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Encyclopedia > Modal logic

In formal logic, a modal logic is any system of formal logic that attempts to deal with modalities. Traditionally, there are three 'modes' or 'moods' or 'modalities' of the copula to be, namely, possibility, probability, and necessity. 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. Typed versions of the lambda calculus extend the standard lambda calculus with types. ... Logic (from ancient Greek λόγος (logos), meaning reason) is the study of arguments. ... Logic (from Classical Greek λόγος logos; meaning word, thought, idea, argument, account, reason, or principle) is the study of the principles and criteria of valid inference and demonstration. ... A modal form is a provision of syntax that indicates the predication of an action, attitude, condition, or state other than that of a simple declaration of fact. ... For other uses, see Copula (disambiguation). ... For the philosophical movement, see Existentialism. ... 1+1=3 and 0=1 are false in all possible worlds. ... Probability is the likelihood or chance that something is the case or will happen. ... This article is about the law definition of necessity. ...


A formal modal logic represents modalities using modal operators. 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. A modal operator is a logical connective, in the language of a modal logic, which forms propositions from propositions. ...


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: Negation (i. ...

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. For the standard formal semantics of the basic modal language, see Kripke semantics. Kripke semantics (also known as possible world semantics, relational semantics, or frame semantics) is a formal semantics for modal logic systems, created in late 1950s and early 1960s by Saul Kripke. ...

Contents

Brief history

The founder of modern formal logic, Gottlob Frege, doubted that modal logic was viable, and he discounted it. 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. In 1937, Robert Feyes, following Gödel, proposed System T modal logic. In 1951, Georg Henrik von Wright proposed System M, which is an elaboration on System T. Also in the 1950s, C.I. Lewis built upon System M to construct his well-known modal systems S1, S2, S3, S4 and S5. By 1965 Saul Kripke solidly established System K, which is the form of modal logic that most scholars use today. Friedrich Ludwig Gottlob Frege (8 November 1848, Wismar – 26 July 1925, IPA: ) was a German mathematician who became a logician and philosopher. ... Rudolf Carnap (May 18, 1891 - September 14, 1970) was a German philosopher. ... Kurt Gödel (IPA: ) (April 28, 1906 Brünn, Austria-Hungary (now Brno, Czech Republic) – January 14, 1978 Princeton, New Jersey) was an Austrian American mathematician and philosopher. ... Friedrich Ludwig Gottlob Frege Friedrich Ludwig Gottlob Frege (November 8, 1848 - July 26, 1925) was a German mathematician, logician, and philosopher who is regarded as a founder of both modern mathematical logic and analytic philosophy. ... Statue of Georg Henrik von Wright in University of Helsinki Georg Henrik von Wright (pronounced, roughly, fon vrikt, IPA: [je:É”rj hÉ›n:rik fÉ”n-vrik:t],) (June 14, 1916 – June 16, 2003) was a Finnish philosopher, who succeeded Ludwig Wittgenstein as professor at the University of Cambridge. ... Clarence Irving Lewis (April 12, 1883 - February 3, 1964) was a pragmatist philosopher. ... Saul Aaron Kripke (born in November 13, 1940 in Bay Shore, New York) is an American philosopher and logician now emeritus from Princeton and teaches as distinguished professor of philosophy at CUNY Graduate Center. ...


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. For other uses, see Latins and Latin (disambiguation). ...


A proposition is said to be

  • possible if it is not necessarily false (regardless of whether it actually is true or false);
  • necessary if it is not possibly false;
  • contingent if it is possibly true and possibly false.

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. 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. On the other hand, it is not possible, in this sense, for there to be an element whose nucleus contains cheese. 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. For a list of set rules, see Laws of science. ... For other uses, see Atom (disambiguation). ... See also: List of elements by atomic number In chemistry and physics, the atomic number (also known as the proton number) is the number of protons found in the nucleus of an atom. ... Cheese is a solid food made from the milk of cows, goats, sheep, and other mammals. ... A line showing the speed of light on a scale model of Earth and the Moon, taking about 1â…“ seconds to traverse that distance. ...


Metaphysical possibility

Philosophers ponder the properties 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 exist). Saul Kripke has argued that every person necessarily has the parents they do have: anyone with different parents wouldn't be the same person. A philosopher is a person devoted to studying and producing results in philosophy. ... This article is about the concept of time. ... This article is about the term God in the context of monotheism and henotheism. ... Saul Aaron Kripke (born in November 13, 1940 in Bay Shore, New York) is an American philosopher and logician now emeritus from Princeton and teaches as distinguished professor of philosophy at CUNY Graduate Center. ...


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". 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)). It has been suggested that Evidence regarding Bigfoot be merged into this article or section. ...


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. Goldbachs conjecture is one of the oldest unsolved problems in number theory and in all of mathematics. ...


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 to the past by certain views of time. ...


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. A subscript is a number, figure, or indicator that appears below the normal line of type, typically used in a formula, mathematical expression, or description of a chemical compound. ...


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. 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". An obligation can be legal or moral. ... The introduction to this article provides insufficient context for those unfamiliar with the subject matter. ... 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. The term doxastic is derived from the ancient Greek doxa which means 'belief.' 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. doxastic logic is a modal logic that is concerned with reasoning about beliefs. ... Reasoning is the mental (cognitive) process of looking for reasons to support beliefs, conclusions, actions or feelings. ... For other uses, see Believe. ... Beginning of Homers Odyssey The Ancient Greek language is the historical stage of the Greek language[1] as it existed during the Archaic (9th–6th centuries BC) and Classical (5th–4th centuries BC) periods in Ancient Greece. ...


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". Michaels the greatest boyfriend in the whole wide world, and Id love to call him in a phonebooth sometime. ... For other uses, see D (disambiguation). ... 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. ... For other uses, see T (disambiguation). ... S5 is one of the first modal logics; it was introduced by Clarence Irving Lewis in 1932. ...


Interpretations of modal logic

Further information: Interpretation (logic)

In the most common interpretation of modal logic, one considers "all 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. Philosophers generally consider logical possibility to be the broadest sort of subjunctive possibility in modal logic. ... Possible Worlds is: Possible Worlds (play) a play by John Mighton Possible Worlds (poetry book) a book of poems by Peter Porter (poet) Possible Worlds (book) a book by J. B. S. Haldane This is a disambiguation page: a list of articles associated with the same title. ...


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). That position is a major tenet of "modal realism". 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. David K. Lewis David Kellogg Lewis (September 28, 1941 – October 14, 2001) is considered to have been one of the leading analytic philosophers of the latter half of the 20th century. ... In the philosophy of language, an indexical behavior or utterance is one whose meaning varies according to certain features of the context in which it is uttered. ... Modal realism is the view, notably propounded by David Lewis, that possible worlds are as real as the actual world. ...


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. Hughes and Cresswell (1996), for example, describe 42 normal and 25 non-normal modal logics. Zeman (1973) describes some systems Hughes and Cresswell omit. Clarence Irving Lewis (April 12, 1883 _ February 3, 1964) was a pragmatist philosopher. ...


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: In logic and mathematics, a propositional calculus (or a sentential calculus) is a formal system in which formulas representing propositions can be formed by combining atomic propositions using logical connectives, and a system of formal proof rules allows to establish that certain formulas are theorems of the formal system. ... Clarence Irving Lewis (April 12, 1883 Stoneham, Massachusetts - February 3, 1964 Cambridge, Massachusetts) was an American academic philosopher. ...

  • 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. Look up duality in Wiktionary, the free dictionary. ...


In many modal logics, the necessity and possibility operators satisfy the following analogs of de Morgan's laws from Boolean algebra: note that demorgans laws are also a big part in circut design. ... Boolean algebra is the finitary algebra of two values. ...

"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. Many modal logics, known collectively as normal modal logics, include the following rule and axiom: In logic and mathematics, a propositional calculus (or a sentential calculus) is a formal system in which formulas representing propositions can be formed by combining atomic propositions using logical connectives, and a system of formal proof rules allows to establish that certain formulas are theorems of the formal system. ... In logic, normal modal logic is a set L of modal formulas such that L contains all propositional tautologies, Kripkes schema: , and L is closed under substitution, detachment rule: from A and A→B infer B, necessitation rule: from A infer . ...

  • 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. A mathematical picture paints a thousand words: the Pythagorean theorem. ... In logic, normal modal logic is a set L of modal formulas such that L contains all propositional tautologies, Kripkes schema: , and L is closed under substitution, detachment rule: from A and A→B infer B, necessitation rule: from A infer . ... Saul Aaron Kripke (born in November 13, 1940 in Bay Shore, New York) is an American philosopher and logician now emeritus from Princeton and teaches as distinguished professor of philosophy at CUNY Graduate Center. ... In logic and mathematics, a propositional calculus (or a sentential calculus) is a formal system in which formulas representing propositions can be formed by combining atomic propositions using logical connectives, and a system of formal proof rules allows to establish that certain formulas are theorems of the formal system. ...


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
  • E:  Diamond p rightarrow Box Diamond p.

These axioms yield the systems:

  • K := K + N
  • T := K + T
  • S4 := T + 4
  • S5 := S4 + B or T + E
  • D := K + D.

K through S5 form a nested hierarchy of systems, making up the core of normal modal logic. D is primarily of interest to those exploring the deontic interpretation of modal logic. In logic, normal modal logic is a set L of modal formulas such that L contains all propositional tautologies, Kripkes schema: , and L is closed under substitution, detachment rule: from A and A→B infer B, necessitation rule: from A infer . ... 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. 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. For other uses, see Aristotle (disambiguation). ... Wikipedia does not yet have an article with this exact name. ... The problem of the futures contingents designs a logical paradox first posed by Diodorus Cronus from the Megarian school of philosophy, under the name of the dominator, and then reactualized by Aristotle in chapter 9 of De Interpretatione. ... De Interpretatione or Hermeneutics (Peri Hermeneias) is a work of the ancient Greek philosopher Aristotle, mainly on the philosophy of language. ... Scholastic redirects here. ... William of Ockham (also Occam or any of several other spellings, IPA: ) (c. ... John Duns Scotus (c. ... For other uses, see Essence (disambiguation). ... It has been suggested that this article or section be merged with Accidental property. ...


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. Clarence Irving Lewis (April 12, 1883 _ February 3, 1964) was a pragmatist philosopher. ... Saul Aaron Kripke (born in November 13, 1940 in Bay Shore, New York) is an American philosopher and logician now emeritus from Princeton and teaches as distinguished professor of philosophy at CUNY Graduate Center. ... Harvard redirects here. ... Kripke semantics (also known as possible world semantics, relational semantics, or frame semantics) is a formal semantics for modal logic systems, created in late 1950s and early 1960s by Saul Kripke. ... Arthur (A.N.) Prior (1914-1969) was one of the foremost logicians of the twentieth century. ...


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. Arthur (A.N.) Prior (1914-1969) was one of the foremost logicians of the twentieth century. ... 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 is Professor Emeritus of Computer Science at Stanford University. ... Dynamic logic may mean: In modal logic: Dynamic logic is used in the context of Artificial Intelligence. ... Amir Pnueli (born April 22, 1941) is an Israeli computer scientist who received the Turing Award in 1996 for seminal work introducing temporal logic into computing science and for outstanding contributions to program and systems verification. ... Linear temporal logic (LTL) is a modal temporal logic with modalities referring to time. ... Computational 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 in the future, any one of which might be actual path that is realised. ...


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, 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 algebras and topology. For a thorough survey of the history of formal modal logic and of the associated mathematics, see Goldblatt (2006). In abstract algebra, a Boolean algebra is an algebraic structure (a collection of elements and operations on them obeying defining axioms) that captures essential properties of both set operations and logic operations. ... In mathematics, a unary operation is an operation with only one operand. ... // Alfred Tarski (January 14, 1902, Warsaw, Russian-ruled Poland – October 26, 1983, Berkeley, California) was a logician and mathematician who spent four decades as a professor of mathematics at the University of California, Berkeley. ... Bjarni Jónsson is a mathematician and logician working in universal algebra and lattice theory. ... In abstract algebra, an interior algebra is an algebraic structure of the signature <A, ·, +, , 0, 1, I> where <A, ·, +, , 0, 1> is a Boolean algebra and I is a unary operator, the interior operator, satisfying the identities: xI ≤ x xII = xI (xy)I = xIyI 1I = 1 xI is called the... In mathematics, the interior of a set S consists of all points which are intuitively not on the edge of S. A point which is in the interior of S is an interior point of S. The notion of interior is in many ways dual to the notion of closure. ... In mathematics, given a partially ordered set (P, ≤), a closure operator on P is a function C : P → P with the following properties: x ≤ C(x) for all x, i. ... For other uses, see Topology (disambiguation). ... In abstract algebra, a Boolean algebra is an algebraic structure (a collection of elements and operations on them obeying defining axioms) that captures essential properties of both set operations and logic operations. ... For other uses, see Topology (disambiguation). ...


References

  • Blackburn, Patrick, Maarten de Rijke, and Yde Venema (2001) Modal Logic. Cambridge Univ. Press. ISBN 0-521-80200-8
  • Blackburn, P., van Benthem, J., and Frank Wolter, eds. (2006) Handbook of Modal Logic. North Holland.
  • Chagrov, Aleksandr, and Michael Zakharyaschev (1997) Modal Logic. Oxford Univ. Press. ISBN 0-19-853779-4
  • Chellas, B. F. (1980) Modal Logic: An Introduction. Cambridge Univ. 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 R.L. Mendelsohn (1998) First Order Modal Logic. Kluwer. ISBN 0-7923-5335-8
  • Garson, James W. (2006) Modal Logic for Philosophers. Cambridge Univ. 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., Dov Gabbay, R. Haehnle, and J. Posegga, eds., Handbook of Tableau Methods. Kluwer: 297-396.
  • Hughes, G. E., and M.J. Cresswell (1996) A New Introduction to Modal Logic. Routledge. ISBN 0-415-12599-5
  • Jónsson, B. and Alfred Tarski, 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 Dana Scott) (1977) An Introduction to Modal Logic, American Philosophical Quarterly Monograph Series, no. 11 (Krister Segerberg, series ed.). Basil Blackwell.

Free and online: Johan (Johannes Franciscus Abraham Karel) van Benthem (June 12, 1949-) is a University Professor (Universiteitshoogleraar) of logic at the Universiteit van Amsterdam (in the ILLC) and professor of philosophy at Stanford University (in the CSLI). ... Analytic tableaux, or, more briefly, just tableaux, are a fundamental concept in automated theorem proving. ... Bjarni Jónsson is a mathematician and logician working in universal algebra and lattice theory. ... // Alfred Tarski (January 14, 1902, Warsaw, Russian-ruled Poland – October 26, 1983, Berkeley, California) was a logician and mathematician who spent four decades as a professor of mathematics at the University of California, Berkeley. ... Edward John Lemmon (1 June 1930-29 July 1966) was a logician and philosopher born in Sheffield, UK. He is most well known for his work on modal logic, particularly his joint text with Dana Scott published posthumously (Lemmon and Scott, 1977). ... Dana Stewart Scott (born 1932) is the emeritus Hillman University Professor of Computer Science, Philosophy, and Mathematical Logic at Carnegie Mellon University; he is now retired and lives in Berkeley, California. ...

Polish notation, less frequently known as Prefix notation, is a form of notation for logic, arithmetic, and algebra. ...

See also

Accessibility relation is a binary relation R between possible worlds which has very powerful uses in both the formal/theoretical aspects of modal logic as well as in its applications to things like epistemology, metaphysics, and value theory. ... Counterpart theory (hereafter CT) is a theoretical framework that uses the counterpart relation (hereafter C-relation) as a replacement for the identity relation between objects in different possible world/times/spaces. ... De dicto and de re are two phrases used to mark important distinctions in intensional statements, associated with the intensional operators in many such statements. ... Description logics (DL) are a family of knowledge representation languages which can be used to represent the terminological knowledge of an application domain in a structured and formally well-understood way. ... doxastic logic is a modal logic that is concerned with reasoning about beliefs. ... Dynamic logic is an extension of modal logic originally intended for reasoning about computer programs and later applied to more general complex behaviors arising in linguistics, philosophy, AI, and other fields. ... Michaels the greatest boyfriend in the whole wide world, and Id love to call him in a phonebooth sometime. ... Hybrid logic refers to a number of extensions to modal logic with more expressive power, though still less than first-order logic. ... In abstract algebra, an interior algebra is an algebraic structure of the signature <A, ·, +, , 0, 1, I> where <A, ·, +, , 0, 1> is a Boolean algebra and I is a unary operator, the interior operator, satisfying the identities: xI ≤ x xII = xI (xy)I = xIyI 1I = 1 xI is called the... Interpretability logics comprise a family of modal logics that extend provability logic to describe interpretability and/or various related metamathematical properties and relations such as weak interpretability, Π1-conservativity, cointerpretability, tolerance, cotolerance, arithmetic complexities. ... Kripke semantics (also known as possible world semantics, relational semantics, or frame semantics) is a formal semantics for modal logic systems, created in late 1950s and early 1960s by Saul Kripke. ... Possible Worlds is: Possible Worlds (play) a play by John Mighton Possible Worlds (poetry book) a book of poems by Peter Porter (poet) Possible Worlds (book) a book by J. B. S. Haldane This is a disambiguation page: a list of articles associated with the same title. ... The problem of the futures contingents designs a logical paradox first posed by Diodorus Cronus from the Megarian school of philosophy, under the name of the dominator, and then reactualized by Aristotle in chapter 9 of De Interpretatione. ... Provability logic, or the logic of provability, is a modal logic where the necessity operator is interpreted as provability in a reasonably rich formal theory such as Peano arithmetic. ... Two dimensionalism is an explanatory approach in analytic philosophy. ... A modal verb (also modal, modal auxiliary verb, modal auxiliary) is a type of auxiliary verb that is used to indicate modality. ...

Notes

External links

The Stanford Encyclopedia of Philosophy (hereafter SEP) is a free online encyclopedia of philosophy run and maintained by Stanford University. ... Edward N. Zalta is a Senior Research Scholar at the Center for the Study of Language and Information. ... John McCarthy (born September 4, 1927, in Boston, Massachusetts, sometimes known affectionately as Uncle John McCarthy), is a prominent computer scientist who received the Turing Award in 1971 for his major contributions to the field of Artificial Intelligence. ...

Acknowledgements

This article includes material from the Free On-line Dictionary of Computing, used with permission under the GFDL. This article does not cite any references or sources. ... GFDL redirects here. ...

Logic Portal
Image File history File links Portal. ... Logic (from Classical Greek λόγος logos; meaning word, thought, idea, argument, account, reason, or principle) is the study of the principles and criteria of valid inference and demonstration. ... The history of logic documents the development of logic as it occurs in various rival cultures and traditions in history. ... In Islamic philosophy, logic played an important role. ... For other uses, see Reason (disambiguation). ... Philosophical logic is the application of formal logical techniques to problems that concern philosophers. ... Philosophy of logic is the branch of philosophy that is concerned with the nature and justification of systems of logic. ... Mathematical logic is a major area of mathematics, which grew out of symbolic logic. ... The metalogic of a system of logic is the formal proof supporting its soundness. ... To meet Wikipedias quality standards, this article or section may require cleanup. ... Reasoning is the mental (cognitive) process of looking for reasons to support beliefs, conclusions, actions or feelings. ... Deductive reasoning is reasoning whose conclusions are intended to necessarily follow from its premises. ... Aristotle appears first to establish the mental behaviour of induction as a category of reasoning. ... Abduction, or inference to the best explanation, is a method of reasoning in which one chooses the hypothesis that would, if true, best explain the relevant evidence. ... Informal logic is the study of arguments as presented in ordinary language, as contrasted with the presentations of arguments in an artificial (technical) or formal language (see formal logic). ... This article is about the word proposition as it is used in logic, philosophy, and linguistics. ... Inference is the act or process of deriving a conclusion based solely on what one already knows. ... Look up argument in Wiktionary, the free dictionary. ... In logic, the form of an argument is valid precisely if it cannot lead from true premises to a false conclusion. ... An argument is cogent if and only if the truth of the arguments premises would render the truth of the conclusion probable (i. ... Traditional logic, also known as term logic, is a loose term for the logical tradition that originated with Aristotle and survived broadly unchanged until the advent of modern predicate logic in the late nineteenth century. ... are you kiddin ? i was lookin for it for hours ... Look up fallacy in Wiktionary, the free dictionary. ... A syllogism (Greek: — conclusion, inference), usually the categorical syllogism, is a kind of logical argument in which one proposition (the conclusion) is inferred from two others (the premises) of a certain form. ... Argumentation theory, or argumentation, embraces the arts and sciences of civil debate, dialogue, conversation, and persuasion. ... Philosophy of logic is the branch of philosophy that is concerned with the nature and justification of systems of logic. ... Platonic realism is a philosophical term usually used to refer to the idea of realism regarding the existence of universals after the Greek philosopher Plato who lived between c. ... Logical atomism is a philosophical belief that originated in the early 20th century with the development of analytic philosophy. ... Logicism is one of the schools of thought in the philosophy of mathematics, putting forth the theory that mathematics is an extension of logic and therefore some or all mathematics is reducible to logic. ... In philosophy, nominalism is the theory that abstract terms, general terms, or universals do not represent objective real existents, but are merely names, words, or vocal utterances (flatus vocis). ... Fictionalism is a doctrine in philosophy that suggests that statements of a certain sort should not be taken to be literally true, but merely a useful fiction. ... Contemporary philosophical realism, also referred to as metaphysical realism, is the belief in a reality that is completely ontologically independent of our conceptual schemes, linguistic practices, beliefs, etc. ... In the philosophy of mathematics, intuitionism, or neointuitionism (opposed to preintuitionism), is an approach to mathematics as the constructive mental activity of humans. ... In the philosophy of mathematics, constructivism asserts that it is necessary to find (or construct) a mathematical object to prove that it exists. ... In the philosophy of mathematics, finitism is an extreme form of constructivism, according to which a mathematical object does not exist unless it can be constructed from natural numbers in a finite number of steps. ... Mathematical logic is a major area of mathematics, which grew out of symbolic logic. ... In mathematics, logic, and computer science, a formal language is a language that is defined by precise mathematical or machine processable formulas. ... In computer science and linguistics, a formal grammar, or sometimes simply grammar, is a precise description of a formal language — that is, of a set of strings. ... In logic and mathematics, a formal system consists of two components, a formal language plus a set of inference rules or transformation rules. ... ... In theoretical computer science formal semantics is the field concerned with the rigorous mathematical study of the meaning of programming languages and models of computation. ... In mathematical logic, a formula is a formal syntactic object that expresses a proposition. ... In logic, WFF is an abbreviation for well-formed formula. ... In mathematics, a set can be thought of as any collection of distinct objects considered as a whole. ... In mathematics, an element (also called a member) is an object contained in a set (or more generally a class). ... In set theory and its applications throughout mathematics, a class is a collection of sets (or sometimes other mathematical objects) that can be unambiguously defined by a property that all its members share. ... This article is about a logical statement. ... In logic, especially in mathematical logic, a rule of inference is a scheme for constructing valid inferences. ... In mathematics, the concept of a relation is a generalization of 2-place relations, such as the relation of equality, denoted by the sign = in a statement like 5 + 7 = 12, or the relation of order, denoted by the sign < in a statement like 5 < 12. Relations that involve two... A mathematical picture paints a thousand words: the Pythagorean theorem. ... Logical consequence is the relation that holds between a set of sentences and a sentence when the latter follows from the former. ... Look up Consistency in Wiktionary, the free dictionary. ... (This article discusses the soundess notion of informal logic. ... Look up completeness in Wiktionary, the free dictionary. ... A logical system or theory is decidable if the set of all well-formed formulas valid in the system is decidable. ... 3SAT redirects here. ... Set theory is the mathematical theory of sets, which represent collections of abstract objects. ... In mathematics, an axiomatic system is any set of axioms from which some or all axioms can be used in conjunction to logically derive theorems. ... Proof theory is a branch of mathematical logic that represents proofs as formal mathematical objects, facilitating their analysis by mathematical techniques. ... In mathematics, model theory is the study of the representation of mathematical concepts in terms of set theory, or the study of the structures that underlie mathematical systems. ... Recursion theory, or computability theory, is a branch of mathematical logic dealing with generalizations of the notion of computable function, and with related notions such as Turing degrees and effective descriptive set theory. ... At the broadest level, type theory is the branch of mathematics and logic that first creates a hierarchy of types, then assigns each mathematical (and possibly other) entity to a type. ... Syntax in logic is a systematic statement of the rules governing the properly formed formulas (WFFs) of a logical system. ... Propositional logic or sentential logic is the logic of propositions, sentences, or clauses. ... A Boolean function describes how to determine a Boolean value output based on some logical calculation from Boolean inputs. ... In logic, the monadic predicate calculus is the fragment of predicate calculus in which all predicate letters are monadic (that is, they take only one argument), and there are no function letters. ... In logic and mathematics, a propositional calculus (or a sentential calculus) is a formal system in which formulas representing propositions can be formed by combining atomic propositions using logical connectives, and a system of formal proof rules allows to establish that certain formulas are theorems of the formal system. ... In logic, a logical connective is a syntactic operation on sentences, or the symbol for such an operation, that corresponds to a logical operation on the logical values of those sentences. ... Truth tables are a type of mathematical table used in logic to determine whether an expression is true or whether an argument is valid. ... ... First-order logic (FOL) is a formal deductive system used by mathematicians, philosophers, linguists, and computer scientists. ... In language and logic, quantification is a construct that specifies the extent of validity of a predicate, that is the extent to which a predicate holds over a range of things. ... In mathematical logic, second-order logic is an extension of first-order logic, which itself is an extension of propositional logic. ... Deontic logic is the field of logic that is concerned with obligation, permission, and related concepts. ... Michaels the greatest boyfriend in the whole wide world, and Id love to call him in a phonebooth sometime. ... 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. ... doxastic logic is a modal logic that is concerned with reasoning about beliefs. ... Classical logic identifies a class of formal logics that have been most intensively studied and most widely used. ... Introduced by Giorgi Japaridze in 2003, Computability logic is a research programme and mathematical framework for redeveloping logic as a systematic formal theory of computability, as opposed to classical logic which is a formal theory of truth. ... For the Super Furry Animals album, see Fuzzy Logic (album). ... In mathematical logic, linear logic is a type of substructural logic that denies the structural rules of weakening and contraction. ... Relevance logic, also called relevant logic, is any of a family of non-classical substructural logics that impose certain restrictions on implication. ... A non-monotonic logic is a formal logic whose consequence relation is not monotonic. ... A paraconsistent logic is a logical system that attempts to deal nontrivially with contradictions. ... Dialetheism is a paraconsistent logic typified by its tolerance of at least some contradictions. ... Intuitionistic logic, or constructivist logic, is the logic used in mathematical intuitionism and other forms of mathematical constructivism. ... Look up paradox in Wiktionary, the free dictionary. ... Antinomy (Greek anti-, against, plus nomos, law) is a term used in logic and epistemology, which, loosely, means a paradox or unresolvable contradiction. ... Is logic empirical? is the title of two articles that discuss the idea that the algebraic properties of logic may, or should, be empirically determined; in particular, they deal with the question of whether empirical facts about quantum phenomena may provide grounds for revising classical logic as a consistent logical... Al Farabi (870-950) was born of a Turkish family and educated by a Christian physician in Baghdad, and was himself later considered a teacher on par with Aristotle. ... Abu Hāmed Mohammad ibn Mohammad al-GhazzālÄ« (1058-1111) (Persian: ), known as Algazel to the western medieval world, born and died in Tus, in the Khorasan province of Persia (modern day Iran). ... For the Christian theologian, see Abd al-Masih ibn Ishaq al-Kindi. ... Fakhr al-Din al-Razi (1149–1209) was a well-known Persian theologian and philosopher from Ray. ... For other uses, see Aristotle (disambiguation). ... Ibn Rushd, known as Averroes (1126 – December 10, 1198), was an Andalusian-Arab philosopher and physician, a master of philosophy and Islamic law, mathematics, and medicine. ... For the lunar crater, see Avicenna (crater). ... Not to be confused with George Boolos. ... Georg Ferdinand Ludwig Philipp Cantor (March 3, 1845[1] – January 6, 1918) was a German mathematician. ... Rudolf Carnap (May 18, 1891, Ronsdorf, Germany – September 14, 1970, Santa Monica, California) was an influential philosopher who was active in central Europe before 1935 and in the United States thereafter. ... ‹ The template below (Expand) is being considered for deletion. ... Dharmakirti (circa 7th century), was an Indian scholar and one of the Buddhist founders of Indian philosophical logic. ... Dignāga (5th century AD), was an Indian scholar and one of the Buddhist founders of Indian philosophical logic. ... Friedrich Ludwig Gottlob Frege (8 November 1848, Wismar – 26 July 1925, IPA: ) was a German mathematician who became a logician and philosopher. ... Gerhard Karl Erich Gentzen (November 24, 1909 – August 4, 1945) was a German mathematician and logician. ... Kanada (also transliterated as Kanad and in other ways; Sanskrit कणाद) was a Hindu sage who founded the philosophical school of Vaisheshika. ... Kurt Gödel (IPA: ) (April 28, 1906 Brünn, Austria-Hungary (now Brno, Czech Republic) – January 14, 1978 Princeton, New Jersey) was an Austrian American mathematician and philosopher. ... The Nyāya SÅ«tras is an ancient Indian text on of philosophy composed by (also Gotama; c. ... | name = David Hilbert | image = Hilbert1912. ... Ala-al-din abu Al-Hassan Ali ibn Abi-Hazm al-Qarshi al-Dimashqi (Arabic: علاء الدين أبو الحسن عليّ بن أبي حزم القرشي الدمشقي ) known as ibn Al-Nafis (Arabic: ابن النفيس ), was an Arab physician who is mostly famous for being the first to describe the pulmonary circulation of the blood. ... Abu Muhammad Ali ibn Ahmad ibn Sa`id ibn Hazm (أبو محمد علي بن احمد بن سعيد بن حزم) (November 7, 994 – August 15, 1069) was an Andalusian Muslim philosopher and theologian of Persian descent [1] born in Córdoba, present day Spain. ... Taqi al-Din Ahmad Ibn Taymiyyah (Arabic: )(January 22, 1263 - 1328), was a Sunni Islamic scholar born in Harran, located in what is now Turkey, close to the Syrian border. ... Saul Aaron Kripke (born in November 13, 1940 in Bay Shore, New York) is an American philosopher and logician now emeritus from Princeton and teaches as distinguished professor of philosophy at CUNY Graduate Center. ... Mozi (Chinese: ; pinyin: ; Wade-Giles: Mo Tzu, Lat. ... For other uses, see Nagarjuna (disambiguation). ... Indian postage stamp depicting (2004), with the implication that he used (पाणिनि; IPA ) was an ancient Indian grammarian from Gandhara (traditionally 520–460 BC, but estimates range from the 7th to 4th centuries BC). ... Giuseppe Peano Giuseppe Peano (August 27, 1858 – April 20, 1932) was an Italian mathematician and philosopher best known for his contributions to set theory. ... Charles Sanders Peirce (IPA: /pɝs/), (September 10, 1839 – April 19, 1914) was an American polymath, physicist, and philosopher, born in Cambridge, Massachusetts. ... Hilary Whitehall Putnam (born July 31, 1926) is an American philosopher who has been a central figure in Western philosophy since the 1960s, especially in philosophy of mind, philosophy of language, and philosophy of science. ... For people named Quine, see Quine (surname). ... Bertrand Arthur William Russell, 3rd Earl Russell, OM, FRS, (18 May 1872 – 2 February 1970), was a British philosopher, logician, mathematician, advocate for social reform, and pacifist. ... Albert Thoralf Skolem (May 23, 1887 - March 23, 1963) was a Norwegian mathematician. ... Shahab al-Din Yahya as-Suhrawardi (from the Arabicشهاب الدين يحيى سهروردى, also known as Sohrevardi) (born 1153 in North-West-Iran; died 1191 in Aleppo) was a persian philosopher and Sufi, founder of School of Illumination, one of the most important islamic doctrine in Philosophy. ... // Alfred Tarski (January 14, 1902, Warsaw, Russian-ruled Poland – October 26, 1983, Berkeley, California) was a logician and mathematician who spent four decades as a professor of mathematics at the University of California, Berkeley. ... Alan Mathison Turing, OBE, FRS (pronounced ) (23 June 1912 – 7 June 1954) was an English mathematician, logician and cryptographer. ... Alfred North Whitehead, OM (February 15, 1861, Ramsgate, Kent, England – December 30, 1947, Cambridge, Massachusetts, U.S.) was an English-born mathematician who became a philosopher. ... Lotfali Askar Zadeh (born February 4, 1921) is a mathematician and computer scientist, and a professor of computer science at the University of California, Berkeley. ... This is a list of topics in logic. ... For a more comprehensive list, see the List of logic topics. ... This is a list of mathematical logic topics, by Wikipedia page. ... Algebra of sets George Boole Boolean algebra Boolean function Boolean logic Boolean homomorphism Boolean Implicant Boolean prime ideal theorem Boolean-valued model Boolean satisfiability problem Booles syllogistic canonical form (Boolean algebra) compactness theorem Complete Boolean algebra connective -- see logical operator de Morgans laws Augustus De Morgan duality (order... Set theory Axiomatic set theory Naive set theory Zermelo set theory Zermelo-Fraenkel set theory Kripke-Platek set theory with urelements Simple theorems in the algebra of sets Axiom of choice Zorns lemma Empty set Cardinality Cardinal number Aleph number Aleph null Aleph one Beth number Ordinal number Well... A logician is a person, such as a philosopher or mathematician, whose topic of scholarly study is logic. ... This is a list of rules of inference. ... This is a list of paradoxes, grouped thematically. ... This is a list of fallacies. ... In logic, a set of symbols is frequently used to express logical constructs. ...

  Results from FactBites:
 
Springer Online Reference Works (1205 words)
The domain of logic in which along with the usual statements modal statements are considered, that is, statements of the type  "it is necessary that &#8230;" ,  "it is possible that …" ;, etc.
Modal logic was formalized for the first time by C.I. Lewis [1], who constructed five propositional systems of modal logic, given in the literature the notations S1–S5 (their formulations are given below).
The majority of systems of modal logic which have been studied are based on classical logic; however, systems based on intuitionistic logic have also been discussed (see, for example, [6]).
Logic - Wikipedia, the free encyclopedia (0 words)
The ambiguity is that "formal logic" is very often used with the alternate meaning of symbolic logic as we have defined it, with informal logic meaning any logical investigation that does not involve symbolic abstraction; it is this sense of 'formal' that is parallel to the received usages coming from "formal languages" or "formal theory".
The boldest attempt to apply logic to mathematics was undoubtedly the logicism pioneered by philosopher-logicians such as Gottlob Frege and Bertrand Russell: the idea was that mathematical theories were logical tautologies, and the programme was to show this by means to a reduction of mathematics to logic.
Logic cut to the heart of computer science as it emerged as a discipline: Alan Turing's work on the Entscheidungsproblem followed from Kurt Gödel's work on the incompleteness theorems, and the notion of general purpose computer that came from this work was of fundamental importance to the designers of the computer machinery in the 1940s.
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