AN INTENSIONAL TYPE THEORY: MOTIVATION ... - Semantic Scholar
The Journal of Symbolic Logic Volume xx, Number x, xxxx xxxx
AN INTENSIONAL TYPE THEORY: MOTIVATION AND CUT-ELIMINATION
PAUL C GILMORE
Abstract. By the theory TT is meant the higher order predicate logic with the following recursively defined types: (1) 1 is the type of individuals and [] is the type of the truth values; (2) [τ1 , . . . , τn ] is the type of the predicates with arguments of the types τ1 , . . . , τn . The theory ITT described in this paper is an intensional version of TT. The types of ITT are the same as the types of TT, but the membership of the type 1 of individuals in ITT is an extension of the membership in TT. The extension consists of allowing any higher order term, in which only variables of type 1 have a free occurrence, to be a term of type 1. This feature of ITT is motivated by a nominalist interpretation of higher order predication. In ITT both well-founded and non-well-founded recursive predicates can be defined as abstraction terms from which all the properties of the predicates can be derived without the use of non-logical axioms. The elementary syntax, semantics, and proof theory for ITT are defined. A semantic consistency proof for ITT is provided and the completeness proof of Takahashi and Prawitz for a version of TT without cut is adapted for ITT; a consequence is the redundancy of cut.
§1. Introduction. Consider a form TT of the simple theory of types in which predicates of any number of arguments are admitted, but no functions. The types of such a predicate logic can be recursively defined as follows: (1) 1 is the type of individuals and [] is the type of the truth values; (2) [τ1 , . . . , τn ] is the type of the predicates with arguments of the types τ1 , . . . , τn , where n ≥ 1. Thus the type [τ1 , . . . , τn ] can be taken to be the type [] when n=0. Apart from notation and the exclusion of functions, the types of TT are the types of Sch¨ utte’s type theory [26]. Although the types of TT are traditionally thought of as necessary for the consistency of the logic, the types can just as well be seen to arise naturally from the predicate and subject distinction of natural languages, for these become the distinction between a predicate and its argument(s). Skepticism has often been expressed that a violation of the type restrictions is the ultimate source of the xxxxx The financial support of the Natural Science and Engineering Research Council of Canada is gratefully acknowledged. c xxxx, Association for Symbolic Logic 0022-4812/xx/xxxx-xxxx/xxxx
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paradoxes. For example, in the concluding paragraph of [6] Church comments on a remark of Frege’s “ . . . Frege’s criticism seems to me still to retain much of its force, and to deserve serious consideration by those who hold that the simple theory of types is the final answer to the riddle of the paradoxes”. Here, as in [27] and [28], an alternative explanation of the paradoxes is offered; namely that they result from a confusion of use and mention when higher order predication is given a nominalist interpretation. A quote from the introduction to [25] is relevant: “Broadly, the result is an outlook which is less Platonic, or less realist in the medieval sense of the word. How far it is possible to go in the direction of nominalism remains, to my mind, and unsolved question, but one which, whether completely solvable or not, can only be adequately investigated by means of mathematical logic.” In the subsections of this introduction to follow, the motivation for the features of ITT that distinguish it from TT is described and some important results obtainable in ITT are sketched. The syntax of ITT is described in §2, the semantics in §3, and the proof theory in §4 in the style of semantic tableaux. A semantic consistency proof is provided in §5. A proof that cut is a redundant rule of the logic is provided in §6; it is an adaptation of the proofs for Sch¨ utte’s type theory provided in [24] and [31]. Finally in §7 two extensions of ITT are described. The first introduces a notation for partial functions with values of type 1 in the traditional manner using definite descriptions; but its development within ITT permits simplifications of the syntax and semantics. The second extension of ITT demonstrates that a solution to the “riddle of the paradoxes” can be found by maintaining a careful distinction between the use and mention of predicate names in a nominalist interpretation of higher order predication. For example, in this consistent extension of ITT Russell’s set can be defined. Details for both logics are provided in [18]. 1.1. Intensional and extensional identity. Following Carnap [5], intensional and extensional identities are defined first before the intension and extension of terms are described. In TT these identities are defined: df
= = (λu, v.∀X.[X(u) → X(v)]) df
=e = (λu, v.∀x.[u( x ) ↔ v( x )]) Here the notation x is used as an abbreviation for x1 , . . . , xn , where n ≥ 0. The type restrictions necessary for the definitions can then be expressed as v:†[u] and X:[†[u]] for the first, and as u, v:[†[x1 ], . . . , †[xn ]] for the second. Here †[cv] denotes the type of a constant or variable cv that is assigned to it by the primitive syntax; and ‘:’ denotes the relationship between a term and its type. Using the usual infix notation for the identities, a theorem of TT is: IE.1) ∀X, Y.[X = Y → X =e Y ], where necessarily †[X] is not 1. The comparable result in the set theory ZF [29] is IEZF) ∀x, y.[x = y → x =e y] where here = is the primitive identity of a first order logic with identity; the sequent is actually a theorem of that logic. The converse of (IEZF), the axiom of extensionality, is the first axiom in the first formulation of ZF [32]. In TT the axiom is the converse of (IE.1), and is often accepted as an axiom of TT:
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EI) ∀X, Y.[X =e Y → X = Y ]. An axiom of extensionality may be acceptable for a pure logic, but it is not for an applied logic trying to meet the needs of computer science. In the design of databases it is essential to understand the distinction between the intension of a predicate and its extension; the former may only be known informally while the latter is provided by data entry. Consider, for example, the extension of an Employee predicate in a company database. It will most likely be maintained by data entry along with a Sex predicate among others. From these two predicates the intension of a predicate MaleEmployee can be defined, and its extension retrieved and printed [15]. By an accident of hiring, however, the two predicates Employee and MaleEmployee may have the same extension; but clearly their intensions must be distinguished. Another example is provided by [22] where a distinction is made between “extensional” and “intensional” occurrences of predicate variables. For this reason the axiom (EI) concluding the intensional identity of predicates from their extensional identity is not accepted in ITT. Instead rules of intensionality are part of the proof theory; they conclude that two predicates are intensionally identical if and only if their names are intensionally identical. 1.2. Nominalism and higher order predication. In a logic of extensions such as TT with (EI) as axiom, a higher order predicate with predicates as arguments is understood to have the extensions of the predicates as arguments. But in a logic in which the intension of a predicate is distinguished from its extension, higher order predication must be re-examined. A nominalist understands a predicate of a universal to be a predicate of a name of the universal. For example, a nominalist understands ‘Yellow is a colour’ to mean ‘Yellow is a colour-word’; the sentence is understood as a description of the use of the word ‘Yellow’ in English. Since computers are consummate nominalists, nominalist interpretations of languages intended for computer applications are needed. But this does require a careful distinction between the use and mention of predicate names, especially when treating abstraction and quantification. For example, in ‘Yellow is a colour-word’ the predicate name ‘Yellow’ is being mentioned while the predicate name ‘colour-word’ is being used. The distinction between the use and mention of predicate names is maintained in the logic ITT as follows: The types of ITT are the same as the types of TT, but the membership of the type 1 of individuals in ITT is an extension of the membership of the same type in TT. The extension consists in adding to the membership of the type 1 any higher order term in which at most variables of type 1 have a free occurrence. A constant C that is a predicate name is necessarily of some type that is not 1 and always has that type in contexts where it is used. But since no variable has a free occurrence in C, C is also of type 1 and it has that type in contexts in which it is being mentioned. Thus C(C) is of type [] if C is of type [1]. A related method of maintaining the distinction between the use and mention of predicate names was used in the predecessors of ITT described in [12], [13], [14] and [16]. The papers prior to [16] attempted to provide a formal set theory that maintains the distinction, but had an awkward notation among its failings. The logic NaDSyL described in [16] is a recent attempt to overcome these
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failings; for example, it uses the notation of the lambda calculus. But a closer study of NaDSyL has revealed that the technique it employs for maintaining the distinction could be employed more simply within a type theory. Mentioning the name of a predicate means that the name is implicitly quoted. This is the reason why higher order terms that are also of type 1 must be restricted to those in which only variables of type 1 have free occurrences. For only such terms can be given a Herbrand interpretation when quoted. For example, let C be a constant of type [1] and x a variable of type 1. Then C(x) is of type [], and also of type 1. As a type 1 term, C(x) is to be interpreted as the function with domain and range the constant type 1 terms: When x is interpreted as a constant type 1 term t, then C(x) is interpreted as the constant type 1 term C(t). Such an interpretation can’t be given for higher order terms not satisfying the restriction. For example, if X is a variable of type [1] and c a constant of type 1, then X(c) is of type [] but not also of type 1. 1.3. Set theory and the lambda calculus. For (IE.1) to be derivable in TT it is necessary that the type of X and Y be other than 1. Although (IE.1) is not derivable in ITT when the the type of X and Y is 1, each instance of the following sequent scheme is derivable when R and S are terms of type τ with dual type 1: IE.2) R = S R =e S. This scheme is similar to the scheme that would result from all possible instantiations of (IEZF) of ZF. As a consequence, the logic ITT combines features from set theory and from a lambda calculus based logic; for this reason it may satisfy the requirements for such a logic described in [20]. Consider for example the following definitions of zero and successor df
0 = (λu.¬u = u) df
S = (λu, v.u = v) where =: [1, 1]. They are definitions in the style of set theory; but all of Peano’s axioms are derivable in ITT including the following pair: S.1) ∀x, y.[S(x) = S(y) → x = y] S.2) ∀x.¬S(x) = 0. Dual typing is critical for their derivations: 0:[1] and S:[1, 1], but also 0:1 and S(x):1 when x:1. The lambda calculus definition of ordered pair from [7] is also available df
: = (λu, v, w.w(u, v)) and the following sequents can be derived: OP.1) ∀x1, y1, x2, y2.[ x1, y1 = x2, y2 → x1 = x2 ∧ y1 = y2], OP.2) ∀x, y.¬ x, y = 0. Details are provided in [17] and [18]. Also provided there is a foundation for recursions. Both well-founded and non-well-founded recursive predicates are defined there using a decidable set of terms called recursion generators; the technique is demonstrated using higher order Horn sequent definitions. This method of defining recursive predicates overcomes some of the complications that arise from their introduction into the applied versions TPS, HOL, and PVS of Church’s formulation of type theory. [1], [19], [23].
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Since non-well-founded predicates can be defined in ITT, there appears to be no obstacle to reproducing within ITT most of the applications of non-wellfounded sets described in [3]. Acknowledgments. Comments by Nino Cocchiarella on an earlier paper and his [8] and [9] encouraged the change from the set theoretic notation of [14] to the lambda notation of [16] and ITT. An observation of Henk Barendregt was critical for the realization that the technique employed in [16] for maintaining the distinction between use and mention could be more simply maintained in ITT. Conversations with Alasdair Urquhart, Peter Apostoli, and Michael Donat have been helpful for details of this paper, as has correspondance with Hendrik Boom, Per Martin-L¨ of, and Joergen Villadsen. The paper has also benefitted from several recommendations made by a referee. §2. The syntax. Type membership for ITT is defined in §2.1, providing the basic syntax for the logic. The lambda reduction relation between terms of ITT is defined in §2.2. In §2.3 some more usual notations for predicate logics are introduced by definitions. 2.1. Type membership. The logic ITT is assumed to have denumerably many constants and variables of each type. In the style of [7], special constants introduce the logical connectives and the quantifiers. The binary logical connective of joint denial, denoted by ↓, is a special constant of type [[],[]]; that is, it is a predicate of two arguments of type []; it is the only primitive logical connective needed since the more usual logical connectives can be defined in terms of it. Similarly a special constant ∀ of type [[τ ]] is introduced for each type τ ; it is the universal quantifier for a type τ variable. The type of each ∀ is not displayed but must be inferred from context. The existential quantifier ∃ is defined in terms of ∀ in the usual way. The notation †[cv] introduced in §1.1 is used in the following definition of type membership; it denotes the unique type assigned to each constant or variable cv by the primitive syntax: (1) (2) (3) (4)
cv:†[cv], cv a constant or variable; ↓:[[], []]; and ∀:[[τ ]] for each type τ . M :[τ, τ ] and N :τ ⇒ (M N ):[ τ ]. M :[ τ ] ⇒ (λx.M ):[†[x],τ ]. M :[ τ ] ⇒ M :1, provided each variable with a free occurrence in M is of type 1.
The unusual clause (4) results from the nominalist interpretation discussed in §1.2. The type 1 assigned to M in (4) is called the dual type of M . Because of this clause a constant c for which †[c] is not of type 1 has 1 as a dual type; nevertheless, †[c] is always to be understood to be the type assigned by the primitive syntax, that is the syntax of TT, and never the dual type 1 assigned by clause (4). No variable has 1 as a dual type nor does a constant c for which †[c] is 1. By a term is meant a member of a type. Note that a term of type 1 is a constant, a variable, or a term of dual type 1 since clauses (1) and (4) are the only ones that yield terms of type 1.
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Let N :†[v]. The substitution notation [N/v]M denotes the result of replacing each free occurrence of v in a term M by N . The notation can result in changes of bound variables within M ; a change is necessary if a free occurrence of v in M is within the scope of an abstractor λy for which y has a free occurrence in N. The count ct[M ] of a term M is defined: ct[cv] = 0 for any constant, including ↓ and ∀, or variable cv; ct[M N ] = ct[M ] + ct[N ] + 1; and ct[λx.M ] = ct[M ] + 1. A proof by induction on ct[M ] of the following lemma will be left to the reader. Lemma (Substitution). Let N :†[v] and M :τ . Then [N/v]M :τ . 2.2. Lambda reductions. The relation of immediate lambda reduction between terms is denoted here by >, and allows α-, β-, and η-reductions [2]. The reflexive and transitive closure of > is denoted by . Many proofs are available for the following theorem of the pure lambda calculus Theorem (Church–Rosser). If M N and M P , then there exists a Q for which N Q and P Q. Clause (4) in the definition of type membership has little effect on the proofs provided in [2] for the theorem as demonstrated in [18]. The theorem ensures that by selecting one of the bound variable variants of a term in normal form as the representative normal form, a unique normal form may be assigned to each term of ITT. 2.3. Formula notations. A formula of ITT is a term of type []. Formulas are the basis for the proof theory for ITT described in §4. But first the sparse notation of the lambda calculus is extended by definitions that introduce notations more common to predicate logics. The application notation is “sugared” by the definitions: df M (N ) = M N df
M (N1 , . . . , Nm , N ) = M (N1 , . . . , Nm )(N ) The prefix notation for ↓ is replaced by an infix notation, and the logical connectives ¬ and ∧ are defined; all other conventional connectives can be defined from them: df [M ↓ N ] =↓ (M, N ) ¬M
df
= [M ↓ M ] df
[M ∧ N ] = [¬M ↓ ¬N ]. A conventional notation for the universal quantifier is defined, and the existential quantifier is defined from it in the usual way: df ∀x.M (x) = ∀(M ) df
∃x.F = ¬∀x.¬F where M :[†[x]] and x has no free occurrence in M . Parenthesis have been dropped here and will continue to be when there is no risk of confusion. §3. Semantics. The definition of a valuation for a type theory requires the prior definition of a domain D(τ ) for each type τ . For the extensional theory TT the standard domains are a nonempty set D(1) of entities for the type 1, the set D([]) with members the truth values represented by + and −, and the
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sets D([τ1 , . . . , τn ]) consisting of all the subsets of the Cartesian product of the sets D(τ1 ), . . . , D(τn ). Thus for these domains only the potential extensions of predicates are provided. To provide a semantics for the intensional type theory ITT, it is necessary to provide potential intensions as well as extensions. This is accomplished in §3.1. But first the variables of type τ are mapped 1-1 onto new constants of type τ . Given a term M , the term M ∗ is obtained from M by replacing each free occurrence of a variable in M by the constant onto which it is mapped. If N is the normal form for a term M of ITT, then N ∗ is the normal form for M ∗ . Although these new constants are not required, their use does simplify the presentation of the semantics. 3.1. Domains, valuations, and models. A domain for an intensional valuation of ITT is a function D defined for each type τ for which: D(1) is { N ∗ , N ∗ N :1 in normal form} D([]) is { N ∗ , s s ∈ S[N ∗ ] and N :[] in normal form} D([ τ ]) is { N ∗ , s s ∈ S[N ∗ ] and N :[ τ ] in normal form} where S[N ∗ ] for D([]) is a non-empty subset of {+, −}, and for D([ τ ]) is a non-empty subset of the set of all subsets of the Cartesian product of the sets D(τ1 ), . . . , D(τn ). A pair N ∗ , N ∗ from D(1) can be understood to be an intension of an individual name and a possible extension for it. This becomes clearer if each constant c for which †[c] is 1 is understood to represent a member of a non-empty set of entities. Then D(1) can be defined to be the union of the set { c, e c:1 and e an entity} and the set { N ∗ , N ∗ N of dual type 1 and in normal form}. The simpler definition of D(1) is used for the proofs to follow. The first member of a pair that is a member of D(τ ), τ = 1, is a potential intension for a predicate of type τ , while the second member is a potential extension for that intension. Thus, for example, members F ∗ , + and G∗ , + of D([]) are extensionally identical but not intentionally if F and G have distinct normal forms. Similarly members P ∗ , s1 and Q∗ , s2 are extensionally identical if s1 and s2 have the same members, but not intensionally if P and Q have distinct normal forms. Note that to define a domain D it is sufficient to define the sets S[N ∗ ]. For example, the standard domain for ITT is the domain for which each S[N ∗ ] in D([]) is {+, −} and each S[N ∗ ] for D([ τ ]) is the set of all subsets of the Cartesian product. A valuation to a given domain D is a function Φ, defined for all τ and M for which M :τ , and such that V.1) Φ(†[c], c) is c, c ∈ D(1), when †[c] is 1; Φ(†[c], c) is c, s ∈ D(†[c]), for each constant c and some s, when †[c] = 1 and c is neither ↓ nor ∀; Φ(†[v], v) ∈ D(†[v]), for each variable v. The value of Φ in (V.1) is an ordered pair so that Φ1 and Φ2 can be defined to be the functions for which Φ(τ, M ) is Φ1 (τ, M ), Φ2 (τ, M ) . The value of
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Φ1 (τ, M ) for the cases (V.2) to (V.6) to follow is defined to be the normal form of [Φ1 (†[x1 ], x1 )/x1 ] · · · [Φ1 (†[xn ], xn )/xn ]M where x1 , . . . , x1 are all the variables with a free occurrence in M . Since no free variable occurs in any Φ1 (†[xi ], xi ), no variable has a free occurrence in Φ1 (τ, M ). In particular, Φ1 (τ, M )∗ is Φ1 (τ, M ). The function Φ2 (τ, M ) is defined in the remaining cases as follows: V.2) Φ2 ([[], []], ↓) is the set of pairs of pairs Φ1 ([], F ), − , Φ1 ([], G), −
. V.3) Φ2 ([[τ ]], ∀) is the set of pairs Φ1 ([τ ], M ), D(τ ) . V.4) Φ2 (1, M ) is Φ1 (1, M ), when M has dual type 1. V.5) Φ2 ([ τ ], M N ), when τ is not empty, is the set of tuples e for which Φ(τ, N ), e ∈ Φ2 ([τ, τ ], M ). Φ2 ([], M N ) is + if Φ(τ, N ) ∈ Φ2 ([τ ], M ) and is − otherwise. For the last condition, Φx is an x-variant of Φ; that is, it is a valuation for which Φx (†[x], y) differs from Φ(†[x], y) at most when y is x. V.6) Φ2 ([†[x], τ ], λx.M ), when τ is not empty, is the set of tuples Φx (†[x], x), e for which e ∈ Φx2 ([ τ ], M ) for some x-variant Φx of Φ. Φ2 ([†[x]], λx.M ) is the set of pairs Φx (†[x], x) for which Φx2 ([], M ) is + for some x-variant Φx of Φ. A proof of the following lemma is needed for the theorem of §3.2. Lemma (Semantic Substitution). Let Φ be a valuation to some domain. Let Q:†[x], where x has no free occurrence in Q, and let P :σ. Let Φx be the x-variant of Φ for which Φx (†[x], x) is Φ(†[x], Q). Then Φx (σ, P ) is Φ(σ, [Q/x]P ). Proof. It may be assumed that x has a free occurrence in P . No matter the form and type τ of P , Φx1 (σ, P ) is the normal form of [Φ1 (†[x], x)/x][Φ1 (†[x1 ], x1 )/x1 ] . . . [Φ1 (†[xn ], xn )/xn ]P where x, x1 , . . . , xn are all the variables with a free occurrence in P . Hence Φx1 (σ, P ) is the normal form of [Φ1 (†[x], Q)/x][Φ1 (†[x1 ], x1 )/x1 ] . . . [Φ1 (†[xn ], xn )/xn ]P which is the normal form of [Φ1 (†[x1 ], x1 )/x1 ] . . . [Φ1 (†[xn ], xn )/xn ][Φ1 (†[x], Q)/x]P since none of the variables x1 , . . . , xn have a free occurrence in Φ1 (†[x], Q). Therefore Φx1 (σ, P ) is Φ1 (σ, [Q/x]P ) as required.
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It follows by (V.4) that Φx2 (1, P ) is Φ2 (1, [Q/x]P ) when P has dual type 1. It is sufficient, therefore to prove that Φx2 (σ, P ) is Φ2 (σ, [Q/x]P ) when σ is not the dual type 1. The proof is by induction on ct[P ]. When ct[P ] is 0, P is necessarily x so the conclusion is immediate. Assume the conclusion of the lemma whenever ct[P ] < ct. Let ct[P ] be ct, and consider the forms that P may take. • P is M N , where M :[τ, τ ], N :τ , and σ is [ τ ]. When τ is not empty, Φx2 (σ, P ) is the set of tuples e for which Φx (τ, N ), e ∈ x Φ2 ([τ, τ ], M ); that is, for which Φ(τ, [Q/x]N ), e ∈ Φ2 ([τ, τ ], [Q/x]M ). Therefore Φx2 ([ τ ], P ) is Φ2 ([ τ ], [Q/x]P ). When τ is empty, Φx2 ([], P ) is + ⇔ Φx (τ, N ) ∈ Φx2 ([τ ], M ) ⇔ Φ(τ, [Q/x]N ) ∈ Φ2 ([τ ], [Q/x]M ) ⇔ Φ([], [Q/x]P ) is +. • P is λy.M , where it may be assumed that y has no free occurrence in Q and is not x. Thus [Q/x](λy.M ) is (λy.[Q/x]M ). Let M :[ τ ] so that σ is [†[y], τ ]. When τ is not empty, Φx2 (σ, P ) is the set of tuples Φyx (†[y], y), e for which e ∈ Φyx 2 ([ τ ], M ); that is the set of tuples Φy (†[y], y), e for which e ∈ Φxy 2 ([ τ ], M ). Hence, by the induction assumption, for which e ∈ Φy2 ([ τ ], [Q/x]M ). Thus Φx2 (σ, P ) is Φ2 (σ, [Q/x]P ) as required. When τ is empty, Φx2 ([†[y]], P ) is the set of pairs Φyx (†[y], y) for which x Φyx 2 ([], M ) is + for some y-variant of Φ . By the induction assumption this is the set of pairs Φy2 (†[y], y) for which Φy2 ([], [Q/x]M ) is + for some y-variant of Φ. Thus Φx2 ([†[y]], P ) is Φ2 ([†[y]], [Q/x]P ) as required. 3.2. Intensional models. An intensional model for ITT with domain D is a valuation Φ to D that satisfies two conditions: IM.1) Φ(τ, P ) ∈ D(τ ), for each type τ and term P of type τ . IM.2) Let P , Q:τ each have dual type 1. Then Φ2 ([], P =1 Q) is ± ⇒ Φ2 ([], P =τ Q) is ±, respectively, where =1 is = of type [1, 1], and =τ is = of type [τ, τ ]. Clause (IM.1) is the condition that originated with [21] for a valuation of type theory to be a model . Clause (IM.2) expresses that the intension of a predicate is identified with its name. For let Φ2 ([], P = Q) be ±, where = :[1, 1]. Then since the terms P and Q occur in a type 1 context, they are being mentioned. From the fact that Φ2 ([], P = Q) is ± it can be concluded that P and Q are identical, respectively, not identical terms and therefore must name intensionally identical, respectively, not identical predicates in the context P = Q when =:[τ, τ ]; that is, Φ2 ([], P = Q) must be ±. Theorem. Any valuation to the standard domain is an intensional model. Proof. Let Φ be a valuation to the standard domain. That Φ satisfies (IM.1) is immediate. To prove that it satisfies (IM.2) it is sufficient to prove that for any type τ it satisfies the following condition:
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Φ2 ([], P =τ Q) is ± ⇒ Φ1 (τ, P ) Φ1 (τ, Q), respectively, Φ1 (τ, P ) Φ1 (τ, Q)
where =τ :[τ, τ ], and expresses that Φ1 (τ, P ) and Φ1 (τ, Q) have the same normal form. That (IM.2) follows from (NF) is a consequence of the definition of domain given in §3.1. A proof of (NF) follows. Φ2 ([], P =τ Q) is ± ⇒ Φ2 ([], ∀(λz.[z(P ) → z(Q)])) is ±, z not free in P or Q ⇒ Φ([[τ ]], (λz.[z(P ) → z(Q)])) ∈, ∈ / Φ2 ([[[τ ]]], ∀) ⇒ Φ2 ([[τ ]], (λz.[z(P ) → z(Q)]) is, is not D([τ ]). But Φ2 ([[τ ]], λz.[z(P ) → z(Q)]) is {Φz ([τ ], z) Φz2 ([], [z(P ) → z(Q)]) is +}. By (IM.1), Φ(τ, P ), Φ(τ, Q) ∈ D(τ ) so that (λv.P ∗ =τ v), {Φ(τ, P )} , (λv.Q∗ =τ v), {Φ(τ, Q)} ∈ D([τ ]). z Let Φ be the z-variant for which Φz ([τ ], z) is (λv.P ∗ = v), {Φ(τ, P )} . Then Φz2 ([], [z(P ) → z(Q)]) is + ⇔ Φz2 ([], z(P )) is − or Φz2 ([], z(Q)) is + ⇔ Φz (τ, P ) ∈ / Φz2 ([τ ], z) or Φz (τ, Q) ∈ Φz2 ([τ ], z) ⇔ Φ(τ, P ) ∈ / Φz2 ([τ ], z) or Φ(τ, Q) ∈ Φz2 ([τ ], z), since z not free in P or Q, ⇔ Φ(τ, P ) is Φ(τ, Q), since Φ(τ, P ) ∈ Φz2 ([τ ], z). ⇔ Φ1 (τ, P ) Φ1 (τ, Q) Thus any valuation to the standard domain satisfies (NF) and therefore both (IM.1) and (IM.2). 3.3. Semantic inferences. Let the valuation Φ be a model of ITT. Since Φ is a function, the value Φ2 (τ, M ) for M :τ is unique. In particular, if F :[] then Φ2 ([], F ) has as its value exactly one of ±. This observation together with the following theorem provides the justification for the proof theory described in §4; a proof of the theorem, which depends on the substitution and semantic substitution lemmas, is left to the reader. A proof is available in [18]. Theorem. Let Φ be a model. ↓) Let F , G:[]. Then Φ2 ([], [F ↓ G]) is + ⇒ Φ2 ([], F ) is − and Φ2 ([], G) is −; Φ2 ([], [F ↓ G]) is − ⇒ Φ2 ([], F ) is + or Φ2 ([], G) is +. ∀)
Let P :[τ ]. Then Φ2 ([], ∀P ) is + ⇒ Φ2 ([], P (t)) is + for all t:τ ; Φ2 ([], ∀P ) is − ⇒ Φ2 ([], P (y)) is − for some y-variant Φy of Φ, y not free in P .
λ)
Let F > G. Then Φ2 ([], F ) is ± ⇒ Φ2 ([], G) is ±.
Int)
Let P ,Q:τ have dual type 1. Then Φ2 ([], P =1 Q) is ± ⇒ Φ2 ([], P =τ Q) is ±, respectively, where =1 :[1, 1], and =τ :[τ, τ ].
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3.4. Sequents and counter-examples. A sequent is an expression Γ Θ where Γ, the antecedent of the sequent, and Θ, the succedent of the sequent are finite, possibly empty, sets of formulas. A sequent Γ Θ is satisfied by a valuation Φ, if Φ2 ([], F ) is − for some F in the antecedent or is + for some F in the succedent. A sequent is valid if it is satisfied by every valuation that is a model. A valuation Φ is a counter-example for a sequent if Φ2 ([], F ) is + for every F in the antecedent and − for every F in the succedent. The proof theory of §4 provides a systematic search procedure for a counterexample for a given sequent Γ Θ. Should the procedure fail to find such a valuation, and if it does fail it will fail in a finite number of steps, then that Γ Θ is valid follows from a theorem of §5. The steps resulting in a failure are recorded as a derivation. Thus a derivation for a sequent is constructed under the assumption that a counter-example Φ exists for the sequent. Signed formulas are introduced to abbreviate assertions about the truth value assigned to a formula by Φ. Thus +F is to be understood as an abbreviation for “Φ2 ([], F ) is +” and −F for “Φ2 ([], F ) is −”, for some conjectured counter-example Φ. Note that Γ Θ has no counter-example if Γ Θ has no counter-example, where Γ ⊆ Γ, Θ ⊆ Θ, and Γ ∪ Θ is not empty.
§4. Proof theory. The proof theory is presented as a logic of sequents using a semantic tree form of the sequent calculus that has evolved from the semantic tableaux derivations of [4]. A similar proof theory for first order logic is described in [30] and in [11]. Semantic rules, in terms of which semantic trees are defined, are described in §4.1; these rules are motivated by the theorem of §3.3. A derivation of a sequent is a closed semantic tree based on the sequent, as these terms are defined in §4.2. 4.1. The semantic rules. There are four + and − pairs of rules for the propositional connective ↓, for the quantifier ∀, for λ and for intensionality. These rules are +↓
+∀
+[F ↓G]
+[F ↓G]
−F
−G
−↓
+F −∀
+∀P
−λ
+F
−∀P
−F −G
+G where F >G
+G
−P (y) y the eigen variable
+P (t) t the eigen term +λ
−[F ↓G]
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+Int
+P =1 Q
+Int
−P =1 Q
+P =τ Q −P =τ Q where P and Q have primary type τ and dual type 1 The last rule has a character different from these logical rules. It has no premiss and two conclusions: Cut +F
−F
Although the Cut rule is redundant, it is neverthless a useful rule since it permits the reuse of derivations.[18] 4.2. Semantic trees and derivations. A semantic tree is a binary tree with nodes that are signed formulas. A semantic tree based on a given sequent is defined as follows: (1) A tree with a single branch consisting of one or more nodes +F and −G, where F is from the antecedent and G is from the succedent of the sequent, is a semantic tree based on the sequent. These are the initial nodes of the tree. (2) Given a semantic tree based on a sequent, a tree obtained from it in any of the following ways is a semantic tree with the same initial nodes based on the sequent: (a) By attaching the conclusion of a single conclusion rule to the leaf of a branch that has a node that is the premiss of the rule, provided that if the rule is −∀ then the eigen variable has no free occurrence in the premiss of the rule or in any node above it. (b) By attaching the two conclusions of the +↓ rule on separate branches to the leaf of a branch that has a node that is the premiss of the rule. (c) By attaching +F and −F on separate branches to the leaf of a branch. A branch of a semantic tree is closed if there is a closing pair of nodes +F and −F on the branch. A semantic tree is closed if each of its branches is closed. A derivation of a sequent is a closed semantic tree based on the sequent. §5. Consistency. Since by the theorem of §3.2 any valuation to the standard domain is an intensional model for ITT, there exist valid sequents. Their relationship with derivable sequents is expressed in the next theorem. Theorem (Consistency). A derivable sequent of ITT is valid. Proof. Consider a derivation for a sequent Γ Θ. Let η be any node of the derivation which does not have an initial node below it. Define Γ[η] and Θ[η] to be the sets of sentences F for which +F , respectively −F , is η itself or is a node above η. Thus if η is the last of the initial nodes of the derivation, Γ[η] ⊆ Γ and Θ[η] ⊆ Θ; hence if Γ[η] Θ[η] is satisfied by a model, so is Γ Θ. Define the height h(η) of a node η to be the maximum of the heights of η on branches on which it occurs, with its height on a branch being the number of
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nodes below it on the branch. Γ Θ will be shown to be valid by induction on h(η). If h(η) = 0, then η is a leaf node of a branch of the derivation. Since the branch is closed, Γ Θ is valid. Assume therefore that h(η) > 0, and that there is a model Φ that is a counter-example for Γ Θ. Necessarily η is immediately above a conclusion η1 or conclusions η1 and η2 of one of the rules of deduction. There are therefore two main cases to consider corresponding to the single conclusion rules +↓, ±∀, ±λ, and ±Int, and to the two conclusion rules −↓ and Cut. • For the single conclusion rules it is sufficient to illustrate the argument with the ±∀ rules with premiss ±∀P and conclusion respectively +P (t) and −P (y), where P :[τ ] and t, y:τ with y not occurring free in any node above the conclusion −P (y) of the rule −∀. For the + case, ∀P ∈ Γ[η], Γ[η1] is Γ[η] ∪ {P (t)}, and Θ[η1] is Θ[η]. Since Φ satisfies Γ[η1] Θ[η1] but does not satisfy Γ[η] Θ[η] it follows that Φ2 ([], ∀P ) is + and Φ2 ([], P (t)) is − which contradicts clause (∀) of the theorem of §3.3. For the − case, Γ[η1] is Γ[η], ∀P ∈ Θ[η] , and Θ[η1] is Θ[η] ∪ {P (y)}. Since Φ satisfies Γ[η1] Θ[η1] but does not satisfy Γ[η] Θ[η] it follows that Φ2 ([], ∀P ) is − and Φ2 ([], P (y)) is +. But further, since y has no free occurrence in a member of Γ[η] ∪ Θ[η], and since Γ[η1] Θ[η1] is valid, Φy2 ([], P (y)) is + for every y-variant Φy of Φ. This contradicts clause (∀) of the theorem. • Let the premiss of an application of −↓ be −[F ↓ G] and the conclusions be +F and +G. Thus [F ↓ G] ∈ Θ[η], Γ[η1] is Γ[η] ∪ {F }, Γ[η2] is Γ[η] ∪ {G}, Θ[η1] is Θ[η], and Θ[η2] is Θ[η]. As before it follows that Φ2 ([], F ) is Φ2 ([], G) is − so that Φ2 ([], [F ↓ G]) is +. This contradicts clause (↓) of the theorem. For the case of cut, let the cut formula be F . In this case Γ[η1] is Γ[η] ∪ {F }, Θ[η1] is Θ[η], Γ[η2] is Γ[η], and Θ[η2] is Θ[η] ∪ {F }. It follows therefore that Φ2 ([], F ) is both + and −, which is impossible. §6. Completeness. Let a derivation of a sequent be understood to be a derivation in which cut is not used, and let an underivable sequent be one which does not have a derivation without cut. The following theorem is a consequence of lemmas stated below in §6.1 and §6.2. Theorem (Completeness without Cut). A counter-example that is a model exists for each underivable sequent of ITT. That cut is redundant is an immediate corollary since by the consistency theorem a derivable sequent is valid no matter whether cut is used in its derivation. 6.1. Underivable sets and semivaluations. A finite set Σ of signed formulas is said to be derivable if the sequent Σ+ Σ− is derivable, where Σ+ is {F | + F ∈ Σ} and Σ− is {F | − F ∈ Σ}. An infinite set Σ of signed sentences is derivable if some finite subset is derivable. An underivable set Σ is a semivaluation if it satisfies the following conditions: (1) +[F ↓ G] ∈ Σ ⇒ −F ∈ Σ and −G ∈ Σ −[F ↓ G] ∈ Σ ⇒ +F ∈ Σ or +G ∈ Σ
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(2) Let M :[τ ]. +∀(M ) ∈ Σ ⇒ +M (t) ∈ Σ, for all t:τ ; −∀(M ) ∈ Σ ⇒ −M (t) ∈ Σ, for some t:τ . (3) Let H > H , where H, H :[]. ±H ∈ Σ ⇒ ±H ∈ Σ. (4) Let P, Q:τ have dual type 1. ±P =1 Q ∈ Σ ⇒ ±P =τ Q ∈ Σ. Semivaluations for first-order logic have also been called downward saturated, or Hintikka sets. A proof of the following lemma is left to the reader. Lemma. Let Σ0 be a finite underivable set of signed sentences. There exists a semivaluation Σ for which Σ0 ⊆ Σ. 6.2. An intensional domain. Let Σ0 be a finite underivable set of signed sentences. By the lemma of §6.1, there exists a semivaluation Σ for which Σ0 ⊆ Σ. Σ∗ is obtained from Σ by replacing each signed formula ±F by ±F ∗ as defined in §3. A domain DΣ for an intensional valuation of ITT is defined from Σ∗ using the notation of §3.1 as follows: DΣ(1) is D(1). DΣ([]) is the set of pairs N ∗ , s for which s ∈ S[N ∗ ], where N :[] is in normal form, and S[N ∗ ] is the set of signs ± for which / S[N ∗ ], +N ∗ ∈ Σ∗ ⇒ + ∈ S[N ∗ ] and − ∈ ∗ ∗ ∗ −N ∈ Σ ⇒ − ∈ S[N ] and + ∈ / S[N ∗ ], and ∗ ∗ ∗ ∗ +N , −N ∈ / Σ ⇒ +, − ∈ S[N ]. DΣ([ τ ]) is the set of pairs N ∗ , s for which s ∈ S[N ∗ ], where N :[ τ ] is in normal form, and S[N ∗ ] is the set of sets s, s ⊆ DΣ(τ1 ) × · · · × DΣ(τn ) for which for all Ni∗ , si ∈ DΣ(τi ), 1 ≤ i ≤ n, +N (N1 , . . . , Nn )∗ ∈ Σ∗ ⇒ N1∗ , s1 , . . . , Nn∗ , sn
∈ s, and −N (N1 , . . . , Nn )∗ ∈ Σ∗ ⇒ N1∗ , s1 , . . . , Nn∗ , sn
∈ / s, Lemma. Each valuation to the domain DΣ is a model. Proof. Let ΦΣ be a valuation to the domain DΣ. Since ΦΣ(1, P ) ∈ DΣ(1) by (V.1) and (V.4), to prove that condition (IM.1) is satisfied it is sufficient to prove for τ = 1, (a) ΦΣ2 ([ τ ], P ) ∈ S[ΦΣ1 ([ τ ], P )] This is proved by induction on ct[P ]. Let ct[P ] be 0. If P is a constant or variable, then (a) follows immediately from (V.1). The other two cases to consider are when P is either ↓ or ∀. • Let P be ↓. Since ΦΣ1 ([[], []], ↓) is ↓, in this case it is sufficient to prove for all N1∗ , s1 , N2∗ , s2 ∈ DΣ([]) that + ↓ (N1∗ , N2∗ ) ∈ Σ∗ ⇒ N1∗ , s1 , N2∗ , s2
∈ ΦΣ2 ([[], []], ↓), respectively, / ΦΣ2 ([[], []], ↓). − ↓ (N1∗ , N2∗ ) ∈ Σ∗ ⇒ N1∗ , s1 , N2∗ , s2
∈ Let N1∗ , s1 , N2∗ , s2 ∈ DΣ([]). Then
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±↓(N1∗ , N2∗ ) ∈ Σ∗ ⇒ −N1∗ ∈ Σ∗ and −N2∗ ∈ Σ∗ , respectively, +N1∗ ∈ Σ∗ or +N2∗ ∈ Σ∗ ⇒ s1 is − and s2 is −, respectively, s1 is + or s2 is + ⇒ N1∗ , s1 , N2∗ , s2
∈ ΦΣ2 ([[], []], ↓), respectively, / ΦΣ2 ([[], []], ↓) N1∗ , s1 , N2∗ , s2
∈ • Let P be ∀. Since ΦΣ1 ([[τ ]], ∀) is ∀, in this case [ τ ] is [[τ ]] to prove for all N ∗ , s ∈ DΣ([τ ]) that +∀N ∗ ∈ Σ∗ ⇒ N ∗ , s ∈ ΦΣ2 ([[τ ]], ∀), respectively, −∀N ∗ ∈ Σ∗ ⇒ N ∗ , s ∈ / ΦΣ2 ([[τ ]], ∀). Let N ∗ , s ∈ DΣ([[τ ]]). Then ±∀N ∗ ∈ Σ∗ ⇒ +N (t)∗ ∈ Σ∗ for all t:τ , respectively, −N (t)∗ ∈ Σ∗ for some t:τ ⇒ t∗ ∈ s for all t:τ , respectively, / s for some t:τ t∗ ∈ ⇒ s is DΣ([τ ]), respectively, s is not DΣ([τ ]) ⇒ N ∗ , s ∈ ΦΣ2 ([[τ ]], ∀), respectively N ∗ , s ∈ ΦΣ2 ([[τ ]], ∀) Assume now that (a) holds whenever ct[P ] < ct. Consider the for which ct[P ] is ct.
15
(1) of §6.1
(V.2) and it is sufficient
(2) of §6.1 df of DΣ([τ ])
(V.3) terms P
• P is M N , where M :[τ, τ ] and N :τ . Assume τ is not empty. In this case it is sufficient to prove under the assumptions Ni∗ , si ∈ DΣ(τi ), 1 ≤ i ≤ n, that +(ΦΣ1 ([ τ ], M N )( N ))∗ ∈ Σ∗ ⇒ N, s ∈ ΦΣ2 ([ τ ], M N ), respectively, −(ΦΣ1 ([ τ ], M N )( N ))∗ ∈ Σ∗ ⇒ N, s ∈ / ΦΣ2 ([ τ ], M N ). By the induction assumption, (a) holds when P is M or N so that ΦΣ(τ, N ) ∈ DΣ(τ ) and for all Ni∗ , si ∈ DΣ(τi ), 1 ≤ i ≤ n, +(ΦΣ1 ([τ, τ ], M )(N, N ))∗ ∈ Σ∗ ⇒ ΦΣ(τ, N ), N, s ∈ ΦΣ2 ([τ, τ ], M ), respectively, −(ΦΣ1 ([τ, τ ], M )(N, N ))∗ ∈ Σ∗ ⇒ ΦΣ(τ, N ), N, s ∈ / ΦΣ2 ([τ, τ ], M ). ∗ ∗ ±(ΦΣ1 ([ τ ], M N )( N )) ∈ Σ ⇒ ±(ΦΣ1 ([τ, τ ], M )(N, N ))∗ ∈ Σ∗ df of S[ ] ⇒ ΦΣ(τ, N ), N, s ∈ ΦΣ2 ([τ, τ ], M ), respectively, ΦΣ(τ, N ), N, s ∈ / ΦΣ2 ([τ, τ ], M ) induction ⇒ N, s ∈ ΦΣ2 ([ τ ], M N ), respectively, / ΦΣ2 ([ τ ], M N ) (V.5) N, s ∈ The case τ is empty is left to the reader. • P is (λx.M ), where M :[ τ ] and x:τ . Assume τ is not empty.
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In this case it is sufficient to prove under the assumptions N ∗ , s ∈ DΣ(τ ) and Ni∗ , si ∈ DΣ(τi ), 1 ≤ i ≤ n, that +(ΦΣ1 ([τ, τ ], λx.M )(N, N ))∗ ∈ Σ∗ ⇒ N, N ∈ ΦΣ2 ([τ, τ ], (λx.M )), respectively, −(ΦΣ1 ([τ, τ ], λx.M )(N, N ))∗ ∈ Σ∗ ⇒ N, N ∈ / ΦΣ2 ([τ, τ ], (λx.M )). ±(ΦΣ1 ([τ, τ ], λx.M )(N, N ))∗ ∈ Σ∗ ⇒ ±(ΦΣ1 ([ τ ], (λx.M )N )(, N ))∗ ∈ Σ∗ df of ΦΣ1 ⇒ ±(ΦΣ1 ([ τ ], [N/x]M )( N ))∗ ∈ Σ∗ (3) of §6.1 ⇒ ±(ΦΣx1 ([ τ ], M )( N ))∗ ∈ Σ∗ ) lemma §3.1 where ΦΣx (τ, x) is ΦΣ(τ, N ) ⇒ N, s ∈ ΦΣx2 ([ τ ], M ), respectively, / ΦΣx2 ([ τ ], M ) induction N, s ∈ ⇒ N, N ∈ ΦΣ2 ([τ, τ ], M ), respectively, / ΦΣ2 ([τ, τ ], M ) (V.6) N, N ∈ Assume τ is empty. In this case it is sufficient to prove that when ΦΣ(τ, N ) ∈ DΣ(τ ), +(ΦΣ1 ([], λx.M )N ))∗ ∈ Σ∗ ⇒ ΦΣ(τ, N ) ∈ ΦΣ2 ([τ ], (λx.M )), respectively, −(ΦΣ1 ([], λx.M )N ))∗ ∈ Σ∗ ⇒ ΦΣ(τ, N ) ∈ / ΦΣ2 ([τ ], (λx.M )). ±ΦΣ1 ([], (λx.M )N ) ∈ Σ∗ ⇒ ±ΦΣ1 ([], [N/x]M )) ∈ Σ∗ (3) §6.1 ⇒ ΦΣ2 ([], [N/x]M )) is ± induction ⇒ ΦΣx2 ([], M )) is ± lemma §3.1 ⇒ ΦΣ(τ, N ) ∈ ΦΣ2 ([τ ], (λx.M ))), respectively, (V.6) ΦΣ(τ, N ) ∈ / ΦΣ2 ([τ ], (λx.M ))) Consider now (IM.2). By (a) ΦΣ2 ([], P =1 Q) ∈ S[ΦΣ1 ([], P =1 Q)] Hence ΦΣ2 ([], P =1 Q) is ± ⇒ ± =1 (ΦΣ1 (1, P ), ΦΣ1 (1, Q)) ∈ Σ∗ df of ΦΣ1 ⇒ ± =τ (ΦΣ1 (τ, P ), ΦΣ1 (τ, Q)) ∈ Σ∗ (4) of §6.1 df of S[ ] ⇒ ΦΣ2 ([], P =τ Q) is ± This completes the proof of the lemma.
Corollary. ±ΦΣ1 ([], F ) ∈ Σ∗ ⇒ ΦΣ2 ([], F ) is ± respectively, for F :[]. By the lemma, ΦΣ is a model of ITT. By the corollary, the underivable set of signed sentences Σ0 obtained from an underivable sequent is satisfied by ΦΣ. This completes the proof of the completeness theorem. §7. Extensions to ITT. A sketch is given in §7.1 of the logic ITTf that introduces a functional notation into ITT. In §7.2 an extension SITT of ITT is described in which such sets as Russell’s can be consistently defined. Full details of the logics are provided in chapters 5 and 6 of [18]. 7.1. ITTf. A notation for total functions with values of type 1 is available in ITT in the form of higher order terms that are also first order terms. The term S defined in §1.3, for example, is used in this way: S(t) is a first order term when
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t is. Also syntactic sugaring provides a functional notation for total functions with values of type other than 1. But partial functions of n arguments can only be represented in ITT by predicates of n + 1 arguments, the extra argument being the value of the function. The logic ITTf is an extension of ITT with an explicit notation for partial or total functions with values of type 1. The types of ITTf are the types of ITT. All the constants and variables of ITT are constants and variables of ITTf of the same type. But ITTf also has available the untyped symbol ι that is used to create function terms. The terms of ITTf are defined by adding the following clauses to the clauses (1)–(3) of §2.1: (4) Let M be a term of type [1]. Then ιM is a function term and term of type 1. (5) Let M be a term of type τ , τ =1, for which each free occurrence of a variable in M is either of type 1, or is an occurrence in a function term with a free occurrence in M . Then M is also a term of type 1. The proof theory of ITTf is obtained from the proof theory of ITT as follows: • The +∀ rule for type 1 eigen terms has a second conclusion −Den(t), where df
Den = (λu.u=u), in order to ensure that the eigen term t is denoting; Farmer in [10] and others use the notation ↓t to express the same thing. • The λ rules are affected by a modification to the definition of β–reduction: A term (λv.P )(t) is not contractible to [t/v]P if t is a term of type 1 in which a function term has a free occurrence. This change is necessary to ensure the proper scope for function terms. • The terms P and Q of an Int rule must satisfy the additional requirement that no function term has a free occurrence in either. • ι rules are added with premiss ±P ([ιM/v]t) and conclusion ±[∃!v.M ]P (t), where the variable v has no free occurrence in a function term with a free occurrence in t. This restriction is necessary to ensure the proper scope for function terms. A semantics for ITTf is provided by a mapping of terms of ITTf into terms of ITT. A proof of a completeness and cut-elimination theorem similar to the theorem of §6 is provided in [18] along with a proof that ITTf is a conservative extension of ITT. 7.2. Consistent uses of Russell’s set. The variables of ITT have two distinct roles to play, as the eigen variable in the conclusion of an application of the −∀ rule, and as abstraction variables in the λ rules. These two roles can be served in formulas by syntactically distinct variables. An abstraction variable is used purely as a placeholder and need never be interpreted. But clearly the eigen variable of an application of the −∀ rule must receive a value. By using distinct variables for these two roles, and interpreting quantification variables as in ITT but interpreting abstraction variables as placeholders, a consistent extension of ITT can be constructed in which such impredicative sets as Russell’s can be defined. Each variable v of ITT has a unique type †[v]; these variables are the quantifiable variables of the extended ITT. An abstraction variable v of the extended ITT on the other hand has a specified type †[v], but also a dual type 1. This allows applications of abstraction not permitted in ITT. For example, it is not
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difficult to derive ¬0(0) in ITT, where 0 has been defined in §1.3. The first occurrence of 0 in 0(0) is of type [1] while the second is of type 1. If v is an abstraction variable for which †[v] is [1], then (λv.¬v(v))(0) is derivable in the extended ITT. df Defining Rus = (λv.¬v(v)) results in Rus(0) being derivable. Similarly df
if V = (λv.v=v), then ¬Rus(V ) is derivable. Note however that neither Rus(Rus) nor ¬Rus(Rus) is derivable since Rus:[[1]] and therefore Rus(Rus) is not well-formed. Thus no contradiction results from Rus being a well-formed predicate. But note also that the predicate Rus of the extended ITT differs significantly from any predicate of ITT since although Rus(0) is derivable, ∃x.Rus(x) is not, since for no quantification variable x is Rus(x) well-formed. Details are provided in [18]. REFERENCES
[1] Peter B. Andrews, Sunil Issar, Daniel Nesmith, and Frank Pfenning, The TPS theorem proving system, 9’th International Conference on Automated Deduction (CADE) (E. Lusk and R. Overbeek, editors). Lecture Notes in Computer Science. vol. 310, Springer-Verlag, 1988, pp. 760-761. [2] H. P. Barendregt, The Lambda Calculus, its syntax and semantics, North-Holland, 1985, Revised Edition. [3] Jon Barwise and Lawrence Moss, Vicious circles, CSLI Publications, 1996. [4] E.W. Beth, Semantic entailment and formal derivability, Mededelingen de Koninklijke Nederlandse Akademie der Wetenschappen, Afdeeling Letterkunde, Nieuwe Reeks, vol. 18 (1955), no.13, pp. 309–342. [5] Rudolph Carnap, Meaning and Necessity, University of Chicago Press 1947. [6] Alonzo Church, Schr¨ oder’s anticipation of the simple theory of types, The Journal of Unified Science (Erkenntnis) , vol. IX (1939), pp. 149–152. [7] Alonzo Church, A formulation of the simple theory of types, Journal of Symbolic Logic, vol. 5 (1940), pp. 56–68. [8] Nino B. Cocchiarella, Logical investigations of predication and the problem of universals, Bibliopolis Press, Naples, 1986. [9] Nino B. Cocchiarella, Conceptual realism versus Quine on classes and higher-order logic, Synthese, vol. 90 (1992), pp. 379–436. [10] William M. Farmer, A partial functions version of Church’s simple theory of types, Journal of Symbolic Logic, vol. 55 (1990), pp. 1269–1290. [11] M.C. Fitting, First-order logic and automated theorem proving, Springer- Verlag, 1996, Second Edition. [12] Paul C. Gilmore, A consistent naive set theory: Foundations for a formal theory of computation, IBM Research Report RC 3413, June 22, 1971. [13] Paul C. Gilmore, Combining unrestricted abstraction with universal quantification, To H.B. Curry: Essays on combinatorial logic, lambda calculus and formalism (J.P. Seldin and J.R. Hindley, editors), Academic Press, 1980, pp. 99–123. This is a revised version of [12]. [14] Paul C. Gilmore, Natural deduction based set theories: A new resolution of the old paradoxes, Journal of Symbolic Logic, vol. 51 (1986), pp. 393–411. [15] Paul C. Gilmore, A foundation for the entity relationship approach: How and why, Proceedings of the 6th Entity Relationship Conference (S.T. March, editor), North-Holland, 1988, pp. 95–113. [16] Paul C. Gilmore, NaDSyL and some applications, Computational logic and proof theory, The Kurt G¨ odel Colloquium 97 (Georg Gottlob, Alexander Leitsch, and Daniele Mundici, editors), Lecture Notes in Computer Science vol. 1289, Springer-Verlag, 1997, pp. 153–166.
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[17] Paul C. Gilmore, An impredicative simple theory of types, presented at the Fourteenth Workshop on Mathematical Foundations for Programming Systems, Queen Mary College, London, May 1998. [18] Paul C. Gilmore, An intensional type theory:Logical foundations for mathematics and computer science, a monograph on ITT in preparation. [19] Michael J.C. Gordon, A proof generating system for higher-order logic, VLSI Specification, Verification and Synthesis, (G. Birtwistle and P. Subrahmanyam, editors), Academic Publishers, Boston, 1987. Also, University of Cambridge Computer Laboratory Technical Report No. 103, pp. 73–128. [20] Michael J.C. Gordon, Set theory, higher order logic or both?, Higher order logic theorem proving and its applications (Joakum von Wright, Jim Grundy, and John Harrison, editors), Lecture Notes in Computer Science, vol. 1125, 1996, pp. 191–202. [21] Leon Henkin, Completeness in the theory of types, Journal of Symbolic Logic, vol. 15 (1953), pp. 81–91. [22] Gopalan Nadathur and Dale Miller, Higher-order logic programming, Handbook of logic in artificial intelligence and logic programming (Dov M. Gabbay, C.J. Hogger and J.A. Robinson, editors), Clarendon Press, Oxford, 1998, pp. 499–590. [23] S. Owre, N. Shankar, and J.M. Rushby, The PVS specification language (beta release), Computer Science Laboratory, SRI International, Menlo Park CA 94025, June 14, 1993. [24] Dag Prawitz, Hauptsatz for higher order logic, Journal of Symbolic Logic, vol. 33 (1968), pp. 452–457. [25] Bertrand Russell, The Principles of Mathematics, George Allen and Unwin Ltd, London, second edition, 1937. [26] K. Sch¨ utte, Syntactical and semantical properties of simple type theory, Journal of Symbolic Logic, vol. 25 (1960), pp. 305–326. [27] Wilfred Sellars, Abstract entities, Rev. of Metaphysics, vol. 16 (1963), pp. 625–671. [28] Wilfred Sellars, Classes as abstract entities and the Russell paradox, Rev. of Metaphysics, vol. 17 (1963), pp. 67–90. [29] Joseph R. Shoenfield, Mathematical logic, Addison-Wesley, 1967. [30] R.M. Smullyan, First-order logic, Springer-Verlag, Berlin, 1994, Revised Edition, Dover Press, New York. [31] Moto-o Takahashi, A proof of cut-elimination theorem in simple type-theory, Journal of the Mathematical Society of Japan, vol. 19, 4 (1967), pp. 399–410. [32] Ernst Zermelo, Investigations in the foundations of set theory, From Frege to G¨ odel (Jean van Heijenoort, editor), Harvard University Press, 1967, pp. 199–215. DEPARTMENT OF COMPUTER SCIENCE UNIVERSITY OF BRITISH COLUMBIA 2366 MAIN MALL VANCOUVER, BC V6T 1Z4 CANADA
E-mail: [email protected]
AN INTENSIONAL TYPE THEORY: MOTIVATION AND CUT-ELIMINATION
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Abstract. By the theory TT is meant the higher order predicate logic with the following recursively defined types: (1) 1 is the type of individuals and [] is the type of the truth values; (2) [τ1 , . . . , τn ] is the type of the predicates with arguments of the types τ1 , . . . , τn . The theory ITT described in this paper is an intensional version of TT. The types of ITT are the same as the types of TT, but the membership of the type 1 of individuals in ITT is an extension of the membership in TT. The extension consists of allowing any higher order term, in which only variables of type 1 have a free occurrence, to be a term of type 1. This feature of ITT is motivated by a nominalist interpretation of higher order predication. In ITT both well-founded and non-well-founded recursive predicates can be defined as abstraction terms from which all the properties of the predicates can be derived without the use of non-logical axioms. The elementary syntax, semantics, and proof theory for ITT are defined. A semantic consistency proof for ITT is provided and the completeness proof of Takahashi and Prawitz for a version of TT without cut is adapted for ITT; a consequence is the redundancy of cut.
§1. Introduction. Consider a form TT of the simple theory of types in which predicates of any number of arguments are admitted, but no functions. The types of such a predicate logic can be recursively defined as follows: (1) 1 is the type of individuals and [] is the type of the truth values; (2) [τ1 , . . . , τn ] is the type of the predicates with arguments of the types τ1 , . . . , τn , where n ≥ 1. Thus the type [τ1 , . . . , τn ] can be taken to be the type [] when n=0. Apart from notation and the exclusion of functions, the types of TT are the types of Sch¨ utte’s type theory [26]. Although the types of TT are traditionally thought of as necessary for the consistency of the logic, the types can just as well be seen to arise naturally from the predicate and subject distinction of natural languages, for these become the distinction between a predicate and its argument(s). Skepticism has often been expressed that a violation of the type restrictions is the ultimate source of the xxxxx The financial support of the Natural Science and Engineering Research Council of Canada is gratefully acknowledged. c xxxx, Association for Symbolic Logic 0022-4812/xx/xxxx-xxxx/xxxx
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paradoxes. For example, in the concluding paragraph of [6] Church comments on a remark of Frege’s “ . . . Frege’s criticism seems to me still to retain much of its force, and to deserve serious consideration by those who hold that the simple theory of types is the final answer to the riddle of the paradoxes”. Here, as in [27] and [28], an alternative explanation of the paradoxes is offered; namely that they result from a confusion of use and mention when higher order predication is given a nominalist interpretation. A quote from the introduction to [25] is relevant: “Broadly, the result is an outlook which is less Platonic, or less realist in the medieval sense of the word. How far it is possible to go in the direction of nominalism remains, to my mind, and unsolved question, but one which, whether completely solvable or not, can only be adequately investigated by means of mathematical logic.” In the subsections of this introduction to follow, the motivation for the features of ITT that distinguish it from TT is described and some important results obtainable in ITT are sketched. The syntax of ITT is described in §2, the semantics in §3, and the proof theory in §4 in the style of semantic tableaux. A semantic consistency proof is provided in §5. A proof that cut is a redundant rule of the logic is provided in §6; it is an adaptation of the proofs for Sch¨ utte’s type theory provided in [24] and [31]. Finally in §7 two extensions of ITT are described. The first introduces a notation for partial functions with values of type 1 in the traditional manner using definite descriptions; but its development within ITT permits simplifications of the syntax and semantics. The second extension of ITT demonstrates that a solution to the “riddle of the paradoxes” can be found by maintaining a careful distinction between the use and mention of predicate names in a nominalist interpretation of higher order predication. For example, in this consistent extension of ITT Russell’s set can be defined. Details for both logics are provided in [18]. 1.1. Intensional and extensional identity. Following Carnap [5], intensional and extensional identities are defined first before the intension and extension of terms are described. In TT these identities are defined: df
= = (λu, v.∀X.[X(u) → X(v)]) df
=e = (λu, v.∀x.[u( x ) ↔ v( x )]) Here the notation x is used as an abbreviation for x1 , . . . , xn , where n ≥ 0. The type restrictions necessary for the definitions can then be expressed as v:†[u] and X:[†[u]] for the first, and as u, v:[†[x1 ], . . . , †[xn ]] for the second. Here †[cv] denotes the type of a constant or variable cv that is assigned to it by the primitive syntax; and ‘:’ denotes the relationship between a term and its type. Using the usual infix notation for the identities, a theorem of TT is: IE.1) ∀X, Y.[X = Y → X =e Y ], where necessarily †[X] is not 1. The comparable result in the set theory ZF [29] is IEZF) ∀x, y.[x = y → x =e y] where here = is the primitive identity of a first order logic with identity; the sequent is actually a theorem of that logic. The converse of (IEZF), the axiom of extensionality, is the first axiom in the first formulation of ZF [32]. In TT the axiom is the converse of (IE.1), and is often accepted as an axiom of TT:
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EI) ∀X, Y.[X =e Y → X = Y ]. An axiom of extensionality may be acceptable for a pure logic, but it is not for an applied logic trying to meet the needs of computer science. In the design of databases it is essential to understand the distinction between the intension of a predicate and its extension; the former may only be known informally while the latter is provided by data entry. Consider, for example, the extension of an Employee predicate in a company database. It will most likely be maintained by data entry along with a Sex predicate among others. From these two predicates the intension of a predicate MaleEmployee can be defined, and its extension retrieved and printed [15]. By an accident of hiring, however, the two predicates Employee and MaleEmployee may have the same extension; but clearly their intensions must be distinguished. Another example is provided by [22] where a distinction is made between “extensional” and “intensional” occurrences of predicate variables. For this reason the axiom (EI) concluding the intensional identity of predicates from their extensional identity is not accepted in ITT. Instead rules of intensionality are part of the proof theory; they conclude that two predicates are intensionally identical if and only if their names are intensionally identical. 1.2. Nominalism and higher order predication. In a logic of extensions such as TT with (EI) as axiom, a higher order predicate with predicates as arguments is understood to have the extensions of the predicates as arguments. But in a logic in which the intension of a predicate is distinguished from its extension, higher order predication must be re-examined. A nominalist understands a predicate of a universal to be a predicate of a name of the universal. For example, a nominalist understands ‘Yellow is a colour’ to mean ‘Yellow is a colour-word’; the sentence is understood as a description of the use of the word ‘Yellow’ in English. Since computers are consummate nominalists, nominalist interpretations of languages intended for computer applications are needed. But this does require a careful distinction between the use and mention of predicate names, especially when treating abstraction and quantification. For example, in ‘Yellow is a colour-word’ the predicate name ‘Yellow’ is being mentioned while the predicate name ‘colour-word’ is being used. The distinction between the use and mention of predicate names is maintained in the logic ITT as follows: The types of ITT are the same as the types of TT, but the membership of the type 1 of individuals in ITT is an extension of the membership of the same type in TT. The extension consists in adding to the membership of the type 1 any higher order term in which at most variables of type 1 have a free occurrence. A constant C that is a predicate name is necessarily of some type that is not 1 and always has that type in contexts where it is used. But since no variable has a free occurrence in C, C is also of type 1 and it has that type in contexts in which it is being mentioned. Thus C(C) is of type [] if C is of type [1]. A related method of maintaining the distinction between the use and mention of predicate names was used in the predecessors of ITT described in [12], [13], [14] and [16]. The papers prior to [16] attempted to provide a formal set theory that maintains the distinction, but had an awkward notation among its failings. The logic NaDSyL described in [16] is a recent attempt to overcome these
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failings; for example, it uses the notation of the lambda calculus. But a closer study of NaDSyL has revealed that the technique it employs for maintaining the distinction could be employed more simply within a type theory. Mentioning the name of a predicate means that the name is implicitly quoted. This is the reason why higher order terms that are also of type 1 must be restricted to those in which only variables of type 1 have free occurrences. For only such terms can be given a Herbrand interpretation when quoted. For example, let C be a constant of type [1] and x a variable of type 1. Then C(x) is of type [], and also of type 1. As a type 1 term, C(x) is to be interpreted as the function with domain and range the constant type 1 terms: When x is interpreted as a constant type 1 term t, then C(x) is interpreted as the constant type 1 term C(t). Such an interpretation can’t be given for higher order terms not satisfying the restriction. For example, if X is a variable of type [1] and c a constant of type 1, then X(c) is of type [] but not also of type 1. 1.3. Set theory and the lambda calculus. For (IE.1) to be derivable in TT it is necessary that the type of X and Y be other than 1. Although (IE.1) is not derivable in ITT when the the type of X and Y is 1, each instance of the following sequent scheme is derivable when R and S are terms of type τ with dual type 1: IE.2) R = S R =e S. This scheme is similar to the scheme that would result from all possible instantiations of (IEZF) of ZF. As a consequence, the logic ITT combines features from set theory and from a lambda calculus based logic; for this reason it may satisfy the requirements for such a logic described in [20]. Consider for example the following definitions of zero and successor df
0 = (λu.¬u = u) df
S = (λu, v.u = v) where =: [1, 1]. They are definitions in the style of set theory; but all of Peano’s axioms are derivable in ITT including the following pair: S.1) ∀x, y.[S(x) = S(y) → x = y] S.2) ∀x.¬S(x) = 0. Dual typing is critical for their derivations: 0:[1] and S:[1, 1], but also 0:1 and S(x):1 when x:1. The lambda calculus definition of ordered pair from [7] is also available df
: = (λu, v, w.w(u, v)) and the following sequents can be derived: OP.1) ∀x1, y1, x2, y2.[ x1, y1 = x2, y2 → x1 = x2 ∧ y1 = y2], OP.2) ∀x, y.¬ x, y = 0. Details are provided in [17] and [18]. Also provided there is a foundation for recursions. Both well-founded and non-well-founded recursive predicates are defined there using a decidable set of terms called recursion generators; the technique is demonstrated using higher order Horn sequent definitions. This method of defining recursive predicates overcomes some of the complications that arise from their introduction into the applied versions TPS, HOL, and PVS of Church’s formulation of type theory. [1], [19], [23].
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Since non-well-founded predicates can be defined in ITT, there appears to be no obstacle to reproducing within ITT most of the applications of non-wellfounded sets described in [3]. Acknowledgments. Comments by Nino Cocchiarella on an earlier paper and his [8] and [9] encouraged the change from the set theoretic notation of [14] to the lambda notation of [16] and ITT. An observation of Henk Barendregt was critical for the realization that the technique employed in [16] for maintaining the distinction between use and mention could be more simply maintained in ITT. Conversations with Alasdair Urquhart, Peter Apostoli, and Michael Donat have been helpful for details of this paper, as has correspondance with Hendrik Boom, Per Martin-L¨ of, and Joergen Villadsen. The paper has also benefitted from several recommendations made by a referee. §2. The syntax. Type membership for ITT is defined in §2.1, providing the basic syntax for the logic. The lambda reduction relation between terms of ITT is defined in §2.2. In §2.3 some more usual notations for predicate logics are introduced by definitions. 2.1. Type membership. The logic ITT is assumed to have denumerably many constants and variables of each type. In the style of [7], special constants introduce the logical connectives and the quantifiers. The binary logical connective of joint denial, denoted by ↓, is a special constant of type [[],[]]; that is, it is a predicate of two arguments of type []; it is the only primitive logical connective needed since the more usual logical connectives can be defined in terms of it. Similarly a special constant ∀ of type [[τ ]] is introduced for each type τ ; it is the universal quantifier for a type τ variable. The type of each ∀ is not displayed but must be inferred from context. The existential quantifier ∃ is defined in terms of ∀ in the usual way. The notation †[cv] introduced in §1.1 is used in the following definition of type membership; it denotes the unique type assigned to each constant or variable cv by the primitive syntax: (1) (2) (3) (4)
cv:†[cv], cv a constant or variable; ↓:[[], []]; and ∀:[[τ ]] for each type τ . M :[τ, τ ] and N :τ ⇒ (M N ):[ τ ]. M :[ τ ] ⇒ (λx.M ):[†[x],τ ]. M :[ τ ] ⇒ M :1, provided each variable with a free occurrence in M is of type 1.
The unusual clause (4) results from the nominalist interpretation discussed in §1.2. The type 1 assigned to M in (4) is called the dual type of M . Because of this clause a constant c for which †[c] is not of type 1 has 1 as a dual type; nevertheless, †[c] is always to be understood to be the type assigned by the primitive syntax, that is the syntax of TT, and never the dual type 1 assigned by clause (4). No variable has 1 as a dual type nor does a constant c for which †[c] is 1. By a term is meant a member of a type. Note that a term of type 1 is a constant, a variable, or a term of dual type 1 since clauses (1) and (4) are the only ones that yield terms of type 1.
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Let N :†[v]. The substitution notation [N/v]M denotes the result of replacing each free occurrence of v in a term M by N . The notation can result in changes of bound variables within M ; a change is necessary if a free occurrence of v in M is within the scope of an abstractor λy for which y has a free occurrence in N. The count ct[M ] of a term M is defined: ct[cv] = 0 for any constant, including ↓ and ∀, or variable cv; ct[M N ] = ct[M ] + ct[N ] + 1; and ct[λx.M ] = ct[M ] + 1. A proof by induction on ct[M ] of the following lemma will be left to the reader. Lemma (Substitution). Let N :†[v] and M :τ . Then [N/v]M :τ . 2.2. Lambda reductions. The relation of immediate lambda reduction between terms is denoted here by >, and allows α-, β-, and η-reductions [2]. The reflexive and transitive closure of > is denoted by . Many proofs are available for the following theorem of the pure lambda calculus Theorem (Church–Rosser). If M N and M P , then there exists a Q for which N Q and P Q. Clause (4) in the definition of type membership has little effect on the proofs provided in [2] for the theorem as demonstrated in [18]. The theorem ensures that by selecting one of the bound variable variants of a term in normal form as the representative normal form, a unique normal form may be assigned to each term of ITT. 2.3. Formula notations. A formula of ITT is a term of type []. Formulas are the basis for the proof theory for ITT described in §4. But first the sparse notation of the lambda calculus is extended by definitions that introduce notations more common to predicate logics. The application notation is “sugared” by the definitions: df M (N ) = M N df
M (N1 , . . . , Nm , N ) = M (N1 , . . . , Nm )(N ) The prefix notation for ↓ is replaced by an infix notation, and the logical connectives ¬ and ∧ are defined; all other conventional connectives can be defined from them: df [M ↓ N ] =↓ (M, N ) ¬M
df
= [M ↓ M ] df
[M ∧ N ] = [¬M ↓ ¬N ]. A conventional notation for the universal quantifier is defined, and the existential quantifier is defined from it in the usual way: df ∀x.M (x) = ∀(M ) df
∃x.F = ¬∀x.¬F where M :[†[x]] and x has no free occurrence in M . Parenthesis have been dropped here and will continue to be when there is no risk of confusion. §3. Semantics. The definition of a valuation for a type theory requires the prior definition of a domain D(τ ) for each type τ . For the extensional theory TT the standard domains are a nonempty set D(1) of entities for the type 1, the set D([]) with members the truth values represented by + and −, and the
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sets D([τ1 , . . . , τn ]) consisting of all the subsets of the Cartesian product of the sets D(τ1 ), . . . , D(τn ). Thus for these domains only the potential extensions of predicates are provided. To provide a semantics for the intensional type theory ITT, it is necessary to provide potential intensions as well as extensions. This is accomplished in §3.1. But first the variables of type τ are mapped 1-1 onto new constants of type τ . Given a term M , the term M ∗ is obtained from M by replacing each free occurrence of a variable in M by the constant onto which it is mapped. If N is the normal form for a term M of ITT, then N ∗ is the normal form for M ∗ . Although these new constants are not required, their use does simplify the presentation of the semantics. 3.1. Domains, valuations, and models. A domain for an intensional valuation of ITT is a function D defined for each type τ for which: D(1) is { N ∗ , N ∗ N :1 in normal form} D([]) is { N ∗ , s s ∈ S[N ∗ ] and N :[] in normal form} D([ τ ]) is { N ∗ , s s ∈ S[N ∗ ] and N :[ τ ] in normal form} where S[N ∗ ] for D([]) is a non-empty subset of {+, −}, and for D([ τ ]) is a non-empty subset of the set of all subsets of the Cartesian product of the sets D(τ1 ), . . . , D(τn ). A pair N ∗ , N ∗ from D(1) can be understood to be an intension of an individual name and a possible extension for it. This becomes clearer if each constant c for which †[c] is 1 is understood to represent a member of a non-empty set of entities. Then D(1) can be defined to be the union of the set { c, e c:1 and e an entity} and the set { N ∗ , N ∗ N of dual type 1 and in normal form}. The simpler definition of D(1) is used for the proofs to follow. The first member of a pair that is a member of D(τ ), τ = 1, is a potential intension for a predicate of type τ , while the second member is a potential extension for that intension. Thus, for example, members F ∗ , + and G∗ , + of D([]) are extensionally identical but not intentionally if F and G have distinct normal forms. Similarly members P ∗ , s1 and Q∗ , s2 are extensionally identical if s1 and s2 have the same members, but not intensionally if P and Q have distinct normal forms. Note that to define a domain D it is sufficient to define the sets S[N ∗ ]. For example, the standard domain for ITT is the domain for which each S[N ∗ ] in D([]) is {+, −} and each S[N ∗ ] for D([ τ ]) is the set of all subsets of the Cartesian product. A valuation to a given domain D is a function Φ, defined for all τ and M for which M :τ , and such that V.1) Φ(†[c], c) is c, c ∈ D(1), when †[c] is 1; Φ(†[c], c) is c, s ∈ D(†[c]), for each constant c and some s, when †[c] = 1 and c is neither ↓ nor ∀; Φ(†[v], v) ∈ D(†[v]), for each variable v. The value of Φ in (V.1) is an ordered pair so that Φ1 and Φ2 can be defined to be the functions for which Φ(τ, M ) is Φ1 (τ, M ), Φ2 (τ, M ) . The value of
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Φ1 (τ, M ) for the cases (V.2) to (V.6) to follow is defined to be the normal form of [Φ1 (†[x1 ], x1 )/x1 ] · · · [Φ1 (†[xn ], xn )/xn ]M where x1 , . . . , x1 are all the variables with a free occurrence in M . Since no free variable occurs in any Φ1 (†[xi ], xi ), no variable has a free occurrence in Φ1 (τ, M ). In particular, Φ1 (τ, M )∗ is Φ1 (τ, M ). The function Φ2 (τ, M ) is defined in the remaining cases as follows: V.2) Φ2 ([[], []], ↓) is the set of pairs of pairs Φ1 ([], F ), − , Φ1 ([], G), −
. V.3) Φ2 ([[τ ]], ∀) is the set of pairs Φ1 ([τ ], M ), D(τ ) . V.4) Φ2 (1, M ) is Φ1 (1, M ), when M has dual type 1. V.5) Φ2 ([ τ ], M N ), when τ is not empty, is the set of tuples e for which Φ(τ, N ), e ∈ Φ2 ([τ, τ ], M ). Φ2 ([], M N ) is + if Φ(τ, N ) ∈ Φ2 ([τ ], M ) and is − otherwise. For the last condition, Φx is an x-variant of Φ; that is, it is a valuation for which Φx (†[x], y) differs from Φ(†[x], y) at most when y is x. V.6) Φ2 ([†[x], τ ], λx.M ), when τ is not empty, is the set of tuples Φx (†[x], x), e for which e ∈ Φx2 ([ τ ], M ) for some x-variant Φx of Φ. Φ2 ([†[x]], λx.M ) is the set of pairs Φx (†[x], x) for which Φx2 ([], M ) is + for some x-variant Φx of Φ. A proof of the following lemma is needed for the theorem of §3.2. Lemma (Semantic Substitution). Let Φ be a valuation to some domain. Let Q:†[x], where x has no free occurrence in Q, and let P :σ. Let Φx be the x-variant of Φ for which Φx (†[x], x) is Φ(†[x], Q). Then Φx (σ, P ) is Φ(σ, [Q/x]P ). Proof. It may be assumed that x has a free occurrence in P . No matter the form and type τ of P , Φx1 (σ, P ) is the normal form of [Φ1 (†[x], x)/x][Φ1 (†[x1 ], x1 )/x1 ] . . . [Φ1 (†[xn ], xn )/xn ]P where x, x1 , . . . , xn are all the variables with a free occurrence in P . Hence Φx1 (σ, P ) is the normal form of [Φ1 (†[x], Q)/x][Φ1 (†[x1 ], x1 )/x1 ] . . . [Φ1 (†[xn ], xn )/xn ]P which is the normal form of [Φ1 (†[x1 ], x1 )/x1 ] . . . [Φ1 (†[xn ], xn )/xn ][Φ1 (†[x], Q)/x]P since none of the variables x1 , . . . , xn have a free occurrence in Φ1 (†[x], Q). Therefore Φx1 (σ, P ) is Φ1 (σ, [Q/x]P ) as required.
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It follows by (V.4) that Φx2 (1, P ) is Φ2 (1, [Q/x]P ) when P has dual type 1. It is sufficient, therefore to prove that Φx2 (σ, P ) is Φ2 (σ, [Q/x]P ) when σ is not the dual type 1. The proof is by induction on ct[P ]. When ct[P ] is 0, P is necessarily x so the conclusion is immediate. Assume the conclusion of the lemma whenever ct[P ] < ct. Let ct[P ] be ct, and consider the forms that P may take. • P is M N , where M :[τ, τ ], N :τ , and σ is [ τ ]. When τ is not empty, Φx2 (σ, P ) is the set of tuples e for which Φx (τ, N ), e ∈ x Φ2 ([τ, τ ], M ); that is, for which Φ(τ, [Q/x]N ), e ∈ Φ2 ([τ, τ ], [Q/x]M ). Therefore Φx2 ([ τ ], P ) is Φ2 ([ τ ], [Q/x]P ). When τ is empty, Φx2 ([], P ) is + ⇔ Φx (τ, N ) ∈ Φx2 ([τ ], M ) ⇔ Φ(τ, [Q/x]N ) ∈ Φ2 ([τ ], [Q/x]M ) ⇔ Φ([], [Q/x]P ) is +. • P is λy.M , where it may be assumed that y has no free occurrence in Q and is not x. Thus [Q/x](λy.M ) is (λy.[Q/x]M ). Let M :[ τ ] so that σ is [†[y], τ ]. When τ is not empty, Φx2 (σ, P ) is the set of tuples Φyx (†[y], y), e for which e ∈ Φyx 2 ([ τ ], M ); that is the set of tuples Φy (†[y], y), e for which e ∈ Φxy 2 ([ τ ], M ). Hence, by the induction assumption, for which e ∈ Φy2 ([ τ ], [Q/x]M ). Thus Φx2 (σ, P ) is Φ2 (σ, [Q/x]P ) as required. When τ is empty, Φx2 ([†[y]], P ) is the set of pairs Φyx (†[y], y) for which x Φyx 2 ([], M ) is + for some y-variant of Φ . By the induction assumption this is the set of pairs Φy2 (†[y], y) for which Φy2 ([], [Q/x]M ) is + for some y-variant of Φ. Thus Φx2 ([†[y]], P ) is Φ2 ([†[y]], [Q/x]P ) as required. 3.2. Intensional models. An intensional model for ITT with domain D is a valuation Φ to D that satisfies two conditions: IM.1) Φ(τ, P ) ∈ D(τ ), for each type τ and term P of type τ . IM.2) Let P , Q:τ each have dual type 1. Then Φ2 ([], P =1 Q) is ± ⇒ Φ2 ([], P =τ Q) is ±, respectively, where =1 is = of type [1, 1], and =τ is = of type [τ, τ ]. Clause (IM.1) is the condition that originated with [21] for a valuation of type theory to be a model . Clause (IM.2) expresses that the intension of a predicate is identified with its name. For let Φ2 ([], P = Q) be ±, where = :[1, 1]. Then since the terms P and Q occur in a type 1 context, they are being mentioned. From the fact that Φ2 ([], P = Q) is ± it can be concluded that P and Q are identical, respectively, not identical terms and therefore must name intensionally identical, respectively, not identical predicates in the context P = Q when =:[τ, τ ]; that is, Φ2 ([], P = Q) must be ±. Theorem. Any valuation to the standard domain is an intensional model. Proof. Let Φ be a valuation to the standard domain. That Φ satisfies (IM.1) is immediate. To prove that it satisfies (IM.2) it is sufficient to prove that for any type τ it satisfies the following condition:
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Φ2 ([], P =τ Q) is ± ⇒ Φ1 (τ, P ) Φ1 (τ, Q), respectively, Φ1 (τ, P ) Φ1 (τ, Q)
where =τ :[τ, τ ], and expresses that Φ1 (τ, P ) and Φ1 (τ, Q) have the same normal form. That (IM.2) follows from (NF) is a consequence of the definition of domain given in §3.1. A proof of (NF) follows. Φ2 ([], P =τ Q) is ± ⇒ Φ2 ([], ∀(λz.[z(P ) → z(Q)])) is ±, z not free in P or Q ⇒ Φ([[τ ]], (λz.[z(P ) → z(Q)])) ∈, ∈ / Φ2 ([[[τ ]]], ∀) ⇒ Φ2 ([[τ ]], (λz.[z(P ) → z(Q)]) is, is not D([τ ]). But Φ2 ([[τ ]], λz.[z(P ) → z(Q)]) is {Φz ([τ ], z) Φz2 ([], [z(P ) → z(Q)]) is +}. By (IM.1), Φ(τ, P ), Φ(τ, Q) ∈ D(τ ) so that (λv.P ∗ =τ v), {Φ(τ, P )} , (λv.Q∗ =τ v), {Φ(τ, Q)} ∈ D([τ ]). z Let Φ be the z-variant for which Φz ([τ ], z) is (λv.P ∗ = v), {Φ(τ, P )} . Then Φz2 ([], [z(P ) → z(Q)]) is + ⇔ Φz2 ([], z(P )) is − or Φz2 ([], z(Q)) is + ⇔ Φz (τ, P ) ∈ / Φz2 ([τ ], z) or Φz (τ, Q) ∈ Φz2 ([τ ], z) ⇔ Φ(τ, P ) ∈ / Φz2 ([τ ], z) or Φ(τ, Q) ∈ Φz2 ([τ ], z), since z not free in P or Q, ⇔ Φ(τ, P ) is Φ(τ, Q), since Φ(τ, P ) ∈ Φz2 ([τ ], z). ⇔ Φ1 (τ, P ) Φ1 (τ, Q) Thus any valuation to the standard domain satisfies (NF) and therefore both (IM.1) and (IM.2). 3.3. Semantic inferences. Let the valuation Φ be a model of ITT. Since Φ is a function, the value Φ2 (τ, M ) for M :τ is unique. In particular, if F :[] then Φ2 ([], F ) has as its value exactly one of ±. This observation together with the following theorem provides the justification for the proof theory described in §4; a proof of the theorem, which depends on the substitution and semantic substitution lemmas, is left to the reader. A proof is available in [18]. Theorem. Let Φ be a model. ↓) Let F , G:[]. Then Φ2 ([], [F ↓ G]) is + ⇒ Φ2 ([], F ) is − and Φ2 ([], G) is −; Φ2 ([], [F ↓ G]) is − ⇒ Φ2 ([], F ) is + or Φ2 ([], G) is +. ∀)
Let P :[τ ]. Then Φ2 ([], ∀P ) is + ⇒ Φ2 ([], P (t)) is + for all t:τ ; Φ2 ([], ∀P ) is − ⇒ Φ2 ([], P (y)) is − for some y-variant Φy of Φ, y not free in P .
λ)
Let F > G. Then Φ2 ([], F ) is ± ⇒ Φ2 ([], G) is ±.
Int)
Let P ,Q:τ have dual type 1. Then Φ2 ([], P =1 Q) is ± ⇒ Φ2 ([], P =τ Q) is ±, respectively, where =1 :[1, 1], and =τ :[τ, τ ].
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3.4. Sequents and counter-examples. A sequent is an expression Γ Θ where Γ, the antecedent of the sequent, and Θ, the succedent of the sequent are finite, possibly empty, sets of formulas. A sequent Γ Θ is satisfied by a valuation Φ, if Φ2 ([], F ) is − for some F in the antecedent or is + for some F in the succedent. A sequent is valid if it is satisfied by every valuation that is a model. A valuation Φ is a counter-example for a sequent if Φ2 ([], F ) is + for every F in the antecedent and − for every F in the succedent. The proof theory of §4 provides a systematic search procedure for a counterexample for a given sequent Γ Θ. Should the procedure fail to find such a valuation, and if it does fail it will fail in a finite number of steps, then that Γ Θ is valid follows from a theorem of §5. The steps resulting in a failure are recorded as a derivation. Thus a derivation for a sequent is constructed under the assumption that a counter-example Φ exists for the sequent. Signed formulas are introduced to abbreviate assertions about the truth value assigned to a formula by Φ. Thus +F is to be understood as an abbreviation for “Φ2 ([], F ) is +” and −F for “Φ2 ([], F ) is −”, for some conjectured counter-example Φ. Note that Γ Θ has no counter-example if Γ Θ has no counter-example, where Γ ⊆ Γ, Θ ⊆ Θ, and Γ ∪ Θ is not empty.
§4. Proof theory. The proof theory is presented as a logic of sequents using a semantic tree form of the sequent calculus that has evolved from the semantic tableaux derivations of [4]. A similar proof theory for first order logic is described in [30] and in [11]. Semantic rules, in terms of which semantic trees are defined, are described in §4.1; these rules are motivated by the theorem of §3.3. A derivation of a sequent is a closed semantic tree based on the sequent, as these terms are defined in §4.2. 4.1. The semantic rules. There are four + and − pairs of rules for the propositional connective ↓, for the quantifier ∀, for λ and for intensionality. These rules are +↓
+∀
+[F ↓G]
+[F ↓G]
−F
−G
−↓
+F −∀
+∀P
−λ
+F
−∀P
−F −G
+G where F >G
+G
−P (y) y the eigen variable
+P (t) t the eigen term +λ
−[F ↓G]
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+Int
+P =1 Q
+Int
−P =1 Q
+P =τ Q −P =τ Q where P and Q have primary type τ and dual type 1 The last rule has a character different from these logical rules. It has no premiss and two conclusions: Cut +F
−F
Although the Cut rule is redundant, it is neverthless a useful rule since it permits the reuse of derivations.[18] 4.2. Semantic trees and derivations. A semantic tree is a binary tree with nodes that are signed formulas. A semantic tree based on a given sequent is defined as follows: (1) A tree with a single branch consisting of one or more nodes +F and −G, where F is from the antecedent and G is from the succedent of the sequent, is a semantic tree based on the sequent. These are the initial nodes of the tree. (2) Given a semantic tree based on a sequent, a tree obtained from it in any of the following ways is a semantic tree with the same initial nodes based on the sequent: (a) By attaching the conclusion of a single conclusion rule to the leaf of a branch that has a node that is the premiss of the rule, provided that if the rule is −∀ then the eigen variable has no free occurrence in the premiss of the rule or in any node above it. (b) By attaching the two conclusions of the +↓ rule on separate branches to the leaf of a branch that has a node that is the premiss of the rule. (c) By attaching +F and −F on separate branches to the leaf of a branch. A branch of a semantic tree is closed if there is a closing pair of nodes +F and −F on the branch. A semantic tree is closed if each of its branches is closed. A derivation of a sequent is a closed semantic tree based on the sequent. §5. Consistency. Since by the theorem of §3.2 any valuation to the standard domain is an intensional model for ITT, there exist valid sequents. Their relationship with derivable sequents is expressed in the next theorem. Theorem (Consistency). A derivable sequent of ITT is valid. Proof. Consider a derivation for a sequent Γ Θ. Let η be any node of the derivation which does not have an initial node below it. Define Γ[η] and Θ[η] to be the sets of sentences F for which +F , respectively −F , is η itself or is a node above η. Thus if η is the last of the initial nodes of the derivation, Γ[η] ⊆ Γ and Θ[η] ⊆ Θ; hence if Γ[η] Θ[η] is satisfied by a model, so is Γ Θ. Define the height h(η) of a node η to be the maximum of the heights of η on branches on which it occurs, with its height on a branch being the number of
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nodes below it on the branch. Γ Θ will be shown to be valid by induction on h(η). If h(η) = 0, then η is a leaf node of a branch of the derivation. Since the branch is closed, Γ Θ is valid. Assume therefore that h(η) > 0, and that there is a model Φ that is a counter-example for Γ Θ. Necessarily η is immediately above a conclusion η1 or conclusions η1 and η2 of one of the rules of deduction. There are therefore two main cases to consider corresponding to the single conclusion rules +↓, ±∀, ±λ, and ±Int, and to the two conclusion rules −↓ and Cut. • For the single conclusion rules it is sufficient to illustrate the argument with the ±∀ rules with premiss ±∀P and conclusion respectively +P (t) and −P (y), where P :[τ ] and t, y:τ with y not occurring free in any node above the conclusion −P (y) of the rule −∀. For the + case, ∀P ∈ Γ[η], Γ[η1] is Γ[η] ∪ {P (t)}, and Θ[η1] is Θ[η]. Since Φ satisfies Γ[η1] Θ[η1] but does not satisfy Γ[η] Θ[η] it follows that Φ2 ([], ∀P ) is + and Φ2 ([], P (t)) is − which contradicts clause (∀) of the theorem of §3.3. For the − case, Γ[η1] is Γ[η], ∀P ∈ Θ[η] , and Θ[η1] is Θ[η] ∪ {P (y)}. Since Φ satisfies Γ[η1] Θ[η1] but does not satisfy Γ[η] Θ[η] it follows that Φ2 ([], ∀P ) is − and Φ2 ([], P (y)) is +. But further, since y has no free occurrence in a member of Γ[η] ∪ Θ[η], and since Γ[η1] Θ[η1] is valid, Φy2 ([], P (y)) is + for every y-variant Φy of Φ. This contradicts clause (∀) of the theorem. • Let the premiss of an application of −↓ be −[F ↓ G] and the conclusions be +F and +G. Thus [F ↓ G] ∈ Θ[η], Γ[η1] is Γ[η] ∪ {F }, Γ[η2] is Γ[η] ∪ {G}, Θ[η1] is Θ[η], and Θ[η2] is Θ[η]. As before it follows that Φ2 ([], F ) is Φ2 ([], G) is − so that Φ2 ([], [F ↓ G]) is +. This contradicts clause (↓) of the theorem. For the case of cut, let the cut formula be F . In this case Γ[η1] is Γ[η] ∪ {F }, Θ[η1] is Θ[η], Γ[η2] is Γ[η], and Θ[η2] is Θ[η] ∪ {F }. It follows therefore that Φ2 ([], F ) is both + and −, which is impossible. §6. Completeness. Let a derivation of a sequent be understood to be a derivation in which cut is not used, and let an underivable sequent be one which does not have a derivation without cut. The following theorem is a consequence of lemmas stated below in §6.1 and §6.2. Theorem (Completeness without Cut). A counter-example that is a model exists for each underivable sequent of ITT. That cut is redundant is an immediate corollary since by the consistency theorem a derivable sequent is valid no matter whether cut is used in its derivation. 6.1. Underivable sets and semivaluations. A finite set Σ of signed formulas is said to be derivable if the sequent Σ+ Σ− is derivable, where Σ+ is {F | + F ∈ Σ} and Σ− is {F | − F ∈ Σ}. An infinite set Σ of signed sentences is derivable if some finite subset is derivable. An underivable set Σ is a semivaluation if it satisfies the following conditions: (1) +[F ↓ G] ∈ Σ ⇒ −F ∈ Σ and −G ∈ Σ −[F ↓ G] ∈ Σ ⇒ +F ∈ Σ or +G ∈ Σ
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(2) Let M :[τ ]. +∀(M ) ∈ Σ ⇒ +M (t) ∈ Σ, for all t:τ ; −∀(M ) ∈ Σ ⇒ −M (t) ∈ Σ, for some t:τ . (3) Let H > H , where H, H :[]. ±H ∈ Σ ⇒ ±H ∈ Σ. (4) Let P, Q:τ have dual type 1. ±P =1 Q ∈ Σ ⇒ ±P =τ Q ∈ Σ. Semivaluations for first-order logic have also been called downward saturated, or Hintikka sets. A proof of the following lemma is left to the reader. Lemma. Let Σ0 be a finite underivable set of signed sentences. There exists a semivaluation Σ for which Σ0 ⊆ Σ. 6.2. An intensional domain. Let Σ0 be a finite underivable set of signed sentences. By the lemma of §6.1, there exists a semivaluation Σ for which Σ0 ⊆ Σ. Σ∗ is obtained from Σ by replacing each signed formula ±F by ±F ∗ as defined in §3. A domain DΣ for an intensional valuation of ITT is defined from Σ∗ using the notation of §3.1 as follows: DΣ(1) is D(1). DΣ([]) is the set of pairs N ∗ , s for which s ∈ S[N ∗ ], where N :[] is in normal form, and S[N ∗ ] is the set of signs ± for which / S[N ∗ ], +N ∗ ∈ Σ∗ ⇒ + ∈ S[N ∗ ] and − ∈ ∗ ∗ ∗ −N ∈ Σ ⇒ − ∈ S[N ] and + ∈ / S[N ∗ ], and ∗ ∗ ∗ ∗ +N , −N ∈ / Σ ⇒ +, − ∈ S[N ]. DΣ([ τ ]) is the set of pairs N ∗ , s for which s ∈ S[N ∗ ], where N :[ τ ] is in normal form, and S[N ∗ ] is the set of sets s, s ⊆ DΣ(τ1 ) × · · · × DΣ(τn ) for which for all Ni∗ , si ∈ DΣ(τi ), 1 ≤ i ≤ n, +N (N1 , . . . , Nn )∗ ∈ Σ∗ ⇒ N1∗ , s1 , . . . , Nn∗ , sn
∈ s, and −N (N1 , . . . , Nn )∗ ∈ Σ∗ ⇒ N1∗ , s1 , . . . , Nn∗ , sn
∈ / s, Lemma. Each valuation to the domain DΣ is a model. Proof. Let ΦΣ be a valuation to the domain DΣ. Since ΦΣ(1, P ) ∈ DΣ(1) by (V.1) and (V.4), to prove that condition (IM.1) is satisfied it is sufficient to prove for τ = 1, (a) ΦΣ2 ([ τ ], P ) ∈ S[ΦΣ1 ([ τ ], P )] This is proved by induction on ct[P ]. Let ct[P ] be 0. If P is a constant or variable, then (a) follows immediately from (V.1). The other two cases to consider are when P is either ↓ or ∀. • Let P be ↓. Since ΦΣ1 ([[], []], ↓) is ↓, in this case it is sufficient to prove for all N1∗ , s1 , N2∗ , s2 ∈ DΣ([]) that + ↓ (N1∗ , N2∗ ) ∈ Σ∗ ⇒ N1∗ , s1 , N2∗ , s2
∈ ΦΣ2 ([[], []], ↓), respectively, / ΦΣ2 ([[], []], ↓). − ↓ (N1∗ , N2∗ ) ∈ Σ∗ ⇒ N1∗ , s1 , N2∗ , s2
∈ Let N1∗ , s1 , N2∗ , s2 ∈ DΣ([]). Then
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±↓(N1∗ , N2∗ ) ∈ Σ∗ ⇒ −N1∗ ∈ Σ∗ and −N2∗ ∈ Σ∗ , respectively, +N1∗ ∈ Σ∗ or +N2∗ ∈ Σ∗ ⇒ s1 is − and s2 is −, respectively, s1 is + or s2 is + ⇒ N1∗ , s1 , N2∗ , s2
∈ ΦΣ2 ([[], []], ↓), respectively, / ΦΣ2 ([[], []], ↓) N1∗ , s1 , N2∗ , s2
∈ • Let P be ∀. Since ΦΣ1 ([[τ ]], ∀) is ∀, in this case [ τ ] is [[τ ]] to prove for all N ∗ , s ∈ DΣ([τ ]) that +∀N ∗ ∈ Σ∗ ⇒ N ∗ , s ∈ ΦΣ2 ([[τ ]], ∀), respectively, −∀N ∗ ∈ Σ∗ ⇒ N ∗ , s ∈ / ΦΣ2 ([[τ ]], ∀). Let N ∗ , s ∈ DΣ([[τ ]]). Then ±∀N ∗ ∈ Σ∗ ⇒ +N (t)∗ ∈ Σ∗ for all t:τ , respectively, −N (t)∗ ∈ Σ∗ for some t:τ ⇒ t∗ ∈ s for all t:τ , respectively, / s for some t:τ t∗ ∈ ⇒ s is DΣ([τ ]), respectively, s is not DΣ([τ ]) ⇒ N ∗ , s ∈ ΦΣ2 ([[τ ]], ∀), respectively N ∗ , s ∈ ΦΣ2 ([[τ ]], ∀) Assume now that (a) holds whenever ct[P ] < ct. Consider the for which ct[P ] is ct.
15
(1) of §6.1
(V.2) and it is sufficient
(2) of §6.1 df of DΣ([τ ])
(V.3) terms P
• P is M N , where M :[τ, τ ] and N :τ . Assume τ is not empty. In this case it is sufficient to prove under the assumptions Ni∗ , si ∈ DΣ(τi ), 1 ≤ i ≤ n, that +(ΦΣ1 ([ τ ], M N )( N ))∗ ∈ Σ∗ ⇒ N, s ∈ ΦΣ2 ([ τ ], M N ), respectively, −(ΦΣ1 ([ τ ], M N )( N ))∗ ∈ Σ∗ ⇒ N, s ∈ / ΦΣ2 ([ τ ], M N ). By the induction assumption, (a) holds when P is M or N so that ΦΣ(τ, N ) ∈ DΣ(τ ) and for all Ni∗ , si ∈ DΣ(τi ), 1 ≤ i ≤ n, +(ΦΣ1 ([τ, τ ], M )(N, N ))∗ ∈ Σ∗ ⇒ ΦΣ(τ, N ), N, s ∈ ΦΣ2 ([τ, τ ], M ), respectively, −(ΦΣ1 ([τ, τ ], M )(N, N ))∗ ∈ Σ∗ ⇒ ΦΣ(τ, N ), N, s ∈ / ΦΣ2 ([τ, τ ], M ). ∗ ∗ ±(ΦΣ1 ([ τ ], M N )( N )) ∈ Σ ⇒ ±(ΦΣ1 ([τ, τ ], M )(N, N ))∗ ∈ Σ∗ df of S[ ] ⇒ ΦΣ(τ, N ), N, s ∈ ΦΣ2 ([τ, τ ], M ), respectively, ΦΣ(τ, N ), N, s ∈ / ΦΣ2 ([τ, τ ], M ) induction ⇒ N, s ∈ ΦΣ2 ([ τ ], M N ), respectively, / ΦΣ2 ([ τ ], M N ) (V.5) N, s ∈ The case τ is empty is left to the reader. • P is (λx.M ), where M :[ τ ] and x:τ . Assume τ is not empty.
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In this case it is sufficient to prove under the assumptions N ∗ , s ∈ DΣ(τ ) and Ni∗ , si ∈ DΣ(τi ), 1 ≤ i ≤ n, that +(ΦΣ1 ([τ, τ ], λx.M )(N, N ))∗ ∈ Σ∗ ⇒ N, N ∈ ΦΣ2 ([τ, τ ], (λx.M )), respectively, −(ΦΣ1 ([τ, τ ], λx.M )(N, N ))∗ ∈ Σ∗ ⇒ N, N ∈ / ΦΣ2 ([τ, τ ], (λx.M )). ±(ΦΣ1 ([τ, τ ], λx.M )(N, N ))∗ ∈ Σ∗ ⇒ ±(ΦΣ1 ([ τ ], (λx.M )N )(, N ))∗ ∈ Σ∗ df of ΦΣ1 ⇒ ±(ΦΣ1 ([ τ ], [N/x]M )( N ))∗ ∈ Σ∗ (3) of §6.1 ⇒ ±(ΦΣx1 ([ τ ], M )( N ))∗ ∈ Σ∗ ) lemma §3.1 where ΦΣx (τ, x) is ΦΣ(τ, N ) ⇒ N, s ∈ ΦΣx2 ([ τ ], M ), respectively, / ΦΣx2 ([ τ ], M ) induction N, s ∈ ⇒ N, N ∈ ΦΣ2 ([τ, τ ], M ), respectively, / ΦΣ2 ([τ, τ ], M ) (V.6) N, N ∈ Assume τ is empty. In this case it is sufficient to prove that when ΦΣ(τ, N ) ∈ DΣ(τ ), +(ΦΣ1 ([], λx.M )N ))∗ ∈ Σ∗ ⇒ ΦΣ(τ, N ) ∈ ΦΣ2 ([τ ], (λx.M )), respectively, −(ΦΣ1 ([], λx.M )N ))∗ ∈ Σ∗ ⇒ ΦΣ(τ, N ) ∈ / ΦΣ2 ([τ ], (λx.M )). ±ΦΣ1 ([], (λx.M )N ) ∈ Σ∗ ⇒ ±ΦΣ1 ([], [N/x]M )) ∈ Σ∗ (3) §6.1 ⇒ ΦΣ2 ([], [N/x]M )) is ± induction ⇒ ΦΣx2 ([], M )) is ± lemma §3.1 ⇒ ΦΣ(τ, N ) ∈ ΦΣ2 ([τ ], (λx.M ))), respectively, (V.6) ΦΣ(τ, N ) ∈ / ΦΣ2 ([τ ], (λx.M ))) Consider now (IM.2). By (a) ΦΣ2 ([], P =1 Q) ∈ S[ΦΣ1 ([], P =1 Q)] Hence ΦΣ2 ([], P =1 Q) is ± ⇒ ± =1 (ΦΣ1 (1, P ), ΦΣ1 (1, Q)) ∈ Σ∗ df of ΦΣ1 ⇒ ± =τ (ΦΣ1 (τ, P ), ΦΣ1 (τ, Q)) ∈ Σ∗ (4) of §6.1 df of S[ ] ⇒ ΦΣ2 ([], P =τ Q) is ± This completes the proof of the lemma.
Corollary. ±ΦΣ1 ([], F ) ∈ Σ∗ ⇒ ΦΣ2 ([], F ) is ± respectively, for F :[]. By the lemma, ΦΣ is a model of ITT. By the corollary, the underivable set of signed sentences Σ0 obtained from an underivable sequent is satisfied by ΦΣ. This completes the proof of the completeness theorem. §7. Extensions to ITT. A sketch is given in §7.1 of the logic ITTf that introduces a functional notation into ITT. In §7.2 an extension SITT of ITT is described in which such sets as Russell’s can be consistently defined. Full details of the logics are provided in chapters 5 and 6 of [18]. 7.1. ITTf. A notation for total functions with values of type 1 is available in ITT in the form of higher order terms that are also first order terms. The term S defined in §1.3, for example, is used in this way: S(t) is a first order term when
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t is. Also syntactic sugaring provides a functional notation for total functions with values of type other than 1. But partial functions of n arguments can only be represented in ITT by predicates of n + 1 arguments, the extra argument being the value of the function. The logic ITTf is an extension of ITT with an explicit notation for partial or total functions with values of type 1. The types of ITTf are the types of ITT. All the constants and variables of ITT are constants and variables of ITTf of the same type. But ITTf also has available the untyped symbol ι that is used to create function terms. The terms of ITTf are defined by adding the following clauses to the clauses (1)–(3) of §2.1: (4) Let M be a term of type [1]. Then ιM is a function term and term of type 1. (5) Let M be a term of type τ , τ =1, for which each free occurrence of a variable in M is either of type 1, or is an occurrence in a function term with a free occurrence in M . Then M is also a term of type 1. The proof theory of ITTf is obtained from the proof theory of ITT as follows: • The +∀ rule for type 1 eigen terms has a second conclusion −Den(t), where df
Den = (λu.u=u), in order to ensure that the eigen term t is denoting; Farmer in [10] and others use the notation ↓t to express the same thing. • The λ rules are affected by a modification to the definition of β–reduction: A term (λv.P )(t) is not contractible to [t/v]P if t is a term of type 1 in which a function term has a free occurrence. This change is necessary to ensure the proper scope for function terms. • The terms P and Q of an Int rule must satisfy the additional requirement that no function term has a free occurrence in either. • ι rules are added with premiss ±P ([ιM/v]t) and conclusion ±[∃!v.M ]P (t), where the variable v has no free occurrence in a function term with a free occurrence in t. This restriction is necessary to ensure the proper scope for function terms. A semantics for ITTf is provided by a mapping of terms of ITTf into terms of ITT. A proof of a completeness and cut-elimination theorem similar to the theorem of §6 is provided in [18] along with a proof that ITTf is a conservative extension of ITT. 7.2. Consistent uses of Russell’s set. The variables of ITT have two distinct roles to play, as the eigen variable in the conclusion of an application of the −∀ rule, and as abstraction variables in the λ rules. These two roles can be served in formulas by syntactically distinct variables. An abstraction variable is used purely as a placeholder and need never be interpreted. But clearly the eigen variable of an application of the −∀ rule must receive a value. By using distinct variables for these two roles, and interpreting quantification variables as in ITT but interpreting abstraction variables as placeholders, a consistent extension of ITT can be constructed in which such impredicative sets as Russell’s can be defined. Each variable v of ITT has a unique type †[v]; these variables are the quantifiable variables of the extended ITT. An abstraction variable v of the extended ITT on the other hand has a specified type †[v], but also a dual type 1. This allows applications of abstraction not permitted in ITT. For example, it is not
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PAUL C GILMORE
difficult to derive ¬0(0) in ITT, where 0 has been defined in §1.3. The first occurrence of 0 in 0(0) is of type [1] while the second is of type 1. If v is an abstraction variable for which †[v] is [1], then (λv.¬v(v))(0) is derivable in the extended ITT. df Defining Rus = (λv.¬v(v)) results in Rus(0) being derivable. Similarly df
if V = (λv.v=v), then ¬Rus(V ) is derivable. Note however that neither Rus(Rus) nor ¬Rus(Rus) is derivable since Rus:[[1]] and therefore Rus(Rus) is not well-formed. Thus no contradiction results from Rus being a well-formed predicate. But note also that the predicate Rus of the extended ITT differs significantly from any predicate of ITT since although Rus(0) is derivable, ∃x.Rus(x) is not, since for no quantification variable x is Rus(x) well-formed. Details are provided in [18]. REFERENCES
[1] Peter B. Andrews, Sunil Issar, Daniel Nesmith, and Frank Pfenning, The TPS theorem proving system, 9’th International Conference on Automated Deduction (CADE) (E. Lusk and R. Overbeek, editors). Lecture Notes in Computer Science. vol. 310, Springer-Verlag, 1988, pp. 760-761. [2] H. P. Barendregt, The Lambda Calculus, its syntax and semantics, North-Holland, 1985, Revised Edition. [3] Jon Barwise and Lawrence Moss, Vicious circles, CSLI Publications, 1996. [4] E.W. Beth, Semantic entailment and formal derivability, Mededelingen de Koninklijke Nederlandse Akademie der Wetenschappen, Afdeeling Letterkunde, Nieuwe Reeks, vol. 18 (1955), no.13, pp. 309–342. [5] Rudolph Carnap, Meaning and Necessity, University of Chicago Press 1947. [6] Alonzo Church, Schr¨ oder’s anticipation of the simple theory of types, The Journal of Unified Science (Erkenntnis) , vol. IX (1939), pp. 149–152. [7] Alonzo Church, A formulation of the simple theory of types, Journal of Symbolic Logic, vol. 5 (1940), pp. 56–68. [8] Nino B. Cocchiarella, Logical investigations of predication and the problem of universals, Bibliopolis Press, Naples, 1986. [9] Nino B. Cocchiarella, Conceptual realism versus Quine on classes and higher-order logic, Synthese, vol. 90 (1992), pp. 379–436. [10] William M. Farmer, A partial functions version of Church’s simple theory of types, Journal of Symbolic Logic, vol. 55 (1990), pp. 1269–1290. [11] M.C. Fitting, First-order logic and automated theorem proving, Springer- Verlag, 1996, Second Edition. [12] Paul C. Gilmore, A consistent naive set theory: Foundations for a formal theory of computation, IBM Research Report RC 3413, June 22, 1971. [13] Paul C. Gilmore, Combining unrestricted abstraction with universal quantification, To H.B. Curry: Essays on combinatorial logic, lambda calculus and formalism (J.P. Seldin and J.R. Hindley, editors), Academic Press, 1980, pp. 99–123. This is a revised version of [12]. [14] Paul C. Gilmore, Natural deduction based set theories: A new resolution of the old paradoxes, Journal of Symbolic Logic, vol. 51 (1986), pp. 393–411. [15] Paul C. Gilmore, A foundation for the entity relationship approach: How and why, Proceedings of the 6th Entity Relationship Conference (S.T. March, editor), North-Holland, 1988, pp. 95–113. [16] Paul C. Gilmore, NaDSyL and some applications, Computational logic and proof theory, The Kurt G¨ odel Colloquium 97 (Georg Gottlob, Alexander Leitsch, and Daniele Mundici, editors), Lecture Notes in Computer Science vol. 1289, Springer-Verlag, 1997, pp. 153–166.
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[17] Paul C. Gilmore, An impredicative simple theory of types, presented at the Fourteenth Workshop on Mathematical Foundations for Programming Systems, Queen Mary College, London, May 1998. [18] Paul C. Gilmore, An intensional type theory:Logical foundations for mathematics and computer science, a monograph on ITT in preparation. [19] Michael J.C. Gordon, A proof generating system for higher-order logic, VLSI Specification, Verification and Synthesis, (G. Birtwistle and P. Subrahmanyam, editors), Academic Publishers, Boston, 1987. Also, University of Cambridge Computer Laboratory Technical Report No. 103, pp. 73–128. [20] Michael J.C. Gordon, Set theory, higher order logic or both?, Higher order logic theorem proving and its applications (Joakum von Wright, Jim Grundy, and John Harrison, editors), Lecture Notes in Computer Science, vol. 1125, 1996, pp. 191–202. [21] Leon Henkin, Completeness in the theory of types, Journal of Symbolic Logic, vol. 15 (1953), pp. 81–91. [22] Gopalan Nadathur and Dale Miller, Higher-order logic programming, Handbook of logic in artificial intelligence and logic programming (Dov M. Gabbay, C.J. Hogger and J.A. Robinson, editors), Clarendon Press, Oxford, 1998, pp. 499–590. [23] S. Owre, N. Shankar, and J.M. Rushby, The PVS specification language (beta release), Computer Science Laboratory, SRI International, Menlo Park CA 94025, June 14, 1993. [24] Dag Prawitz, Hauptsatz for higher order logic, Journal of Symbolic Logic, vol. 33 (1968), pp. 452–457. [25] Bertrand Russell, The Principles of Mathematics, George Allen and Unwin Ltd, London, second edition, 1937. [26] K. Sch¨ utte, Syntactical and semantical properties of simple type theory, Journal of Symbolic Logic, vol. 25 (1960), pp. 305–326. [27] Wilfred Sellars, Abstract entities, Rev. of Metaphysics, vol. 16 (1963), pp. 625–671. [28] Wilfred Sellars, Classes as abstract entities and the Russell paradox, Rev. of Metaphysics, vol. 17 (1963), pp. 67–90. [29] Joseph R. Shoenfield, Mathematical logic, Addison-Wesley, 1967. [30] R.M. Smullyan, First-order logic, Springer-Verlag, Berlin, 1994, Revised Edition, Dover Press, New York. [31] Moto-o Takahashi, A proof of cut-elimination theorem in simple type-theory, Journal of the Mathematical Society of Japan, vol. 19, 4 (1967), pp. 399–410. [32] Ernst Zermelo, Investigations in the foundations of set theory, From Frege to G¨ odel (Jean van Heijenoort, editor), Harvard University Press, 1967, pp. 199–215. DEPARTMENT OF COMPUTER SCIENCE UNIVERSITY OF BRITISH COLUMBIA 2366 MAIN MALL VANCOUVER, BC V6T 1Z4 CANADA
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