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"System T" redirects here. For the system used with the programming language, see Typed lambda calculus. A

In formal logic, a modal logic is any system of formal logic that attempts to deal with modalities. Mathematical logic is a subfield of Logic and Mathematics with close connections to Computer science and Philosophical logic. Logic is the study of the principles of valid demonstration and Inference. In Linguistics, modals are expressions broadly associated with notions of Possibility and Necessity. Traditionally, there are three 'modes' or 'moods' or 'modalities' of the copula to be, namely, possibility, probability, and necessity. In common usage existence is the world of which we are aware through our senses but in Philosophy the word has a more specialized meaning and is often contrasted with A logically possible Proposition is one that can be asserted without implying a logical Contradiction. Probability is the likelihood or chance that something is the case or will happen In Criminal law, necessity may be either a possible justification or an exculpation for breaking the Law. Logics for dealing with a number of related terms, such as eventually, formerly, can, could, might, may, must, are by extension also called modal logics, since it turns out that these can be treated in similar ways.

A formal modal logic represents modalities using modal operators. In Modal logic, a modal operator is an Operator which forms Propositions from propositions For example, "Jones's murder was a possibility", "Jones was possibly murdered", and "It is possible that Jones was murdered" all contain the notion of possibility. In a modal logic this is represented as an operator, Possibly, attaching to the sentence Jones was murdered.

The basic unary (1-place) modal operators are usually written \Box (or L) for Necessarily and \Diamond (or M) 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 Jones was murdered if and only if it is not necessary that Jones was not murdered. In Modal logic, a classical modal logic L is any modal logic containing (as axiom or theorem \Diamond A \equiv \lnot\Box\lnot A and being closed In Logic and Mathematics, negation or not is an operation on Logical values for example the logical value of a Proposition For the standard formal semantics of the basic modal language, see Kripke semantics. Kripke semantics (also known as relational semantics or frame semantics, and often confused with Possible world semantics) is a formal Semantics

Contents

Brief history

The founder of modern formal logic, Gottlob Frege, doubted that modal logic was viable, and he discounted it. Friedrich Ludwig Gottlob Frege ( 8 November 1848, Wismar, Grand Duchy of Mecklenburg-Schwerin  &ndash 26 July 1925 Two of his well-known readers, Rudolph Carnap and Kurt Gödel (1933) broke with Frege on this topic, and chose to pursue the mathematical structure of a logic that deals with the three classic modes. Rudolf Carnap ( May 18, 1891 &ndash September 14, 1970) was an influential German -born philosopher who was active in Kurt Gödel (kʊɐ̯t ˈgøːdl̩ (April 28 1906 – January 14 1978 was an Austrian American Logician, Mathematician and Philosopher Friedrich Ludwig Gottlob Frege ( 8 November 1848, Wismar, Grand Duchy of Mecklenburg-Schwerin  &ndash 26 July 1925 In 1937, Robert Feyes, following Gödel, proposed System T modal logic. A modal logic is any system of formal logic that attempts to deal with modalities. In 1951, Georg Henrik von Wright proposed System M, which is an elaboration on System T. Georg Henrik von Wright (pronounced roughly fon vrikt, IPA hɛnrik fɔn-vrikt ( June 14, 1916 &ndash June 16, 2003) was Also in the 1950s, C.I. Lewis built upon System M to construct his well-known modal systems S1, S2, S3, S4 and S5. Clarence Irving Lewis ( April 12, 1883 Stoneham Massachusetts - February 3, 1964 Cambridge Massachusetts) usually By 1965 Saul Kripke solidly established System K, which is the form of modal logic that most scholars use today. Saul Aaron Kripke (born on November 13, 1940 in Bay Shore New York) is an American philosopher and Logician now Emeritus

Alethic modalities

Modalities of necessity and possibility are called alethic modalities. They are also sometimes called special modalities, from the Latin species. Latin ( lingua Latīna, laˈtiːna is an Italic language, historically spoken in Latium and Ancient Rome. 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.

A proposition is said to be

Clearly if we wish the definitions of these notions to be non-circular, we need to take either possibility or necessity as primitive, or further analyze these notions in terms of others that include neither possibility nor necessity, and which are themselves non-circularly defined.


Physical possibility

Something is physically possible if it is permitted by the laws of nature. A physical law or scientific law is a Scientific generalization based on empirical Observations of physical behavior (i For example, it is possible for there to be an atom with an atomic number of 150, though there may not in fact be one. History See also Atomic theory, Atomism The concept that matter is composed of discrete units and cannot be divided into arbitrarily tiny See also List of elements by atomic number In Chemistry and Physics, the atomic number (also known as the proton On the other hand, it is not possible, in this sense, for there to be an element whose nucleus contains cheese. Cheese is a Food made from Milk, usually the milk of cows, Buffalo, Goats or sheep, by coagulation. While it is logically possible to accelerate beyond the speed of light, it is not, according to modern science, physically possible for objects with mass.

Metaphysical possibility

Philosophers ponder the properties objects have independently of those dictated by scientific laws. Philosophy is the study of general problems concerning matters such as existence knowledge truth beauty justice validity mind and language 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 exist). For other uses see Time (disambiguation Time is a component of a measuring system used to sequence events to compare the durations of God is the principal or sole Deity in Religions and other belief systems that worship one deity. Saul Kripke has argued that every person necessarily has the parents they do have: anyone with different parents wouldn't be the same person. Saul Aaron Kripke (born on November 13, 1940 in Bay Shore New York) is an American philosopher and Logician now Emeritus

Metaphysical possibility is generally thought to be stronger 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. Thus, "It is possible that bigfoot exists" might mean either "It would be possible for such a creature as a bigfoot to exist", or (more likely), "For all I know, bigfoot exists" (It's compatible with what I know that bigfoot exists).

In the former case, the speaker might know that there are not any bigfoots, but is saying that (unlike round squares), there could be some – the existence of bigfoot is not impossible. In the latter case he is saying that there may well be some "right now".

Epistemic logic

Epistemic modalities (from the Greek episteme, knowledge), deal with the certainty of sentences. The operators are translated as "It is certainly true that. . . " and "It may (given the available information) be true that. . . " In ordinary speech both 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". Bigfoot or Sasquatch is alleged to be an Ape -like creature inhabiting remote forests mainly in the Pacific Northwest region of the United States and Canada 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". Goldbach's conjecture is one of the oldest unsolved problems in Number theory and in all of Mathematics. 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.

Temporal logic

There are several analogous modes of speech, which though less likely to be confused with alethic modalities are still closely related. One is talk of time. It seems reasonable to say that possibly it will rain tomorrow, and possibly it won't; on the other hand, if it rained yesterday, if it really already did so, then it cannot be quite correct to say "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. In Philosophy and Logic, accidental necessity, often stated in its Latin form necessitas per accidens, refers to the Necessity attributed

A standard method for formalizing talk of time is to use two pairs of operators, one for the past and one for the future. For the past, let "It has always been the case that. . . " be equivalent to the box, and let "It was once the case that. . . " be equivalent to the diamond. For the future, let "It will always be the case that. . . " be equivalent to the box, and let "it will eventually be the case that. . . " be equivalent to the diamond. If these two systems are used together, it will, obviously, be necessary to indicate, as by subscripts, which box is which. This article is about the terms 'subscript' and 'superscript' as used in typography

Additional binary operators are also relevant to temporal logics, q. v. Linear Temporal Logic. Linear temporal logic (LTL is a modal Temporal logic with modalities referring to time

Deontic logic

Likewise talk of morality, or of obligation and norms generally, seems to have a modal structure. An obligation is a requirement to take some course of action whether legal or moral. Norms are sentences or sentence meanings with practical i e action-oriented (rather than descriptive explanatory or expressive import the most common of which 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 logic is the field of Logic that is concerned with Obligation, Permission, and related concepts

Doxastic logic

Main article: Doxastic logic

Doxastic logic is a modal logic that is concerned with reasoning about beliefs. Doxastic logic is a Modal logic that is concerned with Reasoning about Beliefs The term doxastic is derived from the Ancient Greek Reasoning is the cognitive process of looking for Reasons for beliefs conclusions actions or feelings Belief is the psychological state in which an individual holds a Proposition or Premise to be true The term doxastic is derived from the ancient Greek doxa which means 'belief. The Ancient Greek language is the historical stage in the development of the Hellenic language family spanning the Archaic (c ' Typically, a doxastic logic uses Bx to mean "It is believed that x is the case" and the set \mathbb{B} denotes a set of beliefs.

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" or non-truth-functional sentential operators) that make them all varieties of the same thing. Epistemic logic is arguably best captured in the system "S4"; deontic logic in the system "D", temporal logic in "T" and alethic logic arguably with "S5". Epistemic logic is a subfield of Modal logic that is concerned with reasoning about Knowledge. D is the fourth letter in the Latin alphabet. Its name in English is spelled dee or occasionally de (diː In Logic, the term temporal logic is used to describe any system of rules and symbolism for representing and reasoning about propositions qualified in terms of Time T is the twentieth letter in the modern Latin alphabet. Its name in English is spelled tee or occasionally te (tiː In Logic and Philosophy, S5 is one of five systems of Modal logic proposed by Clarence Irving Lewis and Cooper Harold Langford in

Interpretations of modal logic

Further information: Interpretation (logic)

In the most common interpretation of modal logic, one considers "all logically possible worlds". In Logic an interpretation gives meaning to an artificial or Formal language or to a sentence of such a language by assigning a denotation (extension A logically possible Proposition is one that can be asserted without implying a logical Contradiction. 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. For example, 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 modal claims commits us to. Are we really alleging the existence of possible worlds, every bit as real as our actual world, just not actual? David Lewis 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). David Kellogg Lewis ( September 28, 1941  &ndash October 14, 2001) is considered to have been one of the leading philosophers of the latter In Linguistics and in Philosophy of language, an indexical behavior or utterance symbolically points to (or indicates) some state of affairs That position is a major tenet of "modal realism". Modal realism is the view notably propounded by David Lewis, that Possible worlds are as real as the actual world Most philosophers decline to endorse such a view, considering it ontologically extravagant, and preferring to seek various ways to paraphrase away the ontological commitments implied by our modal claims.

Formal rules

Many systems of modal logic, with widely varying properties, have been proposed since C. I. Lewis began working in the area in 1910. Clarence Irving Lewis ( April 12, 1883 Stoneham Massachusetts - February 3, 1964 Cambridge Massachusetts) usually 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". This is a technical mathematical article about the area of mathematical logic variously known as "propositional calculus" or "propositional logic" The notation of Lewis, much employed since, denotes "necessarily p" by a prefixed "box" ( \Box p ) whose scope is established by parentheses. Clarence Irving Lewis ( April 12, 1883 Stoneham Massachusetts - February 3, 1964 Cambridge Massachusetts) usually Likewise, a prefixed "diamond" (\Diamond p) denotes "possibly p". Regardless of notation, each of these operators is definable in terms of the other:

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". In Logic, De Morgan's laws or De Morgan's theorem are rules in Formal logic relating pairs of dual Logical operators in a systematic manner expressed Boolean algebra (or Boolean logic) is a logical calculus of truth values, developed by George Boole in the late 1830s
"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. This is a technical mathematical article about the area of mathematical logic variously known as "propositional calculus" or "propositional logic" Many modal logics, known collectively as normal modal logics, include the following rule and axiom:

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. In Logic, a normal Modal logic is a set L of modal formulas such that L contains All propositional tautologies; Saul Aaron Kripke (born on November 13, 1940 in Bay Shore New York) is an American philosopher and Logician now Emeritus This is a technical mathematical article about the area of mathematical logic variously known as "propositional calculus" or "propositional logic" 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:

Other well-known elementary axioms are:

These axioms yield the systems:

K through S5 form a nested hierarchy of systems, making up the core of normal modal logic. In Logic, a normal Modal logic is a set L of modal formulas such that L contains All propositional tautologies; D is primarily of interest to those exploring the deontic interpretation of modal logic. Deontic logic is the field of Logic that is concerned with Obligation, Permission, and related concepts

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.

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. Aristotle (Greek Aristotélēs) (384 BC – 322 BC was a Greek philosopher a student of Plato and teacher of Alexander the Great. A syllogism, or logical appeal, (συλλογισμός &mdash "conclusion" "inference" (usually the categorical syllogism) is a kind of The problem of future contingents is a Logical Paradox first posed by Diodorus Cronus from the Megarian school of philosophy, under the name Aristotle 's De Interpretatione (the Latin title by which it is usually known or On Interpretation ( Greek Περὶ Ἑρμηνείας 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. Scholasticism was the dominant form of theology and philosophy in the Latin West in the Middle Ages, particularly in the 12th 13th and 14th centuries William of Ockham (also Occam, Hockham, or any of several other spellings ˈɒkəm (c In Philosophy, essence is the attribute or set of attributes that make an object or substance what it fundamentally is and which it has by necessity The philosophical term accident has been employed throughout the history of philosophy with several distinct meanings

C. I. Lewis founded modern modal logic in his 1910 Harvard thesis and in a series of scholarly articles beginning in 1912. Clarence Irving Lewis ( April 12, 1883 Stoneham Massachusetts - February 3, 1964 Cambridge Massachusetts) usually 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. Saul Aaron Kripke (born on November 13, 1940 in Bay Shore New York) is an American philosopher and Logician now Emeritus Kripke semantics (also known as relational semantics or frame semantics, and often confused with Possible world semantics) is a formal Semantics These are commonly referred to as "possible worlds" semantics. Kripke and A. N. Prior had previously corresponded at some length. Arthur Norman Prior (1914 Masterton, New Zealand – 1969 Trondheim, Norway) was a noted logician.

A. N. Prior created temporal logic, closely related to modal logic, in 1957 by adding modal operators [F] and [P] meaning "henceforth" and "hitherto". Arthur Norman Prior (1914 Masterton, New Zealand – 1969 Trondheim, Norway) was a noted logician. In Logic, the term temporal logic is used to describe any system of rules and symbolism for representing and reasoning about propositions qualified in terms of Time Vaughan Pratt introduced dynamic logic in 1976. Vaughan Ronald Pratt (born 1944 a Professor Emeritus at Stanford University, was one of the earliest pioneers in the field of Computer science. In 1977, Amir Pnueli proposed using temporal logic to formalise the behaviour of continually operating concurrent programs. Amir Pnueli (אמיר פנואלי born April 22, 1941) is an Israeli Computer scientist who received the Turing Award in 1996 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. Linear temporal logic (LTL is a modal Temporal logic with modalities referring to time Computation tree logic (CTL is a branching-time logic, meaning that its model of time is a tree-like structure in which the future is not determined there are different paths The Hennessy - Milner logic is a Temporal logic in computer science

The mathematical structure of modal logic, namely Boolean algebras augmented with unary operations (often called "modal algebras"), began to emerge with J. In Abstract algebra, a Boolean algebra or Boolean lattice is a complemented distributive lattice. In Mathematics, a unary operation is an operation with only one Operand, i 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). Alfred Tarski ( January 14, 1901, Warsaw, Russian ruled Poland – October 26, 1983, Berkeley California Bjarni Jónsson (born 1920 is an Icelandic Mathematician and Logician working in Universal algebra and Lattice theory. 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. In Abstract algebra, an interior algebra is a certain type of Algebraic structure that encodes the idea of the topological Interior of a set In Mathematics, the interior of a set S consists of all points of S that are intuitively "not on the edge of S " A closure operator on a set S is a function cl P ( S) → P ( S) from the Power set of S Topology ( Greek topos, "place" and logos, "study" is the branch of Mathematics that studies the properties of Texts on modal logic typically do little more than mention its connections with the study of Boolean algebras and topology. In Abstract algebra, a Boolean algebra or Boolean lattice is a complemented distributive lattice. Topology ( Greek topos, "place" and logos, "study" is the branch of Mathematics that studies the properties of For a thorough survey of the history of formal modal logic and of the associated mathematics, see Goldblatt (2006).

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Acknowledgements

This article includes material from the Free On-line Dictionary of Computing, used with permission under the GFDL. The Free On-line Dictionary of Computing ( FOLDOC) is an online searchable encyclopedic Dictionary of Computing subjects The GNU Free Documentation License ( GNU FDL or simply GFDL) is a Copyleft License for free documentation designed by the Free Software

Dictionary

modal logic

-noun

  1. (logic) Any formal system that attempts to deal with modalities, such as possibility and necessity, but also obligation and permission.
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