W. W. Tait. Intensional interpretation of functionals of finite type. Journal of Symbolic Logic, 32(2), 1967.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....of uses of defL rule, instead of decreasing it. Unfortunately, the natR and natL rules have the same behavior as weakening during the proof of cut elimination: it duplicates proofs, making impossible to find a reasonable induction measure. The method developed in [McD97] is based on ideas by Tait [Tai67] and MartinL of [ML71] and uses derivations themselves as a measure by defining well founded orderings on derivations, performing the induction relative to those orderings. The argument in [McD97] strongly depends on the fact that FO is a singleconclusion logics. Its vital definition ....

W.W. Tait. Intensional interpretations of functionals of finite type I. Journal of Symbolic Logic, 32(2):198--212, 1967.

....and perhaps more surprising, result of this paper is then that all normal characterisations of (strong head) normalisation are consequences of the strong normalisation of cut elimination. Many strong normalisation results in the context of types use the technique of Computability Predicates [24, 18], which provides a means for proving termination of typeable terms using a predicate defined by induction on the structure of types. This technique has been widely used to study normalisation properties (or similar results) as for example in [20, 12, 15, 22, 19, 1, 2, 17, 7, 4, 16, 5] this list ....

....are straightforward by Definition 15. 4. STRONG NORMALISATION OF DERIVATION REDUCTION In this subsection, we will prove a strong normalisation result for derivation reduction. In order to prove that each derivation in is strongly normalisable with respect to D , a notion of computable [24, 18] derivations will be introduced. We will show that all computable derivations are strongly normalisable with respect to derivation reduction, and then that all derivations in are computable. Definition 19 (Computability Predicate) Comp (D) is defined recursively on types by: Comp (D : B M ....

W.W. Tait. Intensional interpretation of functionals of finite type I. Journal of Symbolic Logic, 32(2):198-- 223, 1967.

....of typable process behaviour, turning possibly diverging computation into a strongly normalising one. As would be imagined by the embeddability of typed l calculi, the proof of SN is non trivial, defying naive structural induction. We adapt methods developed for strongly normalising l calculi [8, 23, 61], combined with process algebraic reasoning techniques [11, 51, 53, 57, 66] As far as we know, this is the first time a compositional principle for ensuring SN has been established for name passing processes with non trivial use of replication. The proof method for SN is applicable to significant ....

....a, although this cycle never arises in actual interaction. How can we then prove termination Simple structural inductions would not be usable for the same reason they do not work in typed l calculi [8, 20] The idea we use is suggested by SN proofs for typed l calculi, due to, among others, Tait [61]. His method employs a semantic interpretation of each type [ s] as a collection of strongly normalising l terms, and shows that all typable terms are indeed in these sets. A key step is to prove that lx : s:M 2 [ s t] for each M : t (for which by induction M 2 [ t] which means, by ....

Tait, W., Intensional interpretation of functionals of finite type, I. J. Symb. Log, 32, 198--212, 1967.

....of typable process behaviour, turning possibly diverging computation into a strongly normalising one. As would be imagined by the embeddability of typed l calculi, the proof of SN is non trivial, defying naive structural induction. We adapt methods developed for strongly normalising l calculi [7, 15, 42], combined with process algebraic reasoning [9, 35, 37, 41, 47] As far as we know, this is the first time a compositional principle for ensuring SN has been established for name passing processes with non trivial use of replication. The proof method for SN is applicable to extensions of the ....

....although this cycle never arises in actual interaction. How can we then prove termination Simple structural inductions would not be usable for the same reason they do not work in typed l calculi [7, 12] The idea we use is suggested by SN proofs for typed l calculi, due to, among others, Tait [42]. His method uses a semantic interpretation of each type [ s] as a collection of strongly normalising l terms, and shows that all typable terms are indeed in these sets. A key step is to prove, for each M : t (for which by induction M 2 [ t] that lx : s:M 2 [ s t] which means, by ....

Tait, W., Intensional interpretation of functionals of finite type, I. J. Symb. Log, 32, 198--212, 1967.

....in the preceding section, the weak normalization property of is a very useful tool in proof theory. In this section we prove the strong normalization property of which is sometimes even more useful. The standard method of proving strong normalization of typed calculi was invented by Tait [104] for simply typed calculus, generalized to secondorder typed calculus by Girard [44] and subsequently simplified by Tait [105] Our presentation follows [8] we consider in this section terms a la Curry. 4.4.1. Definition. i) SN fi = fM 2 j M is strongly normalizing g. ii) For A; B , ....

....of second order typed calculus We end the chapter by extending the proof of strong normalization of simply typed calculus from Chapter 4 to second order typed calculus a la Curry. As mentioned earlier, the standard method of proving strong normalization of typed calculi was invented by Tait [104] for simply typed calculus and generalized to second order typed calculus by Girard [44] Our presentation follows again [8] 12.6.1. Definition. i) The set of type variables is denoted U and the set of second order types is denoted by Pi 2 . ii) A valuation in S is a map : U S: ....

W.W. Tait. Intensional interpretations of functionals of finite type I. Journal of Symbolic Logic, 32(2):190--212, 1967.

....constructor T corresponds to the modality # just as the constructors in the ordinary typed lambda calculus correspond to in propositional formulas. They gave a natural deduction system for PLL and prove a strong normalization theorem by using the method in Prawitz [Pra97] see also Tait [Tai67] and Troelstra [Tro73] Gol81] argued for an application of the logic in Grothendieck s topology. He extracted the principle (#) A is locally true at # i# A is true at all points close to # For instance, two functions f and g are said to be equivalent, or to have the same germ, at a point # in ....

W. Tait, Intensional interpretations of functional of finite type, Journal

....what we shall prove. This will be done by extending the technique used by Crossley Shepherdson in [4] This technique is, in turn, an adaptation of Girard s proof for his system F, see [6] and utilizes the concepts of candidate for reducibility (an extension of R term originally due to Tait[12]) and neutral terms . These concepts together with the stronger induction hypothesis, namely that a term is not only strongly normalizable but is also an element of a candidate for reducibilty (defined below) allow us to prove strong normalization. 5.1 Candidates for Reducibility First we need ....

W.W. Tait, Intensional interpretation of functionals of finite type I, Journal of Symbolic Logic 32 (1967) 198--212.

....between the simple theory of types and the Calculus of Inductive Constructions will require further generalizations of the General Schema allowing for dependent and polymorphic inductive types. The strong normalization proof of our new calculus is based on Tait s computability predicates method [46,24]. In contrast with [28] the whole structure of the proof is made quite modular thanks to a novel formulation of our new version of the General Schema. Here, given a left hand side f( l) we define the (infinite) set of possible right hand sides r such that the rule f( l) r follows the ....

....inductively from a starting set of terms extracted from the left hand side, called the set of accessible subterms, by the use of computability preserving operations. Here, computability refers to Tait s computability predicate method for proving the termination of the simply typed calculus [46], which was later extended by Girard to the polymorphic calculus [22,24] To explain our construction, we need to recall the basics of Tait s method. The starting observation is that it is not possible to prove the termination of fi reduction directly by induction on the structure of terms ....

W. W. Tait. Intensional interpretations of functionals of finite type I. Journal of Symbolic Logic, 32(2), 1967.

....we use the set theoretic semantics of intersection types and saturated sets, which is referred to as the reducibility method. The reducibility method is a generally accepted way for proving the strong normalisation property of various type systems such as the simply typed lambda calculus in Tait [23], and the polymorphic lambda calculus in Tait [24] and Girard [13] All the above mentioned papers characterising evaluation properties of terms and of terms in x by means of intersection types apply variants of this method. 1 The Calculus and the Type Assignment Following [10] we consider the ....

.... atom, but we cannot derive y: xz:z) yy) oe oe: A typing for the last term is for example: y: xz:z) yy) ae) oe oe: 3 All Typeable Terms are Strongly Normalisable The general idea of the reducibility method is to interpret types by suitable sets (saturated and stable sets in Tait [23] and Krivine [15] and admissible relations in Mitchell [18] and [19] of terms (reducible terms) which satisfy the required property (e.g. strong normalisation) and then to develop semantics in order to obtain the soundness of the type assignment. A consequence of soundness, the fact that every ....

W. W. Tait. Intensional interpretations of functionals of finite type I. Journal of Symbolic Logic, 32:198--212, 1967.

....0 E: iff M Gamma [F 1 =y 1 ] F 2 =y 2 ] c d o t s [F n =yn ] E: Proof. Note that in the case of a closed term E with empty contexts, we have P ; E: iff M ; E: as in [29] However, inspired by the example of Tait s strong normalization theorem for the first order typed calculus [46], we facilitate the proof by strengthening the induction hypothesis of what is to be a syntax directed induction on E. We first introduce a standard structural lemma allowing us to normalize derivations in the polytype system for use in a syntax directed proof. See, for example, 38] for the ....

W. W. Tait. Intensional Interpretation of Functionals of Finite Type I. J. Symbolic Logic 32 (1967), pp. 198--212.

....normalizing. The General Schema is a syntactic criterion which ensures the strong normalization of IDTSs. It has been designed so as to allow a strong normalization proof by the technique of computability predicates introduced by Tait for proving the normalization of the simply typed calculus [26, 9]. Hereafter, we only give basic definitions. The reader will find more details in [5] Given a rule with left hand side f( l) we inductively define a set of admissible right hand sides that we call the computable closure of l, starting from the accessible metavariables of l. The main ....

.... (y) by (7) since [y]F (y) is a strict covered subterm of [x]sin(F (x) D( y]F (y) x) by (4) cos(F (x) by (3) D( y]F (y) x) Theta cos(F (x) by (6) and the whole right hand side by (5) 5 Termination proof The termination proof follows Tait s technique of computability predicates [26, 9]. Computability predicates are sets of strongly normalizable terms satisfying appropriate conditions. For each type, we define an interpretation which is a computability predicate and we prove that every term is computable, i.e. it belongs to the interpretation of its type. For precise ....

W. W. Tait. Intensional interpretations of functionals of finite type I. Journal of Symbolic Logic, 32(2), 1967.

.... to show that there are no settheoretical models of System F [Rey84] We consider as a simply typed programming language corresponding to a subset of Martin Lof s Type Theory (MLTT) Mar84] For our system we show the important property of strong normalisation, following the idea of Tait [Tai67]. To show strong normalisation of the simply typed calculus, he defined a set valued interpreting function [ Gamma] on the types. For each type oe the set [ oe] contains the computable terms of type oe. Later Girard extended this idea to the impredicative System F under the name candidates of ....

....using Saturated Sets Even for the simply typed lambda calculus strong normalisation cannot be proven by a mere induction over the term structure, since an application (x:t) s can beta reduce to a term t[x : s] that neither is a subterm of t nor of s. To strengthen the induction hypothesis, Tait [Tai67] introduced the set of computable terms [ oe] SN oe of type oe. e.g. given P SN oe , Q SN we define P ) Q : ft 2 SN oe j 8u 2 P: t u 2 Qg [ oe ] oe] The new obligation Tm oe [ oe] can be proven by induction over the terms (cf. Prop. 11) and the ....

W. W. Tait. Intensional interpretations of functionals of finite type I. Journal of Symbolic Logic, 32(2):198--212, June 1967.

....of typable process behaviour, turning possibly diverging computation into a strongly normalising one. As would be imagined by the embeddability of typed l calculi, the proof of SN is non trivial, defying naive structural induction. We adapt methods developed for strongly normalising l calculi [7, 16, 43], combined with process algebraic reasoning [9, 36, 38, 42, 48] As far as we know, this is the first time a compositional principle for ensuring SN has been established for name passing processes with non trivial use of replication. The proof method for SN is applicable to extensions of the ....

....a, although this cycle never arises in actual interaction. How can we then prove termination Simple structural inductions would not be usable for the same reason they do not work in typed l calculi [7, 13] The idea we use is suggested by SN proofs for typed l calculi, due to, among others, Tait [43]. His method employs a semantic interpretation of each type [ s] as a collection of strongly normalising l terms, and shows that all typable terms are indeed in these sets. A key step is to prove that lx : s:M 2 [ s t] for each M : t (for which by induction M 2 [ t] which means, by ....

Tait, W., Intensional interpretation of functionals of finite type, I. J. Symb. Log, 32, 198--212, 1967.

....involves only primitive recursion and can thus be easily proven terminating, but the point we drive here is that this particular implementation can be proven terminating with our approach. including mutual recursion, datatypes, polymorphism, etc. This naturally leads us to the reducibility method [12]. We are to form a notion of reducibility for the dependent types extended with metrics, in which the novelty lies in the treatment of general recursion. This formation, which is novel to our knowledge, constitutes the main technical contribution of the paper. The third question is about ....

....make is that though it seems evident that the use of R cannot cause non termination, it is not trivial at all to prove every term in T is terminating. Notice that such a proof cannot be obtained in Peano arithmetic. The notion of reducibility is precisely invented for overcoming the difficulty [12]. Actually, every term in T is strongly normalizing, but this obviously is untrue in ML ; 0; Nested Recursive Function Call The program in Figure 6 involving a nested recursive function call implements McCarthy s 91 function. The withtype clause indicates that for every integer x, ....

W. W. Tait. Intensional Interpretations of Functionals of Finite Type I. Journal of Symbolic Logic, 32(2):198--212, June 1967.

....of typable process behaviour, turning possibly diverging computation into a strongly normalising one. As would be imagined by the embeddability of typed l calculi, the proof of SN is non trivial, defying naive structural induction. We adapt methods developed for strongly normalising l calculi [7, 16, 43], combined with process algebraic reasoning [9, 36, 38, 42, 48] As far as we know, this is the first time a compositional principle for ensuring SN has been established for name passing processes with non trivial use of replication. The proof method for SN is applicable to extensions of the ....

....a, although this cycle never arises in actual interaction. How can we then prove termination Simple structural inductions would not be usable for the same reason they do not work in typed l calculi [7, 13] The idea we use is suggested by SN proofs for typed l calculi, due to, among others, Tait [43]. His method employs a semantic interpretation of each type [ s] as a collection of strongly normalising l terms, and shows that all typable terms are indeed in these sets. A key step is to prove that lx : s:M 2 [ s t] for each M : t (for which by induction M 2 [ t] which means, by ....

Tait, W., Intensional interpretation of functionals of finite type, I. J. Symb. Log, 32, 198--212, 1967.

....with the natR and natL rules. Unfortunately, Schroeder Heister s proofs of cut elimination for logics with definitions do not extend to our setting with natural number induction. However, cut elimination does hold, and in Appendix A we give a proof adapting a technique originated by Tait [41] and extended by Martin Lof [22] This proof requires a slightly more complicated construction for the closure theorem. Theorem 3.5 (Closure of FO DeltaIN Derivations under Substitution) Suppose Pi is a FO DeltaIN derivation of Gamma Gamma B, and is an arbitrary substitution. Then ....

....in FO DeltaIN using the definition D(obj 0 ) 8m; m 1 ; m 2 : hm ; m 1 i oe hm ; m 2 i oe m 1 j m 2 ] 8m; m 1 ; m 2 : hm ; abs m 1 )i oe hm ; abs m 2 )i oe (abs m 1 ) j (abs m 2 ) 8m; m 1 ; m 2 : hm m 1 i oe hm m 2 i oe m 1 j m 2 ] 4.5. ADDITIONAL PROPERTIES 41 Theorem 4.10 (Type Preservation) The following are derivable in FO DeltaIN using the definition D(obj 0 ) 8m; n: hm ; ni oe 8t( htypeof m ti oe htypeof n ti) 8m; n: hm ; ni oe 8t( htypeof m ti oe htypeof n ti) Theorem 4.11 (Equivalence of semantics) The following are ....

W.W. Tait. Intensional interpretations of functionals of finite type I. Journal of Symbolic Logic, 32(2):198--212, 1967. BIBLIOGRAPHY 55

....show a subject reduction result in Section 3. In Sections 4 to 8, we present the formal construction needed to show that any typeable term in a typeable CS has an approximant of the same type (Theorem 8. 2) In [3] this approximation result has been obtained for LC, by a computability technique [20]. A particular problem to solve in this paper is that the approach of [3] cannot be automatically translated to a technique to use in CS, because of the absence of abstraction in CS. In order to prove the approximation result for CS, we will modify the type system slightly and introduce, in ....

....of restricting bases to their relevant contents, i.e. to contain only the types actually used for the variables of a term. In the next section, we will prove that derivations in this system are strongly normalizable; for this we will use the well known method of Computability Predicates [20]. Then, in Section 8, we will show that the approximation theorem If B E t:oe, then there exists a 2A C (t) such that B E a:oe, as well as the three normalization properties stated in the introduction of this paper, are consequences of this strong normalization result for r E . ....

[Article contains additional citation context not shown here]

W.W. Tait. Intensional interpretation of functionals of finite type I. Journal of Symbolic Logic, 32(2):198-- 223, 1967.

....Developments (FD) roughly expressing that if the generation of the parts occurring in a transformation sequence is not larger than 1, the sequence is finite. We give two proofs. In Subsection 3. 1 a proofmethod originally employed by Tait to prove termination of the simply typed lambda calculus [Tai67] is adapted yielding FD sec. Its merit is that specialising it to TRSs and the lambda calculus yields the simplest proofs of FD we know of. In Subsection 3.2 a generalisation of a proof due to De Vrijer [Vri85] for the calculus is presented, not only showing finiteness but also providing an ....

W. W. Tait. Intensional interpretations of functionals of finite type I. the Journal of Symbolic Logic, 32(2):198--212, June 1967.

....for x in r. These rules show that rst (r, s of type ) corresponds to if t then r else s, whereas rst is the usual recursor with base r (of type ) and loop s (of type ) to be passed through t times. The first three rules are called conversions, the others we call conversions. As is well known (Tait, 1967; Troelstra and van Dalen, 1988) the rewriting procedure stops for each term with a unique normal form. Hence the congruence given by the conversions is decidable. In our setting, congruent terms are identified. This P mostly saves us from needing an equalitycalculus. However the user importing ....

Tait, W. W.: 1967, `Intensional Interpretation of Functionals of Finite Type I'. The Journal of Symbolic Logic 32(2), 198--212.

.... induction on the structure of the term and extends the technique used by Crossley Shepherdson in [2] This technique is, in turn, an adaptation of Girard s proof for his system F, see [4] and utilizes the concepts of candidate for reducibility (an extension of R term originally due to Tait[9]) and neutral terms . These concepts together with the stronger induction hypothesis, namely that a 7 term is not only strongly normalizable but is also an element of a candidate for reducibilty (defined below) allow us to prove strong normalization. We now outline the proof the strong ....

W.W. Tait, Intensional interpretation of functionals of finite type I, Journal of Symbolic Logic 32 (1967) pp. 198--212.

....B . By Lemma 5.11, P 2 SN B SN A . Then P Q 2 SN A SN fi . ut 5.13. Remark. A similar technique for handling the difficult application case is due to Xi [82] There are many other proofs of strong normalization of simply typed calculus. The following is an incomplete list. Tait [73] proves (strong) normalization of Godel s system T , which extends simply typed calculus with primitive recursion. The proof makes use of the notion of (strong) computability and is quite short but complex. The proof uses the fundamental lemma of perpetuality to show that the set of (strongly) ....

W.W. Tait. Intensional interpretations of functionals of finite type I. Journal of Symbolic Logic, 32(2):190--212, 1967.

....so that the argument shows that adding fixed points to the typed system with products results in a non confluent calculus when the pairing axiom is oriented as a contraction. The calculus with pair types was shown to be strongly normalizing (SN ) by deVrier in 1982 [deV87] adapting Tait s method [Tai67]. The presence of a unit type (terminal object) spoils confluence for the traditional reduction. Say that a system has products if the types include both pairs and a terminal object. Poign e and Voss [PV87] explored a rich calculus including products, and gave proofs of termination and confluence, ....

....2. if t Gamma Gamma [f 1 f 2 ] then each f i is computable, 3. if t Gamma Gamma ht 1 ; t 2 i then t 1 and t 2 are each computable, and 4. if t Gamma Gamma oe i a then A is computable. Let C T denote the computable terms of type T , and set C to be S fC T j T a type.g Tait [Tai67] originated the strategy of using an inductively defined predicate such as computability to prove termination in the calculus. Prawitz [Pra70] pointed out the possibility of basing a notion of computability, there termed validity, based on I terms rather than on E terms (as Tait s method is) ....

W. W. Tait. Intensional interpretation of functionals of finite type I, J. Symbolic Logic 32, pp. 198-212, 1967.

....result which shows that a straightforward introduction of the subset type by just adding formation, introduction and elimination rules does not work for ITT . Proof of theorem 1. The proof of this theorem is by induction on the length of the derivation of a # A and is based on Tait s method [10] for proving normalization. However, in order to cope with subset elimination, where the interpretation by # of the type in the major premiss a # x # A B(x) is just A # , we must show that we in the interpretation of the minor premiss c(x) # C(x) x # A, y # B(x) can do without the ....

W.W. Tait. Intensional interpretation of functionals of finite type I. Journal of Symbolic Logic Vol. 32, No. 2, June 1967, pp. 198-212.

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W. W. Tait. Intensional interpretation of functionals of finite type. Journal of Symbolic Logic, 32(2), 1967.

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W. W. Tait. Intensional interpretation of functionals of finite type I. Journal of Symbolic Logic, 32:198--212, 1967.

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W. W. Tait. Intensional interpretation of functionals of finite type I. Journal of Symbolic Logic, 32:198--212, 1967. 30

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Tait, W.W.: Intensional interpretations of functionals of finite type i. Journal of Symbolic Logic 32(2) (1967) 198--212

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W.W. Tait. Intensional interpretation of functionals of finite type I. Journal of Symbolic Logic, 32:198--212, 1967.

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Tait, W. W., Intensional interpretations of functionals of finite type I, Journal of Symbolic Logic 32 (1967), pp. 198--212.

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W.W. Tait, "Intensional Interpretation of Functionals of Finite Type", J. of Symbolic Logic 32, 1967, 198-212.

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Tait, W. W.: 1967, `Intensional Interpretations of Functionals of Finite Type I'. Journal of Symbolic Logic 32(2), 198--212.

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W.W. Tait. Intensional interpretation of functionals of finite type I. Journal of Symbolic Logic, 32(2):198--223, 1967.

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W. W. Tait. Intensional interpretation of functionals of finite type I. J. Symbolic Logic, 32:198--212, 1967.

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William W. Tait. Intensional interpretations of functionals of finite type I. Journal of Symbolic Logic, 32:198--212, 1967.

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Tait W.W.,"Intensional Interpretation of Functionals of Finite Types I", J. Symbolic Logic 32, 1967, 198-212.

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W. W. Tait. Intensional interpretations of functionals of finite type I. Journal of Symbolic Logic, 32(2):198--212, 1967.

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W. W. Tait. Intensional interpretation of functionals of finite type I. Journal Of Symbolic Logic, 32:198--212, 1967.

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W. W. Tait. Intensional interpretation of functionals of finite type I. Journal Of Symbolic Logic, 32:198--212, 1967.

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W.W. Tait. Intensional interpretations of functionals of finite type I. Journal of Symbolic Logic, 32:198--212, 1967. 33

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W.W. Tait. Intensional interpretations of functionals of finite type I. Journal of Symbolic Logic, 32(2):198--212, 1967.

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W. W. Tait. Intensional interpretations of functionals of finite type I. Journal of Symbolic Logic, 32:198--212, 1967.

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W. W. Tait. Intensional interpretations of functionals of finite type I. Journal of Symbolic Logic, 32(2):198--212, 1967. Cited on page 2.

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William W. Tait. Intensional interpretations of functionals of finite type I. Journal of Symbolic Logic, 32:198--212, 1967.

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William W. Tait. Intensional interpretations of functionals of finite type I. Journal of Symbolic Logic, 32:198--212, 1967.

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W. W. Tait. Intensional interpretations of functionals of finite type I. Journal of Symbolic Logic, 32:198--212, 1967.

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W.W. Tait, "Intensional interpretations of functionals of finite type". Journal of Symbolic logic Volume 32, Number 2, June 1967.

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W. Tait. Intensional interpretation of functionals of finite type I. Information and Computation, 32:198--212, 1967.

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W.W. Tait,"Intensional Interpretation of Functionals of Finite Types I", J. Symbolic Logic 32, 1967, 198-212.

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W. Tait. Intensional interpretation of functionals of finite type I. Information and Computation, 32:198--212, 1967.

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Tait W.W., "Intensional Interpretation of Functionals of Finite Types I", J. Symbolic Logic, 32, 1967, 198-212.

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