BENEDIKT L ¨OWE
Institute for Logic, Language and Computation, Universiteit van Amsterdam Fachbereich Mathematik, Universit¨at Hamburg
Churchill College, University of Cambridge and
ROBERT PASSMANN
Institute for Logic, Language and Computation, Universiteit van Amsterdam and
SOURAV TARAFDER
Department of Mathematics, St. Xavier’s College, Kolkata
Abstract. An algebra-valued model of set theory is called loyal to its algebra if the model and its algebra have the same propositional logic; it is called faithful if all elements of the algebra are truth values of a sentence of the language of set theory in the model. We observe that non-trivial automorphisms of the algebra result in models that are not faithful and apply this to construct three classes of illoyal models: the tail stretches, the transposition twists, and the maximal twists. (Version 7; 2 October 2018)
§1. Background The construction ofalgebra-valued models of set theorystarts from an algebraAand a modelV of set theory and forms anA-valued model of set theory that reflects both the set theory ofV and the logic ofA. This construction is the natural generalisation of Boolean-valued models, Heyting-valued models, lattice-valued models, and orthomodular-valued models (Bell, 2005; Grayson, 1979; Ozawa, 2009; Titani, 1999) and was developed by L¨owe and Tarafder (2015).
L¨owe and Tarafder (2015,§6) used an algebraPS3of paraconsistent logic to
con-struct aPS3-valued model of set theory that exhibits the paraconsistency inherited
from the algebraPS3. For more on the algebraPS3, cf. (Chakraborty and Tarafder,
2016); for more on the set theory in thePS3-valued model, cf. (Tarafder, 2015).
Recently, Passmann (2018) introduced the terms “loyalty” and “faithfulness” while studying the precise relationship between the logic of the algebraAand the logical phenomena witnessed in theA-valued model of set theory. A model is called loyal to its algebra if the propositional logic in the model is the same as the logic of the algebra from which it was constructed and faithful if every element of the algebra is the truth value of a sentence in the model. The model constructed by L¨owe and Tarafder (2015) is both loyal and faithful toPS3. In this paper, we shall
give elementary constructions to produce illoyal models by stretching and twisting Boolean algebras.
After we give the basic definitions in §2., we remind the reader of the construc-tion of algebra-valued models of set theory in §3.. In§4., we introduce our main technique: a non-trivial automorphisms of an algebraAexcludes values from being truth values of sentences in theA-valued model of set theory (Corollary 7). Finally, in §5., we apply this technique to produce three classes of models: tail stretches (§5.2.), transposition twists (§5.3.), and maximal twists (§5.4.).
§2. Basic definitions
Algebras. Let Λ be a set of logical connectives; in this paper, we shall assume that
{∧,∨,0,1} ⊆Λ⊆ {∧,∨,→,¬,0,1}.
An algebra A with underlying set A is called a Λ-algebra if it has one operation for each of the logical connectives in Λ such that (A,∧,∨,0,1) is a distributive lattice;1 we can define≤on
Abyx≤y if and only ifx∧y=x. An elementa∈A is anatom if it is≤-minimal inA\{0}; we write At(A) for the set of atoms inA. If Λ ={∧,∨,→,0,1}, we call Aanimplication algebra and if Λ ={∧,∨,→ ¬,0,1}, we call Aanimplication-negation algebra.
We call a Λ-algebra Awith underlying set A complete if for everyX ⊆A, the ≤-supremum and≤-infimum exist; in this case, we write W
X and V
X for these elements ofA. A complete Λ-algebraAis calledatomic if for everya∈A, there is anX ⊆At(A) such thata=WX.
Boolean algebras, complementation, & Heyting algebras. An algebraB= (B,∧,∨,¬,0,1) is called aBoolean algebra if for allb∈B, we have thatb∧ ¬b=0 andb∨ ¬b=1. As usual, we can define an implication by
x→y:=¬x∨y; (#)
using this definition, we can consider Boolean algebras as implication algebras or implication-negation algebras. An implication algebra (B,∧,∨,→,0,1) is called a Boolean implication algebra if there is a Boolean algebra (B,∧,∨,¬,0,1) such that → is defined by (#) from ∨ and ¬ or, equivalently, if the negation defined by ¬∗x:=x→0satisfies¬∗b∧b=0and¬∗b∨b=1.
On an atomic distributive lattice A= (A,∧,∨,0,1), we have a canonical defi-nition for a negation operation, the complementation negation: since Ais atomic, every elementa∈Ais uniquely represented by a setX ⊆At(A) such thata=WX. Then we define the complementation negation by
¬c(
_
X) :=_{t∈At(A) ;t /∈X}.
In this situation, (A,∧,∨,¬c,0,1) is an atomic Boolean algebra. Moreover, if
(A,∧,∨,¬,0,1) is an atomic Boolean algebra and¬c is the complementation
nega-tion of the atomic distributive lattice (A,∧,∨,0,1), then ¬ = ¬c. Of course, for
1
As usual, we use the same notation for the syntactic logical connectives and the operations on A interpreting them. In the rare cases where proper marking of these
symbols improves readability, we attach a subscriptAto the algebra operations inA,
e.g.,∧A,∨A,
V
A, or
W
every setX, the power set algebra (℘(X),∩,∪,∅, X) forms an atomic distributive lattice and, with the set complementation operator, a Boolean algebra.
If (H,∧,∨,0,1) is a complete distributive lattice, then an implication algebra H = (H,∧,∨,→,0,1) is called a complete Heyting algebra if and only if for all a, b∈H, we have that
a→b=_{x∈H;a∧x≤b}.
In a Heyting algebra H, we can define a negation ¬H by¬Hx:=x→0. Note that Boolean implication algebras are Heyting algebras.
A Heyting algebra is calledlinear if (H,≤) is a linear order; Horn (1969) showed that the formula (p→q)∨(q →p) characterises the variety of Heyting algebras generated by the linear Heyting algebras.
We shall later use the following linear three element Heyting algebra 3:= ({0,1/2,1},∧,∨,→,0,1) with0≤1/2≤1and→defined by
→ 0 1/2 1
0 1 1 1
1/2 0 1 1
1 0 1/2 1 .
Languages. Fix a setSof non-logical symbols, a countable setP of propositional variables, and a countable setV of first-order variables. We denote the propositional logic with connectives Λ and propositional variablePbyLΛand the first-order logic
with connectives Λ, variables in V and constant, relation and function symbols in S byLΛ,S. The subset of sentences of LΛ,S will be denoted by SentΛ,S. Note that
both LΛ and SentΛ,S have the structure of a Λ-algebra and that the Λ-algebraLΛ
is generated by closure under the connectives in Λ from the setP.
For arbitrary sets Λ of logical connectives andSof non-logical symbols, we define NFFΛto be the closure ofP under the logical connectives other than¬and NFFΛ,S
to be the closure of the atomic formulae ofLΛ,S under the logical connectives other
than ¬. These formulas are called thenegation-free Λ-formulas. Clearly, if¬∈/ Λ, thenLΛ= NFFΛ andLΛ,S= NFFΛ,S.
Homomorphisms, assignments, & translations. For any two Λ-algebras A andB, a mapf :A→Bis called a Λ-homomorphismif it preserves all connectives in Λ; it is called a Λ-isomorphismif it is a bijective Λ-homomorphism; isomorphisms from Ato Aare called Λ-automorphisms.
If Aand Bare two complete Λ-algebras andf :A→B is a Λ-homomorphism, we call it complete if it preserves the operations W and V, i.e., f(W
A(X)) = W
B({f(x) ; x∈X}) andf( V
A(X)) = V
B({f(x) ; x∈X}) forX ⊆A.
Since LΛ is generated from P, we can think of any Λ-homomorphism defined
on LΛ as a function on P, homomorphically extended to all of LΛ. If A is a Λ-algebra with underlying set A, we say that Λ-homomorphisms ι : LΛ → A are
A-assignments; ifS is a set of non-logical symbols, we say that Λ-homomorphisms T :LΛ→SentΛ,S areS-translations.
x, y∈D, we havex∧y ∈D. For any designated setD, the propositional logic of (A, D) is defined as
L(A, D) :={ϕ∈ LΛ; ι(ϕ)∈Dfor allA-assignmentsι}.
Note that if Bis a Boolean algebra andD is any filter, thenL(B, D) =CPC, the classical propositional calculus (Blackburn et al., 2001, Theorem 5.11).
Algebra-valued structures and their propositional logic. IfAis a Λ-algebra andSis a set of non-logical symbols, then any Λ-homomorphismJ·K: SentΛ,S→A
will be called an A-valued S-structure. Note that ifS0 ⊆S andJ·K is an A-valued S-structure, then J·KSentΛ,S0 is anA-valuedS0-structure.
We define thepropositional logic of(J·K, D) as
L(J·K, D) :={ϕ∈ LΛ;JT(ϕ)K∈D for allS-translationsT}.
Note that if T is an S-translation and J·K is an A-valued S-structure, then ϕ 7→
JT(ϕ)Kis anA-assignment, so
L(A, D)⊆L(J·K, D). (†) Clearly, ran(J·K) ⊆A is closed under all operations in Λ (since J·K is a homomor-phism) and thus defines a sub-Λ-algebra AJ·K ofA. TheA-assignments that are of the formϕ7→JT(ϕ)Kare exactly theAJ·K-assignments, so we obtain
L(J·K, D) =L(AJ·K, D).
Loyalty & faithfulness. AnA-valued S-structureJ·K is called loyal to(A, D) if the converse of (†) holds, i.e., L(A, D) = L(J·K, D); it is called faithful to A if for everya∈A, there is aϕ∈SentΛ,S such thatJϕK=a; equivalently, ifAJ·K=A. The two notions central for our paper were introduced by Passmann (2018) in a more general setting for classes of so-calledHeyting structures in the sense of Fourman and Scott (1979) (cf. Passmann, 2018, Definitions 2.39 & 2.40). In this paper, we shall not need the level of generality provided in (Passmann, 2018) and stick to the above simplified definitions.
Lemma 1 LetΛbe any set of propositional connectives,S be any set of non-logical symbols,Abe aΛ-algebra, andJ·Kbe anA-valuedS-structure. Then, ifJ·Kis faithful toA, then it is loyal to (A, D)for any designated setD.
Proof. By (†), we only need to prove one inclusion; if ϕ /∈ L(A, D), then let p1, . . . , pn be the propositional variables occurring inϕand letιbe an assignment
such thatι(ϕ)∈/ D. By faithfulness, find sentences σi ∈SentΛ,S such thatJσiK=
ι(pi) for 1≤i≤n. Let T be any translation such thatT(pi) = σi for 1≤i≤n.
ThenJT(ϕ)K=ι(ϕ)∈/ D, and henceTwitnesses thatϕ /∈L(J·K, D). (Cf. Passmann, 2018, Proposition 2.50, for a proof in the more general setting for classes of Heyting
structures.)
Note that the notions of faithfulness and loyalty crucially depend on the choice of S. As mentioned above, if S0 ⊆ S and J·K is an A-valued S-structure, then
J·K0 :=J·K
SentΛ,S0 is anA-valuedS0-structure. Since SentΛ,S0 ⊆SentΛ,S, we have that ifJ·K0 is faithful to
∀x∀y[∀z(z∈x↔z∈y)→x=y] (Extensionality) ∀x∀y∃z∀w(w∈z↔(w=x∨w=y)) (Pairing) ∃x[∃y(∀z(z∈y→0)∧y∈x)∧ ∀w∈x∃u∈x(w∈u)] (Infinity) ∀x∃y∀z(z∈y↔ ∃w∈x(z∈x)) (Union) ∀x∃y∀z(z∈y↔ ∀w∈z(w∈x)) (Power Set) ∀p0· · · ∀pn∀x∃y∀z(z∈y↔z∈x∧ϕ(z, p0, . . . , pn)) (Separationϕ)
∀p0· · · ∀pn−1∀x[∀y∈x∃zϕ(y, z, p0, . . . , pn−1)
→ ∃w∀v∈x∃u∈w ϕ(v, u, p0, . . . , pn−1)] (Collectionϕ)
∀p0· · · ∀pn∀x[∀y∈x ϕ(y, p0, . . . , pn)→ϕ(x, p0, . . . , pn)]
→ ∀zϕ(z, p0, . . . , pn) (Set Inductionϕ)
Fig. 1. The axioms ofZFformulated inL{∧,∨,→,0,1},{∈}.
However, faithfulness tiesJ·Kvery closely to the algebraA: in particular, it cannot hold if the algebraAis bigger than the set SentΛ,S, so for countable languages, no
A-valuedS-structure can be faithful to an uncountable algebraA.
Thus, ifAis an uncountable algebra,San uncountable set of non-logical symbols,
J·Kis anA-valuedS-structure that is faithful toA, andS0 is a countable subset of S, then J·K0:=J·K
LΛ,S0 cannot be faithful toA.
§3. Algebra-valued models of set theory In the following, we give an overview of general construction of an algebra-valued model of set theory following L¨owe and Tarafder (2015). The original ideas go back to Boolean-valued models independently discovered by Solovay and Vopˇenka (1965) and were further gener-alised to other classes of algebras (Grayson, 1979; Ozawa, 2007, 2009; Takeuti and Titani, 1992; Titani, 1999; Titani and Kozawa, 2003). Details can be found in (Bell, 2005).
In the following, we shall use the phrase “V is a model of set theory” to mean thatV is a transitive set such that (V,∈)|=ZF. Of course, the existence of sets like this cannot be proved inZFand requires some (mild) additional metamathematical assumptions. The choice of ZFas the set theory in our base model is not relevant for the constructions of this paper and one can generalise the results to models of weaker or alternative set theories; however, we shall not explore this route in this paper.
Since we are sometimes working in languages without negation, we need to formulate the axioms of ZFin a negation-free context given in Figure 1, following L¨owe and Tarafder (2015,§3).2IfV is a model of set theory andAis any set, then
2
We should like to stress that the negation-free axioms given are classically equivalent to what is usually calledZF, but not exactly the same axioms: e.g., we useCollection
we construct a universe of names by transfinite recursion: Nameα(V, A) :={x; xis a function and ran(x)⊆A
and there isξ < αwith dom(x)⊆Nameξ(V, A)}and
Name(V, A) :={x;∃α(x∈Nameα(V, A))}.
We let SV,A be the set of non-logical symbols consisting of the binary relation
symbol ∈and a constant symbol for every name in Name(V, A) (as usual, we use the name itself as the constant symbol). The languageLΛ,SV,A is usually called the forcing language.
IfAis a Λ-algebra with underlying setA, we can now define a mapJ·KAassigning to each ϕ ∈ LΛ,SV,A a truth value in A by recursion (the definition of Ju ∈ vKA andJu=vKAis recursion on the hierarchy of names; the rest is a recursion on the complexity ofϕ):
J0KA=0,
J1KA=1,
Ju∈vKA= _
x∈dom(v)
(v(x)∧Jx=uKA),
Ju=vKA= ^
x∈dom(u)
(u(x)→Jx∈vKA)∧ ^
y∈dom(v)
(v(y)→Jy∈uKA),
Jϕ∧ψKA=JϕKA∧JψKA,
Jϕ∨ψKA=JϕKA∨JψKA,
Jϕ→ψKA=JϕKA→JψKA,
J¬ϕKA=¬JϕKA
J∀xϕ(x)KA= ^
u∈Name(V,A)
Jϕ(u)KA, and
J∃xϕ(x)KA= _
u∈Name(V,A)
Jϕ(u)KA.
By construction, it is clear that J·KA is an
A-valued SV,A-structure and hence,
by restricting it to SentΛ,{∈}, we can consider it as an A-valued {∈}-structure. Usually, it is the restriction to SentΛ,{∈} that set theorists are interested in: as a consequence, we shall use the notation J·KAto refer to theA-valued{∈}-structure and the notationJ·KName
A :=J·KAto refer to its extension to SentΛ,SV,A.
The results for algebra-valued models of set theory were originally proved for Boolean algebras, then extended to Heyting algebras:
Theorem 2 If V is a model of set theory, B= (B,∧,∨,→,¬,0,1) is a Boolean algebra or Heyting algebra, and ϕis any axiom ofZF, thenJϕKB=1.
Proof. Cf. (Bell, 2005, Theorem 1.33 & pp. 165–166).
Lemma 3 Let H= (H,∧,∨,→,0,1)be a Heyting algebra andV be a model of set theory. ThenJ·KName
∧ 0 1/2 1
0 0 0 0
1/2 0 1/2 1/2
1 0 1/2 1
∨ 0 1/2 1
0 0 1/2 1 1/2 1/2 1/2 1
1 1 1 1
→ 0 1/2 1
0 1 1 1
1/2 0 1 1
1 0 1 1
¬
0 1
1/2 1/2
[image:7.595.138.472.170.235.2]1 0
Fig. 2. Connectives forPS3
Proof. Consider u:={(∅, a)} ∈Name(V, H) and ϕ:=∃x(x∈u)∈SentΛ,SV,H. It
is easy to check thatJϕKNameH =a.
In order to formulate results for implication algebras, L¨owe and Tarafder (2015, p. 197) introduced NFF-ZF, the axiom system of allZF-axioms where the two axiom schemata are restricted to instances of negation-free formulas. They introduced a three-element algebraPS3(L¨owe and Tarafder, 2015, Fig. 2 &§6) and proved the
following result (for the sake of completeness, we give the definition ofPS3in Figure
2):
Theorem 4 If V is a model of set theory and ϕ is any axiom of NFF-ZF, then
JϕKPS3 = 1. Furthermore, J·KPS3 is faithful to PS3 and hence loyal to (A, D) for every designated setD by Lemma 1.
Proof. Cf. (L¨owe and Tarafder, 2015, Corollary 5.2) for the first claim. L¨owe and Tarafder (2015, Theorem 6.2) give a sentenceϕ∈SentΛ,{∈}such thatJϕKPS3 =1/2
which establishes faithfulness.
§4. Automorphisms and algebra-valued models of set theory Given a model of set theory V and any Λ-algebras A and B and a Λ-homomorphism f :A→B, we can define a mapfb: Name(V,A)→Name(V,B) by∈-recursion via
dom(fb(u)) :={fb(v) ;v∈dom(u)}and b
f(u)(fb(v)) :=f(u(v)).
Proposition 5 Suppose that V is a model of set theory, A and B are complete Λ-algebras andf :A→Bis a completeΛ-isomorphism. Letϕ∈ LΛ,{∈} withnfree variables andu1, . . . , un ∈Name(V,A). Then
f(Jϕ(u1, . . . , un)KA) =Jϕ(fb(u1), . . . ,fb(un))KB.
Proof. For atomic formulas, this is easily proved by induction on the rank of the names involved. For non-atomic formulas, the claim follows by induction on the complexity of the formula (where the quantifier cases need the fact that f is a
bijection).
Corollary 6 Suppose that V is a model of set theory, A and B are complete Λ-algebras and f :A→Bis a completeΛ-isomorphism. Letϕ∈SentΛ,{∈}. Then
f(JϕKA) =JϕKB.
Proof. By Corollary 6, ifJϕKA=a, thenf(a) =a. Proposition 8 If A = (A,∧,∨,0,1) is an atomic distributive lattice and a ∈ A\{0,1}, then there is a{∧,∨,¬c,0,1}-automorphismf of Asuch thatf(a)6=a. Proof. Note that the assumptions imply thatA 6={0,1} and hence At(A)6=∅. By atomicity, every permutation π: At(A)→At(A) induces an automorphism of Apreserving∧,∨,¬c,0, and1byfπ(WX) =W{π(t) ;t∈X}forX ⊆At(A). Let a=WX
a. Sincea 6=0, we have Xa 6=∅; since a 6=1, we have Xa 6= At(A). So, pickt0∈Xa andt1∈At(A)\Xa and let πbe the transposition that interchanges
t0andt1. Then
t0≤
_
Xa =a, but
t06≤
_
{π(t) ;t∈Xa}=fπ(
_
Xa) =fπ(a),
whence a6=fπ(a).
Corollary 9 If V is a model of set theory, B is an atomic Boolean (implication) algebra with more than two elements, and D is any filter onB, then J·KB is loyal, but not faithful to (B, D).
Proof. By Proposition 8, all elements except for 0 and 1 are moved by some automorphism of an atomic Boolean (implication) algebra and hence by Corollary 7, for each sentence ϕ ∈ LΛ,{∈}, we have that JϕKB ∈ {0,1}. In particular, this means thatL(J·KB, D) =L({0,1},{1}) =CPC=L(B, D). Clearly, atomicity is not a necessary condition for the conclusion of Corollary 9: the Boolean algebra of infinite and co-infinite subsets of N is atomless and hence non-atomic, but every nontrivial element is moved by an automorphism, so Corollary 7 applies. We do not know whether this result extends to Boolean algebras without this property, e.g., rigid Boolean algebras (cf. van Douwen et al., 1980,§2):
Question 10 Are there (necessarily countable) Boolean algebras B such thatJ·KB is faithful toB for some designated setD?
§5. Stretching and twisting the loyalty of Boolean algebras
5.1. What can be considered a negation? In this section, we start from an atomic, complete Boolean algebra B and modify it, to get an algebra A that gives rise to an illoyal J·KA. The first construction is the well-known construction of tail extensions of Boolean algebras to obtain a Heyting algebra. The other two constructions arenegation twists: in these, we interpretBas a Boolean implication algebra via the definitiona→b :=¬a∨b, and then add a new, twisted negation to it that changes its logic.
5.2. Tail stretches Let B = (B,∧,∨,→,¬,0,1) be a Boolean algebra, and 1∗ ∈/ B be an additional element that we add to the top of B to form the tail stretch Has follows:H :=B∪ {1∗}, the complete lattice structure ofHis the order sum ofBand the one element lattice{1∗}, and →∗is defined as follows:3
a→∗b:=
a→b ifa, b∈B such thata6≤b, 1∗ ifa, b∈B witha≤bor ifb=1∗,
b ifa=1∗.
Lemma 11 The tail stretch H = (H,∧,∨,→∗,0,1∗) is a Heyting algebra with p∨ ¬p /∈L(H,{1∗}), so in particular,L(H,{1∗})6=CPC.
Proof. Ifb6=0∈B, then by definitionb→∗0=¬bwhere¬refers to the negation in B. In particular,b∨ ¬Hb=b∨ ¬b=16=1∗.
Lemma 12 If f : B → B is an automorphism of the Boolean algebra B, then f∗:H→H defined by
f∗(b) :=
f(b) ifb∈B and 1∗ ifb=1∗. is an automorphism ofH.
Proof. Easy to check.
Theorem 13 Let V be a model of set theory, B an atomic Boolean algebra with more than two elements, and Hbe the tail stretch ofB as defined above. Then the
H-valued model of set theory VH is not faithful toH. Furthermore, we have that (p→q)∨(q→p)∈L(J·KH,{1∗})\L(H,{1∗}).
Consequently,VH is illoyal to(
H,{1∗}).
Proof. Since B is atomic with more than two elements, each of the non-trivial elements ofB is moved by an automorphism ofBby Proposition 8. By Lemma 12, these remain automorphisms of H. As a consequence, we can apply Corollary 6 to get that ran(J·KH)⊆ {0,1,1∗} which is isomorphic to the linear Heyting algebra3 and thus the range is a linear Heyting algebra. As mentioned, Horn (1969) proved that (p→ q)∨(q → p) characterises the variety generated by the linear Heyting algebras, so (p→q)∨(q→p)∈L(J·KH,{1∗}). However, since
Bhas more than two elements, we can pick imcomparable a, b∈B. Then a→∗ b andb →∗ aare both elements ofB, and thus (p→q)∨(q→p)∈/ L(H,{1∗}).
We remark that ran(J·KH) ={0,1,1∗}, as can be seen by checking that theJ·K H -value of the sentence formalising the statement “every subset of{∅}is either∅or {∅}” is1.
3
InH, we use the (Heyting algebra) definition¬Hh:=h→
∗
0to define a negation; note that if06=b∈B,¬Hb=¬b, but¬H0=1
∗
6
5.3. Transposition twists LetB= (B,∧,∨,→,¬,0,1) be an atomic Boolean algebra,a, b∈At(B) witha6=b, and πbe the transposition that transposesaand b. SinceBis an atomic Boolean algebra,¬=¬c. Thenfπ as defined in the proof of
Proposition 8 is a {∧,∨,→,¬,0,1}-automorphism ofB. We now define a twisted negation by
¬π(
_
X) :=_{π(t)∈At(B) ;t /∈X}
and let theπ-twistofBbeBπ:= (B,∧,∨,→,¬π,0,1).4We observe that the twisted
negation¬π satisfies the rule of contraposition.
Lemma 14 If either¬ca=W{t∈At(B) ; t6=a} or¬cb=W{t∈At(B) ; t6=b} is not in D, then¬(p∧ ¬p)∈/L(Bπ, D). In particular,L(Bπ, D)6=CPC.
Proof. Without loss of generality, W{t ∈At(
B) ;t 6=b} =¬cb=¬πa /∈ D. Since
a≤ ¬πa, we have thata=¬πa∧a, and so¬π(¬πa∧a) =¬πa /∈D.
Lemma 15 There is an automorphism f of Bπ such thatf(a) =b. In particular, J·KBπ is not faithful toBπ.
Proof. We know that fπ is an automorphism of B. Sinceπ is a transposition, we have thatπ2= id andπ=π−1; using this, we observe that f
π still preserves¬π:
fπ(¬π(
_
X)) =fπ(
_
{π(t)∈At(B) ;t /∈X}) =_{π(π(t))∈At(B) ; t /∈X} =_{t∈At(B) ;t /∈X} =¬π(
_
{π(t)∈At(B) ; t /∈X}) =¬π(fπ(
_ X))).
Thus, fπ is an automorphism of Bπ; clearly, fπ(a) = b. The second claim follows
from Corollary 7.
Now letV be a model of set theory andJ·KBπ theBπ-valued{∈}-structure derived
from V andB.
Lemma 16 If x∈ran(J·KBπ), then¬πx=¬cx.
Proof. Let x = W
X for some X ⊆ At(B). By Corollary 7 and Lemma 15, if x ∈ ran(J·KBπ), then fπ(x) = x. This means that either both a, b ∈ X or both
a, b /∈X. In both cases, it is easily seen that¬πx=¬cx.
Theorem 17 For any filterD,L(J·KBπ, D) =CPC. In particular, if either¬caor
¬cb is not in D, thenJ·KBπ is not loyal to(Bπ, D).
Proof. As mentioned in §2., if we letC := BJ·KBπ = (ran(J·KBπ,∧,∨,→,¬π,0,1), then L(J·KBπ, D) = L(C, D). But Lemma 16 implies that C = (ran(J·KBπ,∧,∨,→ ,¬c,0,1) which is a Boolean algebra (as a subalgebra of B). Thus, L(J·KBπ, D) = L(C, D) =CPC. The second claim follows from Lemma 14.
4
1OO
1OO
Loo //R π // --L Rrr
[image:11.595.189.411.173.222.2]B 0 0 Bπ
Fig. 3. The four-element Boolean algebra and its transposition twist. Negations are indicated by arrows.
As the simplest possible special case, we can consider the Boolean algebra B generated by two atoms L and R; then, there is one nontrivial transpositionπ(L) = R and all nontrivial elements of B are moved by the automorphism fπ. As a
consequence of Corollary 7, all sentences will get either value 0 or value1 under
J·KBπ, and henceL(J·KBπ, D) is classical (cf. Figure 3).
5.4. Maximal twists Again, letB= (B,∧,∨,→,¬,0,1) be an atomic Boolean algebra with more than two elements and define the maximal negation by
¬mb:=
1 ifb6=1and 0 ifb=1
for everyb∈B. We let themaximal twistofBbeBm:= (B,∧,∨,→,¬m,0,1); once
more observe that the maximal negation ¬msatisfies the rule of contraposition.
Lemma 18 If there is some 0 6= b /∈ D, then (p∧ ¬p) → q /∈ L(Bm, D). In
particular,L(Bm, D)6=CPC.
Proof. Letc:=¬cb. Note that the assumption b6=0impliesc6=1. In particular,
¬mc=1, and thusc∧ ¬mc=c. Also
c→b=¬cb→b
=¬c¬cb∨b
=b∨b=b.
Thus, the assignmentιwithp7→c andq7→byields ι((p∧ ¬p)→q) =b /∈D.
Lemma 19 For any b /∈ {0,1}, there is an automorphism f of Bm such that
f(b)6=b. In particular,J·KBm is not faithful to Bm.
Proof. We claim that any automorphismf ofBalso preserves¬m. Supposef is an
automorphism of B. Ifb =1, then clearlyf(¬m1) = f(0) = 0=¬m1=¬mf(1).
Now let b 6= 1. Since f is bijective and f(1) = 1, we have that f(b) 6= 1. So f(¬mb) =f(1) =1=¬mf(b). The second claim follows from Corollary 7.
Theorem 20 For any designated setD,L(J·KBm, D) =CPC. In particular,J·KBm
is not loyal to(Bm, D).
Proof. Lemma 19 gives us that every nontrivial element of B is moved by an automorphism, so we can apply the argument from the proof of Corollary 9: since for eachϕ∈ LΛ,{∈}, we have thatJϕKBm ∈ {0,1}, we get that
L(J·KBm, D) =L({0,1},{1}) =CPC.
As mentioned at the end of§2., our examples show that restricting the language can change faithful models into illoyal ones: for our twisted algebras Bπ and Bm,
the general faithfulness result Lemma 3 holds for J·KName
Bπ and J·K
Name
Bm . However,
Theorems 17 & 20 show that their restrictions J·KBπ and J·KBm are neither faithful nor loyal.
BIBLIOGRAPHY
Bell, J. L. (2005). Set theory, Boolean-valued models and independence proofs, volume 47 ofOxford Logic Guides. The Clarendon Press Oxford University Press, Oxford, third edition.
Blackburn, P., de Rijke, M., and Venema, Y. (2001). Modal Logic, volume 53 of Cambridge Tracts in Theoretical Computer Science. Cambridge University Press. Chakraborty, M. K. and Tarafder, S. (2016). A paraconsistent logic obtained from an algebra-valued model of set theory. In B´eziau, J. Y., Chakraborty, M. K., and Dutta, S., editors,New Directions in Paraconsistent Logic, 5th WCP, Kolkata, India, February 2014, volume 152 of Springer Proceedings in Mathematics & Statistics, pages 165–183. Springer-Verlag.
Dunn, J. M. (1995). Generalized ortho negation. In Wansing, H., editor,Negation. A Notion in Focus, volume 7 ofPerspektiven der Analytischen Philosophie, pages 3–26. De Gruyter.
Fourman, M. P. and Scott, D. S. (1979). Sheaves and logic. In Fourman, M. P., Mulvey, C. J., and Scott, D. S., editors,Applications of sheaves. Proceedings of the Research Symposium on Applications of Sheaf Theory to Logic, Algebra and Analysis held at the University of Durham, Durham, July 9–21, 1977, volume 753 ofLecture Notes in Mathematics, pages 302–401. Springer-Verlag.
Grayson, R. J. (1979). Heyting-valued models for intuitionistic set theory. In Fourman, M. P., Mulvey, C. J., and Scott, D. S., editors,Applications of sheaves, Proceedings of the Research Symposium on Applications of Sheaf Theory to Logic, Algebra and Analysis held at the University of Durham, Durham, July 9–21, 1977, volume 753 ofLecture Notes in Mathematics, pages 402–414. Springer, Berlin. Horn, A. (1969). Logic with truth values in a linearly ordered heyting algebra.
Journal of Symbolic Logic, 34:395–408.
L¨owe, B. and Tarafder, S. (2015). Generalized algebra-valued models of set theory. Review of Symbolic Logic, 8(1):192–205.
Ozawa, M. (2007). Transfer principle in quantum set theory. Journal of Symbolic Logic, 72(2):625–648.
Ozawa, M. (2009). Orthomodular-valued models for quantum set theory. preprint, arXiv 0908.0367.
Passmann, R. (2018). Loyalty and faithfulness of model constructions for constructive set theory. Master’s thesis, Universiteit van Amsterdam.
Takeuti, G. and Titani, S. (1992). Fuzzy logic and fuzzy set theory. Archive for Mathematical Logic, 32(1):1–32.
Tarafder, S. (2015). Ordinals in an algebra-valued model of a paraconsistent set theory. In Banerjee, M. and Krishna, S. N., editors,Logic and Its Applications, 6th International Conference, ICLA 2015, Mumbai, India, January 8–10, 2015, Proceedings, volume 8923 ofLecture Notes in Computer Science, pages 195–206. Springer-Verlag.
Titani, S. and Kozawa, H. (2003). Quantum set theory. International Journal of Theoretical Physics, 42(11):2575–2602.
van Douwen, E. K., Monk, J. D., and Rubin, M. (1980). Some questions about boolean algebras. Algebra Universalis, 11:220–243.
