what is...
Modal Logic?
Walter Cloete
March 2009
The formal study of logic has had its roots in philosophy and compared to the philosophical approach
mathematical logic is relatively young. The discovery
of non-Euclidean geometries and paradoxes in Cantor’s
set theory drew significant attention to the attempt to
define notions such as “proof” and “implication”. Furthermore the work of Gödel, Cohen, Hilbert and others
suggest that describing formal reasoning may not be as
trivial as is often thought and warrants further study.
For the past fifty years or so many logics have been
developed for applications to Computer Science, these
include applications to database theory, artificial intelligence and program verification.
This note outlines some aspects of the system of
logic known as Modal Logic and attempts to motivate
two applications of the theory.
theorems. In propositional logic the corresponding notion is that of a tautology (those formulae that are true
for all valuations) because of this importance of tautologies there are deductive systems (consisting of axioms
and rules) for reasoning about tautologies. Strictly
speaking we only use the term tautology when we are
interested in behaviour under valuations and we talk
about theorems or axioms when reasoning within a deductive system. The two fundamental results when
studying any logic are that of soundness and completeness 1 . Combined these two results state that ‘φ is a
theorem if and only if φ is a tautology’.
One of the results often assumed by mathematicians
that can be proven in propositional logic is the deduction theorem that states ‘If we can prove ψ by assuming
φ then we can deduce φ → ψ’. This is an example of a
meta-theorem since it tells us about theorems.
Propositional Logic
Modal Logic via Kripke semantics
Perhaps the simplest system of logic is Propositional
Logic where atomic statements are each either true (1)
or false (0) and can be combined to form formulae using
the binary connectives ∨ (‘or’), ∧ (‘and’), → (‘implies’)
and the unary connective ¬ (‘not’). The behaviour of
the connectives is summarized in the table below.
p
0
0
1
1
q
0
1
0
1
p∨q
0
1
1
1
p∧q
0
0
0
1
p→q
1
1
0
1
Modal logic attempts to describe additional ‘modes of
truth’ that are not captured by propositional logic.
These include ‘known to be true’, ‘possibly true’ and
‘will be true at some future time’.
The implication operator (→) of propositional logic
can be criticized for giving some unintuitive results. For
example φ → ψ is true when φ is false, regardless of the
value of ψ. Modal logic is the result of an attempt
by C.I. Lewis to describe ‘φ entails ψ’ which we now
capture in the notion ‘necessarily φ implies ψ’.
Modal logic adds the modal operators 2 (‘box’) and
3 (‘diamond’) to the usual propositional logic. Formally we let τ be a non-empty set that we call the
similarity type, and for each α ∈ τ we define a modal
operator 2α together with its dual 3α (dual in the sense
that 3α φ := ¬2α ¬φ). Although interpretation of the
modalities would depend on the application, we can interpret 2α φ as ‘agent α knows φ’, and 3α φ can be
interpreted as ‘agent α cannot refute φ’. If τ is a singleton we omit α and can read 2φ as ‘necessarily φ’,
with this interpretation 3φ is read as ‘possibly φ’.
As for propositional logic there are deductive systems for modal logic, and for many years these provided
the only way to reason about modal formulae. However
in 1959 a 19 year old Harvard University undergraduate named Saul Kripke suggested what is now known
¬p
1
0
It can be shown that some of these operators are redundant, for example all the operators can be defined
in terms of → and ¬.
The assignment of truth values to atomic statements are done via a valuation, this is a function
V : Φ → {0, 1}, where Φ denotes the set of atomic
statements (which is usually fixed in advance). It is
easy to see how a given valuation can be extended inductively to have the set of all formulae as domain. Also
note that if Φ is denumerable so is any set of formulae.
In most of mathematics we are interested in studying mathematical structures and showing that these
structures satisfy certain universal truths which we call
1 Note
2 Here
that Gödel’s incompleteness theorems are not the negation of this kind of completeness.
Kripke frames are not related to Frames as studied in pointless topology.
1
as Kripke semantics.
Given a similarity type τ we
τ
define a Kripke
frame 2 as an ordered pair F := W, (Rα )α∈τ with W
a non-empty set and Rα ⊆ W × W for each α ∈ τ .
The set W is often interpreted as being a set of possible ‘worlds’ and Rα with α ∈ τ is referred to as an
accessibility relation of F. To interpret the truth of formulae we define a Kripke model (or Kripke structure)
of a Kripke frame F as M = hF, V i where the function
V : Φ → P (W ) is called a valuation. This valuation is
a generalization of the valuation in propositional logic
since we interpret V (p) as the set of worlds where p ∈ Φ
is true. We write M, w |= p iff w ∈ V (p).
Given a Kripke Model M = W, (Rα )α∈τ , V and
w ∈ W we can determine whether a propositional logic
formula φ is true in w in a way similar to what we
used in propositional logic. Correspondingly we write
M, w |= φ iff φ is true in w.
Now we inductively define the interpretation of a
formula involving some of the modal operators by saying 2α ψ is true in w iff for every v ∈ W , ψ is true in v
whenever wRα v. Or rather:
Visualizing Kripke Frames
If its set of worlds is denumerable a Kripke frame has a
pictorial representation as a directed graph3 . We consider the following example:
Let M = h{w1 , w2 , w3 , w4 , w5 } , R, V i with R =
{(w1 , w2 ) , (w2 , w3 ) , (w1 , w4 )} and V (p) = {w3 , w4 }.
We obtain the directed graph:
p
p
w1
w2
w3
w5
w4
The reader can now easily confirm the following:
M, w1 |= 3p, M, w2 |= 2p, M |= 22p, M |= 3p ∨ 2p.
Propositional Temporal Logic
◦ M |= φ iff M, w |= φ for every world w of M
A popular application of modal logic is to Temporal
Logic where we are interested in describing truth that
may vary with time. For example the statement ‘the
sun will shine again’ can be restated as ‘there exists a
time t such that t is in the future from now and the sun
shines at t’. So we consider the model M = hT, <, V i
where T is the set of possible points in time and <
represents our ordering of the points in time. So if p
represents the atomic statement ‘the sun shines’ then
3p is ‘the sun will shine again’.
In the context of temporal logic it may be assumed
that the relation(s) are transitive or linear depending
on our perception of time. If we introduce sufficient
structure and operators to our temporal logics we can
model certain properties of computer programs, this is
especially useful where concurrency is concerned.
◦ F, w |= φ iff M, w |= φ for every model M of F
Reasoning about knowledge
◦ F |= φ iff F, w |= φ for every world w of F
Another interesting application of modal logic is modelling the reasoning of agents with restricted information. For example we may want to describe the notion
‘α knows that β knows φ’. It is fairly easy to extend
modal logic to describe such notions. Interested readers
are encouraged to read up on ‘The wise man puzzle’ and
the ‘Muddy children puzzle’. An easy to read but formal treatment of both of these problems can be found
in (author?) [2], while (author?) [3] solves the latter and introduces some Temporal Dynamic Epistemic
Logic.
M, w |= 2α ψ iff for all v ∈ W, wRα v only if M, v |= ψ
M, w |= 3α ψ iff for some v ∈ W, wRα v and M, v |= ψ
We can now have the following four notions of truth
of a modal formula φ:
◦ M, w |= φ has already been defined.
Relating Modal Logic to First Order Logic and
Second Order Logic
Since reasoning with Kripke frames is in some sense reasoning about relations it is common (although not necessary) to assume additional properties of the relations
studied. Interestingly enough, notions such as reflexivity, symmetry and transitivity are first order properties
but can all be represented using modal formulae, however anti-symmetry cannot. A natural question to ask
is when do such representations exist and why are there
some that do not. The second question is partially
answered by noticing that the effect of quantifiers in
modal formulae is always restricted by the accessibility
relations. Those properties that can be expressed using
modal formulae all lie in the two variable fragment of
first-order logic. We also note that the concept of an
arbitrary model is a second order property. The relationship between modal logic and first order logic is an
active area of research.
3 Here
Further Reading
[1] Rod Girle, Modal Logics and Philosophy, Acumen
(2000).
[2] Michael R. A. Huth and Mark D. Ryan, Logic in
Computer Science: Modelling and Reasoning about
systems, Cambridge University Press (2000).
[3] Joshua Sack, Muddy Children, other Logic Puzzles,
and Temporal Dynamic Epistemic Logic,
www.csulb.edu/∼jsack/csulbtalk2.pdf.
we need to relax the usual requirement of irreflexivity
2
3