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modal logic
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{{short description|Type of formal logic}}Modal logic is a type of formal logic primarily developed in the 1960s that extends classical propositional and predicate logic to include operators expressing modality. A modalâ€”a word that expresses a modalityâ€”qualifies a statement. For example, the statement "John is happy" might be qualified by saying that John is usually happy, in which case the term "usually" is functioning as a modal. The traditional alethic modalities, or modalities of truth, include possibility ("Possibly, p", "It is possible that p"), necessity ("Necessarily, p", "It is necessary that p"), and impossibility ("Impossibly, p", "It is impossible that p").BOOK, Formal Logic, Arthur N., Prior, Oxford Univ. Press, 1962, 185, Second, 9780198241560, Other modalities that have been formalized in modal logic include temporal modalities, or modalities of time (notably, "It was the case that p", "It has always been that p", "It will be that p", "It will always be that p"),BOOK, 9783709176641,weblink Temporal Logic, Rescher, Nicholas, Urquhart, Alasdair, Springer-Verlag, 1971, 52, BOOK, Past, Present and Future, Arthur N., Prior, Oxford Univ. Press, 1967, 9780198243113, deontic modalities (notably, "It is obligatory that p", and "It is permissible that p"), epistemic modalities, or modalities of knowledge ("It is known that p")"Knowledge and Belief", by Jaakko Hinntikka, Cornell Univ. Press, 1962 and doxastic modalities, or modalities of belief ("It is believed that p")."Topics in Philosophical Logic", by N. Rescher, Humanities Press, 1968, p. 41A 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".It is fallacious to confuse necessity and possibility. In particular, this is known as the modal fallacy.The basic unary (1-place) modal operators are usually written "â–¡" for "Necessarily" and "â—‡" for "Possibly". Following the example above, if P is to represent the statement of "it will rain today", the possibility of rain would be represented by Diamond P. This reads: It is possible that it will rain today. Similarly Box P reads: It is necessary that it will rain today, expressing certainty regarding the statement.In a classical modal logic, each can be expressed by the other with negation.
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Diamond P leftrightarrow lnot Box lnot P;
In natural language, this reads: it is possible that it will rain today if and only if it is not necessary that it will not rain today. Similarly, necessity can be expressed in terms of possibility in the following negation:
Box P leftrightarrow lnot Diamond lnot P
which states it is necessary that it will rain today if and only if it is not possible that it will not rain today. Alternative symbols used for the modal operators are "L" for "Necessarily" and "M" for "Possibly".So in the standard work A New Introduction to Modal Logic, by G. E. Hughes and M. J. Cresswell, Routledge, 1996, passim.Development of modal logic
In addition to his non-modal syllogistic, Aristotle also developed a modal syllogistic in Book I of his Prior Analytics (chs 8â€“22), which Theophrastus attempted to improve.SEP, logic-ancient, Ancient Logic, Bobzien, Susanne, There are also passages in Aristotle's work, such as the famous sea-battle argument in De Interpretatione Â§9, that are now seen as anticipations of the connection of modal logic with potentiality and time. In the Hellenistic period, the logicians Diodorus Cronus, Philo the Dialectician and the Stoic Chrysippus each developed a modal system that accounted for the interdefinability of possibility and necessity, accepted axiom T (see below), and combined elements of modal logic and temporal logic in attempts to solve the notorious Master Argument.Bobzien, S. (1993). "Chrysippus' Modal Logic and its Relation to Philo and Diodorus", in K. Doering & Th. Ebert (eds), Dialektiker und Stoiker, Stuttgart 1993, pp. 63â€“84. The earliest formal system of modal logic was developed by Avicenna, who ultimately developed a theory of "temporally modal" syllogistic.History of logic: Arabic logic, EncyclopÃ¦dia Britannica. 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 thesisTHESIS, Ph.D. thesis, Clarence Irving Lewis, The Place of Intuition in Knowledge, Harvard University, 1910, and in a series of scholarly articles beginning in 1912. This work culminated in his 1932 book Symbolic Logic (with C. H. Langford),BOOK, Clarence Irving Lewis and Cooper Harold Langford, Symbolic Logic, Dover Publications, 1st, 1932, which introduced the five systems S1 through S5.Ruth C. Barcan (later Ruth Barcan Marcus) developed the first axiomatic systems of quantified modal logic â€” first and second order extensions of Lewis' S2, S4, and S5.JOURNAL, Ruth C. Barcan, A Functional Calculus of First Order Based on Strict Implication, Journal of Symbolic Logic, 11, 1, 1â€“16, Mar 1946, 10.2307/2269159, 2269159, JOURNAL, Ruth C. Barcan, The Deduction Theorem in a Functional Calculus of First Order Based on Strict Implication, Journal of Symbolic Logic, 11, 4, 115â€“118, Dec 1946, 10.2307/2268309, 2268309, JOURNAL, Ruth C. Barcan, The Identity of Individuals in a Strict Functional Calculus of Second Order, Journal of Symbolic Logic, 12, 1, 12â€“15, Mar 1947, 10.2307/2267171, 2267171, The contemporary era in modal semantics 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. Kripke semantics is basically simple, but proofs are eased using semantic-tableaux or analytic tableaux, as explained by E. W. Beth.A. N. Prior created modern temporal logic, closely related to modal logic, in 1957 by adding modal operators [F] and [P] meaning "eventually" and "previously". 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), computation tree logic (CTL), Hennessyâ€“Milner logic, and T.{{clarify|reason=Add a wikilink, give a longer name, or give a reference for the 'T' logic.|date=November 2016}}The mathematical structure of modal logic, namely Boolean algebras 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,JOURNAL, McKinsey, J. C. C., A Solution of the Decision Problem for the Lewis Systems S2 and S4, with an Application to Topology, J. Symb. Log., 1941, 6, 4, 117â€“134, 2267105, 10.2307/2267105, and reached full flower in the work of Alfred Tarski and his student Bjarni JÃ³nsson (JÃ³nsson and Tarski 1951â€“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 algebras and topology. For a thorough survey of the history of formal modal logic and of the associated mathematics, see Robert Goldblatt (2006).Robert Goldbaltt, Mathematical Modal Logic: A view of it evolutionSemantics
Model theory
The semantics for modal logic are usually given as follows:Fitting and Mendelsohn. First-Order Modal Logic. Kluwer Academic Publishers, 1998. Section 1.6 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 u means that the world u is accessible from world w. That is to say, the state of affairs known as u is a live possibility for w. This gives a pair, langle G, Rrangle. Some formulations of modal logic also include a constant term in G, conventionally called "the actual world", which is often symbolized as w*.Next, the frame is extended to a model by specifying the truth-values of all propositions at each of the worlds in G. We do so by defining a relation v between possible worlds and positive literals. If there is a world w such that v(w, P), then P is true at w. A model is thus an ordered triple, langle G, R, v rangle.Then we recursively define the truth of a formula at a world in a model:- if v(w, P) then w models P
- w models neg P if and only if w not models P
- w models (P wedge Q) if and only if w models P and w models Q
- w models Box P if and only if for every element u of G, if w R u then u models P
- w models Diamond P if and only if for some element u of G, it holds that w R u and u models P
- models P if and only if w models P
- reflexive iff w R w, for every w in G
- symmetric iff w R u implies u R w, for all w and u in G
- transitive iff w R u and u R q together imply w R q, for all w, u, q in G.
- serial iff, for each w in G there is some u in G such that w R u.
- Euclidean iff, for every u, t, and w, w R u and w R t implies u R t (note that it also implies: t R u)
- K := no conditions
- D := serial
- T := reflexive
- B := reflexive and symmetric
- S4 := reflexive and transitive
- S5 := reflexive and Euclidean
w models Diamond P if and only if for some element u of G, it holds that u models P and w R u.
If we consider frames based on the total relation we can just say that
w models Diamond P if and only if for some element u of G, it holds that u models P.
We can drop the accessibility clause from the latter stipulation because in such total frames it is trivially true of all w and u that w R u. But note that this does not have to be the case in all S5 frames, which can still consist of multiple parts that are fully connected among themselves but still disconnected from each other.All of these logical systems can also be defined axiomatically, as is shown in the next section. For example, in S5, the axioms P implies BoxDiamond P, Box P implies BoxBox P and Box P implies 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 C. I. Lewis, much employed since, denotes "necessarily p" by a prefixed "box" (â–¡p) whose scope is established by parentheses. Likewise, a prefixed "diamond" (â—‡p) denotes "possibly p". Regardless of notation, each of these operators is definable in terms of the other in classical modal logic:- â–¡p (necessarily p) is equivalent to {{math|Â¬â—‡Â¬p}} ("not possible that not-p")
- â—‡p (possibly p) is equivalent to {{math|Â¬â–¡Â¬p}} ("not necessarily not-p")
"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 â–¡p is likewise a theorem.
- K, Distribution Axiom: {{math|â–¡(p â†’ q) â†’ (â–¡p â†’ â–¡q).}}
- T, Reflexivity Axiom: {{math|â–¡p â†’ p}} (If p is necessary, then p is the case.)
- 4: Box p to Box Box p
- B: p to Box Diamond p
- D: Box p to Diamond p
- 5: Diamond p to Box Diamond p
- K := K + N
- T := K + T
- S4 := T + 4
- S5 := T + 5
- D := K + D.
Structural proof theory
Sequent calculi and systems of natural deduction have been developed for several modal logics, but it has proven hard to combine generality with other features expected of good structural proof theories, such as purity (the proof theory does not introduce extra-logical notions such as labels) and analyticity (the logical rules support a clean notion of analytic proof). More complex calculi have been applied to modal logic to achieve generality.Decision methods
Analytic tableaux provide the most popular decision method for modal logics.Alethic logic
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 it is not necessarily false (regardless of whether it is actually true or actually false);
- necessary if it is not possibly false (i.e. true and necessarily true);
- contingent if it is not necessarily false and not necessarily true (i.e. possible but not necessarily true);
- impossible if it is not possibly true (i.e. false and necessarily false).
- "Somebody or something turned the lights on" is necessary.
- "Friedrich turned the lights on", "Friedrich's roommate Max turned the lights on" and "A burglar named Adolf broke into Friedrich's house and turned the lights on" are contingent.
- All of the above statements are possible.
- It is impossible that Socrates (who has been dead for over two thousand years) turned the lights on.
Physical possibility
Something is physically, or nomically, possible if it is permitted by the laws of physics.{{citation needed|date=January 2016}} For example, current theory is thought to allow for there to be an atom with an atomic number of 126,NEWS, Press release: Superheavy Element 114 Confirmed: A Stepping Stone to the Island of Stability,weblink Lawrence Berkeley National Laboratory, 24 September 2009, even if there are no such atoms in existence. In contrast, while it is logically possible to accelerate beyond the speed of light,JOURNAL, Feinberg, G., 1967, Possibility of Faster-Than-Light Particles, Physical Review, 159, 5, 1089â€“1105, 1967PhRv..159.1089F, 10.1103/PhysRev.159.1089, See also Feinberg's later paper: Phys. Rev. D 17, 1651 (1978) modern science stipulates that it is not physically possible for material particles or information.JOURNAL, Einstein, Albert, Albert Einstein, Zur Elektrodynamik bewegter KÃ¶rper, Annalen der Physik, 17, 891â€“921, 1905-06-30, 1905AnP...322..891E, 10.1002/andp.19053221004, 10,Metaphysical possibility
Philosophers{{who|date=April 2012}} debate if objects have properties independent of those dictated by scientific laws. For example, it might be metaphysically necessary, as some who advocate physicalism have thought, that all thinking beings have bodiesWEB, Stoljar, Daniel, Physicalism,weblink The Stanford Encyclopedia of Philosophy, 16 December 2014, and can experience the passage of time. Saul Kripke has argued that every person necessarily has the parents they do have: anyone with different parents would not be the same person.Saul Kripke. Naming and Necessity. Harvard University Press, 1980. pg 113Metaphysical possibility has been thought to be more restricting than bare logical possibilityBOOK, Thomson, Judith and Alex Byrne, Content and Modality : Themes from the Philosophy of Robert Stalnaker, 2006, Oxford University Press, Oxford, 107,weblink 16 December 2014, 9780191515736, (i.e., fewer things are metaphysically possible than are logically possible). However, its exact relation (if any) to logical possibility or to physical possibility is a matter of dispute. Philosophers{{who|date=April 2012}} 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.Epistemic logic
Epistemic modalities (from the Greek episteme, knowledge), deal with the certainty of sentences. The â–¡ operator is translated as "x knows thatâ€¦", and the â—‡ operator is translated as "For all x knows, it may be true thatâ€¦" 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, it's possible that Bigfoots 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: there is no physical or biological reason that large, featherless, bipedal creatures with thick hair could not exist in the forests of North America (regardless of whether or not they do). Similarly, "it is possible for the person reading this sentence to be fourteen feet tall and named Chad" is metaphysically true (such a person would not somehow be prevented from doing so on account of their height and name), but not alethically true unless you match that description, and not epistemically true if it's known that fourteen-foot-tall human beings have never existed.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 falseâ€”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 (i.e., 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, (i.e., 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.See Goldbach's conjecture â€“ OriginsEpistemic 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" â€“ in the sense of epistemic possibility â€“ 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" â€“ in the sense of metaphysical possibility â€“ 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 â–¡P â†’ â–¡â–¡P be an axiom in these systems? While the answer to this question is unclear,cf. Blindsight and Subliminal perception for negative empirical evidence there is at least one axiom that is generally included in epistemic modal logic, because it is minimally true of all normal modal logics (see the section on axiomatic systems):- K, Distribution Axiom: Box (p to q) to (Box p to Box q).
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 sometimes 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
Or we can trade these operators to deal only with the future (or past). For example,
Diamond_1 P = FP
Box_1 P = GP
or,
Diamond_2 P = P and/or FP
Box_2 P = P and GP
The operators F and G may seem initially foreign, but they create normal modal systems. Note that FP is the same as Â¬GÂ¬P. We can combine the above operators to form complex statements. For example, PP â†’ â–¡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, since 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, â—‡P means "at a future time in the computation it is possible that the computer state will be such that P is true"; â–¡P means "at all future times in the computation P will be true". In another version, â—‡P means "at the immediate next state of the computation, P might be true"; â–¡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".Deontic logics commonly lack the axiom T semantically corresponding to the reflexivity of the accessibility relation in Kripke semantics: in symbols, Boxphitophi. Interpreting â–¡ 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. The consequent is obviously false.Instead, using Kripke semantics, we say that though our own world does not realize all obligations, the worlds accessible to it do (i.e., T holds at these worlds). These worlds are called idealized worlds. P is obligatory with respect to our own world if at all idealized worlds accessible to our world, P holds. Though this was one of the first interpretations of the formal semantics, it has recently come under criticism.See, e.g., JOURNAL, Sven, Hansson, Ideal Worldsâ€”Wishful Thinking in Deontic Logic, Studia Logica, 82, 3, 329â€“336, 2006, 10.1007/s11225-006-8100-3, One other principle that is often (at least traditionally) accepted as a deontic principle is D, BoxphitoDiamondphi, 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 have stolen some money, and another, Q: you have stolen a small amount of money. Now suppose we want to express the thought that "if you have stolen some money, it ought to be a small amount of money". There are two likely candidates,
(1) (K to Box Q)
(2) Box (K to Q)
But (1) and K together entail â–¡Q, which says that it ought to be the case that you have stolen a small amount of money. This surely isn't right, because you ought not to have stolen anything at all. And (2) doesn't work either: If the right representation of "if you have stolen some money it ought to be a small amount" is (2), then the right representation of (3) "if you have stolen some money then it ought to be a large amount" is Box (K to (K land lnot Q)). Now suppose (as seems reasonable) that you ought not to steal anything, or Box lnot K. But then we can deduce Box (K to (K land lnot Q)) via Box (lnot K) to Box (K to K land lnot K) and Box (K land lnot K to (K land lnot Q)) (the contrapositive of Q to K); so sentence (3) follows from our hypothesis (of course the same logic shows sentence (2)). But that can't be right, and is not right when we use natural language. Telling someone they should not steal certainly does not imply that they should steal large amounts of money if they do engage in theft.Ted Sider's Logic for Philosophy, unknown page.weblinkDoxastic 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 â–¡, 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
{{See also|Intensional logic}}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
{{Further| Accessibility relation|Possible worlds}}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.Under this "possible worlds idiom," to maintain that Bigfoot's existence is possible but not actual, one says, "There is some possible world in which Bigfoot exists; but in the actual world, Bigfoot does not exist". However, it is unclear what 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 'possible world' is something of a misnomer â€“ that the term 'possible world' is just a useful way of visualizing the concept of possibility.Kripke, Saul. Naming and Necessity. (1980; Harvard UP), pp. 43â€“5. 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, and neither commit us to the existence of a possible world.Kripke, Saul. Naming and Necessity. (1980; Harvard UP), pp. 15â€“6. 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 â€“ this world.David Lewis, On the Plurality of Worlds (1986; Blackwell) 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.Adams, Robert M. Theories of Actuality. NoÃ»s, Vol. 8, No. 3 (Sep., 1974), particularly pp. 225â€“31.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".Further applications
Modal logics have begun to be used in areas of the humanities such as literature, poetry, art and history.Seeweblink andweblink H. Miller, "Lives Unled in Realist Fiction", Representations 98, Spring 2007, The Regents of the University of California, {{ISSN|0734-6018}}, pp. 118â€“134.Controversies
Nicholas Rescher has argued that Bertrand Russell rejected modal logic, and that this rejection led to the theory of modal logic languishing for decades.BOOK, Rescher, Nicholas, Bertrand Russell Memorial Volume, 1979, George Allen and Unwin, London, 146, George W. Roberts, Russell and Modal Logic, However, Jan Dejnozka has argued against this view, stating that a modal system which Dejnozka calls "MDL" is described in Russell's works, although Russell did believe the concept of modality to "come from confusing propositions with propositional functions," as he wrote in The Analysis of Matter.JOURNAL, Dejnozka, Jan, Ontological Foundations of Russell's Theory of Modality, Erkenntnis, 1990, 32, 3, 383â€“418,weblink 2012-10-22, 10.1007/bf00216469, ; quote is cited from BOOK, Russell, Bertrand, The Analysis of Matter,weblink 1927, 173, Arthur Norman Prior warned Ruth Barcan Marcus to prepare well in the debates concerning quantified modal logic with Willard Van Orman Quine, due to the biases against modal logic.Ruth Barcan Marcus, Modalities: Philosophical Essays, Oxford University Press, 1993, p. x.See also
{{div col|colwidth=15em}}- Accessibility relation
- Conceptual necessity
- Counterpart theory
- David Kellogg Lewis
- De dicto and de re
- Description logic
- Doxastic logic
- Dynamic logic
- Enthymeme
- Hybrid logic
- Interior algebra
- Interpretability logic
- Kripke semantics
- Metaphysical necessity
- Modal verb
- Multi-valued logic
- Possible worlds
- Provability logic
- Regular modal logic
- Relevance logic
- Rhetoric
- Strict conditional
- Two-dimensionalism
Notes
{{Reflist|30em}}References
- This article includes material from the Free On-line Dictionary of Computing, used with permission under the GFDL.
- Barcan-Marcus, Ruth JSL 11 (1946) and JSL 112 (1947) and "Modalities", OUP, 1993, 1995.
- Beth, Evert W., 1955. "Semantic entailment and formal derivability", Mededlingen van de Koninklijke Nederlandse Akademie van Wetenschappen, Afdeling Letterkunde, N.R. Vol 18, no 13, 1955, pp 309â€“42. Reprinted in Jaakko Intikka (ed.) The Philosophy of Mathematics, Oxford University Press, 1969 (Semantic Tableaux proof methods).
- Beth, Evert W., "Formal Methods: An Introduction to Symbolic Logic and to the Study of Effective Operations in Arithmetic and Logic", D. Reidel, 1962 (Semantic Tableaux proof methods).
- Blackburn, P.; van Benthem, J.; and Wolter, Frank; Eds. (2006) Handbook of Modal Logic. North Holland.
- Blackburn, Patrick; de Rijke, Maarten; and Venema, Yde (2001) Modal Logic. Cambridge University Press. {{ISBN|0-521-80200-8}}
- 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: 136â€“58. {{ISBN|0-631-20693-0}}
- Fitting, Melvin; and Mendelsohn, R. L. (1998) First Order Modal Logic. Kluwer. {{ISBN|0-7923-5335-8}}
- James Garson (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.
- â€”â€” (1993) Mathematics of Modality, CSLI Lecture Notes No. 43. University of Chicago Press.
- â€”â€” (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.
- GorÃ©, Rajeev (1999) "Tableau Methods for Modal and Temporal Logics" in D'Agostino, M.; Gabbay, D.; Haehnle, R.; and Posegga, J.; Eds., Handbook of Tableau Methods. Kluwer: 297â€“396.
- Hughes, G. E., and Cresswell, M. J. (1996) A New Introduction to Modal Logic. Routledge. {{ISBN|0-415-12599-5}}
- JÃ³nsson, B. and Tarski, A., 1951â€“52, "Boolean Algebra with Operators I and II", American Journal of Mathematics 73: 891â€“939 and 74: 129â€“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.
- Snyder, D. Paul "Modal Logic and its applications", Van Nostrand Reinhold Company, 1971 (proof tree methods).
- Zeman, J. J. (1973) Modal Logic. Reidel. Employs Polish notation.
- History of logic, EncyclopÃ¦dia Britannica.
Further reading
- Ruth Barcan Marcus, Modalities, OUP 1993.
- D.M. Gabbay, A. Kurucz, F. Wolter and M. Zakharyaschev, Many-Dimensional Modal Logics: Theory and Applications, Elsevier, Studies in Logic and the Foundations of Mathematics, volume 148, 2003, {{ISBN|0-444-50826-0}}. Covers many varieties of modal logics, e.g. temporal, epistemic, dynamic, description, spatial from a unified perspective with emphasis on computer science aspects, e.g. decidability and complexity.
- Andrea Borghini, A Critical Introduction to the Metaphysics of Modality, New York, Bloomsbury, 2016.
External links
- Internet Encyclopedia of Philosophy:
- "Modal Logic: A Contemporary View" â€“ by Johan van Benthem.
- "Rudolf Carnap's Modal Logic" â€“ by MJ Cresswell.
- Stanford Encyclopedia of Philosophy:
- "Modal logic" â€“ by James Garson.
- "Provability Logic" â€“ by Rineke Verbrugge.
- Edward N. Zalta, 1995, "Basic Concepts in Modal Logic."
- John McCarthy, 1996, "Modal Logic."
- Molle a Java prover for experimenting with modal logics
- Suber, Peter, 2002, "Bibliography of Modal Logic."
- List of Logic Systems List of many modal logics with sources, by John Halleck.
- Advances in Modal Logic. Biannual international conference and book series in modal logic.
- S4prover A tableaux prover for S4 logic
- "Some Remarks on Logic and Topology" â€“ by Richard Moot; exposits a topological semantics for the modal logic S4.
- LoTREC The most generic prover for modal logics from IRIT/Toulouse University
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