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Elements of Modal Logic
D.K. Johnston
Department Philosophy
University of Victoria
Contents
1 Languages, logics, and systems.
1
2 Lindenbaum’s Lemma.
3
3 Frames and models.
4
4 The logic K.
4
5 The Fundamental Theorem.
5
6 The logic T.
7
7 Some extensions of T.
8
8 Some logics between K and T.
11
1
Languages, logics, and systems.
We begin our formal definition of the modal language M by specifying an
infinite set of propositional variables:
At = {pn | n ≥ 0 }
Next we specify a set of primitive logical operators:
{⊥, →, }
We can now give a recursive definition of the set M.
1. At ⊆ M
2. ⊥ ∈ M
3. α, β ∈ M ⇒ (α → β) ∈ M
4. α ∈ M ⇒ α ∈ M
We allow the formulas of M to be re-written according to the following
abbreviations:
¬α =def α → ⊥
α ∨ β =def ¬α → β
α ∧ β =def ¬(α → ¬β)
α ↔ β =def (α → β) ∧ (β → α)
3α =def ¬¬α
A logic is any non-empty subset of M. For example, the Propositional
Calculus pc can be defined as the set of all members of M that are tautologies.
Suppose that L is a logic and that Σ is a subset of M. Then we write,
Σ L α
iff there is a finite sequence σ1 . . . σn of members of Σ such that the formula,
(σ1 ∧ . . . ∧ σn ) → α
is a member of L. When the set Σ is empty, we write simply L α. When
the identity of L is obvious from the context, we will often write Σ α.
Note that L α iff α ∈ L. When L α, α is said to be a theorem of L.
1
A system S is a pair (A, R) where A ⊆ M is a non-empty set of axioms
and R is a set of inference rules. An inference rule has the form,
α1 . . . αn β
A set Σ is said to be closed under an inference rule iff β ∈ Σ whenever all of
the αi ’s are in Σ. Each system S determines a logic L(S), which is defined
as the smallest set containing A that is closed under the rules of R.
The logic pc has an associated system. Let Spc = (Apc , Rpc ), where Apc
contains every instance of the formula schemas,
[A1] α → (β → α)
[A2] (α → (β → γ)) → ((α → β) → (α → γ))
[A3] (¬α → ¬β) → (β → α)
and where Rpc contains the single rule modus ponens,
[MP] α, α → β β
It can be proved that L(Spc ) = pc.
Thus, every system is associated with a logic, and every logic is associated
with a system. (If all else fails, we can always form a system with A = L
and with R empty.) For this reason, we will often use the term ‘logic’ to
refer to both logics and systems. For example, we will often talk about the
axioms or rules of a logic, even though, properly speaking, these axioms and
rules belong to the associated system.
We will make immediate use of this loosened terminology. A logic L1 is
an extension of a logic L2 when the axioms of L2 are theorems of L1 , and L1
is closed under the rules of L2 . When this is the case, it is easily shown that
L2 ⊆ L1 . Since every logic is trivially an extension of itself, we say that L1
is a proper extension of L2 when L1 ⊆ L2
A set Σ of formulas is said to be L-consistent iff Σ L ⊥. A set of
formulas is L-inconsistent when it is not L-consistent. When the identity of
L is obvious from the context, we will often say simply that Σ is consistent
or inconsistent.
With these concepts in hand, we state two fundamental results:
Theorem 1 (Deduction Theorem). If L is an extension of pc, then
Σ ∪ {α} L β iff Σ L α → β
Theorem 2. If L is an extension of pc, then Σ is L-consistent iff every
finite subset of Σ is L-consistent.
2
2
Lindenbaum’s Lemma.
A set Σ of formulas is said to be maximal iff, for every α ∈ M, either α ∈ Σ
or ¬α ∈ Σ. Σ is said to be L-maximal consistent iff Σ is both maximal and
L-consistent.
Theorem 3 (Lindenbaum’s Lemma). If L is an extension of pc, then
every L-consistent set is included in a L-maximal consistent set.
Proof. Suppose that Σ is consistent. Let σ0 . . . σi . . . be an enumeration of
M. We define an infinite series of sets Σ0 . . . Σi . . . as follows:
Let Σ+ =
Σ0 = Σ
..
.
Σi ∪ {σi } if Σi ∪ {σi } is consistent
Σi+1 =
Σi otherwise
i≥0 Σi .
We show that Σ+ is both maximal and consistent.
(i) Assume that Σ+ is inconsistent. By the Theorem 2, there must be
some finite subset of Σ+ which is inconsistent. This finite subset must be
included in one of the sets Σi . But then this Σi would be inconsistent, which
is impossible by the definition of Σ+ . Thus Σ+ cannot be inconsistent.
(ii) Assume that for some α ∈ M, neither α nor ¬α are in Σ+ .
Suppose α = σi and ¬α = σj in the ordering of M, with i < j.
∴ Σi ∪ {α} ⊥ and Σj ∪ {¬α} ⊥, by definition of Σ+ .
∴ Σi α → ⊥ by the Deduction Theorem.
∴ Σi ¬α. But Σi ⊆ Σj . ∴ Σj ¬α.
But Σj ¬α → ⊥ by the Deduction Theorem.
∴ Σj ⊥ by [MP].
∴ Σj is inconsistent, contrary to the definition of Σ+ .
∴ Either α or ¬α is in Σ+ .
All of the logics to be studied here are extensions of pc, and so henceforth
this will be taken for granted.
Corollary 3.1. If Σ is a maximal consistent set,
Σ α iff α ∈ Σ
Proof. Obviously, α ∈ Σ ⇒ Σ α, since pc α → α. Now assume that
α ∈ Σ. ∴ ¬α ∈ Σ, since Σ is maximal. ∴ Σ ¬α. ∴ Σ α, since Σ is
consistent. ∴ Σ α ⇒ α ∈ Σ
3
Corollary 3.2. If Σ is an L-maximal consistent set,
L α ⇒ α ∈ Σ
3
Frames and models.
We give a definition of truth for the formulas of M. Let D be any non-empty
set. A valuation V on the domain D is a function that assigns a subset of
D to every n ≥ 0. Let R be a binary relation on D. Then for every α ∈ M,
α is defined recursively as follows:
1. pn = V (n)
2. ⊥ = ∅
3. α → β = −α ∪ β
4. α = {x | ∀y, Rxy ⇒ y ∈ α }
The structure F = (D, R) is called a relational frame. Where V is a valuation
defined on D, the structure M = (D, R, V ) is a model on the frame F. αM
is the truth set of α on the model M. When the identity of M is obvious or
irrelevant, we will often write simply α.
If αM = ∅, then α is satisfied by M. If ∃x ∈ D : x ∈ αM, then α is
falsified by M. If αM = D, then α is valid on M, and we write M α. If
α is valid on every model on a frame F, then α is valid on F, and we write
F α.
Let C be a class of relational frames. A formula α is valid on C iff α is
valid on every F in C. A logic L is sound with respect to C iff every theorem
of L is valid on C; i.e. iff L α ⇒ C α. L is complete with respect to C iff
L includes every formula that is valid on C; i.e. iff C α ⇒ L α. If L is
both sound and complete with respect to C, then L is determined by C; i.e.
C α iff L α.
4
The logic K.
The modal logic K is defined as the smallest extension of pc that contains
every instance of the axiom schema,
[K] α ∧ β → (α ∧ β)
and which is closed under both
4
[RR] α → β α → β
[RN] α α
Theorem 4. K is sound with respect to the class of relational frames.
Proof. This involves proving that the axioms of K are valid on every relational frame, and that the inference rules of K preserve validity.
(i) Assume that x ∈ α ∧ β on some model.
∴ x ∈ α ∩ β.
∴ ∀y, Rxy ⇒ y ∈ α and y ∈ β.
∴ ∀y, Rxy ⇒ y ∈ α ∧ β.
∴ x ∈ (α ∧ β).
(ii) Assume that F α → β, and that x ∈ α for some model on F.
∴ ∀y, Rxy ⇒ y ∈ α.
But ∀y, y ∈ α ⇒ y ∈ β, since F α → β.
∴ ∀y, Rxy ⇒ y ∈ β.
∴ x ∈ β.
(iii) Assume that F α.
∴ ∀xy, Rxy ⇒ y ∈ α, for all M on F.
∴ ∀x, x ∈ α.
∴ F α.
5
The Fundamental Theorem.
Where L is a logic, let DL be the set of all L-maximal consistent sets. The
binary relation RL is defined on DL as follows:
RL xy iff ∀α, α ∈ x ⇒ α ∈ y
We also define a valuation,
VL (n) = {x | pn ∈ x }
The model ML = (DL , RL , VL ) is the canonical model of the logic L.
Theorem 5 (Fundamental Theorem). If L is an extension of K, then
for all α ∈ M,
x ∈ αML iff α ∈ x
5
Proof. The proof is by induction on the syntactical structure of α.
(i) (α = pn ) This step follows trivially from the definition of VL .
(ii) (α = ⊥) This step is also trivial: ⊥ = ∅ by the truth definition, and
⊥ will not be in any maximal consistent set.
(iii) (α = β → γ)
By the truth definition, x ∈ β → γ iff x ∈ β or x ∈ γ.
∴ x ∈ β → γ iff β ∈ x or γ ∈ x, by the induction hypothesis.
∴ x ∈ β → γ iff ¬β ∈ x or γ ∈ x.
But pc ¬β → (β → γ) and pc γ → (β → γ).
∴ x ∈ β → γ iff (β → γ) ∈ x.
(iv) (α = β)
First, assume that β ∈ x.
∴ ∀y, RL xy ⇒ β ∈ y by definition of RL .
∴ ∀y, RL xy ⇒ y ∈ β by the induction hypothesis.
∴ x ∈ β.
Next, assume that β ∈ x.
We show that ∃y: RL xy and y ∈ β.
Let Γ = {γ | γ ∈ x }.
Lemma 5.1. Γ ∪ {¬β} is consistent.
Proof. Assume Γ ∪ {¬β} ⊥.
∴ Γ β by the Deduction Theorem.
But Γ = ∅ by [RN].
∴ ∃γ1 . . . γn ∈ Γ: (γ1 ∧ . . . ∧ γn ) → β.
∴ (γ1 ∧ . . . ∧ γn ) → β by [RR].
But γ1 ∧ . . . ∧ γn → (γ1 ∧ . . . ∧ γn ) by [K].
∴ γ1 ∧ . . . ∧ γn → β.
But γ1 . . . γn ∈ x by definition of Γ.
∴ β ∈ x, contrary to assumption.
∴ Γ ∪ {¬β} is consistent.
By Lindenbaum’s Lemma, Γ ∪ {¬β} is in some y ∈ DL .
∴ β ∈ y. ∴ y ∈ β by the induction hypothesis.
But RL xy by definition of Γ. ∴ x ∈ β.
Corollary 5.1. K is complete with respect to the class of relational frames.
Proof. Suppose that K α. ∴ ¬α is K-consistent. ∴ ¬α ∈ x for some x ∈ DK ,
by Lindenbaum’s Lemma. ∴ α fails on MK , by the Fundamental Theorem.
∴ α is not valid on the class of relational frames.
6
Corollary 5.2. K is determined by the class of relational frames.
Proof. This follows directly from Theorem 4 and Corollary 5.1.
Corollary 5.3. If L is an extension of K, then
ML α iff L α
6
The logic T.
The modal logic T is formed by adding every instance of the schema,
[T] α → α
to the axioms of K. First we show that T is a proper extension of K. Let
M = (D, R, V ) be a model where
D = {x, y}
R = {(x, y)}
V (0) = {y}
Then x ∈ p0 , but x ∈ p0 . Therefore [T] fails on M. But every theorem
of K will be valid on M by Theorem 4. Therefore K [T].
A relational frame F = (D, R) is reflexive iff
∀x ∈ D, Rxx
Theorem 6. T is determined by the class of reflexive relational frames.
Proof. By Theorem 4, we know that every theorem of K is valid on any
relational frame, and that the rules [RR] and [RN] preserve validity. So, for
soundness of T, we need only prove:
Lemma 6.1. [T] is valid on any reflexive relational frame.
Proof. Suppose that F is reflexive, and that x ∈ α for some M on F.
∴ ∀y, Rxy ⇒ y ∈ α. But Rxx. ∴ x ∈ α. ∴ x ∈ α → α.
For completeness, it suffices to show:
Lemma 6.2. FT is reflexive.
Proof. Assume x ∈ DT , and that α ∈ x for some α. But (α → α) ∈ x
by Corollary 3.2. ∴ α ∈ x. ∴ α ∈ x ⇒ α ∈ x. ∴ RT xx.
7
Corollary 5.3 implies that every non-theorem of T will fail on MT , and so
T α ⇒ FT α. But since FT is reflexive, it follows that no non-theorem of
T will be valid on the class of reflexive relational frames.
The argument used to prove the completeness of T can be generalised as
follows:
Theorem 7. Let C be a class of relational frames, and let L be an extension
of K. Then FL ∈ C implies that L is complete with respect to C.
7
Some extensions of T.
The modal logic S4 results from the addition of every instance of the schema
[4] α → α
to the axioms of T. We show that S4 is a proper extension of T. Let
M = (D, R, V ) be a model where
D = {x, y, z}
R = {(x, x), (x, y), (y, y), (y, z), (z, z)}
V (0) = {x, y}
The frame F = (D, R) will look like this:
xZ
/y
X
/z
Z
Since R is reflexive, every theorem of T will be valid on M. But x ∈ p0 ,
while x ∈ p0 because z ∈ p0 . Thus [4] fails at x on M, and so T [4].
A relation R is transitive iff:
∀xyz, Rxy & Ryz ⇒ Rxz
Theorem 8. S4 is determined by the class of reflexive and transitive relational frames.
Proof. By Theorem 6, we know that every theorem of T is valid on any
reflexive relational frame. We also know that the rules [RR] and [RN] preserve validity on any relational frame. Therefore, soundness of S4 follows
immediately from:
8
Lemma 8.1. [4] is valid on any transitive relational frame.
Proof. Assume F is transitive and that x ∈ α for some M on F.
∴ ∃y: Rxy & y ∈ α. ∴ ∃z: Ryz & z ∈ α.
But Rxz since R is transitive. ∴ x ∈ α.
Lemma 8.2. FS4 is transitive.
Proof. Assume RS4 xy and RS4 yz. Assume α ∈ x for some α.
But (α → α) ∈ x. ∴ α ∈ x.
∴ α ∈ y, since RS4 xy. ∴ α ∈ z, since RS4 yz.
∴ α ∈ x ⇒ α ∈ z. ∴ RS4 xz.
FS4 is reflexive because S4 [T]. Therefore, there is a reflexive and transitive
relational frame that falsifies all non-theorems of S4.
The modal logic B results from the addition of every instance of the
schema
[B] 3α → α
to the axioms of T. B can be shown to be a proper extension of T, by
constructing a reflexive relational frame where the axiom [B] fails.
A relation R is symmetric iff:
∀xy, Rxy ⇒ Ryx
Theorem 9. B is determined by the class of reflexive and symmetric relational frames.
Lemma 9.1. [B] is valid on any symmetric relational frame.
Lemma 9.2. FB is symmetric.
Proof. Assume RB xy and that α ∈ y for some α.
∴ 3α ∈ x since RB xy. But (3α → α) ∈ x. ∴ α ∈ x.
∴ α ∈ y ⇒ α ∈ x. ∴ RB yx.
The modal logic S5 results from the addition of every instance of the
schema
[5] 3α → α
to the axioms of T. S5 can be shown to be a proper extension of T, by
constructing a reflexive relational frame where the axiom [5] fails.
A relation R is euclidean iff:
∀xyz, Rxy & Rxz ⇒ Ryz
9
Theorem 10. S5 is determined by the class of reflexive and euclidean relational frames.
Lemma 10.1. [5] is valid on any euclidean relational frame.
Lemma 10.2. FS5 is euclidean.
Proof. Assume RS5 xy and RS5 xz. Assume α ∈ y for some α.
∴ 3α ∈ x since RS5 xy. But (3α → α) ∈ x.
∴ α ∈ x. ∴ α ∈ z since RS5 xz.
∴ α ∈ y ⇒ α ∈ z. ∴ RS5 yz.
Theorem 11. S5 is a proper extension of both S4 and B.
Proof. The following lemmas, together with Theorem 7, suffice to prove that
S5 is an extension of both of these logics.
Lemma 11.1. If R is reflexive and euclidean, then R is transitive.
Lemma 11.2. If R is reflexive and euclidean, then R is symmetric.
A reflexive transitive frame can be constructed on which the axiom [5] will
fail. Similarly, a reflexive symmetric frame can be constructed on which [5]
will fail. Therefore, S5 is a proper extension of both S4 and B.
Finally, we note that it is possible to construct both (a) a reflexive transitive frame where [B] fails; and (b) a reflexive symmetric frame where [4]
fails. It follows that the logics S4 and B do not include one another.
The relationships between the various logics we have examined so far
can now be set out in the following diagram:
> S5 `BB
BB
~~
~
BB
~
BB
~~
~
~
S4
B `@@
|>
@@
||
@@
|
@@
||
||
TO
K
10
8
Some logics between K and T.
Consider the following axioms:
[D] α → 3α
[X1] α → α
[X2] (α → α)
[X3] α → α
[X4] 3α → 3α
[X5] α → (3α → α)
Since each one of these axioms can be shown to fail on some relational frame,
they are not theorems of K. It can also be shown that these axioms are valid
on any reflexive relational frame, which implies that they are theorems of T.
Therefore, the extensions of K generated by these axioms will be ‘between’
the logics K and T. To simplify matters, we will give each one of these logics
the name of its characteristic axiom.
A relation is serial iff ∀x, ∃y: Rxy.
Theorem 12. D is determined by the class of serial relational frames.
Proof. Soundness is trivial. For completeness, assume x ∈ DD .
Define Γ = {γ | γ ∈ x } and assume Γ ⊥.
∴ ∃γ1 . . . γn ∈ Γ: γ1 ∧ . . . ∧ γn−1 → ¬γn .
∴ (γ1 ∧ . . . ∧ γn−1 ) → ¬γn by [RR].
But γ1 ∧ . . . ∧ γn−1 → (γ1 ∧ . . . ∧ γn−1 ) by [K].
∴ γ1 ∧ . . . ∧ γn−1 → ¬γn .
But γ1 . . . γn ∈ x. ∴ ¬γn ∈ x.
∴ ¬3γn ∈ x. But (γn → 3γn ) ∈ x by [D]. ∴ 3γn ∈ x.
∴ x is inconsistent, which is absurd.
∴ Γ is consistent, and so is included in some y ∈ DD .
∴ ∃y: RD xy. ∴ RD is serial.
The remaining axioms are determined by classes of relational frames that
are defined by the following conditions:
(x1) ∀x, ∃y: Rxy & Ryx
(x2) ∀xy, Rxy ⇒ Ryy
11
(x3) ∀xy, Rxy ⇒ ∃z: Rxz & Rzy
(x4) ∀x, ∃y: Rxy & ∀z, Ryz ⇒ Rxz
(x5) ∀xy, Rxy ⇒ Rxx
By examining the implications that hold between these relational conditions, we can deduce the inclusion relationships that hold between the logics
themselves:
6 T hP
nnn> `AAPPP
nn }}
nnn }}}
n
n
}
nn
}}
nnn
X4
X1 aC
O
CC
CC
CC
C
AA PPPP
AA PPP
AA
PPP
P
X2
O
D `AA
AA
AA
AA
K
12
> X3
}}
}
}
}}
}}
= X5
{{
{
{
{{
{{