Two Notions of Semantics of the Simple Theory of Types

U.U.D.M. Project Report 2014:25

Examensarbete i matematik, 15 hp

Handledare och examinator: Inger Sigstam Juni 2014

Two Notions of Semantics of the Simple Theory of Types

Mattias Granberg Olsson

En popul¨ arvetenskaplig sammanfattning

Enkel typteori ¨ar ett formellt system f¨or h¨ogre ordningens logik. Den version vi h¨ari be- traktar ¨ar den som studerades av Church och Henkin i [3] respektive [4]. I [7] studerade vi uppbyggnaden av formler i spr˚aket och det formella h¨arledningssystemet, kortfattat uttryckt syntaxen f¨or enkel typteori. M˚alet f¨or denna magisteruppsats ¨ar att f¨ortydliga och mer detaljerat redog¨ora f¨or resultatet fr˚an Henkins artikel [4] som behandlar tolk- ningen av formler, det vill s¨aga systemets semantik. Formler tolkas som objekt i modeller, och s¨arskild vikt l¨aggs vid de formler som d¨ari tolkas som n˚agot av sanningsv¨ardena 0 (falskt) och 1 (sant), allts˚a p˚ast˚aenden. Tv˚a ¨onskv¨arda egenskaper ¨ar d˚a att allt som kan bevisas ¨ar sant i alla modeller (sundhet med avseende p˚a semantiken) samt att allt som

¨ar sant i alla modeller ocks˚a kan bevisas (fullst¨andighet med avseende p˚a semantiken).

Vi definierar tv˚a varianter av semantik till enkel typteori, standardmodeller och generella modeller, och visar att b˚ada ¨ar sunda, men att endast den senare ¨ar fullst¨andig.

Alla formler i enkel typteori ¨ar tilldelade en typ (d¨arav namnet), som avg¨or hur de f˚ar sammanfogas med andra formler. P˚a samma s¨att ¨ar modellerna (av b˚ada slagen) indelade i typdom¨aner, och varje formel tolkas som ett objekt i typdom¨anen motsvarande formelns typ. Typerna ¨ar konstruerade p˚a s˚a s¨att att det finns en typ i f¨or individer (de underliggande, primitiva objekten f¨or v˚art studium, e.g. naturliga tal), en typ o f¨or p˚ast˚aenden (sanningsv¨arden), samt f¨or alla par α, β av typer en typ hαβi (eller αβ kort och gott) f¨or funktioner fr˚an typdom¨an β till typdom¨an α. Formler av typ oα kommer d¨arvidlag att tolkas som (un¨ara) predikat p˚a (dvs. p˚ast˚aenden om) objekt av typ α, och kan s˚aledes inte appliceras p˚a objekt av n˚agon annan typ (p˚ast˚aenden om individer kan exempelvis inte appliceras p˚a andra p˚ast˚aenden). I standardmodeller utg¨or objekten av typ αβ alla funktioner mellan β och α, medan generella modeller till˚ater vilken m¨angd som helst av s˚adana funktioner, s˚a l¨ange den ¨overgripande konstruktionen blir meningsfull och inneh˚aller likhetspredikatet.

Formlernas uppbyggnad ¨ar enkel och elegant, inga termer finns som skymmer sikten:

alla variabler x och konstanter c ¨ar formler; sammans¨attningen [ϕψ] av en formel ϕ av typ αβ med en formel ψ av typ β ¨ar en formel av typ α, och tolkas som det v¨arde

“funktionen” ϕ antar f¨or input ψ; och λ-abstraktionerna [λxϕ] av en formel ϕ ¨ar formler av typ αβ, d¨ar ϕ har typ α och x har typ β. Dessa senare tolkas, i b˚ade standard- och generella modeller, som den funktion som till ett objekt d av typ β tilldelar det objekt av typ α som erh˚alls om man tolkar ϕ med x utbytt mot d.

Vi kan exemplifiera skillnaden mellan standard- och generella modeller enligt f¨oljande:

I en standardmodell kommer objekten av typ oi att utg¨ora alla delm¨angder av individ- dom¨anen, medan s˚a icke n¨odv¨andigtvis ¨ar fallet i en generell modell. Detta f˚ar till f¨oljd att en axiomatisering av de naturliga talen entydigt definierar dessa i standarmodeller.

En f¨oljd av fullst¨andighet med avseende p˚a en semantik (givet vissa andra antaganden som h¨ar ¨ar uppfyllda) ¨ar dock att detta ¨ar om¨ojligt, d˚a godtyckligt stora modeller av dessa kan skapas i en fullst¨andig semantik. Allts˚a ¨ar standardsemantiken ofullst¨andig.

Bevissystemet f¨or enkel typteori har blott ett f˚atal regler: vi f˚ar byta namn p˚a bundna variabler (α-konvertering); byta ett utryck p˚a formen [[λxϕ]ψ] mot det uttryck som erh˚alls om alla x i ϕ byts mot ψ, och vice versa (β-konvertering); allkvantifiera en fri

variabel i en formel (generalisering) eller ers¨atta densamma med vilken formel det vara m˚a (substitution); samt fr˚an en implikation (e.g. ϕ→ ψ) och dess antecedent (ϕ) sluta oss till dess succedent (ψ) (modus ponens). D¨arut¨over antas vissa formler som axiom (dessa ¨ar ur semantisk synvinkel rimliga, d˚a de kommer att vara sanna i alla modeller).

Dessutom f˚ar vi naturligtvis anv¨anda de utsagor som antagits som premisser f¨or beviset.

Eftersom alla standardmodeller ocks˚a ¨ar generella modeller r¨acker det att visa att en bevisbar formel ¨ar sann i alla generella modeller f¨or att visa att b˚ada typerna av semantik

¨

ar sunda. Att alla axiom ¨ar sanna ¨ar en enkel sak att verifiera. Vidare tolkas formler likadant oavsett namn p˚a de bundna variablerna, s˚a α-konvertering bevarar sanning.

Formler med (minst) en fri variabel anses vara sanna enbart om de ¨ar sanna f¨or alla tolkningar av den fria variabeln i fr˚aga, vilket ¨ar detsamma som att dess alltillslutning

¨ar sann; generalisering bevarar sanning. P˚a liknande s¨att f¨oljer att substitution bevarar sanning. Det ¨ar dessutom relativt enkelt att visa att en formell implikation tolkas som en (klassisk) implikation, varf¨or ¨aven modus ponens bevarar sanning. Slutligen tolkas β-konvertering i princip som funktionsapplikation, vilket bevarar sanning.

Fullst¨andighet bevisas till sist genom att en konsistent (mots¨agelsefri) m¨angd utsagor Γ utvidgas till en maximalkonsistent dito ∆ (maximal m.a.p. ⊆). Tv˚a slutna formler ϕ och ψidentifieras om likheten ϕ ≡ ψ ¨ar bevisbar fr˚an ∆, och typdom¨aner konstrueras utifr˚an dessa f¨or att precis motsvara tolkningen av sammans¨attning som funktionsapplikation, vilket ger en modell f¨or ∆. Allts˚a har varje konsistent m¨angd utsagor en modell, s˚a om en utsaga ϕ ¨ar sann i alla modeller till Γ , och Γ ∪{¬ϕ} d¨armed saknar modell, m˚aste s˚aledes den senare vara inkonsistent. Detta ¨ar m¨ojligt enbart om ϕ kan bevisas fr˚an Γ .

Two Notions of Semantics of the Simple Theory of Types

Mattias Granberg Olsson 2014-06-25

The simple theory of types, as formulated by Church, is considered from a semantic perspective. Two kinds of models are defined, standard and gen- eral, and the deductive system is proven sound with respect to both notions of semantics thus conceived. Proofs are given for the Model Existence The- orem and Completeness Theorem of general models, while the semantics of standard models is shown to be incomplete.

Contents

Introduction 1

Some Notational Remarks . . . 1

Acknowledgements . . . 2

1. Preliminaries: Syntax 3 Formulae and Variables . . . 3

Formal Deduction . . . 5

2. Models 7 Frames. . . 7

General and Standard Models . . . 12

Truth . . . 16

3. Soundness 31 4. (In)Completeness 40 The Notion of Consistency. . . 40

Formulae for Variables . . . 47

The Model Existence Theorem . . . 54

Completeness and incompleteness . . . 64

A. Formal theorems 69

Introduction

In a desire to better understand the foundations of mathematics, I, through the supervi- sor of my bachelor’s thesis, came to consider the simple theory of types which, it could be argued, can fill this role. Said thesis ([7]), therefore, concerned the simple theory of types, principally as formulated by Church ([3]) but with a few alterations due to Henkin ([4]), though merely as a formal theory. However, due to time constraints, only the syntactic part of the theory was presented, though in great detail. Accordingly, this one-year master’s thesis concerns the semantics of this theory, the results of which (most notably the Completeness (4.28) and Incompleteness (4.30) theorems) are primarily due to Henkin ([4]), though we will include a technical correction by Andrews ([1]).

The reader is encouraged to have access to a copy of [7], since a great many of the results and definitions therein are used, implicitly and explicitly, throughout this thesis.

That being said, the most fundamental definitions and results thereof will be given in section1. The other sections, meanwhile, contain the lion’s share of the subject matter;

in section2the basic definitions concerning models and interpretations, as well as some initial consequences thereof, are given, while section3contains the soundness theorem(s).

Section 4, finally, treats the completeness and incompleteness results alluded to above.

The thesis closes with an appendix containing a number of formal derivations, which would have made the, already slightly opaque, main text nearly unreadable had they been included.

For convenience, we will work in a standard mathematical setting, that is our meta theory will be ZFC. It would be of some interest to investigate whether the full strength of said theory is needed, since it seems plausible that we cannot do without neither choice nor replacement. Such an investigation, however, falls way outside the scope of this thesis.

Some Notational Remarks

For any n, m ∈ N ∪ {ω} we will use the following notations:

[n, m] ={k ∈ N ∪ {ω} | n ≤ k ≤ m}

[n, m[ ={k ∈ N ∪ {ω} | n ≤ k < m}

]n, m[ ={k ∈ N ∪ {ω} | n < k < m}

]n, m] ={k ∈ N ∪ {ω} | n < k ≤ m}.

We will also use the abbreviations¬¬¬, ∧∧∧, ∨∨∨, ∀∀∀, ∃∃∃, ⇒, ⇐ and ⇔ in the meta language, corresponding to the intended meaning of the formal symbols ¬, ∧, ∨, ∀, ∃, →, ← and ↔ in the obvious and usual fashion. Furthermore we will use many of the usual conventions, which most readers will probably be familiar with, as well as the extra conventions that strings will be denoted by a, b, c, d, e, f, type symbols by lowercase Greek letters at the beginning of the alphabet, well formed formulae by lowercase Greek letters and assignments and sets of well formed formulae by uppercase Greek letters.

Arbitrary variable symbols will be denoted by lowercase boldface letters towards the

end of the alphabet, e.g. x, y, z, while x, y, z are used to denote specific such (i.e. there is (for every type) a specific variable by the name x). Furthermore ⊂ will mean strict and ⊆ non-strict set containment. A statement like a, b ∈ A (such as the one in the beginning of this section) is to be interpreted as a ∈ A and b ∈ A. Concerning sequences and indexed families (finite or not), we will use the convention (as opposed to strings) of treating them as sets, that is we will write a ∈{ak}k∈I ⊆ A when we wish to convey that a = a_k for some k ∈ I and a_k ∈ A for all k ∈ I. However, should the a_k come from different sets Ak, we will write {ak}k∈I ∈ Q

k∈IA_k as usual. For finite sequences (with a meta theoretically well-defined arity) we will not distinguish between sequence and tuple notation, e.g. {k}⁴_k=1= (1, 2, 3, 4).

Some comments as to the notation concerning functions will also be in order. If A and B are sets, we will write f : A −→ B to mean that f is a function from A to B; if in addition f happens to be injective we will denote this by f : A  B. If f is only a partial function from A to B (that is, a function from some D ⊆ A to B) we will write f : A B instead. Finally, if f : A −→ A is a function whose image is contained within its domain we will denote the iterations of f by

fⁿ⁺¹ = f◦ fⁿ for all n ∈ N.

Finally we will, for the benefit of readers acquainted with first order logic, use a terminology akin to the standard one in that area. Moreover, for convenience we will often refer to the simple theory of types as formulated herein (and in [3], [4] and [7]) by simply “type theory”, though this is usually taken to mean a much wider class of related theories.

Acknowledgements

I would like to express my thanks to my supervisor Inger Sigstam, who accepted this role despite the numerous delays associated with my bachelor’s thesis.

1. Preliminaries: Syntax

For convenience we will here briefly recall the basics of the syntax of the simple theory of types, in the form that we shall use it. A (much) more thorough exposition of this subject is given in [7], which we shall refer to occasionally (some of the below definitions are in fact literally pasted from that work).

Formulae and Variables

The type symbols (or types) T are given by:

• i, o ∈ T ,

• ∀∀∀α, β ∈ T : (hαβi ∈ T ).

Here i and o are the types of individuals and propositions, respectively, whereas αβ (short for hαβi) is the type of functions from objects of type β to those of type α. We will also have use for the shorthand notation α⁰=hααihααi, where α ∈ T .

The primitive symbols S of the language of type theory are:

• the improper symbols K ={λ, [, ]};

• the proper symbols B, consisting of – the variables X ,

– the logical constants L ={Noo, A_hooio} ∪ {Πohoαi| α ∈ T } ∪ {ιαhoαi| α ∈ T }, – the (non-logical) constant symbols C, an arbitrary set of symbols.

Here N is the formal negation, A the disjunction, Π_ohoα the universal quantifier and ι_αhoαi a choice (or description) operator, for the type α. As the subscripts of these might suggest, every proper symbol is assigned a type. This partitions the sets X and C into countable sets X_α and C_α, where α ∈ T . We postulate that m_α, n_α, p_α, q_α, x_α, y_α, z_α∈ X_α denote distinct variables, for every α ∈ T .

The well-formed formulae W are constructed inductively from the primitive symbols:

• Proper symbols are well-formed formulae of the same type as the symbol.

• If ϕ ∈ W has type α and x ∈ X has type β, then [λxϕ] ∈ W and has type αβ.

• If ϕ ∈ W has type αβ and ψ ∈ W has type β, then [ϕψ] ∈ W and has type α.

In particular every well-formed formula has a type, given by the function T : W −→ T . That being said we will, in order to facilitate reading, often indicate the type of a formula by an index, as in N_oo. However, if the type is irrelevant, we will often simply omit it.

We denote the set of all well formed formulae of type α by W_α.

The following abbreviations, where ϕ, ψ ∈ W and x ∈ X , will ease the reading and interpretation of formulae:

• (¬ϕo) = [N_ooϕ_o].

• (ϕ_o∨ ψo) = [[A_hooioϕ_o]ψ_o].

• (ϕ_o∧ ψo) = (¬((¬ϕo)∨ (¬ψo))).

• (ϕ_o→ ψo) = ((¬ϕo)∨ ψo).

• (ϕ_o↔ ψo) = ((ϕ_o → ψo)∧ (ψo→ ϕo)).

• (∀x_αϕ_o) = [Π_ohoαi[λx_αϕ_o]].

• (∃x_αϕ_o) = (¬(∀xα(¬ϕo))).

• ( ι x_αϕ_o) = [ι_αhoαi[λx_αϕ_o]].

• Q_hoαiα = [λx_α[λy_α(∀f_oα([f_oαx_α]→ [foαy_α]))]].

• (ϕ_α≡ ψ_α) = [[Q_hoαiαϕ_α]ψ_α].

• (ϕ_α6≡ ψ_α) = (¬(ϕα≡ ψ_α)).

• ⊥ = (∀p_op_o).

• > = (¬⊥).

• Id_αα= [λx_αx_α].

• π_hαβiα= [λx_α[λy_βx_α]].

• 0_α⁰ = [λf_ααIdαα].

• 1_α⁰ = [λf_αα[λx_α[f_ααx_α]]].

• 2_α⁰ = [λf_αα[λx_α[f_αα[f_ααx_α]]]].

• S_α⁰_α⁰ = [λn_α⁰[λf_αα[λx_α[f_αα[[n_α⁰f_αα]x_α]]]]].

• N_oα⁰ = [λn_α⁰(∀f_oα⁰([f_oα⁰0_α⁰]→ ((∀xα⁰([f_oα⁰x_α⁰]→ [foα⁰[S_α⁰_α⁰x_α⁰]]))→ [foα⁰n_α⁰])))].

These will be subject to the aforementioned convention of omitting types, when possible.

We define the sets VF(ϕ) and VB(ϕ) of free and bound variables of ϕ, respectively, by induction on ϕ as follows

• For every s ∈ B, VF(s) =

{s} if s ∈ X

∅ otherwise and VB(s) = ∅.

• For every ψ ∈ W and x ∈ X

VF([λxv]) = VF(v)\ {x}

and

VB([λxv]) = VB(v)∪{x}.

• For all α, β ∈ T and ϕ_αβ, ψ_β∈ W

VF([vw]) = VF(v)∪ VF(w) and

VB([vw]) = VB(v)∪ VB(w).

Thus, in short, variables are bound by λ. The operators SF(x)(ϑ) and SB(x)(y) of substitution of free occurrences of x ∈ X by ϑ ∈ W and bound occurrences of x ∈ X by y ∈ X , respectively, are similarly defined, as is the operator S(x)(ϑ), substitution of all occurrences of x ∈ X by ϑ ∈ W (their composition). Worthy of note is also the notation W ={ϕ ∈ W | VF(ϕ) = ∅} for the set of closed well-formed formulae, which we will partition by typing using indices, as before.

Formal Deduction

The deductive system of type theory is made up by axioms and rules of inference.

1.1 Definition (Rules of inference). The rules of inference are the following:

α-conversion Let α, β ∈ T and x, y ∈ X_α. For any well-formed formulae ϕ_β, ψ_o such that aϕb = ψ for some a, b ∈ S^∗ where x is not free in ϕ and y does not occur in ϕ, we may from ψ infer a S(x)(y)(ϕ)b.

β-contraction Let α, β ∈ T . For any x ∈ X_α, any well-formed formulae ϕ_β, ψ_α and η_o such that neither x nor any free variable of ψ is a bound variable of ϕ, and a[[λxϕ]ψ]b = ηfor some a, b ∈ S^∗, we may from η infer a S(x)(ψ)(ϕ)b.

β-expansion From any well-formed formula ϕo we may infer ψo, where ψ is any well- formed formula from which ϕ could be inferred by β-contraction.

Substitution Let α ∈ T . For any well-formed formulae ϕ_oα, ψ_α and any variable x_α such that x is not free in ϕ, we may from [ϕx] infer [ϕψ]

Modus ponens For any well-formed formulae ϕ_o, ψ_o ∈ W we may from (ϕ → ψ) and ϕ infer ψ.

Generalisation Let α ∈ T . For any well-formed formula ϕ_oα and any variable x_α such that x is not free in ϕ, we may from [ϕx] infer [Π_ohoαiϕ_oα].

1.2 Definition (Axioms). The following are the axioms of simple type theory:

1. ((p_o∨ po)→ po).

2. (po→ (po∨ qo)).

3. ((po∨ qo)→ (qo∨ po)).

4. ((p_o→ qo)→ ((ro∨ po)→ (ro∨ qo))).

5. For all α ∈ T : ([Π_ohoαif_oα]→ [foαx_α]).

6. For all α ∈ T : ((∀xα(p_o∨ [foαx_α]))→ (po∨ [Πohoαif_oα])).

7. (∃xi(∃y_i(x_i 6≡ y_i))).

8. ([N_oi⁰x_i⁰]→ ([Noi⁰y_i⁰]→ (([Si⁰i⁰x_i⁰]≡ [S_i⁰_i⁰y_i⁰])→ (xi⁰ ≡ y_i⁰)))).

9. For all α ∈ T : ([f_oαx_α]→ ((∀yα([f_oαy_α]→ (xα≡ y_α)))→ [foα[ι_αhoαif_oα]])).

10. For all α, β ∈ T : ((∀x_β([f_αβx_β]≡ [g_αβx_β]))→ (fαβ≡ g_αβ)).

10^o. ((xo↔ yo)→ (xo≡ y_o)).

11. For all α ∈ T : ([foαx_α]→ [foα[ι_αhoαif_oα]]).

We will not consider all of the above axioms as integral to our theory, but will mainly be concerned with 1–6 and 10–11. The rest are presented mainly for completeness.

1.3 Definition (Formal Proof). Let Γ ⊆ W_o, ϕ ∈ W_o and n ∈ N. A formal proof of length n + 1 of ϕ on the assumptions Γ is a string P ∈ (Wo)ⁿ⁺¹ such that P(n) = ϕ, where for every 0 ≤ k ≤ n, P(k) is either a formal axiom, a formula of Γ , or can be inferred from{P(l) | 0 ≤ 1 < k}. We write

Γ ` ϕ

when there is a formal proof (of any length) of ϕ on the assumptions Γ . If ∅ ` ϕ we call ϕ a formal theorem and write

` ϕ

1.4 Theorem (The Deduction Theorem). In a simple theory of types with (at least) axioms 1–6, we have for every Γ ⊆ W_o, ϕ ∈ W_o and ψ ∈ W_o that

Γ ∪{ϕ} ` ψ ⇒ Γ ` (ϕ → ψ).

2. Models

As explained earlier, we will henceforth consider the simple theory of types with axioms 1–6, 10, 10^oand 11. The following definitions and results are due to Henkin [4]. However, since his exposition is quite brief, while brevity is not the aim of this thesis, we will introduce some minor auxiliary concepts. We begin by defining the structures forming the backbone of our models, and proceed by defining the two different classes of models as certain subclasses thereof. Finally we fix the notion of truth (in models), and show the validity of the axioms.

Frames

In order to systematically interpret well-formed formulae, we would need a set of each type in which to do so for the formulae of that type. We also need to have interpretations of the constants (logical or not) under consideration. The following definition makes these ideas precise.

2.1 Definition. By a frame we will mean a sequence D ={Dα}α∈T of sets. Given α ∈ T we will call Dα the domain (of D) of type α. A structured frame is a tuple M = (D, C) consisting of a frame D ={Dα}α∈T and a sequence C ={kc}c∈L∪C such that

• D_o={0, 1}

• For all α, β ∈ T , D_αβ ⊆ D^D_α^β

• For every α ∈ T and every c_α∈ L ∪ C, k_c∈ D_α

• k_N∈ D_oo is such that

k_N(n) = 1 − n for all n ∈{0, 1}.

• k_A: D_hooio is such that

k_A(n)(m) = 1 − (1 − n)· (1 − m) for all n, m ∈{0, 1}.¹

• For all α ∈ T , k_Πohoαi∈ D_ohoαi is such that

k_Πohoαi(f) =



1 if f(d) = 1 for all d ∈ Dα

0 otherwise for all f ∈ D_oα.

1Thus Doocontains at least the identity and constantly affirmative (if 1 and 0 are regarded as true and false, respectively, which is intended) function.

• For all α ∈ T , k_ιαhoαi ∈ D_αhoαi is such that f(k_ιαhoαi(f)) = 1

for every f ∈ D_oαfor which there is a d ∈ D_α such that f(d) = 1.²

We will call a frame or structured frame degenerate if D_α = ∅ for some α ∈ T .

Considered as structures where we wish to interpret the well-formed formulae, however, non-degenerate structured frames does not really suffice, since we are not guaranteed that the interpretations of all well-formed formulae are contained in the respective type domain. This issue will be treated in the next subsection. Later, we will show that it is in fact enough to require that all closed well-formed formulae has an interpretation (4.21).

In [7] a general version of the following theorem was exhibited.

2.2 Theorem (Type-recursion). For every nonempty set M, every a, b ∈ M and every function h : M × M −→ M, there is a unique function f : T −→ M satisfying

• f(o) = a,

• f(i) = b,

• f(αβ) = h((f(α), f(β))), for all α, β ∈ T .

Indeed, we have an even stronger recursion property, which we will need to construct frames.

2.3 Theorem (Grounded Type-recursion). Let H be a (class) function and a, b be arbitrary (sets). There is a unique set M and function f : T −→ M such that f is surjective and

• f(o) = a,

• f(i) = b,

• f(αβ) = H((f(α), f(β))), for all α, β ∈ T .

Proof. Define L : T −→ P(T ) by recursion (2.2) as follows:

2Henkin defines the interpretations of the symbols in L directly when defining valuations (see below), where he proclaims V(Φ)(ιαhoαi)to be “some fixed function whose value for any argument f of Doα

is one of the elements of Dαmapped into T by f” (where T denotes the truth-value “Truth”, which is 1in our definition). Depending on how you interpret “fixed” (i.e. whether the interpretation is fixed with respect to Φ or not), the valuations described may or may not be uniquely defined. Therefore, we have chosen the corresponding definition from Andrews [2] for his essentially equivalent system, which corresponds to interpreting “fixed” as “uniquely determined by the structure”.

• L(o) ={o},

• L(i) ={i},

• L(αβ) = L(α) ∪ L(β), for all α, β ∈ T .

Sublemma 2.3.1. L(α) is finite for every α ∈ T . Proof. By induction.

• L(o) and L(i) are singletons, and thus finite.

• Assume γ, δ ∈ T are such that L(γ) and L(δ) are finite. Then L(γδ) = L(γ) ∪ L(δ) is finite as well.

Hence L(α) is finite for every α ∈ T .

Let R ={(α, β) ∈ T²| α ∈ L(β)}, so that L(β) = {α ∈ T | αRβ} for every β ∈ T . Sublemma 2.3.2. R is wellfounded, i.e. every nonempty A ⊆ T has an R-minimal element, that is an element α ∈ A such that for every β ∈ A for which βRα, β = α.

Proof. Take γ ∈ A. Then L(γ) ∩ A 6= ∅ but finite, whence there is an R-minimal α∈ L(γ) ∩ A. Now if β ∈ A is such that βRα, then β ∈ L(γ) ∩ A and thus β = α. Hence αis R-minimal in A.

Sublemma 2.3.3. There are functions p₁ : T \ {o, i} −→ T and p2 : T \ {o, i} −→ T such that

γ =hp₁(γ)p₂(γ)i for all γ ∈ T \ {o, i}.

Proof. Let π1, π₂:T²−→ T be the standard projections (i.e. q = (π1(q), π₂(q))for all q∈ T²). Define p : T −→ T² by recursion as

• p(o) = (o, o).

• p(i) = (i, i). (These cases are highly irrelevant.)

• For all α, β ∈ T , p(αβ) = (α, β).

Now p₁ = π₁◦ p  (T \ {o, i}) and p2= π₂◦ p  (T \ {o, i}) has the desired properties.

Let

F((A, B)) =

















H(B(p₁(A)), B(p₂(A))) if A ∈ T \ {o, i} and

Bis a function with L(A) ⊆ dom(B)

a if A = o

b if A = i

∅ otherwise

so that F is a (class) function. By well-founded recursion (see for example [5]), there is a unique (class) function G on T such that

G(α) = F((α, G L(α)))

for all α ∈ T . By the axiom of replacement M = G[T ] is a set. Let f : T −→ M be defined by f(α) = G(α) for all α ∈ T . Then

f(o) = F((o, f ∅)) = a f(i) = F((i, f ∅)) = b f(αβ) = F((αβ, f L(αβ)))

= H((f L(αβ)(p1(αβ)), f L(αβ)(p2(αβ))))

= H((f(α), f(β))) as desired.

Now if g were any other such function, then g(α) = F((α, g  L(α))), whence g(α) = G(α) = f(α), for all α ∈ T , by uniqueness of G. Thus g[T ] = f[T ] = M, whereby f and Mabove are unique.

Remark 1. Given a frame D = {Dα}α∈T, since the set D_α is uniquely determined for every α ∈ T , the axiom of replacement also gives that {Dα| α ∈ T } is a set, whereby S

α∈T D_α is a set as well.

Thus, given a structured frame we can consider functions into its “total universe”, so to speak. This is exactly what we need to define the interpretation, or evaluation, of formulae. Our first step along that road is the interpretation of variables, which we consider next.

2.4 Definition (Assignment). Let M = ({Dα}α∈T,{kc}c∈L∪C)be a structured frame. An assignment with respect to M is a function Φ : X −→ Sα∈T D_α such that Φ(x_α)∈ D_α for every α ∈ T . We denote by AMthe set of all assignments with respect to M.

Remark 2. Since X_α 6= ∅ for all α ∈ T , M is a non-degenerate structured frame if and only if AM 6= ∅. Moreover, if M = (D, C) and N = (E, B) are structured frames with the same underlying frame, then A_M= A_N.

2.5 Definition. Let M = ({Dα}α∈T,{kc}c∈L∪C) be a non-degenerate structured frame, Φ∈ A_M, α ∈ T , d ∈ Dα and x ∈ Xα. The assignment Φ^x_d ∈ A_M is defined via

Φ^x_d(y) =



d if y = x Φ(y) otherwise for all y ∈ X .

The following easy lemma will be of use later.

2.6 Lemma. For all non-degenerate structured frames M = ({Dα}α∈T,{kc}c∈L∪C), Φ ∈ A_M, α, β ∈ T , d ∈ Dα, e ∈ Dβ and distinct xα, y_β∈ X :

1. Φ^x_d^x_e = Φ^x_e. 2. Φ^x_d^y_e = Φ^y_e^x_d. Proof. 1. Clearly

Φ^x_d^x_e(z) =



e if z = x Φ^x_d(z) otherwise

=



e if z = x Φ(z) otherwise

= Φ^x_e(z) for all z ∈ X .

2. Similarly, since x 6= y,

Φ^x_d^y_e(z) =

e if z = y Φ^x_d(z) otherwise

=







e if z = y d if z = x Φ(z) otherwise

=



d if z = x Φ^y_e(z) otherwise

= Φ^y_e^x_d(z) for all z ∈ X .

Having thus specified how to assign a meaning to variables, we can extend this to all formulae inductively. Recall, however, that we are not guaranteed that a valuation as below exist even in every non-degenerate structured frame. In the next section we will therefore limit our attention to those frames where this is possible.

2.7 Definition (Valuation). Let M = ({Dα}α∈T,{kc}c∈L∪C) be a non-degenerate struc- tured frame. A valuation for M is a function V : AM −→ (Sα∈T D_α)^W which has the following properties:

• For every Φ ∈ A_M, α ∈ T and every ϕ ∈ Wα we have that V(Φ)(ϕ) ∈ Dα.

• For every Φ ∈ A_M and every x ∈ X , we have that V(Φ)(x) = Φ(x).

• For every Φ ∈ A_M and every c ∈ L ∪ C, we have that V(Φ)(c) = k_c.

• For every Φ ∈ A_M and all α, β ∈ T , x ∈ X_β and ϕ_α ∈ W, V(Φ)([λxϕ]) ∈ D_αβ is such that

V(Φ)([λxϕ])(d) = V(Φ^x_d)(ϕ) for all d ∈ D_β.

• For every Φ ∈ A_M and all α, β ∈ T and ϕαβ, ψ_β ∈ W, we have that V(Φ)([ϕ_αβψ_β]) = V(Φ)(ϕ_αβ)(V(Φ)(ψ_β)).

Given a valuation V for M, for each ϕ ∈ W and Φ ∈ A_Mwe define the evaluation of ϕ with respect to Φ as the object

V(Φ)(ϕ) ∈ D_{T (ϕ)}. General and Standard Models

With a method of evaluating formulae now at our disposal, we will define the more gen- eral class of structures (whence the name) which will constitute models for our system.

2.8 Definition (General model). A general model is a non-degenerate structured frame M = ({Dα}α∈T,{kc}c∈L∪C) with the following properties:

• For every α ∈ T there is a function q ∈ D_hoαiα such that

q(a)(b) =

1 if a = b 0 if a 6= b for all a, b ∈ Dα3.

• There is a valuation function for M.

2.9 Lemma. For every general model M there is a unique valuation function.

3The existence of such a function is necessary to ensure that the formal equality is interpreted as the meta theoretical equality, which Andrews [1] pointed out (and proved). Without this requirement we will have models in which extensionally equal functions are considered distinct.

Proof. Let M = ({Dα}α∈T,{kc}c∈L∪C)be a general model. We know there is at least one valuation function by definition, so it suffices to show that if V and V⁰ are valuation functions, then V = V⁰. This follows from the fact that given ϕ ∈ W, V(Φ)(ϕ) = V⁰(Φ)(ϕ)for all Φ ∈ A_M, which will be proved by induction on ϕ.

• Consider s ∈ B. Since B = L ∪ X ∪ C we see that V(Φ)(s) =



Φ(s) if s ∈ X k_s if s ∈ L ∪ C

= V⁰(Φ)(s) for all Φ ∈ AM.

• Now consider [λx_αψ_β], where α, β ∈ T , x ∈ Xα and ψ ∈ Wβ is such that V(Φ)(ψ) = V⁰(Φ)(ψ)

for all Φ ∈ A_M. Then, given such a Φ ∈ A_M, we see that V(Φ)([λxψ])(d) = V(Φ^x_d)(ψ)

= V⁰(Φ^x_d)(ψ)

= V⁰(Φ)([λxψ])(d)

for all d ∈ D_α. Thus V(Φ)([λxψ]) = V⁰(Φ)([λxψ]) for all Φ ∈ A_M.

• Finally consider [ψ_αβϑ_β]∈ W, where α, β ∈ T and ψ, ϑ ∈ W are such that V(Φ)(η) = V⁰(Φ)(η)

for all Φ ∈ A_M and both η ∈{ψ, ϑ}. Then

V(Φ)([ψϑ]) = V(Φ)(ψ)V(Φ)(ϑ)

= V⁰(Φ)(ψ)V⁰(Φ)(ϑ)

= V⁰(Φ)([ψϑ]) for all Φ ∈ AM.

Hence V(Φ)(ϕ) = V⁰(Φ)(ϕ)for all Φ ∈ A_M as desired.

Thus we will henceforth when speaking about “the valuation” in the context of a general model mean the unique valuation for that model. Unless stated otherwise, it will always be denoted by V.

Having defined a way of interpreting formulae, we can work our way towards the definition of truth in the next subsection. Along the way we will make a detour to single out the other class of models we will consider, namely standard models (2.12).

2.10 Lemma. Let M = ({Dα}α∈T,{kc}c∈L∪C) be a general model and ϕ ∈ W. If Φ, Ψ ∈ A_M are such that Φ  VF(ϕ) = Ψ  VF(ϕ), then V(Φ)(ϕ) = V(Ψ)(ϕ). In particular, the evaluation of a closed well-formed formula is independent of the assignment.

Proof. By induction on ϕ ∈ W.

• Let s ∈ B. If s ∈ X then VF(s) ={s}, wherefore

V(Φ)(s) = Φ(s) = Ψ(s) = V(Ψ)(s)

for all Φ, Ψ ∈ A_M such that Φ  VF(s) = Ψ  VF(s). Otherwise, s ∈ L ∪ C, whence VF(s) = ∅ and

V(Φ)(s) = k_s = V(Ψ)(s) for all Φ, Ψ ∈ A_M.

• Suppose α, β ∈ T , ψ ∈ W_α and x ∈ X_β are such that V(Φ)(ψ) = V(Ψ)(ψ)

for all Φ, Ψ ∈ AM such that Φ  VF(ψ) = Ψ  VF(ψ). Assume Φ, Ψ ∈ AMare such that Φ  VF([λxψ]) = Ψ  VF([λxψ]). Thus

Φ(y) = Ψ(y) for all y ∈ VF(ψ)\ {x}, so that

Φ^x_d(y) =



d if y = x Φ(y) otherwise

=



d if y = x Ψ(y) otherwise

= Ψ^x_d(y) for all y ∈ VF(ψ) and all d ∈ D_β. Consequently

V(Φ)([λxψ])(d) = V(Φ^x_d)(ψ) = V(Ψ^x_d)(ψ) = V(Ψ)([λxψ])(d) for all d ∈ Dβ, whence V(Φ)([λxψ]) = V(Ψ)([λxψ]).

• Let α, β ∈ T and ψ_αβ, ϑ_β∈ W be such that V(Φ)(η) = V(Ψ)(η)

for all Φ, Ψ ∈ A_M with the property that Φ  VF(η) = Ψ  VF(η), for both η∈{ψ, ϑ}. Then

V(Φ)([ψϑ]) = V(Φ)(ψ)(V(Φ)(ϑ)) = V(Ψ)(ψ)(V(Ψ)(ϑ)) = V(Ψ)([ψϑ]) for all Φ, Ψ ∈ A_M such that Φ  VF([ψϑ]) = Ψ  VF([ψϑ]), since we know that VF([ψϑ]) = VF(ψ)∪ VF(ϑ).

Hence V(Φ)(ϕ) = V(Ψ)(ϕ) for all Φ, Ψ ∈ A_M such that Φ  VF(ϕ) = Ψ  VF(ϕ), as claimed.

This allows us to define the denotation of a closed formula as an object independent of the choice of assignment.

2.11 Definition (Denotation). Let M be a general model. For every ϕ ∈ W, the denotation of ϕ is the evaluation of ϕ with respect to some, and thus every, assignment Φ∈ A_M.

As pointed out earlier, general models will be our prime semantic objects. However, a certain subclass thereof seems to have particularly simple properties.

2.12 Definition (Standard model). A standard model is a non-degenerate structured frame M = ({Dα}α∈T,{kc}c∈L∪C) such that

D_αβ= D^D_α^β for all α, β ∈ T .

2.13 Lemma. Every standard model is a general model.

Proof. Let M = ({Dα}α∈T,{kc}c∈L∪C) be a standard model; then M is a non-degenerate structured frame. For every α ∈ T the function q : Dα−→ D^Do^α defined by

q(a)(b) =



1 if a = b 0 if a 6= b

for all a, b ∈ Dα is an element of (D^D_o^α)^Dα = D^D_oα^α = D_hoαiα. Furthermore, we define F : W −→ (Sα∈T D^A_α^M) (our valuation to be, with arguments in reversed order) as the unique such function which satisfies the following criteria:

• For all α ∈ T and every ϕ ∈ W_α, F(ϕ) : A_M−→ Dα.

• For every s ∈ B,

F(s)(Φ) =



Φ(s) if s ∈ X k_s if s ∈ L ∪ C for all Φ ∈ A_M.

• For all α, β ∈ T , ϕ_α ∈ W and x ∈ X_β

F([λxϕ])(Φ)(d) = F(ϕ)(Φ^x_d) for all Φ ∈ A_M and d ∈ D_β.

• For all α, β ∈ T and ϕ_αβ, ψ_β∈ W let

F([ϕψ])(Φ) = F(ϕ)(Φ)(F(ψ)(Φ)).

Such a function exists by Theorem 2.8 of [7]. Thus let V : AM −→ S_α∈T D_α
W

be defined via

V(Φ)(ϕ) = F(ϕ)(Φ) for all Φ ∈ AM and ϕ ∈ W, whence

V(Φ)(ϕ)∈ D_α

for all Φ ∈ AM, α ∈ T and ϕ ∈ Wα. Furthermore, for every Φ ∈ AM: V(Φ)(s) = F(s)(Φ) =



Φ(s) if s ∈ X k_s if s ∈ L ∪ C for all s_α∈ B;

V(Φ)([λxψ])(d) = F([λxψ])(Φ)(d) = F(ψ)(Φ^x_d) = V(Φ^x_d)(ψ) for all α, β ∈ T , ψ ∈ W_α, x ∈ X_β and d ∈ D_β; and

V(Φ)([ψϑ]) = F([ψϑ])(Φ) = F(ψ)(Φ)(F(ϑ)(Φ)) = V(Φ)(ψ)(V(Φ)(ϑ))

for all α, β ∈ T and ψ_αβ, ϑ_β∈ W. Hence V is a valuation for M, whereby M is a general model, as desired.

These two classes of models, general models and standard models, give rise to, as we shall see, two quite different notions of semantics; we will speak of general semantics and standard semantics, respectively. This difference is characterised by theorems 4.28 and 4.30, which also clarify why we do not restrict ourselves to the study of standard models, even though they seem to be, if anything, more well-behaved than the general. On the contrary, since everything true of all general models is true of all standard models in particular, we will henceforth mostly consider general models.

Truth

To simplify things later, and since it is worthy of note on its own merits, we now show that many of the aforementioned abbreviations behave largely as we would expect (and indeed intend) them to. The following lemma will thus be widely applicable, even though this will not always be made explicit.

2.14 Lemma. Let M = ({Dα}α∈T,{kc}c∈L∪C) be a general model. The valuation V for M has the following properties, for every Φ ∈ A_M, ϕ, ψ ∈ W, α ∈ T , x ∈ X_α and a∈ D_α:

1. V(Φ)((¬ϕo)) = 1 if and only if V(Φ)(ϕ_o) = 0.

2. V(Φ)((ϕo∨ ψo)) = 1 if and only if V(Φ)(ϕo) = 1or V(Φ)(ψo) = 1.

3. V(Φ)((ϕo∧ ψo)) = 1 if and only if V(Φ)(ϕo) = 1and V(Φ)(ψo) = 1.

4. V(Φ)((ϕ_o → ψo)) = 1 if and only if V(Φ)(ϕ_o) = 0 or V(Φ)(ψ_o) = 1 if and only if V(Φ)(ϕ_o) = 1⇒ V(Φ)(ψo) = 1.

5. V(Φ)((ϕ_o↔ ψo)) = 1 if and only if V(Φ)(ϕ_o) = V(Φ)(ψ_o).

6. V(Φ)((∀xαϕ_o)) = 1 if and only if V(Φ^x_d)(ϕ_o) = 1 for all d ∈ Dα. 7. V(Φ)((∃xαϕ_o)) = 1 if and only if V(Φ^x_d)(ϕ_o) = 1 for some d ∈ Dα.

8. If V(Φ^x_d)(ϕ_o) = 1for some d ∈ D_α and V(Φ)((ιx_αϕ_o)) = a, then V(Φ^x_a)(ϕ_o) = 1.

9. V(Φ)(ϕ ≡ ψ) = 1 if and only if V(Φ)(ϕ) = V(Φ)(ψ).

10. V(Φ)(ϕ 6≡ ψ) = 1 if and only if V(Φ)(ϕ) 6= V(Φ)(ψ).

11. V(Φ)(⊥) = 0.

12. V(Φ)(>) = 1.

13. V(Φ)(Id_αα) =id_Dα

14. V(Φ)(0_α⁰)(g) =id_Dα for all g ∈ D_αα.

15. V(Φ)(S_α⁰_α⁰)(F)(g) = g◦ F(g) for all F ∈ D_α⁰and g ∈ D_αα. 16. V(Φ)(N_oα⁰)(V(Φ)(0_α⁰)) = 1.

17. V(Φ)(N_oα⁰)(V(Φ)(S_α⁰_α⁰)(F)) = 1 for all F ∈ D_α⁰ for which V(Φ)(N_oα⁰)(F) = 1.

Proof. 1.

V(Φ)((¬ϕo)) = 1

⇔ V(Φ)([Nooϕ_o]) = 1

⇔ V(Φ)(Noo)(V(Φ)(ϕ_o)) = 1

⇔ 1 − V(Φ)(ϕo) = 1

⇔ V(Φ)(ϕo) = 0.

2.

V(Φ)((ϕ_o∨ ψo)) = 1

⇔ V(Φ)([[Ahooioϕ_o]ψ_o]) = 1

⇔ V(Φ)([Ahooioϕ_o])(V(Φ)(ψ_o)) = 1

⇔ V(Φ)(Ahooio)(V(Φ)(ϕ_o))(V(Φ)(ψ_o)) = 1

⇔ 1 − (1 − V(Φ)(ϕo))· (1 − V(Φ)(ψ_o)) = 1

⇔ (1 − V(Φ)(ϕo))· (1 − V(Φ)(ψ_o)) = 0.

Furthermore, (1 − V(Φ)(ϕ_o))· (1 − V(Φ)(ψ_o)) = 0if and only if 1 − V(Φ)(ϕ_o) = 0 or 1 − V(Φ)(ψo) = 0, if and only if V(Φ)(ϕo) = 1or V(Φ)(ψo) = 1.

3. Since (ϕo ∧ ψo) = (¬((¬ϕo)∨ (¬ψo))) we have that V(Φ)((ϕo ∧ ψo)) = 1 if and only if V(Φ)(((¬ϕo) ∨ (¬ψo))) = 0, if and only if V(Φ)((¬ϕo)) = 0 and V(Φ)((¬ψo)) = 0, if and only if V(Φ)(ϕo) = 1and V(Φ)(ψo) = 1, by above.

4. Since (ϕo→ ψo) = ((¬ϕo)∨ψo)we have that V(Φ)((ϕo→ ψo)) = 1if and only if V(Φ)((¬ϕo)) = 1or V(Φ)(ψ_o) = 1, if and only if V(Φ)(ϕ_o) = 0or V(Φ)(ψ_o) = 1, if and only if V(Φ)(ϕ_o) = 1⇒ V(Φ)(ψo) = 1.

5. Since (ϕo ↔ ψo) = ((ϕ_o→ ψo)∧ (ψo→ ϕo)), V(Φ)((ϕo ↔ ψo)) = 1 if and only if V(Φ)((ϕo→ ψo)) = 1 and V(Φ)((ψo→ ϕo)) = 1, if and only if

V(Φ)(ϕ_o) = 0 or V(Φ)(ψ_o) = 1 and

V(Φ)(ψ_o) = 0 or V(Φ)(ϕ_o) = 1, if and only if

V(Φ)(ϕ_o) = 0 and V(Φ)(ψ_o) = 0 or

V(Φ)(ψ_o) = 1 and V(Φ)(ϕ_o) = 1, if and only if V(Φ)(ϕo) = V(Φ)(ψ_o).

6. First

V(Φ)((∀x_αϕ_o)) = 1

⇔ V(Φ)([Πohoαi[λx_αϕ_o]]) = 1

⇔ V(Φ)(Πohoαi)(V(Φ)([λx_αϕ_o])) = 1

⇔ kΠ_ohoαi(V(Φ)([λx_αϕ_o])) = 1.

Furthermore, k_Πohoαi(V(Φ)([λx_αϕ_o])) = 1 if and only if V(Φ)([λxαϕ_o])(d) = 1for all d ∈ Dα, if and only if V(Φ^x_d^α)(ϕ_o) = 1 for all d ∈ Dα.

7. Since (∃xαϕ_o) = (¬(∀xα(¬ϕo))), we have that V(Φ)((∃xαϕ_o)) = 1 if and only if V(Φ)((∀x_α(¬ϕo))) = 0, if and only if V(Φ^x_d^α)((¬ϕo)) = 0for some d ∈ D_α, if and only if V(Φ^x_d^α)(ϕ_o) = 1for some d ∈ Dα.

8. Assume that V(Φ)(( ι

xαϕ_o)) = a and that d ∈ Dα is such that V(Φ^x_d^α)(ϕ_o) = 1.

Then

V(Φ)([λx_αϕ_o])(d) = V(Φ^x_d^α)(ϕ_o) = 1.

Thus

V(Φ^x_a^α)(ϕ_o) = V(Φ)([λx_αϕ_o])(a)

= V(Φ)([λx_αϕ_o])(V(Φ)(( ιx_αϕ_o)))

= V(Φ)([λx_αϕ_o])(V(Φ)([ι_αhoαi[λx_αϕ_o]]))

= V(Φ)([λx_αϕ_o])(V(Φ)(ι_αhoαi)(V(Φ)([λx_αϕ_o])))

= V(Φ)([λx_αϕ_o])(k_ιαhoαi(V(Φ)([λx_αϕ_o])))

= 1

as well, by the definition of kιαhoαi.

9. Let a, b ∈ Dα. If a = b let d ∈ Doα. Then d(a) = d(b), i.e. either d(a) = 0 or d(b) = 1, while V(Φ^x_a^y_b^f_d)([fx]) = d(a)and V(Φ^x_a^y_b^f_d)([fy]) = d(b). Thus

V(Φ^x_a^y_b^f_d)(([fx]→ [fy])) = 1.

Since d ∈ D_oα was arbitrary, we have that

V(Φ)(Q_hoαiα)(a)(b) = V(Φ)([λx[λy(∀f_oα([fx]→ [fy]))]])(a)(b)

= V(Φ^x_a^y_b)((∀f_oα([fx]→ [fy])))

= 1.

If on the other hand a 6= b, let q ∈ D_hoαiα be such that q(a)(b) = 0 but q(a)(a) = 1, as guaranteed in the definition of general models. Then V(Φ^x_a^y_b^f_q(a))([fx]) = q(a)(a) = 1and V(Φ^x_a^y_b^f_q(a))([fy]) = q(a)(b) = 0, so that

V(Φ^x_a^y_b^f_q(a))(([fx]→ [fy])) = 0.

Thus

V(Φ)(Q_hoαiα)(a)(b) = V(Φ)([λx[λy(∀f_oα([fx]→ [fy]))]])(a)(b)

= V(Φ^x_a^y_b)((∀x_α([fx]→ [fy])))

= 0.

Hence

V(Φ)(Q_hoαiα)(a)(b) =

1 if a = b 0 if a 6= b.

Since V(Φ)(ϕ ≡ ψ) = V(Φ)(Qhoαiα)(V(Φ)(ϕ))(V(Φ)(ψ)), the claim follows.

10. V(Φ)(ϕ 6≡ ψ) = 1⇔ V(Φ)(ϕ ≡ ψ) = 0 ⇔ V(Φ)(ϕ) 6= V(Φ)(ψ).

11. Since V(Φ^p₀^o)(p_o) = 0, we have

V(Φ)(⊥) = V(Φ)((∀p_op_o)) = 0 by 6.

12. V(Φ)(>) = V(Φ)((¬⊥)) = 1.

13. For every d ∈ Dα we have that

V(Φ)(Id_αα)(d) = V(Φ)([λx_αx_α])(d) = V(Φ^x_d^α)(x_α) = Φ^x_d^α(x_α) = d.

Hence V(Φ)(Id_αα) =id_Dα. 14. For every g ∈ Dαα

V(Φ)(0_α⁰)(g) = V(Φ)([λf_ααIdαα])(g) = V(Φ^f_g^αα)(Idαα) =id_Dα by above.

15. Let F ∈ Dα⁰ and g ∈ Dαα. Then

V(Φ)(S_α⁰_α⁰)(F)(g)(d) = V(Φ)([λn_α⁰[λf_αα[λx_α[f_αα[[n_α⁰f_αα]x_α]]]]])(F)(g)(d)

= V(Φⁿ_F^{α 0})([λf_αα[λx_α[f_αα[[n_α⁰f_αα]x_α]]]])(g)(d)

= V(Φⁿ_F^{α 0f}_g^αα)([λx_α[f_αα[[n_α⁰f_αα]x_α]]])(d)

= V(Φⁿ_F^{α 0f}_g^ααx_d^α)([f_αα[[n_α⁰f_αα]x_α]])

= V(Φⁿ_F^{α 0f}_g^ααx_d^α)(f_αα)(V(Φⁿ_F^{α 0f}_g^ααx_d^α)([[n_α⁰f_αα]x_α]))

= Φⁿ_F^{α 0f}_g^ααx_d^α(f_αα)(V(Φⁿ_F^{α 0f}_g^ααx_d^α)([n_α⁰f_αα])(V(Φⁿ_F^{α 0f}_g^ααx_d^α)(x_α)))

= g(V(Φⁿ_F^{α 0f}_g^ααx_d^α)(n_α⁰)(V(Φⁿ_F^{α 0f}_g^ααx_d^α)(f_αα))(Φⁿ_F^{α 0f}_g^ααx_d^α(x_α)))

= g(Φⁿ_F^{α 0f}_g^ααx_d^α(n_α⁰)(Φⁿ_F^{α 0f}_g^ααx_d^α(f_αα))(d))

= g(F(g)(d))

= (g◦ F(g))(d) for all d ∈ D_α, whereby

V(Φ)(S_α⁰_α⁰)(F)(g) = g◦ F(g) as desired.

16. Notice first that for all P ∈ D_oα⁰ and all Ψ ∈ A_M, if P(V(Φ)(0_α⁰)) = 0 and Ψ(f_oα⁰) = P then

V(Ψ)(([f_oα⁰0_α⁰]→ ((∀xα⁰([f_oα⁰x_α⁰]→ [foα⁰[S_α⁰_α⁰x_α⁰]]))→ [foα⁰n_α⁰]))) = 1, since

V(Ψ)([f_oα⁰0_α⁰]) = V(Ψ)(f_oα⁰)(V(Ψ)(0_α⁰))

= Ψ(f_oα⁰)(V(Ψ)(0_α⁰))

= P(V(Φ)(0_α⁰))

= 0