A
modal logic is any system of
formal logic that attempts
to deal with
modalities. Modals qualify
the truth of a judgment. For example, if it is true that "John is
happy," we might qualify this statement by saying that "John is
very happy," in which case the term "very" would be a
modality. Traditionally, there are three "modes" or "moods" or
"modalities" represented by modal logic, namely,
possibility,
probability, and
necessity.
A formal modal logic represents modalities using
modal operators. For example, "It might rain
today" and "It is possible that rain will fall today" both contain
the notion of possibility. In a modal logic this is represented as
an operator,
Possibly, attached to the sentence
It
will rain today.
The basic unary (1-place) modal operators are usually written \Box
for
Necessarily and \Diamond for
Possibly. In a
classical modal logic, each
can be expressed by the other and
negation:
- \Diamond P \leftrightarrow \lnot \Box \lnot P;
- \Box P \leftrightarrow \lnot \Diamond \lnot P.
Thus it is
possible that it will rain today if and only if
it is
not necessary that it will
not rain today.
For the standard formal semantics of the basic modal language, see
Kripke semantics.
Development of modal logic
Although
Aristotle's logic is almost
entirely concerned with the theory of the
categorical syllogism, there are
passages in his work, such as the famous
sea-battle argument in
De Interpretatione § 9,
that are now seen as anticipations of modal logic and its
connection with potentiality and time. Modal logic as a self-aware
subject owes much to the writings of the
Scholastics, in particular
William of Ockham and
John Duns Scotus, who reasoned informally
in a modal manner, mainly to analyze statements about
essence and
accident.
C. I.
Lewis founded modern modal logic in his
1910 Harvard thesis and in a series of scholarly articles beginning
in 1912. This work culminated in his 1932 book
Symbolic
Logic (with C. H. Langford), which introduced the five systems
S1 through
S5.
The contemporary era in modal logic began
in 1959, when Saul Kripke (then only a
19 year old Harvard
University undergraduate) introduced the now-standard Kripke semantics for modal logics.
These are commonly referred to as "possible worlds" semantics.
Kripke and
A. N. Prior had
previously corresponded at some length.
A. N.
Prior created
temporal logic, closely related to modal
logic, in 1957 by adding modal operators [F] and [P] meaning
"henceforth" and "hitherto".
Vaughan
Pratt introduced
dynamic
logic in 1976. In 1977,
Amir Pnueli
proposed using temporal logic to formalise the behaviour of
continually operating concurrent programs. Flavors of temporal
logic include propositional dynamic logic (PDL), propositional
linear temporal logic (PLTL),
linear temporal logic (LTL),
computational tree logic (CTL),
Hennessy-Milner logic, and
T.
The mathematical structure of modal logic, namely
Boolean algebra augmented with
unary operations (often called
"modal algebras"), began to emerge with J. C. C. McKinsey's 1941
proof that
S2 and
S4 are decidable, and reached
full flower in the work of
Alfred
Tarski and his student
Bjarni
Jonsson (Jonsson and Tarski 1951-52). This work revealed that
S4 and
S5 are models of
interior algebra, a proper extension of
Boolean algebra originally designed to capture the properties of
the
interior and
closure operators of
topology. Texts on modal logic typically do little
more than mention its connections with the study of
Boolean algebra and
topology. For a thorough survey of the history of
formal modal logic and of the associated mathematics, see
Goldblatt (2006).
Formalizations
Semantics
The semantics for modal logic are usually given like so:First we
define a
frame, which consists of a non-empty set,
G, whose members are generally called possible worlds, and
a binary relation,
R, that holds (or not) between the
possible worlds of
G. This binary relation is called the
accessibility relation. For example,
w R
v means that the world
v is accessible from world
w. That is to say, the state of affairs known as
v is a live possibility for
w. This gives a pair,
<<EM>G,
R>.
Next, a frame is extended to a
model by specifying the
truth-values of all propositions at each particular world in
G. We do so by defining a relation ⊨ between possible
worlds and propositional letters. If there is a world
w
such that
w ⊨ P, then P is true at
w. A model is
thus an ordered triple, <<EM>G,
R, ⊨>.
Then we define
truth in a model:
- w ⊨ ¬P if and only if w ⊭ P
- w ⊨ (P & Q) if and only if w ⊨ P and
w ⊨ Q
- w ⊨ \BoxP if and only if for every element v
of G, if w R v then v
⊨ P
- w ⊨ \DiamondP if and only if for some element
v of G, if w R v then
v ⊨ P
According to these semantics, a truth is
necessary with
respect to a possible world
w if it is true at every world
that
w can see, and
possible if it is true at
some world that
w can see. Possibility thereby depends
upon the accessibility relation
R, and this allows us to
express the relative nature of possibility. For example, we might
say that at our own world it is not possible to travel faster than
the speed of light, but that given other circumstances it might
have been possible to do so. If we accept this principle, then we
reject the axiom that \lnot \DiamondP → \lnot \Diamond \Diamond P
(or equivalently, that \BoxP → \Box\BoxP). We reject the principle
that the impossibility of something at every possible world we can
see implies the impossibility of that thing at the worlds that can
be seen from the worlds we can see.
There are several systems like this one that have been espoused,
which are variously based around different characterizations of the
accessibility relation (also called
frame conditions). The
accessibility relation is:
- reflexive iff w R w, for
every w in G
- symmetric iff w R v implies
v R w, for all w and v
in G
- transitive iff w R v and
v R q together imply w
R q, for all w,v, q in
G.
- serial iff, for each w in G there is
some v in G such that w R
v.
The logics that stem from these frame conditions are:
- K := no conditions
- D := serial
- T := reflexive
- S4 := reflexive and transitive
- S5 := reflexive, symmetric and transitive
S5 is the strongest logic. Above, we saw that a sentence is
S5-valid if it is valid in all frames where R is an equivalence
relation (reflexive, symmetric and transitive). But provably, this
produces the same set of valid sentences as merely requiring that
it be valid in all frames where R is a total relation, i.e. where
all frames see all other frames. Thus, the S5 system's semantics
can be defined without the use of an accessibility relation, R, by
simply dropping that clause from the definition of validity. S5,
therefore, is the only system where \BoxP means "true in all
possible worlds". All of these logics can be defined axiomatically,
as is shown in the next section. In S5, the axioms P →
\Box\DiamondP, \BoxP → \Box\BoxP, and \BoxP → P (corresponding to
symmetry,
transitivity and
reflexivity,
respectively) hold, whereas at least one of these axioms does not
hold in each of the other, weaker logics.
Axiomatic Systems
The first formalizations of modal logic were axiomatic. Numerous
variations with very different properties have been proposed since
C. I.
Lewis began working in the area in 1910.
Hughes and Cresswell (1996), for example, describe 42 normal and 25
non-normal modal logics. Zeman (1973) describes some systems Hughes
and Cresswell omit.
Modern treatments of modal logic begin by augmenting the
propositional calculus with two unary
operations, one denoting "necessity" and the other "possibility".
The notation of
Lewis, much
employed since, denotes "necessarily
p" by a prefixed
"box" ( \Box p ) whose scope is established by parentheses.
Likewise, a prefixed "diamond" (\Diamond p) denotes "possibly
p". Regardless of notation, each of these operators is
definable in terms of the other:
- \Box p (necessarily p) is equivalent to \neg \Diamond
\neg p ("not possible that not-p")
- \Diamond p (possibly p) is equivalent to \neg \Box
\neg p ("not necessarily not-p")
Hence \Box and \Diamond form a
dual pair of
operators.
In many modal logics, the necessity and possibility operators
satisfy the following analogs of
de
Morgan's laws from
Boolean
algebra:
- "It is not necessary that X" is
logically equivalent to "It is possible that not
X".
- "It is not possible that X" is
logically equivalent to "It is necessary that not
X".
Precisely what axioms and rules must be added to the
propositional calculus to create a
usable system of modal logic is a matter of philosophical opinion,
often driven by the theorems one wishes to prove; or, in computer
science, it is a matter of what sort of computational or deductive
system one wishes to model. Many modal logics, known collectively
as
normal modal logics, include
the following rule and axiom:
- N, Necessitation Rule: If p
is a theorem (of any system invoking
N), then \Box p is likewise a theorem.
- K, Distribution Axiom: \Box (p
\rightarrow q) \rightarrow (\Box p \rightarrow \Box q).
The weakest
normal modal logic,
named
K in honor of
Saul
Kripke, is simply the
propositional calculus augmented by
\Box , the rule
N, and the axiom
K.
K is weak in that it fails to
determine whether a proposition can be necessary but only
contingently necessary. That is, it is not a theorem of
K
that if \Box p is true then \Box \Box p is true, i.e., that
necessary truths are "necessarily necessary". If such perplexities
are deemed forced and artificial, this defect of
K is not
a great one. In any case, different answers to such questions yield
different systems of modal logic.
Adding axioms to
K gives rise to other well-known modal
systems. One cannot prove in
K that if "
p is
necessary" then
p is true. The axiom
T
remedies this defect:
- T, Reflexivity Axiom: \Box p
\rightarrow p (If p is necessary, then p is the
case.) T holds in most but not all modal logics.
Zeman (1973) describes a few exceptions, such as
S1^{0}.
Other well-known elementary axioms are:
- 4: \Box p \rightarrow \Box \Box p
- B: p \rightarrow \Box \Diamond p
- D: \Box p \rightarrow \Diamond p
These axioms yield the systems:
- K := K + N
- T := K + T
- S4 := T + 4
- S5 := S4 + B
- D := K + D.
K through
S5 form a nested hierarchy of systems,
making up the core of
normal modal
logic. But specific rules or sets of rules may be appropriate
for specific systems. For example, in deontic logic, \Box p
\rightarrow \Diamond p (If it ought to be that
P, then it
is permitted that
P) seems appropriate, but we should
probably not include that p \rightarrow \Box \Diamond p.
The commonly employed system
S5 simply makes all modal
truths necessary. For example, if
p is possible, then it
is "necessary" that
p is possible. Also, if
p is
necessary, then it is necessary that
p is necessary.
Although controversial, this is commonly justified on the grounds
that
S5 is the system obtained if every possible world is
possible relative to every other world. Other systems of modal
logic have been formulated, in part because
S5 does not
describe every kind of metaphysical modality of interest. This
suggests that talk of possible worlds and their semantics may not
do justice to all modalities.
Alethic modalities
Modalities of necessity and possibility are called
alethic
modalities. They are also sometimes called
special
modalities, from the
Latin species.
Modal logic was first developed to deal with these concepts, and
only afterward was extended to others. For this reason, or perhaps
for their familiarity and simplicity, necessity and possibility are
often casually treated as
the subject matter of modal
logic. Moreover it is easier to make sense of relativizing
necessity, e.g. to legal, physical, nomological, epistemic, and so
on, than it is to make sense of relativizing other notions.
In
classical modal logic, a
proposition is said to be
- possible if and only if it is not
necessarily false (regardless of whether it is actually true
or actually false);
- necessary if and only if it is not
possibly false; and
- contingent if and only if it is not
necessarily false and not necessarily true (ie.
possible but not necessarily true).
In classical modal logic, therefore, either the notion of
possibility or necessity may be taken to be basic, where these
other notions are defined in terms of it in the manner of
De Morgan duality.
Intuitionistic modal logic treats
possibility and necessity as not perfectly symmetric.
For those with difficulty with the concept of something being
possible but not true, the meaning of these terms may be made more
comprehensible by thinking of multiple "possible worlds" (in the
sense of
Leibniz) or "alternate universes";
something "necessary" is true in all possible worlds, something
"possible" is true in at least one possible world. These "possible
world semantics" are formalized with
Kripke semantics.
Physical possibility
Something is physically possible if it is permitted by the
laws of physics. For example, current theory
allows for there to be an
atom with an
atomic number of 150, though there may not in
fact be any such atoms in existence. Similarly, while it is
logically possible to accelerate beyond the
speed of light, that is not, according to
modern science, physically possible for material particles or
information.
Metaphysical possibility
Philosophers ponder the properties that
objects have independently of those dictated by scientific laws.
For example, it might be metaphysically necessary, as some have
thought, that all thinking beings have bodies and can experience
the passage of
time, or that
God exists (or does not).
Saul
Kripke has argued that every person necessarily has the parents
they do have: anyone with different parents would not be the same
person.
Metaphysical possibility is generally thought to be more
restricting than bare logical possibility (i.e., fewer things are
metaphysically possible than are logically possible). Its exact
relation to physical possibility is a matter of some dispute.
Philosophers also disagree over whether metaphysical truths are
necessary merely "by definition", or whether they reflect some
underlying deep facts about the world, or something else
entirely.
Confusion with epistemic modalities
Alethic modalities and epistemic modalities (see below) are often
expressed in English using the same words. "It is possible that
bigfoot exists" can mean either "Bigfoot
could exist,
whether or not bigfoot does in fact exist" (alethic), or more
likely, "For all I know, bigfoot exists" (epistemic).
Epistemic logic
Epistemic modalities (from the Greek
episteme, knowledge), deal with the
certainty of
sentences. The \Box operator is translated as "x knows that…", and
the \Diamond operator is translated as "For all x knows, it may be
true that…" In ordinary speech both metaphysical and epistemic
modalities are often expressed in similar words; the following
contrasts may help:
A person, Jones, might reasonably say
both: (1) "No, it is
not possible that
Bigfoot exists; I
am quite certain of that";
and, (2) "Sure, Bigfoot
possibly
could exist". What Jones means by (1) is that
given all the available information, there is no question remaining
as to whether Bigfoot exists. This is an epistemic claim. By (2) he
makes the
metaphysical claim that it is
possible
for Bigfoot to exist,
even though he does not (which
is not equivalent to "it is
possible that Bigfoot exists –
for all I know", which contradicts (1)).
From the other direction, Jones might say, (3) "It is
possible that
Goldbach's
conjecture is true; but also
possible that it is
false", and
also (4) "if it
is true, then it is
necessarily true, and not possibly false". Here Jones means that it
is
epistemically possible that it is true or false, for
all he knows (Goldbach's conjecture has not been proven either true
or false), but if there
is a proof (heretofore
undiscovered), then it would show that it is not
logically
possible for Goldbach's conjecture to be false—there could be no
set of numbers that violated it. Logical possibility is a form of
alethic possibility; (4) makes a claim about whether it is
possible (ie, logically speaking) that a mathematical truth to have
been false, but (3) only makes a claim about whether it is
possible, for all Jones knows, (ie, speaking of certitude) that the
mathematical claim is specifically either true or false, and so
again Jones does not contradict himself. It is worthwhile to
observe that Jones is not necessarily correct: It is possible
(epistemically) that Goldbach's conjecture is both true and
unprovable.
Epistemic possibilities also bear on the actual world in a way that
metaphysical possibilities do not. Metaphysical possibilities bear
on ways the world
might have been, but epistemic
possibilities bear on the way the world
may be (for all we
know). Suppose, for example, that I want to know whether or not to
take an umbrella before I leave. If you tell me "it is
possible
that it is raining outside" – in the sense of epistemic
possibility – then that would weigh on whether or not I take the
umbrella. But if you just tell me that "it is
possible for
it to rain outside" – in the sense of
metaphysical
possibility – then I am no better off for this bit of modal
enlightenment.
Some features of epistemic modal logic are in debate. For example,
if
x knows that
p, does
x know that it
knows that
p? That is to say, should \Box P \rightarrow
\Box \Box P be an axiom in these systems? While the answer to this
question is unclear, there is at least one axiom that
must
be included in epistemic modal logic, because it is minimally true
of all modal logics (see
the section on axiomatic
systems):
- K, Distribution Axiom: \Box (p
\rightarrow q) \rightarrow (\Box p \rightarrow \Box q).
But this is disconcerting, because with
K, we can
prove that we know all the logical consequences of our beliefs: If
q is a logical consequence of
p, then \Box (p
\rightarrow q). And if so, then we can deduce that (\Box p
\rightarrow \Box q) using
K. When we translate
this into epistemic terms, this says that if
q is a
logical consequence of
p, then we know that it is, and if
we know
p, we know
q. That is to say, we know all
the logical consequences of our beliefs. This must be true for all
possible modal interpretations of epistemic cases where \Box is
translated as "knows that". But then, for example, if
x
knows that prime numbers are divisible only by themselves and the
number one, then
x knows that
8683317618811886495518194401279999999 is prime (since this number
is only divisible by itself and the number one). That is to say,
under the modal interpretation of knowledge, anyone who knows the
definition of a prime number knows that this number is prime. This
shows that epistemic modal logic is an idealized account of
knowledge, and explains objective, rather than subjective knowledge
(if anything).
Temporal logic
Temporal logic is an approach to the semantics of expressions with
tense, that is, expressions with
qualifications of when. Some expressions, such as '2 + 2 = 4', are
true at all times, while tensed expressions such as 'John is happy'
are only true sometimes.
In temporal logic, tense constructions are treated in terms of
modalities, where a standard method for formalizing talk of time is
to use
two pairs of operators, one for the past and one
for the future (P will just mean 'it is presently the case that
P'). For example:
- FP : It will sometime be the case
that P
- GP : It will always be the case
that P
- PP : It was sometime the case that
P
- HP : It has always been the case
that P
There are then at least three modal logics that we can develop. For
example, we can stipulate that,
- \Diamond P = P is the case at some time
t
- \Box P = P is the case at every time t
and we can add two other operators to talk about the future by
itself (these would have to be distinguished from the first set by
some subscript). Either,
- \Diamond P = FP
- \Box P = GP
or,
- \Diamond P = P and/or
FP
- \Box P = P and
GP
The second and third interpretations may seem odd, but such
assignments do create classical modal systems. Note that
FP is the same as
¬G'¬P. We
can combine the above operators to form complex
statements. For example,
PP → \Box
PP says (effectively),
Everything that is past and true is
necessary.
It seems reasonable to say that possibly it will rain tomorrow, and
possibly it won't; on the other hand, seeing as how we can't change
the past, if it is true that it rained yesterday, it probably isn't
true that it may not have rained yesterday. It seems the past is
"fixed", or necessary, in a way the future is not. This is
sometimes referred to as
accidental
necessity. But if the past is "fixed", and everything that is
in the future will eventually be in the past, then it seems
plausible to say that future events are necessary too
Similarly, the
problem of
future contingents considers the semantics of assertions about
the future: is either of the propositions 'There will be a sea
battle tomorrow', or 'There will not be a sea battle tomorrow' now
true? Considering this thesis led
Aristotle to reject the
principle of bivalence for assertions
concerning the future.
Additional binary operators are also relevant to temporal logics,
q.v. Linear Temporal
Logic.
Versions of temporal logic can be used in
computer science to model computer
operations and prove theorems about them. In one version, \Diamond
P means "at a future time in the computation it is possible that
the computer state will be such that P is true"; \Box P means "at
all future times in the computation P will be true". In another
version, \Diamond P means "at the immediate next state of the
computation, P might be true"; \Box P means "at the immediate next
state of the computation, P will be true". These differ in the
choice of
Accessibility
relation. (P always means "P is true at the current computer
state".) These two examples involve nondeterministic or
not-fully-understood computations; there are many other modal
logics specialized to different types of program analysis. Each one
naturally leads to slightly different axioms.
Deontic logic
Likewise talk of morality, or of
obligation and
norms generally, seems to have a modal
structure. The difference between "You must do this" and "You may
do this" looks a lot like the difference between "This is
necessary" and "This is possible". Such logics are called
deontic, from the Greek for
"duty". One characteristic feature of deontic logics is that they
lack the axiom
T semantically corresponding to the
reflexivity of the accessibility relation in
Kripke semantics: in symbols,
\Box\phi\to\phi. Interpreting \Box as "it is obligatory that",
T informally says that every obligation is true.
For example, if it is obligatory not to kill others (i.e. killing
is morally forbidden), then
T implies that people
actually do not kill others. This consequence is obviously
false.
However, in
Kripke semantics for
deontic logic,
T is supposed to hold at
accessible worlds (relative to the actual world w). These worlds
are to be thought of as idealized in the sense that all obligations
(in w) are fulfilled there. Hence a sentence A is obligatory just
in case A holds at all idealized worlds. So in order to discuss
obligations under Kripke semantics, there must be some world where
\Box\phi\to\phi, where everything that ought to be the case, is the
case. Though this was one of the first interpretations of the
formal semantics, it has recently come under criticism. See e.g.
Sven Hansson, "Ideal Worlds--Wishful Thinking in Deontic Logic",
Studia Logica, Vol. 82 (3), pp. 329-336, 2006.
One other principle that is often (at least traditionally) accepted
as a deontic principle is
D, \Box\phi\to\Diamond\phi,
which corresponds to the seriality (or extendability or
unboundedness) of the accessibility relation. It is an embodiment
of the Kantian idea that "ought implies can". (Clearly the "can"
can be interpreted in various senses, e.g. in a moral or alethic
sense.)
Intuitive problems with deontic logic
When we try and formalize ethics with standard modal logic, we run
into some problems. Suppose that we have a proposition
K:
you kill the victim, and another,
Q: you kill the victim
quickly. Now suppose we want to express the thought that "if you do
kill the victim, you ought to kill him quickly". There are two
likely candidates,
- (1) (K \rightarrow \Box Q)
- (2) \Box (K \rightarrow Q)
But (1) says that if you kill the victim, then it ought to be the
case that you kill him quickly. This surely isn't right, because
you ought not to have killed him at all. And (2) doesn't work
either. If the right representation of "if you kill the victim then
you ought to kill him quickly" is (2), then the right
representation of (3) "if you kill the victim then you ought to
kill him slowly" is \Box (K \rightarrow \lnot Q). Now suppose (as
seems reasonable) that you should not kill the victim, or \Box
\lnot K. But then we can deduce \Box (K \rightarrow \lnot Q), which
would express sentence (3). So if you should not kill the victim,
then if you kill him, you should kill him slowly. But that can't be
right, and is not right when we use natural language. Telling
someone they should not kill the victim certainly does not imply
that they should kill the victim slowly if they do kill him.
Doxastic logic
Doxastic logic concerns the logic of belief (of some set
of agents). The term doxastic is derived from the
ancient Greek doxa which means
"belief". Typically, a doxastic logic uses \Box, often written "B",
to mean "It is believed that", or when relativized to a particular
agent s, "It is believed by s that".
Other modal logics
Significantly, modal logics can be developed to accommodate most of
these idioms; it is the fact of their common logical structure (the
use of "intensional" sentential operators) that make them all
varieties of the same thing.
The ontology of possibility
In the most common interpretation of modal logic, one considers
"
logically possible worlds". If a
statement is true in all
possible
worlds, then it is a necessary truth. If a statement happens to
be true in our world, but is not true in all possible worlds, then
it is a contingent truth. A statement that is true in some possible
world (not necessarily our own) is called a possible truth.
Whether this "possible worlds idiom" is the best way to interpret
modal logic, and how literally this idiom can be taken, is a live
issue for metaphysicians. The possible worlds idiom would translate
the claim about Bigfoot as "There is some possible world in which
Bigfoot exists". To maintain that Bigfoot's existence is possible,
but not actual, one could say, "There is some possible world in
which Bigfoot exists; but in the actual world, Bigfoot does not
exist". But it is unclear what it is that making this claim commits
us to. Are we really alleging the existence of possible worlds,
every bit as real as our actual world, just not actual?
Saul Kripke believes that this is a misnomer -
that the term 'possible world' is just a useful way of visualizing
the concept of possibility. For him, the sentences "you could have
rolled a 4 instead of a 6" and "there is a possible world where you
rolled a 4, but you rolled a 6 in the actual world" are not
significantly different statements.
David Lewis, on the other hand,
made himself notorious by biting the bullet, asserting that all
merely possible worlds are as real as our own, and that what
distinguishes our world as
actual is simply that it is
indeed our world –
this world (see
Indexicality). That position is a major tenet
of "
modal realism". Some philosophers
decline to endorse any version of modal realism, considering it
ontologically extravagant, and prefer to seek various ways to
paraphrase away these ontological commitments.
Robert Adams holds that 'possible
worlds' are better thought of as 'world-stories', or consistent
sets of propositions. Thus, it is possible that you rolled a 4 if
such a state of affairs can be described coherently.
Computer scientists will generally pick a highly specific
interpretation of the modal operators specialized to the particular
sort of computation being analysed. In place of "all worlds", you
may have "all possible next states of the computer", or "all
possible future states of the computer".
See also
References
- Blackburn, Patrick; de Rijke, Maarten; and Venema, Yde (2001)
Modal Logic. Cambridge University Press. ISBN
0-521-80200-8
- Blackburn, P.; van
Benthem, J.; and Wolter, Frank; Eds. (2006) Handbook of Modal Logic. North Holland.
- Chagrov, Aleksandr; and Zakharyaschev, Michael (1997) Modal
Logic. Oxford University Press. ISBN 0-19-853779-4
- Chellas, B. F. (1980) Modal Logic: An Introduction.
Cambridge University Press. ISBN 0-521-22476-4
- Cresswell, M. J. (2001) "Modal Logic" in Goble, Lou; Ed.,
The Blackwell Guide to Philosophical Logic. Basil
Blackwell: 136-58. ISBN 0-631-20693-0
- Fitting, Melvin; and Mendelsohn, R. L. (1998) First Order
Modal Logic. Kluwer. ISBN 0-7923-5335-8
- Garson, James W. (2006) Modal Logic for Philosophers.
Cambridge University Press. ISBN 0-521-68229-0. A thorough
introduction to modal logic, with coverage of various derivation
systems and a distinctive approach to the use of diagrams in aiding
comprehension.
- Girle, Rod (2000) Modal Logics and Philosophy. Acumen
(UK). ISBN 0-7735-2139-9. Proof by refutation trees. A good introduction to
the varied interpretations of modal logic.
- Goldblatt, Robert (1992) "Logics of Time and
Computation", 2nd ed., CSLI Lecture Notes No. 7. University of
Chicago Press.
- —— (1993) Mathematics of Modality, CSLI Lecture Notes
No. 43. University of Chicago Press.
- —— (2006) " Mathematical Modal Logic: a View of its Evolution", in
Gabbay, D. M.; and Woods, John; Eds., Handbook of the History
of Logic, Vol. 6. Elsevier BV.
- Goré, Rajeev (1999) "Tableau Methods for Modal and Temporal
Logics" in D'Agostino, M.; Gabbay, Dov; Haehnle, R.; and Posegga,
J.; Eds., Handbook of Tableau Methods. Kluwer:
297-396.
- Hughes, G. E., and Cresswell, M. J. (1996) A New
Introduction to Modal Logic. Routledge. ISBN
0-415-12599-5
- Jónsson, B. and Tarski, A., 1951-52, "Boolean Algebra with
Operators I and II", American Journal of Mathematics 73:
891-939 and 74: 129-62.
- Kracht, Marcus (1999) Tools and Techniques in Modal
Logic, Studies in Logic and the Foundations of Mathematics No.
142. North Holland.
- Lemmon, E. J. (with Scott, D.)
(1977) An Introduction to Modal Logic, American
Philosophical Quarterly Monograph Series, no. 11 (Krister
Segerberg, series ed.). Basil Blackwell.
- Lewis, C. I. (with Langford, C. H.) (1932). Symbolic Logic.
Dover reprint, 1959.
- Prior, A. N. (1957) Time and Modality. Oxford
University Press.
Free and online:
Notes
- Fitting and Mendelsohn. First-Order Modal Logic.
Kluwer Academic Publishers, 1998. Section 1.6
- Saul Kripke. Naming and Necessity. Harvard University
Press, 1980. pg 113
- See Goldbach's conjecture >
Origins
- Ted Sider's Logic for Philosophy Currently page 228, but subject
to change.
- Kripke, Saul. Naming and Necessity. (1980; Harvard
UP), pp. 43-5.
- Kripke, Saul. Naming and Necessity. (1980; Harvard
UP), pp. 15-6.
- David Lewis, On the Plurality of Worlds (1986;
Blackwell)
- Adams, Robert M. Theories of Actuality. Noûs, Vol. 8,
No. 3 (Sep., 1974), particularly pp. 225-31.
Further reading
- Dov M. Gabbay, Many-dimensional modal logics:
theory and applications, Gulf Professional Publishing, 2003,
ISBN 0444508260. Covers many varieties of modal logic, e.g.
temporal, epistemic, dynamic, description, spatial from a unified
perspective with emphasis on computer science aspects, e.g.
decidability and complexity.
External links
Acknowledgements
This article includes material from the
Free On-line Dictionary of
Computing, used with under the
GFDL.