Applied Logic
Lecture 2 part 1 - Modal logic
Marcin Szczuka
Institute of Informatics, The University of Warsaw
Monographic lecture, Spring semester 2016/2017
Marcin Szczuka (MIMUW)
Applied Logic
2017
1 / 27
Lecture plan
1
Language of modal logic
2
Structural semantics, modal logical consequence
Kripke model
Semantics in modal sense
3
Example
4
Three satisfaction relations
Properties
Marcin Szczuka (MIMUW)
Applied Logic
2017
2 / 27
Modality
(After Wikipedia - the free Encyclopedia)
In linguistics, modals are expressions broadly associated with notions of
possibility and necessity. Modals have a wide variety of interpretations
which depend not only upon the particular modal used, but also upon where
the modal occurs in a sentence, the meaning of the sentence independent
of the modal, the conversational context, and a variety of other factors. For
example, the interpretation of an English sentence containing the modal
“must” can be that of a statement of inference or knowledge (roughly,
epistemic) or a statement of how something ought to be (roughly, deontic).
The following pair of examples illustrate the interpretative difference:
1
John didn’t show up for work. He must be sick.
2
John didn’t show up for work. He must be fired.
Modal logic is a type of formal logic that extends the standards of formal
logic to include the elements of modality (for example, possibility and
necessity). Traditionally, there are three “modes” or “moods” or “modalities”
represented by modal logic, namely, possibility, probability, and necessity.
Marcin Szczuka (MIMUW)
Applied Logic
2017
3 / 27
Language of modal logic
The language of (propositional) modal logic is an extension of the one we
have for propositional case. We add a battery of new 1-ary connectives
that are called modal operators or, informally, box connectives.
Originally there was only one such connective, . For various reasons
nowadays we consider a whole family of connectives i labelled by
elements of a set I (i ∈ I), which is called a signature of modal system.
Language of modal logic with signature I consists of:
propositional variables V AR = {p, q, r, ...}
Propositional connectives: ¬, ∨, >, ⊥, →, ∧
Family of modal connectives
{i : i ∈ I}
together with parentheses ’(’ and ’)’ which help to establish precedence of
operations.
Marcin Szczuka (MIMUW)
Applied Logic
2017
4 / 27
Modal formulaæ
Formulæ (properly constructed) in modal language with signature I are
defined recursively as:
[Atomic formulæ:] Propositional variables
V AR = {p, q, r, ...} and >, ⊥ are formulæ.
[Propositional formulæ:] If φ i ψ are formulæ then
¬φ, (φ ∨ ψ), (φ ∧ ψ), (φ → ψ)
are also (properly constructed) formulæ.
[Modal formulaæ:] If φ is a formula then
i φ
is a formula for each label i ∈ I.
The family of all properly constructed formulæ we denote by F ORM .
An example of modal formula: 1 ((p ∧ q) ∨ 2 ¬p) → p
Marcin Szczuka (MIMUW)
Applied Logic
2017
5 / 27
Modal systems
Initially modal systems contained a single modal connective. The formula
φ in such case may have various interpretations:
It is obligatory that φ holds.
It is known that φ.
...
The modal connective can be viewed as (a special version of) general
quantifier. We can also introduce its counterpart, which roughly
corresponds to existence quantifier. This is done by setting
♦i φ =def ¬i ¬φ.
♦i φ can be interpreted as “it is possible that φ holds”.
One example of binary modal logic temporal logic), which has two modal
connectives: and . These connectives are usually interpreted as: φ –
“φ will happen”; φ – “φ already happened”.
Marcin Szczuka (MIMUW)
Applied Logic
2017
6 / 27
Special modal formulæ
Several particular formulæ (or rather formula templates) play a prominent
rôle in modal logic. They are named, mostly due to historical reasons, as
follows:
D(φ) : φ → ♦φ
T(φ) : φ → φ
B(φ) : φ → ♦φ
4(φ) : φ → φ
5(φ) : ♦φ → ♦φ
P(φ) : φ → φ
Q(φ) : ♦φ → φ
R(φ) : φ → φ
G(φ) : ♦φ → ♦φ
L(φ) : T(φ) → φ
M(φ) : ♦φ → ♦φ
Additionally, we utilise a distributive property of necessity w.r.t. implication
K(φ, θ) :
Marcin Szczuka (MIMUW)
(φ → θ) ⇒ (φ → θ)
Applied Logic
2017
7 / 27
Lecture plan
1
Language of modal logic
2
Structural semantics, modal logical consequence
Kripke model
Semantics in modal sense
3
Example
4
Three satisfaction relations
Properties
Marcin Szczuka (MIMUW)
Applied Logic
2017
8 / 27
Lecture plan
1
Language of modal logic
2
Structural semantics, modal logical consequence
Kripke model
Semantics in modal sense
3
Example
4
Three satisfaction relations
Properties
Marcin Szczuka (MIMUW)
Applied Logic
2017
9 / 27
Transition systems, frames, Kripke structures
In propositional system all we needed to define the semantic consequence
was a set B = {0, 1}. In modal case this becomes much trickier, as we
have to reflect the modality. in other words we have to create semantics in
such a way, that it corresponds to use of modal connectives.
It was only in 1960s when Saul Kripke popularised the use of directed
transitional structures (directed graphs) as a model for modal systems. It
brought significant clarifications to the way people used to work with modal
languages and systems. In a sense, we can say that the modal logic is a
study of such transitional structures, and the modal language, with all its
attachments, is merely a tool designed to help in expressing some facts
about these structures.
Definition – Kripke model
Kripke model is a pair K = hΓ, vali, where Γ = (W, E) is a directed graph
describing transitions (transition/reachability relation) between states (or
worlds) from W and val : W × V AR → {0, 1} is a valuation.
Marcin Szczuka (MIMUW)
Applied Logic
2017
10 / 27
Lecture plan
1
Language of modal logic
2
Structural semantics, modal logical consequence
Kripke model
Semantics in modal sense
3
Example
4
Three satisfaction relations
Properties
Marcin Szczuka (MIMUW)
Applied Logic
2017
11 / 27
Structural semantics
Semantics of formula φ in a state s ∈ W of Kripke model K is defined by
relation (K, val, s) |= φ between:
model K
valuation val
state s (element in structure of model K)
modal formula φ
If the model K is established and common for all considered formulæ, we
will simplify the notation to s |= φ.
The semantics we will try to define now will be, as previously, a function
assigning logical value to formula. This time, however, this function will
depend on state (vertex, world), i.e., the place in transitional structure.
For the next step we will need the notion of accessibility. The state q ∈ W
−
is said to be accessible from state s ∈ W if →
sq ∈ E. By R(s) we denote set
of all states accessible (reachable) from s.
Marcin Szczuka (MIMUW)
Applied Logic
2017
12 / 27
Structural semantics – definition
For each state s ∈ W we define inductively w.r.t. composition of formula
φ, a function [[.]]s : F ORM → {0, 1}, such that:
(Const) [[>]]s = 1; [[⊥]]s = 0
(Var) for variables p ∈ V AR
[[p]]s = val(s, p)
(¬) for any formula φ ∈ F ORM
[[¬φ]]s = 1 − [[φ]]s
(∗) for any formulæ φ, ψ ∈ F ORM
[[φ ∗ ψ]]s = [[φ]]s ∗ [[ψ]]s
where ∗ is a logical connective (e.g., ∨, ∧, ⇒)
() for any formula φ ∈ F ORM
Y
[[φ]]s =
[[φ]]s0
s0 ∈R(s)
where R(s) is the set of states accessible from s.
Marcin Szczuka (MIMUW)
Applied Logic
2017
13 / 27
Modal logical (semantic) consequence
Modal consequence relation
Relation
s |= φ
holds if and only if [[φ]]s = 1.
Formally, the relation K, val, s |= φ translates to a sentence “s forces φ”,
but we can read it also as:
φ is satisfied in the model (K, val, s);
φ is true at s;
(K, val, s) models φ.
One can easily arrive at the following proposition:
If φ is a propositional tautology then φ is satisfied at each state of the
(given) Kripke model.
Marcin Szczuka (MIMUW)
Applied Logic
2017
14 / 27
Lecture plan
1
Language of modal logic
2
Structural semantics, modal logical consequence
Kripke model
Semantics in modal sense
3
Example
4
Three satisfaction relations
Properties
Marcin Szczuka (MIMUW)
Applied Logic
2017
15 / 27
Simple modal semantics example
Let V AR = {p}. Consider a simple Kripke model with four (4) states
{a, b, c, d} and accessibility defined by the following directed graph:
- b
a
6@
I
@
@
@
d
Marcin Szczuka (MIMUW)
@
@
?
- c
Applied Logic
2017
16 / 27
Example – valuation
Consider the following valuation:
val(a, p) = val(c, p) = 1;
val(b, p) = val(d, p) = 0;
Directly from the above, we obtain:
a |= p
b |= ¬p
c |= p
Furthermore, using the graph from previous slide we get: R(a) = {b},
R(a) = {c}, R(d) = {a, c}, R(c) = {a}. Hence, following semantical
consequences hold:
a |= ¬p
Marcin Szczuka (MIMUW)
b |= p
Applied Logic
d |= p
2017
17 / 27
Example continued
We can now redraw initial accessibility graph labelling the states (vertices)
with corresponding, satisfied formulæ.
p, ¬p
- b ¬p, p
a
6@
I
@
@
@
¬p, p
Marcin Szczuka (MIMUW)
d
@
@
?
- c
p, p
Applied Logic
2017
18 / 27
Example - formulæ with two modalities
Let us consider formulæ with two modal connectives. Observe, that:
a |= p
b |= p
c |= ¬p
Moreover, state d is quite interesting since it provides access to all other
states in the following way:
a
- b
-c
- a...
R c
@
- a
- b
- c...
d
@
@
Marcin Szczuka (MIMUW)
Applied Logic
2017
19 / 27
Example continued
For state d we obtain:
d |= p
d |= ¬2 p
d |= ¬3 p
d |= 4 p
and so on ...
Meanwhile, for state a there exists only one path of length 3:
a
-b
-c
-a
Hence, for any φ:
a |= 3 φ
⇔ a |= φ
giving
a |= (3 φ ↔ φ).
For the same reason formula (3 φ ↔ φ) is true at states b and c. This
gives us
d |= (4 φ ↔ φ)
Marcin Szczuka (MIMUW)
Applied Logic
2017
20 / 27
Modal connective ♦
Corollary – semantics of ♦
Let (K, val) be a Kripke model. Then:
a |= ♦φ
⇔
(∃x ∈ R(a))[x |= φ]
Proof:
a |= ♦φ ⇔ a |= ¬¬φ
⇔ ¬[a |= ¬φ]
⇔ ¬(∀x ∈ R(a))[x |= ¬φ]
⇔ (∃x ∈ R(a))¬[x |= ¬φ]
⇔ (∃x ∈ R(a))[x |= φ]
Marcin Szczuka (MIMUW)
Applied Logic
2017
21 / 27
Lecture plan
1
Language of modal logic
2
Structural semantics, modal logical consequence
Kripke model
Semantics in modal sense
3
Example
4
Three satisfaction relations
Properties
Marcin Szczuka (MIMUW)
Applied Logic
2017
22 / 27
Three satisfaction relations
Unlike the propositional case, in modal system we will have not one, but
three satisfaction relations.
|=p
|=v
|=u
They are defined in the following way:
The relation |=p is simply the rewriting of relation |= from previous
slides. Relation ’|=p ’ denotes a pointed (state-attached) model with
valuation.
Relation |=v is used for models with valuation. We obtain |=v from
|=p by generalising (quantifying) over all states
K, val |=v φ
⇔
(∀s ∈ W ) {K, val, s |= φ}
Relation |=u is called structural, since it is obtained from |=v by
generalisation over all valuations
K |=u φ
Marcin Szczuka (MIMUW)
⇔
(∀val) {K, val |=v φ}
Applied Logic
2017
23 / 27
Three relations – Example
Recall the previous example.
For the model used there the following holds:
K, val |=v P → 3 P
but, (e.g. for valuation val(P, {a, b, c, d} = {0, 0, 0, 1})) it is not true that
K |=v ¬P → 3 P
Thus:
not [K |=u ¬P → 3 P ]
On the other hand, structural satisfaction holds for:
K |=u 4 P ↔ P
Marcin Szczuka (MIMUW)
Applied Logic
2017
24 / 27
Lecture plan
1
Language of modal logic
2
Structural semantics, modal logical consequence
Kripke model
Semantics in modal sense
3
Example
4
Three satisfaction relations
Properties
Marcin Szczuka (MIMUW)
Applied Logic
2017
25 / 27
Properties of structural satisfiability
If the structure (graph) K = (W, E) represents accessibility relation which
is, respectively:
1
Reflexive (Zwrotna)
2
Transitive (Przechodnia)
3
Pathetic (Dyskretna)
4
Dense (Gęsta)
then, in each respective case, for each formula φ the corresponding
compound formula
1
φ → φ
2
φ → 2 φ
3
φ → φ
4
2 φ → φ
is structurally satisfied in K (with relation |=u ).
Marcin Szczuka (MIMUW)
Applied Logic
2017
26 / 27
Relation properties
Dense relation
A binary relation R is said to be dense if, for all R-related x and y, there is
a z such that x and z and also z and y are R-related.
Formally:
∀x ∀y xRy ⇒ (∃z xRz ∧ zRy).
For example, a strict partial order is a dense order iff is a dense
relation.
Marcin Szczuka (MIMUW)
Applied Logic
2017
27 / 27