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On Reductions of Hintikka Sets for Higher-Order Logic

April 2020

April 2020

Authors:

Alexander Steen

University of Greifswald

Christoph Benzmüller

Otto-Friedrich-Universität Bamberg

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On Reductions of Hintikka Sets for Higher-Order Logic

Alexander Steen1, Christoph Benzmüller2

1University of Luxembourg, FSTM, alexander.steen@uni.lu

2Freie Universität Berlin, FB Mathematik und Informatik,c.benzmueller@fu-berlin.de

January 28, 2021

Abstract

Steen’s (2018) Hintikka set properties for Church’s type theory based on primitive equality are reduced

to the Hintikka set properties of Brown (2007). Using this reduction, a model existence theorem for Steen’s

properties is derived.

1 Introduction and Preliminaries

Abstract consistency properties and Hintikka sets play an important role in the study (e.g., of Henkin-complete-

ness) of proof calculi for Church’s type theory [7, 2], aka. classical higher-order logic (HOL). Technically quite

diﬀerent deﬁnitions of these terms have been used in the literature, since they depend on the primitive logical

connectives assumed in each case. The deﬁnitions of Benzmüller, Brown and Kohlhase [3, 4], for example, are

based on negation, disjunction and universal quantiﬁcation, while Steen [8], in the tradition of Andrews [1],

works with primitive equality only. Despite their conceptual relationship, important semantical corollaries that

are implied by these syntax related Hintikka properties (such as model existence theorems) can hence not be

directly transferred between formalisms.

Brown [6, 5] presents generalised abstract consistency properties in which the primitive logical connectives

can vary. In this paper we show that the properties of Steen can be reduced to those of Brown.1Theorem 1

that we establish in this paper paves way for the convenient reuse of (e.g., model existence) results from the

work of Brown [6, 5] in the context of Steen’s setting.

Paper structure. In the remainder of this introduction we recapitulate some relevant (syntactic) notions on

HOL, mainly to clarify our notation; for further details on HOL as relevant for this paper. §2 we present the

Hintikka properties as used by Brown, and in §3 we give the related properties as used by Steen. In §3 we then

show various lemmas that are implied in Steen’s setting, and it are (some of ) those lemmas which prepare the

main reduction result of this paper (Theorem 1), which is given in §4. A model existence theorem for Hintikka

sets as deﬁned by Steen is then derived in §5.

Equality conventions. Diﬀerent notions of equality will be used in the following: If a concept is deﬁned (as

an abbreviation), the symbol := is used. Primitive equality, written =τ, refers to a logical constant symbol

from the HOL language such that sτ=τtτis a term of the logic (assuming sτand tτare, where τis a

type annotation), cf. details further below. Leibniz-equality, written ˙=, is a deﬁned term; usually it stands

for λXτ. λYτ.∀Poτ .(P X)⇒(P Y ), where ⇒is a (primitive) logical connective. Meta equality ≡denotes

set-theoretic identity between objects. Finally, ≡?, for ?⊆ {β, η }is used for syntactic equality modulo β−,η−

and βη-conversion, respectively (as in the related work, α-conversion is taken as implicit).

Syntax of HOL. The set Tof simple types is freely generated from the base types oand ιby juxtaposition.

The types oand ιrepresent the type of Booleans and individuals, respectively. A type ντ represents the type

of a total function from objects of type τto objects of type ν.

1In an earlier reduction attempt we tried to reduce Steen’s [8] abstract consistency properties to those of Benzmüller, Brown and

Kohlhase [3]. This attempt had gaps that could not easily be closed (as was pointed out by an unknown reviewer). The mapping

that we applied in this initial reduction attempt between the two formalisms replaced primitive equations (in Steen’s Hintikka sets)

by Leibniz equations (to obtain Hintikka sets in the style of Benzmüller, Brown and Kohlhase). It thereby introduced additional

universally quantiﬁed formulas which in turn triggered the applicability of abstract consistency conditions whose validity could

not be ensured by referring to those of Steen. In this work we therefore instead map to the structurally better suited, generalised

conditions of Brown [6, 5], which enables us to circumvent the problems induced by our previous reduction attempt to the work of

Benzmüller, Brown and Kohlhase.

Let Στbe a set of constant symbols of type τ∈ T and let Σ := Sτ∈T Στbe the union of all typed symbols,

called a signature. Let further Vdenote a set of (typed) variable symbols. From these the terms of HOL are

constructed by the following abstract syntax (τ , ν ∈ T ):

s, t ::= cτ∈Σ|Xτ∈ V | (λXτ. sν)ντ |(sν τ tτ)ν

The terms are called constants,variables,abstractions and applications, respectively. The set of all terms of

type τover a signature Σis denoted Λτ(Σ), and Λc

τ(Σ) is used for closed terms, respectively. The notion of free

and bound variables are deﬁned as usual, and a term tis called closed if tdoes not contain any free variables.

It is assumed that the set Vcontains countably inﬁnitely many variable symbols for each type τ∈ T .

The type of a term is written as subscript but may be dropped by convention if clear from the context

(or if not important). Also, parentheses are omitted whenever possible, and application is assumed to be left-

associative. Furthermore, the scope of an λ-abstraction’s body reaches as far to the right as is consistent with

the remaining brackets. Nested applications s t1. . . tnmay also be written in vector notation s tn.

Variants of HOL often diﬀer regarding the choice of the primitive logical connectives in the signature Σ.

In any case, s6=tis used in the remainder as an abbreviation for ¬(s=t). Also, for simplicity, binary

logical connectives may be written in inﬁx notation; e.g., the term po∨qoformally represents the application

(∨ooo poqo). Furthermore, binder notation is used for universal and existential quantiﬁcation: The term ∀Xτ. so

is used as a short-hand for Πτ(λXτ. so), where Πτis a constant symbol. Finally, Leibniz-equality, denoted ˙=,

is deﬁned as ˙= := λXτ. λYτ.∀Poτ .(P X)⇒(P Y ). A Σ-formula sois a term from so∈Λo(Σ) of type oand a

Σ-sentence if it is a closed Σ-formula. The reference to Σmay be omitted if clear from the context.

In the following, variables are denoted by capital letters such as Xτ, Yτ, Zτ, and, more speciﬁcally, the

variable symbols Po, Qoand Fντ , Gν τ are used for predicate or Boolean variables and variables of functional

type, respectively. Analogously, lower case letters sτ, tτ, uτdenote general terms and fντ , gντ are used for terms

of functional type.

Semantics of HOL. The semantics of HOL, including the notions of Σ-models and Σ-Henkin models, is not

discussed here; cf. [3, 8, 6, 5] for details.

2 Hintikka sets as deﬁned by Brown [6]

In the formulation of HOL as employed by Brown [6], the set of primitive logical connectives is not ﬁxed and

may be chosen arbitrarily from the set {>o,⊥o,¬oo,∧ooo,∨ooo ,⊃ooo ,≡ooo} ∪ {Πo(oτ )|τ∈ T } ∪ {Σo(oτ)|τ∈

T } ∪ {=τ

oτ τ |τ∈ T }. All remaining constant symbols from Σare called parameters. In this presentation, the

notation from Brown is adapted and we apply the conventions from above. For example, we denote general

terms with lower case symbols sand tinstead of upper case letters as used by Brown, and instead of wﬀ τ(Σ),

which Brown uses to denote the set of internal terms of type τ, we use Λτ(Σ).

Brown distinguishes between internal terms, the elements of Λτ(Σ), and external propositions (meta-level

propositions) in a set prop(Σ), which are deﬁned as follows [6, Def. 2.1.20] (we use the color red to visually

highlight Brown’s meta-level connectives and their associated Hintikka set properties in the remainder of this

paper):

•If s∈Λo(Σ), then s∈prop(Σ).

•If α∈ T and s, t ∈Λα(Σ), then [s=

·t]∈prop(Σ).

•>

·∈prop(Σ).

•If s∈prop(Σ), then [¬

·s]∈prop(Σ).

•If s, t ∈prop(Σ), then [s∨

·t]∈prop(Σ).

•If s∈prop(Σ), then [∀

·Xαs]∈prop(Σ).

Closed propositions s∈prop(Σ) are also called sentences; we then write s∈sent(Σ).

Brown introduces the following Hintikka set properties for external propositions.

Properties of extensional Hintikka sets H[6, Def. 5.5.4]:

∇c:s /∈ H or ¬

·s /∈ H.

∇βη :If s∈ H, then s↓∈ H.

∇⊥:¬

·>

·/∈ H.

∇¬:If ¬

·¬

·s∈ H, then s∈H.

∇∨:If s∨

·t∈ H, then s∈ H or t∈ H.

∇∧:If ¬

·(s∨

·t)∈ H, then ¬

·s∈ H and ¬

·t∈ H.

∇∀:If ∀

·Xτs∈ H, then [t/x]s∈ H for every closed term t∈Λc

τ(Σ).

∇∃:If ¬

·(∀

·Xτs)∈ H, then there is a parameter pτ∈Στsuch that ¬

·([p/X]s)∈ H.

∇]:If s∈ H, then s]∈ H. Also, if ¬

·s∈ H, then ¬

·s]∈ H.

∇m:If ¬

·h sn∈ H and h tn∈ H, then there is an iwith 1≤i≤nsuch that ¬

·si=

·ti∈ H.

∇dec :If pis a parameter and ¬

·(p sn)=

·ι(p tn)∈ H, then there is an i,1≤i≤n, s.t. ¬

·si=

·ti∈ H.

∇b:If ¬

·(s=

·ot)∈ H, then {s, ¬

·t}⊆Hor {¬

·s, t}⊆H.

∇f:If ¬

·(f=

·ντ g)∈ H, then there is a parameter pτ∈Στsuch that ¬

·(f p =

·νg p)∈ H.

∇o

=:If s=

·ot∈ H, then {s, t}⊆Hor {¬

·s, ¬

·t}⊆H.

∇→

=:If f=

·ντ g∈ H, then (f u =

·νg u)∈ H for every closed term u∈Λc

τ(Σ).

∇r

=:¬

·(s=

·is)/∈ H.

∇u

=:Suppose (s=

·it)∈ H and ¬

·(u=

·iv)∈ H. Then ¬

·(s=

·iu)∈ H or ¬

·(t=

·iv)∈ H. Also, ¬

·(s=

·iv)∈ H

or ¬

·(t=

·iu)∈ H.

The collection of all sets of sentences satisfying all these properties is called Hintβfb(Σ).

3 Hintikka sets as deﬁned by Steen [8]

In the formulation of HOL as employed by Steen [8], the equality predicate, denoted =τ, for each type τ, is

assumed to be the only logical connective present in the signature Σ, i.e., {=τ|τ∈ T } ⊆ Σ. All (potentially)

remaining constant symbols from Σare called parameters. Such signatures are also referred to as Σ=. A formu-

lation of HOL based on equality as sole logical connective originates from Andrew’s system Q0, cf. [1] and the

references therein. The usual logical connectives are deﬁned as follows (technically our formulation is a modi-

ﬁcation of the one used by Andrews [1], since the order of terms in deﬁning equations is swapped in many cases):

>o:= =o

ooo =ooo

o(ooo)(ooo)=o

ooo

⊥o:= (λPo. P ) =oo (λPo.>)

¬oo := λPo. P =o⊥

∧ooo := λPo. λQo.(λFooo. F > >) =o(ooo)(λFooo. F P Q)

∨ooo := λPo. λQo.¬(¬P∧ ¬Q)

⇒ooo := λPo. λQo.¬P∨Q

⇔ooo := λPo. λQo. P =oQ

Πτ

o(oτ):= λPoτ . P =oτ λXτ.>

The (primitive and deﬁned) connectives of this formulation of HOL are written in blue in the following, as

are the properties below.

Properties for acceptable Hintikka sets [8, Def. 3.15]

∇c:s /∈ H or ¬s /∈ H.

∇βη : If s≡βη tand s∈ H, then t∈ H.

∇r

=:(s6=s)/∈ H.

∇s

=: If u[s]p∈ H and s=t∈ H then u[t]p∈ H.

∇+

b: If s=t∈ H, then {s, t}⊆H or {¬s, ¬t}⊆H.

∇−

b: If s6=t∈ H, then {s, ¬t} ⊆ H or {¬s, t}⊆H.

∇+

f: If fν τ =gντ ∈ H, then f s=g s ∈ H for any closed term s∈Λc

τ(Σ).

∇−

f: If fν τ 6=gντ ∈ H, then f w6=g w ∈ H for some parameter w∈Στ.

∇m: If s, t are atomic and s, ¬t∈ H, then s6=t∈ H.

∇d: If h sn6=h tn∈ H, then there is an iwith 1≤i≤nsuch that si6=ti∈ H.

The collection of all sets satisfying these properties is called H. Every element H ∈ His called acceptable.

Deﬁnition 1. A set Hof formulas is called saturated iﬀ s∈ H or ¬s∈ H for every closed formula s.

Derived properties

Lemma 1 (Basic properties).Let H ∈ H. Then it holds that

(a) ⊥/∈ H

(b) ¬>/∈ H

(d) If so=>∈Hor >=so∈ H (so6=⊥ ∈ H or ⊥6=so∈ H), then {so,>} ⊆ H ({so,¬⊥} ⊆ H)

(e) If so=⊥∈Hor ⊥=so∈ H (so6=> ∈ H or >6=so∈ H), then {¬so,¬⊥} ⊆ H ({¬so,>} ⊆ H)

(f) If ¬⊥ ∈ H, then >∈H

(g) If > ∈ H, then ¬⊥ ∈ H

(h) If s=t∈ H and t=u∈ H, then s=u∈ H.

Proof. Let H ∈ Hbe an acceptable Hintikka set.

(a) Assume ⊥∈H. By deﬁnition of ⊥it holds (λP. P)=(λP . >)∈ H. Hence, by ~

∇+

fand ~

∇βη , it follows

that w=>∈Hfor any closed term w. Taking w≡¬>we obtain ¬>=>∈H. But then, ~

∇+

bgives us a

contradiction to ~

∇c. Hence ⊥/∈ H.

(b) Assume ¬> ∈ H. By deﬁnition of ⊥it holds (=6==)∈ H, which contradicts ~

∇r

=. Hence, ¬>/∈ H.

∇+

bgives us that either {>,⊥} ⊆ H or {¬>,¬⊥} ⊆ H. Either case is

impossible by either (a) or (b) of this lemma. Hence, >=⊥/∈ H.

(d) Let so=> ∈ H or >=so∈ H. In both cases it follows by ~

∇+

bthat either {s, >} ⊆ H or {¬s, ¬>} ⊆ H.

Since the latter case contradicts (b) from above, it follows that {s, >} ⊆ H. The negative cases are

analogous using ~

∇−

(e) Let so=⊥ ∈ H or ⊥=so∈ H. In both cases it follows by ~

∇+

bthat either {s, ⊥} ⊆ H or {¬s, ¬⊥} ⊆ H.

Since the former case contradicts (a) from above, it follows that {¬s, ¬⊥} ⊆ H. The negative case is

analogous using ~

∇−

(f) Let ¬⊥ ∈ H. Then, by deﬁnition of ⊥,(λP. P )6=(λP. >)∈ H. By ~

∇−

fand ~

∇βη it holds that p6=> ∈ H

for some parameter p. By ~

∇−

bit follows that either {p, ¬>} ⊆ H or {¬p, >} ⊆ H. Since the former case

is ruled out by (b) from above, the latter case yields the desired result.

(g) Let > ∈ H, that is, =o=ooo=o∈ H. By ~

∇+

fand ~

∇βη it follows that (so=to)=(so=to)∈ H for every two

closed formulas s, t. For s≡t≡¬⊥it follows that (¬⊥=¬⊥)=(¬⊥=¬⊥)∈ H, and hence, by ~

∇+

b, either

¬⊥=¬⊥∈Hor ¬⊥6=¬⊥∈H. Since the latter case is ruled out by ~

∇r

=, it follows that ¬⊥=¬⊥∈H.

Again, by ~

∇+

b, it follows that either ¬⊥∈Hor ¬(¬⊥)∈ H. The latter case is impossible by ~

∇r

=since

¬(¬⊥)≡(⊥6=⊥)and hence ¬⊥ ∈ H.

(h) Let s=t∈ H and t=u∈ H. By ~

∇s

=it follows directly that s=u∈ H.

Lemma 2 (Properties of usual connectives).Let H ∈ H. Then it holds that

(a) If ¬¬so∈ H, then s∈ H

(b) If (so∨to)∈ H, then s∈ H or t∈ H.

(d) If ΠαF∈ H, then F s ∈ H for every closed term s.

(e) If ¬ΠαF∈ H, then ¬(F w)∈ H for some parameter w∈Σ.

Proof. Let H ∈ Hbe an acceptable Hintikka set.

(a) Let ¬¬so∈ H. By deﬁnition of ¬and ~

∇βη it holds (s6=⊥)∈ H. Hence, by ~

∇−

b, either {s, ¬⊥} ⊆ H,

or {¬s, ⊥} ⊆ H. As the latter case is impossible by Lemma 1(a), it follows that {s, ¬⊥} ⊆ H and, in

particular, that s∈ H.

(b) Let (so∨to)∈ H. By deﬁnition of ∨,¬and ~

∇βη it holds λP. P > >6=λP. P (¬s) (¬t)∈ H. Hence, by

∇−

fand ~

∇βη , it follows that (p> >)6=p(¬s) (¬t)for some parameter p∈Σ. By ~

∇deither (i) >6=¬s∈ H

or (ii) >6=¬t∈ H. Hence, by ~

∇−

b, applied to both cases, it holds that either (i) ¬¬s∈ H, or (ii) ¬¬t∈ H

(because ¬>/∈ H by Lemma 1(b)). It follows that s∈ H or t∈ H by (a) of this lemma.

∇βη it holds (λP. P > >)=(λP. P s t)∈ H. Hence, by ~

∇+

and ~

∇βη , it follows that (w> >)=(w s t)∈ H for every closed term w. By ~

∇βη , using w≡λx. λy. x

and w≡λx. λy. y, it holds >=s∈ H and >=t∈ H, respectively. Application of Lemma 1(d) yields the

desired result.

(d) Let Παs∈ H. By deﬁnition of Παand ~

∇βη it holds s=λx. >∈ H. Hence, by ~

∇+

fand ~

∇βη , it follows

that s t=> ∈ H for every closed term t. Application of Lemma 1(d) yields the desired result.

(e) Let ¬Παs∈ H. By deﬁnition of Παand ~

∇βη it holds that s6=(λx. >)∈ H. Hence, by ~

∇−

fand ~

∇βη , it

follows that (s p)6=> ∈ H for some parameter p. Application of Lemma 1(e) yields the desired result.

Lemma 3 (Properties of Leibniz equality).Let H ∈ H. Then it holds that

(a) If s˙= t∈ H, then s=t∈ H.

(b) If ¬(s˙= t)∈ H, then s6=t∈ H.

(d) If u[s]p∈ H and s˙= t∈ H, then u[t]p∈ H.

(e) If s˙= t∈ H, then t˙= s∈ H.

(f) If s˙= t∈ H and t˙= u∈ H, then s˙= u∈ H.

Proof. Let H ∈ Hbe an acceptable Hintikka set.

(a) Let (s˙= t)∈ H. By deﬁnition of ˙= and ~

∇βη we have λP. (P s)⇒(P t)=λP. >∈ H, and hence, by

∇+

fand ~

∇βη , it holds that (w s)⇒(w t)=>∈Hfor every closed term w. Then, (w s)⇒(w t)∈ H by

Lemma 1(d). By deﬁnition of ⇒and ~

∇βη it holds that ¬(w s)∨(w t)∈ H and hence, by Lemma 2(b),

that ¬(w s)∈ H or (w t)∈ H. For w≡(λX. s=X)it follows by ~

∇βη that ¬(s=s)∈ H or (s=t)∈ H.

Since the former case contradicts ~

∇r

=, it follows that (s=t)∈ H.

(b) Let ¬(s˙= t)∈ H. By deﬁnition of ˙= and ~

∇βη we have λP. (P s)⇒(P t)6=λP. >∈ H. Hence, by

∇−

fand ~

∇βη , it holds that (p s)⇒(p t)6=> ∈ H for some parameter p. By Lemma 1(e) it follows that

¬(p s)⇒(p t)∈ H. Then, by Lemma 2(a) and 2(c), it follows that ¬¬(p s)∈ H and ¬(p t)∈ H.

Moreover, (p s)∈ H by Lemma 2(a). By ~

∇mit then follows that (p s)6=(p t)∈ H, and ﬁnally, by ~

∇d,

that s6=t∈ H.

∇r

=. Hence, ¬(s˙= s)/∈ H.

(d) Let u[s]p∈ H and s˙= t∈ H. By (a) above it holds that s=t∈ H and thus by ~

∇s

=it follows that u[t]p∈ H.

(e) Let (s˙= t)∈ H. By deﬁnition of ˙= and ~

∇βη we have λP. (P s)⇒(P t)=λP. >∈ H. Hence, by

∇+

fand ~

∇βη , it holds that (w s)⇒(w t)=>∈Hfor every closed term w. Then, (w s)⇒(w t)∈ H by

Lemma 1(d). By deﬁnition of ⇒and ~

∇βη it holds that ¬(w s)∨(w t)∈ H and hence by Lemma 2(b)

that ¬(w s)∈ H or (w t)∈ H. For w≡(λX. t ˙= s), it follows by ~

∇βη that ¬(t˙= s)∈ H or (t˙= s)∈ H.

Assume ¬(t˙= s)∈ H. Since by Lemma 3(a) it holds that s=t∈ H, it follows by ~

∇s

=that ¬(t˙= t)∈ H,

which contradicts (c) above. Hence, (t˙= s)∈ H.

(f) Let (s˙= t)∈ H and (t˙= u)∈ H. By (d) above it follows that (s˙= u)∈ H.

Lemma 4 (Suﬃcient conditions for saturatedness).Let H ∈ H. Then it holds that

(a) If > ∈ H, then His saturated.

(b) If ¬s∈ H for some closed term s, then His saturated.

(d) If s∧t∈ H for some closed terms s, t, then His saturated.

(e) If ΠτP∈ H for some closed term Pτ→o, then His saturated.

(f) If s˙= t∈ H for some closed terms s, t, then His saturated.

Proof. Let H ∈ Hbe an acceptable Hintikka set.

(a) Let > ∈ H, that is, =o=ooo=o∈ H. By ~

∇+

fand ~

∇βη it follows that (so=to)=(so=to)∈ H for every two

closed formulas s, t. For s≡t≡cfor some closed term c, it follows that (c=c)=(c=c)∈ H and thus, by

∇+

band ~

∇r

=, it holds that c=c∈H. By ~

∇+

bit follows that c∈ H or ¬c∈ H. Hence, His saturated.

(b) If ¬s∈ H for some closed term s, then s=⊥∈H. By ~

∇+

b, it follows that either {s, ⊥} ⊆ H or

{¬s, ¬⊥} ⊆ H. Since the former case is ruled out by Lemma 1(a), it follows that ¬⊥ ∈ H. By Lemma 1(f )

it follows that >∈Hand by (a) above it follows that His saturated.

∇βη we have ¬(¬s∧¬t)∈ H. An

application of (b) yields the desired result.

(d) If s∧t∈ H for some closed terms s, t, then by deﬁnition of ∧and ~

∇βη it holds (λg. g s t)=(λg. g > >)∈ H.

By ~

∇+

fand ~

∇βη it follows that s=>∈H(take λx. λy. x). By ~

∇+

b, it follows that either {s, >} ⊆ H or

{¬s, ¬>} ⊆ H. Since the latter case is ruled out by Lemma 1(b), it follows that > ∈ H. An application

of (a) above yields the desired result.

(e) If Πτs∈ H for some closed terms s, then by deﬁnition of Πτand ~

∇βη it holds that s=(λx. >)∈ H.

By ~

∇+

fand ~

∇βη it follows that (s w)=> ∈ H for every closed term w. By ~

∇+

b, it follows that either

{(s w),>} ⊆ H or {¬(s w),¬>} ⊆ H. Since the latter case is ruled out by Lemma 1(b), it follows that

> ∈ H. An application of (a) above yields the desired result.

(f) Let s˙= t∈ H. By deﬁnition of Πτand ~

∇βη it holds that ΠλP. (P s)⇒(P t)∈ H. An application of (e)

yields the desired result.

Corollary 1. Let H ∈ Hand let s6=t∈ H or ¬(s˙= t)∈ H for some closed terms s, t. Then, His saturated.

Proof. As (s6=t)≡¬(s=t), both cases are a special instance of Lemma 4(b).

Lemma 5 (Saturated sets properties).Let H ∈ Hand let Hbe saturated. Then it holds that

(a) If s=t∈ H, then s˙= t∈ H.

(b) If s=t∈ H then t=s∈ H.

(d) s˙= s∈ H for every closed term s.

Proof. Let H ∈ Hand let Hbe saturated.

(a) Let s=t∈ H and assume s˙= t /∈ H. Since His saturated we have ¬(s˙= t)∈ H. Then, by Lemma 3(b),

it follows that s6=t∈ H, and thus {s=t, s6=t}⊆H, which contradicts ~

∇c. Hence, s˙= t∈ H.

(b) Let s=t∈ H and assume t=s /∈ H. Since His saturated we have t6=s∈ H. Then, by ~

∇s

=, it follows that

t6=t∈ H which contradicts ~

∇r

=. Hence, t=s∈ H.

Since this contradicts ~

∇r

=it follows that s=s∈ H.

(d) Let sbe a closed term of some type and assume that s˙= s /∈ H. Since His saturated we have that

¬(s6=s)∈ H. Since this is impossible by Lemma 3(c) it follows that s˙= s∈ H.

Lemma 6 (Properties of negated equalities).Let H ∈ H. It holds that

(a) If s6=t∈ H, then t6=s∈ H.

(b) If ¬(s˙= t)∈ H, then ¬(t˙= s)∈ H

Proof. Let H ∈ H.

(a) Let s6=t∈ H and assume that t6=s /∈ H. By Corollary 1 it follows that His saturated and hence we have

that t=s∈ H. By saturation and Lemma 5(b) it follows that s=t∈ H, and thus {s6=t, s =t}⊆H,

which contradicts ~

∇c. Hence, t6=s∈ H.

(b) Let ¬(s˙= t)∈ H and assume ¬(t˙= s)/∈ H. By Corollary 1 it follows that His saturated and hence we

have that t˙= s∈ H. Then, by Lemma 3(d), it follows that s˙= t∈ H, and thus {¬(s˙= t), s ˙= t}⊆H,

which contradicts ~

∇c. Hence, ¬(t˙= s)∈ H.

that (s˙= t)∈ H. Then, by Lemma 3(a), it follows that s=t∈ H, and thus {s=t, s 6=t} ⊆ H, which

contradicts ~

∇c. Hence, ¬(s˙= t)∈ H.

Deﬁnition 2 (Leibniz-free).Let Sbe a set of formulas. Sis called Leibniz-free iﬀ s˙= t /∈Sfor any terms s, t.

Corollary 2 (Impredicativity Gap).Let H ∈ H.His saturated or Leibniz-free.

Proof. Assume that His not Leibniz-free. Then, there exists some formula s˙= t∈ H. An application of

Lemma 4(f) yields the desired result.

Summary of properties of equality and Leibniz-equality. The following table contains an overview of

the implied properties of =and ˙= , respectively. A property that holds unconditionally is marked with X, a

property that holds for saturated Hintikka sets is marked with sat..

Property ?≡=?≡˙=

s?s∈ H sat.sat.

¬(s?s)/∈ H X X

If s?t∈ H and t?u∈ H, then s?u∈ H X X

If s?t∈ H, then t?s∈ H sat.X

If u[s]p∈ H and s?t∈ H, then u[t]p∈ H X X

4 Reduction of H(Steen) to Hintβfb (Brown)

We reduce the notion of Hintikka sets of Steen to the notion of Hintikka sets by Brown. Conceptually, every

formula from a set H ∈ His ﬁrst translated to its red equivalent under a signature Σ⊆ {¬} ∪ {=τ|τ∈ T }

containing no further primitive logical connectives. Note that this involves mapping a primitive connective to a

primitive connective (i.e. =to =) as well as mapping a deﬁned connective to a primitive connective (i.e. ¬to ¬).

In a second step, the red formulas are translated into their meta-counterparts, i.e. into external propositions.

Formally, we deﬁne the mapping as follows: Let Σbe given as introduced above. We deﬁne H.:=

{(s[¬\¬])[=\=]|s∈H}, where s[l\r]denotes that term in Λc(Σ)that is obtained by replacing all occurrences

of lin sby r. Obviously, if s∈Λc

o(Σ)then (s[¬\¬])[=\=]∈Λc

o(Σ), and thus H.⊆Λc

o(Σ).

As the Hintikka properties of Brown are deﬁned in terms of external propositions, we proceed by constructing

a set H]

.⊆prop(Σ)from H.by enriching H.with its external counterparts: To that end, for any term

so∈Λo(Σ), we denote by s]∈prop(Σ)the term that is constructed by replacing the primitive connective at

head position by its external equivalent2, i.e. we have that (¬s)]is equal to ¬

·sand that (s=ot)]is equal to

·ot. If there is no connective at head position, the term is left unchanged, i.e., (p sn)]equals p snand (X sn)]

equals X snwhenever p∈Σis a parameter and Xis a variable.

Then, H]

.is deﬁned inductively as follows. For H ∈ H, let H]

.be the smallest set of external propositions

over Σsuch that

2This corresponds to the original deﬁnition of s]by Brown [6, Def. 2.1.38].

(1) if so∈ H, then s∈ H]

., and

(2) if so∈ H, then s]∈ H]

., and

(3) if ¬

·s∈ H]

., then ¬

·s]∈ H]

Intuitively, this translation adopts the translated object-level terms of H(clause (1)), and additionally aug-

ments the set corresponding meta-level terms by replacing connectives at head positions by its meta-connective

equivalent (clause (2)). Also, connectives directly under a (meta-)negation are considered (clause (3)).3Deep

translations are then implicitly provided by the Hintikka closure properties.

We provide simple examples of unsaturated Hintikka sets H,H.and H]

•Let H:= {s|s≡βη tfor t∈ {a=ιb, a =ιa, b =ιb, b =ιa}}. Then H.={s|s≡βη tfor t∈ {a=ιb, a =ι

a, b =ιb, b =ιa}} and H]

.=H.∪ {s|s≡βη tfor t∈ {a=

·ιb, a =

·ιa, b =

·ιb, b =

·ιa}}.

•Let H:= {s|s≡βη tfor t∈ {a=ιp(¬), a =ιa, p(¬)=ιp(¬), p(¬)=ιa}}. Then H.={s|s≡β η tfor t∈

{a=ιp(¬), a =ιa, p(¬)=ιp(¬), p(¬)=ιa}} and H]

.=H.∪ {s|s≡βη tfor t∈ {a=

·ιp(¬), a =

·ιa, p(¬)=

·ι

p(¬), p(¬)=

·ιa}}.

We now show that if H ∈ Hthen H]

.∈Hintβfb (i.e., H]

.fulﬁls all ~

∇from §2).

Theorem 1 (Reduction of Hto Hintβfb).If H ∈ H, then there exists an extensional Hintikka set H0∈Hintβfb

such that H]

.≡ H0.

Proof. Let H ∈ Hbe a Hintikka set according §3, i.e., fulﬁlling all ~

∇properties.

We verify every ~

∇-property for H]

∇c: Assume that both s∈ H]

.and ¬

·s∈ H]

.. Then, by deﬁnition, s∈Hand ¬s∈H. As this contradicts ~

∇c,

it follows that s/∈ H]

.or ¬

·s/∈ H]

∇βη : Let s∈ H]

.. Then, by deﬁnition, s∈ H. Since s≡βη s↓it follows by ~

∇βη that s↓∈ H. By deﬁnition, we

then have s↓∈ H]

∇⊥:¬

·>

·/∈ H]

.is vacuously true, as >