smorynski.1973.investigationsofintuitionisticformalsystemsbymeansofkripkeframes

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INVESTIGATIONS OP INTUITIONISTIC FORMAL SYSTEMSBY MEANS OP KRIPKE MODELS

BYCRAIG ALAN SMORYNSKI

B. S., University of Illinois, 1969

THESIS

Submitted in partial fulfillment of the requirementsfor the degree of Doctor of Philosophy in Mathematicsin the Graduate College of theUniversity of Illinois, 1973

Chicago, Illinois


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UNIVERSITY OF ILLINOIS AT CHICAGO CIRCLETHE GRADUATE C O L L E G E

1 AUGUST 1975

I HEREBY RECOMMEND THAT THE THESIS PREPARED UNDER MYSUPERVISION rv CRAIG ALAN SMORYNSKI

INVESTIGATIONS OP INTUITIONISTIC FORMAL SYSTEMSENTITLED.BY MEANS OF KRIPKE MODELS

BE ACCEPTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FORTHE DEGREE r>F DOCTOR OF PHILOSOPHY

pJlj-yC c jyU.A/iyIn Charge of Thesis

Head of Department

Recommendation concurred inf

tRequired for doctor's degree but n o t for master's.

Committeeon

Final Examination

1)517


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iii

PREFACE

The advantage of a preface is that, in it, one can saythings that would be out of place in the text or even in theIntroduction. But such things may be questionable even in apreface and, besides, it is getting to be overly fashionable totell anecdotes about how one discovered such and such result.Suffice it to say that we obtained all sorts of fine results onthe Kripke models over the last four years and are reporting themhere, if not for the first time, at least for the first time intheir entirety.

Most of the actual research was done during our firsttwo years, near the end of which we were informed that we hadenough material for a thesis. Even for one who likes to write,the thought of taking a half-dozen or so short papers and a lotof scratch paper and organizing the mess into a coherent wholeis rather upsetting. We decided that it would be easiest to writethe third Chapter first (whence the reader should considerhimself warned that the references in the third Chapter to theearlier Chapters are not very specific). That was finished beforeleaving Stanford.

When we left Stanford, we took a six month vacation fromschool and a year's vacation from our thesis. Most of the creditfor our ever finishing it must go to A.S. Troelstra, whoindicated that, at the speed we were writing, our first publicationwould be in 1984. At Troelstra's suggestion, we published the


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i vthird Chapter (slightly modified) and wrote a long version of thefourth.Chapter for his forthcoming "book, MetamathematicalInvestigation of Intuitlonistic Arithmetic and Analysis.(It willhe the fifth Chapter of the "book.) Writing the fourth Chapter ofour thesis was then a simple task. We waited, however, until wehad written the first two Chapters so that the references to theearlier Chapters could be quite specific. Because this Chapterwas originally written for Troelstra's book, in which Troelstrawrote four comprehensive Chapters on Heyting's Arithmetic, wefelt that those Chapters gave sufficient background and, thus, i)we did not make Chapter 4 as self-contained as the other Chapters,and ii) we gave fewer references to the literature than is ourcustom. (Two specific references we ought to have made concernLuckhardt's Theorem (Lemma 2 of Section II A). This result canbe found on page 150 of his Extensional Godel FunctionalInterpretation (Springer, Berlin, 1973) and is a strengthening ofKreisel's original result that

J-/\x(AviA)o-n VxA - Ax(AV>A)o V/xAis not a derived rule for HA (page 122 of "Survey of proof theory,II" in J.E.Penstad, ed., Proceedings of the Second ScandinavianLogic Symposium (North-Holland, Amsterdam, 1971)).)

As things turned out, only Chapter 1 was difficult towrite and we finished it by typing it up as we wrote so that wewould be very unlikely to throw everything away and start over.This may make the exposition a little rough at spots but it gotthe job done.

As with most research, we received a great deal of help


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Vfrom a great many people. Those who have contributed more or lessdirectly include S. Feferman, D. Gabbay, W.A. Howard,D.H.J, de Jongh, G. Kreisel, G. Mints, and R. Statman. We shouldalso thank M. Pitting for introducing us to the fascinating(Kreisel says "addictive") study of Intermediate Logics andH. Ono for keeping us informed on recent developments. We mustalso thank J. Myhill for solving a problem of ours and apologizeto him for our ignorance (the problem having been solved a decadeago).

Our greatest thanks must go to our teachers, S. Feferman,W.A. Howard, W.W. Tait, and (especially) G. Kreisel, who havebeen most influential in shaping our attitudes towardmathematical logic.


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vi

CONTENTS

PREFACE iiiTABLE OF CONTENTS viINTRODUCTION 1Chapter 1. The Propositional Calculus 15

I. INTRODUCTION TO THE KRIPKE MODELS 15A. Introduction 15B. The Completeness Theorem 20C. Immediate Applications of the Completeness

Theorem 26II. SOME MODEL THEORY 27

A. Trees .27B. The Extension Theorem 32C. Minimization 36D. Finite Minimal Models .38E. Strong Homomorphisms . ^0

III. THE GEOMETRY OF INTERMEDIATE LOGICS 45A. Examples 4 - 5B. Geometry 52C. The Finite Model Property 5D. LC and Extensions. 59E. Height 63F. Slices 67G. Width 69H. Local Width 7^


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vii.76.79.86.86.86.90.95100103106106108115118121123124126126126127

130133139139,141

1. CounterslicesJ. Some Commente on the Counterslices2. The Predicate CalculusTHE KRIPKE MODELS FOR THE PREDICATE CALCULUS..A. Definition and ExamplesB. Kripke Models as Classical ModelsC. The Completeness TheoremD. The Aczel SlashE. The Operation( ) i> {"& .KRIPKE MODEL THEORYA. TreesB. Complete Sequences RevisitedC. Models with Constant DomainsD. Herbrand TheoremsE. Normal ModelsF. Function SymbolsG. Final Comment.3. Elementary Intuitionistic TheoriesQUANTIFIER ELIMINATIONA. IntroductionB. Proof of Theorem 1C. Applications of Theorem 1: Quantifier

EliminationsD. Applications of the Quantifier EliminationCOINCIDENCE WITH A CLASSICAL THEORY: CRITERIAA. Quantifier EliminationB. A Coincidence Criterion


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viiiC. Another Criterion 142D. Almost Coincidence 146

III. UNDECIDABILITY 149A. Introduction 149B. The Monadic Predicate Calculus 150C. The Theory of Equality 154D. Some Extensions of the Theory of Equality 162E. The Theory of a Successor Function 169P. The Theory of Dense Linear Order 170G. Further Examples 176H. Comments 17@

IV. THE COMPLETENESS PROBLEM l8lA. Discussion 181B. Completeness of Mg in E 183

Chapter 4. Heyting's Arithmetic 186I. THE MODEL-THEOPETIC TREATMENT OP HEYTING'S

ARITHMETIC 186A. Heyting's Arithmetic 186B. The Operation( ) )' 189C. Formulae Preserved Under( ) ?( )' 193D. de Jongh's Theorem 197E. de Jongh's Theorem for One Propositional

Variable 201F. Another Theorem of de Jongh.( Digression) 202

II. THE GODEL INTERPRETATION FORMULAE 205A. Markov's Principle 205B. The Independence of MP 209C. Proof-Theoretic Closure Properties 215


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ixD. The Operation( )> (+ j)' 216E. The Independence-of-Premise Postulate 219F. Mutual Independence of MP and IPq 220G. Final Comments on MP and IPq 222

III. DEFINABILITY CONSIDERATIONS 223A. The Operations( )? ( )* 223B. Definability 225C. i Substitution Instances in de Jongh's

Theorem 230D. Uniform TJ2 Substitutions in de Jongh's

Theorem 236E. de Jongh's Theorem for MP 239F. Further Applications 240

Appendices 244Chapter 1, Section IV 244

Appendix A; Lattices 244Appendix B: Formulae in One Variable...... 259

Chapter 4, Section IV 269Appendix: The Godel-Rosser-Mostowski-Kripke-MyhillTheorem. 269

References 273Vita 282


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1

INTRODUCTION

0. As indicated by the title, the table of contents, andthe size of this paper, the present work gives a fairly completetreatment of the Kripke models and their applications tointuitionistic formal systems.

Basically, a Kripke model is a partially ordered set ofclassical models. Kripke models were introduced to give a modelfor intuitionistic logic here the partial order can he takento represent either varying stages of time or states of knowledge.In the propositional case, "both interpretations are quiteconvincing. For the pure predicate calculus, the Kripke semanticsmay also be fairly convincing. But when one chooses a concretetheory, the models are clearly unrelated to the intuitionisticallyintended interpretation of the given theory. Nonetheless, thereis a strong formal connection between intuitionistic theoriesand their Kripke models amely the Strong Completeness Theorem.This completeness result, combined with the relatives ease inhandling the models, makes the Kripke models an extremely usefultool for the study of intuitionistic formal systems.1. As a glance at the table of contents will show, theprfesent work is divided into four Chapters on the PropositionalCalculus, the Predicate Calculus, Elementary IntuitionisticTheories, and Heyting's Arithmetic. In the first two Chapters, westudy the Kripke models themselves and prove several resultsneeded for later applications. In the first Chapter, we also


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2present applications of the Kripke models to the study ofIntermediate logics. The third and fourth Chapters apply resultsof the first two Chapters to elementary intuitionistic theories(primarily the decision problem) and to Heyting's Arithmetic,respectively.

Let us briefly discuss the contents of the individualChapters.

The first Chapter treats the propositional case. Westart with the Completeness Theorem and its immediate applications.We then present some techniques which have proven themselves tobe of use in studying the propositional calculus. In Section III,we apply these techniques to obtain numerous results (old andnew) on Intermediate logics. We also include two Appendices as afourth Section. The first of these discusses the relationsbetween the Kripke models and lattices. This is included not onlyfor the sake of completeness, but also because most discussionsof the equivalence of the Kripke models with the lattices onlydiscuss one or the other of the two approaches. The secondappendix discusses formulae in one variable and proves some resultsneeded in the fourth Chapter. It also allows one to compare thedifference in geometric restrictions on height and on width.

The geometrical aspect of the Kripke models is their mostimportant aspect in the propositional case. Since we discuss thisad nauseum in Section III of Chapter 1, we shall only considerone example here: the Jaskowski sequence. In his book (cited inthe bibliography). Mostowski gives several references to work onproving the completeness of Jaskwski's sequence of lattices.


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3Eg, a significant portion of Rose's dissertation is devoted toreproving this completeness, and, further, Gal, Rosser, and Scottdevoted a paper to one of Rose's lemmas. When one looks at Kripkemodels, Jaskowski's lattices become the Jaskowski trees, and,although there are things to prove, one can literally see whyone has completeness for this and many related sequences.

Unlike the other Chapters, the second Chapter, on thefirst-order case, contains little of anything new. It was writtenprimarily to round out the exposition and to establish someresults needed in the later Chapters. One point that is novel isour treatment of several results (eg. Compactness, Lowenheim-Skolem) which are usually proven directly by mimicing the proofsgiven in the classical case. In this manner, since classical modelsform a special case of the Kripke models, one may be tempted toconsider the result for the Kripke models as a generalization ofthe classical result. A classical model of the theory of denselinear order; a Kripke model of the theory is not a dense linearorder. Combined with other evidence, we see that Kripke models arenot, a priori, of interest in themselves and the generalizationsof results of classical model theory to Kripke model theory,however useful, are not significant as generalizations. Thus,nothing is gained by proving these results directly and we preferto obtain them by a trivial reduction to the classical case.

In addition to putting these results in their properplaces, there is another reason for giving the trivial proofswhere possible: With less effort, we obtain stronger results. For


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4example, one can compare the strong form of the Lowenheim-SkolemTheorem obtained by the reduction with the form obtained by anappeal to the proof of the Completeness Theorem (where a momentsthought is still required). A direct proof of the strong formwould amount to copying the proof for the classical case, usingthe fact, without admitting it, that a Kripke model is aclassical model. For another example, consider the existence ofarithmetically definable models, of r.e. theory; Arithmetizingthe completeness proof givesaA odel; reduction to theclassical case yieldsaA model.

The third Chapter surveys the decision (and related)problem(s)for intuitionistic theories. We begin by presentingLifshits' Theorem on quantifier elimination, giving some examples,using Kripke models to give simple decidability proofs for theseexamples, and giving some applications. In the second Section, wegive some simple sufficient conditions for recognizing that anintuitionistically formulated theory actually coincides with itsclassical extension.

In section III, we present a number of undecidabilityresults for intuitionistic theories whose classical extensions aredecidable. In the propositional case, the geometric configurationof the underlying partially ordered set and our ability to decidewhere to force the atoms were the two parameters we had to work

with. In the first order case, we can also manipulate the,domains.This latter parameter, together with our ability to force atomicformulae where we please, gives us a large amount of freedom inconstructing models. Thus, even the simple theories turn out to


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be undecidable. Indeed, the simpler the theory, the easier it isto guarantee that the model constructed is a model of the giventheory. It follows that the undecidability results are hereditaryundecidability results.

As a survey, this Chapter is incomplete. The mayoromissions are a discussion of reduction classes and several morerecent results of Gabbay.

The fourth, and most important, Chapter is devoted tothe study of Heyting's Arithmetic. Here, a simple closureproperty of the class of models of Heyting's Arithmetic is usedto give new proofs of some standard results, some not so standardresults, and some new results. The Highlights of the Chapter areour simple proof of the prepositional case of a Theorem ofde Jongh and our (more difficult) proofs of a couple ofrefinements.

De Jongh's Theorem is an analog to a trivial result inclassical arithmetic. Let A(p1,...,pn) be a proposltional scheme.Then every substitution instance, A(B.j,...,B ), within Peano'sarithmetic is pro-viable iff A(p1,...,pn) is a tautology. The proofis quite trivial-- by Hilbert-Post Completeness, the schemeA(p-j,...Pn) is consistent iff A is a tautology. In theintuitionistic case,where one doesn't have Hilbert-Post Completeness, the corresponding result, that every substitution instanteis provable in Heyting's Arithmetic iff the original propositionalformula is an intuitionistic tautology, is far from obvious andwas first proven by de Jongh by combining the use of forcing andrealizability. We present an extremely simple proof of this result


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6in Section I. In Section III, we use a more sophisticated argumentto simplify the substitution instances to being and to givea model-theoretic proof of Friedman's uniform TT result.(It

-,. Amight be added that Friedman has recently announced uniform z, ^instances, thus obtaining the best result possible.)2. Having briefly described some of the results, we oughtto discuss the relation between our work and work in theliterature.

The largest overlap in results said proofs occurs in thefirst Chapter, where, having begun our work in comparativeisolation, we reproved many results which had already beenpublished.Segerberg's Minimization Theorem is a good example. Aswith this example, some results are used so often in the textthat we had to include them. Some examples are included becausewe prefer our treatment. Eg. our proof of the completeness ofJaskowski's sequence is very similar to Gabbay's but, ourproof, not depending on the symmetry of the models, allows us torecognize at a glance the completeness for other sequences oftrees (eg. sequences as irregular as those trees pictured inMostowski's book cited in the bibliography). Similarly, ourcharacterization of Hosoi's slices is slightly stronger thanOno's.

Many of the results on Intermediate Logics are well-known, but were originally obtained by the use of pseudo-boolean techniques. The reader familiar with such proofs maywish to compare them with the model-theorectic proofs presented


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here. Of course, there "being a partial equivalence of the twotechniques, some of the alternative proofs are trivial variants.

As commented in 1, above, the Kripke model can be viewedas a classical model and in Chapter 2 we obtain some results bya direct reduction to the classical case. We also remarked thatsome of these results can also be given proofs analogous to thoseof their classical counterparts. Thus, in the quantified case,the Kripke theory looks like classical model theory ut onlyat first. With Kripke models, there are new topics to be discussed g. Glivenko and Explicit Definability Theorems. For the former,we mimic, but extend, Fitting and Gabbay to obtain a much weakerresult than those obtained proof-theoretically. For the latter,we present bezel's treatment, but also diverge from it to give themost simple-minded treatment possible with Kripke models.

Our coincidence criterion, in the third Chapter, relatesforcing and model completeness in the same fashion that Barwiseand Robinson relate forcing and model completeness. This, alongwith the result on almost coincidence, however, is the only placewhere our treatment resembles Barwise and Robinson's work onforcing and model theory. (For one thing, this is the only placewhere generic structures have been of importance in our work.)

Our technique for proving undecidability results is tointerpret the intuitionistic monadic predicate calculus within thetheory we wish to prove undecidable. Lifshits gave a syntacticproof that such an interpretation works for the theory of equalityOur proofs are semantic and most closely resemble the Rabin-Scott


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method. Gabbay's technique (We only present one example of thistechnique.) is a combination of such a Rabin-Scott argument withthe use of a reduction class.

For arithmetic, our proofs of de Jongh's Theorem and itsrefinements resemble de Jongh's proof in that we both use thegeometry of Kripke countermodels to propositional formulae. Ourbridges to arithmetic, however, are different we use forcingand de Jongh uses a combination of forcing and realizability.Friedman's proof uses a slash-theoretic generalization ofrealizability.

In Chapter 4, with the exception of Section III, wherewe use some logic, our methods are fairly standard, more or lessalgebraic, tricks. Thus, rather than discussing proofs, it mightbe best to say something about the range of applicability of ourmethods. First, most of our results require us to assume that, inwhatever extension T of HA we consider, the axioms must, looselyspeaking, be generated from Harrop sentences, induction principles,and reflection principles. Thus, we cannot treat systems such asHA + CT or HA + EOT. (One might also notice that the concretemodels which we construct are all on finite trees, whence theydo not contradict any laws of classical logic. This can be gottenaround to some extent by placing Kripke models instead of classicalmodels at the terminal nodes. The crucial thing here is that thetheory T be preserved under our operations for constructing newmodels i.e. T be generated by the Harrop sentences and inductionand reflection principles.)


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Two interesting axiom schemes, well-known from Godel'sDiatectica Interpretation, are Markov's Principle (also calledthe Principle of Constructive Choice) and the Independence ofPremises Postulate. Kripke models are quite applicable to Markov'sPrinciple, although the class of extensions T = HA + MP to whichwe can apply our methods is narrowed considerably: T must be truein the standard model. The Independence of Premises Postulate,however, is simply not as amenable to study by means of theKripke models as Markov's Principle and we only prove its mutualindependence with Markov's Principle. Aside from our de JonghTheorem for Markov's Principle, realizability interpretations canbe used to obtain approximately the same results as obtained herefor this scheme (the mayor difference being the classes ofextensions T2A + MP to which the methods are applicable).(See eg. Troelstra's paper cited in the bibliography).

The relation between our method and Friedman's generalization of the Kleene slash if not clear. First, we might expectsome sort of relationship since our approach generalizes Aczel'sapproach to a slash predicate. (This terminology is atrocioussince Kleene's slash was based on realizability, we ought,perhaps, to refer to slashes as (function free) realizabilityinterpretations.). Second, both techniques are useful not onlyfor proving Explicit Definability Theorems, but also forcharacterizing the intuitionistic proposifcional calculus in termsof disjunctive properties. (See de Jongh's thesis, cited in thebibliography to Chapter 1, for a slash-theorectic characterization


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of the intuitionistic propositional calculus. With Kripke models,the intuitionistic propositional calculus is the only IntermediateLogic closed under the operation 5^*(X 3- )1 for finite S-'.C If the elements of * have origins and we bound the cardinalityof yt we get more logics.] ) hus, both techniques can beused for proving de Jongh's Theorem. Further, we both give thesame closure properties of the class, P , of theories T -2 HAfor which we can prove Explicit Definability. (These closureproperties assert that tf5 is closed under the addition of Harropsentences, taking unions, and the following operation:

Te TP , A has only x free, and T h An for all n>T +1\xAx$P.)Perhaps a combined use of forcing and the Kleene slash (a la deJongh's combination of forcing and recursive realizability) willclarify matters somewhat. (Observe that the Aczel slash is notof this form.)

Historically, set theoretic semantic methods (latticetheoretical, topological) have no been useful in studyingHeyting's Arithmetic the greatest difficulty probably being in

s

the construction of non-trivial models. (This is certainly themayor obstacle to the use of Kripke models.)o a certain extent,this is still true for the Kripke models unless we apply theCompleteness Theorem and quote results from other sources, wecannot even give a model of HA contradicting the law of theExcluded Middle. Nonetheless, we can easily give some non-trivialmodels. The situation changes, of course, when one considers theTheory of Species, Here, non-trivial models are easy to construct


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1 1

simply because we now have a new easily manipulable parameter.The arithmetic portions of such models, however, are usuallytrivial and, partially for this reason, we have not studiedmodels of the Theory of Species. We might mention, however, thatPrawitz has proven a Completeness Theorem for the Theory ofSpecies with full Comprehension with respect to a class of second-order Kripke models (in Intuitlonism and Proof Theory, eds.Myhill, Kino, and Vesley (North-Holland, Amsterdam, 1970) ).3. The third, and especially, the fourth Chapters give usan uneasy feeling about Kripke models. As mentioned above, theKripke model, with its partially ordered set of worlds, issupposed to represent either some temporal or positivisticidealization of intuitionistic logic. One feels, eg., that, thus,a Progression ought to be a natural Kripke model. Unfortunately,this just doesn't happen to be the casei we must assign domainsto each element of the underlying partially ordered set and, inthe case of arithmetic, the natural domain to assign is the setof natural numbers. But this, combined with the decidability ofatomic formulae, reduces our model to the standard classical model.The net result is that, to get a non-trivial model, we do not usenew positive information at later stages, but new negativeinformation (supplied by the addition of non-standard integers).This is most unpalatable.

Prom the basic motivations given for the definition of aKripke model, it follows that a natural way of obtaining modelsof Heyting's Arithmetic is to choose a partially ordered system


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of non-standard models of classical Peano's Arithmetic.Unfortunately, there is no guarantee that the structures soobtained will indeed be models of Heyting's Arithmetic and weare required to use overly sophisticated techniques (in Chapter4 Section III) to make such a simple idea work and then onlywith limited success. (Eg. we still cannot find, withoutreferring to the Completeness Theorem, a model of Heyting'sArithmetic and the negation fo some instance of the law of theExcluded Middle.)

Thus, we have two reasons for dissatisfaction with theKripke models in their present form. If we consider the truthdefinitions for the various connectives, we see that the mostquestionable interpretations are those of 1 ,3 and Ax. Thereis room for a relnterpretation of these connectives and quantifier nd, thus, the possibility of new models.

The use of pseudo-boolean-valued Kripke models (like theuse of boolean-valued models in classical logic) will yield someresults (eg. an improvement on the recursion-theoretic complexitiesof countermodels), but it will still not allow one to avoid theuse of non- standard integers or obtain Progressions as models.A more promising alternative is given by de Jongh's combinationof forcing and realizability.

Whenever one makes some attempt at a constructive'semantics- by focussing on some temporal aspect of the logic,using fuzzy truth values, realizing formulae recursively, orwhatever one will probably obtain a model of Heyting's Arithmetic


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The usefulness of such an approach depends to a large extent onthe sort of constructive principles one is trying to model. Forthe pure predicate calculus, temporal logic may be close enoughto the intended interpretation, "but for an applied theory, suchas HA, it is far too simplistic. (Thus, eg., we have to useresults on definable models of classical arithmeticcomplementing the temporal interpretation with a mild flavor ofexplicitness.) Combining the use of forcing with some morepowerful constructive attitude (than mere explicitness) ought toyield something mare palatable and more technically useful.4. A final comment concerns the non-intuitionistic characterof many of the proofs. For the sake of philosophical homogeneity,one may object that proofs of intuitionistically meaningfulresults ought to be intuitionistic. While the proofs given hereare set theoretic in nature, many of them can be made arithmeticby means of the Hilbert-Bernays Completeness Theorem. (Observethat such proofs are not carried out in Peano's Arithmetic, butrather in Peano's Arithmetic augmented by some consistencystatements (which can be recognized as valid on constructivegrounds).) Then, by a result of Kreisel's, if the result is T (eg. an independence result), the proof can be carried outwithin Heyting's Arithmetic (augmented by the necessaryconsistency statements). Thus, intuitionistic proofs can beobtained for many of the results presented here.[There is, however, something to be gained by giving (set-

theoretic) non-intuitionistic proofs: The less object A resembles


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object B, the less the possibility of mistaking object A forobject B. Though one might simply pass it off at thoughtlessness,there is a growing tendency in logic today to make such mistakes.A typical example is the confusion between an intuitionisticformal system and systems with the Explicit Definability Property(occasionally reffered to as "formally intuitionistic" systems ).While the Explicit Definability Property is a property shared bya great many intuitionistic formal systems, it can hardly beconsidered characteristic: The set of true sentences ofarithmetic has the Explicit Definability Property ut is it"formally intuitionistic"? . The fact that the model-theoreticproofs given here bear no resemblance to intuitionistic proofsmay help the reader remember that the results obtained are formalresults about formal systems and do not deal with theintuitionistic notions directly. (Unfortunately, even thisrealization cannot be taken for granted.)^ .


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Chapter 1, The Propositional Calculus 15I. INTRODUCTION TO THE KRIPKE MODELSA. Introduction

In [26 Kripke introduced a set-theoretic semanticsfor intuitionistic logic. Despite the fact that his models forintuitionistic logic are not themselves natural objects of study,they have proven to be of great use in attacking formal problemsin intuitionistic mathematics. In this Chapter, we present andstudy these models and some of their applications in thepropositional case. The quantified theory and arithmetic will betreated in subsequent Chapters.

There are three standard ways of motivating the definitionof a Kripke model for the intuitionistic propositional calculus:i) via modal logic; ii) via tense logic; and iii) via forcing,i) lies beyond the scope of this work and we will not consider it.Neither ii) nor iii) is completely convincing, especially inconsidering models of arithmetic. Nonetheless, we shall take them asour starting point.

ii) proceeds as follows: We first note the obvioustemporal content of some discussion of intuitionistic principles,eg. in [15], page 115, where Heyting discusses "infinitelyproceeding sequences, depending upon the solving of problems."Here, Heyting defines a mathematical proposition p to have beentested if either ip or np as been proved. Whit this notion,he defines an infinite sequence by the rule: "As long as p has notbeen tested, ... choose an = 2~n, but if p is tested between thechoice of am and that of am+-j then ... choose am+ = 2 nfor


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for every q." Here we have various stages of time (the timeswhen we choose am and am+-|) and certain additional items ofinformation at the succeeding stages (namely, knowledge whenchoosing am+ that p has "been tested).

The problem now is to formulate a set-theoretic modeltheory for the underlying tense logic. For this, we need firsta partially ordered set (K, )of stages of time. Supposedly,the stages are not linearly ordered, because, at any given time,we may imagine the future branching off into different directions .e. we may imagine different possible futures. For laterreference, we will call a partially ordered set,(K, ), apropositional model structure (pms).

Our next task is to determine, for a staged in time anda formula A, when A is true at , written " oC | f - A" and,anticipating, read " oC forces A." Clearly oC \\- ABiff

ot 11- A and ok |h B. Also < * < 1 - A vB iff oC tl- A or ol I*- B.Implication is tricky: If o|l-AoB, 3 and ( 3 |(- A, then,since we must have P it- A B', we know (3 If- B. The converse is notobvious, since we may notknow at & that every (3 which forcesA also forces B; but a) we are not interested here in episteraological questions, and b) we need a usable condition to define

oH. | f- AaB, so we agree that o


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1 7

is a triple,(K,6 ih), where(K, )s a pins and I V - is aforcing relation defined on (K, l)satisfying:

i) for atomic A, t L II- A and ot & p imply fi II- A;ii) oC ih AaB iff ot ||- A and | ( - B;iii) < K It- A v B iff ot Ih A or < * |l- B;iv) d t- AoB iff Vf3 ( | J - B) andv) oc i f - iA iff V 50*-( {3 A).A word about notation: Iripke models shall always be denoted

by K = (K,, II-), with possible super - or subscripts. The timestages, 1 K, will always be denoted by small greek letters,*

, y > an(l will be called nodes(to avoid conflict with theuse of the word "elements" when domains are added for thepredicate calculus in Chapter 2).

Prom this definition immediately follow several useful facts(which will be used without mention in the sequel):

a) for all A, if oc l | - A and ot 6 p f then { 3 | J - A;b) any relation defined on atomic formulae satisfying i)

extends uniquely to a forcing relation on all formulae;c) the relation x . | | - A depends only on those[it ; and,

from this,d) if there are no p>ot, then o c ll- A iff A follows

tautologically from the atomic subformulae of A forced bytogether with the negations of those atomic subformulae not forcedby o t .

A node


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18path through the pms as terminating). As we shall see shortly,the existence of terminal nodes can be fairly useful. To obtainthis existence, we shall turn to forcing.

Uote: Since the tense logic approach naturally yieldscomplete sequences (as stages at eternity), there is no actualneed to introduce forcing at this point. Nonetheless, it is aconvenient point to introduce this approach.

Following 23 and C l, a complete sequence in the modelK = (K,^ |h) is an increasing sequence, s = {s^r , such that,f o r e ac h f o r m u l a A t h e r e i s an n s u c h t h a t sn I I - A o r sn lh ~ I A .The existence of complete sequences is guaranteed by thefollowing Lemma of Cohen's:Lemma 1. Given K and oi e K, there is a complete sequence s suchthat Sq = oL .

The proof is well-known and we omit it.To obtain the terminal nodes promised above, we add to

K its complete sequences to obtain a model Kc = (Kc, \hc):i)Kc = K U{s: s is complete over K};ii) iff oc , 3 K and 6 3 or o4 e K and^ s

and for some n c


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18path through the pms as terminating). As we shall see shortly,the existence of terminal nodes can be fairly useful. To obtainthis existence, we shall turn to forcing.

Note: Since the tense logic approach naturally yieldscomplete sequences (as stages at eternity), there is no actualneed to introduce forcing at this point. Nonetheless, it is aconvenient point to introduce this approach.

Following C2] and C6l, a complete sequence in the modelK = (K,^ |l-) is an increasing sequence, s = {s^r , such that,for each formula A there is an n such that sn II- A or sni|- ~)A.The existence of complete sequences is guaranteed by thefollowing Lemma of Cohen's:Lemma 1. Given K and oL K, there is a complete sequence s sucht h a t S q = oC .

The proof is well-known and we omit it.To obtain the terminal nodes promised above, we add to

K its complete sequences to obtain a model Kc = (Kc, c, lh):i) Kc = K U {s: s is complete over K}ii) iff oc , 3 6 K and ^$r o(. e K and^ s

and for some n c


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19

Proof: By induction on the length of A. For atomic A,this follows by definition. For A = B a C , ByC, the proof istrivial.

Let A = B3C and let l h A, t ( 3.Iff6K, (3 If- Bimplies (* 1 1 - C, and hence p ll-cB implies pjl-cC. Supposep is asequence s and s )f-B . Then for large n sn I I - B. Butrf cs iffot *=sn for some n. Thus, for large n, snft- B C, and sn | | - G.By induction hypothesis, s ^(-CG. Thus s \l-B implies s \V,C andwe may conclude U-CA.

Now suppose |I/A. Then there is a such that (3 i|-B,P W/C. Hence ( 3|hB and ( & \c0 and


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20sequence with bq=^.Then, obviously, s II-CA, whence sn I f - A forsome n. Hence^ ll/ 1A and, since 3 was arbitrary, ot ~| TA. QEDCorollary 2. 1A is always forced iff ~\A is a tautology.

Proof: Notice that for any A, if oc l|-in, thentt-lA. QED

A final Corollary, due to Godel, isCorollary 3. e t A contain only the connectives1A, A is alwaysforced iff A is a tautology.

The proof is an easy induction on the length of A, whichwe omit.

The use of complete sequences to prove Glivenko Theoremsis due to Fitting( whose treatment differs slightly fromours. The present treatment is basically that of C s ] .

We shall give further applications of complete sequencesand terminal nodes in the succeeding Sections. First, however,we turn to the problem of the formal connection between thepresent models and the intuitionistic propositional calculusi.e. the question of completeness.B. The Completeness Theorem

The completeness of the intuitionistic propositionalcalculus, denoted here by I, with respect to validity in Kripkemodels was established by Kripke in 126].Completeness withrespect to logical consequence within the models was establishedindependently by Aczel til),itting ([6)>and Thomason (C42 ).The proof, in the propositional case, is, as pointed out in[6l,a straightforward generalization of the use of maximal filters in


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21the Lindenbaum algebra. Instead of using maximal filters, we useprime filters, and, since the prime filters have inclusionrelations among them, we have a natural candidate for a Kripkemodel.

To " b e precise, this is the mechanism behind thecompleteness proof a discussion of lattices and prime filtersbeing postponed to an Appendix. Here we deal with theories andtheir prime extensions, (This is not merely a renaming, as theresult we prove shortly is weaker than the result for lattices.)

In order to give the proof, we will need a list of axiomsand rules of inference. For convenience, we will use the followingsequent calculus from[26l(T being finite, unordered sets offormulae, with possible repetition):Axiom: T,A > A, fa;Rules:

> A, ^ r2>B, P* A * 1p p > ^ ^ A ^ B-*r > A, f a ^ P >B, fa ^ p - | , > r 2 B >&2P > AvB, P >AvB, f^ Y2 * AvB * V | A2P A B P 1 M, r 2 B >&2, s,

|V-

P > A^B P . j , P &2f.A-. 1 r-A. A ,


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22Cut rf1, A > Tg -~*A &2.

^ 2 * 1 2As usual, we write I T *fa or simply | - P } a if

there is a derivation of p? from the above rules and axioms.We also write ^ A for * > A, and S I r P for an arbitrary setS of formulae, when we have a derivation ofT* a from axiomsA, for A6S. Notice that, by using an admissible weakeningrule and cut, S - f1-> iff ( - Sq,J** or some finite Sq?=S.

Let us digress a moment to motivate the choice of rules.Suppose we are given a formula A which we believe to be false,say A = (B3C)v(C3B).Let us write f oC |J- A for all ACPand o C WA for any A6 A . Then we want > ( B 3 C ) v ( C 3B). Butoc B=>C, C B.ocNow oL l[/B30iff for some{3 t (3 ||B and M/C. Thus B C. Butwe have p \ - G sB, so there is a such that C B. Thus,

B CVwe have constructed a countermodel to A = (B3C)(C5B).

Now, if we look at the rules, we notice that they formthe reverse process: When we go from f1 fa to P ' a forp^ol, we can give derivations, using the above rules, ofTfromP a1 > Continuing, our construction will only be blockedwhen we haveP, A /\, A. If every attempt is blocked (Notethat we may branch as follows: If F *AAB, then either there is


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23such thatPAr aysuch thatPB,)e may read off a

proof ofT/\. If an attempt is not "blocked, we have acountermodel forT> h. This is perhaps clearer if one usesthe rules in 1343.)

A readable proof of this completeness theorem may hefound in[6] Chapter 2. This proof yields completeness ofthe cut-free miles and the standard proof-theoretic corollaries.Model-theoretically, it gives completeness of I for finite treemodels. However, as we will require completeness with respectto logical consequence as well as with respect to validity, wewill present the alternative completeness proof. These corollarieswill then follow by a model-theoretic constructions.

First, we will need several definitions. That A is alogical consequence of a set S of formulae, written S r A, shallmean that for all models K and all oL K, if oc j | - B for all B6S,then 1 1 - A. Logical validity of A shall mean 0 = A and shallbe written = A. Similarly one defines validity with respect to aclass of models and5 for any theory T extending I, we shall referto the result T - A iff|=A (relative to a class of models) as a(weak) completeness theorem and the result that for all S, T+S }-Aiff SIsA (relative to the given models), as a strong completenesstheorem. The result we wish to prove isTheorem 2. For all S, SMff SNA i.e. I is strongly completefor the class of all Kripke models.

To prove this, let us make two more definitions: By atheory, T, we shall mean a consistent set of sentences closedunder the rules of inference. T is prime if T J - AvB implies T (- A


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24or T t - B.

Our first Lemma is the analog of the Lindenbaum Lemma:Lemma 2. Suppose T V j ^ A. Then there is a prime theory T2T suchthat T (-/ A.

Proof: We obtain T as the union of an increasing sequence,{Tn) of theories. First, let(a B enumerate all disjunctionsand let Tn be T. For k+1, let Av,yB be the first untreated0 nk nkdisjunction such that T, I - A V B .* nk k

By the ruleVeither T,+A V/ A or T,+B l/A. Letk K nkTfc+i be the deductive closure of Tj+An if the first alternativekholds and that of T +B if not. By closure under cut, thatkTk+1^followsfrom"the fact Tk+AnW A in"thesecond case.)

Letting T =^n,we see immediately that I is a theoryand T)t* A. If T V Tn AmvBm for rl3i"trarily large n,and for some k, = An Bnk" Henoe k+1 > " Am or k+1 ^ Bm'whence T - Am or T Bm. 1 m mThus T. is prime. QEDLemma 3. Let T be prime and let Tf range over all prime extensionsof T. Then,

i) T B >C iff for all T', T' B implies T1 ] 0, andii) T'.V IB iff for ll T1, T1 B.Proof: i) Let T ^ BC and let T' extende T. Then T' - B C.

If, in addition, T' - B, then, by cut, T' r C.Conversely, suppose T BaC. By :>, T+B|/ C. Lemma 2

yields a prime T1 such that T+BST1,T' / C.ii) For "(B, consistency requires that T^ ib implies


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25

T1 B. The converse follows from >1. QEDWe may now prove Theorem 2 quite easily. Let K be the

set of prime extensions of I (relative to the language, whichwe have tacitly assumed is countable); let ^ be the set theoreticinclusion relation; and let \ \ - be defined, for atomic A, by T II- Aiff T V- A. Lemma 3 easily yields the followingLemma 4. For all prime T, A, T \ \ - A iff T V- A.

Now suppose St/ A. Then for some prime theory T, Sc T / Aand so T I V - B for all B6S and T 1 1 / A. Hence S and Theorem 2is half-proven. For the converse, interpretTas assertingthat for all oc , if oc \ \ - A for all A T , then o4 B for someBC . This assertion is true for all axioms P,A > A,/\ and forsequents >A for ACS(when we restrict our attention to thoseOC such that 06 \(- A for all ACS). One then verifies that thisproperty is preserved by the rules of inference. Thus, if S - A,then sl=A. This completes the proof of Theorem 2.

The reader interested in the relations between Kripkemodels and lattices is invited to skip ahead to the Appendix onlattices.

Let us say that a node C is an origin of a pms,(K, ),if ot=pfor all (3K. Then Theorem 2 easily yieldsTheorem 3. I is strongly complete with respect to the class ofKripke models with origins.

To see this, notice that S A iff there is a model Kand anot^K such that \ \ - B for all BeS andoC t / A. Restrictingthe model tothoseyields h result.


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26

We shall find it convenient in the sequel to assume.unless otherwise noted, that every model has an origin/which wewill denote byd .C. Immediate Applications of the Completeness Theorem

The first application of a Strong Completeness Theoremis usually Compactness:Theorem 4. S I s A iff SqI=A for some finite Sq6 S.

The proof of this is standard.Recalling our applications of complete sequences, we

have the Glivenko Theorem( 2~])nd its corollaries:Theorem 5. Let P be a set of formulae.

i) P y. VIA iff A is a tautological consequence ofPii) taiff 1A is a tautological consequence ofP;andiii) for A containing only the connectives1 A, (. A iff

A is a tautological consequence of P.Another application is the Disjunction Theorem("I).

Theorem 6. If V AvB, then - A or B.Proof: Let K1,K2 modelswith origins d 1, L 2,

respectively, such that oC ^ \\/ A and . DefineKq =(K0,*Q, ll-Q) by

i) Kq = {) 2 X {2} u ,0C0 a new l:)3ectii)(t , i) ^ , ; ) ) iff i=j and oC - 3 ;iii)"*or a11tKe o;iv) for atomic C,(e , i) C iff oC 0; andv)for atomic C, Cq 1(-/q C.


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27First, property iv) is easily seen to hold for all C. Thus( ) 1)H-/q A and(oi 2) B. Hence *q can force neitherA nor B and q AvB. Thus A and B imply I-/ AvB. QED

Aczel ( L i l ) pointed out that a muchstronger result thanTheorem 6 can be obtained from the completeness proof itself. Inthe case of the propositional calculus, however, we shall concernourselves more with the, model theory than with the proof theory.

II. SOME MODEL THEORYA. Trees

Trees are always of interest to logicians and we shall" b e no exception. If we attempt to construct a countermodel toa formula A, eg. by the method briefly described in Section Iabove, we will probably construct a finite tree. Thus, trees forma natural class of Kripke models.

Let us formally define a tree to be a partially orderedset (K,-), with origin q, such that:

i) for every K,and for every , there is aleast Y with ^^ Y .Such ay is called a successor of oC ;

ii) for every** k, there is a finite sequencet oc 2 **#ot-n=oc ucl1 that i+1is a succesor of c* .Further, if (3^ 6 , then for some i.

By ii) the set of (not necessarily immediate) predecessorsof any nodeocform a finite linearly ordered set, as opposed to amore general notion of tree where the finiteness condition is


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28dropped (eg. in18}, Theorem 12). The notion of tree consideredhere is isomorphic to that of a tree of finite sequences.

We first wish to establish the completeness of I withrespect to trees (i.e. models whose pms's are trees). To do this,we simply split a model into a tree: Let (K, , 11-) be a Kripkemodel and let K r be the set of all non-trivial finite sequencesTf =( f o> ' n-1 sucJl that i)^T0 is origin of(K, ),and ii) for i< , i ~Tr istlie "fcree ordering of finitesequences and is defined by

tr lV-Tr A iff I I - A,A atomic, and where 1(Tf) is the length of Tf , i.e. 1(Tf0,...,

) n. Kripke (C2 )showed that the relation stated holdsfor all A:Theorem 1. For any formula A, Tf H-Tr A iff I V - A.

Proof: By induction on the complexity of A. For A atomic,the assertion follows by definition. The cases A = BAC, BVC aretrivial.

Thus, let A = B3C and suppose first that If ^^l-B=>CTr6.nd 6 TrB- then*1($)-1 But i(if).i "f(s)-l

and so II- C. Thus 6H-TrC and If \lrTr B C.Conversely, let \ W - B =C. Then, for some


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29Corollary. I is strongly complete with respect to the class oftrees.

If K is finite (i.e. K is finite), then KTr=(KTr, Tr,|l-Tris also finite since the sequences were chosen to be strictlyincreasing. Thus, once we've proven that I is complete withrespect to the class of finite models,weknow that it is completewith respect to the finite trees.Theorem 2. Let K be a model and let oi liy A. Then there is afinite sub-pms (K*, *) of (K, ^), containing the origin, and aforcing relation |(-* on(K*, *)such that lh* B iffo .

Now define K* to be the union of a sequence, {Kn\, ofsets defined as follows

K0- t-'oJ Suppose we have defined Kn and P Kn* e'fc Y i T e amaximal set of such that

i) ^ ^ for all i;ii) S(p) S(6 or all i;iii) if p and 6 S , then S(S) S( p)r


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30

S(S) S( S ); andiv) S( S.)^ S(6p or i j.

Se n+1 = Kn^ ^ f eKn ~ Kn-1 " 'where _1 isempty.

As mentioned, K* =^n and is the restriction of ^to K*. The finiteness of K* follows easily from the finitenessof S. Define \ V - * by U-* B iff


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all nodes. The fact that V-Y1(A=>B) = A B, Corollary 2, andthe Compactness Theorem (Theorem 4, Section I) are easilycombined to give a proof of the more general version given byTheorem 5, Section I.

For the sake of Chapter 3, we mention the followingCorollary 3. I is decidable.

To see this, notice that we can get an upper bound onthe size of K* effectively from the number of subformulae of A.To decide a formula A, we then need only test its truth in allpins's with cardinality less than this bound. Since there areonly finitely many such pms's (and finitely many forcing relationson them relative to the subformulae of A), we can decide A afterfinitely many steps.

The reader who has read the alternate proof of theCompleteness Theorem, refereed to above, will notice that theproof yields a relatively simple decision procedure (see also[27])It also yields, if IV A, a simple procedure for constructing afinite tree in which l | - / A. That something is gained by thepresent approach will be clear later, when we use the fact thatcertain geometric properties of K are preserved in the passageto Kf Until then, we beg the reader's indulgence.

Notice that((K*)Tr,-*Tr)is a subtree of(KTr,-Tr)but that the forcing relation IV-*Tr is not the restriction ofJ|-Tr to K*Tr,except for subformulae of A. In general, we cannotexpect that, if a formula is falsified in a tree, it can befalsified in a finite subtree: I is complete with respect to thefull binary tree, but not for the class of finite binary trees.


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32We leave this as an exercise (which requires a few techniquesnot presented yet).B. The Extension Theorem

A strong converse to the problem discussed at the end ofSection A does hold: Every forcing relation on a subtree (Kq- q)of a tree)s "the restriction of a forcing relation on(K , )to Kq. This is the content of the following Theorem, whichgeneralizes a result of Gabbay(Lei ):Theorem 3. let (Kq, q) be a subtree of the tree(K , ).let l\-Qbe a forcing relation defined on (Kq, q). Then there is a forcingrelation 1 1 - . j defined on (K , such that, for all andall formulae A, oi ||-q A iff oL | | - 1 A.Note: By a subtree we do not mean merely a tree which is a sub-ordering of another tree, as is evident by the exercise stated atthe end of Section A. The successors of a node ^ KQ must besuccessors of in the tree (K1 ). For convenience, we alsorequire the origins of (Kq,-q) and (K^-j) to coincide.

Proof: We will define Iby aking the nodes of K^Kqbehave like complete sequences over Kq. For each choosea complete sequence s of Kq such that s^(0) .Letcx-eK^KQ.Then there is a maximum ( * KQ such that * .Define, foratomic A, A iff s^ H-oA(i,e* forcing in the completionmodel KqC). For and atomic A, let


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33i) for


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343. A modified Jaskowski Sequence.

Jf Jf

' VJ1

Let us close this Section with some remarks. First, Iis not complete with respect to any element of any of the abovesequences. Indeed, it is not complete with respect t o any finitemodel, as we will see in Section III below. We will also see thatthe completeness fails when "finite" is replaced by "finitedimensional" or "finite measure", where these terms stand fornatural geometric measures of complexity of a partial order.

A second remark concerns a useful property of the modifiedJaskowski trees (a property shared, incidentally, by the Diagonaltrees): Every node of J* is determined by the terminal nodeslying beyond it. From this, we can prove the following lemma:Lemma 1. LetotoCn k * 1 6 terminal nodes of J*. Supposewe have a forcing relation, | l - , defined on J* such that, for eachi j, there is a formula A such that I t - A nd C ^ \ l / Aij*Then, if S is a set of nodes of J* such that


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35imply pCS, here is a formula A, built up from.the -^..'s, forwhich S s ; oC.ll- A}-. In particular, for any01- there is anA uch that {: 3 > oC.^ =[3: 3 II-

Proof: First, if C. . 11/lA. ., then oC |f-TA . LetJ XJ J J-JAi = iij* Tllen Ai andot^lj- lAi for i j.

Now, since oc is determined by those ol 5s ot, it is alsodetermined by thoseoc. oc. Let A_, = A. for 4 * andJ x 01 0Aoc0=AiDAr

We must show that p A iff^ =ot. For ot q this istrivial. Let ot jt&Q. If \ V - A , the set of terminal nodes notbeyond includes those not beyond oc(otherwise (3 1 - / ~A^ forsome i). Hence the set of terminal nodes beyond (1 is included inthe set of terminal nodes beyond t ..Let be one of these.Since the set of predecessors of &^ is linearly ordered, either

{ 3 or p


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36fact, every node is the least node which forces some formula, andevery set, closed under is the set of all nodes forcing someformula. It is this latter strong property, combined with thecompleteness of I for , that will be used in Chapter 4.C. Minimization

The problem of minimization, or the removal of redundancies, is familiar t o us from Automata Theory (See eg.where the problem is easily solved by identifying states thatperform the same task. Basically, the same trick will yield theresult for Kripke models we collapse the structure onto itself,by making identifications and changing the order where necessary,t o obtain a (in certain respects) minimal model equivalent t o theoriginal one. Here, we say two Kripke models, , K^, areequivalent iff for any ci K - j there is a fieK such that fA: ot 11-^= ft lh2$nd, conversely, for any there is asuch that ^A: ( . V l~ 2 = f e ' - ^ h - |

Let T be a set of formulae, transitive in the sense thatit contains all subformulae of every formula in it, and let K begiven. For O^K, let (odlT = k T:d f - A . et :

k3 and let be the usual set theoretic inclusion relation.For atomic A, define foOT 1 V -T A iff A U]T. The model == (Krp, I j - j , ) yields a relative solution of the minimizationproblem:Theorem 4.([39l ). For A T, C4ji Ihp A iff A 6 L(\

iff o ( A.


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37Proof: By induction on the length of A. For A atomic,

this follows by definition. For A = BaC, B vC, the proofs aretrivial.

Thus, let A = BOC and assume that t lh B C. Ifthen BrC eTplT. ButCp^T | | - T B implies B T ,

whence Eplrp5 andCplT B imply f > ft- C, i.e. ceCp] #ThusCpiT C. Hence A.

Conversely, one easily shows that Of- IM BOC impliesW/T B=C.

The case A = TB is treated similarly. QEDLet T = F, the set of all formulae. Then is called the

Minimal Model of K. Note that, forc


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38iff oc R 3 andCoO R 1 ) - R A iff At0^ (for atomic

A T), we haveTheorem 5. Let KR = (0*-3 R: CcK} R,||-R) and let A T.Then .oc A iff It- A.

The proof is identical to that of Theorem 4and we omitit.

Theorem 5 can he used as the basis of justifying certainreductions on trees which, when applied to a finite tree, resultin another finite tree equivalent to the original. While thishas some interesting applications, we will find minimal modelsto he of greater use in the present work than minimal trees. Wethus refer the reader to[23]&nd [24 for these reductions andtheir applications.(C23l also considers reductions on finitemodels and proves that reductions terminate in minimal models.)

D. Finite Minimal ModelsBy their irredundancy, minimal models possess some useful

properties. One of these is that they often reflect, geometrically,properties of the theories which they are models of. This isespecially true of finite minimal models.

Before we can clarify this remark, we must first establisha few basic properties of finite minimal models and introduce thenotion of an Intermediate Logic.T.pmma P. Let K be a finite minimal model. Then, for any subsetSckclosed under ^(i.e. oc S and ^ S imply p S), there isa formula Ag such that S = {


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39construct for C 6 a formula A , such that oc |f- A n andr o c p &p

A&p ' 1 1 1 ( 1 is " t h e rea der .Theorem 6. Let K be a finite minimal model of the scheme ofsubstitution instances of a formula A. Let | | - ' be another forcingrelation on (K, ). Then(K, I } - ' ) is also a model of the schemedetermined by A.

Proof: Let 11/' A1 for some instance A' of A. Foreach atom B in A', associate the set B = {oc : oC U-* B}.Replace B in A' throughout by A to obtain another instance A1'of A. One easily sees that < . q \V/- A'1, a contradiction. QED

A related and more or less obvious result isTheorem 7. Let (K, be given (with cardinality at most that ofthe set of atomic formulae). Then there is a forcing relation, ||-,defined on it such that K = (K, |l-) is minimal (i.e. isomorphicto its minimal model).

To see this, let, for each o< , A be an atom which isforced only at c and above.

Now suppose T is a theory given by a collection of axiomschemata. Theorems 6 and 7 easily yield theCorollary. T is complete for its finite minimal models iff T iscomplete for a class of finite pms's.

To see that we need the assumption on T that it beaxiomatized by a set of schemata, we prove the rather obviousTheorem 8. If T is the theory of a class of pms's, then T isclosed under the rule of substitution (i.e. T is axiomatized byschemata).

Proof: Suppose A is a formula valid in the given class


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40of pms's and suppose A' is a substitution instance, obtained fromA " b y replacing the atoms A^ ,... A in A by formulae . B ,respectively. Suppose also that K = (K, |f-) is a model onone of the pms's of the given class and foe f$ .

If Kj, is finite, we also haveiii) VotfK \/x6Kj[f oc ^ x > 3peK(po


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41

*Vi-l * " " * * ^ 9 ThuS i F Fand, every node y = 0( "being equivalent to a y we see that, if

V I t - A _ , then Y D" ArY-s forsome i.. ThusEPJ I,'j) J[3ll ). Let Kq = (Kq, ^Qf ||-q) andK-j = (K , .j) be as just described. Then, for all nodes

o


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42Theorem holds by definition for atomic formulae. The connectivesA,vare handled with no difficulty.

Let A = BoC.First assume B- C, f B=C.Then there is a P-| --j foe such that P | l - 1 B, lt/1 C. Byproperty iii) of a strong homomorphism, Pifp q for some00 q04" By induction hypothesis, (iQ B, C, whenceoc I H / q B^C, a contradiction.

Conversely, let Ih^Q hen, for some (3 ^ c ,we have {5 \ ) T Q B, p I^/q C. The induction hypothesis yieldsf (3 1K1 B, f(i IM1 C , w h en ce f w . B^G.

The case A = "|B is treated similarly. QEDAs an application, we prove a slightly weakened version

of the Extension Theorem of Section B:Corollary 1. Let(K1, )e a sub-tree of (K2, and supposethat for every nodeoC e there is a terminal node ^3 e suchthat p --1^ Then every forcing relation, I V - - ) on (K , )extends to a forcing relation on(K2,-2)

Proof: Choose, for each * - , a terminal node t^ * - .Map 3 of Kg into by p > P , if [ 3 e ,

p ^ oc ifo


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43applications of the Extension Theorem given in Section B.

Using Theorem 10, we can also prove the followingCorollary 2. I is complete for the full binary tree.

We will show that there is a strong homomorphism of thefull binary tree onto each of the modified Jaskowski trees. Welet B denote the full binary tree (i.e. the tree of all finitesequences of O's and 1's). If J* is the n-th modified Jaskowskitree, we let Jn1'**,,Jnm deno-te m distinct isomorphic copies ofit. l j - is the one-node pms and, with the obvious interpretation,JS+i iB

Also, B is isomorphic to

We show by induction on n how to find a strong homomorphism ontoJ*. For n = 1, simply map the full binary tree onto the singlenodec


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44

\*cf* 0 y nn'oio Jn1//t

X.JS2

n1


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45III. THE GEOMETRY 0? INTERMEDIATE LOGICS

A. ExamplesA t the end of the last Section, we defined an Intermediate

Logic to b e any extension T of I by axiom schemata. By ourdefinition of theory, T is consistent and, hence ICT C, whereC is the classical propositional calculus. To see this, observethat if T C, then T h A1 for all substitution instances A1 ofsome non-tautology A. Since A is not a tautology, there is asubstitution instance A ' obtained by replacing each atom of A byone of the formulae D AID, DDD, such that A ' is contradictory,i.e. C 1 - tA1, whence I "lA 1 , whence T V - nA'. Thus, theIntermediate Logics are precisely those Logics intermediate betweenI and C.

The set of Intermediate Logics is partially ordered underinclusion and, in fact, forms a distributive lattice under thisordering. Eg. if T1 and T2 are Intermediate Logics axiomatized byschema t a ^ A ^ - j - a n d f B - j ? j , r e s p e c t i v e l y , t h e n T . | a 2 (T^Tg) i saxiomatized by the schemata ?AivBj XJ ( AiAB;j3 Ixj Properties of this lattice are discussed in eg. f l 8 l and [193.

If T is an Intermediate Logic and T' is an extension ofT by the addition of the scheme of substitution instances of af o r m u l a A , w e w r i t e T ' = T + A . T h e e x p r e s s i o n T1' = T + T ' i ssimilarly defined. Further, we will denote certain specialIntermediate Logics by special symbols.

W e consider some examples:1. LC. The system LC of M. Dummett ( + 1 ), also known


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46as u> ^ )> isobtained by adding the scheme(Au B)v(B-DA) to I.Theorem 1. ([ C16] ). LG is strongly complete for the classof (models defined on) linear pms's.

Proof" We must show two things: i) LC is valid in thisclass of models; and ii) this is a representative class of models.

i) Let (K, be linear and let \ \ - be a forcing relationdefined on it. Suppose c ( . q \|/ (ADB)V(BOA)for some formulaeA, B. Then o(Q lb* A B, (0 BDA, and there are ^such that oC If- A,0( l|/ B, f- B,^ A. By linearity, el ^^or 0( - but the larger node must force both A and B, acontradiction.

ii) To show that the linear models are a representativesample of models of LC, we shall show that, if K is a model of LC,then its minimal model, K , is linear. Thus, let K be a model ofLC and let K be such that and are incomparable.Since the ordering on classesM,s given by set theoreticinclusion, this incomparability means that there are formulae A,B such that A WFCplp .

Thus,


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47and c(lh A for some atom all other atoms failing to be forced.Sinced f - A, it follows that0WiA. It being given that

A, it follows that A viA. By linearity, we havea model of LC and hence LC ^ C. To see that LC ^ I either i) usethe Disjunction Theorem (Theorem 6, Section I)to deduce that, ifLC = I, then I - ADB or I - BaA for all A, B, which is absurd;or ii) use Theorem 7, Section II, by which there is a forcingrelation making a non-linear model minimal.

2. KC. The Logic KC is obtained from I by adding thescheme nA y-YiA, The following result was communicated to us byM. Fitting:Theorem 2. KG is strongly complete for the class of directedpms*s.

(A pms (K, is directed if for any o(,^ e K there isaYK such that ^Vnd ^t >)

We shall also proveTheorem 21. KC is strongly complete for the class of mps's withmaximum nodes.

Proof of Theorems 2 and 2': We shall prove i) KC is validin every directed pms; and ii) if K = (K, ||) is a model of KC,then (Kc)-p has a maximum node, where Kc is the completion modelof K (See Section I-A) and (K)-p is the minimization of thiscompletion.

i) Let K = (K, be a model with (K, directedand suppose o(q ||-/ ~iA Then (q ||-/ "lA,-*lA and there are

( *? - ^ o such that ^ A P nA. B y t h e directedness o f(K,^), there is a .But then til- A,-iA, a contradictio


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48Thus KC is valid in K.

ii) Let K be a mo del of KC and let s b e a complete sequenceover K . T h e n s o(Q an d w e h a ve q \ - l A >s -\A,

o( Q t f c - - ? T A > s|- C- | 1 A s | f -C ASince v ~ n A , we have s | - A iff oiQ If- -jiA. T hus , i f s^and s2 are complete sequences, fs^p = C82~^f and since every nodeof K lies below a complete sequence, th ere is a unique maximum nodein (K)p.

i) and ii) obviously prove th e Th eorems. QEDIf one recall s th e d iscussion of Kripke mod els as a

modelling of tense logi c , then Theorems 1, 2, and 2' make intuitivesense. If time is linear, then either A must be known at least asearly as B, whence B^A, or B is known at least as early as A,whence Az>B. There is a unique eternity iff every negation isd e ci de d , i . e . -iA vnA is always true. In particular, if time islinear, th ere is a unique eternity:Corollary 1. LC is strongly complete for the cl ass of l inear pms'swit h maximum nodes.

By Corollary 1, I C^LC^C. To see t hat LC4KC, applyTheorem 7, Section II, to th e non-linear pms:

T hus , a l l inclusions listed are proper.The minimal mo dels of LC are all linear. We have shown

that certain geometric properties of models correspond naturallyto KC namely directedness and the existence of maximum nodes.


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49The minimal models of KC need not f h owever, have these properties:Consider t h e model K wit h two infinite ascending sequences o.j,..,

andnd . . . , beyond o(Q, suc h that I Ip n IH .A2 n+-j where , . . . is the list of atomic formulae:

Ai Ag A j

Clearly t his mo del is minimal (no two nodes force even the sameatomic formulae). However, any complete sequence forces al l atomicformulae, wh ence any two complete sequences are equivalent andare identified in(K0)^. Thus K is a minimal model of KC is neitherdirected nor has a complete sequence.

3 . KP . The system K P . , named after Kreisel and Putnam whointroduced it( ), is obtained by adding to I the scheme(1A3BVC)O(iA3B)v(lA3C)"]. The completeness proof forIC was simple al l minimal models were linear. KC was not quiteas trivial we had to minimize t he completion mo dels. KP offersan even more complicated example, both in the description of themodels and in th e c ompleteness proof.

First, let us describe th e family of models for w h i c hKP is complete. For t his , we need some notation. Given a pms (K,^),wit h o(fe-K, let (K^ ,^) be the restriction of (K, to those(9 ^ (,, and for ECK,let E+ = y: JxE(x y) ,

E" = y: *3xE y - x)Eg . K^ = We define the following curious property on


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(K, ): (*) Vol^&K([e?( "^E+)~ is empty or anti-directed}.

Anti-directed just means that, if we reverse t he or der , t he setis directed , i .e . for l ( there is a such thatLemma 1. Let (K, have property (*) and let K = (K, J | - ) bea mo del on th is pms. Then K is a mo de l of K P .

P roof: Let ^0 y-(n 3B vC) J " ( - | A " 5 B ) V(1A00) ]. Thenfor some o B and lh nA DC, a contradiction.

Let Kp - (E+)~ be a non-empty and anti-directed. Since- i A exactly fore/ in this set , and since

j - / tAsB, ^Jf-/iAi?C, there must be o ( . T ~ (E+)~ suc ht hat oC B, oC)M C,

y\- o. y \ y b-B u t , there is a K^ (E+)~ suc h t h at - ^ A s wit h Tf I I -IA, hence J f - BVC. If Stir B, then YII- B,a contradiction. If J" I I - C, then ||- C, again a contradictionThus p ust force one of t he implications nA 15 B,^A ^>C. QE D

As seen in t he proof of Lemma 1, t he curious condition( *) simply asserts th at th e set of nodes beyond anywhich ouldconcievably force iA cannot be split into two subsets forcing B


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51and C respectively. The linear mod els of LO and t he directed modelsof KC are examples of mod els satisfying th e non-splitting condition(*). A non-trivial example of partial orderings satisfying (*) aregiven by the Medvedev Orders. (See .) If X is a non-empty set,then collection of non-empty subsets of X ordered under reverseinclusion, (P(X) -{*]. ), s the Medvedev ordering determinedby X. (Equivanlently, one coul d take (P(X) - )) Medvedeord erings strongly satisfy condition (*). Let SX be non-empty andlet E C yX: 0 Y ~$ (i.e. E-^: Y^J . ThenE+ =[y (I) - 0$ * * 3zE(YC Z)J andE4"" = f Y P ( X ) - 0 $ : T)ZE(YrZ * 0) . h u s , deleting (E+)~from ^Y: Y - sf ields t he set of subsets of S disjoint from a l lelements of E. If t his set is non-empty, its union is a minimumelement of it a n d , hence, it is anti-directed.

In f7l , Gabbay proved th e foll owingProposition 1. KP is complete for the c lass of mps's satisfyingcondition (*).

Observe that t his reads "complete" and not "stronglycomplete". Gabbay's proof is as fol low s: One takes a m o d e l K ofKP and minimizes relative to a transitive set, T, of formulaecontaining a formula A not forced in K. T happens to be infinite,but has only finitely many non-equivalent formulae. The model soconstructed is a finite minimal one and forces enough axioms ofKP to guarantee that if satisfies condition (*). This yieldscompleteness, but not strong com pleteness, and th e question ofstrong c ompleteness is open. Since we wi l l not need to use t hisresult, we omit the proof of the Proposition. The interested reader


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52is referred toB . Geometry

By the Completeness Theorem (Theorem 2, Section I), everyIntermed iate Logic T is strongly complete for a class of Kripkemodels. The examples given above were also complete for classesof pins's. It is not known whet her or not th is is a general propertyof Intermediate L o g i c s - * - - i . e . whether or not every IntermediateLogic is complete for a class of pins's.

As evidenced by the examples c onsidered in Section A,above, there are connections between th e geometric forms of pms'sand formulae valid on them or, perhaps better, formulae notvalid on t he m . In t h e example of L G , al l minimal mod els had to beof a certain form . For KC this failed and for KP t he problem isopen. Thus, we cannot describe t his connection in terms of t hegeometry of t he minimal mod els. Furt her , one may have completenessfor two distinct geometric forms eg. KC is complete wit h respectto t he directed pms's and wit h respect to pms's possessingmaximum nodes. If T is complete for a class of pms's, the mostnatural geometric property to associate wit h T is whateverproperty characterizes the largest class of pms's for w h i c h T iscomplete, i.e. in w h i c h T is vali d . Let us reconsider t he examplesof Section A:

1 . L C . LC is naturally associated wit h t he linear pms'sboth because its minimal models are linear and because of thefollowingTheorem 3. LC is valid in (K, i) iff (K, ) is linear.


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53P roof: Th at linearity implies the validity of 1C was

shown in Section A. Conversely, suppose (K, is not linear, sayo( , t K are such t hat . et A, B be atoms and d efine | | -

* > y Yt A iffY ,T C l l - b iff tP ,

C for any a * f c o n i C o " f c h r t h a . i i A orObviously, (ADB) (BPA). QE D

2. KC. KG is valid in directed pms's, but its minimalmodels need not be directed. However, we haveTheorem 4. KC is valid in (K, iff (K, ) is directed.

Proof: Again, half of t his has already been established,let (K, -) fail to be directed , say^,^ have no common superiorn o d e , and let A be an atom. Define Ih by

J I I - A iff O t ,T t 1 1 / B for any atom B other than A.

Then o i l | A, (? | - A and (oiA V HA. QE D3. K P . For KP we have

/

Theorem 5. KP is valid in (K, iff (K, satisfies condition(*) of Section A.

P roof: Let (K, be a pms whic h does not satisfy (*).Then there are o(K and E- u c h that K^ (E+)~ is non-emptyand not anti-directed. Let - (E+)~ be suc h t hat forno Ka- (E+)~ is p.r. efine a m o de l on (K, byletting, for atoms A, B, C.

i) |\- B iff JV( f(?S"AfK 0S+)-))+ii) C iff Siiii) | - A iff S&s+' nd


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54iv) no other atomic formula is ever f o r ce d .

Then, by iii) J* \ \ - nA for all


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55set of noil-theorems as follows: By Theorem 8, Section II, anecessary and sufficient condition that the scheme of substitutioninstances of A fail to be valid in a finite pms (K, is that Aitself fail to be valid in (K, ). Since T is finitely axiomatizedwe can decide wh ic h finite pins's T is valid in. Since we canenumerate th ese, we can enumerate the formulae refutable in them.Finally, since the set of theorems of T and the set of non-theorems of T are recursively enumerable, the set of theorems ofT is recursive and T is decidable. QE D

For example, KP has t he FMP and is finitely axiomatized.Thus KP is decidable. LG and KG are finitely axiomatized and oncewe s how t hat they have the FMP, we can conclud e th at they are alsodecidable.

To conveniently discuss the FMP, we introduce the basiclanguage of partial order. We have variables x^ , x 2, . . . , rangingover no des, a binary relation symbol ^enoting the or der , t hesymbol for equality, t he usual logical connectives A,V, ),and quantifiers At V * The connectives and quantifiers areinterpreted classically and we assume the axioms

^x(x^x),Axy(x-y y-xoX=y), Axyz(x^yay-z:>x-z),as well as t he axioms of equality. We may add a constant symbol^ o denote c(Q and the axiom Ax(5g-x).Two important c lasses of sentences are the classes ofuniversal sentences and positive sentences. A sentence A, i . e . aformula A all of whose variables are quantified, is universal ifit is of t he form , A^ . . xnB(x^ . . . XJ^)where B contains no quantifiers. A sentence A is positive if it


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56is constructed using only the connectives a,v and the quantifiersA.V -- i . e . A is positive if it contains no 1 rD.Theorem 8. i) Let T h e complete for a class of pms's w h i c h aredefined by a set of positive sentences in an extension of thebasic language by the addition of individual constants to denotespecial nodes (eg. c q to denote or, in the case of KC, aconstant o( - j to denote the maximum node). Then T has the F M P .

ii) Let T be complete for a class of pms's which aredefined by a set of universal sentences in an extension of t hebasic language by the addition of finitely many individualconstants. Then T has the F M P .

Proof: i) As is easily seen (See eg. , Chapter 5,exercises.), positive sentences are preserved under h omomorphicimages, i.e. quotients. Let K be a mo del on one of these pms'sand let iw A. If we let S be th e set of subformulae of A, thenthe minimization Kc of K relative to the set S of formulae is su c h o t hat (Kg, ^g) also satisfies the sentences defining pms's of T.Hence K is a mo del of T, 1nd it suffices to prove thatthe set Kg is finite. B u t , a node /k]g of Kg is determined by theset of formulae of S w h i c h forces. Since S is finite, there areonly many suc h sets.

ii) Universal sentences are preserved by passage tosubstructure i . e . if A is universal and h o l d s of (K, ^), then Ais also true of (K*, *), for the substructure(K*,4*)of (K, =)as defined in the proof of Theorem 2, Section II. As in part i),t his wi l l yield t he result . If t he basic language is augmented byfinitely many individ ual constants, these constants wi l l denote


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57certain nodes of (K, ). The construction of (K*, *) will thenbe modified by also including these nodes in K*, along wit hwhatever ot her nodes are required . QEDCorollary 1. 1G has t he F M P .Corollary 2. KC has the F M P .Corollary 3. LC and KC are decidable.To prove these Corol laries, observe that the linear pins's aredefined by the axiom, A y(x^y V y^x),w hi c h is bot h universal and positive. Thus, we may conclude t hatLC has t he FMP by eith er part of Theorem 8. Directed pins's aredefined by the axiom,

A y V z xz y z),w h i c h is positive. Hence Theorem 8-i) yields t he FMP for KC. Ont he other h a n d , if we a d d a constant o ( . ^ and th e axiom,

/\x(x^ 1)we define the pms's possessing maximum nodes. This axiom is bot hpositive and universal, whence we can conclude that KC has t heFMP from either part of Theorem 8.

The decidability of LC and KC th en foll ows from Theorem 7In these cases, h owever, a impler procedure is obtainable. If,for example, LC A, then we can obtain an upper bound on t hesize of t he mo dels we h ave to inspect in our search for acounterexample. Eg . in minimizing with respect to the set S ofsubformulae of A, Kg has most 2n nodes, wh ere S has cardinality n.Also, by t he simple geometric forms of the mo de ls , we d o not haveto enumerate all finite pras's and test t hem for t he validity of


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58

the logics in question the enumeration of t he finite linearpms's and finite directed pms's are easy tasks.

Theorem 8 offers a very weak converse to the Corollaryto Theorem 6, b y w h i c h every Logic w h i c h has t he F M P is completefor a class of pms's. As mentioned, it is an open problem whetheror not every logic is complete for a class of pms's. Jankov, in21"} , produced a large number of Logics failing to possess theFMP. Fine, in 51 , gave a simple example of a decidable finitelyaxiomatized Intermediate Logic complete for a class of pms's whichdid not have the F M P .

The converse to the Corollary to Theorem 6 is, t h us ,false in general. The FMP is a sufficient, but not a necessary,condition for a logic to be complete for a class of pms's. Coupledwith finite axiomatizability, the FMP guarantees decidability.Furt her , finite minimal models reflect geometrically the formulaet h at are valid in t he m . T h us , the FMP is an extremely usefulproperty for an Intermediate Logic to h ave and special interestattaches itself to finding sufficient cond itions for a Logic tohave the F M P . An interesting example is th e foll owing Theorem ofOno, w h i c h we state without proof:Proposition 2.( 3 3 " ) . Let T " b e in a finite slice. Then T hast h e F M P iff T is complete for a class of pms's.

We define w h at it means for an Intermediate Logic to bein a finite slice in Section F, below. For t h e time being, we canstate this result as foll ows: If th e models of T are short enough,t hen the completeness of T for a class of pms's guarantees the


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59FMP for T. T h u s , Proposition 2 is similar to Theorem 8 n t hatthey both assert th at, if T is complete for a class of pms's ofa sufficiently simple geometric form , t hen T has the F M P .

One of th e most useful theorems yield ing the FMP is dueto McKay and is based on the following Proposition of Diego:Proposition 3.( ). Let be atoms. Then there areonly finitely many non-equivalent formuale constructed from theseatoms using only th e c onnective " D.

McKay generalized this toProposition 4.( 281 ). Let A ^ , . . . , A ^ be atoms. Then th ere areonly finitely many non-equivalent formul ae constructed from theseatoms using only t he connectives

The proofs of these Propositions lie beyond the scope oft his Chapter. We wi l l , h owever, cite the application referred toabove:Proposition 5 ( 2 8 ] ). Let T be axiomatized by a finite set offormulae (where we use a rule of substitution) w h i c h are built upfrom their atoms using onl

smorynski.1973.investigationsofintuitionisticformalsystemsbymeansofkripkeframes

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