# Adequate conditionals and Kripke’s theory of truth

Johannes Stern
Department of Philosophy
University of Bristol
<EMAIL_ADDRESS>
###### Abstract

In this paper we show how to introduce a conditional to Kripke’s theory of
truth that respects the deduction theorem for the consequence relation
associated with the theory. To this effect we develop a novel supervaluational
framework, called strong Kleene supervaluation, that we take to be a promising
framework for handling the truth-conditions of non-monotone notion in the
presence of semantic indeterminacy more generally.

##  1 Introduction

After almost 50 years of Kripke’s Outline of a Theory of Truth Kripke’s
inductive definition of a self-applicable truth predicate remains one of the
most prominent semantic approaches to truth. Of course, a lot of work has been
carried out since: Gupta, Herzberger and Belnap have given us the Revision
theory of truth (Herzberger,, 1982; Gupta,, 1982; Gupta and Belnap,, 1993)
while others have extended Kripke’s proposal to tackle the revenge problem
(cf., e.g., Glanzberg,, 2004; Simmons,, 2018; Schlenker,, 2010); while still
others have applied it to intensional languages (cf., e.g., Halbach and
Welch,, 2009; Stern,, 2014, 2016, 2021). An important limitation of Kripke’s
proposal is that it is difficult to introduce conditionals or restricted
quantifiers such as ${\sf Every}$ or ${\sf Most}$ to Kripke’s theory of truth,
that is, to formalize forms of conditional reasoning within the Kripkean
framework. The problem is due to the fact that the truth conditions of such
notions are non-monotone and this seems to be non-accidental: changing the
truth-conditions to restore monotonicity seems to trigger a loss of central
characteristic features of these notions.111By monotonicity we refer to the
idea that an increase in semantic information leads to an increase of
sentences receiving a classical truth value. There have been important
contributions to this problem and in particular the problem of introducing
conditionals to Kripke’s theory of truth, and, more generally, theories of
naive truth by, e.g., Yablo, (2013); Beall, (2009); Bacon, (2013); Rossi,
(2016); Iacona and Rossi, (2024) and, notably, Field, (2008, 2016, 2021), but
all these proposals sacrifice important logical properties of the conditional
in favor of the so-called transparency of the truth predicate.222We refer to
Section 2 for a precise introduction of this terminology. The aim of this
paper is to develop a general strategy for simultaneously introducing
different conditionals, restricted quantifiers and, more generally, non-
monotone notions that, in contrast, preserves the salient logical properties
but sacrifices the transparency of the truth predicate. For the purpose of
this paper we focus on conditionals, although our strategy applies more
widely. We show how to construct Kripkean truth models for languages that
include logically adequate conditionals. We shall argue that this requires the
conditional to respect the deduction theorem for the consequence relation
associated with the truth theory.

To this effect we develop a supervaluational semantics, which in contrast to
orthodox supervaluational semantics, quantifies over partial precisification
instead of classical precisification. On this semantics a conditional is true
in a given partial model $M$, if it is true in all available partial models
that are semantically compatible with $M$. The crux of the matter is then to
specify the notions of available partial model and semantic compatibility.
This leads us to the framework of strong Kleene supervaluations and to strong
Kleene supervaluation structures. A strong Kleene supervaluation structure
consists of a non-empty domain, a set of strong Kleene interpretations and an
admissibility relation on these interpretations: a conditional sentence will
be true in a strong Kleene structure relative to an interpretation $J$ iff it
is true at all admissible interpretations $J^{\prime}$ extending $J$. Our
strategy for handling conditionals shares important aspects with Kripke
semantics for intuitionistic logic and, in particular, the truth conditions of
the intuitionistic conditional. Ultimately, our approach amounts to
introducing an intuitionistic conditional to strong Kleene logic and strong
Kleene supervaluation for a conditional, as we introduce it in this paper,
turns out to be notational variant of constant domain Kripke semantics for
${\sf N3}$-Nelson logic as introduced by Thomason, (1969).333We refer to
Wansing, (1991, 2001) for an introduction and discussion of Nelson logic.

The main task of the paper is to show how to construct strong Kleene
supervaluation structures for a language with a self-applicable truth
predicate that comprises interpretations at which $\varphi$ and
$\scaleobj{0.9}{\mathrm{T}}\raise
0.0pt\hbox{$\ulcorner$}\hbox{$\varphi$}\raise 0.0pt\hbox{$\urcorner$}$ receive
the same truth value for every sentence $\varphi$. Establishing the existence
of such interpretations shows that we can apply Kripke’s theory of truth to
non-classical logics that have an adequate conditional: by constructing a
Kripkean truth model for a language with a conditional we show how to
introduce an ${\sf N3}$-conditional to Kripke’s theory of truth. Incidentally,
our construction thus answers a question posed by Albert Visser who asks
whether we can “extend (…) Kripke’s solution [Kripke’s theory of truth, JS]
with, say, Thomasonian implication?” (Visser,, 1989, p. 645). However, we do
not stop there. We also show how strong Kleene supervaluation can be combined
with ordering semantic to provide adequate truth conditions for indicative and
subjunctive conditionals in terms of strict or variably strict implication,
that is, our strategy can be employed to give truth conditions for various
natural language conditionals in the context of Kripke’s theory of truth.
Whilst in this paper we focus on various conditionals we think of strong
Kleene supervaluation as a general strategy and framework for handling non-
monotone notions such as restricted quantifiers in the presence of semantic
indeterminacy broadly understood.

#### Plan of the paper

We start our investigation by a brief discussion of Kripke’s theory of truth
and the challenge to Kripke’s theory of truth posed by conditionals and
conditional reasoning more generally (Section 2). In Section 3 we discuss the
dilemma Curry’s paradox poses to truth theorist: either truth is not
transparent or the conditional does not respect the deduction theorem. We take
the latter to imply that the truth theory would lack and adequate conditional,
which we take to be a non-negiotable requirement on any satisfactory truth
theory. We then turn to ideas motivating the framework of strong Kleene
supervaluation which we introduce in detail in Section 4. In Section 5 we
expand supervaluation structures for arbitrary first-order languages to
provide an interpretation of the language extended by a self-applicable truth
predicate. Section 6 shows how to single out so-called Kripkean truth
structures amongst the strong Kleene supervaluation structures of the language
with a self-applicable truth predicate. Kripkean truth structures are truth
structures that contain interpretations at which $\varphi$ and
$\scaleobj{0.9}{\mathrm{T}}\raise
0.0pt\hbox{$\ulcorner$}\hbox{$\varphi$}\raise 0.0pt\hbox{$\urcorner$}$ will
have the same truth value for every sentence $\varphi$ of the language with
the self-applicable truth predicate. In Section 7 we examine the properties of
the truth predicate and the conditional in Kripkean truth structures, whilst
Section 8 discusses the recursion-theoretic complexity of the construction. Up
to this point our semantics was concerned with a logically adequate
conditional for a non-classical logic, that is, a non-classical conditional
that assumes the role material implication has for classical logic in the non-
classical logic under consideration. In the final section of the paper,
Section 9 we show how the framework can be extended to a semantics for
indicative and/or subjunctive conditionals. This amounts to combining strong
Kleene supervaluation with ordering semantics and to show how the construction
of Kripkean truth structures can be carried out in the extended framework.

##  2 Conditionals and Kripke’s Theory of Truth

Kripke’s theory of truth (Kripke,, 1975) seeks to define the interpretation of
an object-linguistic, self-applicable truth predicate. Of course, due to
Tarski, (1935) we know that no classical model can satisfy the
$\scaleobj{0.9}{\mathrm{T}}$-scheme, that is, in no classical model is it
possible that $\varphi$ and $\scaleobj{0.9}{\mathrm{T}}\raise
0.0pt\hbox{$\ulcorner$}\hbox{$\varphi$}\raise 0.0pt\hbox{$\urcorner$}$ have
the same semantic value for every sentence $\varphi$.444That is, modulo the
type of construction given by Gupta, (1982). Kripke’s insight was that the
latter (but not the former) can be achieved if we resort to non-classical
models: relative to a non-classical models $\mathcal{M}$ one can find an
interpretation of the truth predicate $S$ such that for all sentences
$\varphi$

(Naivity) $\displaystyle(M,S)\Vdash\scaleobj{0.9}{\mathrm{T}}\raise
0.0pt\hbox{$\ulcorner$}\hbox{$\varphi$}\raise 0.0pt\hbox{$\urcorner$}\text{
iff }(M,S)\Vdash\varphi.$

Models for which Naivity holds will be called models of _naive_ truth.
Throughout this paper it is important to distinguish Naivity from
Transparency. Transparent truth models are models of naive truth that satisfy
the following stronger property:

(Transparency) $\displaystyle(M,S)\Vdash\psi(\scaleobj{0.9}{\mathrm{T}}\raise
0.0pt\hbox{$\ulcorner$}\hbox{$\varphi$}\raise
0.0pt\hbox{$\urcorner$}/\chi)\text{ iff }(M,S)\Vdash\psi(\varphi/\chi).$

Roughly put Transparency is Naivity plus $M$-extensionality, that is, the
assumption that the truth value of a sentence in the model $(M,S)$ only
depends on the truth values of its subsentences in $(M,S)$. As we discuss
below, Transparency holds for some but not all variants of Kripke’s theory of
truth.

The standard strategy of showing the existence of naive (transparent) truth
models is by defining a suitable monotone operation on $\mathcal{P}({\sf
Sent})$ relative to some ground model $M$. Such an operation is also called a
Kripke jump. There is some flexibility as to which logic and evaluation scheme
is employed, but the most prominent variant of the Kripke jump is the
operation

$\mathcal{K}_{3}(S)=\\{\varphi\,|\,(M,S)\Vdash_{\sf K3}\varphi\\},$

that is, the operation $\mathcal{K}_{3}$ applied to a set of sentences $S$
yields the set of sentences that are true in the model $(M,S)$ according to
the strong Kleene scheme. A cardinality argument then establishes that the
operation must have a fixed point, indeed, a minimal fixed point and if the
operation has been suitably defined these fixed points yield attractive
interpretations of truth predicate over $M$: if $S$ is fixed point, then
$(M,S)$ is a naive truth model.555Monotonicity guarantees the existence of
fixed points. It does not guarantee the existence of non-trivial fixed points.
We will come back to this issue at a later stage. Instead of using a simple
cardinality argument the existence of fixed point can also be shown using an
algebraic argument, that is, the Knaster-Tarski fixed-point theorem and
variants thereof (cf. Visser,, 1989). This approach does not only establish
the existence of a minimal fixed point, but also provides valuable information
about the structure of the different fixed points of the operation.

One principal downside of Kripke’s theory of truth is already implicit in the
formulation of its positive result, that is, the existence of a naive truth
model: the equivalence of $\varphi$ and $\scaleobj{0.9}{\mathrm{T}}\raise
0.0pt\hbox{$\ulcorner$}\hbox{$\varphi$}\raise 0.0pt\hbox{$\urcorner$}$
relative to a model $(M,S)$ cannot be stated in the objectlanguage. There is
no conditional that enables us to express the metalinguistic ‘iff’ in the
objectlanguage. Of course, this is no mere coincident as such a conditional
would re-introduce the Liar paradox. More importantly, the partial
logics/semantics employed in the context of Kripke’s theory of truth do not
comprise an attractive conditional that adequately captures conditional
reasoning in language and logic. It is arguably for that reason that
(Feferman,, 1984, p. 95) famously complained that in such logics “no sustained
ordinary reasoning can be carried on” (Feferman,, 1984). A theory of truth
without an adequate conditional, that is, a conditional that enables us to
carry out conditional reasoning is left wanting. So the question arises how an
adequate conditional can be integrated with Kripke’s theory of truth and what
it takes for a conditional to be adequate.

##  3 Conditionals, Truth and the Deduction Theorem

In natural and formal languages there is a plethora of different conditionals
and before we can start answering the above question we need to clarify the
scope of our investigation. To this effect we distinguish between an adequate
conditional for a given logic, which we take to be key for carrying on
conditional reasoning and conditionals such as indicative or subjunctive
conditionals that are important from the perspective of natural language
and/or philosophical semantics. Initially, our focus will be on the former,
i.e., on introducing a suitable logical conditional to Kripke’s theory of
truth. However, as we show in Section 9 that once we have introduced such a
logical conditional, it is rather straightforward to extend the framework by
indicative and subjunctive conditionals.

Arguably, conditional reasoning is aptly characterized by two rules, that is,
by the conditional introduction and the conditional elimination rule:

Conditional Introduction

If, given a set of formulas $\Gamma$, the formula $\psi$ follows from the
formula $\varphi$ in the logic $L$, then the conditional
$\varphi\rightarrow\psi$ follows from $\Gamma$ in $L$.

Conditional Elimination

If $\varphi\rightarrow\psi$ follows from a set of formulas $\Gamma$ in $L$,
then $\psi$ follows from $\Gamma$ and $\varphi$.

If a logic $L$ is equipped with a conditional for which both conditional
introduction and conditional elimination hold, then the logic $L$ has a
conditional that respects the deduction theorem. Let $L$ be a logic and
$\vDash_{L}$ the consequence relation of $L$. Then $\rightarrow$ satisfies the
deduction theorem for $L$ iff

$\displaystyle\Gamma,\varphi\vDash_{L}\psi$ $\displaystyle\,\,{\rm
iff}\,\,\Gamma\vDash_{L}\varphi\rightarrow\psi.$

On the view we endorse in this paper a conditional that violates either
conditional introduction or conditional elimination in a logic $L$ does not
adequately capture conditional reasoning in $L$; it is not an adequate
conditional for $L$. Accordingly, a conditional is adequate conditional for a
logic $L$, if and only if, it respects the deduction theorem for $L$.

Yet, a conditional that respects the deduction theorem for the given
consequence relation of a logic $L$ imposes severe limitation on the
properties a truth predicate can have in theories, if the consequences
relation of that logic is a structural consequence relation. Let $\vDash_{L}$
be the consequence relation of some logic. Call $\vDash_{L}$ a naive
consequence relation iff

$\displaystyle(\scaleobj{0.9}{\mathrm{T}}\text{\sc l})$
$\displaystyle\Gamma,\varphi\vDash_{L}\Delta\text{ iff
}\Gamma,\scaleobj{0.9}{\mathrm{T}}\raise
0.0pt\hbox{$\ulcorner$}\hbox{$\varphi$}\raise
0.0pt\hbox{$\urcorner$}\vDash_{L}\Delta$
$\displaystyle(\scaleobj{0.9}{\mathrm{T}}\text{\sc r})$
$\displaystyle\Gamma\vDash_{L}\varphi,\Delta\text{ iff
}\Gamma\vDash_{L}\scaleobj{0.9}{\mathrm{T}}\raise
0.0pt\hbox{$\ulcorner$}\hbox{$\varphi$}\raise 0.0pt\hbox{$\urcorner$},\Delta$

for all sentences $\varphi$ and sets of sentences $\Gamma,\Delta$. Moreover,
$\vDash_{L}$ is a structural consequence relation iff it has the following
properties for $\Gamma,\Delta\subseteq{\sf Frml}$ and $\varphi\in{\sf
Frml}$:777We assume a multiconclusion formulation of the consequence relation.
Yet nothing hinges on this assumption. Everything we say goes through if we
work with single conclusions.

$\displaystyle({\sf Ref})$
$\displaystyle\Gamma,\varphi\vDash_{L}\varphi,\Delta;$ $\displaystyle({\sf
Weak})$ $\displaystyle\text{if }\Gamma\vDash_{L}\Delta,\text{ then
}\Gamma,\Gamma^{\prime}\vDash_{L}\Delta,\Delta^{\prime};$ $\displaystyle({\sf
Contr})$ $\displaystyle\text{if }\Gamma,\varphi,\varphi\vDash_{L}\Delta,\text{
then }\Gamma,\varphi\vDash_{L}\Gamma;$ $\displaystyle({\sf Trans})$
$\displaystyle\text{if
}\Gamma\vDash_{L}\varphi,\Delta\,\&\,\Gamma,\varphi\vDash_{L}\Delta\text{ then
}\Gamma\vDash_{L}\Delta.$

The proposition below shows that we cannot have both the Deduction theorem,
and a structural and naive consequence relation. This is one guise of so-
called Curry’s paradox:

######  Proposition 1 (Curry).

Let $\vDash_{L}$ be a structural consequence relation for a (non-trivial)
logic $L$ and $\rightarrow$ a conditional for which the deduction theorem
$({\sf Ded})$ holds relative to $\vDash_{L}$. Then $\vDash_{L}$ cannot be a
naive consequence relation.

###### Proof.

Let $\kappa$ be the sentence $\scaleobj{0.9}{\mathrm{T}}\raise
0.0pt\hbox{$\ulcorner$}\hbox{$\kappa$}\raise
0.0pt\hbox{$\urcorner$}\rightarrow\bot$. Then

  1. 1.

$\kappa\vDash\kappa\hfill({\sf Ref})$

  2. 2.

$\kappa\vDash\scaleobj{0.9}{\mathrm{T}}\raise
0.0pt\hbox{$\ulcorner$}\hbox{$\kappa$}\raise
0.0pt\hbox{$\urcorner$}\rightarrow\bot\hfill 1,\text{ Def. }\kappa$

  3. 3.

$\scaleobj{0.9}{\mathrm{T}}\raise 0.0pt\hbox{$\ulcorner$}\hbox{$\kappa$}\raise
0.0pt\hbox{$\urcorner$}\vDash\scaleobj{0.9}{\mathrm{T}}\raise
0.0pt\hbox{$\ulcorner$}\hbox{$\kappa$}\raise
0.0pt\hbox{$\urcorner$}\rightarrow\bot\hfill
2,(\scaleobj{0.9}{\mathrm{T}}\text{\sc l})$

  4. 4.

$\scaleobj{0.9}{\mathrm{T}}\raise 0.0pt\hbox{$\ulcorner$}\hbox{$\kappa$}\raise
0.0pt\hbox{$\urcorner$},\scaleobj{0.9}{\mathrm{T}}\raise
0.0pt\hbox{$\ulcorner$}\hbox{$\kappa$}\raise
0.0pt\hbox{$\urcorner$}\vDash\bot\hfill 3,{\sf Ded}$

  5. 5.

$\scaleobj{0.9}{\mathrm{T}}\raise 0.0pt\hbox{$\ulcorner$}\hbox{$\kappa$}\raise
0.0pt\hbox{$\urcorner$}\vDash\bot\hfill 4,({\sf Contr})$

  6. 6.

$\vDash\scaleobj{0.9}{\mathrm{T}}\raise
0.0pt\hbox{$\ulcorner$}\hbox{$\kappa$}\raise
0.0pt\hbox{$\urcorner$}\rightarrow\bot\hfill 5,{\sf Ded}$

  7. 7.

$\vDash\kappa\hfill 6,\text{ Def. }\kappa$

  8. 8.

$\vDash\scaleobj{0.9}{\mathrm{T}}\raise
0.0pt\hbox{$\ulcorner$}\hbox{$\kappa$}\raise 0.0pt\hbox{$\urcorner$}\hfill
7,(\scaleobj{0.9}{\mathrm{T}}\text{\sc r})$

  9. 9.

$\vDash\bot\hfill 5,8,({\sf Trans})$

∎

Proposition 1 is deeply troubling. It shows that we cannot have it both ways:
either $\varphi$ and $\scaleobj{0.9}{\mathrm{T}}\raise
0.0pt\hbox{$\ulcorner$}\hbox{$\varphi$}\raise 0.0pt\hbox{$\urcorner$}$ must
somewhat come apart or we must accept a conditional that, at least from a
logical perspective, is inadequate and arguably not up for the task of
capturing conditional reasoning. Some theorist have taken Proposition 1 to
show that we should adopt so-called substructural logics, that is, logics with
consequence relations that fail to meet one of $({\sf Ref})$, $({\sf Contr})$,
or $({\sf Trans})$.888The structural rule of $({\sf Weak})$ was not used in
the proof of Proposition 1. As interesting as these approaches may be, we
shall not pursue it here, but take the structural rules to be non-
negotiable.999Readers interested in substructural logics are referred to
Paoli, (2013); Restall, (2002), and to Beall et al., (2018) for an
introduction to substructural “solutions” to the semantic paradoxes.

A number of truth-disquotationalist have argued that truth is a logical notion
that is fully characterized by the truth introduction and elimination
rules.101010See, e.g., Beall, (2021) for a pronounced formulations of this
view. Taking this idea seriously we can, somewhat tongue in cheek, give the
following pointed formulation of the dilemma posed to us by Curry’s paradox:
either truth is not a logical notion or there is no logically adequate
conditional. Perhaps surprisingly, in light of this dilemma many philosophers
have come down in favor of the logicality of truth and have sacrificed salient
properties of the conditional, that is, in most cases the rule of conditional
introduction.111111These include Field, (2008, 2016, 2021), Yablo, (2013),
Rossi, (2016); Iacona and Rossi, (2024), Beall, (2009), and Bacon, (2013) to
name only few. In this paper, we investigate the alternative option and
develop a conditional for which the deduction theorem holds relative to a
suitable non-naive consequence relation.

In the discussion of Curry’s paradox we left open how we are to understand the
consequence relation of the given logic $L$. We now focus on the standard
semantic definition of the consequence relation as truth preservation relative
to a suitable model class, i.e., $\Gamma\vDash_{L}\Delta$ iff for all
$M\in\mathfrak{M}_{L}:$

$\displaystyle\text{ if }M\Vdash\bigwedge\Gamma,$ $\displaystyle\text{ then
}M\Vdash\bigvee\Delta.$

In light of Curry’s paradox we can conclude that we can only find an adequate
conditional for $\vDash_{L}$ iff the model class $\mathfrak{M}_{L}$ also
contains non-naive truth models, that is, models for which there are sentences
$\psi$ such that $\psi$ and $\scaleobj{0.9}{\mathrm{T}}\raise
0.02205pt\hbox{$\ulcorner$}\hbox{$\psi$}\raise 0.02205pt\hbox{$\urcorner$}$
are not semantically equivalent. However, Curry’s paradox does not rule out
the possibility that $\mathfrak{M}_{L}$ contains some naive truth models.
Indeed, we shall show how to construct a naive truth model $M$ with an
associated model class $\mathfrak{M}$ for a logic with an adequate conditional
such that $M\in\mathfrak{M}$ and in all non-naive models
$M^{\prime}\in\mathfrak{M}$ the truth predicate will still satisfy desirable
truth-like properties.

The principal upshot of Curry’s paradox then is that the conditional cannot
express truth preservation relative to a collection of naive truth models,
which, in turn, rules out the possibility of a transparent truth predicate. As
a direct consequence we also know that the truth conditions of a sentence
$\varphi\rightarrow\psi$ in a model $M$ cannot be given in terms of truth
preservation in $M$. More generally, due to Curry’s paradox the truth
conditions of $\varphi\rightarrow\psi$ cannot be specified solely in terms of
the truth values of $\varphi$ and $\psi$ in $M$, for otherwise the conditional
would constitute a transparent context in a naive truth model, which, again,
is ruled out by Curry’s paradox. The upshot then is that the truth conditions
of a conditional $\varphi\rightarrow\psi$ in a model $M$ need to resort to a
collection of models associated with the given model $M$ and this collection
needs to contain non-naive models. But how should the collection of models
relevant for the evaluation of a conditional in $M$, i.e., the models
associated with $M$ be determined?

###  3.1 Towards Truth-conditions

Up to this point we have not focused on a particular logic but from now we
focus on strong Kleene logic ${\sf K3}$. Our ultimate aim is to introduce an
adequate conditional to ${\sf K3}$ and then show how to construct a naive
truth model in the resulting logic/framework. ${\sf K3}$ allows for truth
value gaps but rules out truth value gluts. This shows in the fact that
explosion is a valid rule of inference for ${\sf K3}$.121212A sequent calculus
for ${\sf K3}$ is given in the appendix to this paper. In light of our
discussion at the end of the previous section, the key question is to
determine the class of models associated with a given partial model $M$. An
immediate answer is to single out those models that are semantically
compatible with $M$. A model $M^{\prime}$ is semantically compatible with $M$,
if $M^{\prime}$ is not in conflict with the semantic information provided by
$M$, that is, if a formula $\varphi$ receives a classical truth value in $M$,
then it will receive the same truth value in $M^{\prime}$. Put differently a
model is compatible with $M$ iff it provides at least as much semantic
information as $M$ and disagrees with $M$ at most on those sentences that do
not receive a classical truth value in $M$ . Let us write $M\leq M^{\prime}$
to indicate that $M^{\prime}$ is semantically compatible with $M$—notice that
semantic compatibility is not a symmetric notion. Then at a first brush we
propose the following truth conditions for a conditional
$\varphi\rightarrow\psi$ in a model $M$:

$\displaystyle M\vDash\varphi\rightarrow\psi$ $\displaystyle\,\,{\rm
iff}\text{ for all }M^{\prime}\text{ s.~{}t. }M\geq M^{\prime}:\text{ if
}M^{\prime}\Vdash\varphi,\text{ then }M^{\prime}\Vdash\psi.$

Our talk of semantic compatibility may remind readers of supervaluational
semantics where, indeed, one quantifies over models that are semantically
compatibility with the given partial model. However, in orthodox
supervaluational semantics one quantifies over classical models, whereas we
think of semantic compatibility as a relation between partial models. In what
is to come we shall generalize the above truth conditons, that is, strong
Kleene supervaluation in the following two ways:

  1. 1.

we do not quantify over all semantically compatible models but only those of a
given set $X$ of models of the language;

  2. 2.

we generalize the idea of semantic compatibility to that of an admissibility
relation on the set $X$.

We are ultimately led to so-called strong Kleene supervaluation structures
(cf. Section 4) and the truth conditions for the conditional in such
structures look very much like the truth conditions of an intuitionistic
conditional in a Kripke frame for intuitionistic logic. Indeed, our proposal
can be seen as introducing an intuitionistic conditional to strong Kleene
logic ${\sf K3}$. It turns out that the resulting logic is known as three-
valued Nelson logic N3 (see, e.g. Thomason,, 1969; Wansing,, 1991, 2001) and
enjoys a soundness and completeness result relative to a constant domain
Kripke semantics for intuitionistic logic (see Thomason,, 1969). Importantly,
the deduction theorem holds for the ${\sf N3}$-conditional. Of course, it then
remains to show how to find naive truth models. Before we turn to introducing
strong Kleene supervaluation in earnest, we repeat a point made in the
Introduction to this paper, namely, that strong Kleene supervaluation serves
as a general framework for obtaining adequate truth conditions for non-
monotone notions in the presence of semantic indeterminacy. For example, as we
discuss elsewhere, it can be applied to obtain a neat semantics for $\langle
1,1\rangle$-quantifiers in partial semantics. Yet, as mentioned, in this paper
we focus our attention on conditionals.

##  4 Strong Kleene Supervaluation

In this section we introduce the formal framework. We start by introducing the
formal syntax of our language $\mathcal{L}$. We assume the language to consist
of

  * •

a denumerable set of variables $v_{0},v_{1},v_{2}\ldots$ denoted ${\sf
Var}_{\mathcal{L}}$;

  * •

a denumerable set of individual constants $c_{0},c_{1},c_{2},\ldots$ denoted
${\sf Cons}_{\mathcal{L}}$;

  * •

a denumerable set of function symbols
$f^{n_{0}}_{0},f^{n_{1}}_{1},f^{n_{2}}_{2},\ldots$ with denoted ${\sf
Funct}_{\mathcal{L}}$ such that $n_{j}$ denotes the arity of $f_{j}$ with
$n_{j},n,j\in\omega$;

  * •

a denumerable set of function symbols
$P^{n_{0}}_{0},P^{n_{1}}_{1},P^{n_{2}}_{2},\ldots$ with denoted ${\sf
Pred}_{\mathcal{L}}$ such that $n_{j}$ denotes the arity of $P_{j}$ with
$n_{j},n,j\in\omega$;

  * •

the identity symbol $=$;

  * •

the logical vocabulary $\neg,\wedge,\rightarrow$.

######  Definition 2 (Term, Formula).

The set of well-formed terms ${\sf Term}_{\mathcal{L}}$ is specified as
follows

$t_{1},\ldots,t_{n}::=v_{j}\,|\,c_{k}\,|\,f_{m}^{n}(t_{1},\ldots,t_{n})$

with $n,j,k,m\in\omega$, $v_{j}\in{\sf Var}_{\mathcal{L}}$, $c_{k}\in{\sf
Cons}_{\mathcal{L}}$, and $f^{n}_{m}\in{\sf Funct}_{\mathcal{L}}$. The set of
well-formed formulas ${\sf Form}_{\mathcal{L}}$ is specified as follows:

$\varphi::=(s=t)\,|\,P_{j}^{m}(t_{1},\ldots,t_{m})\,|\,\neg\varphi\,|\,\varphi\wedge\varphi\,|\,\varphi\rightarrow\varphi\,|\,\forall
x\varphi$

with $j,m\in\omega$, $s,t,t_{1},\ldots,t_{m}\in{\sf Term}_{\mathcal{L}}$, and
$x\in{\sf Var}_{\mathcal{L}}$.

We shall make use of further logical symbols which, however, we conceive of
mere notational abbreviations. In particular, we set

$\displaystyle\varphi\vee\psi:=\neg(\neg\varphi\wedge\neg\psi)$
$\displaystyle\exists x\varphi:=\neg\forall x\neg\varphi.$

With the syntax in place we start introducing the semantics and the notion of
a (strong Kleene) supervaluation structure which will consist of a non-empty
domain and set $X$ of interpretation functions ordered by an admissibility
relation $H$. First, we specify the notion of a (partial) interpretation on a
non-empty domain.

######  Definition 3 (Interpretation, Ordering).

Let $D\neq\emptyset$. A function $I$ for some language $\mathcal{L}$ is called
a ${\sf k}$-interpretation iff

  1. (i)

$I(t)\in D$ for all $t\in{\sf Const}_{\mathcal{L}}$;

  2. (ii)

$I(f^{n})\in^{D^{n}}\\!\\!D$ for all $f\in{\sf Funct}_{\mathcal{L}}$;

  3. (iii)

$I(P^{n})\in\mathcal{P}(D^{n})\times\mathcal{P}(D^{n})$;

  4. (iv)

$I^{+}(P^{n})\cap I^{-}(P)=\emptyset$.

The set of all interpretations over $D$ is denoted by ${\sf Int}_{D}$. For
$I,J\in{\sf Int}_{D}$ we say $I\leq J$ iff $I^{+}(P)\subseteq J^{+}(P)$ and
$I^{-}(P)\subseteq J^{-}(P)$ for all $P\in{\sf Pred}$.

######  Definition 4 (Supervaluation structure).

A tuple $\mathfrak{M}=(D,X,H)$ is called a supervaluation structure iff

  1. (i)

$D$ is a nonempty domain of individuals;

  2. (ii)

$X\subseteq{\sf Int}_{D}$ such that for all $J,I\in X$

     * •

$J(c)=I(c)$ for $c\in{\sf Const}_{\mathcal{L}}$;

     * •

$J(f)(d_{1},\ldots,d_{n})=I(f)(d_{1},\ldots,d_{n})$ for $f\in{\sf
Func}_{\mathcal{L}}$, $d_{1},\ldots,d_{n}\in D$, and $n\in\omega$;

  3. (iii)

$H\subseteq X\times X$ such that

     * •

$H$ is reflexive and transitive;

     * •

if $(x,y)\in H$, then $x\leq y$.

It remains explaining the notion of truth in a supervalutation structure
relative to a given interpretation. To this effect we need to appeal to
variable assignments.

######  Definition 5 (Assignment).

Let $D\neq\emptyset$. An assignment on $D$ is a function $\beta:{\sf
Var}\rightarrow D$ that assigns to the variables of the language objects in
$D$. Let $\beta$ be an assignment and $d\in D$. Then $\beta(x:d)$ such that

$\displaystyle\beta(x:d)(v):=\begin{cases}d,&\text{ if }x\doteq v\\\
\beta(v),&\text{otherwise}.\end{cases}$

is an $x$-variant of $\beta$.

We now specify the denotation of a term in a supervaluation structure relative
to an assignment.

######  Definition 6 (Denotation in a supervaluation structure).

Let $\mathfrak{M}=(D,X,H)$ be a supervaluation structure and $\beta$ an
assignment on $F$. Then for $J\in X$ the denotation of a term $t$ under the
assignment $\beta$ is defined as follows:

$\displaystyle J^{\beta}(t):=\begin{cases}\beta(t),&\text{ if }t\in{\sf
Var}_{\mathcal{L}}\\\ J(t),&\text{ if }t\in{\sf Const}_{\mathcal{L}}\\\
J(f^{n})(J^{\beta}(t_{1}),\ldots,J^{\beta}(t_{n})),&\text{ if }t\doteq
f^{n}(t_{1},\ldots,t_{n}).\end{cases}$

Notice that $J^{\beta}(t)=I^{\beta}(t)$ for all $J,I\in X$ and variable
assignments $\beta$.

######  Definition 7 (Truth in supervaluation structure).

Let $\mathfrak{M}=(D,X,H)$ be a supervaluation structure with $J\in X$ and
$\beta$ a variable assignment. Then a formula $\varphi$ is true in
$\mathfrak{M}$ at $J$ under assignment $\beta$
($\mathfrak{M},J\Vdash\varphi[\beta]$) iff

$\displaystyle J^{\beta}(t_{1}),\ldots,J^{\beta)}(t_{n})\rangle\in
J^{+}(P^{n}),$ $\displaystyle\text{ if
}\varphi\doteq(P^{n}t_{1},\ldots,t_{n});$ $\displaystyle
J^{\beta}(t_{1}),\ldots,J^{\beta}(t_{n})\rangle\in J^{-}(P^{n}),$
$\displaystyle\text{ if }\varphi\doteq(\neg P^{n}t_{1},\ldots,t_{n});$
$\displaystyle J^{\beta}(s)=J^{\beta}(t),$ $\displaystyle\text{ if
}\varphi\doteq(s=t);$ $\displaystyle J^{\beta}(s)\neq J^{\beta}(t),$
$\displaystyle\text{ if }\varphi\doteq(\neg(s=t));$
$\displaystyle\mathfrak{M},J\Vdash\psi[\beta]$ $\displaystyle\text{ if
}\varphi\doteq\neg\neg\psi;$
$\displaystyle\mathfrak{M},J\Vdash\psi[\beta]\,\&\,\mathfrak{M},J\Vdash\chi[\beta],$
$\displaystyle\text{ if }\varphi\doteq(\psi\wedge\chi);$
$\displaystyle\mathfrak{M},J\Vdash\neg\psi[\beta]\text{ or
}\mathfrak{M},J\Vdash\neg\chi[\beta],$ $\displaystyle\text{ if
}\varphi\doteq(\neg(\psi\wedge\chi));$ $\displaystyle\forall
J^{\prime}((J,J^{\prime})\in
H\Rightarrow\mathfrak{M},J\not\Vdash\psi[\beta]\text{ or
}\mathfrak{M},J\Vdash\chi[\beta],$ $\displaystyle\text{ if
}\varphi\doteq(\psi\rightarrow\chi));$
$\displaystyle\mathfrak{M},J\Vdash\psi[\beta]\,\&\,\mathfrak{M},J\Vdash\neg\chi[\beta],$
$\displaystyle\text{ if }\varphi\doteq\neg(\psi\rightarrow\chi);$
$\displaystyle\mathfrak{M},J\Vdash\psi[\beta(x:d)]\text{ for all }d\in D,$
$\displaystyle\text{ if }\varphi\doteq\forall x\psi;$
$\displaystyle\mathfrak{M},J\Vdash\neg\psi[\beta(x:d)]\text{ for some }d\in
D,$ $\displaystyle\text{ if }\varphi\doteq\neg\forall x\psi.$

We say that $\varphi$ is true at $J$ $(\mathfrak{M},J\Vdash\varphi)$ iff
$\mathfrak{M},J\Vdash\varphi[\beta]$ for all assignments $\beta$. Finally, a
formula is said to be true in $\mathfrak{M}$ ($\mathfrak{M}\Vdash\varphi$) iff
$\mathfrak{M},J\Vdash\varphi$ for all $J\in X$.

As mentioned previously truth in a supervaluation structure as specified in
Definition 7 coincides with truth in Kripke frames for constant domain Nelson
logic ${\sf N3}$ (Thomason,, 1969; Wansing,, 1991) with classical
identity.131313For a presentation of the various logics appealed to in this
paper in form of sequent calculi we refer the reader to Appendix A. As a
guideline we point out that if $\Gamma\Rightarrow\Delta$ is a valid ${\sf
K3}$-sequent, then $\bigwedge\Gamma\rightarrow\bigvee\Delta$ is an ${\sf
N3}$-logical truth, where $\Gamma$ and $\Delta$ are finite sets of formulae.

Thus, unsurprisingly, we obtain the following monotonicity/persistence
property.

######  Lemma 8.

Let $\mathfrak{M}=(D,X,H)$ be a supervaluation structure, $\beta$ a variable
assignment, and $J,J^{\prime}\in X$ such that $(J,J^{\prime})\in H$.Then for
all $\varphi\in\mathcal{L}$

$\displaystyle\text{if }\mathfrak{M},J\Vdash\varphi[\beta],\text{ then
}\mathfrak{M},J^{\prime}\Vdash\varphi[\beta].$

###### Proof.

The claim is established by an induction on the positive complexity of
$\varphi$. The only interesting case is that of the conditional. The claim
follows because due to the transitivity of $H$ we obtain:

$\\{I\in X\,|\,(J^{\prime},I)\in H\\}\subseteq\\{I\in X\,|\,(J,I)\in H\\}.$

∎

The main difference between the Nelson logic and the present framework is that
in our case interpretations serve, in some sense, a double purpose: they serve
as index/parameter in the semantic explanation whilst, at the same time,
providing the interpretation of the descriptive vocabulary. In contrast, in
Kripke frames these two roles are kept apart. Worlds serve as parameter in the
semantic explanation, whilst an interpretation on the frame determines the
extension of the descriptive vocabulary relative to a world. Indeed, this
seems to be a good general characterization of the conceptual difference
between many forms of supervaluation and possible world semantics.

Nelson logic can be understood as extending strong Kleene logic ${\sf K3}$ by
an intuitionistic conditional and fittingly the Nelson conditional does not
allow for contraposition.141414In ${\sf K3}$ the sequent
$\varphi,\neg\varphi\Rightarrow\bot$ is provable, but it’s contraposition
$\top\Rightarrow\varphi,\neg\varphi$ is not ${\sf K3}$-valid. The failure of
contraposition is not a mere accident but a consequence of the extensional
(local) refutation condition of the conditional and double negation
elimination: if contraposition is introduced to the framework then a formula
$\varphi\rightarrow\psi$ would be equivalent to $\neg\varphi\vee\psi$ which
would collapse ${\sf N3}$ into classical logic:

  1. 1.

$\neg(\varphi\rightarrow\varphi)\leftrightarrow\varphi\wedge\neg\varphi$ by
truth conditions

  2. 2.

$\neg\neg(\varphi\rightarrow\varphi)\leftrightarrow\neg(\varphi\wedge\neg\varphi)$
contraposition

  3. 3.

$\varphi\rightarrow\varphi\leftrightarrow\varphi\vee\neg\varphi$ double
negation intro/elim; De Morgan

  4. 4.

$\varphi\vee\neg\varphi$

The last inference follows from the fact that $\varphi\rightarrow\varphi$ is
an ${\sf N3}$-logical truth and modus ponens. However, from the perspective of
${\sf K3}$ the failure of contraposition seems the right outcome, as the
consequence relation of ${\sf K3}$ does not admit contraposition. However,
since the deduction theorem holds for the ${\sf N3}$-conditional the above
argument also shows that ${\sf N3}$ and thus strong Kleene supervaluation are
hyperintensional and, indeed, not self-extensional in the sense of Odintsov
and Wansing, (2021): despite the fact that $\neg(\varphi\rightarrow\psi)$ and
$\varphi\wedge\neg\psi$ are interderivable in ${\sf N3}$ they cannot be
substituted salva veritate in every context. From our perspective the moral of
this observation is that strong Kleene supervaluation forces us to take
bilateralism seriously: the proposition expressed by a sentence is a tuple
consisting of the truth- and the falsemakers of the sentence, where the set of
falsemakers cannot be retrieved from the set of truthmakers of a sentence.
From this bilateral perspective ${\sf N3}$ ceases to be hyperintensional in
the following sense: if two sentences express the same semantic content they
can be substituted salva veritate in every context in ${\sf N3}$. Adopting the
aforementioned perspective we propose to revise Odintsov and Wansing’s
criterion for self-extensionality: a bilateral logic is self-extensional iff
for formulas $\varphi,\psi,$ and $\theta(p)$:

$\displaystyle\text{if }\varphi\dashv\vdash\psi\text{ and
}\neg\varphi\dashv\vdash\neg\psi,\text{ then
}\theta(\varphi/p)\dashv\vdash(\psi/p.)$

According to this revised criterion ${\sf N3}$ (and thus strong Kleene
supervaluation) are self-extensional and in this sense not
hyperintensional.151515Unsurprisingly, ${\sf N3}$ will still be
hyperintensional in the sense of Cresswell, (1975).

To be sure, the asymmetry between the global/intensional truth conditions of
the conditional and it’s local/extensional falsity condition seem to be a
downside to the otherwise attractive ${\sf N3}$-conditional. Is it possible to
give alternative falsity conditions for conditionals, that is, falsity
conditions that are global (extensional) like the truth conditions of the
conditional? One idea that comes to mind, inspired by the falsification
condition of the intuitionistic conditional, takes a conditional to be false
at a given interpretation iff for every interpretation it can “see” an
admissible interpretation which provides a counterexample to the truth of the
conditional:

$\displaystyle\mathfrak{M},J\Vdash\neg(\varphi\rightarrow\psi)[\beta]$
$\displaystyle\text{ iff }\forall J^{\prime}(J,J)\in H\Rightarrow\exists
J^{\prime\prime}((J^{\prime},J^{\prime\prime})\in
H\,\&\,\mathfrak{M},J^{\prime\prime}\Vdash\varphi\,\&\,\mathfrak{M},J^{\prime\prime}\Vdash\neg\psi)).$

If such falsity conditions were adopted one might even try to work over
symmetric strong Kleene logic ${\sf KS3}$ and obtain a conditional that
respects contraposition.171717${\sf KS3}$ is our label for the logic thaat
result from intersecting ${\sf K3}$ and ${\sf LP}$. See Nicolai and Stern,
(2021) for a ue of this label and a presentation of the semantics and proof-
theory of the logic. Presumably the resulting logic would amount to a partial
counterpart to Leitgeb’s (2019) paraconsistent logic HYPE, i.e., the logic
${\sf QN}^{*}$ (cf. Odintsov and Wansing,, 2021). However, as Nelson logic
${\sf N3}$ is well understood and has many attractive properties we leave
further discussion of alternate falsification conditions for the conditional
and the resulting logics for another occasion.181818We also note that at least
upon first glance it seems that the lower bound of the recursion theoretic
complexity of the interpretation of the naive truth predicate would increase
from $\Pi^{1}_{1}$ to $\Pi^{1}_{2}$ on the revised falsity condition of the
conditional.

This completes the introduction of strong Kleene supervaluation structures for
an arbitrary first-order language. The next step is to show how to extend the
framework to a langauge with a self-applicable truth predicate and,
ultimately, to construct naive truth models. From Curry’s paradox we know that
within a strong Kleene supervaluation structure there must be interpretations
at which the truth predicate is not naive, yet we can find attractive
interpretations at which the truth predicate is naive. In sum, truth will be
locally but not globally naive. Moreover, in addition we can make sure that
the truth predicate has certain desirable properties globally, and on our view
we thus arrive at an attractive truth theory.

##  5 Truth Structures

Having introduced the general framework of strong Kleene supervaluation we now
start constructing supervaluation structures, which, relative to a set of
designated interpretations, yield attractive interpretations of a language
with an objectlinguistic, self-applicable truth predicate. We continue working
with a first-order language $\mathcal{L}$, but in contrast to Section 4 we
need to make some assumptions regarding the interpretation of the language. As
we shall require names for the expressions of the language and also want to be
able to talk about substitutions of expressions, we need our language to
comprise syntactic vocabulary. We also need to make sure that this vocabulary
is interpreted in the intended way. To this effect we assume that the language
$\mathcal{L}$ and an associated supervaluation structure
$\mathfrak{M}=(D,X,H)$ satisfy the following requirements:

  * •

$\mathcal{L}$ extends the language of some syntax theory $\mathcal{L}_{S}$,
e.g., the language of arithmetic;

  * •

for each $d\in D$ there exists a $c\in{\sf Const}_{\mathcal{L}}$ such that
$J(c)=d$ and for all $\varphi\in\mathcal{L}_{S}$; $J\in X$ and assignment
$\beta$:

$\text{for all }c\in{\sf
Const}_{\mathcal{L}}(\mathfrak{M},J\Vdash\varphi(c)[\beta])\text{ iff
}\mathfrak{M},J\Vdash\forall x\varphi(x/c)[\beta];$

  * •

for all $\varphi\in\mathcal{L}_{S}$; $J,J^{\prime}\in X$ and assignments
$\beta$.

    1. (i)

$\mathfrak{M},J\Vdash\varphi[\beta]\text{ iff
}\mathfrak{M},J^{\prime}\Vdash\varphi[\beta]$

    2. (ii)

$\mathfrak{M},J\Vdash\varphi\vee\neg\varphi[\beta]$

The language $\mathcal{L}_{\rm T}$ extends $\mathcal{L}$ by introducing an
additional one-place predicate constant, that is, the truth predicate
$\scaleobj{0.9}{\mathrm{T}}$.

Given a supervaluation structure for $\mathcal{L}$ we need to provide an
interpretation for the truth predicate relative to every $J\in X$. To this
effect we introduce valuation functions, that is, functions that assign an
interpretation to the truth predicate relative to an interpretation.

######  Definition 9 (Basic Valuations).

A valuation on a supervaluation structure $\mathfrak{M}=(D,X,H)$ is a function
$f:X\rightarrow\mathcal{P}({\sf Sent}_{\mathcal{L}_{\rm T}})$. The set of all
valuation on $\mathfrak{M}$ is denoted by ${\sf Val}_{\mathfrak{M}}$. A
valuation function is basic for $\mathfrak{M}$ iff

  1. (i)

$f$ is consistent, that is, $f(J)\cap\overline{f(J)}=\emptyset$ for all $J\in
X$ where

$\overline{f(J)}:=\\{\varphi\,|\,\neg\varphi\in f(J)\\};$

  2. (ii)

for all $J\in X$ and $\varphi\in\mathcal{L}$:

$\displaystyle\text{if }\varphi\in f(J),$ $\displaystyle\text{ then
}\mathfrak{M},J\Vdash\varphi$

  3. (iii)

for all $J,J^{\prime}\in X$, if $(J,J^{\prime})\in H$, then $f(J)\subseteq
f(J^{\prime})$.

The set of basic valuation functions for a given model $\mathfrak{M}$ is
denoted by $\mathcal{B}_{\mathfrak{M}}$.

In short, a valuation is deemed basic, if (i) it is consistent, (ii) does not
declare $\mathcal{L}$-sentences true at an interpretation, unless they are
made true in the underlying supervaluation structure at that interpretation,
and if (iii) the valuation respects the ordering of the interpretations in
$X$. The next task is to define an ordering on the basic valuations and to
introduce the idea of the admissibility condition for basic valuations

######  Definition 10 (Ordering, Admissibility condition).

Let $\leq\subseteq{\sf Val}_{\mathfrak{M}}\times{\sf Val}_{\mathfrak{M}}$ be
an ordering on ${\sf Val}_{\mathfrak{M}}$ defined as follows:

$\displaystyle f\leq g$ $\displaystyle:\leftrightarrow\forall J\in
X(f(J)\subseteq g(J))$

An admissibility condition is a function $\Phi:{\sf
Val}_{\mathfrak{M}}\rightarrow\mathcal{P}(\mathcal{B}_{\mathfrak{M}})$ that
yields a set of basic valuation functions for every member of ${\sf
Val}_{\mathfrak{M}}$ such that

  * •

$\Phi(f)=\emptyset$, if $f\not\in\mathcal{B}_{\mathfrak{M}}$;

  * •

if $f\leq g$ and $f\in\mathcal{B}_{\mathfrak{M}}$, then
$\Phi(g)\subseteq\Phi(f)$;

  * •

if $g\in\Phi(f)$, then $f\leq g;$

for all $f,g\in{\sf Val}_{\mathfrak{M}}$. For all
$f,g\in\mathcal{B}_{\mathfrak{M}}$ the partial order
$\leq_{\Phi}\subset\mathcal{B}_{\mathfrak{M}}\times\mathcal{B}_{\mathfrak{M}}$
is defined as follows.

$\displaystyle f\leq_{\Phi}g$ $\displaystyle:\leftrightarrow g\in\Phi(f).$

Notice that by definition $\leq_{\Phi}$ is transitive, i.e., for $f,g,h\in{\sf
Val}_{\mathfrak{M}}$, if $f\leq_{\Phi}g$ and $g\leq_{\Phi}h$, then
$f\leq_{\Phi}g$.

Together, an interpretation for the language $\mathcal{L}$ and a valuation
function yield an interpretation function for the language $\mathcal{L}_{\rm
T}$.

######  Definition 11 (Truth interpretation, Admissibility relation).

Let $J\in X$ and $f\in\mathcal{B}_{\mathfrak{M}}$, then $J_{f}$ is a called a
truth-interpretation for the language $\mathcal{L}_{\rm T}$. We have

$\displaystyle J_{f}(P):=\begin{cases}f(J),&\text{ if
}P\doteq\scaleobj{0.9}{\mathrm{T}};\\\ J(P),&\text{ otherwise}.\end{cases}$

We may then define an admissiblity relation on
$X\times\mathcal{B}_{\mathfrak{M}}$ for truth interpretations on the basis of
$H$ and $\leq_{\Phi}$, that is, for all $I,J\in X$ and $f,g\in{\sf
Val}_{\mathfrak{M}}$

$\displaystyle(I_{f},J_{g})\in H_{\Phi}$
$\displaystyle:\leftrightarrow(I,J)\in H\,\&\,f\leq_{\Phi}g.$

Having defined an admissibility relation for truth interpretations we can
introduce the notion of a truth structure, that is, a supervaluation structure
for the language $\mathcal{L}_{\rm T}$.

######  Definition 12 (Truth Structure, Grounded Truth Structure).

Let $\mathfrak{M}=(D,X,H)$ be a supervaluation structure for $\mathcal{L}$.
Then the tuple $(D,X\times Y,H_{\Phi})$ with
$Y\subseteq\mathcal{B}_{\mathfrak{M}}$ and where $H_{\Phi}$ restricted to
$X\times Y$ is called a truth structure over $\mathfrak{M}$. If
$Y\subseteq\mathcal{B}_{\mathfrak{M}}$ has a $\leq$-minimal valuation
function, that is, an $f$ in $Y$ such that $f\leq g$ for all $g\in Y$ and
$Y\cap\Phi(f)\neq\emptyset$, then $(D,X\times Y,H_{\Phi})$ is called a
grounded truth structure and $Y$ is called a grounded set of valuations. The
set of all grounded sets of valuations is denoted by ${\sf
Adm}_{\mathfrak{M}}$.

If $\leq_{\Phi}$ is reflexive on $Y$ for a given truth structure/grounded
truth structure $\mathfrak{M}_{\scaleobj{0.9}{\mathrm{T}}}$, then
$\mathfrak{M}_{\scaleobj{0.9}{\mathrm{T}}}$ is a supervaluation structure for
the language $\mathcal{L}_{\rm T}$.191919So, strictly speaking, truth
structures are not supervaluation structure for $\mathcal{L}_{\rm T}$.
However, as we discuss in Section 6, given a reflexive truth-interpretation
$J_{f}$, one can obtain a supervaluation structure for $\mathcal{L}_{\rm T}$
by constructing the $J_{f}$-generated substructure of the given truth
structure. Yet, we are not guaranteed that the predicate
‘$\scaleobj{0.9}{\mathrm{T}}$’ has desirable, truth-like properties in
$\mathfrak{M}_{\scaleobj{0.9}{\mathrm{T}}}$: a truth structure places only
minimal constraints on the interpretation of the truth predicate. In
particular, we are not guaranteed that there is a truth interpretation $J_{f}$
such that for all $\mathcal{L}_{\rm T}$-sentence $\varphi$:

$\displaystyle\mathfrak{M}_{\scaleobj{0.9}{\mathrm{T}}},J_{f}\Vdash\varphi$
$\displaystyle\text{ iff
}\mathfrak{M}_{\scaleobj{0.9}{\mathrm{T}}},J_{f}\Vdash\scaleobj{0.9}{\mathrm{T}}\raise
0.0pt\hbox{$\ulcorner$}\hbox{$\varphi$}\raise 0.0pt\hbox{$\urcorner$}.$

To construct (reflexive) truth structures for which there is a designated set
of truth interpretations that satisfy the modified version of convention
$\scaleobj{0.9}{\mathrm{T}}$ will be the goal of the next section. However,
before we turn to this task we prove two general monotonicity properties of
truth structures.

######  Lemma 13 ($J$-monotonicity).

Let $(D,X\times Y,H_{\Phi})$ be a truth structure. Then for $J,J^{\prime}\in
X$, $f\in Y$, assignments $\beta$ and all $\varphi\in\mathcal{L}_{\rm T}$: if
$(J,J^{\prime})\in H$, then

$\displaystyle{\rm if}\,((D,X\times Y,H_{\Phi}),J_{f}\Vdash\varphi[\beta]$
$\displaystyle\,{\rm then}\,(D,X\times
Y,H_{\Phi}),J^{\prime}_{f}\Vdash\varphi[\beta]).$

###### Proof.

The only interesting case is that of the conditional. Notice that there are
two cases: (i) there is a $g\in Y$ with $f\leq_{\Phi}g$ and (ii) there exists
no $g\in Y$ such that $f\leq_{\Phi}g$. In the latter case all conditionals are
true at both $J_{f}$ and $J_{f}^{\prime}$ as no truth-interpretation will be
accessible from either $J_{f}$ or $J_{f}^{\prime}$ (by definition of
$H_{\Phi}$) and the claim holds trivially. For case (i), the reasoning is as
for Lemma 8 since $H_{\Phi}$ is transitive. ∎

######  Lemma 14 ($(Y,f)$-monotonicity).

Let $Z\subseteq Y\subseteq\mathcal{B}_{\mathfrak{M}}$ and $f\in Y$ and $g\in
Z$. Then for all $J\in X$ assignments $\beta$ and $\varphi\in\mathcal{L}_{\rm
T}$: if $f\leq g$, then

$\displaystyle{\rm if}\,(D,X\times Y,H_{\Phi}),J_{f},\Vdash\varphi[\beta],$
$\displaystyle\,{\rm then}\,(F,X\times Z,H_{\Phi}),J_{g}\Vdash\varphi[\beta].$

###### Proof.

Again the only interesting case is the conditional, but the claim follows
since by Definition 11, $f\leq g$ implies $\Phi(g)\subseteq\Phi(f)$. Then by
transitivity $(J_{f},J^{\prime}_{k})\in H_{\Phi}$ for all $J^{\prime}\in X$
and $k\in Y$, if $(J_{g},J^{\prime}_{k})\in H_{\Phi}$.∎

##  6 Kripkean Truth Structures

Let $\mathfrak{M}=(D,X,H)$ be a supervaluation structure of $\mathcal{L}$. We
now wish to construct a grounded truth structure $(D,X\times Y_{f},H_{\Phi})$
($f$ is the minimal element of $Y$) such that for all $J\in X$ and all
$\mathcal{L}_{\rm T}$-sentences:

$\displaystyle(D,X\times Y_{f},H_{\Phi}),J_{f}\Vdash\varphi$
$\displaystyle\,\,{\rm iff}\,\,(D,X\times
Y_{f},H_{\Phi}),J_{f}\Vdash\scaleobj{0.9}{\mathrm{T}}\raise
0.0pt\hbox{$\ulcorner$}\hbox{$\varphi$}\raise 0.0pt\hbox{$\urcorner$}.$

The rough idea is that the truth interpretations $J_{f}$ are models of naive
truth, while the truth interpretation $J_{g}$ with $g>f$ constitute first-
order models of $\mathcal{L}_{\rm T}$ that are compatible with the semantic
information given by $J_{f}$ and thus relevant for evaluating the conditional
connective. However, these models may not be naive truth models.

We start the construction by defining an operation
$\theta^{\Phi}_{\mathfrak{M}}:\mathcal{P}(\mathcal{B}_{\mathfrak{M}})\times\mathcal{B}_{\mathfrak{M}}\rightarrow{\sf
Val}_{\mathfrak{M}}$ which maps valuation functions onto valuation functions
relative to a truth model based on $\mathfrak{M}$. For
$Y\subseteq\mathcal{B}_{\mathfrak{M}}$ and $f\in Y$ the operation $\theta$
yields a new valuation function such that for all $J\in X$

$\displaystyle[\theta^{\Phi}_{\mathfrak{M}}(Y,f)](J)$
$\displaystyle:=\\{\varphi\in{\sf Sent}_{\mathcal{L}_{\rm T}}\,|\,(D,X\times
Y,H_{\Phi}),J_{f}\Vdash\varphi\\}.$

######  Lemma 15.

Let $\mathfrak{M}=(D,X,H)$ be an arbitrary supervaluation structure for
$\mathcal{L}$ and $f\in Y\in\mathcal{P}(\mathcal{B}_{\mathfrak{M}})$. Then
$\theta^{\Phi}_{\mathfrak{M}}$ is well-defined on
$\mathcal{B}_{\mathfrak{M}}$, if $\Phi(f)\cap Y\neq\emptyset$, that is,

$\text{if }f\in Y\,\&\,\Phi(f)\cap Y\neq\emptyset,\text{ then
}\theta^{\Phi}_{\mathfrak{M}}(Y,f)\in\mathcal{B}_{\mathfrak{M}}.$

###### Proof.

We need to show that $\theta^{\Phi}_{\mathfrak{M}}(Y,f)$ satisfies condition
(i-iii) of Definition 9. Since $f\in Y\subseteq\mathcal{B}_{\mathfrak{M}}$ and
$\Phi(f)\cap Y\neq\emptyset$ (i) and (ii) are immediate. (iii) is a direct
consequence of Lemma 13 ∎

From now on we restrict our attention to grounded truth structures $(D,X\times
Y_{f},H_{\Phi})$, that is, truth structures for which there is $\Phi$-minimal
$f\in Y_{f}$ with $\Phi(f)\cap Y_{f}\neq\emptyset$. By employing the operation
$\theta^{\Phi}_{\mathfrak{M}}$ we define a second operation
$\Theta^{\Phi}_{\mathfrak{M}}:\mathsf{Adm}_{\mathfrak{M}}\rightarrow\mathcal{P}(\mathcal{B}_{\mathfrak{M}})$
on grounded sets of valuations such that for
$Y_{f}\in\mathsf{Adm}_{\mathfrak{M}}$:

$\displaystyle\Theta^{\Phi}_{\mathfrak{M}}(Y_{f})$ $\displaystyle:=\\{g\in
Y_{f}\,|\,\theta^{\Phi}_{\mathfrak{M}}(Y_{f},f)\leq g\\}.$

Intuitively, the operation $\theta^{\Phi}_{\mathfrak{M}}$ will be used to
construct the minimal fixed point, while the operation
$\Theta^{\Phi}_{\mathfrak{M}}$ determines the truth interpretations that are
compatible with the semantic information provided and thus relevant for
evaluating the conditional. We note that $\Theta^{\Phi}_{\mathfrak{M}}$ is
monotone and decreasing, i.e.,

$\displaystyle{\rm Monotonicity:}$ $\displaystyle\text{if }Y_{f}\subseteq
Z_{g}\text{ then
}\Theta^{\Phi}_{\mathfrak{M}}(Y_{f})\subseteq\Theta^{\Phi}_{\mathfrak{M}}(Z_{g}),$
$\displaystyle{\rm Decreasing:}$
$\displaystyle\Theta^{\Phi}_{\mathfrak{M}}(Y_{f})\subseteq Y_{f}$

for $Y_{f},Z_{g}\in{\sf Adm}_{\mathfrak{M}}$.

Notice however that for arbitrary $Y_{f}\in{\sf Adm}_{\mathfrak{M}}$ we are
not guaranteed that $\Theta^{\Phi}_{\mathfrak{M}}(Y_{f})\in{\sf
Adm}_{\mathfrak{M}}$: for one, it is possible that
$\theta^{\Phi}_{\mathfrak{M}}(Y_{f},f)\not\in Y_{f}$ and thus, possibly,
$\Theta^{\Phi}_{\mathfrak{M}}(Y_{f})$ may lack a minimal element. For another
it is possible that $\Theta^{\Phi}_{\mathfrak{M}}(Y_{f})=\emptyset$. In short,
in both cases $\Theta^{\Phi}_{\mathfrak{M}}(Y_{f})$ may not be a grounded
truth set.

Ultimately we want to show the existence of non-trivial fixed points of
$\theta^{\Phi}_{\mathfrak{M}}$ and $\Theta^{\Phi}_{\mathfrak{M}}$ for suitable
admissibility conditions $\Phi$, i.e., $f\in Y_{f}\in{\sf Adm}_{\mathfrak{M}}$
(and thus $Y_{f}\neq\emptyset$) such that

$\displaystyle\theta^{\Phi}_{\mathfrak{M}}(Y_{f},f)=f$
$\displaystyle\Theta^{\Phi}_{\mathfrak{M}}(Y_{f})=Y_{f}.$

In turns out that in the event that $\theta^{\Phi}_{\mathfrak{M}}(Y_{f},f)\in
Y_{f}$, the existence of a fixed point $Y_{f}$ of
$\Theta^{\Phi}_{\mathfrak{M}}$ implies that
$\theta^{\Phi}_{\mathfrak{M}}(Y_{f},f)=f$, i.e., that $f$ is a fixed point of
$\theta$ relative to the grounded truth set $Y_{f}$. More precisely, for all
$J\in X$ $f(J)$ would be the minimal fixed point over $J$ relative to set of
admissible valuation $Y_{f}$

######  Lemma 16.

Let $Y_{f}\in{\sf Adm}_{\mathfrak{M}}$ with
$\theta^{\Phi}_{\mathfrak{M}}(Y_{f},f)\in Y_{f}$. Then

$\displaystyle\Theta^{\Phi}_{\mathfrak{M}}(Y_{f})=Y_{f}\text{ iff
}\theta^{\Phi}_{\mathfrak{M}}(Y_{f},f)=f.$

Our strategy for finding fixed points will be to start from a suitable
grounded truth set with minimal valuation function $f$and then, by
consecutively and simultaneously applying $\theta$ and $\Theta$, construct the
minimal fixed point of $\theta$ whilst simultaneously sieving out unsuitable
valuation functions until we end up with a fixed point of $\Theta$. To spell
out this idea in a rigorous fashion we define iterative applications of
$\Theta^{\Phi}_{\mathfrak{M}}$ to some set $Y_{f}\in{\sf Adm}$ with minimal
valuation function $f$ by transfinite recursion on the ordinals and,
simultaneously, iterative applications of $\theta^{\Phi}_{\mathfrak{M}}$ to
the pair ($Y_{f},f)$:

$\displaystyle\theta^{\alpha}(Y_{f},f)$ $\displaystyle:=\begin{cases}f,&\text{
if }\alpha=0;\\\ \theta(\Theta^{\beta}(Y_{f}),\theta^{\beta}(Y_{f},f)),&\text{
if }\alpha=\beta+1;\\\ \bigcup_{\beta<\alpha}\theta^{\beta}(Y_{f},f),&\text{
if $\alpha$ is limit.}\end{cases}$

$\displaystyle\Theta^{\alpha}(Y_{f})$
$\displaystyle:=\begin{cases}Y_{f},&\text{ if }\alpha=0;\\\
\\{g\in\Theta^{\beta}(Y_{f})\\}\,|\,\theta^{\alpha}(Y_{f},f)\leq g\\},&\text{
if }\alpha=\beta+1\text{ and }\Theta^{\beta}(Y_{f})\in{\sf
Adm}_{\mathfrak{M}};\\\ \emptyset,&\text{ if }\alpha=\beta+1\text{ and
}\Theta^{\beta}(Y_{f})\not\in{\sf Adm}_{\mathfrak{M}};\\\
\\{g\in\bigcap_{\beta<\alpha}\Theta^{\beta}(Y_{f})\,|\,\theta^{\alpha}(Y_{f},f)\leq
g\\},&\text{ if $\alpha$ is limit.}\end{cases}$

Since $\Theta^{\Phi}_{\mathfrak{M}}$ is monotone and decreasing, we can employ
a standard cardinality argument to show that for all $Y_{f}\in{\sf
Adm}_{\mathfrak{M}}$ there is an $\xi$ such that
$\Theta^{\xi}(Y_{f})=\Theta^{\alpha}(Y_{f})$ for all $\alpha\geq\xi$.
Unfortunately, this falls short from establishing the existence of non-trivial
fixed points of $\Theta^{\Phi}_{\mathfrak{M}}$ for it may turn out that
$\Theta^{\xi}(Y_{f})=\emptyset$, which implies $\theta^{\xi+1}(Y_{f},f)={\sf
Sent}$. This can happen because (i) for ordinal $\alpha\leq\xi$ the set
$\Theta^{\alpha}(Y_{f})\neq\emptyset$ lacks a minimal element, i.e.,
$\Theta^{\alpha}(Y_{f})\not\in{\sf Adm}_{\mathfrak{M}}$ or, alternatively, if
for some limit ordinal $\kappa\leq\xi$ there exists no
$g\in\bigcap_{\beta<\kappa}\Theta^{\beta}(Y_{f})$ such that
$\theta^{\kappa}(Y_{f},f)\leq g$. So in both cases the $\theta$-operation has
moved us outside the range of available interpretations of the truth predicate
albeit for different reasons. Indeed this may happen even if
$\Theta^{\xi}(Y_{f})\neq\emptyset$, that is, in case we have found a fixed
point of $\Theta$ but $\theta^{\xi}(Y_{f},f)\not\in\Theta^{\xi}(Y_{f})$. This
shows that finding a fixed point of $\Theta$ does not guaranteed that we have
found a Kripkean truth model for it may be that
$\theta^{\xi}(Y_{f},f)\not\in\Theta^{\xi}(Y_{f})\neq\emptyset$:
$\theta^{\xi}(Y_{f},f)$ may not be a fixed point of $\theta^{\Phi}$. In this
case we would have $\theta(\theta^{\xi}(Y_{f},f),Y_{f})(J)=\emptyset$ for all
$J\in X$ and

$\\{g\in\Theta^{\xi}(Y_{f})\,|\,\theta^{\xi}(Y_{f},f)\leq
g\\}=\\{g\in\Theta^{\xi}(Y_{f})\,|\,\theta^{\xi+1}(Y_{f},f)\leq g\\}$

even though

$\theta^{\xi+1}(Y_{f},f)\neq\theta^{\xi}(Y_{f},f).$

Summing up, in constructing the the fixed point of $\theta$ and $\Theta$ we
need to make sure that the $\theta$ operation does not take us outside the
grounded truth set $Y_{f}$, i.e., one needs to find fixed points $Z_{g}$ such
that $\theta^{\Phi}(Z_{g},g)\in\Theta^{\Phi}(Z_{g})$. Then Lemma 16 guarantees
that we we have found an evaluation function $f$ and suitable grounded truth
set $Y_{f}$ (with $f\in Y_{f}$) such that the two are fixed points of
$\theta^{\Phi}$ and $\Theta^{\Phi}$ respectively.

By imposing suitable condition on the starting set $Y_{f}$ of the definition
and the admissible condition $\Phi$ we can guarantee that the process of
"sieving out" leads to a fixed point of $\Theta^{\Phi}$ with the desired
properties. For example, Conditions (a)-(d) below amount to sufficient
conditions on $Y_{f}$ (and $f$) and $\Phi$ that guarantee that none of the
problems mentioned above can arise:

  1. (a)

$Y_{f}$ is upward closed, i.e., if $f\leq g$ for
$g\in\mathcal{B}_{\mathfrak{M}}$, then $g\in Y_{f}$;

  2. (b)

$f\leq\theta(Y_{f},f)$;

  3. (c)

for all $g\in Y_{f}$, if $g\leq\theta(Y_{f},g)\in Y_{f}$, then
$\Phi(\theta(Y_{f},g))\cap Y_{f}\neq\emptyset$;

  4. (d)

$\Phi$ is compact on $Y_{f}$.

With (a)-(c) in place we know that the fixed-point construction may only go
wrong at limit stages and, indeed, for certain choices of admissibility
conditions the construction will go wrong at limit ordinals as the following
example illustrates.

######  Example 17 ($\omega$-consistency).

Let $\mathfrak{M}=(D,X,H)$ be a classical model, i.e., $X=\\{J\\}$ where $J$
is a classical interpretation and $H$ is $\\{\langle J,J\rangle\\}$. For
$f\in{\sf Val}_{\mathfrak{M}}$ let $\Phi_{\omega}$ be defined as follows:

$\displaystyle\Phi_{\omega}(f)$ $\displaystyle:=\begin{cases}\emptyset,&\,{\rm
if}\,f\not\in\mathcal{B}_{\mathfrak{M}}\\\
\\{g\in\mathcal{B}_{\mathfrak{M}}\,|\,f\leq g\,\&\,g(J)\text{ is
$\omega$-consistent},&\,{\rm otherwise}.\end{cases}$

Now let ${\sf f_{mc}}$ be a function such that for $n\in\omega$ formula
$\varphi$

$\displaystyle{\sf f_{mc}}(n,\\#\varphi):=\begin{cases}\\#\varphi,&\text{ if
}n=0;\\\ \\#\scaleobj{0.9}{\mathrm{T}}{\sf f}_{\sf
mc}^{\bullet}(\overline{m},\raise
0.0pt\hbox{$\ulcorner$}\hbox{$\varphi$}\raise 0.0pt\hbox{$\urcorner$}),&\text{
if }n=m+1.\end{cases}$

and where ${\sf f}_{\sf mc}^{\bullet}$ is a function symbol representing the
p.r function ${\sf f_{mc}}$. Moreover, let $Y_{f}=\mathcal{B}_{\mathfrak{M}}$
whith $f(J)=\emptyset$ and let $\delta$ be the sentence $\neg\forall
x\scaleobj{0.9}{\mathrm{T}}{\sf f}_{\sf mc}^{\bullet}(x,\raise
0.02205pt\hbox{$\ulcorner$}\hbox{$\delta$}\raise
0.02205pt\hbox{$\urcorner$})$. Then it is easy to see that

$\scaleobj{0.9}{\mathrm{T}}{\sf f}_{\sf mc}^{\bullet}(\overline{n},\raise
0.02205pt\hbox{$\ulcorner$}\hbox{$\delta$}\raise
0.02205pt\hbox{$\urcorner$})\in\theta^{\omega}(Y_{f},f)$

for all $n\in\omega$. Thus $\theta^{\omega}(Y_{f},f)$ is
$\omega$-inconsistent, which means that $\Theta^{\omega}(Y_{f})=\emptyset$ and
thus
$\Phi_{\omega}(\theta^{\omega}(Y_{f},f))\cap\Theta^{\omega}(Y_{f})=\emptyset$.
This implies that $\theta^{\omega+1}(Y_{f},f)(J)={\sf Sent}$ and thus
$\theta^{\omega+1}(Y_{f},f)\not\in\mathcal{B}_{\mathfrak{M}}$. See McGee,
(1991) and Campbell-Moore, (2021) for further discussion.

Problems like the one presented in Example 17 can be blocked by Condition (d),
i.e., by requiring that the admissibility condition is compact.

######  Definition 18 (Compactness).

Let $Y_{f}\in{\sf Adm}_{\mathfrak{M}}$ be a grounded truth set and $\Phi$ an
admissibility condition. For $X\subseteq Y_{f}$ set
$\Phi(X)=\\{\Phi(f)\,|\,f\in X\\}$. We say that $\Phi$ is compact on $Y_{f}$
iff for all $X\subseteq Y_{f}$:

$\displaystyle\text{if
}\Phi(f_{1})\cap\ldots\cap\Phi(f_{n})\neq\emptyset\text{ for all $n\in\omega$
and $f_{1},\ldots f_{n}\in X$},$ $\displaystyle\text{ then
}\bigcap\Phi(X)\neq\emptyset.$

If the admissibility condition is compact, that is, Condition (d) is met in
addition to Conditions (a)-(c), there will be fixed points of $\theta^{\Phi}$
and $\Theta^{\Phi}$.

######  Proposition 19.

Let $Y_{f}=\\{g\in\mathcal{B}_{\mathfrak{M}}\,|\,f\leq g\\}$ and let $\Phi$ be
compact on $Y_{f}$. If $f\leq\theta^{\Phi}_{\mathfrak{M}}(Y_{f},f)$ and for
all $g\in Y_{f}$:

$\displaystyle g\leq\theta(Y_{f},g)\in Y_{f}$
$\displaystyle\Rightarrow\Phi(\theta(Y_{f},g))\cap Y_{f}\neq\emptyset,$

then there exists $\xi$ such that for all $\alpha\geq\xi$

$\displaystyle\theta^{\xi}(Y_{f},f)\in\Theta^{\xi}(Y_{f})=\Theta^{\alpha}(Y_{f}).$

In the proof of the proposition we shall make use of the following auxiliary
lemma.

######  Lemma 20.

Let $Y_{f}\in{\sf Adm}_{\mathfrak{M}}$ and for $g\in Y_{f}$ let
$Y^{g\leq}_{f}:=\\{h\in Y_{f}\,|\,g\leq h\\}.$ Then

$\text{if }\Phi(g)\cap Y_{f}\neq\emptyset\text{ then
}\theta(Y_{f},g)=\theta(Y^{g\leq}_{f},g).$

Recall that by definition $f\leq h$ for all $h\in Y_{f}$. So, in particular,
we know that $f\leq g$, which means that Lemma 20 is a Lemma about
$g$-generated $\leq$-substructures The lemma highlights that as in Kripke
frames for ${\sf N3}$ truth at a world is local notion—only the worlds that
are—potentially—accessible from a given world will be relevant.222222We are
not guaranteed that $f\leq_{\Phi}g$ for all $g\in Y_{f}$ such that $f\leq g$,
so we actually still consider valuation functions that are not relevant for
the evaluation of the conditional.

###### Proof.

By induction on the complexity of $\varphi$ we show that

$\displaystyle(D,X\times Y_{f},H_{\Phi}),J_{g}\Vdash\varphi$
$\displaystyle\text{ iff }(D,X\times
Y^{g\leq}_{f},H_{\Phi}),J_{g}\Vdash\varphi.$

The only interesting case is the case of the conditional. The left-to-right
direction is immediate since $Y^{g\leq}_{f}\subseteq Y^{f}$. Since
$\Phi(g)\cap Y_{f}\neq\emptyset$ it suffices to observe that

$\\{h\in Y_{f}\,|\,g\leq_{\Phi}h\\}\subseteq Y^{g\leq}_{f}$

to establish the converse direction. ∎

Ultimately, Lemma 20 tells us that for suitable choices of the set $Y_{f}$ we
can ignore the operation $\Theta^{\Phi}$ and rather than defining the minimal
fixed point of $\theta^{\Phi}$ via a simultaneous inductive definition,
inductively define the fixed point in the parameter $Y_{f}$.

###### Proof of Proposition 19.

By induction on $\alpha$ we show that for all $\beta,\alpha$

$\text{if }\beta<\alpha,\text{ then
}\theta^{\beta}(Y_{f},f)\leq\theta^{\alpha}(Y_{f},f)\text{ and
}\theta^{\alpha}(Y_{f},f)\in\Theta^{\alpha}(Y_{f}).$

As induction hypothesis we may assume that
$\theta^{\gamma}(Y_{f},f)\leq\theta^{\beta}(Y_{f},f)$ for all $\beta,\gamma$
with $\gamma<\beta<\alpha$. If $\alpha=0$ there is nothing to show. Let
$\alpha=\beta+1$. We conduct a secondary induction on $\beta$. If $\beta=0$
then $\alpha=1$ and the claim follows by assumption. Let $\beta=\gamma+1$.
Then by IH of the primary induction
$\theta^{\gamma}(Y_{f},f)\leq\theta^{\beta}(Y_{f},f)$ and
$\theta^{\beta}(Y_{f},f)\in\Theta^{\beta}(Y_{f})\subseteq Y_{f}$. Then by
assumption we know that
$\Phi(\theta^{\beta}(Y_{f},f))\cap\Theta^{\beta}(Y_{f})\neq\emptyset$ and
$\Theta^{\beta}(Y_{f})\subseteq\Theta^{\gamma}(Y_{f})$. By Lemma 14 and Lemma
20 this implies

$\theta^{\beta}(Y_{f},f)=\theta(\Theta^{\gamma}(Y_{f}),\theta^{\gamma}(Y_{f},f))\leq\theta(\Theta^{\beta}(Y_{f}),\theta^{\beta}(Y_{f},f))=\theta^{\alpha}(Y_{f},f).$

Moreover, by Lemma 15 we also know that

$\theta^{\alpha}(Y_{f},f)\in\Theta^{\alpha}(Y_{f})\in{\sf
Adm}_{\mathfrak{M}}.$

Let $\beta$ be a limit ordinal. We show that
$\Phi(\theta^{\beta}(Y_{f},f))\cap Y_{f}\neq\emptyset$. Notice that by the
definition of $\Phi$ (for arbitrary $\Phi$) this shows that
$\Phi(\theta^{\beta}(Y_{f},f))\in Y_{f}$. Let
$\Phi^{\beta}:=\\{\Phi(\theta^{\gamma}(Y_{f},f))\,|\,\gamma<\beta\\}$. Then by
IH $\bigcap X\neq\emptyset$ for every finite subset $X$ of $\Phi^{\beta}$ and
by compactness

$\bigcap\Phi^{\beta}\neq\emptyset.$

It is then straightforward to verify that
$\theta^{\beta}(Y_{f},f)\leq\theta(\Theta^{\beta}(Y_{f}),\theta^{\beta}(Y_{f},f)=\theta^{\alpha}(Y_{f},f)\in\Theta^{\alpha}(Y_{f})\in{\sf
Adm}_{\mathfrak{M}}$.232323Assume by reductio that this is not the case. Then
there must be a sentence $\varphi\in\theta^{\gamma}(Y_{f},f)(J)$ for some
$\gamma<\beta$, interpretation $J$ such that
$\varphi\not\in\theta(\Theta^{\gamma}(Y_{f},f),\theta^{\gamma}(Y_{f},f)(J)$.
Contradiction.

Finally, let $\alpha$ be a limit ordinal. Then the claim follows by definition
of $\theta^{\alpha}(Y_{f},f)$ and the fact that $\theta^{\alpha}(Y_{f},f)\in
Y_{f}$.

Since the sequence of $\theta^{\alpha}$ for $\alpha\in{\sf ON}$ is progressive
we may infer that there is an ordinal $\xi$ such that
$\theta(\Theta^{\xi}(Y_{f}),\theta^{\xi}(Y_{f},f))=\theta^{\xi}(Y_{f},f)\in\Theta^{\xi}(Y_{f})$,
that is, $\theta^{\xi}(Y_{f},f)=\theta^{\alpha}(Y_{f},f)$ for all
$\alpha\geq\xi$. This also implies that
$\Theta(\Theta^{\xi}(Y_{f}))=\Theta^{\xi}(Y_{f})$ and thus
$\Theta^{\xi}(Y_{f})=\Theta^{\alpha}(Y_{f})$ for $\xi\leq\alpha$. ∎

Proposition 19 is an abstract fixed-point result in the sense that it does not
show that we can find grounded sets of valuation functions $Y_{f}$ and
admissibility conditions $\Phi$ that satisfy conditions (a)-(d). However, if
there are, we know that fixed points exist and naive truth models alongside.

######  Remark 21.

Our proof of the abstract fixed-point result consisted in simultaneously
inductively defining the interpretation of the truth predicate at an
interpretation as well as the grounded set of valuation functions of the
Kripkean truth structure. It is worth highlighting that as a consequence of
Lemma 20 it turns out that if the inductive definition is successful, we may
take the initial set of grounded valuation functions $Y_{f}$ as a parameter in
the inductive definition of the fixed point of $\theta$.

This observation also allows us to provide an algebraic perspective of the
abstract fixed point result: instead of showing the existence of fixed points
of $\theta^{\Phi_{\sf e}}_{\mathfrak{M}}$ by conducting an inductive
definition, one can employ a variant of the Knaster-Tarski fixed-point result:
to this effect one shows that $(Y^{+}_{f},\leq)$ with

$Y^{+}_{f}:=\\{g\in\mathcal{B}_{\mathfrak{M}}\,|\,f\leq g\,\&\,\Phi_{\sf
e}(g)\cap\mathcal{B}_{\mathfrak{M}}\neq\emptyset\\}$

is a CCPO, i.e., a consistent and complete partial order (cf. Visser,, 1989,
p. 556).242424Note that the compactness of $\Phi_{\sf e}$ is crucial to this
effect. One then needs to show that the operation $\theta^{\Phi_{\sf
e}}_{Y^{+}_{f}}$ with

$\theta^{\Phi_{\sf e}}_{Y^{+}_{f}}(g):=\theta^{\Phi_{\sf
e}}_{\mathfrak{M}}(Y^{+}_{f},g)$

for all $g\in Y^{+}_{f}$ is monotone on $Y^{+}_{f}$, i.e., one needs to show
that for all $g,h\in Y^{+}_{f}$:

  1. (i)

$\theta^{\Phi_{\sf e}}_{Y^{+}_{f}}(g),\theta^{\Phi_{\sf e}}_{Y^{+}_{f}}(h)\in
Y^{+}_{f}$;

  2. (ii)

if $g\leq h$, then $\theta^{\Phi_{\sf e}}_{Y^{+}_{f}}(g)\leq\theta^{\Phi_{\sf
e}}_{Y^{+}_{f}}(h)$.

The crucial point is to show (i), as (ii) then follows from Lemma 14. Notice
that it follows form the assumption in Proposition 19 that

$\displaystyle g\leq\theta(Y_{f},g)\in Y_{f}$
$\displaystyle\Rightarrow\Phi(\theta(Y_{f},g))\cap Y_{f}\neq\emptyset,$

for all $g\in Y_{f}$. Once one has established that $(Y^{+}_{f},\leq)$ is a
CCPO and that $\theta^{\Phi_{\sf e}}_{Y^{+}_{f}}$ is monotone on $Y^{+}_{f}$
the existence of fixed points of $\theta^{\Phi_{\sf e}}_{Y^{+}_{f}}$ follows
by the generalisation of Knaster-Tarski discussed in, e.g., Visser, (1989).
Moreover, $({\sf Fix}^{\Phi_{\sf e}}_{Y^{+}_{f}},\leq)$ is itself a CCPO with

${\sf Fix}^{\Phi_{\sf e}}_{Y^{+}_{f}}:=\\{g\in Y^{+}_{f}\,|\,\theta^{\Phi_{\sf
e}}_{Y^{+}_{f}}(g)=g\\}.$

We refer to (Visser,, 1989, p. 556-664) for a general discussion of algebraic
fixed-fixed point constructions.

###  6.1 Applications of Proposition 19

We now wish to provide fixed-point results for specific admissibility
condition. We start by introducing different admissibility conditions.

######  Definition 22.

Let $f\in\mathcal{B}_{\mathfrak{M}}$ and let $L$ be some logic. Then $f$ is
$L$-saturated iff for all $J\in X$, all $\Delta\subseteq\mathcal{L}_{\rm T}$
all $\Gamma\subset f(J)$ such that

$\displaystyle{\rm if}\,\vdash_{L}\Gamma\Rightarrow\Delta,$
$\displaystyle\,\,{\rm then}\,\,\Delta\cap f(J)\neq\emptyset.$

Moreover for $f\in{\sf Val}_{\mathfrak{M}}$ we set:

$\displaystyle\Phi_{\sf c}(f)$ $\displaystyle:=\begin{cases}\emptyset,&\text{
if }f\not\in\mathcal{B}_{\mathfrak{M}};\\\
\\{g\in\mathcal{B}_{\mathfrak{M}}\,|\,f\leq g\\},&\text{
otherwise.}\end{cases}$ $\displaystyle\Phi_{\sf K3}(f)$
$\displaystyle:=\begin{cases}\emptyset,&\text{ if
}f\not\in\mathcal{B}_{\mathfrak{M}};\\\
\\{g\in\mathcal{B}_{\mathfrak{M}}\,|\,f\leq g\,\&\,g\text{ is {\sf
K3}-saturated}\\},&\text{ otherwise.}\end{cases}$ $\displaystyle\Phi_{\sf
N3}(f)$ $\displaystyle:=\begin{cases}\emptyset,&\text{ if
}f\not\in\mathcal{B}_{\mathfrak{M}};\\\
\\{g\in\mathcal{B}_{\mathfrak{M}}\,|\,f\leq g\,\&\,g\text{ is {\sf
N3}-saturated}\\},&\text{ otherwise.}\end{cases}$ $\displaystyle\Phi_{\sf
Nve}(f)$ $\displaystyle:=\begin{cases}\emptyset,&\text{ if
}f\not\in\mathcal{B}_{\mathfrak{M}};\\\
\\{g\in\mathcal{B}_{\mathfrak{M}}\,|\,f\leq g\,\&\,g\text{ is ${\sf
K3\scaleobj{0.9}{\mathrm{T}}}$-saturated}\\},&\text{ otherwise.}\end{cases}$

The different logics are displayed in form of their sequent calculi in
Appendix A. The logic ${\sf K3\scaleobj{0.9}{\mathrm{T}}}$ is the naive logic
of truth in the sense of, e.g., Kremer, (1988), i.e., the logic obtained from
${\sf K3}$ by the addition of naive truth rules.

The next step would be to show the existence of fixed points of the operation
$\theta_{\sf e}$ and $\Theta_{\sf e}$ for ${\sf e}\in\\{{\sf c},{\sf K3},{\sf
N3},{\sf Nve}\\}$, that is, to show that there is an
$f\in\mathcal{B}_{\mathfrak{M}}$ and $Y_{f}:=\\{g\leq
f\,|\,g\in\mathcal{B}_{\mathfrak{M}}\\}\in{\sf Adm}_{\mathfrak{M}}$ such that
there exists $g\in Z_{g}\subseteq Y_{f}$ with $\theta^{\Phi_{\sf
e}}_{\mathfrak{M}}(Z_{g},g)=g$ and $\Theta^{\Phi_{\sf
e}}_{\mathfrak{M}}(Z_{g})=Z_{g}$. Indeed, if we let $f$ be the valuation
function $f(J)=\emptyset$ for all $J\in X$, then one can show that $f$ and
$Y_{f}$ satisfy all the conditions for applying Proposition 19 for ${\sf
e}\in\\{{\sf c},{\sf K3},{\sf N3},{\sf Nve}\\}$. However, we directly prove a
stronger fixed-point result, that is, we show that we can fixed points $Z_{g}$
of $\Theta^{\Phi_{\sf e}}$ such that $h(J)$ is a fixed point of the
conditional free fragment of $\mathcal{L}_{\rm T}$ ($\mathcal{L}_{\rm T}^{-}$)
for every $h\in Z_{g}$. To this effect we let
$\mathcal{B}^{\scaleobj{0.9}{\mathrm{T}}}_{\mathfrak{M}}\subseteq\mathcal{B}_{\mathfrak{M}}$
be the set ov valuation functions such that for all
$f\in\mathcal{B}^{\scaleobj{0.9}{\mathrm{T}}}_{\mathfrak{M}}$ and all $J\in
X$:

  1. (i)

$f(J)$ consistently extends a ${\sf K3}$-fixed point $S$, i.e.,
$f(J)\cap\mathcal{L}_{\rm T}^{-}$ is ${\sf
K3}\scaleobj{0.9}{\mathrm{T}}$-saturated;

  2. (ii)

$f(J)$ is ${\sf K3}$-saturated;

Our aim is to find a set
$Y_{f}^{\scaleobj{0.9}{\mathrm{T}}}\subseteq\mathcal{B}^{\scaleobj{0.9}{\mathrm{T}}}_{\mathfrak{M}}$
such that for ${\sf e}\in\\{{\sf K3},{\sf N3},{\sf Nve}\\}$ (notice that
$\Phi_{\sf c}$ and $\Phi_{\sf K3}$ coincide over
$\mathcal{B}^{\scaleobj{0.9}{\mathrm{T}}}_{\mathfrak{M}}$):

  * •

$\Phi_{\sf e}$ is compact on $Y_{f}^{\scaleobj{0.9}{\mathrm{T}}}$;

  * •

$f\leq\theta^{\Phi_{\sf e}}(Y^{\scaleobj{0.9}{\mathrm{T}}}_{f},f)\in
Y_{f}^{\scaleobj{0.9}{\mathrm{T}}}$;

  * •

for all $g\in Y^{\scaleobj{0.9}{\mathrm{T}}}_{f}$:

$\text{ if }g\leq\theta^{\Phi_{\sf
e}}(Y^{\scaleobj{0.9}{\mathrm{T}}}_{f},g)\in
Y^{\scaleobj{0.9}{\mathrm{T}}}_{f},\text{ then }\Phi_{\sf e}(\theta^{\Phi_{\sf
e}}(Y^{\scaleobj{0.9}{\mathrm{T}}}_{f},g))\cap Y_{f}\neq\emptyset.$

Once we find such a set, the existence of fixed points follows from
Proposition 19.

Let $\mathcal{K}:\mathcal{P}({\sf Sent})\rightarrow\mathcal{P}({\sf Sent})$ be
an operation such that for all $J\in X$, $g\in{\sf Val}_{\mathfrak{M}}$ and
all $\varphi\in\mathcal{L}_{\rm T}$:

$\displaystyle\varphi\in\mathcal{K}(g(J))$ $\displaystyle\text{ iff
}\begin{cases}J(t)=J(s),&\text{ if }\varphi\doteq(t=s);\\\ J(t)\neq
J(s),&\text{ if }\varphi\doteq(\neg t=s);\\\ \langle
J(t_{1}),\ldots,J(t_{n})\rangle\in J^{+}(P^{n}),&\text{ if
}\varphi\doteq(Pt_{1},\ldots,t_{n});\\\ \langle
J(t_{1}),\ldots,J(t_{n})\rangle\in J^{-}(P^{n}),&\text{ if }\varphi\doteq(\neg
Pt_{1},\ldots,t_{n});\\\ \psi\in g(J),&\text{ if
}\varphi\doteq\scaleobj{0.9}{\mathrm{T}}t\,\&\,J(t)=\psi;\\\ \neg\psi\in
g(J),&\text{ if
}\varphi\doteq\neg\scaleobj{0.9}{\mathrm{T}}t\,\&\,J(t)=\psi;\\\ \psi\in
g(J)\,\&\,\chi\in g(J),&\text{ if }\varphi\doteq(\psi\wedge\chi);\\\
\neg\psi\in gl(J)\text{ or }\neg\chi\in g(J),&\text{ if
}\varphi\doteq\neg(\psi\wedge\chi);\\\ \psi(c)\in g(J)\text{ for all }c\in{\sf
Const}_{\mathcal{L}_{\rm T}},&\text{ if }\varphi\doteq\forall x\psi;\\\
\psi(c)\in g(J)\text{ for some }c\in{\sf Const}_{\mathcal{L}_{\rm T}},&\text{
if }\varphi\doteq\neg\forall x\psi;\end{cases}$

Since $\mathcal{K}$ is monotone on $\mathcal{B}$, and $\mathcal{K}(g(J))$ is
consistent for all $J\in X$ and $g\in\mathcal{B}$, we know that there is an
$f_{\sf k}\in\mathcal{B}$ such that $f_{\sf k}(J)$ is the minimal
$\mathcal{K}$ fixed point over the set $\\{\varphi\,|\,(D,J)\Vdash_{\sf
K3}\varphi\\}$ for all $J\in X$. Moreover, by the properties of $\mathcal{K}$
clearly $f_{\sf k}\in\mathcal{B}^{\scaleobj{0.9}{\mathrm{T}}}_{\mathfrak{M}}$.

######  Lemma 23.

Let ${\sf e}\in\\{{\sf c},{\sf K3},{\sf N3},{\sf Nve}\\}$ and
$Y^{\scaleobj{0.9}{\mathrm{T}}}_{f_{\sf
k}}:=\\{g\in\mathcal{B}^{\scaleobj{0.9}{\mathrm{T}}}_{\mathfrak{M}}\,|\,f_{\sf
k}\leq g\\}$ with $f_{\sf
k}\in\mathcal{B}^{\scaleobj{0.9}{\mathrm{T}}}_{\mathfrak{M}}$ as discussed
above. Then

  1. (a)

$f_{\sf k}\leq\theta^{\Phi_{\sf e}}(Y^{\scaleobj{0.9}{\mathrm{T}}}_{f_{\sf
k}},f_{\sf k})\in Y_{f_{\sf k}}^{\scaleobj{0.9}{\mathrm{T}}}$;

  2. (b)

for all $g\in Y^{\scaleobj{0.9}{\mathrm{T}}}_{f}$, if $g\leq\theta^{\Phi_{\sf
e}}(Y^{\scaleobj{0.9}{\mathrm{T}}}_{f},g)\in
Y^{\scaleobj{0.9}{\mathrm{T}}}_{f}$, then $\Phi_{\sf e}(\theta^{\Phi_{\sf
e}}(Y^{\scaleobj{0.9}{\mathrm{T}}}_{f},g))\cap Y_{f}\neq\emptyset.$

###### Proof.

(a) is straightforward to verify. Notice that $\theta_{\Phi_{\sf e}}(Z_{g},g)$
is ${\sf N3}$-saturated for all $g\in Z_{g}\in{\sf Adm}_{\mathfrak{M}}$ for
${\sf e}\in\\{{\sf c},{\sf K3},{\sf N3},{\sf Nve}\\}$, and that for
$\varphi\in\mathcal{L}_{\rm T}^{-}$:

$\displaystyle\varphi\in f_{\sf k}(J)$ $\displaystyle\,{\rm
iff}\,\varphi\in\theta^{\Phi_{\sf e}}(Y^{\scaleobj{0.9}{\mathrm{T}}}_{f_{\sf
k}},f_{\sf k}).$

For (b) notice that for ${\sf e}\in\\{{\sf c},{\sf K3},{\sf N3}\\}$, the claim
is immediate since $\theta^{\Phi_{\sf e}}_{\mathfrak{M}}(Y_{f_{\sf
k}}^{\scaleobj{0.9}{\mathrm{T}}},g)\in\Phi(\theta^{\Phi_{\sf
e}}(Y^{\scaleobj{0.9}{\mathrm{T}}}_{f_{\sf k}},g))$. For ${\sf e}={\sf Nve}$
we observe that $\theta^{\Phi_{\sf e}}_{\mathfrak{M}}(Y_{f_{\sf
k}}^{\scaleobj{0.9}{\mathrm{T}}},g)(J)\subseteq\mathcal{K}([\theta^{\Phi_{\sf
e}}_{\mathfrak{M}}(Y_{f_{\sf k}}^{\scaleobj{0.9}{\mathrm{T}}},g)(J)])$ for all
$J\in X$. Moreover, since by assumption $\theta^{\Phi_{\sf
e}}_{\mathfrak{M}}(Y_{f_{\sf k}}^{\scaleobj{0.9}{\mathrm{T}}},g)(J)$ is
consistent if follows that $\mathcal{K}([\theta^{\Phi_{\sf
e}}_{\mathfrak{M}}(Y_{f_{\sf k}}^{\scaleobj{0.9}{\mathrm{T}}},g)(J)])$ is
consistent. This means that there is a consistent set
$S_{J}\supseteq\theta^{\Phi_{\sf e}}_{\mathfrak{M}}(Y_{f_{\sf
k}}^{\scaleobj{0.9}{\mathrm{T}}},g)(J)$ such that $\mathcal{K}(S_{J})=S_{J}$.
Define a valuation function $g^{\prime}$ such that for all $I\in X$:

$g^{\prime}(I):=S_{I}.$

Then we have $g^{\prime}\geq\theta^{\Phi_{\sf e}}_{\mathfrak{M}}(Y_{f_{\sf
k}}^{\scaleobj{0.9}{\mathrm{T}}},g)$ and $g^{\prime}$ is ${\sf
K3}\scaleobj{0.9}{\mathrm{T}}$-saturated and $g^{\prime}\in
Y^{\scaleobj{0.9}{\mathrm{T}}}_{f_{\sf k}}$:

$\Phi_{\sf e}(\theta^{\Phi_{\sf e}}(Y^{\scaleobj{0.9}{\mathrm{T}}}_{f_{\sf
k}},g))\cap Y^{\scaleobj{0.9}{\mathrm{T}}}_{\sf k}\neq\emptyset.$

∎

It remains to show that $\Phi_{\sf e}$ is compact on
$Y^{\scaleobj{0.9}{\mathrm{T}}}_{\mathfrak{M}}$ before we can apply
Proposition 19.

######  Lemma 24.

Let $Y^{\scaleobj{0.9}{\mathrm{T}}}_{f_{\sf k}}$ be defined as in Lemma 23 and
${\sf e}\in\\{{\sf c},{\sf K3},{\sf N3},{\sf Nve}\\}$. Then $\Phi_{\sf e}$ is
compact on $Y^{\scaleobj{0.9}{\mathrm{T}}}_{f_{\sf k}}$.

###### Proof.

Let $Z$ be an arbitrary subset of $Y^{\scaleobj{0.9}{\mathrm{T}}}_{f_{\sf k}}$
and assume that $\Phi_{\sf e}(f_{1})\cap\ldots\cap\Phi_{\sf
e}(f_{n})\neq\emptyset$ for all $n\in\omega$ and $f_{1},\ldots,f_{n}\in Z$.

Now let ${\sf e}\in\\{{\sf K3,N3,K3\scaleobj{0.9}{\mathrm{T}}}\\}$. We show
that $\bigcup\\{f(J)\,|\,f\in Z\\}$ is ${\sf e}$-consistent. Assume for
reductio that $\bigcup\\{f(J)\,|\,f\in Z\\}$ is not ${\sf e}$-consistent for
some $J\in X$, that is, there is a finite set of formulas
$\Gamma\subseteq\bigcup\\{f(J)\,|\,f\in Z\\}$ such that $\Gamma\vdash_{\sf
e}\bot$. But then there must be an $f_{1},\ldots,f_{n}\in Z$ for some
$n\in\omega$ such that $\Gamma\subseteq f_{1}(J)\cup\ldots\cup f_{n}(J)$. This
implies that $\Phi_{\sf e}(f_{1})\cap\ldots\cap\Phi_{\sf e}(f_{n})=\emptyset$:
contradiction. Moreover, since every $f(J)$ is ${\sf K3}$-saturated and
$\bigcup\\{f(J)\,|\,f\in Z\\}$ is ${\sf K3}$-saturated and by essentially the
same reasoning $\bigcup\\{f(J)\,|\,f\in Z\\}\cap\mathcal{L}_{\rm T}^{-}$ is
${\sf K3}\scaleobj{0.9}{\mathrm{T}}$ saturated. This guarantees that we can
extend $\bigcup\\{f(J)\,|\,f\in Z\\}$ to an ${\sf e}$-saturated set $g(J)$ for
${\sf e}\in\\{{\sf N3},{\sf K3}\scaleobj{0.9}{\mathrm{T}}\\}$ such that
$g(J)\cap\mathcal{L}_{\rm T}^{-}$ is ${\sf K3}\scaleobj{0.9}{\mathrm{T}}$
saturated and then define an admissible valuation function $g$ accordingly. By
construction $g\in\bigcap_{f\in Z}\Phi(f)$. ∎

We have now gathered all relevant facts to establish the existence of fixed
points of $\theta^{\Phi_{\sf e}}$ and $\Theta^{\Phi_{\sf e}}$ for ${\sf
e}\in\\{{\sf c},{\sf K3},{\sf N3},{\sf Nve}\\}$.

######  Proposition 25.

Let $\mathfrak{M}$ be a supervaluation structure and let
$Y^{\scaleobj{0.9}{\mathrm{T}}}_{f_{\sf k}}$ be defined as in Lemma 23 and
${\sf e}\in\\{{\sf c},{\sf K3},{\sf N3},{\sf Nve}\\}$. Then there exists $g\in
Z_{g}\subseteq Y^{\scaleobj{0.9}{\mathrm{T}}}_{f}$ such that
$\theta^{\Phi_{\sf e}}_{\mathfrak{M}}(Z_{g},g)=g$ and $\Theta^{\Phi_{\sf
e}}_{\mathfrak{M}}(Z_{g})=Z_{g}$.

###### Proof.

By Proposition 19, and Lemmas 23 and 25.∎

######  Corollary 26.

Let $\mathfrak{M}=(D,X,H)$ be a supervaluation structure and ${\sf
e}\in\\{{\sf c},{\sf K3},{\sf N3},{\sf Nve}\\}$. Then there exists a grounded
truth set
$Y_{f}\subseteq\mathcal{B}^{\scaleobj{0.9}{\mathrm{T}}}_{\mathfrak{M}}$ and
admissible valuation function $f$ such that for all $\varphi\in{\sf
Sent}_{\mathcal{L}_{\rm T}}$

$\displaystyle(D,X\times Y_{f},H_{\Phi_{\sf e}}),J_{f}\Vdash\varphi$
$\displaystyle\text{ iff }(D,X\times Y_{f},H_{\Phi_{\sf
e}}),J_{f}\Vdash\scaleobj{0.9}{\mathrm{T}}\raise
0.0pt\hbox{$\ulcorner$}\hbox{$\varphi$}\raise 0.0pt\hbox{$\urcorner$}$

for all $J\in X$.

Corollary 26 establishes the existence of Kripkean truth structures. However,
strictly speaking they do not provide us with supervaluation structures for
the language $\mathcal{L}_{\rm T}$ that comprise Kripkean truth models. The
reason is that a relation $H$ on $X$ was deemed an admissibility relation only
if it is reflexive. For arbitrary Kripkean truth structures we are not
guaranteed that $H_{\Phi_{\sf e}}$ is reflexive on $X\times Y_{f}$ for ${\sf
e}\in\\{{\sf K3,N3,Nve}\\}$ as $Y_{f}$ may contain valuation functions $g$
such that $g(J)$ is not ${\sf e}$-saturated and thus $(J_{g},J_{g})\not\in
H_{\Phi_{\sf e}}$. Yet, it is straightforward to obtain a Kripkean
supervaluation structure for $\mathcal{L}_{\rm T}$ from a Kripkean truth
structure $(D,X\times Y_{f},H_{\Phi_{\sf e}})$ by forming the $f$-generated
$\leq_{\Phi_{\sf e}}$-substructure.

######  Corollary 27 (Kripkean supervaluation structure).

Let $(D,X\times Y_{f},H_{\Phi_{\sf e}})$ be a Kripkean truth structure for
$\mathcal{L}_{\rm T}$ and ${\sf e}\in\\{{\sf K3,N3,Nve}\\}$. Set

$Y^{\leq_{\Phi_{\sf e}}}_{f}:=\\{g\in Y_{f}\,|\,f\leq_{\Phi_{\sf e}}g\\}.$

Then $(D,X\times Y^{\leq_{\Phi_{\sf e}}}_{f},H_{\Phi_{\sf e}})$ is a Kripkean
truth structure such that $H_{\Phi_{\sf e}}$ is reflexive on $X\times
Y^{\leq_{\Phi_{\sf e}}}_{f}$ and $\theta^{\Phi_{\sf
e}}_{\mathfrak{M}}(Y^{\leq_{\Phi_{\sf e}}}_{f},f)=f$.

Before we turn to the properties of the conditional in Kripkean Truth
Structures we reflect on some constraints and limitations of the framework of
the construction. In particular, we reflect again on the idea of transparent
truth.

######  Remark 28 (Transparency).

One may ask why the $\Phi_{\sf Nve}$ construction does not lead to transparent
truth models (in contradiction to Curry’s paradox). After all, the conditional
gets, in this case, evaluated with respect to models with naive valuations
functions. The short answer is that a naive valuation does not guarantee that
the model is a naive truth model.Indeed, for every $\Phi_{\sf Nve}$-fixed
point $(Y_{f},f)$ there must be a $g\in\Phi_{\sf Nve}(f)\cap Y_{f}$ such that
$(\mathfrak{M}_{\scaleobj{0.9}{\mathrm{T}}},J_{g})$ is not a naive truth model
for all $J\in X$. This can be illustrated by looking at the Curry sentence
$\kappa$ for it is easy to see that if $\kappa\in g(J)$ for $J\in X$, then
$(\mathfrak{M}_{\scaleobj{0.9}{\mathrm{T}}},J_{g})$ is not a naive truth
model. Suppose otherwise. Then since by assumption
$(\mathfrak{M}_{\scaleobj{0.9}{\mathrm{T}}},J_{g})\Vdash\scaleobj{0.9}{\mathrm{T}}\raise
0.0pt\hbox{$\ulcorner$}\hbox{$\kappa$}\raise 0.0pt\hbox{$\urcorner$}$ and by
naivity

$\displaystyle(\mathfrak{M}_{\scaleobj{0.9}{\mathrm{T}}},J_{g})\Vdash\scaleobj{0.9}{\mathrm{T}}\raise
0.0pt\hbox{$\ulcorner$}\hbox{$\kappa$}\raise 0.0pt\hbox{$\urcorner$}$
$\displaystyle\,{\rm
iff}\,(\mathfrak{M}_{\scaleobj{0.9}{\mathrm{T}}},J_{g})\Vdash\kappa$

it follows that
$(\mathfrak{M}_{\scaleobj{0.9}{\mathrm{T}}},J_{g})\Vdash\kappa$. By definition
of $\Phi_{\sf Nve}$ we know that $g\in\Phi_{\sf Nve}(g)$ and thus, by
Definition 7, we obtain

$\displaystyle{\rm
if}\,(\mathfrak{M}_{\scaleobj{0.9}{\mathrm{T}}},J_{g})\Vdash\scaleobj{0.9}{\mathrm{T}}\raise
0.0pt\hbox{$\ulcorner$}\hbox{$\kappa$}\raise 0.0pt\hbox{$\urcorner$},$
$\displaystyle\,{\rm
then}\,(\mathfrak{M}_{\scaleobj{0.9}{\mathrm{T}}},J_{g})\Vdash\bot.$

This yields a contradiction and we infer that if $\kappa\in g(J)$ for some
naive, non-trivial valuation, then $(\mathfrak{M},J_{g})$ is not a naive truth
model. Accordingly, if $f$ is a $\Phi_{\sf Nve}$-fixed point, then
$\kappa\not\in f(J)$ for $J\in X$. Also, since $f$ is a fixed-point this
implies by naivity that
$(\mathfrak{M}_{\scaleobj{0.9}{\mathrm{T}}},J_{f})\not\Vdash\kappa$ and by
Definition 7

$\exists J^{\prime}_{g}((J_{f},J^{\prime}_{g})\in H_{\Phi_{\sf
Nve}}\,\&\,(\mathfrak{M}_{\scaleobj{0.9}{\mathrm{T}}},J^{\prime}_{g})\Vdash\scaleobj{0.9}{\mathrm{T}}\raise
0.0pt\hbox{$\ulcorner$}\hbox{$\kappa$}\raise 0.0pt\hbox{$\urcorner$}).$

The latter implies that $\kappa\in g(J^{\prime})$ and thus
$(\mathfrak{M}_{\scaleobj{0.9}{\mathrm{T}}},J^{\prime}_{g})$ is not a naive
truth model, which proves our initial claim. It is then straightforward to see
that $f$ is not transparent since $\kappa\rightarrow\kappa\in f(J)$ for all
$J\in X$, but $\scaleobj{0.9}{\mathrm{T}}\kappa\rightarrow\kappa\not\in f(J)$,
as
$(\mathfrak{M}_{\scaleobj{0.9}{\mathrm{T}}},J^{\prime}_{g})\Vdash\scaleobj{0.9}{\mathrm{T}}\raise
0.0pt\hbox{$\ulcorner$}\hbox{$\kappa$}\raise 0.0pt\hbox{$\urcorner$})$ and
$(\mathfrak{M}_{\scaleobj{0.9}{\mathrm{T}}},J^{\prime}_{g})\not\Vdash\kappa)$.

In the above remark we highlighted that the existence of naive truth models
requires for $J\in X$ the existence of admissible precisifications $f$ such
that $\kappa\in f(J)$. This in turn shows that we cannot find non-trivial
fixed points if the following admissibility condition is assumed:

$\displaystyle\Phi_{\sf N3-Nve}(f)$
$\displaystyle:=\begin{cases}\emptyset,&\text{ if
}f\not\in\mathcal{B}_{\mathfrak{M}};\\\ \\{g\in{\sf
Val}^{\mathcal{B}}_{\mathfrak{M}}\,|\,f\leq g\,\&\,g\text{ is ${\sf
N3}$-naive}\\},&\text{ otherwise.}\end{cases}$

######  Lemma 29.

Let $\mathfrak{M}=(D,X,H)$ be a supervaluation structure. Then there is no
$Y\subseteq\mathcal{B}_{\mathfrak{M}}$ and $f\in\mathcal{B}_{\mathfrak{M}}$
such that $f\leq\theta^{\Phi_{\sf
N3-Nve}}_{\mathfrak{M}}(Y,f)\in\mathcal{B}_{\mathfrak{M}}$.

###### Proof.

Let $J\in X$. Then there are two cases: either (i) $\kappa\in f(J)$ or (ii)
$\kappa\not\in f(J)$. In case (i) we know that
$\kappa\not\in[\theta^{\Phi_{\sf N3-Nve}}_{\mathfrak{M}}(f)](J)$ and thus
$f\not\leq\theta^{\Phi_{\sf N3-Nve}}_{\mathfrak{M}}(f)$. (ii) Assume then that
$\kappa\not\in f(J)$. Notice that there is no ${\sf N3-Nve}$ valuation
function $g$ such that $\kappa\in g(J)$ for some $J\in X$. Assume otherwise.
Then by naivity $\scaleobj{0.9}{\mathrm{T}}\raise
0.0pt\hbox{$\ulcorner$}\hbox{$\kappa$}\raise 0.0pt\hbox{$\urcorner$}\in g(J)$
and by modus pones $\bot\in g(J)$. The latter implies that
$g\not\in\mathcal{B}_{\mathfrak{M}}$. But since for all $g\in\Phi_{\sf
N3-Nve}(f)$ we have $\kappa\not\in g(J)$ for $J\in X$ it follows
$\kappa\in[\theta^{\Phi_{\sf N3-Nve}}_{\mathfrak{M}}(f)](J)$ and
$[\theta^{\Phi_{\sf
N3-Nve}}_{\mathfrak{M}}(f)](J)\not\in\mathcal{B}_{\mathfrak{M}}$ since by
definition $[\theta^{\Phi_{\sf N3-Nve}}_{\mathfrak{M}}(f)](J)$ is ${\sf
N3}$-closed. ∎

The moral of Lemma 29 is that fixed points are only to be found iff valuations
conceive of the conditional as an unanalysed unit,252525At least for those
conditionals that cannot be decided in ${\sf K3}$ or conceive of truth as a
non-naive predicate. The Curry paradox forces us to opt for one of these
options.

##  7 The truth predicate and the conditional in Strong Kleene Supervaluation

The properties of $\scaleobj{0.9}{\mathrm{T}}$ and $\rightarrow$ on a Kripkean
supervaluation structure $\mathfrak{M}_{\scaleobj{0.9}{\mathrm{T}}}=(D,X\times
Y_{f},H_{\Phi_{\sf e}}$ will obviously depend on the admissibility condition
$\Phi_{\sf e}$ employed. Before we discuss the differences between the
different admissibility conditions, we investigate the rules and principles
that turn out true on all admissibility conditions we discuss. However,
building on Proposition 25 we assume that
$Y_{f}\subseteq\mathcal{B}_{\scaleobj{0.9}{\mathrm{T}}}$, i.e., that the
valuations function of the structure are Kripkean fixed points for
$\mathcal{L}_{\rm T}^{-}$ and ${\sf K3}$-saturated. This will guarantee that
the $\scaleobj{0.9}{\mathrm{T}}$-scheme will hold for $\mathcal{L}_{\rm
T}^{-}$-sentences. Moreover, by Corollary 27 we may assume $H_{\Phi_{\sf e}}$
to be reflexive on $X\times Y_{f}$. This, in turn, guarantees that all the
logical truths of Nelson logic ${\sf N3}$ will be true at any truth
interpretation of the structure. In particular, modus ponens will hold for all
$J_{g}\in X\times Y_{f}$ if
$\mathfrak{M}_{\scaleobj{0.9}{\mathrm{T}}},J_{g}\Vdash\varphi$ and
$\mathfrak{M}_{\scaleobj{0.9}{\mathrm{T}}},J_{g}\Vdash\varphi\rightarrow\psi$,
then $\mathfrak{M}_{\scaleobj{0.9}{\mathrm{T}}},J_{g}\Vdash\psi$. In contrast,
the deduction theorem will only hold globally, that is, we have:

$\displaystyle\Gamma,\varphi\vDash_{\mathfrak{M}_{\scaleobj{0.9}{\mathrm{T}}}}\psi$
$\displaystyle\,\,{\rm
iff}\,\,\Gamma\vDash_{\mathfrak{M}_{\scaleobj{0.9}{\mathrm{T}}}}\varphi\rightarrow\psi.$

Let us now turn to the properties of the truth predicate in the Kripkean truth
structures and collect some important facts:

  1. (i)

for all $\varphi\in\mathcal{L}_{\rm T}$

$\displaystyle\mathfrak{M}_{\scaleobj{0.9}{\mathrm{T}}},J_{f}\Vdash\scaleobj{0.9}{\mathrm{T}}\raise
0.0pt\hbox{$\ulcorner$}\hbox{$\varphi$}\raise 0.0pt\hbox{$\urcorner$}\,\,$
$\displaystyle{\rm
iff}\,\,\mathfrak{M}_{\scaleobj{0.9}{\mathrm{T}}},J_{f}\Vdash\varphi$
$\displaystyle\mathfrak{M}_{\scaleobj{0.9}{\mathrm{T}}},J_{f}\Vdash\neg\scaleobj{0.9}{\mathrm{T}}\raise
0.0pt\hbox{$\ulcorner$}\hbox{$\varphi$}\raise 0.0pt\hbox{$\urcorner$}\,\,$
$\displaystyle{\rm
iff}\,\,\mathfrak{M}_{\scaleobj{0.9}{\mathrm{T}}},J_{f}\Vdash\neg\varphi$

  2. (ii)

for all $\varphi\in\mathcal{L}_{\rm T}$, if
$\mathfrak{M}_{\scaleobj{0.9}{\mathrm{T}}}\Vdash\varphi\vee\neg\varphi$ or if
$\varphi\in\mathcal{L}\cup\mathcal{L}_{\rm T}^{-}$, then

$\displaystyle\mathfrak{M}_{\scaleobj{0.9}{\mathrm{T}}}\Vdash\varphi\,\,{\rm
iff}\,\,\mathfrak{M}_{\scaleobj{0.9}{\mathrm{T}}}\Vdash\scaleobj{0.9}{\mathrm{T}}\raise
0.0pt\hbox{$\ulcorner$}\hbox{$\varphi$}\raise 0.0pt\hbox{$\urcorner$}$
$\displaystyle\mathfrak{M}_{\scaleobj{0.9}{\mathrm{T}}}\Vdash\neg\varphi\,\,{\rm
iff}\,\,\mathfrak{M}_{\scaleobj{0.9}{\mathrm{T}}}\Vdash\neg\scaleobj{0.9}{\mathrm{T}}\raise
0.0pt\hbox{$\ulcorner$}\hbox{$\varphi$}\raise 0.0pt\hbox{$\urcorner$}$
$\displaystyle\mathfrak{M}_{\scaleobj{0.9}{\mathrm{T}}}\Vdash\varphi\leftrightarrow\scaleobj{0.9}{\mathrm{T}}\raise
0.0pt\hbox{$\ulcorner$}\hbox{$\varphi$}\raise 0.0pt\hbox{$\urcorner$}$
$\displaystyle\mathfrak{M}_{\scaleobj{0.9}{\mathrm{T}}}\Vdash\neg\varphi\leftrightarrow\neg\scaleobj{0.9}{\mathrm{T}}\raise
0.0pt\hbox{$\ulcorner$}\hbox{$\varphi$}\raise 0.0pt\hbox{$\urcorner$}$

  3. (iii)

Let $\Gamma,\Delta\subseteq\mathcal{L}_{\rm T}$ and $\Gamma\vDash_{\sf
N3}\Delta$ or
$\Gamma\vDash_{\mathfrak{M}_{\scaleobj{0.9}{\mathrm{T}}}}\Delta$. Then:

$\displaystyle{\rm
if}\,\mathfrak{M}_{\scaleobj{0.9}{\mathrm{T}}},J_{f}\Vdash\scaleobj{0.9}{\mathrm{T}}\raise
0.0pt\hbox{$\ulcorner$}\hbox{$\gamma$}\raise 0.0pt\hbox{$\urcorner$}\text{ for
all }\gamma\in\Gamma,$ $\displaystyle\,{\rm
then}\,\mathfrak{M}_{\scaleobj{0.9}{\mathrm{T}}},J_{f}\Vdash\scaleobj{0.9}{\mathrm{T}}\raise
0.02205pt\hbox{$\ulcorner$}\hbox{$\delta$}\raise
0.02205pt\hbox{$\urcorner$}\text{ for some }\delta\in\Delta.$

(i) simply states that $J_{f}$ is a Kripkean fixed-point on
$\mathfrak{M}_{\scaleobj{0.9}{\mathrm{T}}}$. (ii) tells us that the
$\scaleobj{0.9}{\mathrm{T}}$-scheme holds on
$\mathfrak{M}_{\scaleobj{0.9}{\mathrm{T}}}$ on the
$\mathfrak{M}_{\scaleobj{0.9}{\mathrm{T}}}$-determinate/stable fragment of
$\mathcal{L}_{\rm T}$, while (iii) tells us that the minimal fixed point is
closed under ${\sf N3}$-consequences.

Notice that (ii) provides us with some insights about the behaviour of
$\mathcal{L}_{\rm T}^{-}$ pathological sentence (in Kripke’s sense) such as
the Liar sentence $\lambda$. By the diagonal lemma we have
$\mathfrak{M}_{\scaleobj{0.9}{\mathrm{T}}}\Vdash\lambda\leftrightarrow\neg\scaleobj{0.9}{\mathrm{T}}\raise
0.02205pt\hbox{$\ulcorner$}\hbox{$\lambda$}\raise
0.02205pt\hbox{$\urcorner$}$, but also by construction:

$\displaystyle\mathfrak{M}_{\scaleobj{0.9}{\mathrm{T}}}\Vdash\lambda\leftrightarrow\scaleobj{0.9}{\mathrm{T}}\raise
0.02205pt\hbox{$\ulcorner$}\hbox{$\lambda$}\raise
0.02205pt\hbox{$\urcorner$};$
$\displaystyle\mathfrak{M}_{\scaleobj{0.9}{\mathrm{T}}}\Vdash\lambda\leftrightarrow\neg\lambda;$
$\displaystyle\mathfrak{M}_{\scaleobj{0.9}{\mathrm{T}}}\Vdash\scaleobj{0.9}{\mathrm{T}}\raise
0.02205pt\hbox{$\ulcorner$}\hbox{$\lambda$}\raise
0.02205pt\hbox{$\urcorner$}\leftrightarrow\neg\scaleobj{0.9}{\mathrm{T}}\raise
0.02205pt\hbox{$\ulcorner$}\hbox{$\lambda$}\raise
0.02205pt\hbox{$\urcorner$};$
$\displaystyle\mathfrak{M}_{\scaleobj{0.9}{\mathrm{T}}}\Vdash\lambda\leftrightarrow\bot.$

Of course, in classical logic this would lead to a contradiction, but not so
in our framework: in our case this simply reflects the fact that neither
$\lambda$ nor $\neg\lambda$ can be a member of a consistent $\mathcal{L}_{\rm
T}^{-}$-fixed point.

Thus far we have focused on general rules and principles that are valid in
truth structures and the Kripkean truth models independently of the
admissibility condition assumed. However, depending on the admissibility
condition assumed different truth principles will come out as true on the
structure $\mathfrak{M}_{\scaleobj{0.9}{\mathrm{T}}}$, as these truth
principles can be seen as expressing particular closure conditions for the
interpretations of the truth predicate. For ease of presentation we take the
language of ${\sf PA}$, $\mathcal{L}_{\sf PA}$, to be the language of our
syntax theory, that is, we handle syntax via arithmetization. We assume an
extension of the usual signature $\\{{\sf 0},{\sf S},{\sf+},{\sf\times}\\}$ by
finitely many function symbols for specific primitive recursive functions and
$\mathcal{L}^{\rightarrow}_{\sf PA}$ the extension of $\mathcal{L}_{\sf PA}$
with the conditional connective. We adopt Feferman’s (1960) dot-notation in
representing syntax and let the set of $\mathcal{L}_{\rm T}$-sentence,
formulas, and terms represent themselves. Importantly, the function symbol
$\dot{\eta}$ (with $\eta$ being a placeholder for the argument of the
function) represents the function which takes Gödel numbers of expressions as
arguments and outputs the Gödel number of the numeral of the Gödel number.
With these preliminaries out of the way we focus on the following truth
principles:

  1. (a)

$\forall x\left({\sf
Sent}(x)\rightarrow(\scaleobj{0.9}{\mathrm{T}}x\wedge\scaleobj{0.9}{\mathrm{T}}\oalign{$\neg
x$\cr\hfil.\hfil}\rightarrow\bot)\right)$;

  2. (b)

$\forall x\left({\sf
Sent}(x)\rightarrow(\scaleobj{0.9}{\mathrm{T}}\oalign{$\neg$\cr\hfil.\hfil}x\leftrightarrow\neg\scaleobj{0.9}{\mathrm{T}}x)\right)$;

  3. (c)

$\forall x\left({\sf
Sent}(x)\rightarrow(\scaleobj{0.9}{\mathrm{T}}\oalign{$\neg$\cr\hfil.\hfil}\oalign{$\neg$\cr\hfil.\hfil}x\leftrightarrow\scaleobj{0.9}{\mathrm{T}}x)\right)$;

  4. (d)

$\forall x\forall y\left({\sf Sent}(x)\wedge{\sf
Sent}(y)\rightarrow(\scaleobj{0.9}{\mathrm{T}}(x\oalign{$\wedge$\cr\hfil.\hfil}y)\leftrightarrow\scaleobj{0.9}{\mathrm{T}}x\wedge\scaleobj{0.9}{\mathrm{T}}y)\right)$;

  5. (e)

$\forall x\forall y\left({\sf Sent}(x)\wedge{\sf
Sent}(y)\rightarrow(\scaleobj{0.9}{\mathrm{T}}\oalign{$\neg$\cr\hfil.\hfil}(x\oalign{$\wedge$\cr\hfil.\hfil}y)\leftrightarrow\scaleobj{0.9}{\mathrm{T}}\oalign{$\neg$\cr\hfil.\hfil}x\vee\scaleobj{0.9}{\mathrm{T}}\oalign{$\neg$\cr\hfil.\hfil}y)\right)$;

  6. (f)

$\forall x\forall v\left({\sf Frml}^{1}(x)\wedge{\sf
Var}(v)\rightarrow(\scaleobj{0.9}{\mathrm{T}}\oalign{$\forall$\cr\hfil.\hfil}vx\rightarrow\forall
y\scaleobj{0.9}{\mathrm{T}}x(\dot{y}/v))\right)$;

  7. (g)

$\forall x\forall v\left({\sf Frml}^{1}(x)\wedge{\sf
Var}(v)\rightarrow(\neg\forall
y\scaleobj{0.9}{\mathrm{T}}x(\dot{y}/v)\rightarrow\scaleobj{0.9}{\mathrm{T}}\oalign{$\neg$\cr\hfil.\hfil}\oalign{$\forall$\cr\hfil.\hfil}vx)\right)$;

  8. (h)

$\forall x\forall y\left({\sf Sent}(x)\wedge{\sf
Sent}(y)\rightarrow(\scaleobj{0.9}{\mathrm{T}}(x\oalign{$\rightarrow$\cr\hfil.\hfil}y)\wedge\scaleobj{0.9}{\mathrm{T}}x\rightarrow\scaleobj{0.9}{\mathrm{T}}y)\right)$;

  9. (i)

$\forall x\forall y\left({\sf Sent}(x)\wedge{\sf
Sent}(y)\rightarrow(\scaleobj{0.9}{\mathrm{T}}\oalign{$\neg$\cr\hfil.\hfil}(x\oalign{$\rightarrow$\cr\hfil.\hfil}y)\leftrightarrow\scaleobj{0.9}{\mathrm{T}}x\wedge\scaleobj{0.9}{\mathrm{T}}\oalign{$\neg$\cr\hfil.\hfil}y)\right)$;

  10. (j)

$\forall x\left({\sf
Sent}(x)\rightarrow(\scaleobj{0.9}{\mathrm{T}}\oalign{$\scaleobj{0.9}{\mathrm{T}}$\cr\hfil.\hfil}\dot{x}\leftrightarrow\scaleobj{0.9}{\mathrm{T}}x)\right)$;

  11. (k)

$\forall x\left({\sf
Sent}(x)\rightarrow(\scaleobj{0.9}{\mathrm{T}}\oalign{$\neg$\cr\hfil.\hfil}\oalign{$\scaleobj{0.9}{\mathrm{T}}$\cr\hfil.\hfil}\dot{x}\leftrightarrow\scaleobj{0.9}{\mathrm{T}}\oalign{$\neg$\cr\hfil.\hfil}x)\right)$.

Principle (a) says that truth predicate is consistent, (b) that a sentence is
false iff it is not true, while (c) to (g) assert that the truth predicate is
closed under strong Kleene logic. Principles (h) and (i) state that the truth
predicate is closed under modes ponens and the compositional rules for negated
conditionals. (j) and (k) tell us that interpretation of the truth predicate
is closed under the ${\sf K3}\scaleobj{0.9}{\mathrm{T}}$ truth-introduction
rules. We arrive at the following picture:

######  Lemma 30.

Let $\mathfrak{M}_{\scaleobj{0.9}{\mathrm{T}}}=(D,X\times Y_{f},H_{\Phi_{\sf
e}})$ be a Kripkean truth structure. Then the following principles are true in
$\mathfrak{M}_{\scaleobj{0.9}{\mathrm{T}}}$:

  * •

(a)-(g), if ${\sf e}\in\\{{\sf K3},{\sf N3},{\sf Nve}\\}$;

  * •

(a)-(i), if ${\sf e}={\sf N3}$;

  * •

(a)-(g), (j) and (k), if ${\sf e}={\sf Nve}$;

Lemma 30 is an immediate consequence of the properties forced on the
interpretation of the truth predicate by the various admissibility conditions.
Interestingly, it highlights again the fact that the interplay of the truth
predicate and the conditional always requires us to choose between a the
logicality of the conditional vs the logicality of truth: as an immediate
consequence of Curry’s paradox, if both (h) and (j) hold for a given structure
$\mathfrak{M}_{\scaleobj{0.9}{\mathrm{T}}}$, then
$\mathfrak{M}_{\scaleobj{0.9}{\mathrm{T}}}$ cannot contain Kripkean truth
models.

Attentive readers may note the absence of the converse directions of
Principles (f) and (g). The converse direction of (f) would require all truth
interpretations to be $\omega$-complete, whereas the converse direction of (g)
requires interpretations to have witnesses for every existential
statement.272727These two properties coincide in classical logic for maximally
consistent sets of sentences, but they do not coincide for saturated sets of
non-classical logics. Enforcing these properties onto to the interpretations
of the truth predicate leads to non-compact admissibility conditions, which
means that Proposition 19 will no longer be applicable. However, in stark
contrast to the classical case (cf. Example 17) we can still find fixed
points, if we adopt admissibility conditions that require admissible
interpretation to be $\omega$ ${\sf e}$-saturated, that is, interpretation
$\omega$-complete saturated sets that have the Henkin property. We leave the
discussion of such non-compact admissibility conditions to a sequel paper in
which we also develop and investigate axiomatic theories of truth in three-
valued Nelson logic ${\sf N3}$.

##  8 Complexity

We now turn to investigating the recursion-theoretic complexity of the
Kripkean fixed points over a structure $\mathfrak{M}=(D,X,H)$. Recall that, as
remarked at the end of Section 6 we can conceive of our construction as an
inductive definition in the parameter $Y_{f}$ rather than a simultaneous
inductive definition of a fixed point of $\theta$ and $\Theta$. We now show
that for every $\theta_{\mathfrak{M}}$-fixed point $(Y_{f},f)$ and all $J\in
X$ the set $f(J)$ is $\Pi^{1}_{1}$-hard in the parameters $Y_{f}$,
$\mathfrak{M}$, and $\Phi$. As a corollary this also tells us that the minimal
fixed point of $\theta_{\mathfrak{M}}$ will be $\Pi^{1}_{1}$-complete if the
relevant parameters are arithmetically definable.

When investigating the recursion-theoretic complexity it is convenient to
think of $\mathcal{L}$ as an extension of the language of arithmetic
$\mathcal{L}_{\sf PA}$ (see Section 7 for details) extended by further,
possibly partial, predicate constants $P_{1},P_{2},\ldots$. The interpretation
of the truth predicate will then consist of a set of Gödel numbers of
sentences of $\mathcal{L}_{\rm T}$, i.e., a set of integers.

To show that for every $\theta_{\mathfrak{M}}$-fixed point $f$ and all $J\in
X$ the set $f(J)$ is $\Pi^{1}_{1}$-hard, we show that every $\Pi^{1}_{1}$-set
$A$ is $1$-reducible to $f(J)$, i.e., there is a recursive $1-1$ function
$\pi$ such that

$\displaystyle n\in A\Leftrightarrow\pi(n)\in f(J).$

Indeed, following the outlines of Welch, (2015) (see also Fischer et al.,,
2015) we establish a stronger claim, that is, we show that
$[\theta_{\mathfrak{M}}(f)](J)$ is $\Pi^{1}_{1}$-hard for any
$\theta_{\mathfrak{M}}$-sound valuation function pair $(Y_{f},f)$, where in
this context $(Y_{f},f)$ is called $\theta_{\mathfrak{M}}$-sound iff $Y_{f}$
and $f$ satisfy the conditions of Proposition 19, that is, iff starting with
the set $Y_{f}$ and valuation function $f$ we can inductively define a fixed
point of $\theta$.

######  Lemma 31.

For ${\sf e}\in\\{{\sf c},{\sf K3},{\sf N3},{\sf Nve},\\}$ let
$\mathfrak{M}_{\scaleobj{0.9}{\mathrm{T}}}=(D,X\times Y_{f},H_{\Phi_{\sf e}})$
be a truth structure with $Y_{f}\in{\sf Adm}_{\mathfrak{M}}$ and $(Y_{f},f$
$\theta^{\Phi_{\sf e}}_{\mathfrak{M}}$-sound. Let $A$ be $\Pi^{1}_{1}$-set of
integers. Then there exists a recursive $1-1$-function $\pi$ such that for all
$J\in X$:

$\displaystyle n\in A\Leftrightarrow\pi(n)\in[\theta^{\Phi_{\sf
e}}_{\mathfrak{M}}f](J).$

###### Proof.

See Appendix B. ∎

######  Proposition 32.

Let $\mathfrak{M}_{\scaleobj{0.9}{\mathrm{T}}}=(D,X\times Y_{f},H_{\Phi_{\sf
e}})$ be a Kripkean truth structure and $f$ a fixed point of
$\theta_{\mathfrak{M}}$. Then $f(J)$ is a $\Pi^{1}_{1}$-hard set of integers
for ${\sf e}\in\\{{\sf c},{\sf K3},{\sf N3},{\sf Nve}\\}$.

###### Proof.

Direct corollary of Lemma 31.∎

Proposition 32 provides us with a lower bound of the fixed points of the
$\theta$-operator relative to a Kripkean Truth Structure. Notice, however,
that the $\theta$-operator can be defined using a $\Pi^{1}_{1}$-formula. This
means that the minimal fixed point defined on the set $\mathcal{B}$ starting
from the minimal valuation function $f_{0}$ with $f_{0}(J)=\emptyset$ for all
$J\in X$ will be a $\Pi^{1}_{1}$-complete set, if the relevant parameters can
be arithmetically defined.

As discussed in Fischer et al., (2015) Lemma 31 implies that an
$\mathbb{N}$-categorical axiomatization of the strong Kleene supervaluation
fixed points is out of reach: it is not possible to single out these fixed
point relative to an intended model using a r.e. set of sentences. However, it
is possible to, following the outlines of Stern, (2018), provide a variant
construction that shares many characteristics of the present framework, but
which is of reduced recursion-theoretic complexity and lends itself to
$\mathbb{N}$-categorical axiomatizations in the sense of Fischer et al.,
(2015). We leave a thorough discussion of the variant construction for another
occasion.

##  9 Modal strong Kleene supervaluation: indicative and subjunctive
conditionals

In the preceding sections we focussed on constructing a model for naive truth
for a language comprising a conditional that can be employed for conditional
reasoning in logic and mathematics, that is, the conditional is to be for the
partial logic what material implication is for classical logic. However,
indicative and subjunctive natural language conditionals are not, at least in
most cases, governed by the rules and principles of material implication or
the ${\sf N3}$-conditional. This shows in the paradoxes of material
implications many of which apply to the ${\sf N3}$-conditional.282828See,
e.g., Egré and Cozic, (2016); von Fintel, (2011) for overview on the semantics
of (natural language) conditionals.

Assuming classical logic the most prominent semantic analysis of indicative
and subjunctive conditionals conceives of them in terms of strict or variably-
strict implication.292929The notion of strict implication dates back to Lewis,
(1914), whereas the notion of variably-strict implication originates with
Stalnaker, (1968) and Lewis, (1973). Both strict and variably-strict
implication amount to forcing the corresponding material implication to be
true in all words of a given set. The difference between strict and variably-
strict implication is in how the set is specified and the set itself. In the
same way we can define strict and variable-strict implication in a suitable
possible world framework, one can define the strict and variably-strict ${\sf
N3}$-conditional on the basis of the ${\sf N3}$-conditional and thereby obtain
a semantic interpretation for indicative and subjunctive conditionals. To
provide an adequate semantics for natural language conditionals thus requires
working in a modal semantics, more specifically, a version of possible world
semantics enriched with an world-relative ordering relation on the set of
possible worlds. In this section we show how to apply strong Kleene
supervaluation to such semantics, that is, we show how to build supervaluation
structures relative to so-called ordering frames and show how Kripkean truth
structures can be constructed relative to these modal strong Kleene
supervaluation structures.

From now on our base language will be the language $\mathcal{L}^{\rhd}$ which
extends $\mathcal{L}$ by a conditional connective $\rhd$ and a one-place modal
operators $\Box$. The set of well-formed formulas ${\sf
Form}_{\mathcal{L}^{\rhd}}$ is specified by

$\displaystyle\varphi$
$\displaystyle::=(s=t)\,|\,P^{m}_{j}t_{1},\ldots,t_{m}\,|\,\Box\varphi\,|\,\neg\varphi\,|\,\varphi\wedge\varphi\,|\,\varphi\rightarrow\varphi)\,|\,\varphi\rhd\varphi\,|\,\forall
x\varphi$

with $j,m\in\omega$, $s,t,t_{1},\ldots,t_{m}\in{\sf Term}_{\mathcal{L}}$, and
$x\in{\sf Var}_{\mathcal{L}}$. In addition to the notational abbreviations
(defined symbols) we introduced for $\mathcal{L}$ we define the strict
implication connective :

$\displaystyle(\strictif)$
$\displaystyle\varphi\strictif\psi:\leftrightarrow\Box(\varphi\rightarrow\psi).$

A key notion in the semantics for natural-language conditionals such as
indicative conditionals is that of an Ordering frame:

######  Definition 33 (Ordering frame).

Let $F=(W,R,\preceq,D)$ be a tuple with $W,D\neq\emptyset$, $R\subseteq
W\times W$ and $\preceq\,\subseteq\leftidx{{}^{W}}{(W\times W)}$. Then $F$ is
called an ordering frame iff for all $w\in W$:

  * •

$\preceq_{w}$ is a weak total order on the set $\\{v\in W\,|\,wRv\text{ or
}v=w\\}$.303030Usually more assumption are made on behalf of $\preceq$, but
these are irrelevant for our construction: further conditions can be freely
imposed to the framework.

While a strong Kleene supervaluation structure for $\mathcal{L}$ was built on
a non-empty domain, strong Kleene supervaluation structures for the modal
language $\mathcal{L}^{\rhd}$, i.e., modal supervaluation structures will be
constructed relative to an ordering frame. In this setting interpretation
functions will assign interpretations to the descriptive vocabulary—henceforth
denoted by ${\sf Descr}_{\mathcal{L}}$—relative to a possible world, i.e.,
relative to elements of $W$.

######  Definition 34 (Interpretation, Ordering).

Let $F$ be an ordering frame. A function $I$ with domain $W\times{\sf
Descr}_{\mathcal{L}}$ for some language $\mathcal{L}$ is called a modal
interpretation iff $I_{w}$ is an ${\sf k}$-interpretation for all $w\in W$
(cf. Definition 3). The set of all interpretations over some ordering frame
$F$ is denoted by ${\sf Int}_{F}$. For $I,J\in{\sf Int}_{F}$ we say $I\leq J$
iff $I^{+}(w,P)\subseteq J^{+}(w,P)$ and $I^{-}(w,P)\subseteq J^{-}(w,P)$ for
all $w\in W$ and $P\in{\sf Pred}$.

######  Definition 35 (Modal supervaluation structure).

A tuple $\mathfrak{M}=(F,X,H)$ is called a modal supervaluation structure iff

  1. (i)

$F$ is an ordering frame;

  2. (ii)

$X\subseteq{\sf Int}_{F}$ such that $J(t,w)=I(t,v)$ for $J,I\in X$ and $w,v\in
W$ and all closed terms $t$;

  3. (iii)

$H\subseteq X\times X$ such that

     * •

$H$ is transitive and reflexive;

     * •

if $(x,y)\in H$, then $x\leq y$.

(i) imposes that the interpretation of closed terms is rigid. This is to keep
things as simple as possible. If this assumption is dropped one needs to deal
with the usual complication that arise in quantified modal logic due to non-
rigid terms.313131See Garson, (2001) for discussion and Halbach, (2021) for
some further complications that arise for modal languages that contain a truth
predicate. Consequently, variable assignments are not world relative, that is,
the notion of an assignment is that of Definition 5.

The denotation of a term is defined as in Definition 6 except for the
additional (but innocuous) world parameter. With these definitions in place we
can specify the truth conditions for formulas of the language
$\mathcal{L}^{\rhd}$. We shall not repeat the truth conditions for the logical
connective already spelled out in Definition 7. These remain unchanged but
relativized to a world parameter, e.g., the truth condition of sentence of the
form $\varphi\rightarrow\psi$ will be:

$\displaystyle\mathfrak{M},J,w\Vdash\varphi\rightarrow\psi[\beta]$
$\displaystyle\,{\rm iff}\,\forall J^{\prime}((J,J^{\prime})\in
H\,\&\,\mathfrak{M},J^{\prime},w\Vdash\varphi[\beta]\Rightarrow\mathfrak{M},J^{\prime},w\Vdash\psi[\beta]).$

######  Definition 36 (Truth in a modal supervaluation structure).

Let $\mathfrak{M}=(F,X,H)$ be a modal supervaluation structure with $J\in
X,w\in W$. Then a formula $\varphi\in\mathcal{L}^{\rhd}$ is true in
$\mathfrak{M}$ at $w$ and $J$ under assignment $\beta$
($\mathfrak{M},w,J\Vdash\varphi[\beta]$) iff

$\displaystyle\forall v\in
W(wRv\Rightarrow\mathfrak{M},v,J\Vdash\psi[\beta]),$ $\displaystyle\,{\rm
if}\,\varphi\doteq\Box\psi;$ $\displaystyle\exists v\in
W(wRv\,\&\,\mathfrak{M},v,J\Vdash\neg\psi[\beta]),$ $\displaystyle\,{\rm
if}\,\varphi\doteq\neg\Box\psi;$ $\displaystyle{\sf Imposs}_{w}(\psi)\,{\rm
or}\,\exists v(wRv\,\&\,\mathfrak{M},v,J\Vdash\psi[\beta]\,\&\,\forall
u(u\preceq_{w}v\Rightarrow\mathfrak{M},u,J\Vdash\psi\rightarrow\chi[\beta]),$
$\displaystyle{\rm if}\,\varphi\doteq\psi\rhd\chi;$ $\displaystyle\exists
v(wRv\,\&\,\mathfrak{M},v,J\Vdash\psi[\beta])\,\&\,$ $\displaystyle\forall
v(wRv\,\&\,\mathfrak{M},v,J\Vdash\psi[\beta]\Rightarrow\exists
u(u\prec_{w}v\,\&\,wRv\,\&\,\mathfrak{M},u,J\Vdash\psi\wedge\neg\chi[\beta],$
$\displaystyle{\rm if}\,\varphi\doteq\neg(\psi\rhd\chi).$

${\sf Imposs}_{w}(\psi)$ is short for $\forall
v(wRv\Rightarrow\mathfrak{M},v,J\Vdash\neg\psi)$. Finally, we write
$\mathfrak{M},w,J\Vdash\varphi$ iff $\mathfrak{M},w,J\Vdash\varphi[\beta]$
holds for all assignments $\beta$ and $\mathfrak{M},w\Vdash\varphi$
($\mathfrak{M}\Vdash\varphi$) iff $\mathfrak{M},w,J\Vdash\varphi$ for all
$J\in X$ (and $w\in W$).323232To be precise, for Definition 36 to be well-
formed the truth-conditions for $\psi\rhd\chi$ and $\neg(\psi\rhd\chi)$,
should say  $\,\forall J^{\prime}((J,J^{\prime})\in
H\Rightarrow(\mathfrak{M},u,J^{\prime}\Vdash\psi[\beta]\Rightarrow\mathfrak{M},u,J^{\prime}\Vdash\chi[\beta])$
and  $\mathfrak{M},u,J\Vdash\psi[\beta]\text{ and
}\mathfrak{M},u,J\Vdash\psi\neg\chi[\beta]$ where we have written
$\mathfrak{M},u,J\Vdash\psi\rightarrow\chi[\beta]$ and
$\mathfrak{M},u,J\Vdash\psi\wedge\neg\chi[\beta]$ respectively.

Definition 36 gives truth-conditions for the $\rhd$ that are widely endorsed
in the literature (cf., e.g., Egré and Cozic,, 2016; von Fintel,, 2011, for
discussion). We take it that to the extent that $\rightarrow$ is an adequate
conditional connective for mathematical reasoning, $\rhd$ (or, possibly, ) are
adequately representing indicative and subjunctive natural language
conditionals.

###  9.1 Modal Truth Structures

With the semantics for $\mathcal{L}^{\rhd}$ in place modal truth structures
for the language $\mathcal{L}^{\rhd}_{\scaleobj{0.9}{\mathrm{T}}}$ can be
constructed along the lines of Section 5 with the exception that all
definitions need to be generalized to allow for a world parameter. As a
consequence the construction follows the pattern of the generalization of
Kripke’s theory of truth to modal languages proposed in Halbach and Welch,
(2009); Stern, (2014, 2016, 2024). We shall state the most important
definitions and propositions, but will omit proofs, as these are immediate
generalizations of those of Sections 5 and 6.

In the modal setting valuation functions are no longer functions from ${\sf
Int}_{F}$ to $\mathcal{P}({\sf Sent})$ but functions from ${\sf Int}_{F}\times
W$ to $\mathcal{P}({\sf Sent})$:

######  Definition 37 (Admissible Valuations—modal case).

A valuation on a supervaluation structure $\mathfrak{M}=(F,X,H)$ is a function
$f:W\times X\rightarrow\mathcal{P}({\sf Sent}_{\mathcal{L}_{\rm T}})$. The set
of all valuation on $\mathfrak{M}$ is denoted by ${\sf Val}_{\mathfrak{M}}$. A
valuation function is admissible for $\mathfrak{M}$ iff

  1. (i)

$f$ is consistent, that is, $f(w,J)\cap\overline{f(w,J)}=\emptyset$ for all
$w\in W$ and $J\in X$ where

$\overline{f(w,J)}:=\\{\varphi\,|\,\neg\varphi\in f(w,J)\\};$

  2. (ii)

for all $J\in X,w\in W$ and $\varphi\in\mathcal{L}$:

$\displaystyle\text{if }\varphi\in f(w,J),$ $\displaystyle\text{ then
}\mathfrak{M},w,J\Vdash\varphi$

  3. (iii)

for all $J,J^{\prime}\in X$, if $(J,J^{\prime})\in H$, then $f(w,J)\subseteq
f(w,J^{\prime})$ for all $w\in W$.

The set of admissible valuation functions for a given model $\mathfrak{M}$ is
denoted by $\mathcal{B}_{\mathfrak{M}}$.

Of course, the addition of the world parameter also means that the definition
of the ordering on ${\sf Val}_{\mathfrak{M}}$ has to be modified: for
$f,g\in{\sf Val}_{\mathfrak{M}}$ set

$\displaystyle f\leq g$ $\displaystyle:\leftrightarrow\forall J\in X\forall
w\in W(f(w,J)\subseteq g(w,J)).$

In contrast, the definition of the admissibility condition $\Phi$ and the
ordering $\leq_{\Phi}$ remain unaltered and are simply those of Definition 10.
We can then define a (grounded) modal truth structure as in Definition 12 but
where the truth structure is built on a modal supervaluation structure, i.e.,
a tuple consisting of a frame, a set of interpretations on the frame and an
admissibility relation.

######  Definition 38 (Truth Structure, Grounded Truth Structure—modal case).

Let $\mathfrak{M}=(F,X,H)$ be a supervaluation structure. Then the tupel
$(F,X\times Y,H_{\Phi})$ with $Y\subseteq\mathcal{B}_{\mathfrak{M}}$ and where
$H_{\Phi}$ is restricted to $X\times Y$ is called a truth structure over
$\mathfrak{M}$. If $Y\subseteq{\sf Val}_{\mathfrak{M}}$ has a $\leq$-minimal
valuation function, that is, an $f$ in $Y$ such that $f\leq g$ for all $g\in
Y$ and $Y\cap\Phi(f)\neq\emptyset$, then $(F,X\times Y,H_{\Phi})$ is called a
grounded truth structure and $Y$ is called a grounded set of valuations. The
set of all grounded sets of valuations is denoted by ${\sf
Adm}_{\mathfrak{M}}$.

We can then use the construction of Section 6 to generalize Propositions 19,
25 and Corollaries 26, 27 to show the existence of Kripkean truth structure
for the language $\mathcal{L}^{\rhd}_{\scaleobj{0.9}{\mathrm{T}}}$. To this
effect we let $\mathcal{B}_{\scaleobj{0.9}{\mathrm{T}}}$ be the set of
valuation functions $f$ such that for all $J\in X$:

  * •

there a exists a valuation function $f_{\Box}:W\rightarrow\mathcal{P}({\sf
Sent}_{\mathcal{L}_{\Box}}$ such that $f_{\Box}$ s a modal strong Kleene fixed
point on the frame $F$ in the sense of Halbach and Welch, (2009); Stern,
(2014, 2016) and $f(w,J)\cap\mathcal{L}_{\Box}=f_{\Box}(w)$ and where
$\mathcal{L}_{\Box}$ is the $\rightarrow$\- and $\rhd$-free fragment of
$\mathcal{L}^{\rhd}_{\scaleobj{0.9}{\mathrm{T}}}$.

  * •

$f(w,J)$ is ${\sf K3}$-saturated for all $w\in W$.

We obtain the following results:

######  Proposition 39 (Modal Fixed Points).

Let $\mathfrak{M}=(F,X,H)$ be a modal supervaluation structure and ${\sf
e}\in\\{{\sf c},{\sf K3},{\sf N3},{\sf Nve}\\}$. Let
$f\in\mathcal{B}^{\scaleobj{0.9}{\mathrm{T}}}_{\mathfrak{M}}$ be minimal in
$\mathcal{B}^{\scaleobj{0.9}{\mathrm{T}}}_{\mathfrak{M}}$ and set
$Y_{f}=\mathcal{B}^{\scaleobj{0.9}{\mathrm{T}}}_{\mathfrak{M}}$. Then there
exists $g\in Z_{g}\subseteq Y_{f}$ such that $\theta^{\Phi_{\sf
e}}_{\mathfrak{M}}(Z_{g},g)=g$ and $\Theta^{\Phi_{\sf
e}}_{\mathfrak{M}}(Z_{g})=Z_{g}$.

######  Proposition 40 (Kripkean Truth Structures for
$\mathcal{L}^{\rhd}_{\scaleobj{0.9}{\mathrm{T}}}$).

Let $\mathfrak{M}=(F,X,H)$ be a modal supervaluation structure and ${\sf
e}\in\\{{\sf c},{\sf K3},{\sf N3},{\sf Nve}\\}$. Then there exists a grounded
truth set $Y_{f}$ and admissible valuation function $f$ such that for all
$\varphi\in{\sf Sent}_{\mathcal{L}_{\rm T}^{\rhd}}$

$\displaystyle(F,X\times Y_{f},H_{\Phi_{\sf e}}),J_{f},w\Vdash\varphi$
$\displaystyle\text{ iff }(D,X\times Y_{f},H_{\Phi_{\sf
e}}),J_{f},w\Vdash\scaleobj{0.9}{\mathrm{T}}\raise
0.0pt\hbox{$\ulcorner$}\hbox{$\varphi$}\raise 0.0pt\hbox{$\urcorner$}$

for all $J\in X$ and $w\in W$.

As in the non-modal case we can find a reflexive Kripkean truth structure by
constructing a $\Phi_{\sf e}$-generated substructure:

######  Corollary 41.

Let ${\sf e}\in\\{{\sf K3},{\sf N3},{\sf Nve}\\}$ and $(F,X\times
Y_{f},H_{\Phi_{\sf e}})$ be a modal Kripkean Truth structure and let
$Y^{\leq_{\Phi_{\sf e}}}_{f}:=\\{g\in Y_{f}\,|\,f\leq_{\Phi_{\sf e}}g\\}$.
Then the structure $(F,X\times Y^{\leq_{\Phi_{\sf e}}}_{f},H_{\Phi_{\sf e}})$
is a modal Kripkean truth structure such that $H_{\Phi_{\sf e}}$ is reflexive
on $X\times Y^{\leq_{\Phi_{\sf e}}}_{f}$.

The propositions establish that the strong Kleene supervaluation framework can
be used to construct Kripkean truth models for natural language conditionals
as well as “logical” conditionals, i.e., conditionals that are use to
formalize conditional reasoning in logic and mathematics.This suggests that
the framework is well suited for interpreting interesting and expressively
rich fragments of natural language, i.e., fragments that allow for self-
reference but also enable us to express different conditionals and conditional
reasoning.

##  10 Conclusion and Outlook

In this paper we set out to apply a Kripke’s theory of truth to languages that
comprise adequate logical conditionals as well as indicative and subjunctive
conditionals. This led us to the framework of strong Kleene supervaluation and
modal strong Kleene supervaluation for which we showed the existence of
Kripkean truth models. Importantly, the framework is much wider: strong Kleene
supervaluation can be applied not only to conditionals but to other non-
monotone notions. Indeed, as we discuss in companion paper the framework lends
itself particular well to giving a semantics and truth theory for $\langle
1,1\rangle$-quantifiers.

There remain open questions and many possible further research avenues to
explore. For one, as mentioned in Section 7 in future work we intend to
provide fixed point results for non-compact admissibility relations according
to which admissible precisifications are required to be $\omega$-complete
saturated sets that have the Henkin property. This would guarantee that both
the universal and the existential quantifier commute with the truth predicate
and open up the possibility of attractive axiomatic truth theories formulated
in ${\sf N3}$ Nelson logic. We will discuss such theories in a sequel to this
study. For another, it seems interesting to investigate variants of our
framework that use alternative falsity conditions for the conditional, in
particular, the falsity condition employed in intuitionistic logic. Arguably,
this would re-establish symmetry between the ruth and falsity condition of the
conditional. Moreover, if symmetric strong Kleene logic ${\sf KS3}$ is
assumed, the resulting conditional should allow for contraposition. However,
the details require carefully study and are left for future research.

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