Moorean Phenomena in Epistemic Logic?
Wesley H. Holliday and Thomas F. Icard, III
Department of Philosophy
Stanford University
Stanford, California, USA
Abstract
A well-known open problem in epistemic logic is to give a syntactic characterization of the successful
formulas. Semantically, a formula is successful if and only if for any pointed model where it is true, it
remains true after deleting all points where the formula was false. The classic example of a formula that is
not successful in this sense is the “Moore sentence” p ∧ ¬p, read as “p is true but you do not know p.”
Not only is the Moore sentence unsuccessful, it is self-refuting, for it never remains true as described. We
show that in logics of knowledge and belief for a single agent (extended by S5), Moorean phenomena are
the source of all self-refutation; moreover, in logics for an introspective agent (extending KD45), Moorean
phenomena are the source of all unsuccessfulness as well. This is a distinctive feature of such logics, for with
a non-introspective agent or multiple agents, non-Moorean unsuccessful formulas appear. We also consider
how successful and self-refuting formulas relate to the Cartesian and learnable formulas, which have been
discussed in connection with Fitch’s “paradox of knowability.” We show that the Cartesian formulas are
exactly the formulas that are not eventually self-refuting and that not all learnable formulas are successful.
In an appendix, we give syntactic characterizations of the successful and the self-refuting formulas.
Keywords: epistemic logic, successful formulas, Moore sentence
1
Introduction
According to the epistemic interpretation of modal logic, the points in a modal
model represent ways the world might be, consistent with an agent’s information.
In this context, “learning” a formula amounts to eliminating those points in the
model where the formula is false. The resulting submodel represents the agent’s
information state after learning has occurred. Some formulas—though not all—
remain true whenever they are learned. A well-known open problem [12,4,5,6,7,2]
in epistemic logic is to give a syntactic characterization of these successful formulas.
Partial results have been obtained (Section 2), but a full solution has proven elusive.
The classic example of an unsuccessful formula is the Moore sentence p ∧ ¬p,
read as “p is true but you do not know p.” This example is a second-person variation
of G.E. Moore’s famous puzzle [16] involving the paradoxical first-person assertion,
“p is true but I do not believe p.” Hintikka devoted a chapter of his seminal 1962
monograph Knowledge and Belief [13] to an analysis of such sentences, including the
second-person Moore sentence. Hintikka observed its unsuccessfulness as follows:
“You may come to know that what I said was true, but saying it in so many words
? In L. Beklemishev, V. Goranko and V. Shehtman, eds., Advances in Modal Logic, Volume 8, 178-199,
College Publications, 2010. Minor changes to the published version have been made in this version.
Holliday and Icard
has the effect of making what is being said false” [13, p. 69]. Yet the formal question
of unsuccessfulness did not arise for Hintikka. Only with the advent of Dynamic
Epistemic Logic (see, e.g., [7]) and the idea of learning as model reduction have the
Moorean phenomena and unsuccessfulness been formally related.
The reason the Moore sentence is unsuccessful is that it can only be true in a
model if p is true at some point and false at some other point accessible from the
first. When the agent learns the sentence, all points where p is false are eliminated
from the model, including all witnesses for ¬p, so the sentence becomes false.
This shows that the Moore sentence is not only unsuccessful, it is self-refuting, for
it always becomes false when learned. Related to the self-refuting property of the
Moore sentence is the fact that the sentence cannot be known. Indeed, the Moore
sentence is at the root of Fitch’s famous “paradox of knowability” [11,3,9]: if there is
an unknown truth, then there is an unknowable truth. For if p is true but unknown,
then the Moore sentence p ∧ ¬p is true, but the Moore sentence cannot be known,
because (p ∧ ¬p) is inconsistent with standard assumptions about knowledge.
While the Moore sentence is conspicuously unsuccessful, other unsuccessful formulas are less conspicuous. The formula ¬ (p ∨ q) ∨ (p ∧ (p ∨ ♦q)) is also unsuccessful, as we show in Example 5.12 below, but is the reason Moorean? While
on the surface this formula looks unlike a Moore sentence, it is in fact possible to
transform the formula to reveal its Moorean character. Indeed, we will prove that
for a wide range of logics, such a transformation is possible for every unsuccessful
formula. However well-disguised, their nature is always Moorean.
In Section 2 we establish notation, give the definitions of successful, self-refuting,
etc., and review what is already known in the literature on successful formulas. In
Section 3 we show that in logics of knowledge and belief for a single agent (extended
by S5), Moorean phenomena are the source of all self-refutation; moreover, in logics
for an introspective agent (extending KD45), Moorean phenomena are the source
of all unsuccessfulness as well. This is a distinctive feature of such logics, for as we
show in Section 4, in logics for a non-introspective agent or multiple agents, nonMoorean unsuccessful formulas appear. Finally, in Section 5 we relate successful
and self-refuting formulas to the Cartesian and learnable formulas, which have been
discussed in connection with Fitch’s paradox, and to the informative, eventually
self-refuting, super-successful formulas, which we introduce here. In Appendix A
we give syntactic characterizations of the successful and the self-refuting formulas.
2
Preliminaries and Previous Results
Throughout we work with a fixed, unimodal language with and its dual ♦, and an
infinite set Prop of propositional variables. The expression ♦+ ϕ abbreviates ♦ϕ ∧ ϕ.
We use the standard semantics of modal logic, where a model is a triple hW, R, V i
with W any set of points, R ⊆ W × W any relation, and V : Prop → ℘(W ) a
valuation function. A pointed model is a pair M, w with M a model and w ∈ W .
The satisfaction relation between pointed models and formulas is defined as usual.
It is typical to take KD45 as a logic of belief and S5 as a logic of knowledge.
Apart from the assumptions that one does not believe or know anything inconsistent, and that what is known is true, these logics assume positive (axiom 4) and
2
Holliday and Icard
negative (axiom 5) introspection: if one believes something, then one believes that
one believes it; if one does not believe something, then one believes that one does
not believe it; and mutatis mutandis for knowledge. Most work on successful formulas has assumed S5. Since our results apply to a wider range of logics, we assume
through Section 3 that we are working with at least KD45, so all of our models
will be serial, transitive, and Euclidean. We call such models quasi-partitions.
Where M0 is a submodel (not necessarily proper) of M, we write M0 ⊆ M.
Rather than studying formulas preserved under arbitrary submodels, we will study
formulas preserved under a special way of taking submodels, as in the following.
Definition 2.1 Given a model M = hW, R, V i, the relativization of M to ϕ is the
(possibly empty) submodel M|ϕ = hW|ϕ , R|ϕ , V|ϕ i of M, where W|ϕ = {w ∈ W :
M, w ϕ}, R|ϕ is R restricted to W|ϕ , and V|ϕ (p) = V (p) ∩ W|ϕ .
Definition 2.2 A formula ϕ is successful (in logic L) iff for every pointed model
(of L), M, w ♦+ ϕ implies M|ϕ , w ϕ. A formula is unsuccessful (in L) iff it is
not successful. A formula is self-refuting (in L) iff for every pointed model (of L),
M, w ♦+ ϕ implies M|ϕ , w 2 ϕ. 1
In the standard definitions of successful and self-refuting formulas, where L is
assumed to be S5, the precondition only requires that ϕ be true at w. Since we are
also working with KD45, we additionally require that ϕ be true at an accessible
point, so that M|ϕ is a quasi-partition provided M is. Our definition reduces to the
standard one in the case of S5. In either case, unsatisfiable formulas are self-refuting
and successful. In the case of KD45, a satisfiable formula such as p ∧ ¬p, read as
“p is true but you believe ¬p,” is self-refuting and successful, since ♦+ (p ∧ ¬p) is
unsatisfiable. While it may be more intuitive to require the satisfiability of ♦+ ϕ for
a successful ϕ, we will follow the standard definition in not requiring satisfiability.
The following lemma relates success and self-refutation across different logics.
Lemma 2.3 Let L be a sublogic of L0 . If ϕ is unsuccessful in L0 , then ϕ is unsuccessful in L, and if ϕ is self-refuting in L, then ϕ is self-refuting in L0 .
Proof. Immediate from Definition 2.2, given that models of L0 are models of L.2
An obstacle to giving a simple syntactic characterization of the set of successful
formulas is its lack of closure properties. Successful formulas are not closed under
negation (take ¬p ∨ p), conjunction (take p and ¬p), or implication (use ϕ → ⊥)
[6]. We show in Proposition 5.11 that they are also not closed under disjunction.
Conversely, if a negated formula is successful, the unnegated formula may be unsuccessful (take ¬ (p ∧ ♦¬p)), if a conjunction is successful, some of the conjuncts
may be unsuccessful (take (p ∧ ♦¬p) ∧ ¬p), and if a disjunction is successful, some
or even all of the disjuncts may be unsuccessful (Proposition 5.9 and Example 5.2).
By contrast, the formulas preserved under arbitrary submodels are well-behaved.
The following result was proved independently by van Benthem and Visser [20,8].
A formula is universal iff it can be constructed using only literals, ∧, ∨, and .
1
The term ‘successful’ is used by Gerbrandy [12], by analogy with the success postulate of belief revision,
while the term ‘self-refuting’ is used by van Benthem [3]. Self-refuting formulas have also been called strongly
unsuccessful [2].
3
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→ p)
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and
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∧
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(q
→
p)
∧
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and
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set
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literal
∈
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and
every
set
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literal
l
∈
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that
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quazi-negation
o
literals, one
from
in A, such
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Definition 6. Given a specification
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and Γformula
ϕ inthat
DNF,
define
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,
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q "C,
q
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qProof.
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that is
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and
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M
|ϕ
|ϕ
M|♦p = M|ψ . This is symptomatic
of 6.
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fact,
established
Section
4,
that
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logics
then
for any
(M,
w)each
with
!in
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vaone
∈specification
W
|and
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vsuch
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each
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in
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ni
Definition
Given
Definition
a specification
Given
"C,
N,
A#
formula
"C,
N,
ϕ
and
DNF,
formula
ϕ
literals,
one
from
ΓM,
in6.w
A,
such
that
Σ
does
the
quazi-negation
Proof. If ϕ contains only formulas of
the
form
Kα
!
ϕ |ϕare
ϕ
Hence
M
,
w
!
ϕ.
without both axioms 4 and
5,
there
non-Moorean
sources
!
! ! of unsuccessfulness.
!
M
of
l
and
d
does
not
contain
the
quazi-negation
of
"C,
N,
A#
"C,"C,
N
N,
, A#
A#where
= "C,
Nform
N
isfor
, the
A#
where
setL(M,
of
N
is
the
set
lϕ∈
all
for
literals
which
∈ |NwR
f
l isand
d=does
not
contain
the
quazi-negation
any
literal
Σ.
then
any
w)
with
M,
wof
!in
ϕ,
v ∈ Wlthere
Ifofϕ
a conjunction
in
normal
and
(ϕ)all
$=ofliterals
∅,
then
isN
successful.
qdoes
is a disjunct d inisϕasuch
disjunct
that dd in
ϕ such
not
that
d
does
∼
l
and
not
contain
there
is
∼
a
selection
l
and
there
Σ
of
is
a
Hence
Mcontain
,
w
!
ϕ.
|ϕ
p ∧ ♦q ∧ # (q
→literals,
p) ∧ ♦Γone
(q ∧A,
p) such
!
literals,
each
that
ΓΣ
in does
A,
notthere
that
contain
Σ
not
thei
Proof. one
If ϕ from
contains
onlyinformulas
of
Kα
and
(αquazi-negation
propositional),
Nfrom
andeach
every
Γ ∈such
A,
is does
a!the
literal
lcontain
∈ Γ that
! thesetform
"¬Kα
Mliteral
M
Mliteral
|ϕ
|ϕ
If
ϕ
is
a
conjunction
in
normal
form
and
L
(ϕ)
$=Σ
ofthen
does
not
of
l
contain
and
d
does
the
not
quazi-negation
contain
the
of
quazi-negation
any
in
of
Σ.
any
in
Ifl and
ϕ for
is daany
conjunction
in
normal
form
and
L
(ϕ)
=
$
∅,
then
ϕ
is
successful.
∧ ϕ,
♦q ∧v #
∧ ♦ (q
(M, w) with M,
∈ (q
W → p)
| wR
v ∧=p) v ∈ W
| wRM
l. wp !
p
Hence M|ϕ , w ! ϕ.
If the
ϕ contains
of
the
form
Kαf
Proof. If ϕ Fig.
contains
only Definition
formulas
form Kα
andformulas
6.
Given
a only
specification
N, A#
and
!propositional)
!
"¬Kα
! (α"C,
1.
pProof.ϕof
M M|
M
M M, w! ! ϕ, vM
|ϕ
! w)
then
for
any
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formula. Yet other connections with universal formulas do hold. Qian [18] showed
Proof.
Kα an
q If ϕ contains only formulas of
p qno nested modal
p operators
! the form
that a formula containing
is successful in K iff it can
M
then for any (M, w) with M, w ! ϕ, v ∈ W | wRM
be transformed using a certain algorithm into a universal formula; moreover, a
Hence M|ϕ , w ! ϕ.
“homogeneous” formula, containing no nested modal operators and no propositional
q
q
variables outside the scope of modal operators, is successful
to
p ∧ ♦qin∧K
# iff
(q it
→isp)equivalent
∧ ♦ (q ∧ p)
a universal formula. The restriction to K is essential, for the formula ♦p is successful
in extensions of KD45 (not in K), but it is not equivalent
to any universal formula.
p
As far as we know, there are no other published results on the syntax of successful formulas in the basic modal language. However, successful formulas are often
studied in the more general context of Dynamic Epistemic
Logic (DEL) [1,12,7],
q
where model changing operations are formalized in the logic itself by adding dynamic operators to the language. In the simplest fragment of DEL extending S5,
known as Public Announcement Logic (PAL), there are sentences of the form [ϕ]ψ,
read as “after the announcement of ϕ, ψ is true,” for which satisfaction is defined:
M, w [ϕ]ψ iff M, w ϕ implies M|ϕ , w ψ.
An advantage of the PAL setting is that it allows for simple definitions of success
and self-refutation [6]. Successful formulas are those for which [ϕ]ϕ is valid (in S5),
2
The result in [4] is given for the case of multi-agent epistemic logic with a common knowledge operator,
assuming M is finite. The proof for the single-agent case is trivial and requires no assumption of finiteness.
4
Holliday and Icard
and self-refuting formulas are those for which [ϕ]¬ϕ is valid (in S5). While we will
not deal directly with this language, our work will apply indirectly to PAL given the
following result, due to Plaza [17], which relates PAL to the basic modal language.
Theorem 2.6 Every formula in the language of PAL is equivalent (over partitions)
to a formula in the basic modal language.
Direct results on successful formulas in PAL with multiple knowledge modalities
and the common knowledge operator have been obtained by van Ditmarsch and
Kooi in [6], which also presents an analysis of the role of (un)successful formulas in
a variety of scenarios involving information change.
Another benefit of the PAL definitions of successful and self-refuting is that they
give an upper bound on the complexity of checking a formula for these properties.
The following result including a lower bound for the success problem is due to Johan
van Benthem in correspondence.
Theorem 2.7 The success problem for S5 is coNP-complete.
Proof. For an upper bound, ϕ is successful iff [ϕ] ϕ is valid, and the validity problem
for single-agent PAL is coNP-complete [14]. For a lower bound, we use the following
reduction of the validity problem for S5, which is coNP-complete, to the success
problem for S5: ϕ is valid iff for a new variable p, ψ ≡ p ∧ (♦¬p ∨ ϕ) is successful.
From left to right, if M, w ψ then M|ψ , w p, and since ϕ is valid, M|ψ , w ϕ, so
M|ψ , w ψ. From right to left, take any pointed model M, w. Extend the language
with a new variable p and extend M to M0 with one new point v, related to all other
points, with the same valuation as w for all variables of the old language. Make
p true everywhere except at v. Then since M0 , w p ∧ ♦¬p, we have M0 , w ψ,
and since ψ is successful, we have M0|ψ , w ψ. But v is eliminated in M0|ψ , so
M0|ψ , w 2 ♦¬p. Hence M0|ψ , w ϕ. We conclude that M, w ϕ, for M0|ψ and M
differ only with respect to the valuation of p, which ϕ does not contain.
2
A similar argument shows that ϕ is valid iff for a new variable p, p ∧ (♦¬p ∨ ¬ϕ)
is self-refuting, so the self-refutation problem is coNP-complete given the upper
bound from PAL. From left to right the reduction is obvious, while from right to
left the same extension of M to M0 works. Similar arguments also show that the
success and self-refutation problems are PSPACE-complete for multimodal S5.
3
The Moorean Source of Unsuccessfulness
To show that Moorean phenomena are the source of all unsuccessfulness in logics for
an introspective agent, we proceed via normal forms for such logics. The following
normal form derives from Carnap [10], and Proposition 3.2 is standard.
Definition 3.1 A formula is in normal form iff it is a disjunction of conjunctions
of the form δ ≡ α ∧ β1 ∧ ... ∧ βn ∧ ♦γ1 ∧ ... ∧ ♦γm where α and each γi are
conjunctions of literals and each βi is a disjunction of literals.
Proposition 3.2 For every formula ϕ, there is a formula ϕ0 in normal form such
that ϕ and ϕ0 are equivalent in K45.
5
Holliday and Icard
Theorem 1.7.6.4 of [15] gives the analogue of Proposition 3.2 for S5. Inspection of
the proof shows that the necessary equivalences hold in K45.
We use the following notation and terminology in this section and Appendix A.
Definition 3.3 Given δ ≡ α ∧ β1 ∧ ... ∧ βn ∧ ♦γ1 ∧ ... ∧ ♦γm in normal form, we
define δ α ≡ α, δ α ≡ α ∧ β1 ∧ ... ∧ βn , and similarly for δ α♦ , δ , δ ♦ , and δ ♦ .
Definition 3.4 Where χ is a conjunction or disjunction of literals, let L (χ) be the
set of literals in χ. A set of literals is open iff no literal in the set is the negation of
any of the others.
Definition 3.5 A conjunction δ ≡ α∧β1 ∧...∧βn ∧♦γ1 ∧...∧♦γm in normal form
is KD45-clear iff: (i) L(α) is open; (ii) there is an open set of literals {l1 , ..., ln } with
li ∈ L (βi ); and (iii) for every γk there is a set of literals {l1 , ..., ln } with li ∈ L (βi )
such that {l1 , ..., ln } ∪ L (γk ) is open. A disjunction in normal form is KD45-clear
iff at least one of its disjuncts is KD45-clear.
In the following, by “clear” and “satisfiable” we mean KD45-clear and satisfiable
in a quasi-partition, respectively. In the case of S5-clear, we must require in clause
(ii) of Definition 3.5 that {l1 , ..., ln } ∪ L(α) is open, in which case (i) is unnecessary.
In both cases, the following lemma holds with the appropriate definition of clarity.
Lemma 3.6 A formula in normal form is satisfiable iff it is clear.
Proof. [Sketch] Suppose ϕ in normal form is satisfiable. Then there is some disjunct
δ of ϕ satisfied at a pointed model. Read off the appropriate open sets of literals
from the current point for clarity condition (i), from some accessible point for (ii),
and from witnesses for each ♦γk in δ for (iii). In the other direction, suppose there
are open sets of literals as described. Construct a model where δ α is true at the
root point w, which is possible by clarity condition (i). For each ♦γk , add a point v
accessible from w and extend the valuation such that all conjuncts of γk and some
disjunct of each βi are true at v, which is possible by (iii). If there are no ♦γk
formulas in δ, add an accessible point v and extend the valuation such that some
disjunct of each βi is true at v, which is possible by (ii). Then ϕ is true at w.
2
The following definition fixes the basic class of Moore conjunctions, as well as
a wider class of Moorean conjunctions. The intuition is that a Moore conjunction
simultaneously asserts a lack of information about another fact being asserted, and a
Moorean conjunction is one that behaves like a Moore conjunction in some context.
We write ∼ ϕ for the negation of ϕ in negation normal form, with ¬ applying
only to literals.
Definition 3.7 Let δ be a conjunction in normal form.
(i) δ is a Moore conjunction iff δ ∧ ♦δ α is not clear or there is a ♦γk conjunct in
δ such that δ ∧ ♦ (δ α ∧ γk ) is not clear.
(ii) δ is a Moorean conjunction iff there is a ♦γk conjunct in δ such that δ ∧ ♦δ α ∧
(∼ δ α ∨ ∼ γk ) is clear.
Assuming S5, δ ∧ ♦δ α may be replaced by δ in both (i) and (ii).
6
Holliday and Icard
The definition of a Moore conjunction generalizes from the paradigmatic case of
p ∧ ♦¬p to include formulas such as p ∧ ♦q ∧ (q → ¬p). The formulas p ∧ ¬p and
p ∧ ¬p are also Moore conjunctions, since for these δ the formula δ ∧ ♦δ α is not
clear, but for the same reason they are not Moorean conjunctions. An example of
a Moorean conjunction that is not a Moore conjunction is p ∧ ♦q. We consider this
formula Moorean because in a context where the agent knows that q implies ¬p,
so (q → ¬p) holds, learning p ∧ ♦q has the same effect as learning p ∧ ♦¬p. By
contrast, the formula p ∧ ♦q ∧ ♦(p ∧ q) rules out the Moorean context with its last
conjunct, and it is not Moorean according to Definition 3.7(ii).
In the proofs of Lemma 3.9 and Theorem 3.13 below, we will assume without
loss of generality that all quasi-partitions considered are chained, i.e., ∀w, v ∈ W :
w 6= v ⇒ wRv ∨ vRw, 3 in which case the following basic facts hold.
Lemma 3.8 Where M is a chained quasi-partition and δ is a conjunction in normal form:
(i) M, w δ ⇒ M δ ♦ ;
(ii) M, w δ ⇒ M|δ = M|δα and M|δ δ α ;
(iii) M|δ , w δ ⇒ M|δ δ.
We now prove the main lemma used in the proof of Theorem 3.13.
Lemma 3.9 Let δ be a conjunction in normal form. The following hold for both
KD45 and S5.
(i) δ is self-refuting if and only if it is a Moore conjunction.
(ii) δ is unsuccessful if and only if it is a Moorean conjunction.
Proof. (⇐ (i)) Suppose δ is a Moore conjunction. Case 1: δ ∧ ♦δ α is not clear. By
Lemmas 3.6 and 3.8(i), δ ∧ ♦δ α is clear iff ♦+ δ is satisfiable, so in this case ♦+ δ is
unsatisfiable and hence δ is self-refuting. Case 2: δ ∧♦δ α is clear, so suppose M, w ♦+ δ. For the ♦γk in δ such that δ ∧ ♦ (δ α ∧ γk ) is not clear, ¬δ ∨ (¬δ α ∨ ¬γk )
is valid by Lemma 3.6, so M, w (¬δ α ∨ ¬γk ) given M, w δ. Then since
(¬δ α ∨ ¬γk ) is universal, it is preserved under submodels by Theorem 2.4, so
M|δ , w (¬δ α ∨ ¬γk ). From Lemma 3.8(ii), M|δ , w δ α , so M|δ , w ¬γk .
Since ♦γk is a conjunct in δ, M|δ , w 2 δ. Since M was arbitrary, δ is self-refuting.
((i) ⇒) We prove the contrapositive. 4 Suppose δ is not a Moore conjunction.
Then δ∧♦δ α is clear, and for every ♦γk in δ, δ∧♦ (δ α ∧ γk ) is clear (∗). If there are no
♦γk conjuncts in δ, then δ is a universal formula with ♦+ δ satisfiable, so it is not selfrefuting by Theorem 2.4. Suppose there are ♦γk conjuncts in δ. We claim that δ 0 ≡
δ ∧ βn+1 ∧ ... ∧ βn+j is clear where {βn+1 , ..., βn+j } = L (δ α ). Given assumption
(∗) and clarity condition (iii) for each δ ∧ ♦ (δ α ∧ γk ), we have that for all ♦γk in δ
there is a set {l1 , ..., ln } with li ∈ L (βi ) such that {l1 , ..., ln } ∪ L (δ α ∧ γk ) is open.
Taking {ln+1 , ..., ln+j } = L (δ α ), {l1 , ..., ln+j } ∪ L (γk ) = {l1 , ..., ln } ∪ L (δ α ∧ γk ) is
open, which gives clarity conditions (i), (ii) and (iii) for δ 0 . Since δ 0 is clear, suppose
N , w δ 0 . From the fact that δ 0 ↔ (δ ∧ δ α ) we have N , w δ ∧ δ α . Given
3
Thanks to Dr. Yanjing Wang for catching the omission of w 6= v in the published version.
The following argument establishes something stronger than we need for Lemma 3.9, but we use it to
establish Corollary 5.3 below. Compare the (⇒) direction of the proof of Theorem A.3 in Appendix A.
4
7
Holliday and Icard
N , w δ α and the assumption that N is a chained quasi-partition, N δ α ; given
N , w δ and Lemma 3.8(i), N δ ♦ . Hence N δ, in which case N , w ♦+ δ and
N|δ = N . It follows that N|δ , w δ, so δ is not self-refuting.
((ii) ⇐) Suppose δ is a Moorean conjunction. Since for some ♦γk in δ, χ ≡
δ ∧ ♦δ α ∧ (∼ δ α ∨ ∼ γk ) is clear, there is a model with M, w χ. Given M, w δ ∧ ♦δ α , we have M, w ♦+ δ by Lemma 3.8(i). Given M, w (∼ δ α ∨ ∼ γk ), by
the same reasoning as in Case 2 of ((i) ⇐), M|δ , w 2 δ. Therefore δ is unsuccessful.
((ii) ⇒) We prove the contrapositive. Suppose M, w ♦+ δ and δ is not a
Moorean conjunction. Then for all ♦γk in δ, χk ≡ δ ∧ ♦δ α ∧ (∼ δ α ∨ ∼ γk ) is not
clear. To show M|δ , w δ, it suffices to show M|δ , w δ ♦ , since δ α is universal and
therefore preserved under submodels. Consider some ♦γk conjunct in δ. It follows
from our assumption that χk is unsatisfiable, whence (δ ∧ ♦δ α ) → ♦ (δ α ∧ γk ) is
valid. Then from M, w ♦+ δ we obtain M, w ♦ (δ α ∧ γk ), so there is a v with
wRv and M, v (δ α ∧ γk ). By Lemma 3.8(ii), v is retained in M|δ . Since γk
is propositional, M|δ , v γk and hence M|δ , w ♦γk . Since ♦γk was arbitrary,
M|δ , w δ ♦ and hence M|δ , w δ. Since M was arbitrary, δ is successful.
2
Lemma 3.9 gives necessary and sufficient conditions for the successfulness of a
conjunction in normal form. We now introduce an apparently stronger notion.
Definition 3.10 A formula ϕ is super-successful (in L) iff for every pointed model
(of L), M, w ♦+ ϕ implies M0 , w ϕ for every M0 such that M|ϕ ⊆ M0 ⊆ M.
If ϕ is super-successful and M, w ϕ, then as points that are not in M|ϕ are
eliminated from M, ϕ remains true at w. Since we take the elimination of points
as an agent’s acquisition of new information, this means that ϕ remains true as the
agent approaches, by way of the incremental acquisition of new information, the
epistemic state of M|ϕ wherein the agent knows ϕ. Intuitively, we can say that a
super-successful formula remains true while an agent is “on the way” to learning it.
We will use the next lemma in the proof of Theorem 3.13.
Lemma 3.11 If δ is a successful conjunction in normal form, δ is super-successful.
Proof. Suppose δ is not super-successful, so there is a pointed model such that
M, w ♦+ δ and an M0 such that M|ϕ ⊆ M0 ⊆ M and M0 , w 2 δ. Since
M0 ⊆ M and δ α is preserved under submodels, we must have M0 , w 2 δ ♦ . But
then M|δ , w 2 δ ♦ given that M|δ ⊆ M0 and δ ♦ is preserved under extensions. Hence
M|δ , w 2 δ, so δ is unsuccessful.
2
We now lift the definition of Moore and Moorean to arbitrary formulas.
Definition 3.12 Let ϕ be an arbitrary formula.
(i) ϕ is a Moore sentence iff any normal form of ϕ is a disjunction of Moore
conjunctions.
(ii) ϕ is a Moorean sentence iff any normal form of ϕ contains a Moorean conjunction as a disjunct.
The following theorem gives necessary conditions for self-refuting and unsuccessful formulas. In Appendix A we strengthen this result with conditions that are
sufficient as well as necessary.
8
Holliday and Icard
Theorem 3.13 Let ϕ be an arbitrary formula.
(i) If ϕ is self-refuting in any sublogic of S5, then ϕ is a Moore sentence.
(ii) If ϕ is unsuccessful in any extension of KD45, then ϕ is a Moorean sentence.
Proof. By Lemma 2.3 it suffices to show the consequent of (i) for ϕ that is selfrefuting in S5 and the consequent of (ii) for ϕ that is unsuccessful in KD45. Since
ϕ is self-refuting (resp. unsuccessful) iff any equivalent normal form of ϕ is selfrefuting (resp. unsuccessful), let us assume that ϕ is already in normal form.
(i) Suppose ϕ is not a Moore sentence, so by Definition 3.12 there is a disjunct
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every
set"C,
ΓbN,
∈ A,
there
is trick
a literal
l ∈with
ΓNthat
isfour
not
the
quazi-nega
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=unsuccessful
"C,
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, A#used
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is�♦
the
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bandTheorem
a,
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the
modifications
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using
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Definition
6.5necessary
ϕ issame
iff any
form
ofliterals
ϕ6.2.
isqu
M
l.
is
a
disjunct
d
in
ϕ
such
that
d
does
not
contain
∼
l
and
t
for
simplicity
we
do
not
write
them
out.
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we
say
case
necessary
modifications
add
fourunsuccessful,
existential
th
pensation.
pthe
∧the
♦q
∧ #Suppose
(qin→
p)
∧ϕ
♦one
(q
∧from
p)
We now consider theχ sources
in
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for
an
and
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If ϕ
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(ϕ)
∅,A,
in
then
normal
ϕthat
isform
successful.
and
Lnot
(ϕ)
$=
∅p
Proof.
(⇒)
Proof.
is
Suppose
unsuccessful,
ϕHolliday
so
there
is
aquantifiers
so
model
there
with
isto
aap
M
m
strictly
definition
χ
of
clarity
so
does
the
not
definition
apply.
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of
clarity
itdoes
does
is easy
not
to
literals,
each
Γis$=
inwithout
such
Σ
cont
2 , so
2 ,(⇒)
mean
that
the
modified
formula
is
clear.
for
simplicity
we
do
not
write
them
out.
When
we
say
that
χ
is
clea
introspection (logics without axioms
4
and
5)
and
in
logics
for
multiple
agents.
Definition
6.
Given
a
specification
"C,
N,
A#
and
formula
ϕ
in
DNF
�♦
�
of, w
linto
and
d�Given
does
not
contain
quazi-negation
of any
M|ϕform
, w � using
ϕ.bϕ Given
Mthe
M,
�w
ϕ.
ϕ, used
we
M,
can
wread
�σ̂ϕ,the
off
we
can
w
read
theoff
disjunct
from
wδlite
th
into Proof.
normal
same
normal
trick
form
with
using
the
infrom
same
Definition
trick
used
6.2.
Since
with
σ̂an
in
|ϕ
! Proof.
! contains
If"C,
contains
only
of
If
the
ϕ
form
Kα
only
and
formulas
¬Kα
(α
of
propositional),
the
form
Kα
pϕthe
N,
A#the
=interesting
"C, Nformulas
, A#formula
where
N
is
the
set
of
all
literals
l
∈
N
for
whic
mean
that
modified
is
clear.
From an epistemic perspective,
most
(normal)
proper
sublogics
!conjuncts
"
!
! four
6.5
ϕ
is∈Γassociated
unsuccessful
iff
any
normal
form
χ(M,
strictly
the
ofM
clarity
not
apply.
However
ofany
disjuncts
Swith
ofNdisjuncts
associated
and
conjuncts
disjuncts
and
such
disjuncts
M,
w
su
! M
case then
theSfor
necessary
modifications
case
the
add
necessary
four
existential
modifications
defin
M
M
2 , so
|ϕ
and
every
set
∈
A,in
there
isdoes
a∼=
literal
lthere
Γthat
that
isM
no
with
M,
wthen
!definition
ϕ,with
for
W
(M,
|w)
wR
vquantifiers
M,
v!add
∈
ϕ,
W∈vto
∈|ϕisthe
W
|aexisten
wR
| of
wR
aw)Theorem
is a disjunct
d in
ϕ
such
that
dvany
does
not
contain
lwand
selecti
of S5 are obtained by dropping
axiom
5 and
adding
something
weaker
itswith
place.
�♦
pensation.
into
normal
form
using
the
same
trick
used
with
σ̂
in
Definiti
Now
suppose
Now
for
contradiction
suppose
for
contradiction
that
there
is
that
a
θ
∈
there
T
such
is
a
that
θ
∈
for
T
for simplicity
we
do
not
write
for
them
simplicity
out.
we
When
do
not
we
say
write
that
them
χ
is
out.
clear,
When
strict
ws
Theorem
6.5
ϕ l.
is each
unsuccessful
iff, wany
form
of ϕ isthe
Moorean
ϕ.3, from
Hence
Msuch
! ϕ.normal
literals,
one
in
A,
that
Σ does
notand
contain
quazi-n
|ϕx, w
|ϕ
Indeed, logics such as S4,Hence
S4.x M
for
= !2,
4, If
etc.,
been
proposed
as
logics
ϕ ishave
aΓconjunction
in
normal
form
L (ϕ) $= ∅,
then
α
α
Proof.
(⇒)
ϕ
is
so
issome
a model
modifications
add
four
existential
quantifie
M,the
w
χ3 .dthe
Hence
M,necessary
w
for
� Suppose
χall.isthat
Hence
♦γ
inquazi-negation
for
θ,0unsuccessful,
M,
all ♦γ
w
�
in
♦
(σ
θ,any
M,
∧isγthere
w
) �for♦in
(σ
∧ σγθ∈)
meanL that
modified
mean
clear.
formula
pensation.
θ the
θversa.
θclear.
0 qcomparable
l�case
and
does
not
the
of
literal
Σ.
of knowledge. Call logics
and
Lof
ifformula
L
is contain
a 3sublogic
of Lmodified
ora vice
b
α
αM
Definition
6.
Given
specification
"C,
N,
A#
and
p ∧ ♦q
# (q
→
p)
∧
♦�Proof.
(q
∧we
p)
pIfbe
∧ϕϕ
∧w
#
(q
→
p)
∧
♦
(qread
p)
M
w
ϕ.
Given
M,
�! vϕ,
weformulas
from
wKα
the
disj
for
simplicity
do
write
them
out.
When
we
χ
one∧ such
♦γ
and
such
let
v♦γ
anot
and
witness
let
with
be
a!can
M,
witness
v∧! �
σoff
with
γM,
vGiven
�that
σform
∧
contains
only
of
the∧
form
and
¬
|ϕ
θ, one
θ ♦q
θ .say
!isthe
"fo
N
and
every
set
Γ
∈
A,
there
is
a
literal
l
∈
Γ
that
is
not
the
quazi-neg
Theorem
6.5
ϕ
is
unsuccessful
Theorem
iff
any
6.5
normal
ϕ
is
unsuccessful
form
of
ϕ
iff
Moorean
any
normal
without
"C,
N,
A#
=
"C,
N
,
A#
where
N
is
set
of
all
literals
a,
b
�♦
�♦
b
M
M th
Proposition 4.1 For any normal,
proper
sublogic
L
of
S5,
comparable
to
S4.4,
S
of
disjuncts
with
associated
conjuncts
and
disjuncts
such
mean
that
the
modified
formula
is
clear.
the fact that the
σ ∈then
fact
S, for
we
that
have
σ(M,
∈M,
S,
we� have
σ w. M,
relation
any
w)w
with
M,
!Then
ϕ,w v�since
∈σW .the
|Then
wR
vsinc
=
l.S5) that is unsuccessful
5
is apensation.
disjunct
d in form
ϕ such
that
d does
contain
∼ l and t
pensation.
there is a formula (consistent
in
Moorean.
p with
p normal
aIf
ϕ isNow
a conjunction
in
and
L (ϕ)
$= ∅,not
then
ϕθ is∈
successful.
Hence
M
,L
w but
! ϕ.is(anot
suppose
for
contradiction
that
there
is
a
T
such
equivalence
relation
equivalence
(a
quasi-partition
relation
quasi-partition
would
suffice)
would
we
have
suffice)
M,
v
we
�
σ
|ϕ
Proof.
ϕ Γisiff
unsuccessful,
so does
there
aMm
literals,
from each
in
A, such
that form
Σ
Theorem
6.5 (⇒)
ϕ one
is Suppose
unsuccessful
any
normal
ofnot
ϕisiscont
α ∧ϕ
Definition
6.
Given
a
specification
"C,
N,
A#
and
formula
in
DNF
M,
w
�
χ
.
Hence
for
all
♦γ
in
θ,
M,
w
�
♦
(σ
γ
)
for
It
follows
that
It
M,
follows
v
�
σ
that
∧
γ
M,
and
v
therefore
�
σ
∧
γ
and
M,
v
therefore
�
ϕ
∧
γ
,
M,
in
which
v
�
ϕ
∧
case
γth
3
θ
θ
θdoes
θϕ,
θs,
Proof. First, we claim that
≡ ♦p
∧pensation.
♦¬p
isSuppose
unsuccessful
in
S4.4
and
inwe
any
b ϕProof.
bϕ.is
of
and
dand
not
contain
the
quazi-negation
of with
any
M
,Holliday
wl ♦q
�
Given
M,
whence
�Holliday
can
read
off
from
w lite
Proof.
If ϕϕ
contains
only
formulas
the
form
Kα
and
(α
proposi
(⇒)
unsuccessful,
so
there
a"θ¬Kα
model
M
and is
Icard
Icard
|ϕp
! ϕ
! ∧of
∧
∧
#
(q
→
p)
♦
(q
∧
p)
!
!
α
θ
θ
"C,
N,
A#
=
"C,
N
,
A#
where
N
is
the
set
of
all
literals
l
∈
N
for
whic
one
such
and
let
vis!2,
be
witness
with
M,
v=M
�disjuncts
σ∈
∧�γMγθ|ϕ
. aM
G
M|ϕ
in M
.�for
Since
γ(M,
is♦γ
propositional,
Since
γw
propositional,
we
have
we
v �M
have
and
therefore
, vW
sublogic of S4.4 by Lemma
In
the
S4.4
model
M
in
ϕaisand
true
at
the
θ. with
|ϕ
|ϕ
Holliday
then
w)
M,
ϕ,
vread
∈Icard
WM
| ,wR
vθthe
v|ϕ
|
a,q b 2.3.
a,
Sin
ofM
disjuncts
with
associated
conjuncts
and
q b�Figure
M
ϕ.any
Given
M,
w
ϕ,
we
can
off
from
wγ♦
disjunct
δθsu
an
|ϕis, w
�♦
aγthe
disjunct
d
in
ϕ
such
that
d
does
not
contain
∼
l
and
there
is
a
selecti
♦
the
fact
that
σ
∈
S,
we
have
M,
w
�
σ
.
Then
since
the
χ
,
so
strictly
the
definition
of
clarity
does
not
apply.
However,
it
is
easy
, soSince
strictly
definition
of
clarity
does
not
apply.
However,
it
is
easy
to
put
χ
left point, but in M|ϕ , theχ2right
point
is
eliminated,
so
ϕ
becomes
false
at
the
left
was
arbitrary,
Since
we
conclude
arbitrary,
that
we
conclude
M
,
w
�
that
θ
.
M
Since
,
w
θ
�
∈
θ
T
.
by
2γθ was
b
Hence
M
,
w
!
ϕ.
θ
|ϕ
|ϕ
|ϕ
Now
suppose
for
contradiction
that
there
isDefinition
a θ quazi-n
∈6.2.TwSiSs
p into
S ofliterals,
disjuncts
with
associated
conjuncts
and
disjuncts
such
that
M,
�♦
one
from
each
Γconjunction
in
A,
such
that
Σ
does
not
contain
normal
form
using
the
same
trick
used
with
σ̂ �♦
in (ϕ)
into normal
form
using
theϕ
same
trick
used
with
σ̂
in
Definition
6.2.
Since
inand
this
χequivalence
, so
strictly
the
definition
of
clarity
does
not
apply.
However,
itwith
is
easy
to
put
χ� ϕ
αform
αthe
2M,
If
ϕ
is
a
in
normal
L
=
$
∅,
then
(⇒)
Suppose
is
Proof.
unsuccessful,
(⇒)
Suppose
so
there
ϕ
is
is
a
unsuccessful,
model
M,
so
w
there
point. Note that the Proof.
formula
is
already
in
normal
form
and
is
not
Moorean.
relation
(a
quasi-partition
would
suffice)
we
have
α
we
have
w
we
�
have
χ
,
which
M,
w
gives
�
χ
,
M,
which
w
�
gives
θ
and
M,
hence
w
�
θ
M
and
,
w
hen
�)
1add
1 quantifiers
|ϕ
anormal
�♦
M,
wcontradiction
�
χ
.
Hence
for
all
♦γ
in
θ,
M,
w
�
♦
(σ
∧
γthe
case
the
necessary
modifications
add
four
existential
quantifiers
tofor
case Now
the necessary
four
existential
to
the
definition,
forform
that
there
is
a
θ
∈
T
such
that
3
θ
θ
intomodifications
using
the
same
trick
used
with
σ̂
in
Definition
6.2.
Since
in
this
ofsuppose
l and
d does
not
contain
the
quazi-negation
of
any
literal
in
Σ.
Holliday
and
Icard
Holliday
and
Icard
�
�
M|ϕ ,forwsimplicity
�
ϕ.p ∧
Given
M,
w→
�that
M
ϕ,simplicity
we
,(q
w
can
ϕ.
read
Given
off
from
M,
w
wout.
�the
ϕ,
disjunct
we
can
read
δ definition,
and
off
from
♦q
#
(q
p)
∧
♦
p)
Itdo∧, follows
M,
v∧�
�
σ
∧
γthere
and
therefore
M,
vtosay
�M,
ϕ
∧
in
wTs
αin
|ϕ
for
we
do
not
When
we
that
χγ�
clear,
we
not
write
them
out.
When
we
say
that
χthem
is some
clear,
strictly
we
have
M
w
have
�
θ
.
M
For
,
w
suppose
�
θ
.
For
suppose
is
there
�β
in
is
θ
some
and
�β
v
in
θvwrite
θis,sets
α
case
the
necessary
modifications
add
four
existential
quantifiers
the
|ϕ
|ϕ
one
such
♦γ
and
let
be
a
witness
with
v
σ
∧
Proof.
If
ϕ
contains
only
formulas
of
the
form
Kα
and
M, w �Proof.
χ3 . Hence
for
all
inis θ,unsuccessful,
M,
� ♦ (σso!∧there
γθ ) for
some
σ "M
∈¬w
andw
Icard
θ♦γθ Holliday
θ
(⇒)
Suppose
ϕ
is
a
model
mean
that
the
modified
formula
is
clear.
that thewith
modified
formula
is
clear.
for
simplicity
we
do
not
write
them
out.
When
we
say
that
χ
is
clear,
strictly
we
S of χmean
disjuncts
associated
S
of
conjuncts
disjuncts
and
with
disjuncts
associated
such
conjuncts
that
M,
and
w
�
disjun
δ
∧
χ
in
M
.
Since
γ
is
propositional,
we
have
M
,
v
�
γ
and
th
1=
M
M
q
�♦
M
,
v
�
¬β.
M
Then
,
v
�
since
¬β.
v
Then
was
retained
since
v
was
in
M
retained
,
we
have
in
M
M,
,
v
we
�
θ
α
|ϕ
|ϕ
χlet
strictly
the
definition
ofhave
clarity
does
not
easy
strictly
the ♦γ
definition
ofthen
clarity
does
apply.
However,
itM,
isM,
easy
put
|ϕsuch
|ϕ
|ϕσ
|ϕit issinc
for
any
w)
with
M,
w
!vϕ,
vapply.
∈σχ∧WγHowever,
wR
vM
2 , so
2 , so
the
fact
σ(M,
∈
S,
we
w
�
one
and
vthat
be
anot
witness
with
�to
. |Then
Given
θ.w
mean, θthat
the
modified
formula
is�♦
clear.
p6.5
♦
�♦ in
M
w
�
ϕ.
Given
M,
w
�
ϕ,
we
can
read
off
from
the
disjun
Now into
suppose
for
contradiction
Now
that
suppose
there
for
is
contradiction
a
θ
∈
T
such
that
that
there
for
all
is
♦γ
a
θ
Theorem
6.5
ϕ
is
unsuccessful
iff
any
normal
form
of
ϕ
is
Moorean
wit
Theorem
ϕ
is
unsuccessful
iff
any
normal
form
of
ϕ
is
Moorean
without
cominto
normal
form
using
the
same
trick
used
with
σ̂
Definition
6.2.
Si
normal
form
using
the
same
trick
used
with
σ̂
in
Definition
6.2.
Since
in
this
|ϕ
χ
,
so
strictly
the
definition
of
clarity
does
not
apply.
However,
it
is
easy
to
put
χ
Since
γ
was
arbitrary,
we
conclude
that
M
,
w
�
θ
.
Since
2which
isthat
a conjunction
in
normal
form
and
Lβ�(ϕ)
∅,M,
then
ϕ |ϕ
is
σ ∈Iffact
S,ϕin
σσ θ∈
case
S,
inwe
which
vrelation
�,βwcase
given
M,
�
w�♦
χ1Then
.$=Hence
wM
�
χ
.v Hence
� β,we
aθM
c
|ϕ
M
!
ϕ.(a
1, successful.
|ϕ
equivalence
quasi-partition
would
suffice)
the
∈Hence
S,M,
have
M,
wM,
�v σ
.given
since
the
relation
�♦the
αadd
pensation.
pensation.
case
the
necessary
four
existential
quantifiers
tothat
theα
the
add
four
existential
quantifiers
to
definition,
6.5
ϕM,
is
unsuccessful
iff
any
normal
form
of
ϕ
is�
Moorean
without
cominto
normal
form
using
the
same
trick
used
with
σ̂all
in
Definition
Since
in
this
α 6.2.
STheorem
ofmodifications
disjuncts
with
associated
conjuncts
and
disjuncts
such
M, wcase
� We
χ3 .necessary
Hence
for
all
♦γ
in
w
�
θ,
χ
M,
.θwhich
w
Hence
�
♦
(σ
for
∧
γ♦
♦γ
in
θ,
M,
σ
w∈hence
�
S.
♦
Con
(σ
♦modifications
�
Fig.
2.
we
have
M,
�
χ
,
gives
M,
w
θwe
and
3M
θw
θ)
1
have
shown
We
M
have
,
w
shown
�
α
∧
∧
,
θ
w
�
,
i.e.,
α
∧
M
θout.
∧
,θfor
θw�
,�some
i.e.,
θ,tosay
in
M
which
�case
θ,
It
follows
that
M,
v
�
σ
∧
γ
and
therefore
M,
v
�
ϕ
∧
γM
for
simplicity
we do
not
write
When
that
χ, w
is
clear,
for simplicity
wepensation.
do not
write
them
out.
When
weadd
sayfour
that
χthem
is clear,
strictly
we
|ϕ
|ϕ
|ϕ
|ϕ
case
the
necessary
modifications
existential
quantifiers
the
definition,
equivalence
relation
(a
quasi-partition
would
suffice)
we
have
M,
v
�
θ
θσ,s
α
Proof.
If
ϕ
contains
only
formulas
of
the
form
Kα
and
¬Kα
(α
proposi
b
b
�
Now
suppose
for
contradiction
that
there
is
a
θ
∈
T
such
th
one such
♦γqθtheand
let
v
be
a
one
witness
such
♦γ
with
and
M,
let
v
�
v
σ
be
∧
a
γ
witness
.
Given
with
M,
M,
w
�
v
χ
p
∧
♦q
∧
#
(q
→
p)
∧
♦
(q
∧
p)
!
"
!
θ
θ
θ
mean
that
the
modified
formula
is
clear.
meananother
that
modified
formula
is
clear.
have
M
,
w
�
θ
.
For
suppose
there
is
some
�β
in
θ
and
for
simplicity
we
do
not
write
them
out.
When
we
say
that
χ
is
clear,
strictly
we
contradiction.
another
contradiction.
It θis
follows
that
It∈for
follows
every
∈
for
TαM
there
θM
∈
aθ|ϕa�
♦
T|2
|ϕv|ϕ
M
in
M
.a,with
Since
γand
is!therefore
propositional,
we
have
,∈�♦
vW
�is
γcase
follows
M,
�
σthat
M,
v| of
�that
ϕθM
∧
inevery
b ∧ γM,
bany
|ϕ
thenM,
for
(M,
w)
w
ϕ,
vin
W
wR
vγ(σ
vwhich
Next, we claim that
isa,Itbathat
model
ofthat
any
logic
L
a�
proper
extension
θ ,=
�♦
w
�
χ
.
Hence
for
all
♦γ
θ,
M,
w
�
♦
∧
γ
)
for
som
mean
that
the
modified
formula
is
clear.
theM
fact
σ
∈
S,
we
have
the
M,
fact
w
that
σ
σ
∈
.
Then
S,
we
since
have
the
M,
w
relation
�
σ
in
.
The
M
3
θ
θ
♦.wit
M
,not,
v,γwθ�!is
¬β.
Then
vϕwe
retained
M
,Lemma
we� θhave
6.5χϕsince
is unsuccessful
iff
any
normal
form
ofby
ϕ ,is
Moorean
Theorem
is
iff
any
form
ofis
iswas
Moorean
without
comM,
w6.5
�. ϕχ
,unsuccessful
M,
wTheorem
�
M,
,normal
whence
�
so
χM,
w
clear
�
χM
so
Lemma
χvis�
clear
3.5.
3.
|ϕwhence
|ϕ
a
3M
3w
γpropositional,
was
conclude
that
M
S
Hence
ϕ.
M
Since
we
have
γin
therefore
M
S4.4 and a proper sublogic
ofin
S5.
Suppose
there
is
athere
theorem
ϕ
of
Lby
with
θχ
|ϕ
αw
|ϕ
θvisand
|ϕ
|ϕ ,wso
�♦
α
Proof.
(⇒)
Suppose
iswitness
unsuccessful,
there
a vmodel
with
M,
w
Proof.
(⇒)
Suppose
ϕSince
isquasi-partition
unsuccessful,
soarbitrary,
is any
aϕ
model
with
M,
�is
ϕMoorean
but
pso
one
such
♦γ
and
let
v
be
a
with
M,
�
σ
∧
γ
.
Gi
pensation.
pensation.
equivalence
relation
(a
equivalence
would
relation
suffice)
(a
quasi-partition
we
have
M,
would
�
σ
suffic
∧
σ
Theorem
6.5
ϕ
is
unsuccessful
iff
normal
form
of
ϕ
without
comθ
θ
α
σ
∈
S,
in
which
case
M,
v
�
β
given
M,
w
�
χ
.
Hence
M
,
0
♦
1
|ϕ
Suppose
(⇐)
ϕSuppose
is
Suppose
without
is,the
compensation,
without
an
appropria
svs
M
,Moorean
w
�ϕ ϕ.
Given
wMoorean
�disjunct
ϕ, variable
weM
can
read
off
the
δThen
and
,Since
w � (⇐)
ϕ. Given
M,
�¬p)
ϕ,have
we
can
read
off
from
w
δ|ϕ
sets
and
we
M,
�ϕ
χM,
which
gives
M,
wcompensation,
�M,
θwdisjunct
and
M 2 ϕ. Change ϕ to ϕ M
by|ϕsubstituting
(pwarbitrary,
∧
for
any
propositional
q�♦
(⇒)
isw
unsuccessful,
so
there
isand
model
�θ
ϕ∈but
γthe
was
we
conclude
that
,aw
� Tfrom
θ. with
. wso
Since
by
|ϕ
1
θ Proof.
pensation.
fact
that
σ
∈
S,
we
have
M,
w
�
σ
Then
since
the
♦
�
It follows
that
M,
v
�
σ
∧
γ
It
and
follows
therefore
that
M,
v
�
σ
ϕ
∧
∧
γ
γ
,
and
in
which
therefore
case
M,
v
is
v
ret
�
b
θ
θ
θ
S
of
disjuncts
with
associated
conjuncts
and
disjuncts
such
that
M,
w
�
S ofb disjuncts
with
associated
conjuncts
and
disjuncts
such
that
M,
w
�
δ
∧
χ
∧
χ
.
M
,#
w
�
ϕ.→
M,
w
ϕ,
can
read
from
disjunct
sets
T and
1 M
2 δ and
αthe
pM
∧ ♦q
∧
(qmodel
p)
∧given
♦
(q�∧
p)
We
have
shown
M
,θwe
w�Lemma
α
∧ off
θw
∧such
θθwLemma
, and
i.e.,
,such
θ,
in
|ϕthe
Let have
be
Let
M
the
by
model
given
3.5
by
that
3.5
M,
wwM
�� χ.
that
For
M
cδ
|ϕ
|ϕsome
we
M,
whave
� Given
χ
which
gives
M,
�
hence
,w
�wh
M
,w
�
.�TFor
suppose
there
is
�β
in
1 ,a,be
|ϕ
|ϕ
θ
θ
5 S4.4 is S4 plus ϕ → (♦ϕ
b
a,
b
Now
suppose
for
contradiction
that
there
is
a
θ
∈
T
such
that
for
all
b
Now
suppose
for
contradiction
that
there
is
a
θ
∈
such
that
for
all
♦γ
in
θ,
equivalence
relation
(a
quasi-partition
would
suffice)
we
have
M
S
of
disjuncts
with
associated
conjuncts
and
disjuncts
such
that
M,
w
�
δ
∧
χ
∧
χ
.
θ
1 |ϕ|ϕ
in →
Mϕ).
γM
isextension
propositional,
in M
M
. wSince
have
γM
,propositional,
v a�there
γθ and
therefore
we
M
M
,2ther
w
, vin
�
ItSince
is a proper
of S4.3
and
ofis
[21].
|ϕ . assume
|ϕwe
|ϕS4.2
another
Italso
that
for
every
θ such
∈
T
�
α∧
α for
,all
w
assume
�contradiction.
Then
,therefore
�that
ϕ.
exists
Then
disjunct
exists
θin
in
aγθθhave
ϕ
q M,
w
�¬β.
χ|ϕ
♦γis
in
M,
wsuch
�
♦ (σ
)disjunct
for
some
σthat
∈M
S.
M, whave
� χ3 . M
Hence
♦γ
in
M,
w
♦there
(σ
∧
γthere
)follows
for
σ
∈TS.
Consider
M
v θ,contradiction
�M,
Then
since
was
retained
in
M
we
|ϕsuppose
aw
,for
�
θunsuccessful,
.ϕ.
For
suppose
is
�β
and
Now
there
avθ,
θsome
∈
that
for
all
♦γ
in,in
θ,θ
3 . �Hence
θsome
θ ,for
θall
θv
|ϕ
|ϕ
♦
|ϕfollows
It
that
v
�
σ
∧
γ
and
therefore
M,
v
�
ϕ
∧
γ
,
in
whi
Proof.
(⇒)
Suppose
ϕ
is
unsuccessful,
so
there
is
a
model
with
M,
w
(⇒)
Suppose
ϕ
is
so
there
is
a
model
with
M,
w
�
ϕ
but
SinceProof.
γ
was
arbitrary,
we
conclude
Since
γ
that
was
arbitrary,
M
,
w
�
we
θ
.
conclude
Since
θ
∈
that
T
by
M
assump
,
w
�
p
θ 3.5.
θ♦γ
|ϕ M,
one
and
v�
a.�.so
with
M,
� then
σLemma
∧�
γ
.χ
Given
M,
w
one θsuch
♦γθ and
let
v� be
witness
with
M,
vinif
�let
σθθαθM,
∧
γχ
Given
M,
w
�∈
χsome
and
M,
,θ whence
M,
w
χβαin
is
clear
by
M,
ww
χ�3∈
.aχ
Hence
for
all
θ,
wT
♦then
(σ
∧if
γgiven
for
σαw
∈
S.θgiven
Consider
9¬β.
2v
θhave
θT
3we
and
must
have
and
we
∈such
must
T
.♦γ
For
∈
∈
/be|ϕ
,witness
For
θ)model
/|ϕM,
T
M,
,we
we
hav
w
θ case
θM
1M,
σ
S,
in
which
M,
v
�
given
w
�
χ
.
Hence
M
v
�
Then
since
v
was
retained
,
have
v
�
θ
M
,
w
�
ϕ.
Given
M,
w
�
ϕ,
we
can
read
off
from
w
the
disjunct
δ
and
s
M|ϕ ,M
w �|ϕϕ.,we
Given
M,
w
�
ϕ,
can
read
off
from
w
the
disjunct
δ
and
sets
T
and
1
α
α
α
Proof.
(⇒)
Suppose
ϕ
is
unsuccessful,
so
there
is
a
with
M,
w
�
ϕ
but
�♦
�♦
|ϕ
α
M
.which
Since
γ α♦
we
have
M
, M,
vsince
�αww
γ�
and
ther
the
fact
that
∈w
S,wsince
we
have
M,
�∧
σinγM
. Given
Then
the
relation
in
the fact
that
σin
∈ α♦
S,
we
have
M,
w
�
σbepropositional,
Then
the
relation
isα♦
we have
M,
w
�
χ
,
we
gives
have
M,
�
�
θ
χ
and
,
hence
M
,
w
θ
.
We
an
one
such
♦γ
let
vis
a. σM,
witness
with
M,
vwhich
�wσ
M,
χ
and
θ� �
|ϕ
|ϕan
α♦
2θ
1
1
θ and
θgives
|ϕ
(⇐)
Suppose
ϕ
is
Moorean
without
compensation,
so
an
a
♦
�
S
of
disjuncts
with
associated
conjuncts
and
disjuncts
such
that
M,
w
�
δ
S of σ
disjuncts
with
associated
conjuncts
and
disjuncts
such
that
M,
w
�
δ
∧
χ
∧
χ
.
M,
w
�
θ
,
in
M,
which
w
�
θ
case
,
in
M
which
,
w
�
case
θ
,
M
or
M,
,
w
w
�
θ
�
♦
,
(σ
or
M,
∧
¬β)
w
�
for
♦
(
s
M
,
w
�
ϕ.
Given
M,
w
�
ϕ,
we
can
read
off
from
w
the
disjunct
δ
and
sets
T
and
1
2
�♦
�♦
α
�♦
|ϕ
|ϕ
|ϕ
∈
S,
in
which
case
M,
v
�
β
given
M,
w
�
χ
.
Hence
M
,
v
�
β,
a
c
♦
We
M
,quasi-partition
αThen
∧vthat
,∧relation
M
,Since
w
�
relation
(a
would
have
M,
� σθ,
equivalence relation
(a quasi-partition
would
suffice)
have
�θ σsince
∧θthe
σsuffice)
γi.e.,
1 ∧M
the
thathave
σequivalence
∈arbitrary,
S,shown
we have
M,
wwe
�w
σ�� .M,
isv an
θ . we
|ϕ
�
|ϕ
|ϕ
Since
γFor
we
w
θinδM
.M
θ
θ was
Now
suppose
for
that
isand
a|ϕ
∈vT�in
such
that
for
suppose
contradiction
that
there
is|ϕ
a,given
θcontradiction
∈
Tsome
that
forthere
all3.5
inθ, M,
θ,
haveNow
M|ϕ
,w
�for
θSLet
.fact
suppose
have
M
there
w
is
�conclude
θsuch
.which
For
�β
suppose
in
θ♦γ
there
is
some
such
with
associated
disjuncts
that
χ
θv
1
2 . all
�♦
α
|ϕ∧ χ
model
Lemma
such
M,
w
�v�
Itthe
follows
that
M,
γand
therefore
� ϕ ∧ γwthat
,�
in ∧
which
case
It follows
that M,
vof�disjuncts
σM
∧ γ be
and
therefore
M,
vconjuncts
�v♦
ϕ�∧σγ ∧,by
in and
case vsuch
isM,
retained
�
N and ϕevery set Γ ∈ A,ϕ thereNisand
a literal
∈ ΓΓ that
not the
o
everyl set
∈ A, isthere
is a quazi-negation
literal l ∈ Γ tha
! !
!
l.
l.
"C,
A#every
= "C,
"C,
NΓ!N,
,∈A#A,
A#where
= "C,
N
, the
A# where
set
ofΓall
Nthat
literals
is the
set
l ∈the
ofNall
forliterals
which lthere
∈ Noff
Nl.N,
and
set
there
is N
aisliteral
l! ∈
is not
quazi-negation
l.
isl.a disjunct d inisϕasuch
disjunct
that dd in
does
ϕ such
not contain
that d does
∼ l and
not there
contain
is a∼selection
l and there
Σ of
is a
!A# and formula
! define
Definition
6.
Given
Definition
aΓA,
specification
6.
Given
"C,
aΓ"C,
specification
N,
"C,
N,does
A#
ϕ aϕ
in
and
DNF,
formula
ϕtha
N
andone
every
set
Γ
∈
there
is
a
literal
l
∈
Γ
that
is
not
the
quazi-negation
oi
N
and
every
set
Γ
∈
A,
there
is
literal
l
∈
Γ
literals,
from
literals,
each
one
in
A,
from
such
each
that
Σ
in
does
A,
such
not
that
contain
Σ
not
contain
the
Definition
6.
Given
a
specification
N,
A#
and
formula
in
DNF,
defin
Definition
6.
Given
a
specification
"C,
N,
A#
an
ϕ
ϕ
!
! !
!
! of all
ϕ
ϕ
"C,
N,
A#
=
"C,
"C,
N
N,
,
A#
A#
where
=
"C,
N
N
is
,
the
A#
where
set
N
literals
is
the
set
l
∈
of
N
all
for
literals
which
l
there
∈
N
!
!
!
!
set
Γl N
∈ ,A,
isquazi-negation
aand
literal
∈
Γof
that
isliteral
not the
of
! define
l.
Definition
6.
Given
adthere
specification
"C,
A#
and
formula
in
DNF,
ofN
l and
and
dA#does
not
of
contain
and
does
thel.
not
contain
the
of
quazi-negation
any
Σ.
any
literal
Σf
"C,
N,every
=
"C,
A#
where
N
isA#
thel=N,
set
all
literals
lin
Nliteral
for
which
ther
"C,
N,
"C,
N
where
Nϕof
is
the
all
N
every
set
Γ
∈, A#
A,
there
is∈
aquazi-negation
lset
∈ of
Γin
tha
ϕ d inisϕasuch
!
! such
isl.
a
disjunct
disjunct
that
d
d
in
does
ϕ
not
contain
that
d
does
∼
l
and
not
there
contain
is
a
∼
selection
l
and
there
Σ
of
is
"C,
A# = "C,
A# where
is the
of ϕ
allsuch
l d∈does
Nisfor
which
there
is aN,disjunct
d inNϕ ,such
that is
not set
contain
∼literals
l and
anot
selection
Σ∼oa
disjunct
d in
thatthere
contain
l.dNadoes
literals,
oneone
from
literals,
each
Γone
A,
from
such
each
that
Γ
Σ
in6.
does
A,does
such
not
contain
does
the
not
contain
Given
ain
specification
"C,
N,
A#
ϕquazi-negation
DNF,
defin
Given
aΓformula
specification
"C,
N,does
A#
an
isDefinition
a disjunct
d6.in
ϕ each
such
dDefinition
does
not
contain
∼and
lthat
and
there
is
ain
selection
Σthe
of
literals,
Γthat
in
A,
such
that
Σ
not
contain
the
quazi-negatio
literals,
one
each
inΣ
A,
such
that
Σ
n
ϕ from
ϕ from
! a specification
!
Holliday
and
Icard
! literals
! in
! define
Definition
6.
Given
"C,
N,
A#
and
formula
ϕ
DNF,
of
l
and
d
does
not
of
l
contain
and
d
does
the
quazi-negation
not
contain
the
of
quazi-negation
any
literal
in
of
Σ.
any
literal
in
Σ
"C,
N,
A#
=
"C,
N
,
A#
where
N
is
the
set
of
all
l
∈
N
for
which
ther
"C,
N,
A#
=
"C,
N
,
A#
where
N
is
the
set
of
all
N
and
every
set
Γ
∈
A,
there
is
a
literal
l
∈
Γ
tha
one
from
Γ
in
A,
such
that
Σ
does
not
contain
the
quazi-negation
Definition
6.
Given
a
specification
"C,
N,
A#
an
Ifliterals,
ϕ
a conjunction
If not
ϕeach
is
in
a
normal
conjunction
form
and
in
normal
L
(ϕ)
=
$
form
∅,
then
and
ϕ
L
(ϕ)
is
successful.
=
$
∅,
then
ϕ
is
suc
of isl and
contain
the
quazi-negation
of
any
literal
in
Σ.
of
l
and
d
does
not
contain
the
quazi-negation
of
ϕd does
!
ϕ set
"C,
A#d does
= "C,
Nϕ!contain
,such
A# where
is A#
the
of ϕ
all
l d∈does
N!isΣ.
for
which
there
!∼literals
disjunct
d in
that
dNquazi-negation
not
contain
l and
there
anot
selection
Σ
is
adoes
disjunct
d"C,
in
that
contain
∼o
l.
ofislaN,
and
not
the
ofsuch
literal
in
"C,
N,
=
N
,any
A#
where
N
is
the
set of
all
0 lisand
isliterals,
a disjunct
d
in
ϕ
such
that
d
does
not
contain
∼
there
is
a
selection
Σ
of
other than p. Since L is Proof.
normal
and
therefore
closed
under
substitution,
ϕ
also
from
each
in
A,
that
Σ
does
not
contain
the
quazi-negatio
literals,
oneform
from
Γ that
inform
A,d (α
such
that
Σ
does
If ϕone
contains
Proof.
only
If Γϕ
formulas
contains
of
the
formulas
Kα
of
and
Kα
propositional),
and"
¬Kα
is asuch
disjunct
d in
ϕeach
such
not
contain
∼n
!only
! the
"¬Kα
!M does
!M(α
M
M
M
M
literals,
one
from
each
Γ
in
A,
such
that
Σ
does
not
contain
the
quazi-negation
Definition
6.
Given
a
specification
"C,
N,
A#
an
|ϕ
|ϕ
!wR
If
ϕ
is
a
conjunction
If
ϕ
is
in
a
normal
conjunction
form
and
in
normal
L
(ϕ)
=
$
form
∅,
then
and
ϕ
L
(ϕ)
is
successful.
=
$
∅,
then
ϕ
is
suc
of
l
and
d
does
not
contain
the
quazi-negation
of
any
literal
in
Σ.
of
l
and
d
does
not
contain
the
quazi-negation
of
a theorem of L and therefore
of
S5.
Moreover,
since
for
all
variables
q
other
than
then
for
any
(M,
then
w)
with
for
any
M,
(M,
w
!
w)
ϕ,
with
v
∈
W
M,
w
|
!
wR
ϕ,
v
∈
=
W
v
∈
|
wR
W
v
|
=
v
∈
N ϕand
Γeach
∈
A,Γthen
there
a!successful.
literal
Γ $=
tha
literals,
one
in A,
that
does
nvW
If ϕ is a conjunction in normal
form
and
Lset
(ϕ)
$=
∅,
ϕissuch
isform
If
is aevery
conjunction
normal
and lΣ
L∈(ϕ)
ϕ from
! in
of
l
and
d
does
not
contain
the
quazi-negation
of
any
literal
in
Σ.
"C,
N,
A#
=
"C,
N
,
A#
where
N
is
the
set
of
all
p, M ¬q, the substitution
of
(p
∧
¬p)
for
q
preserves
(un)satisfiability
in
M,
so
Hence
, w !Hence
ϕ.
Mnormal
ϕ.
l. form
|ϕ , w !
If ϕ isM
a |ϕ
conjunction
in
L (ϕ)
$= contain
∅, then ϕ
successful.
of
l andand
d does
not
theis quazi-negation
of
!
N
and
every
set
ΓϕKα
∈
A,the
there
a(α
literal
l¬Kα
∈!Γ (α
tha
Proof.
If ϕ
Proof.
If ϕ
formulas
contains
of
the
form
formulas
of
and
¬Kα
form
Kα
propositional),
and
is
disjunct
dform
inKα
such
that
dis(α
does
not
contain
∼
!only
!
"
!
"
M 2 ϕ0 . Hence ϕ0 is not
aProof.
theorem
S4.4.only
But
by
a aresult
of
Zeman
[21],
for
If contains
ϕofcontains
only
formulas
of
the
and
¬Kα
propositional)
Proof.
If
ϕ
contains
only
formulas
of
the
form
Kα
M
M|ϕ
M
M
M
M
!
"
!
!
|ϕ
!
l.
then
for
any
(M,
then
w)
with
for
any
M,
(M,
w
!
w)
ϕ,
with
v
∈
W
M,
w
|
!
wR
ϕ,
v
∈
=
W
v
∈
|
wR
W
v
|
wR
=
v
∈
M
M
M
M
M
N
and
every
set
Γ
∈
A,
there
is
a
literal
l
∈
Γ
tha
literals,
one
from
each
Γ
in
A,
such
that
Σ
does
nvW
|ϕ
ϕ∧
isfor
aone
conjunction
in
and
LS5
(ϕ)
∅,
ϕ! is
successful.
If
ϕ
is
in
normal
L
Definition
6.
Given
athen
specification
N,
A#| $=
an
If#
ϕ
contains
formulas
of
Kα
and
p Proof.
∧If
♦q
(q
→
pp)∧∧
♦q
♦only
∧
(q
#
∧normal
(q
p)is→
∧
(q
p)
then
any
(M,
w)
with
M,
w
!form
ϕ,
vconjunction
∈∧
W
| $=
wR
vM,
=w
v(α
∈propositional),
Wand
|(ϕ)
wR
any formula ψ, containing
exactly
variable,
that
a p)
theorem
ofform
but
not
then
for
any
(M,
w)
with
!form
ϕ,
v"C,
∈W
wR
!♦a the
"a¬Kα
ϕ
M
M
!
!
M
M
Hence
M
,
w
!
Hence
ϕ.
M
,
w
!
ϕ.
|ϕ
|ϕ
l.
|ϕ
|ϕ
ϕ S4.4
isforaM
conjunction
in normal
and
Lw(ϕ)
$=
∅,in
then
successful.
of
l ϕ,
and
does
not
contain
theisvN
quazi-negation
of
"C,
N,
"C,
where
is
set
of
then
any
with
M,
wN
!form
vdconjunction
∈
|N
wR
vnormal
=ϕ
W the
!| wR
Hence
, wS5.
!w)
ϕ.Hence
Hence
M
,W
!Given
ϕ.
theorem of S4.4, adding ψIf
to
gives
L is
S5,
Since
|ϕ(M,
If
ϕ
isa A#
acontradiction.
and
$=an
|ϕ=
and
every
set
Γ
∈, A#
A,
there
is form
a∈
literal
∈(ϕ)
Γ all
tha
!6.
Definition
ais
specification
"C,l L
N,
A#
N
and
every
set
Γ
∈
A,
there
is
a
literal
l
∈
Γ
that
not
the
quazi-negation
of
Proof.
If
ϕ
contains
only
formulas
of
the
form
Kα
and
¬Kα
(α
propositional)
Proof.
If
ϕ
contains
only
formulas
of
the
form
Kα
Hence
M
,
w
!
ϕ.
is
a
disjunct
d
in
ϕ
such
that
d
does
not
contain
∼
ϕ
!
"
!
!
p
p ϕ is unsuccessful
|ϕ
!
!
M models any logic between
S4.4
and
S5,
in
these
logics.
2
l.
M|ϕ
M
M
M
M
"C,
N,
A#
=
"C,
N
,
A#
where
N
is
the
set
of
all
Definition
6.
Given
a
specification
"C,
N,
A#
an
Proof.
If
ϕ
contains
only
formulas
of
the
form
Kα
and
¬Kα
(α
propositional),
p l.∧then
♦q
∧
#
(q
→
p
p)
∧
∧
♦q
♦
∧
(q
#
∧
(q
p)
→
p)
∧
♦
(q
∧
p)
for
any
(M,
w)
with
M,
w
!
ϕ,
v
∈
W
|
wR
v
=
v
∈
W
|
wR
then
for
any
(M,
w)
with
M,
w
!
ϕ,
v
∈
W
|
wR
!
"
!
literals,
from
each
Γ in
A, such
does
n
Proof.
Ifone
ϕ (q
contains
only
formulas
ofthat
the Σ
form
Kα
p ∧ ♦q ∧ # (q → p) ∧ ♦ (q ∧ p)p
∧a ♦q
∧
#
→
p)
∧
∧
p)
ϕ
M|ϕ
M∼
! ♦
! !
M
M(qthat
|ϕ
is
disjunct
d
in
ϕ
such
d
does
not
contain
"C,
N,
=
NwR
,with
A#
where
set
of |all
for
any
(M,
w)
with
M,
wHence
!ϕlϕ,
∈
|w)
=the
vN
∈ϕ,is
W the
| wR
M
Hence
M
,w
!p)
ϕ.
M
,W
w"C,
!Given
ϕ.
Proposition 4.1 shows then
that
S5
is
unique
among
the
typical
of
knowledge
|ϕ
If
isforA#
avlogics
conjunction
ina vnormal
form
and
L
(ϕ)
$=
|ϕ!does
of
and
d
not
contain
quazi-negation
of
p
∧
♦q
∧
#
(q
→
∧
♦
(q
∧
p)
!6.
then
any
(M,
M,
w
!
v
∈
W
wR
Definition
specification
"C,
N,
A#
an
!
andevery
every,set
setpN
A,
there
isaΓaaliteral
literal
Γ
not
quazi-negation
of
and
ΓΓ
∈∈A,
isliterals,
l l N,
∈from
isis
not
quazi-negation
ofqu
and
every
set
∈
A,one
isthat
aeach
literal
l the
∈the
is not
the
Γformula
in
A,
that
Σ does
Definition
6.
Given
athere
specification
"C,
A#
and
ϕthat
innot
DNF,
define
Hence
M
ϕ.
is
disjunct
d∈Γin
ϕthat
that
dΓsuch
does
contain
∼n
ϕthere
pqNN
|ϕ w !
!such for
!
insofar as all of its unsuccessful
formulas
are! Moorean.
The
counterexample
Hence
M
,w
! ϕ.
ϕ
"C,
"C,
N
,
A#
where
N
is
the
set
of
all
q
p
p
!N, A#
|ϕ=
l.l.
l.
of
and
d#
contain
the
quazi-negation
ofn
"C,
N,
A#
where
N
is ∧
the
set
ofnot
all
literals
l ∈such
N
for
which
there
literals,
one
from
each
Γ
in
A,
that
Σ
does
Proof.
If
ϕdoes
contains
only
formulas
of
the
form
Kα
♦qA#
∧ #=
(q"C,
→N
p) ,∧
♦ (q
∧ p)p
∧al ♦q
(q
→
∧
♦know
(qthat
∧
p)
the weaker logics shows that
negative
introspection,
one
can
!not
pp ∧without
isdoes
disjunct
dcome
in p)
ϕ∼to
such
d does
contain
∼
M
is
a
disjunct
d
in
ϕ
such
that
d
not
contain
l
and
there
is
a
selection
Σ
of
of
land
and
d#
does
contain
the
quazi-negation
pN
∧ ♦q
# (qknow
→
p)Γwhether
∧∈♦A,
(q there
∧ p)
then
for
any
(M,
w)that
with
M,
wthe
!form
ϕ,
vand
∈ lW
|of
wR
!
!setnot
p by being truly told, “You
do∧not
orliterals,
not
p,”
aspecification
surprising
case
N
every
Γ
∈
A,
there
is
a
literal
∈
Γ
tha
and
every
set
is
a
literal
l
∈
Γ
is
not
quazi-negation
ooi
If
ϕ
is
a
conjunction
in
normal
L
(ϕ)
=
$
p
∧
♦q
∧
(q
→
p)
∧
♦
(q
∧
p)
!A#
one
from
each
Γ
in
A,
such
that
Σ
does
n
Definition
6.
Given
a
specification
"C,
N,
A#
and
formula
ϕ
in
DNF,
define
!
Definition
6.
Given
a
specification
"C,
N,
and
formula
ϕ
in
DNF,
define
Definition
6.
Given
a
"C,
N,
A#
and
formula
ϕ
N
and
every
set
Γ
∈
A,
there
is
a
literal
l
∈
Γ
that
is
not
the
quazi-negation
N
and
every
set
Γ
∈
A,
there
is
a
literal
l
∈
Γ
tha
literals,
one
from
each
Γ
in
A,
such
that
Σ
does
not
contain
the
quazi-negation
Hence
, w !Proposition
ϕ.!
ϕϕ
ϕ
q"C,
pq N,
p
! ! M|ϕ
! ! , A#
!weaker
l.
l.
of unsuccessfulness. Theq"C,
following
proposition,
although
than
of
l
and
d
does
not
contain
the
quazi-negation
of
"C,
N,
A#
=
"C,
N
where
N
is
the
set
of
all
literals
l
∈
N
for
which
there
A#
=
"C,
N
,
A#
where
N
is
the
set
of
all
literals
l
∈
N
for
which
there
N,
A#
=
"C,
N
,
A#
where
N
is
the
set
of
all
literals
l
∈
N
q
l.l and d does not contain the
l.
If
ϕ is aIfconjunction
in normal
form
L
(ϕ) Kα
$= f
quazi-negation
of any
literal
in
Σ.ofand
pof
Proof.
ϕ
contains
only
formulas
the
form
q
p
4.1, shows that non-Moorean
unsuccessful
formulas
appear
if
we
weaken
logics
of
disjunctddinin
suchthat
thatdddin
does
notcontain
contain
andthere
thereisisa∼aselection
isisaadisjunct
does
not
∼∼l land
ΣΣof
isϕϕ
asuch
disjunct
ϕ such
that
d does
not
contain
l!selection
and there
isofa
M
ϕliteral
isfor
a any
conjunction
in
normal
form
and
L (ϕ)
$=
pIf
∧
♦q
∧
#
→
p)
∧
♦
∧l! p)
!(q6.
Definition
Given
aa(q
specification
N,
A#
an
Definition
6.
Given
aainin
specification
"C,
N,
A#
and
formula
ϕ
in
DNF,
defin
then
(M,
w)
with
M,
w
!
ϕ,
vis"C,
∈not
W
|of
wR
Nliterals,
andways
every
set
N
Γ
and
∈
A,
every
there
set
is
a
Γ
∈
A,
there
l
∈
Γ
is
that
a
literal
is
not
the
∈
Γ
quazi-negation
that
the
qu
one
from
each
Γ
A,
such
that
Σ
does
not
contain
the
quazi-negation
knowledge and belief in other
as
well.
literals,
one
from
each
Γ
A,
such
that
Σ
does
not
contain
the
quazi-negation
Definition
6.
Given
specification
"C,
N,
A#
and
formula
ϕ
in
DNF,
defin
literals,
one
from
each
Γ
in
A,
such
that
Σ
does
not
contain
the
Definition
6.
Given
specification
"C,
N,
A#
an
Proof.
If
ϕ
contains
only
formulas
of
the
form
Kα
ϕ
ϕ
!
!
!
!
!
ϕ
ϕ
"C,
N,
A#
=
"C,
N
,
A#
where
N
is
the
set
of
all
"C,
N,
A#
=
"C,
N
,
A#
where
N
is
the
set
of
all
literals
l
∈
N
for
which
ther
q
q
Hence
M
,
w
!
ϕ.
!
!
!
!
|ϕ=set
M
l.of"C,
l.
If
ϕ
is
a
conjunction
in
normal
form
and
L
(ϕ)
=
$
l
and
d
does
not
contain
the
quazi-negation
of
any
literal
in
Σ.
of
l
and
d
does
not
contain
the
quazi-negation
of
any
literal
in
Σ.
N,
A#
=
"C,
N
,
A#
where
N
is
the
of
all
literals
l
∈
N
for
which
ther
of
l
and
d
does
not
contain
the
quazi-negation
of
any
literal
in
Σ
"C,
N,
A#
"C,
N
,
A#
where
N
is
the
set
of
all
then
forIf
any
(M,
w)
with
M,
!successful.
ϕ,of vthe
∈W
|Kα
wR
contains
only
formulas
formΣ
ϕa isdisjunct
a conjunction
inornormal
form
and
L
(ϕ)
$=such
∅,
then
ϕw
isdoes
Proposition 4.2 For anyqIfissublogic
L ofd in
KTB
KD5,
there
isϕ acontain
formula
(conpqProof.
!
is
a
disjunct
d
in
ϕ
that
d
not
contain
∼
ϕ
such
that
d
does
not
∼
l
and
there
is
a
selection
M
is a disjunct d in ϕ such thatHence
d adoes
not
∼
l and
there
is anot
selection
ΣwR
is
disjunct
d in
ϕ such
that
d does
contain
∼oo
M
w
! ϕ.
|ϕ ,contain
then
for
any
(M,
w)
with
M,
w
!
ϕ,
v
∈
W
|
sistent with S5) that is unsuccessful
in
L
but
is
not
Moorean.
literals,
one
from
each
Γ
in
A,A#
such
that
Σ
does
ni
Definition
6.from
Given
Definition
a specification
6.A,
"C,
athat
N,
A#
and
ϕ
in
and
DNF,
formula
define
ϕ
literals,
one
each
ΓΓ in
that
Σ
the
quazi-negation
pGiven
∧such
♦q ∧
#
(q
→does
p)
∧not
♦formula
(qcontain
∧N,
p)
Proof.
If
ϕspecification
contains
only
formulas
of
the
form
Kα
literals,
from
each
in
such
ΣΓ
not
contain
the
quazi-negatio
literals,
one
from
each
Γ"C,
in
such
does
!that
Proof.
Ifϕ one
ϕ contains
the
Kα
and
¬Kα
(α
propositional),
N A,
and
set
∈ϕ.
A,
isA,a! literal
l! ∈Σ
ΓM
that ni
ϕ formulas
Hence
M
,form
w
!does
! only
! every
! of
! there
|ϕ
!
"literal
of
l
and
d
does
not
contain
the
quazi-negation
of
"C,
N,
A#
=
"C,
"C,
N
N,
,
A#
A#
where
=
"C,
N
N
is
,
the
A#
where
set
of
all
N
literals
is
the
set
l
∈
of
N
all
for
literals
which
l
there
∈
N
f
of
l
and
d
does
not
contain
the
quazi-negation
of
any
in
Σ.
then
for
any
(M,
w)
with
M,
w
!
ϕ,
v
∈
W
|
wR
M
M
M
M
|ϕ | ϕ
|ϕ
conjunction
in
normal
form
L(ϕ)
(ϕ)
$=
∅,♦
then
ϕ
successful.
IfIf
ϕ
aa♦p
conjunction
form
and
L(q
∅,
then
isisvsuccessful.
ofϕislis
and
d∧does
not
contain
the
of
any
in
Σ.
ϕ isin
anormal
conjunction
in
normal
and
L
(ϕ)
$= W
∅, then
is of
suc
lquazi-negation
and
d#
the
quazi-negation
pof
∧
♦q
∧and
→not
p)
∧contain
(q
∧,ϕ
then
for
any
(M,
w)
with
M,
!
ϕ,
vFigure
∈does
W
|form
wR
v|ϕliteral
=p)
∈
wR
Proof. For KTB consider
≡
♦qIfand
the
model
M
in
3.$=
In
M
l. w
qpdoes
is a disjunct d inisϕasuch
disjunct
that dd in
ϕ such
not
contain
that
d
does
∼
l
and
not
contain
there
is
∼
a
selection
l
and
there
Σ
of
is
a
Hence
M
,
w
!
ϕ.
|ϕ
Hence
M|ϕ , w so
! ϕ.
the left and right points are
eliminated,
ϕ becomes false
at the
center
psuch
∧each
♦q
∧Γ#
(qdoes
→ such
p)point.
∧♦
(q ∧Σp)does
!
literals,
from
literals,
each
ΓIfone
in
A,
from
that
Σ
in
A,
not
that
contain
quazi-negation
not
the
Proof.Ifone
Ifϕϕcontains
contains
only
formulas
of
the
form
Kα
and
¬Kα
(α
propositional),
Proof.
only
formulas
of
the
form
Kα
and
¬Kα
(α
propositional),
N
and
every
Γ ∈
A,
there
is
a!the
literal
lcontain
∈"¬Kα
Γ that
Proof.
ϕ
contains
only
formulas
the
form
Kα
and
(αi
Definition
6.
Given
aof
specification
"C,
N,
A#
! set
!ϕ
""
!M
!
!and
p
M
M
M
M
M
M
M
M
M
|ϕ
|ϕliteral
|ϕΣ
!of
! vis
If
ϕ
is
a
conjunction
in
normal
form
and
L
(ϕ)
$=
ofthen
lIf and
d
does
not
of
l
contain
and
d
does
the
not
quazi-negation
contain
the
quazi-negation
any
literal
in
of
Σ.
any
in
If
ϕ
is
a
conjunction
in
normal
form
and
L
(ϕ)
=
$
∅,
then
ϕ
successful.
p
∧
♦q
∧
#
(q
→
p)
∧
♦
(q
∧
p)
for
any
(M,
w)
with
M,
w
!
ϕ,
v
∈
W
|
wR
v
=
∈
W
|
wR
then
for
any
(M,
w)
with
M,
w
!
ϕ,
v
∈
W
|
wR
=
v
∈
W
|
wR
l.
then
for
any
(M,
w)
with
M,
w
!
ϕ,
v
∈
W
|
wR
v
=
v
∈|ϕv
A#
"C,LN(ϕ)
, A#
where
Nϕ is
the and
set of
all lite
and
$= in
∅, then
successful.
is a=
conjunction
normal
form
L (ϕ)
$=
p ∧ϕ
♦qis∧a#conjunction
(q → p) ∧ ♦in
(qnormal
∧"C,
p)If
pqN,ϕform
HenceM
M|ϕ|ϕ, w
, w!!
Hence
ϕ.ϕ. M|ϕ ,iswa!disjunct
Hence
ϕ.
d in ϕ such that d does not contain ∼ l a
Proof.
If
ϕ
contains
formulas
of
the
form
Kα
Proof.
If
ϕ
contains
only
formulas
of
the
form
Kα
and
(α
propositional)
Definition
6.
Given
aΓonly
specification
N,does
A#
and
!
!
""¬Kα
!! that
p
Proof.
If
ϕ
contains
only
formulas
of
the
form
Kα
and
¬Kα
(α"C,
propositional)
Proof.
If
ϕ
contains
only
formulas
of
the
form
Kα
literals,
one
from
each
in
A,
such
Σ
M not
M|ϕ
M|
M
M
!
ϕ
!
q
pthen for any (M, w) with M,
!
!
then
for
any
(M,
w)
with
M,
w
!
ϕ,
vvthen
∈
W
||lite
wR
w
!!A#
ϕ,
vvnormal
∈
W
|| ∅,
wR
=
vv$=
∈
W
||M
wR
M
M
M$=
Mv
"C,
N,
=
"C,
N
,
A#
where
N
is
the
set
of
all
|ϕ
is
afor
in♦
a∧
conjunction
form
and
in
L
(ϕ)
form
then
and
ϕ
L
(ϕ)
is
successful.
∅,
ϕ
is
suc
∧♦q
♦q∧
∧conjunction
→If
∧is♦
(q
p)
pIfp∧ϕ
##
(q(q→
(qnormal
∧∧(q
p)
then
any
(M,
w)
with
M,
w
ϕ,
∈
W
wR
v
=
∈
W
wR
pp)p)
∧ϕ∧
♦q
#
→
p)
∧
♦
(q
∧
p)
then
for
any
(M,
w)
with
M,
w
!
ϕ,
∈
W
wR
of lqand d does not contain the quazi-negation of any
Hence
M
,
w
!
ϕ.
Hence
M
,
w
!
ϕ.
|ϕin
is
a
disjunct
d
ϕ
such
that
d
does
not
contain
∼
l
a
Fig.
3. ϕ.
Hence M|ϕ
,
w
!
Hence
M
,
w
!
ϕ.
|ϕ
|ϕ
Proof.
If
ϕ
contains
Proof.
only
If
ϕ
formulas
contains
of
only
the
form
formulas
Kα
of
and
the
¬Kα
form
(α
Kα
propositional),
and
¬Kα
(α
literals,
one
from
each
Γ
in
A,
such
that
Σ
does
not
!
!
"
!
"
!
q
ppq
p
M
M
M|ϕ
|ϕ v
For KD5 consider ψ then
≡p¬q
∨ (p
∧
♦♦q)
and
theof
M
inW
Figure
4.
Only
∧
(q
→
p)
∧M
♦
∧
then
w)p)
for
M,
(M,
wmodel
ϕ,♦q
with
v∧
∈#
w |!
wR
ϕ,
=
Wp)
v∈
| wR
WM
v |=
wRM
v∈
♦q
∧
(q
∧
(q
∧
p)
lpp!and
contain
quazi-negation
p∧
∧for
♦qany
∧#
#(M,
(q →
→
p)with
∧♦
♦any
(q If
∧
p)
∧w)
∧does
#M,
(qnot
→
p)
♦vv(q
(q∈the
∧
p)
ϕ
is
a♦qdconjunction
in∧normal
form
and L (ϕ) of
$= any
∅, t
the right point is eliminated
in M|ϕ
so!ψHence
Hence
ϕ.becomes
M|ϕfalse
, w !at
ϕ.the left point.
|ψ , w
p
p
Proof.
Kα an
qq p
q
p If ϕ contains only formulas of
! the form
M $= ∅,M
p ∧ ♦q ∧ # (q → pp)∧∧♦q♦ ∧
(q#∧(q
p)
→
p)a∧any
♦ (q(M,
∧ p)w) with
If
ϕ isfor
conjunction
in normal
L (ϕ)
then
M, w !form
ϕ, and
v∈W
| wR t
Hence M|ϕ , w ! ϕ.
Proof.
Kα an
qq If ϕ contains only formulas of
p qq
p
! the form
M
then
for
any
(M,
w)
with
M,
w
!
ϕ,
v
∈
W
|
wRM
p ∧ ♦q ∧ # (q → p) ∧ ♦ (q ∧ p)
Fig. 4.
Hence M|ϕ , w ! ϕ.
q in KD45 to qthe normal pform ¬q ∨ (p ∧ ♦q), which is
The formula ψ is equivalent
p ∧ ♦q ∧ #6(q → p) ∧ ♦ (q ∧ p)
not Moorean, since neither disjunct is a Moorean conjunction.
2
There are other formulas that one may wish to categorize
as Moorean, perhaps
pq
even as Moore sentences, but which are inconsistent with KD45.
Example 4.3 The formula p ∧ ♦¬p is a kind of “higher order” Moore sentence,
which is satisfiable on intransitive frames but is also qself-refuting over such frames.
Similarly, the formula ♦p ∧ ♦¬♦p is satisfiable yet self-refuting on non-Euclidean
frames. The first formula says that the agent is not aware of what he believes, while
the second says that he is not aware of what he does not believe. These formulas
are the very witnesses of a failure to validate axioms 4 and 5, respectively.
6
Note that while the right disjunct is not a Moorean conjunction, in S5 (but not in KD45) it implies
the Moorean conjunction p ∧ ♦q. This is an instance of the more general fact that in S5 some successful
formulas imply unsuccessful formulas.
10
a, b
a, b
b
p
Holliday and Icard
a, b
A natural question is whether bthere are non-Moorean
sources of unsuccessfulness
in languages more expressive
Consider, for example,
p than the basic modal language.
p
a language with multiple modalities, as in multi-agent epistemic logic. Without
giving a formal definition of Moorean in the multi-agent case, it is nonetheless clear
a, b case than in the singlethat there are more waysa,tob be Moorean in the multi-agent
b
agent case, even assuming quasi-partitions for peach relation. For example, there
are self-refuting, indirect Moore sentences such as a p ∧ ♦b ¬p, which imply singleagent Moore sentences (in this case, p ∧ ♦b ¬p) in multi-agent S5. There are also
self-refuting, higher order Moore sentences such as a p∧♦b ¬a p and ♦a p∧♦b ¬♦a p,
p
which resemble the higher order, single-agent Moore sentences consistent with logics
weaker than KD45, noted in Example 4.3. However, not all unsuccessful formulas
in the multi-agent context have a Moorean character.
a, b
b
a, b
a, b
b
a, b
Example 4.4 In the model M in Figure 5, the formula ϕ ≡ ♦a p ∧ ♦a ♦b ¬p is true
p
at the left point but false at the right point, since ♦a p is false at the right point.
p
Fig. 5.
As a result, in the model M|ϕ the right point is eliminated, in which case ϕ becomes
false at the left point because ♦a ♦b ¬p becomes false there.
The formula ♦a p ∧ ♦a ♦b ¬p does not resemble any single-agent Moorean formula,
yet it is nonetheless unsuccessful. Given the connection that we have observed
between introspection and Moorean phenomena in the single-agent case, we can see
why there should be non-Moorean unsuccessful formulas in the multi-agent case:
agents do not have introspective access to each other’s knowledge or beliefs.
5
Related Classes of Formulas
Theorem 3.13 shows that in a wide range of logics, every self-refuting formula is
a Moore sentence (i) and every unsuccessful formula is a Moorean sentence (ii).
However, neither the converse of (i) nor the converse of (ii) holds in general. In
this section, we relate the failures of the converses of (i) and (ii) to other classes of
formulas. In Theorems A.3 and A.6 in Appendix A, we overcome these failures and
give syntactic characterizations of the self-refuting and unsuccessful formulas as the
strong Moore sentences and strong Moorean sentences, respectively.
For simplicity we assume in this section that we are working with S5 only.
5.1
Informative, Cartesian, and eventually self-refuting formulas
Definition 5.1 A formula ϕ is (potentially) informative iff there is a pointed model
such that M, w ϕ and M|ϕ 6= M. Otherwise ϕ is uninformative. A formula ϕ is
always informative iff for all pointed models such that M, w ϕ, M|ϕ 6= M.
Note that if a formula is not always informative, then it is not self-refuting, for there
is a model such that M, w ϕ but M|ϕ = M, so M|ϕ , w ϕ. This observation
11
Holliday and Icard
explains some of the counterexamples to the converse of Theorem 3.13(i), Moore
sentences that are not self-refuting.
Example 5.2 The formula (p ∧ ♦¬p) ∨ (q ∧ ♦¬q) is a Moore sentence, but it is
not always informative (take a model with only two connected points, one of which
satisfies p but not q and the other of which satisfies q but not p) and hence not
self-refuting. The formula (p ∧ ♦¬p) ∨ (¬p ∧ ♦p) is a Moore sentence, but it is
uninformative and hence successful, but not self-refuting. These examples show that
neither the self-refuting nor the unsuccessful formulas are closed under disjunction.
From our previous results, we have the following corollary.
Corollary 5.3 A conjunction in normal form is always informative iff it is selfrefuting.
Proof. By inspection of the proof of Lemma 3.9(i).
2
However, a Moore sentence that is always informative may not be self-refuting.
Example 5.4 The formula ϕ ≡ (p ∧ ♦¬p)∨(p ∧ q ∧ ♦¬q) is always informative, for
if M, w (p ∧ ♦¬p), then there is a witness to ♦¬p that is eliminated in M|ϕ , and if
M, w (p ∧ q ∧ ♦¬q), then either the witness to ♦¬q does not satisfy (p ∧ ♦¬p), in
which case it does not satisfy ϕ and is eliminated in M|ϕ , or it does satisfy (p ∧ ♦¬p),
in which case there is a witness to ♦¬p that is eliminated in M|ϕ . In either case,
M|ϕ 6= M, so ϕ is always informative. However, ϕ is not self-refuting, as shown
by the partition model M with W = {w, v, u}, V (p) = {w, v}, and V (q) = {w, u}.
We have M, w p ∧ q ∧ ♦¬q, and given W|ϕ = {w, v}, also M|ϕ , w p ∧ q ∧ ♦¬q.
Interestingly, the formula ϕ is self-refuting within two steps; given M, w ϕ and
M|ϕ , w ϕ, we have M|ϕ |ϕ , w 2 ϕ. Let δ1 ≡ (p ∧ ♦¬p) and δ2 ≡ (p ∧ q ∧ ♦¬q)
be the disjuncts of
ϕ. All ¬p-points are eliminated in M|ϕ , so M|ϕ ¬δ1. Hence
M|ϕ |ϕ = M|ϕ |δ . But since M|ϕ , w δ2 and δ2 is self-refuting, M|ϕ |δ , w 2
2
2
δ2 . Since δ1 is existential, M|ϕ |δ , w 2 δ1 , so we conclude that M|ϕ |ϕ , w 2 ϕ.
2
Example 5.4 points to the interest of self-refutation “in the long run.”
Definition 5.5 Given
a model M, we define M|n ϕ recursively by M|0 ϕ = M,
M|n+1 ϕ = M|n ϕ |ϕ . A formula ϕ is self-refuting within n steps iff for all pointed
models, if M, w ϕ, then M|m ϕ , w 2 ϕ for some m ≤ n; ϕ is eventually self-refuting
iff for all pointed models, if M, w ϕ, then there is an n such that M|n ϕ , w 2 ϕ.
Using Definition 5.5, we can generalize Corollary 5.3. First, recall Fitch’s paradox, mentioned in Section 1, which we can restate as: if all truths are knowable,
then all truths are (already) known. One proposal for avoiding the paradox [19] is
to restrict the claim that all truths are knowable to the claim that all Cartesian
truths are knowable, in which case it does not follow that all truths are known.
Definition 5.6 ϕ is Cartesian iff ϕ is satisfiable. 7
7
The term ‘Cartesian’ is due to Tennant [19], though his definition is in terms of consistency rather than
satisfiability. The term ‘knowable’ seems more natural, but ‘knowable’ is used in a different sense in the
literature [9], for what we call ‘learnable’ in Definition 5.13.
12
Holliday and Icard
The class of formulas ϕ for which ϕ is satisfiable seems a natural object of study
in its own right. The following proposition establishes a connection between these
formulas, defined “statically,” and the other formulas classes that we have defined
“dynamically” in terms of model transformations. It also generalizes Corollary 5.3.
Proposition 5.7 The following are equivalent:
(i) ϕ is always informative.
(ii) ϕ is not Cartesian.
(iii) ϕ is eventually self-refuting.
Proof. Suppose ϕ is always informative and M, w ϕ. Without loss of generality,
assume M is chained (as above Lemma 3.8). Since ϕ is always informative, M|ϕ 6=
M. Hence there is a point v such that wRv and M, v 2 ϕ, in which case M, w 2 ϕ.
Since M was arbitrary, ϕ is unsatisfiable, so ϕ is not Cartesian.
Suppose ϕ is not Cartesian, and without loss of generality, assume ϕ is in normal
form. We prove by induction that for all n ≥ 0, if M|n ϕ , w ϕ, then there are
♦γ1 , ..., ♦γn+1 distinct subformulas of ϕ with M|n+1 ϕ ¬♦γ1 ∧ ... ∧ ¬♦γn+1 (∗). It
follows that since ϕ has some finite number n of distinct ♦γk subformulas, we must
have M|m ϕ , w 2 ϕ for some m ≤ n. For the base case, assume M, w ϕ. Since ϕ is
not Cartesian, M|ϕ , w 2 ϕ. Let v be such that wRv and M|ϕ , v 2 ϕ, and note that
M, v ϕ, for otherwise v would not have been retained in M|ϕ . From the fact that
universal formulas are preserved under submodels, it follows that for some ♦γ1 in ϕ,
we have M, v ♦γ1 but M|ϕ , v 2 ♦γ1 . Hence M|ϕ ¬♦γ1 by Lemma 3.8(i). For
the inductive step, assume that M|n+1 ϕ , w ϕ. Since w is retained in M|n+1 ϕ , we
have M|n ϕ , w ϕ, so by the induction hypothesis there are ♦γ1 , ..., ♦γn+1 distinct
subformulas of ϕ for which (∗) holds. By the same reasoning as before, since ϕ is
Cartesian, M|n+2 ϕ , w 2 ϕ, so there is some z with wRz and some ♦γn+2 in ϕ such
that M|n+1 ϕ , z ♦γn+2 but M|n+2 ϕ ¬♦γn+2 . Moreover, since ♦γk formulas are
preserved under extensions, M|n+2 ϕ ¬♦γ1 ∧ ... ∧ ¬♦γn+1 as well. Finally, given
M|n+1 ϕ , z ♦γn+2 and (∗), ♦γn+2 is distinct from ♦γ1 , ..., ♦γn+1 .
Suppose ϕ is not always informative. Then there is a model with M, w ϕ and
M|ϕ = M, in which case M|n ϕ = M for all n, so ϕ is not eventually self-refuting.2
Corollary 5.8 ϕ is eventually self-refuting iff it is self-refuting within n steps, with
n bounded by the number of distinct diamond formulas in a normal form of ϕ.
Proof. By inspection of the proof of the previous proposition.
5.2
2
Successful, super-successful, and learnable formulas
The converse of Theorem 3.13(ii) fails because a formula ϕ in normal form that contains a Moorean conjunction as a disjunct may be successful. For example, it is easy
to see that if the disjunction of the non-Moorean conjunctions in ϕ is a consequence
of the disjunction of the Moorean conjunctions in ϕ, then ϕ is not only successful
but super-successful, given Lemma 3.11. Examples of such ϕ include (p ∧ ♦¬p) ∨ p
and (p ∧ ♦¬p) ∨ ♦¬p. There are also successful formulas ψ that contain Moorean
conjunctions as disjuncts, but do not meet the condition of ϕ. These formulas nevertheless manage to be successful by a kind of compensation: when one disjunct
13
Holliday and Icard
of ψ goes from true at M, w to false at M|ψ , w, another disjunct compensates by
going from false at M, w to true at M|ψ , w. The formula (p ∧ ♦¬p) ∨ p exhibits
this kind of compensation against a Moore conjunction, while the formula in the
proof of the following proposition does so against a Moorean conjunction.
Proposition 5.9 Not all successful formulas are super-successful.
Proof. The formula ϕ ≡ (p ∧ ♦q) ∨ p is successful but not super-successful. Suppose M, w ϕ, and without loss of generality assume that M is chained. Case
1: M, w p. Then M|ϕ = M, so M|ϕ , w ϕ. Case 2: M, w 2 p. Then
M|ϕ = M|p∧♦q , and given M|p∧♦q p, we again have M|ϕ , w ϕ. Hence ϕ is
successful. But ϕ is not super-successful, as shown by the partition model N with
W = {w, v, u}, V (p) = {w}, and V (q) = {u}. We begin with N , w ϕ, for while
N , w 2 p, it holds that N , w p ∧ ♦q. Moreover, given W|ϕ = {w}, we still have
N|ϕ , w ϕ, for while N|ϕ , w 2 p ∧ ♦q, it holds that N|ϕ , w p. However, in the
extension N 0 of N|ϕ with W 0 = {w, v}, we now have N 0 , w 2 ϕ. The fact that
u∈
/ W 0 gives N 0 , w 2 p ∧ ♦q, and the fact that v ∈ W 0 gives N 0 , w 2 p.
2
Proposition 5.10 A formula ϕ is super-successful iff for a propositional variable
p that does not occur in ϕ, ϕ ∨ p is successful.
Proof. (⇒) Suppose ϕ is super-successful and M, w ϕ ∨ p, where p does not
occur in ϕ. If M, w p, then M|ϕ∨p , w p, and if M, w ϕ, then given M|ϕ ⊆
M|ϕ∨p and the assumption that ϕ is super-successful, M|ϕ∨p , w ϕ. In either case,
M|ϕ∨p , w ϕ ∨ p, so ϕ ∨ p is successful.
(⇐) Suppose ϕ is not super-successful, so there is an M = hW, R, V i with w ∈ W
such that M, w ϕ, and an M0 = hW 0 , R0 , V 0 i such that M|ϕ ⊆ M0 ⊆ M and
M0 , w 2 ϕ. Let M∗ = hW, R, V ∗ i be the same as M except that V ∗ (p) = W 0 \ W|ϕ .
Then since p does not occur in ϕ, we have M∗ , w ϕ∨p given M, w ϕ. Moreover,
∗
since W|ϕ∨p
= W 0 , we have M∗|ϕ∨p , w 2 ϕ given M0 , w 2 ϕ. From the assumption
that M, w ϕ, it follows that w ∈ W|ϕ , in which case w ∈
/ V ∗ (p) and hence
∗
∗
M|ϕ∨p , w 2 p. But then M|ϕ∨p , w 2 ϕ ∨ p, so ϕ ∨ p is unsuccessful.
2
By Proposition 5.10, complexity and syntactic characterization results for successful
formulas carry over immediately to super-successful formulas.
From the previous propositions we obtain a surprising failure of closure.
Corollary 5.11 The set of successful formulas is not closed under disjunction.
Proof. Immediate from Propositions 5.9 and 5.10.
2
Example 5.12 The formula ¬p ∧ ¬q is successful, and by the proof of Proposition
5.9, (p ∧ ♦q) ∨ p is also successful. However, the disjunction of these formulas,
χ ≡ (¬p ∧ ¬q) ∨ ((p ∧ ♦q) ∨ p) is unsuccessful, as shown by the model N in the
proof of Proposition 5.9. We begin with N , w χ, since while N , w 2 ¬p ∧ ¬q (and
hence N|χ , w 2 ¬p ∧ ¬q), we have already seen that N , w (p ∧ ♦q) ∨ p. However,
N|χ = N 0 , and we have already seen that N 0 , w 2 (p ∧ ♦q) ∨ p.
By contrast, the set of super-successful formulas is closed under disjunction, since
M|ϕ∨ψ is an extension of M|ϕ and M|ψ .
14
Holliday and Icard
Another consequence of the previous results concerns the relation between successful and learnable formulas. Like the Cartesian formulas, the learnable formulas
have been discussed in connection with Fitch’s paradox [3,9].
Definition 5.13 A formula ϕ is (always) learnable iff for all pointed models, if
M, w ϕ, then there is some ψ such that M|ψ , w ϕ. 8
Formulas that are learnable (and satisfiable) are Cartesian according to Definition 5.6, but the converse does not hold [3]. For example, the formula p ∧ ♦q
is Cartesian, since (p ∧ ♦q) is satisfiable, but it is not (always) learnable, for if
M, w (p → ¬q) and M0 ⊆ M, then M0 , w 2 (p ∧ ♦q).
All successful formulas are learnable [9], since if M, w ϕ and ϕ is successful,
then not only M|ϕ , w ϕ but also M|ϕ , w ϕ. For if v is retained in M|ϕ ,
then M, v ϕ, in which case M|ϕ , v ϕ by the successfulness of ϕ. Hence for a
successful ϕ, we can always take ψ in Definition 5.13 to be ϕ itself. On the other
hand, it is natural to ask whether some unsuccessful formulas are learnable as well.
Corollary 5.14 Not all learnable formulas are successful.
Proof. Let δ1 ∨ δ2 be an unsuccessful disjunction with δ1 and δ2 successful, as given
by Corollary 5.11. If M, w δ1 ∨ δ2 , then M, w δi for i = 1 or i = 2. Since δi
is successful, we have M|δi , w δi , in which case M|δi , w (δ1 ∨ δ2 ). Since M
was arbitrary, δ1 ∨ δ2 is (always) learnable.
2
6
Conclusion
From a technical point of view, we have studied the question of when a modal
formula is preserved under relativizing models to the formula itself. In the epistemic
interpretation of modal logic, this preservation question takes on a new significance:
it concerns whether an agent retains knowledge of what is learned, which requires
that what is learned remain true. We have shown (Theorem 3.13) that for an
introspective agent, the only true sentences that may become false when learned
are variants of the Moore sentence. For an agent without introspection or multiple
agents without introspective access to each other’s knowledge or beliefs, there are
non-Moorean sources of unsuccessfulness (Propositions 4.1 and 4.2, Example 4.4).
In connection with our study of Moorean phenomena, we have observed a number
of related results. We saw that the sentences that always provide information to an
agent, no matter the agent’s prior epistemic state, are exactly those sentences that
cannot be known—and will eventually become false if repeated enough (Proposition
5.7); we saw that there are sentences that always remain true when they are learned,
but whose truth value may oscillate while an agent is on the way to learning them
(Proposition 5.9); and we saw that there are sentences that sometimes become false
when learned directly, but which an agent can always come to know indirectly by
learning something else (Corollary 5.14).
In Appendix A, we return to the problem with which we began. We give syntactic characterizations of the self-refuting and unsuccessful formulas as the strong
Moore sentences and strong Moorean sentences, respectively.
8
The term ‘learnable’ is used by van Benthem [3]. Balbiani et al. [9] use the term ‘knowable’.
15
Holliday and Icard
Acknowledgement
We wish to thank Hans van Ditmarsch and the anonymous referees for their helpful
comments on an earlier draft of this paper, and Johan van Benthem for stimulating
our interest in the topic of successful formulas.
References
[1] Baltag, A., L. Moss and S. Solecki, The logic of public announcements, common knowledge and private
suspicions, in: I. Gilboa, editor, Proceedings of the 7th Conference on Theoretical Aspects of Rationality
and Knowledge (TARK 98), Morgan Kaufmann, 1998 pp. 43–56.
[2] Baltag, A., H. van Ditmarsch and L. Moss, Epistemic logic and information update, in: P. Adriaans
and J. van Benthem, editors, Philosophy of Information, North-Holland, 2008 pp. 361–456.
[3] van Benthem, J., What one may come to know, Analysis 64 (2004), pp. 95–105.
[4] van Benthem, J., One is a lonely number: on the logic of communication, in: Z. Chatzidakis, P. Koepke
and W. Pohlers, editors, Logic Colloquium ’02, ASL & A.K. Peters, 2006 pp. 96–129.
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[7] van Ditmarsch, H., W. van der Hoek and B. Kooi, “Dynamic Epistemic Logic,” Springer, 2008.
[8] Andréka, H., I. Németi and J. van Benthem, Modal languages and bounded fragments of predicate logic,
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after an announcement’, The Review of Symbolic Logic 1 (2008), pp. 305–334.
[10] Carnap, R., Modalities and quantification, The Journal of Symbolic Logic 11 (1946), pp. 33–64.
[11] Fitch, F. B., A logical analysis of some value concepts, The Journal of Symbolic Logic 28 (1963),
pp. 135–142.
[12] Gerbrandy, J., “Bisimulations on Planet Kripke,” Ph.D. thesis, University of Amsterdam (1999), ILLC
Dissertation Series DS-1999-01.
[13] Hintikka, J., “Knowledge and Belief: An Introduction to the Logic of the Two Notions,” College
Publications, 2005.
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Proceedings of the Fifth International Joint Conference on Autonomous Agents and Multiagent Systems
(AAMAS 06) (2006), pp. 137–143.
[15] Meyer, J.-J. Ch. and W. van der Hoek, “Epistemic Logic for AI and Computer Science,” Cambridge
Tracts in Theoretical Computer Science 41, Cambridge University Press, 1995.
[16] Moore, G. E., A reply to my critics, in: P. Schilpp, editor, The Philosophy of G.E. Moore, The Library
of Living Philosophers 4, Northwestern University, 1942 pp. 535–677.
[17] Plaza, J., Logics of public communications, in: M. Emrich, M. Pfeifer, M. Hadzikadic and Z. Ras,
editors, Proceedings of the 4th International Symposium on Methodologies for Intelligent Systems:
Poster Session Program, Oak Ridge National Laboratory, ORNL/DSRD-24, 1989 pp. 201–216.
[18] Qian, L., Sentences true after being announced, in: Proceedings of the Student Session of the 1st North
American Summer School in Logic, Language, and Information (NASSLLI), Stanford University, 2002.
[19] Tennant, N., “The Taming of The True,” Oxford: Clarendon Press, 1997.
[20] Visser, A., J. van Benthem, D. de Jongh and G. R. R. de Lavalette, NNIL, A study in intuitionistic
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Logic 12 (1971), pp. 341–357.
16
Holliday and Icard
A
Appendix
In this appendix, we address the problem of giving syntactic characterizations of
the successful and self-refuting formulas. Given Theorem 2.6, Lemma 3.6, and the
PAL definitions of successful and self-refuting, there is a trivial characterization of
both classes: ϕ is successful iff the result of reducing the PAL formula ¬[ϕ]ϕ to
normal form in the basic modal language is not clear; ϕ is self-refuting iff the result
of reducing the PAL formula ¬[ϕ]¬ϕ to normal form in the basic modal language
is not clear. Although by Proposition 2.7, the PAL definitions of successful and
self-refuting lead to optimal methods for checking for these properties, they do
not provide much insight beyond the semantic definitions of the formula classes. In
Theorems A.3 and A.6 below, we give syntactic characterizations of the self-refuting
and unsuccessful formulas that reveal more than the trivial characterization.
We state the following results for S5 with the standard definitions of successful
(∀M, w : M, w ϕ ⇒ M|ϕ , w ϕ) and self-refuting (∀M, w : M, w ϕ ⇒
M|ϕ , w 2 ϕ). With minor changes the results also hold for KD45, given the
modified definitions using precondition ♦+ ϕ instead of ϕ.
We will continue to use the abbreviations δ α , δ ♦ , etc., from Definition 3.3. We
will also use δ̂ ♦ to denote a conjunct of δ ♦ , i.e., a βi or ♦γk in δ. As before,
∼ ϕ is the negation of ϕ in negation normal form, with ¬ applying only to literals.
Definition A.1 ϕ is a strong Moore sentence iff for any normal form ϕ∗ of ϕ, no
disjunct of ϕ∗ is compensated in ϕ∗ . A disjunct δ ≡ α∧β1 ∧...∧βn ∧♦γ1 ∧...∧♦γm
of ϕ∗ is compensated in ϕ∗ iff there is a disjunct δ 0 of ϕ∗ , a subset S of the disjuncts
of ϕ∗ , a sequence of disjuncts σγ1 , ..., σγm (not necessarily distinct) from S, and for
every σ ∈
/ S a conjunct σ̂ ♦ of σ ♦ , such that χ ≡ δ 0 ∧ δ α ∧ χ ∧ χ♦ is clear, where
χ ≡
^
σ∈S
σ ♦ ∧
^
σ ∈S
/
χ♦ ≡
∼ σ̂ ♦ ∧
^
k≤m
^
σ∈S,j≤n
(∼ σ α ∨ βj ) ;
♦ σγαk ∧ γk .
Note that χ is in normal form, so the definition of clarity properly applies. The
reason for the use of ∼ σ̂ ♦ is that ∼ σ ♦ may be a disjunction, in which case χ
would not be in normal form. Hence we pull the disjunction out of the formula and
add an existential quantifier to the condition, since σ ♦ is false iff there is some
conjunct σ̂ ♦ of σ ♦ that is false.
As we will see in the proof of Theorem A.3, the existence of a compensated
disjunct in ϕ is equivalent to there being a model in which ϕ has a “successful
update,” in the sense that M, w ϕ and M|ϕ , w ϕ. For ϕ to have a successful
update in M, there must be a disjunct δ 0 of ϕ true at a point w in M and a
disjunct δ of ϕ (possibly distinct from δ 0 ) true at w in M|ϕ . What is required for δ
to be true at w in M|ϕ is that its propositional part δ α is true at w in M, that its
universal part δ is already true at w in M or becomes true at w in M|ϕ , and that
its existential part δ ♦ remains true at w in M|ϕ . This is exactly what χ captures.
Proposition A.2 Every strong Moore sentence is a Moore sentence.
17
Holliday and Icard
Proof. By Definitions A.1 and 3.12, it suffices to show that for ϕ in normal form, if
a disjunct δ of ϕ is not compensated in ϕ, then δ is a Moore conjunction. To prove
the contrapositive, suppose δ is not a Moore conjunction. Then by Definition 3.7,
δ ∧♦δ α is clear and δ ∧♦ (δ α ∧ γk ) is clear for every ♦γk conjunct in δ. It follows that
V
V
♦ (δ α ∧ γk ), let
♦ (δ α ∧ γk ) is clear and hence satisfiable. Given M, w δ∧
δ∧
k≤m
k≤m
V ♦ V
S be the set of disjuncts in ϕ such that M, w σ ∧
¬σ ♦ . For every σ ∈
/ S,
σ∈S
σ
∈S
/
V ♦ V
pick a conjunct σ̂ ♦ of σ ♦ such that M, w σ ∧
∼ σ̂ ♦ . Since δ ∈ S, let
σ∈S
σ ∈S
/
the sequence σγ1 , ..., σγm of disjuncts from S be such that σγi ≡ δ for 1 ≤ i ≤ m.
We claim that for the χ as in Definition A.1 based on these choices, M, w χ.
V
♦ (δ α ∧ γk ) and our choice of σγ1 , ..., σγm , we have
From the fact that M, w k≤m
V
M, w χ♦ . Since M, w δ , it is immediate that M, w (∼ σ ∨ βj ).
σ∈S,j≤n
V
V ♦
∼ σ̂ ♦ , this gives M, w χ . Finally, setting
σ ∧
Together with M, w σ∈S
σ ∈S
/
δ 0 ≡ δ, we have M, w χ. Hence χ is clear, so δ is compensated in ϕ.
2
We can now generalize Theorem 3.13(i).
Theorem A.3 ϕ is self-refuting if and only if ϕ is a strong Moore sentence.
Proof. ϕ is self-refuting iff any equivalent normal form of ϕ is self-refuting, so let
us assume ϕ is already in normal form.
(⇐) We prove the contrapositive. Suppose ϕ is not self-refuting, so there is a
pointed model such that M, w ϕ and M|ϕ , w ϕ. Given M|ϕ , w ϕ, there is a
disjunct δ of ϕ such that M|ϕ , w δ. We claim that δ is compensated in ϕ, so ϕ is
not a strong Moore sentence. It suffices to show the satisfiability of an appropriate
χ as in Definition A.1.
Since M, w ϕ, there is a disjunct δ 0 of ϕ such that M, w δ 0 . This gives the
first conjunct of χ. Since M|ϕ , w δ, we have M, w δ α because δ α is propositional
and hence preserved under extensions. This gives the second conjunct of χ.
Next we claim M, w χ . Let S be the set of the disjuncts of ϕ such that
V ♦ V
¬σ ♦ , and let σ̂ ♦ be a false conjunct of each σ ♦ for σ ∈
/ S.
M, w σ ∧
σ∈S
σ ∈S
/
V
For reductio, suppose M, w 2
(∼ σ α ∨ βj ), where β1 ∧ ... ∧ βn ≡ δ .
σ∈S,j≤n
Then there is some v with wRv and M, v σ α ∧ ¬βj for some σ ∈ S and j ≤ n.
Since σ ∈ S, we have M, w σ ♦ . It follows by Lemma 3.8(i) that M, v σ ♦ ,
in which case M, v σ given M, v σ α . Hence v is retained in M|ϕ . But then
given M|ϕ , v 2 βj , we have M|ϕ , w 2 βj , which contradicts the assumption that
M|ϕ , w δ. We conclude that M, w χ , which gives the third conjunct of χ.
Finally, we claim M, w χ♦ . Given M|ϕ , w δ ♦ , take an arbitrary ♦γk in δ, and
let v be such that wRv and M|ϕ , v γk . It follows that M, v σ for some disjunct
σ of ϕ, which we label as σγk , for otherwise v would not be retained in M|ϕ . Since
γk is propositional, M, v γk given M|ϕ , v γk . Therefore M, w ♦ σγαk ∧ γk .
Then from the fact that M, v σγ♦
, we have M, w σγ♦
by Lemma 3.8(i), so
k
k
σγk ∈ S. Since ♦γk was arbitrary, M, w χ♦ , which gives the final conjunct of χ.
(⇒) Again we prove the contrapositive. Suppose δ is not a strong Moore sen18
Holliday and Icard
tence, so there is some δ that is compensated in ϕ, for which an appropriate χ as in
Definition A.1 is clear. Where M, w χ, we claim that M|ϕ , w δ. We will show
M|ϕ , w δ α , M|ϕ , w δ , and M|ϕ , w δ ♦ separately.
Given M, w δ, we have M|ϕ , w δ α since δ α is propositional.
Next, for any v retained in M|ϕ , we have M, v σ α for some disjunct σ of ϕ.
It must be that σ ∈ S, for otherwise M ¬σ given M, w χ and Lemma 3.8(i).
V
Then from the fact that M, w (∼ σ α ∨ βj ), we have M|ϕ , v βj for all
σ∈S,j≤n
j ≤ n. Since v was arbitrary, M|ϕ , w δ .
Finally, given M, w χ♦ , for any ♦γk in δ we have M, w ♦ (σγk ∧ γk ) with
.
σγk ∈ S. Let v be such that wRv and M, v σγαk ∧ γk . Since σγk ∈ S, M, w σγ♦
k
♦
It follows by Lemma 3.8(i) that M, v σγk , in which case M, v σγk given M, v σγαk . Hence v is retained in M|ϕ . Since γk is propositional, we have M|ϕ , v γk
given M, v γk , whence M|ϕ , w ♦γk . Since γk was arbitrary, M|ϕ , w δ ♦ .
We conclude that M|ϕ , w δ, in which case M|ϕ , w ϕ. Given M, w χ, we
have M, w δ and hence M, w ϕ, so ϕ is not self-refuting.
2
Finally, we will prove an analogous generalization of Theorem 3.13(ii).
Definition A.4 ϕ is a strong Moorean sentence iff for any normal form ϕ∗ of ϕ,
there is a disjunct δ and non-empty sets S and T of disjuncts of ϕ∗ , with for every
θ ∈ T , a ♦γθ in θ, such that χ ≡ δ ∧ χ1 ∧ χ2 ∧ χ3 is clear, where
t(θ) ≡ θα♦ ∧
χ1 ≡
^
θ∈T
t (θ) ∧
^
θ∈T
/
χ3 ≡
^
σ∈S, β in θ
(∼ σ α ∨ β) ;
¬t (θ) ; χ2 ≡
^
σ∈S,θ∈T
^
σ∈S
σ ♦ ∧
^
σ ∈S
/
¬σ ♦ ;
(∼ σ α ∨ ∼ γθ ) .
The formula χ is not yet in normal form, due to the ¬t(θ) conjuncts in χ1 and the
¬σ ♦ conjuncts in χ2 , so strictly the definition of clarity does not apply. However,
it is straightforward to put χ into normal form using the same method involving
σ̂ ♦ as in Definition A.1, together with some distribution of ∧ and ∨. Since in this
case the necessary modifications add four existential quantifiers to the definition,
for simplicity we do not write them out. When we say that χ is clear, strictly we
mean that the modified formula is clear.
As we will see in the proof of Theorem A.6, the clarity of χ is equivalent to
there being a model in which ϕ has an unsuccessful update. For ϕ to have an
unsuccessful update in M, there must be some disjunct δ in ϕ such that M, w δ,
but no disjunct δ 0 such that M|ϕ , w δ 0 . To ensure that there are no such δ 0 ,
we need only keep track of those disjuncts θ of ϕ whose propositional part θα and
existential part θ♦ are true in M and whose universal part θ was already true in
M or becomes true in M|ϕ , since all other disjuncts will be false in M|ϕ . This is
the purpose of χ1 . For each such θ, we must ensure that there is a diamond formula
♦γθ in θ that becomes false in M|ϕ , because none of its witnesses satisfy any of the
disjuncts that are satisfied somewhere in M. This is the purpose of χ3 and χ2 .
19
Holliday and Icard
Proposition A.5 Every strong Moorean sentence is a Moorean sentence.
Proof. Assume ϕ is a strong Moorean sentence, so an appropriate χ as in Definition
A.4 is clear. Note that the distinguished disjunct δ of ϕ∗ must be a member of both
S and T . Then given that δ ∧ χ3 is clear, we have that δ ∧ (∼ δ α ∨ ∼ γδ ) is clear,
where ♦γδ is a conjunct of δ. Hence δ is a Moorean conjunction by Definition 3.7,
in which case ϕ is a Moorean sentence by Definition 3.12.
2
Theorem A.6 ϕ is unsuccessful if and only if ϕ is a strong Moorean sentence.
Proof. As before, let us assume that ϕ is already in normal form.
(⇐) Suppose ϕ is a strong Moorean sentence, so an appropriate χ as in Definition
A.4 is clear. Consider a pointed model such that M, w χ. For reductio, assume
M|ϕ , w ϕ. Then there exists a disjunct θ in ϕ such that M|ϕ , w θ.
We claim that θ ∈ T . For if θ ∈
/ T , then given M, w χ1 there are two
cases. Case 1: M, w 2 θα♦ . Then M|ϕ , w 2 θα♦ since θα♦ is existential. Case
2: M, w ♦ (σ α ∧ ¬β) for some σ ∈ S and β in θ. Let v be such that wRv
and M, v σ α ∧ ¬β. Since σ ∈ S, we have M, w σ ♦ , in which case M, v σ
given Lemma 3.8(i) and M, v σ α . Hence v is retained in M|ϕ , and since β
is propositional, M|ϕ , v 2 β, so M|ϕ , w 2 β and M|ϕ , w 2 θ . In both cases,
M|ϕ , w 2 θ, a contradiction. Therefore θ ∈ T .
Given M|ϕ , w θ, for every ♦γ in θ there is a v with wRv and M|ϕ , v γ.
Since v was retained in M|ϕ , we have M, v σ for some σ ∈ S. Then given
M, w χ3 and θ ∈ T , we have M, v ¬γ. Since γ is propositional, M|ϕ , v ¬γ, a
contradiction. We conclude that M|ϕ , w 2 ϕ, so ϕ is unsuccessful.
(⇒) Suppose ϕ is unsuccessful, so there is a pointed model with M, w ϕ but
M|ϕ , w 2 ϕ. To show that ϕ is a strong Moorean sentence, it suffices to show that
an appropriate χ as in Definition A.4 is satisfiable. Given M, w ϕ, we can read
off from w the disjunct δ and sets S and T such that M, w δ ∧ χ1 ∧ χ2 . It only
remains to show M, w χ3 . For reductio, suppose there is a θ ∈ T such that for all
V
(∼ σ α ∨ ∼ γ), i.e., M, w ♦ (σ α ∧ γ) for some σ ∈ S. Then
♦γ in θ, M, w 2
σ∈S
we claim that M|ϕ , w θ. As before, we take the three parts of θ separately.
For θ♦ , consider any ♦γ in θ and let v be such that wRv and M, v σ α ∧ γ.
Given M, w χ2 and the fact that σ ∈ S, we have M, w σ ♦ , so M, v σ by
Lemma 3.8(i). Hence v is retained in M|ϕ . Since M, v γ and γ is propositional,
M|ϕ , v γ and therefore M|ϕ , w ♦γ. Since ♦γ was arbitrary, M|ϕ , w θ♦ .
For θ , take any v retained in M|ϕ such that wRv, and we have M, v σ α for
some σ ∈ S. It follows that for any β in θ, we have M, v β given M, w χ1 .
Since β is propositional, M|ϕ , v β. Since v and β were arbitrary, M|ϕ , w δ .
Finally, for θα , since by assumption θ ∈ T and M, w χ1 , we have M, w θα
and hence M|ϕ , w θα .
We have shown M|ϕ , w θ and hence M|ϕ , w ϕ, which contradicts our initial
assumption. It follows that for every θ ∈ T there is a ♦γθ in θ such that M, w χ3 .2
20