A definite clause grammatical inversion of extended Montague semantics
R. I. Bainbridge · White Rose eTheses Online (University of Leeds, The University of Sheffield, University of York) · 1987
If in addition Q=Bthen h is a homomorphism from ye r to , E r Given a homomorphic meaning assignment, we may conclude:(U21) h(ý) =the meaning of ý' in L under the interpretation , e r iff C' e MEL A 3C(Ce L) E ACSACR ).Although there is as yet no introduction of set theoretic entities to serve as meanings, the rule by rule hypothesis, ie. the convention that to every syntactic rule constructing an from there should correspond a semantic rule for constructing the meaning of an from the meanings of , is clearly entailed by the commitment to homomorphic assignment, thus: f 86.That is to say, the operations share a common number of argument places.-191-"The rule by rule hypothesis must be included as an essential Montagovian characteristic.Passing reference has already been made to a problem with Montague's formulations.Montague insists that for all sequences in the domain of Fy and all sequences in the domain of Fy., if F I( ) -FY .() then y-y' and -.This use of Occam's razor eliminates unnecessary duplications, but fails to confront the question: what m-tuples may legitimately occur as arguments of an m placed operation F Clause U9 tells us only that according to a given rule members of categories are permitted, not that FY is restricted to the domain CSk x ... X CS m;thus subsequent rules in S may permit m tuples from alternative domains.As we have seen, there is in PTQ indeed a many: 1 relationship between syntax rules and functional operations.Each of S14, S15, and S16 invokes f10, the domain of which is variously PT X Pt' PT X PCN or P. I, X PIV: but the price paid for this prevarication is that the correspondence between f10 and its translation schemas is not 1: 1.If interpretation is to be direct, U20 specifies a 1: 1 correlation between syntactic and semantic rules, whilst it will transpire that U47, makes similar demands in the case of indirect interpretation.Arguably therefore Montague's specification in UG should be augmented by:(U22) If Fý l ) and F '() are both well formed then either Sk = S. or y *y'.r1mY J1 1m n in Constraining the structural operations to specific domains in order to protect U20 would effectively enforce a 1: 1 correspondence between structural operations and syntax rules.Some authors have adopted the convention of employing a common index for syntax rules and operations "for ease of reference" (sic Janssen).It now transpires that such uniformity should actually be mandatory if UG principles are to be inviolate. Model Theoretic SemanticsAn intensional model was defined in $2.1.4as M- .The orthodox interpretation function I is now defunct, but the remaining apparatus survives as:Model base M- .Montague's definition of the set Type of types is exactly as presented in §2.1.1,and the set den(a, Al) of for all members of _' , &XS.In fact the generalisation is only apparent because constants turn out to be immune to changes in context.For any lexical item ßk, T E XT we have j(ßk, r) e den(T.M).A logically possible model for IL then becomes: , e r is a Fregean interpretation and:(U36) if v c-Vara then fl u)(w, 1, J) -j(v).(U37) If ae Cona thenj(a)(w, Q) -J(a)(w, t, j') for allj' that are a variant to j.(U38)GO(a4ß)(w, týJ) -a(w, t, j)[ß(w, t, j)]" (U39)G1(a, ß)(w, t, J) -1 iff a(w, Q) -ß(w, t, j).(U40) G2(a)(w, t, J) -11 e den(t, M WxT such that, for all w', t' e WxT, TI(w', t') -a(w', t', J).(U41) G3(a)(w, 11J) -a(w, t, J)(w, t).(U42) G (a, ß)(w, Q) -Tj E den(T, t)den(6' M such that, for all ýE den(a, M), ij(ý) -ß(w, t, j) for allj' which differ from j in at most that j'(a) -C. Concessionary TranslationAs a concession to the faint hearted Montague suggests that, although direct construction of a (Fregean) model for a (fragment of) natural language is possible, and was indeed attempted in EFL it may be "somewhat more perspicuous to translate the fragment into another language for which an interpretation is already available."This strategy he proceeds to justify.Let L-«, XS, S, 80>7e I'Se AR> and L'-«, XS, S', SO>W i", Be A', R>.Then a translation base TB - , i, E I .from which we may conclude that:(U51) If tI e vSE 0CS and i' e US.E CS, and 1IRý and 1'Rý' and k(i) = 11' then ý' is the translation of C on the basis of TB."M-PARSER applies the analytical M-rules to the S-tree ... in a top to bottom fashion.... Successful application of a rule S', results in a tuple .M-PARSER is then applied to ul, ..., un.Each application of M-PARSER to u1 gives a (possibly empty) set of D-trees for uff.For each tuple of D-trees in the Cartesian products of these sets a D-tree y is constructed." [LI] Part of Landsbergen's earlier paper is devoted to proving that te M-GENERATOR(d) 44 de M-PARSER(t).Assuming this to be the case, let S-PARSER(s) -the S-tree assigned by Ga to a sentence s.LOGTREE(d) -the LD-tree corresponding to D-tree d.LOGTREE'(e) -the D-tree corresponding to LD-tree e. Then Landsbergen is able to define the function: ANALYSIS(s) -de {e: 3t3d(te S-PARSER(s)nde M-PARSER(t)Aee LOGTREE(d)} which maps sentences to LD-trees, and a reverse function: GENERATION(e) -def {s: 3t3d(de LOGTREE'(e)AtE M-GENERATOR(d)nse LEAVES(t)}.mapping LD-trees to surface phrases.Two M-grammars Gi and Gi are described by Landsbergen as logically isomorphic iff: Ve(3s(se GENERATIONi.(e))H3s'(s'e GENERATION{e)))ie. iff for each ID-tree assigned to a sentence s by G1 there is a sentence s' to which Gassigns the same LD-tree.According to Landsbergen's thesis, two languages are logically isomorphic if they may be described by logically isomorphic M-grammars; moreover inter translation between logically isomorphic languages using the common LD-trees as interlingua becomes feasible.Rules such as Montague's S3, which in PTQ contain meta variables, are accommodated in Mgrammar by recourse to rule schemes.A rule having a parameter p, ie.one of form -211-S1-«C1 gyp, Alp>, , Sr> may be defined in terms of a schema: where Pi is a parameter set, Ili ) g Pi., and Ai(p, ) -ur.Landsbergen requires that Cip( ) =defp e Ii( ) and Al, p( )'defA1(P, ).