the set of entities) (0 - falsehood, 1 - truth) (the set of total functions from S to D 1 (the set of total functions Wherk no confusion can arise, the subscripts il and S are omitted /n symbols for sets of possible denotations. The rules for evaluatkng an expression Of the logic are given in PTQ. An evaluation is performed with respect to a model 4, a point of reference i in S, and a variable assignment 9. This function y assigns a denotation of the appropriate type to each variable in the expression, that is, for any variable u of type a '7 (u) E Da.</Paragraph> <Paragraph position="3"> The result ofi evaluating an expression a of type a is a possible denotation of type a, i.e,,a member of D cz This value is denoted by a d, i ,'t? and is called the denotation or extension of a with respect to , i, and g.</Paragraph> </Section> <Section position="5" start_page="0" end_page="0" type="metho"> <SectionTitle> 111. SPECIFYING A MODEL </SectionTitle> <Paragraph position="0"> The first step is to give the set A of entities and the set S of points of reference. These two sets uniquely determine for each type a each set Da of possible denotations of expressions of type a. To complete the model, the meaning function F must be specified. The values of F f~r constants of tgpe a are functi~ons from S to Dh.</Paragraph> <Paragraph position="1"> We mow begin to build the intensional model to be used in our examples. Because we wish to write out the sets D c7: explicitly, we construct a finite model.</Paragraph> <Paragraph position="2"> While it would be po-ible td write functions explicitly in ordered pair notation, the result would be long and cumbersome, because in general the elements of the pairs are themselves functions. We overcome this difficulty by ihtroducing names for functions and by taklng advantage of the type system of the model.</Paragraph> <Paragraph position="3"> We use the type system to provide an order in which to consider the denotation sets and their elements. The ordering is such that at each Stage the new functions can be specified as ordered pairs of names already intxduced. The meaning function F is also specifled using these function names.</Paragraph> <Paragraph position="4"> Let the set of points of reference S be 1, 121 and the set of entities, A or D , be {Jo, ~n}.</Paragraph> <Paragraph position="5"> C It is important to distinguish words in the English vocabulary from conlstants in the logic front elements of the 1node1; for exfirnple, John and walk are English words, j and wtrZk' are logical constants, and Jo is gn entity, ana'element of De. English w~rds are given in italics. Logical constants that are direct translations of English words are ?rimed.</Paragraph> <Paragraph position="6"> The function F, which defines the relafionship between constants and elements of the mod&l, assigns to each congtant of type e a function from indices to entities. For example, for the ldgical constant j, F(j) E D = D $' where %s, &) e&quot; In our 0, e> finite model, there are only four m~mbers of D. and we use <s;e>,' do ,al ,a2 ,a3 as their names. They are defined by: If E't,i)+ =. ,u2, then ,i is* assigned { (7 1 L1r1) (12 JO) 1 , that is, the functipn whose value at index I1 is Llh and whose value at index 72 is Yo.</Paragraph> <Paragraph position="7"> Words in both of the syntactic categories CIV.and IV translate inko logical constants of type C+..u&quot;,c>., t), so the values of for these constants are functions from inaces to elements of type (<b,e?,t). For example-,</Paragraph> <Paragraph position="9"> and PYu'nicoriz,') (i I and F (r~alk ') (i) E D &s, e):t> In the model, there are 16 meinbers of 9 and we use ,e>, t> BOt - *,rB15 a$ their names; they are defined by: There are 16L = 256 members of D 0, <cbe>, t>) and they wlll not be enumerated here.</Paragraph> <Paragraph position="10"> The set of- entities, A = D , is also known as the set of e individuals. The members of the set D. of possible .Cs, a) denotations of, type <a, a) are called individual concepts. The members of D are functions from (<t?, L?> , t > <s3 t\ > to { 0 ,I], and thus a member df ? can be viewed as a st7?! J.( <<i> t3> J t> i l t for uhich the function is 1 (true) . A; clement of il is a !~PG~~,+ 13 t :i - , : / L?,. p;,. t, .?- f::. < t:> <&: L>> J t >></Paragraph> </Section> <Section position="6" start_page="0" end_page="7" type="metho"> <SectionTitle> IV, REASONABLE MODELS </SectionTitle> <Paragraph position="0"> Not a11 models that can be constructed are reasonable.</Paragraph> <Paragraph position="1"> Montague's meaning postulates are restricticns on models; they limit the choices for the values of the meaning function P, that is, they constrain the possible meaning of certain English words.</Paragraph> <Section position="1" start_page="0" end_page="7" type="sub_section"> <SectionTitle> Meanins Postulate 1 </SectionTitle> <Paragraph position="0"> We 'now examine how Meaning Postulate 1 affects the choice of values of ' for constants of type e. As an example, we consider F (,/). Meaning Postulate 1 fbr ,i is (314 U (I,) The denotation of this meaning postulate is [ (314) 0 (zc~J) ] n,i,g .</Paragraph> <Paragraph position="1"> This denotation must be 1 if the model is to be reasonable.</Paragraph> <Paragraph position="2"> Following the recursive defiflitio,~ in PTQ [pa 2581:</Paragraph> <Paragraph position="4"> is , where ' is a @assignment like 9 except that () is .u. I O7,it,,7 I [ u (a=j)~ . is 1 if? (u=j) is, 1, for all i E S.</Paragraph> <Paragraph position="5"> Dz,itFg # 1 is 1 iff 1.1</Paragraph> <Paragraph position="7"> I.@., there exists v E 1) l: such that for all if E S,</Paragraph> <Paragraph position="9"> V. In other words, Meaning Postulate 1 requires that the value of E' for the argument ,i be a constant funCtion, i .e., a function that has tk salne value at all points of. reference.</Paragraph> <Paragraph position="10"> Tl~us /+' (J) cannot be just any member of. 11 < 3 y i' > ; it must be cither MGaning Postulate 2 For Meaning Postulate 2 similar analysis applies. This meaning postulate restricts the choice of values of ' for'the constants of type (Zs, e>, t > which are translations of extensional comqon nouns. For example, Meaning Postulate 2 for !cnicorn ' is I3 [iciiicor~i ' (x) -> (310 .c = 1 ] . A reasonable model must make it true. Again, following the recursive definition:</Paragraph> <Paragraph position="12"> 'O,is 1 iff there exists a v E D such</Paragraph> <Paragraph position="14"> is 1 where 9' is a d -assignmen< like g .except that 9 ' (10 is v .</Paragraph> <Paragraph position="16"> (^ ) is the function h such that for all it' c S, For all I ' E St whenever (r) (I)] ( () is 1, then there exists v L such that 3 (s) is X*iV(v) .</Paragraph> <Paragraph position="17"> t The consequent of the result says that %he individual concept that is the value of (2) must be a constant function, i.e.,it must evaluate to the same entity u at every point of referbnce. In our model, only do and a3 are constant individual concepts.</Paragraph> <Paragraph position="18"> Meaning Postulate 2 restricts the possible denotations for these logical coristants of type , , t to subsets of 'the individual concepts that are constant functions. In any reasonable model, ,we must choose F so tha* that is, ;tnicorrthobd can be true only ~~~ase individual^ concepLs which yield the same entity at evdry point of reference, L We use A* as the h-operator of the metalanguage. Thus, hxi&quot;(v) denotes the function from S to D with the conjtant value V.</Paragraph> <Paragraph position="19"> the constants that are translations of extensional intransitive verbs. Meaning Postulate 3 for l~tr ik ' 'ip (3M) (Vr ) Cl [ua l ir ' (x ) [&quot;iV] (*c) ] where M and .c are variables of type s c t and <s, n), respectively. , A reasonable model must make this meaning postulate true.</Paragraph> <Paragraph position="20"> [ (x [walk (2) [&quot;All (%) ] ] m~i'g is 1 iff there exists an mc~ such that [ (VX) [walk1 (XI t, [V~l] 'ivr) I I Q4%,L41 4% (e, t?> is 1 where g' is a @-assignment 1ike.g except thht '(b) is m. [ (Vx) u [waZkl.(x) t-, [&quot;MI (&quot;rc)I I d ,i,gr is 1 iff for all x E D</Paragraph> <Paragraph position="22"> is 1 where gfl is a a-assignment like g ' except that 9.&quot; (x) is x. m ,i,g 11 ,</Paragraph> <Paragraph position="24"> which is [M , ir UL '~9.~~ (-,i 1 )$) which is [gU(M) (i I)] (g&quot; fx) (i ') ) which is [p71([t)] (\(!'I).</Paragraph> <Paragraph position="25"> Thus the meaning pos4tulate is. true just in case there csists vil E L such that for a11 \ i,? (c:,t3, , (: (i ')I (i) {SJ kt*, 0) I is [(I ((I' for all : Z :: Thus, dcpen~is only on the cntity and thc point of reference. If : *+ t is true at a &quot; t point. of reference for an individual concept, then' *A:: i is true at that point sf reference for every individual concept that gives the same entity at that point of reference. Relating this to our model, we see that since</Paragraph> <Paragraph position="27"> Meanipg Postulate 3 requires: Thus, tre must choose F so that V, USING a MODEL We now explicitly construct a particular reasonable model in which tc evalqate some formulas.</Paragraph> <Paragraph position="28"> It must satisfy the constraints on F developed above. We assign to,the constant j the entity Jo at all points of reference:</Paragraph> <Paragraph position="30"> We s.tipul'ate that there are no unicorns at point 0% reference 75 and one unicorn, entity Un, at point of reference 12:</Paragraph> <Paragraph position="32"> At pgint of reference 7 entities Jo and Un walk and at point of reference 12, only 30 walks: Our first etraLuation u'sing this model w141 be for the expression which is ,the translation of John: Finding the.. denotation of. this expression: 'AP [&quot;PI (rd I '~'~'9 is A function h j* such that h (p) is</Paragraph> <Paragraph position="34"> where g ' is a -assignment like g exoept that is p+ whikh is [P 0Z3i9g' (i)] (h+iI[j which is [gl(P) (it)] (X*il[F(j) (i')]) which is [p (i)] (F(j)).</Paragraph> <Paragraph position="35"> Thus, the expression denotes the function ii .* = X*p [ [p (i) ] (F (j) ) ] J So hj, is the set of properties which, when evaluated at point of reference i, are true of the individual .concept that is the value of F for j.</Paragraph> <Paragraph position="36"> In this example of a reasonable model, P (,j) is no and the possible denotations, p (i) , for which ro gets true are B8 through fils.</Paragraph> <Paragraph position="37"> At point of reference TI, the value of the function which is the denotation of the expression that translates John is</Paragraph> <Paragraph position="39"> Now, we evaluate the direct tralslation of the sentence At point of reference 11, the denotation of this expression is:</Paragraph> <Paragraph position="41"> The model can be also used LG illastrate the general fact that the d'enotations of an expression and any of its reductions are the same. We have shown elsewhere [Friedman, 19781 that each expression of the logic has a unique reduced form in which no further contractioh is possible. We now evaluate the reduced form of the trahsldtion of John walks, [walk '.(&quot;j) ] .</Paragraph> <Paragraph position="42"> At point of r.eference t! : As another example, consider the logical expression We now consider extending our model to accommodate sentences with adverbs Such* as s 1~1dZy. &quot;We have that</Paragraph> <Paragraph position="44"> 0, (Ls, 3 t , t , as, C?, t>?> There are not only too many possible denot'aations to consider enumerating them, but in each possible denotation, there are.too many components to consider fully specifying it (see below) . Consequently, in this example, only that portion of F (s Zoz~Zy ' ) which is needed will be specified.</Paragraph> <Paragraph position="45"> The denotation for the sentence Joh?~ w~zzks slot~Zt~ is: [~F(sZOZ~ZZJ') (i)] (P(~~zlii'))] (F(j)) At point of reference 11, Jo and UII are both walking, and choosiqg that JO is walking slowly: and VIT. SIZE OF A MODEL The smallest interesting model for PTQ has two points of reference and two entities. For such a model, the size of the sets of possible denotrjtions and the size of their elements (in ordered pair representetion) are as given in the following table. .Set of possible Denotations Denotations These size computatiops have relevance to a possible computer implementation. For a typicai large-scale computer, the number of addressable memory locations is on the order of 224 (about 16 million), yet 9 of the 16 sets of possible denotations have more inembers than this. This means that a full model cannot be explicitly represented in a computes. However) in evaluating expressions only some of the possible denotations are actually used. I~nplernentation on a cpmputes might allow a partial specification of a model, with only those possible denotations that are actually denoted by the expressiohs considered. In this context it is also interesting to note that the number of neurons in thelhuman brain is currently estipated to be about 2&quot; (10 billion).</Paragraph> <Paragraph position="47"> We have explicitly constructed an intensional model to illustrate the basic notions of the model-theoretic semantics of PTQ. We showed how the meaning postulates constrain the model to be reasonable and evaluated some simple formulas.</Paragraph> <Paragraph position="48"> The number of possible denotations in a small model was shown to present a problem to be solved if explicit representations of models are to be used. This problem is discussed further in Friedman, Moran, an& Warren 119781.</Paragraph> <Paragraph position="49"> nepart rnent of Gomp uter and ~uprxi $2, l~;ric.~: 1 ~-)btainnd f ~grn En glis h senC,-.nr;~3~'~ r ~~v?iLr~n+:cd in an intensiona'l moiel to lot arninn t'~~\i:: truth value.-,. In this paper we j,y7y:r- 11 ,- 3 n i~t eracfiv~ f~ornpu tcr systen for specic yi:;; Eini tt? int siqn31 mudel an 3 nvirbixt in J 177ic.11 f crrn 1~s. We first qive an .>v~rvii?w of &quot;.P svsttxn an13 its intended uses. we ?%en 7rwii.r- s Ietailcd description of the system k.3 o'lw what is w?ail~d in -1ctu3lly carrying ont t5Fq 3pprr3~h. ?xD{?nsive exam~l?s 3rt7 given in the</Paragraph> <Paragraph position="51"> qcran. is rnsponsihle FOC th~ de~ign and coding of the program.</Paragraph> <Paragraph position="52"> This rcsedrch is sunnorte I in part by N3tiona1 Science roundation</Paragraph> <Paragraph position="54"> Zodel- th~aar~t ic seniant ics is on~ irn~a~tant approach to analyzillq the rnoar:i~;q of zxpressio~~s uf 11dtura1- Ianguaqe.</Paragraph> <Paragraph position="55"> Irarski% .vdLuati~n ~t.th0.1 tor f crrnula~ af ma ttlematieal logic can be ~xt,eridr\i to ilatur'aJ-1nnjuaqt. i.ittl~.r direcrly or by an intcr!nrLiinta staur of 1~1ti1, into a mhthtmaticaL logic.</Paragraph> <Paragraph position="56"> This formnl trrlt h*cot;Ji tional tr~d tmant proqides a precise neans of assiq ninu a?anltitqs to n3tura1-Ianquage sentences. I modelthcoretic semantics, the rnctani~q of a Ecrnula is defined by a recursive process of evaluat ic~ ir. a model.</Paragraph> <Paragraph position="57"> ;he formal system presente:I by Fichafd M.ontagile in --I The ~rgy'z tr~ntgnwnt ,f quar.t.ificatiqc ..------I---- z&&arv fnqlish (PTQ) applies mo3~1- tb-eorat ic sewant ics to Fnqlish and forms the basis of the work described in tbis paper, In IT2 tht meaning of an Snglish phrase is obtained by intar ~rt;tj,o.'~ a correspondinq 1agica1 forinuld in appropriate mcidel. FTQ uses a tensed intensional loqic: the fragment of Lnylish emphasizes quantifiers and intensioca--1 constructsa PTQ raisss a cumber of Interesting questions if one actuallv attenpts to use tht? i3~as in a computer systemr For example, there dr2 va~icus ways ir, which a model might be represented, There are also r,r-oblems in Eirdinq algorithms for the yrocesses suggested bv Pya. We have Fade sn.3cifi.c choicestin our solutiOns to these questians acd *re pursui'ng them to find their limits,</Paragraph> <Paragraph position="59"> info~mation ;1rlcl then completes the interpretation, The systern also clsntains facilities t~) a110w 1: use;' ta ~~tld tb ar delete from the model, The syst~m i.s it tor ust in linquistic rzsearch and for exploci~~q pat6ntial apflic3tions af :;onta~.tie ar3mmar. Xontague qrds111ar j)rt?.s~v!ts FORI*~ difticulties bacause of the complaxity of the loqic and its sem~tntics. ;he loi~ic of PTQ is a typcd l=imbd;3 C~~CU~US with irrter,si~31.~dl typesu Inten~retation is based on possible- ori ids semantics. The aeaniny of a formula Hepends on th;3 actaal world in which it is avaluated, althauqh the actual world docs GOT; appsar explicitly in the Eormula, Informally, we miqht cocsider the hctual world as a hiddea free variabl? in the fcrmula, Scmct accus tomell la vs, in particular subs.titutivit y of equ-1.1s anfi univer-sal instarltiation, fail to ,hold. Fesults dre thus often counter-i~ltuitive. 3ur computer a implementation of inEUGrprvfa+ion has 6Gen valuable to us xn our own res2arch and as art 3 in verifyirg cr disproving conjectures that we havi; four:3 in thz literature.</Paragraph> <Paragraph position="60"> Nat ural-1a11guqe quest-ion answer ins systems trapslate an Epqlish sentence to intern'al repr~s~ntation and respond after processing h reprssentation against stored data, PTQ is a formal framework in ~hich an Eny lish sentence has a representation as a loqical formula, and forinulas have valuss defined hsing a lagical modsl, This formal framework provides an approach $0 the nat ura1-languaqe qrlestf 3n-answering pr~~larn, if computer representations and alyerIthms can be tound for its implements tion, Our implementation is of interest in this cant-axt; we plah to explore its applications further. (For another way of lookinq at ::ontaque8s fbram'e,work cosputationally, set3 HoPbs and Rascnsehei.11 (1 Q77) ,) This docup~r, dvscrihes the nodel;-builinq and interpretation system and how to use it. The underlying intensional loyic is presented in Secti~n 2;. The op~ration of the prograp is qiven in Sect ions ILZ 2nd ZV. Sect ion V exp3 ains tne csmmands used :in interacting with th~ system.</Paragraph> <Paragraph position="61"> Zxamplas arc qi~en in the text and three sample .tuns are provided in the A ppenrlices. 111 dl1 axamples lines that are entered by the user dre lower-cgse left- justified and prefixed by , whereas output lines are uoper-case, prefixed by > and indented, .c nr. 9 introductory presenta tion of the modeltheocetic senlantics OX PT(' (Friedman, Moran, and War'ren, this fiche), a simpla lrodel in this notation is de.veloped and use.1: Appendix P shows the use of the coinputer program to develop the model. and interpret formu-las in it., h user may want a modgl which satisfies a particular set or^ formulas; Appendix B shows the development of such a model duricg the interpretation of those formulas. In Appendix C, we develop a modal as a countere~ample to a formula criven as a thoorom in I'TQ (p. 265). A programmers description of the interpretd tion systi.. is providei in P@r-an (N-J farth~ominq) , ReatSera who n:iyht wish to adapt this pcsyrda to typad lambd,r calculi with other types ace raferrird to ':oran (N- 7, fort b.coming) I'll@ syst+m P:uil;ls mcdfls anfiint-erpr~ ts f o~mulas ir, them, Before b7i~cussLnc; the pdrticukazs ~f our proqram, go Dreser,t here th? ur,ier?ying for921 tbeorv. ,he? t~rsid intensional logic in the interprstati.on syste~ is that of :ontague (1 37: 1573). *. fi e haw faun? 1 (1975) hi~lpfuL as 3 saurct. for 3 formal trea+,me;:+ of i:lt~r:;ion.i1 l~qic, but have modifled his natation to meet dur needs,</Paragraph> <Paragraph position="63"> ,h~ ~ygg~ of intac~i~~al lcaic are defined as follows: (1) F an.i'&quot;'I!~ sra elercart3rv types, (2) rhc'nsv~r a and b are types, <a,b> is ty?e, an3 (3) wherever a is a tvp?, <S,a> is a type.</Paragraph> <Paragraph position="64"> For each ty+ thcr2 is a set of constants a a set. of variables.</Paragraph> <Paragraph position="65"> The follawin~, ar 1 mly tk~ follcwing, arm geazinaful ---I --. expressions ----- null (3E1s) : 1) I5vt.r~ emstant of type a is a 2P of type -. a.</Paragraph> <Paragraph position="66"> 2) ,very v?zi;~hlc of type g is a 32' of type 2.</Paragraph> <Paragraph position="67"> 3) If A is a 3,i of type - a ir~d V is a variabl~ of type (LAa-ADA V,A) is a HE of type <b,a>.</Paragraph> <Paragraph position="68"> b ~f A is a MF of type <b,c> and E? is a YE of type -, then (A B) is a MF: of type s.</Paragraph> <Paragraph position="69"> If A is a of eype <c,<b,d>> and 8 and C are ME'S of types $ and ,I c then (A B C) is a ME of type - d. (This 3xpression and ((A C) R) will lave the sank</Paragraph> <Paragraph position="71"> if ? is a of type TS and V is a variable, then reference, respectively. A fram &sed 2; 5 is an indexed family of danotatlbn-sets, (D-a) a E typx?, uhere ti) DE=A [ii) D - TS = {3,1) (iii) D-<a, b> is a-subset of 3 a bada I (iv) D-<S,a> is a subset of D - a</Paragraph> <Paragraph position="73"> For the elementary type d the set of pOss~a&e denotations is the set of entities. For the ~lementary type IS the denotation-set has the two elements, (false) ar.4 1 (true) . For a complex ty?e <a,b>, the dtnotation-set D - <a,b> is a set of total f~ndtions</Paragraph> <Paragraph position="75"> the special. C~SG D - a - is handled by the qen~ral rula.</Paragraph> <Paragraph position="76"> A ----- rimdel I.. 3 war on - 4 --- and rrrr- ixJ is a pdir C (2 - a) a saps F> where ,, I 1) (D-3) BE;~&~s is a Frame based an A apd IxJ, where I is a set of p~ssihle wcsl~!s and J is an ordered set of nomznts in time.</Paragraph> <Paragraph position="77"> 2) F is the hsaninq function which assigns to ezcn LOQIC~L constant ~f type 2 an elzment of the denotation-set</Paragraph> <Paragraph position="79"> Evaluation is a rcla tionship hatueen. meaningPS u1 sxpresslons and models. We A%fi~!e recursivaly the value or i--rr-r-rr- denotation dTB:Y,i,j,j] of a meaningful expressior! B with respect to a model 3 and .i, j, and g, where is 3 mole1 on A and IxJ, i EI, j EJ, and, g is a= variable assignment, q assiqns to edcn variable of type 2 an elem~nt of P - a.</Paragraph> <Paragraph position="80"> 1) If E is a constant, then d[B:H,i,j,q] is F(B)(<i,j>), 2) If B is a v'ariable, then d[9;?l,ir j,g1 is g (B) .</Paragraph> <Paragraph position="81"> 3) If H EU?:I - a and V is a variable of type b, then d[ (LABBD.4 v B) , j g] is thaf fulr~tion h with domain D-b such that whenever x is i h that donlain, h Qx) is d[~;~,i, j,qt Ir where qt is a variable assignment like CJ except EM the possible differance that gt (V) is x, 4) ~f ncYE - <arb> and C ~b!r-a, tben (I[ (B C);M,i,j,j] is the valus of the function Bit j, for the arqument d[ C ; :I, i ,.j .J 1.</Paragraph> <Paragraph position="82"> 5) ~f E E~~-<dj<crb3>, c E 32- c and DEMX-d, then d[(B C 9) it, j, is d[ {(I? D) C) ;J.i, j,g].</Paragraph> <Paragraph position="83"> 5) If R, C E II?-a, d[ (EQUPL B ) Hi, j g is 7 if and only if d[s;a,i,j,gJ is d[~;~j,i,j,g].</Paragraph> <Paragraph position="84"> 7) If P, C) EME-~S~ then d[(NOP P),,, is 1 if and only if d[I';:~:,i,j,g] is Similarly for AND, OR, IMPLIES, IFF, If P EYE - IS, then d[ (NTCF:SADILY P) ;M,i,j,g ] is 1 if and only'if d[P;Y,il,j',q] is ? for all I'PE I and jt& J. ~[(~UTUFS P),j is 1 if and only if d[P:M,i,jq,q] is '4 for some j' such that j d[(PAST P);i.l,i,j,g] is 1 if and only if d[P;E?,i,j!,g] is 1 for some j1 such that j1 < j, 9) If PC Mg - TS and V is a variable- of type a, then a[ (THEFL-IS V P) ;?!,i,j,q] is 1 if and only if there exists Y ED .I a such that ,j, is 1, where gq is as in 3. Similarly for .d[ TPOP-ALL V PI:E,i,j,q]. In a genegal model (4-xroddl) , the denotation-sets of! the frame are less ccnstrainel. As (350~~, 5 - 5 = k a,n.d I) - TS = {(?,I). However, for .3 complex tY FG <a&> we have</Paragraph> <Paragraph position="86"> denotations for all taxpressions be in the model. This means in effect that thz functions h which are the denotathns of LASBCAexpressions and IS IT-~~xpressicr,~ IRUS t $xist in the appropxiate denotatior!-sets, The definition of evaiuatior, in a 3-model is exactly as in a sta~darrl luu3sl. Not+, howwer, that LAMEDAabstraction and auantification arc restricted to the denotation-sets of the model,</Paragraph> </Section> <Section position="2" start_page="7" end_page="7" type="sub_section"> <SectionTitle> Named Modeis </SectionTitle> <Paragraph position="0"> ----- ------The models used in the computer system do not correspond exactly to the standard or qeneral models of formal logic. We therefore introduc2 a r,cw formal defihition of 'named models', This dezinition allows a model that map be smaller than a ymod@l, but expands toward a q-model when new functions are A named mo;iel (n-model) is based on a frame which need not be closed undzr valuation, The valuation function d is a partial function, undefin~d where the valuation function as defined above take$ a value not in the model. 3 .Ir <a ,b> D I bD--a art& evecy element has a unique name. A named model need not be a g-model; howkver, a lcoverinq q-model' must exist, The coverinq g- model is the closure of the nMmodel unbr the condition that the f unckions requir-d for evaluation exist. LAM BDA-abstract ion and quantification in an n model art. restricted to the named elements in the denotatlon-sets,</Paragraph> </Section> </Section> <Section position="7" start_page="7" end_page="7" type="metho"> <SectionTitle> Dyn8mi.c Barned Yo3els </SectionTitle> <Paragraph position="0"> - ---- --- -1---The models use3 in our interpretation system arc dynamic nmodels r They are not closed under the valuation function; however, whenever a denotation of type <a,b> is created by evaluation of an expression with LAMBCA or IIT, the dyrqmic n-model is imnediately expanded. The function is named, added to the appropriate denotation-set, and all functions with zhis denotation-set as domain re expanded to include values for the n4w arqument, LAKBDA-aQstcaction and quantification in a dynamic n-model are r3strictet.l to the current named elements in the dsnata.t:ion- se t s. chis has tht? consequence that they cannot be precisely defined inde~endentlv of the order of evaluation of a formula, A dynamic rr-anole1 may, b~ thought or either as one expanding n-model or as 3 sequence of n-mc.;l.~ls. i:t is not yet clear which view wiLl he most productive.</Paragraph> <Section position="1" start_page="7" end_page="7" type="sub_section"> <SectionTitle> Finite Kodels </SectionTitle> <Paragraph position="0"> ------ -----I While the ictensic~al loqic has in6iditely many types, only a finite number (xcur ir. ex~r~ssions of PTQ and hhus only a finite number of r:-.ts ~f possible llznotaticns arc neel2d. In the current versiori ot *.ha interpretation system the sets A and IxJ, and h~nce all d~~a.otaticn-sets~ are finite, In the finite case the nntions of standard n~odul and g-model are -the same, under somt weak conditions. we have introduced the distinction becquse we think of our dynamic models as 'potentially icfinite' , and fi~d the not,&or. of c:overins d a-*mcd,el, - sugyestive. One might think of the g-model as a representation of 'reality1 and an n-model as a finite r~~presentation of the svst~rn's knowledge, A dynamic n-model is a representation +-hat is forced tc expand toward reality as the system needs f uifcher knowle~lye to carry out its- tasks., We plan alsewhere to ci>nsid%r t h~seq rnodsls further, both formally and as possible psychological models.</Paragraph> <Paragraph position="1"> Models &g ug astern C, ------This system accepts only finite models. A finite model can reveal the principles ir~volved in intcxprtj~dtion of formulas, and most ~f the interesting problems it1 the eva3uation of formulas can arise there. fu usinq th~ system in resea~ch, we have encountered no problems related to size, However, even a small stgndard or nil model is too large to b@ practical, The closure property of these models causes the inclusion of elements that may not he needed, Dynamic named models are practical.</Paragraph> <Paragraph position="2"> They need include only those elements whose use is ant; cipated, and elements can be added to the model to meet neu or unanticipated uses, The functions in standard a~d general models are total, The functions in named niodels are also total, although they may he incompletely (partially) apecif ied, that is, the value of the function may be giver, for only some of the arguments. The unspecified values fqr a function are regagded as determined, but unknown to the system. If the interpretation of an expression causes a fuxction to 6b ap~li~d to ar. argument for which its value is unspecified, the interpratatio~ is suspendeti and the user is prompted for the needed value. This approach using partially-specif ied total f u nctions and dynamic mode.1~ contrasts with a~proaches using partial functions in static models (Kutschera, 1975) .</Paragraph> <Paragraph position="3"> .In a nnnied ml, a D - set is a subset' of the set of all possible fupctions of its type, with s dynaiuic model, the user can enter n function of type <a,h> that. has a valua not in I! I b; this caris+s the elament to be a3tl'~d to D - b. Similarly, the a\ldition of at: ?l~nt.nt to D-a causes the expansior, of the specifications of a11 current functicnS ct type. <a,b?.</Paragraph> <Paragraph position="4"> In d Iynamic model, it may be reasonable to have two functions with thz sane specific3tian; 2n as yet uqspecifled value may be different for the two functions, or they may have rliffer~~lt valutacs for some aI11-clument not ye+ in th~ model, ?h~ dt3110tatior: of a LAMEDA-expr~ssion or ar, IKT-expression is a function an3 if t'ndt functi~n is not an element of the mod~l it is added. If an element is added to D-a during the interpretation of a LAFIRD3-expression of type <a&> or an argument to which it is applied, the body of the LAEBDAexpressior! is in terprgted for this new - alement and the specification of the denotat lor, m- the ZA2:ECA-expression is expanded to, includz this argumenb and its computed value.</Paragraph> <Paragraph position="5"> This ability to expand t-he modal dynani,cally during the interprstation of formulas qives tile US-?'~ a second means of constructinq picdels: starti~g with a minimar outline of the model, the user interpro ts expressions describing the desired model. when pro~npted for unspecified values, the user can respond with elltries that will make the expression true.</Paragraph> <Paragraph position="6"> The model can also be expanded under' the direct control of the ust;r, Elenlctnts can be addlad to the model or pnspecified values of functions can he ent:~re\L</Paragraph> </Section> </Section> <Section position="8" start_page="7" end_page="7" type="metho"> <SectionTitle> I?iPLE?!FNTATIQN OF NAMEC EODLLS </SectionTitle> <Paragraph position="0"> III. ------------ -- ----y ------Names for ~YES~ gid gsgss Am -,a,, e- null Semantic types play a significdnt role in the interactions batweer the user and the system, but the names of complex types as constructed in PTQ are cumhersou\a. Fcr exantple, words in the syntact.$c category TR'Vr/ZE (e.g. lint and @aboutt) ar2 translated into logical constants of type:</Paragraph> <Paragraph position="2"> To facilitate interaction? the system uses new type names that are mean iny ful, simple, and easily distinguished from each other.</Paragraph> <Paragraph position="3"> The conventiou is to use the names of the syntactic categofries as names for the corresponding semantic typez, i.e., for syntactic category x, the name of the correspondinq type, f (x) , will also be x. This can be done here without confusion becaose the syntactic categcrias do not play any pa~t in this svstem. Where PTQ uses special symbols, e.g., iV, IV, for the cogpound syntactic catqories, 5/@, - ,V/TZ, we also use these special symbols, Flpwaver, there are types that ja not corcesponil to syntactic cat~.qories, e.rl., the types fat the intensions of the typed that corraspond to syntactic catoqories; the name ue use for one of those is the corflhi~~ation of the names forits component types, u. y., <S,IAV/TE>, It is possiL:le to have several names for the same type, for example, E(C%) = E(:V). In such a case, anothsr (neutral) name is ~eederl - tile conv?rttion is that this name is formed by combinimq the types using an equal sign as a slzparator (2. q., Sn'=Z'V), Type narn?s such as CY and IV will be referre? to as names when it is necessary to distinguish them from the SU~~YE~ --I-- --oaer type 11ame5 ( CN-3V OX: 2) Ihe user can usually refx to a t-vpe lrsinq &tiler a subtype or type name (in the case of EU(CN] = Y(XV), hy Ch, IV CX CN=IV), whichever is most natural, When the uscr is prompted for a type, the preferred resnonse is one of these new type nan;es. However, the user may enter any equivaknt form, For exanple, IAV=?V/IV, <<S,IV>,IV>, ant! <<S,<<S,Tb,TS>>,<<S,I>,TS>> a,l1 name the same type, Since a type and its suhtyues ah1 have the same comFonents, these e'quivalent forms refer to the type, Names are also assigned to th2 sets ojf possible denotations of each tv~5. 1h~ riame for a sst of possible denotations of a type is formed by concatenatinq D and the name for that type.</Paragraph> <Paragraph position="4"> The Cull type name must be used hem, not a subtype name, That is, the nams D - Cti is not recoqniz~d by the syqtem; it expects</Paragraph> <Paragraph position="6"> PU, g Logical Constants and Variable g&;~~fjggg me -c --I- YIIICIIIW..I.(t- 3*-- ------w The system contains the tensed intensional logic used in PTY, Th~rr is also an extensional version without intensionality, modality, or tense; we do not discuss it in this paper, TWutypes arlsl those occurring in formulas that are translations of English sentences or: are meaning postulates. The types and s~~btypes for intensicnal models are: The logical coEstants are formed by capitalizing the words of which they are the cranslaticns. For: examrle, the WOE,\ N~alktg translates to the constant dtWBLK't, li'trt? logical constants are; TY'PE F*: J, I:, E3& 3.</Paragraph> <Paragraph position="7"> TYPE CY=TV: SIIRTYPP: CC: XAN, WO::Ah', PACK, IS, PFN, UNI:CC3h-, PEICE, TSM Fi FATrlFE SUBTYPE ZV: 3, WALK, TALK, EISE, CHAKGE;, TYPE IAV=IV/XV: SUI3i=YP,E 1AV: EXPTCLY, SLCtWLY, VOLUKTAFILY, ALLEGEDLY, SUBTYFE IVfIV: ?FY0TOp UISH-TC TYPE 2V: FIflD, LXE, SAT, LCVE, EAT,&quot;, SEEK, CONCSIVE. TYPZ ZAV/TE: Ih, ?.EC3T, TY PE ~V/IS: ~~L;LVETT~~AT, ASSLS?&IEiA T.</Paragraph> <Paragraph position="8"> 79 permit an unr+stricttd n~tffber of variables, certain letters are dasianat~d as variabls prefixes, and a variable of type I* a is a variable prafix of type - a follc~ed by 2x6 or more digits. The variable pr+fix ,, TYPE. ?: U, V, TYFL <S,Z>: X, Y.</Paragraph> <Paragraph position="9"> TYPZ <S,C>J=ZTJ><: F, 0, TYPE <S,<Z,iS>>: E.</Paragraph> <Paragraph position="10"> P &quot;PE <3,2>: 3.</Paragraph> <Paragraph position="11"> TYPE <S,<Z,<Z,7S>>>: S, 'TYPE <S,<d,<<T,TS>,<F,TS>>>>: G.</Paragraph> <Paragraph position="12"> TXP? <Z,TS>; KO Changqs to thz built-in logical constacts and variable prefixes can be acco~plishei wLth the ccmnands in Section V. The loqic is bullt into the system with a saries of declarations yivinq the types, logical constants, and variable prefix~s, and an ord~ring of the types. qhis ordering, which need not contain all the types, gives the sequence in which the system will prompt the user to enter the sets of possible aeaotatiahs, The built-in loqic can easily he modifies or raplac6d bf chnnginq th~sta decld-saiions (Ncrrac, N-7, Eo~t hcominy) Sets of ~osskble Denotaticns of Corng&?i? 212s C* -- .I- -I-- L- ----..I)-----A set of poss'ible denotations cf a complex typa <a,h> is a set of functions of type <a,h>* EacK' function is entered by givinq its name and specification. eve^ thouqh functio~s are understwd to be total, functions can be entered with partial specifications. For each set of functions, the proqram prints: 1) the name ot the set P_<;l,b>, 2) the ndse of the set D-a that is the domaic, and the names of alewnts in the domain, 3) tne name of the set D - f: that is the range, and the names of elements iri the range. These are the names to be used as values in entering the fu~ction, 4) tha logical constants whose possible den'bt ations or denotatio~s of their intensicns are in this set. Follbving t-his prelimina~y information, the program asks for the name for the first element :o be entered. This name can be chose& alm~st arbi trarily. There are a few illegal names, but they are refused by the system. Tile following cannot bc used: the name sf ~cdther clerner3 of sng set; the nae of any of the rn*rv~ model conrponcar.ts (2.q. 'T&quot;NL~II~;S'~ } ; tha nant? for a type, a set ~f: pOssiI11v d~notarions, cr a uoint of referaxe; the nalne of nuaEet OF a a comnan:l; a n.3rnr in with th.2 character ='* system atom (T, E;X) .</Paragraph> <Paragraph position="13"> After the Ilant9 for an elemqnt is +r.t.crt?d, the prlograax leals the user throuylik its specifi-cation, and then asks for the name of the next element. indicate the end of a set of el~ments~ tne user responds ritn XIL* *hen the specification of a set is tzrminated, tne pro~ram displays its Elements and then requests specificltion ci +[LC next set,</Paragraph> <Paragraph position="15"> ,qfter the Iiafi~ fcr a function has been Pr;r;er9d, tk.0 proqram guid~s ths uzer throiijii its .cp~cifacatior.. For sach element in the ~Q~RS~R, the proqram prints its raa% ar,d its specification and the user rcs~owds with the raac of tJln corCespGr,ding value.</Paragraph> <Paragraph position="16"> ff the function is to be unspeci-fied for. this argument, the user enters NIL, Pny element entered as a value should already be in the model, However, if en element that is not yet in the model is needed, the user may enter the name intended for that element.</Paragraph> <Paragraph position="17"> The program detects this as a potential error and asks the Jser for instructions (see F below) r</Paragraph> <Paragraph position="19"> When the specificat ion of a function is completed, the model is checked oAr an uquival~.nt fU11ctlon. TFI such' a situation, there would be two names. for the Sam+ Olernent and, in functions for which thiq element is an argilment, a valuq will be specifieii for each name, Tf these values are nct the same, the interpretation of an .expressi3n Can depend cc the name used. To prevsnt this, onEU of: the equivalent elements should be deleted or</Paragraph> <Paragraph position="21"> During th~ sp~cificntion of a function, tile value for a particular argument is entered by jiving the name of an element of the 'range, If the uses enters a name which is not in the model, the program 3sks for clarificatian, f f the user- chooses to specify the tzlc>mrnt immmediately, the specification of the function is sus;,pended vhei-le it is eritereu. For exampie:* the spacifiqtion of the ftlnction continues uninterrupted, When the entry of functions into a set is finiShed, the user is prompted to enter the sp~cifications of the 3eferred rle~ents.</Paragraph> <Paragraph position="23"> If in entering a model or adding to one, the us& enters elements ir&o a set which is the domain of some previously specified tunction, the specif lcatzon of that function will need to be expanded to include the r.ew elecnrnbsb The system will suspend its curr~nt projtzt and prompt for this# swpansion.</Paragraph> <Paragraph position="25"> . as --a In a model, the meaning function F assigns to each logical constant a function that is the denotatioir of the intension of that constant. If th? constant is of type ,I a the value of F is of type <S,a>. After entering the .sot D rn <S,a>, the user is prompted to enter the value of F for each of the constants of Yo check is malie that a constant satisfies any meaning postulates, since it car, be interestin7 to investigate both models that do an'd do not satisfy them, The user car, find out whether a meaning postulate is satisfied by interpreting it in the modd, 'as is don2, for exam~le, in Appendix C.</Paragraph> </Section> class="xml-element"></Paper>