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102-431: Possible worlds are one of the foundational concepts in modal and intensional logics . Formulas in these logics are used to represent statements about what might be true, what should be true, what one believes to be true and so forth. To give these statements a formal interpretation, logicians use structures containing possible worlds. For instance, in the relational semantics for classical propositional modal logic,

204-417: A possible world . A formula's truth value at one possible world can depend on the truth values of other formulas at other accessible possible worlds . In particular, ◊ P {\displaystyle \Diamond P} is true at a world if P {\displaystyle P} is true at some accessible possible world, while ◻ P {\displaystyle \Box P}

306-508: A bachelor's degree in mathematics. During his sophomore year at Harvard, he taught a graduate-level logic course at nearby MIT . Upon graduation he received a Fulbright Fellowship , and in 1963 was appointed to the Society of Fellows . Kripke later said, "I wish I could have skipped college. I got to know some interesting people but I can't say I learned anything. I probably would have learned it all anyway just reading on my own." His cousin

408-472: A succinct representation of a large number of possible worlds. Possible worlds are often regarded with suspicion, which is why their proponents have struggled to find arguments in their favor. An often-cited argument is called the argument from ways . It defines possible worlds as "ways things could have been" and relies for its premises and inferences on assumptions from natural language , for example: The central step of this argument happens at (2) where

510-464: A tautology , representing the principle that only true statements can count as knowledge. However, this formula is not a tautology in deontic modal logic, since what ought to be true can be false. Modal logics are formal systems that include unary operators such as ◊ {\displaystyle \Diamond } and ◻ {\displaystyle \Box } , representing possibility and necessity respectively. For instance

612-431: A class of finite frames. An application of this notion is the decidability question: it follows from Post's theorem that a recursively axiomatized modal logic L which has FMP is decidable, provided it is decidable whether a given finite frame is a model of L. In particular, every finitely axiomatizable logic with FMP is decidable. There are various methods for establishing FMP for a given logic. Refinements and extensions of

714-533: A class of modal algebras, and a finite modal algebra can be transformed into a Kripke frame. As an example, Robert Bull proved using this method that every normal extension of S4.3 has FMP, and is Kripke complete. Kripke semantics has a straightforward generalization to logics with more than one modality. A Kripke frame for a language with { ◻ i ∣ i ∈ I } {\displaystyle \{\Box _{i}\mid \,i\in I\}} as

816-483: A clean notion of analytic proof ). More complex calculi have been applied to modal logic to achieve generality. Analytic tableaux provide the most popular decision method for modal logics. 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

918-579: A combined epistemic-deontic logic could use the formula [ K ] ⟨ D ⟩ P {\displaystyle [K]\langle D\rangle P} read as "I know P is permitted". Systems of modal logic can include infinitely many modal operators distinguished by indices, i.e. ◻ 1 {\displaystyle \Box _{1}} , ◻ 2 {\displaystyle \Box _{2}} , ◻ 3 {\displaystyle \Box _{3}} , and so on. The standard semantics for modal logic

1020-530: A formula. For instance, consider a model M {\displaystyle {\mathfrak {M}}} whose accessibility relation is reflexive . Because the relation is reflexive, we will have that M , w ⊨ P → ◊ P {\displaystyle {\mathfrak {M}},w\models P\rightarrow \Diamond P} for any w ∈ G {\displaystyle w\in G} regardless of which valuation function

1122-447: A model M {\displaystyle {\mathfrak {M}}} : According to this semantics, a formula is necessary with respect to a world w {\displaystyle w} if it holds at every world that is accessible from w {\displaystyle w} . It is possible if it holds at some world that is accessible from w {\displaystyle w} . Possibility thereby depends upon

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1224-581: A paradox for skepticism about meaning . Much of his work remains unpublished or exists only as tape recordings and privately circulated manuscripts. Saul Kripke was the oldest of three children born to Dorothy K. Kripke and Myer S. Kripke . His father was the leader of Beth El Synagogue, the only Conservative congregation in Omaha , Nebraska ; his mother wrote Jewish educational books for children. Saul and his two sisters, Madeline and Netta, attended Dundee Grade School and Omaha Central High School . Kripke

1326-420: 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 is 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

1428-478: A posteriori , such as that water is H 2 O. A 1970 Princeton lecture series, published in book form in 1980 as Naming and Necessity , is considered one of the most important philosophical works of the 20th century. It introduced the concept of names as rigid designators , designating (picking out, denoting, referring to) the same object in every possible world, as contrasted with descriptions . It also established Kripke's causal theory of reference , disputing

1530-526: A prefixed "box" (□ p ) whose scope is established by parentheses. Likewise, a prefixed "diamond" (◇ p ) denotes "possibly p ". Similar to the quantifiers in first-order logic , "necessarily p " (□ p ) does not assume the range of quantification (the set of accessible possible worlds in Kripke semantics ) to be non-empty, whereas "possibly p " (◇ p ) often implicitly assumes ◊ ⊤ {\displaystyle \Diamond \top } (viz.

1632-584: A role similar to the Lindenbaum–Tarski algebra construction in algebraic semantics. A set of formulas is L - consistent if no contradiction can be derived from them using the axioms of L , and modus ponens . A maximal L-consistent set (an L - MCS for short) is an L -consistent set which has no proper L -consistent superset. The canonical model of L is a Kripke model ⟨ W , R , ⊩ ⟩ {\displaystyle \langle W,R,\Vdash \rangle } , where W

1734-457: A single accessibility relation R , and subsets D i  ⊆  W for each modality. Satisfaction is defined as: Carlson models are easier to visualize and to work with than usual polymodal Kripke models; there are, however, Kripke complete polymodal logics which are Carlson incomplete. In Semantical Considerations on Modal Logic , published in 1963, Kripke responded to a difficulty with classical quantification theory . The motivation for

1836-412: A system that uses the world-relative interpretation and preserves the classical rules. But the costs are severe. First, his language is artificially impoverished, and second, the rules for the propositional modal logic must be weakened. Kripke's possible worlds theory has been used by narratologists (beginning with Pavel and Dolezel) to understand "reader's manipulation of alternative plot developments, or

1938-466: A transparent way of modeling certain concepts such as the evidence or justification one has for one's beliefs. Topological semantics is widely used in recent work in formal epistemology and has antecedents in earlier work such as David Lewis and Angelika Kratzer 's logics for counterfactuals . The first formalizations of modal logic were axiomatic . Numerous variations with very different properties have been proposed since C. I. Lewis began working in

2040-467: Is Eric Kripke , known for creating the television show The Boys . After briefly teaching at Harvard, Kripke moved in 1968 to Rockefeller University in New York City, where he taught until 1976. In 1978 he took a chaired professorship at Princeton University . In 1988 he received the university's Behrman Award for distinguished achievement in the humanities. In 2002 Kripke began teaching at

2142-463: Is complete with respect to C if L  ⊇ Thm( C ). Semantics is useful for investigating a logic (i.e., a derivation system) only if the semantical entailment relation reflects its syntactical counterpart, the consequence relation ( derivability ). It is vital to know which modal logics are sound and complete with respect to a class of Kripke frames, and for them, to determine which class it is. For any class C of Kripke frames, Thm( C )

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2244-458: Is possible that it is raining outside" – in the sense of epistemic possibility – then that would weigh on whether or not I take the umbrella. But if you just tell me that "it is possible for it to rain outside" – in the sense of metaphysical possibility – then I am no better off for this bit of modal enlightenment. Some features of epistemic modal logic are in debate. For example, if x knows that p , does x know that it knows that p ? That

2346-442: 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

2448-467: Is a normal modal logic (in particular, theorems of the minimal normal modal logic, K , are valid in every Kripke model). However, the converse does not hold generally. There are Kripke incomplete normal modal logics, which is unproblematic, because most of the modal systems studied are complete of classes of frames described by simple conditions. A normal modal logic L corresponds to a class of frames C , if C  = Mod( L ). In other words, C

2550-803: Is a partially ordered Kripke frame, and ⊩ {\displaystyle \Vdash } satisfies the following conditions: Intuitionistic logic is sound and complete with respect to its Kripke semantics, and it has the Finite Model Property. Intuitionistic first-order logic Let L be a first-order language. A Kripke model of L is a triple ⟨ W , ≤ , { M w } w ∈ W ⟩ {\displaystyle \langle W,\leq ,\{M_{w}\}_{w\in W}\rangle } , where ⟨ W , ≤ ⟩ {\displaystyle \langle W,\leq \rangle }

2652-491: Is a semantics for modal logic involving possible worlds , now called Kripke semantics . He received the 2001 Schock Prize in Logic and Philosophy. Kripke was also partly responsible for the revival of metaphysics and essentialism after the decline of logical positivism , claiming necessity is a metaphysical notion distinct from the epistemic notion of a priori , and that there are necessary truths that are known

2754-401: Is a topological space and V {\displaystyle V} is a valuation function which maps each atomic formula to some subset of X {\displaystyle X} . The basic interior semantics interprets formulas of modal logic as follows: Topological approaches subsume relational ones, allowing non-normal modal logics . The extra structure they provide also allows

2856-481: Is a Kripke frame, and ⊩ {\displaystyle \Vdash } is a relation between nodes of W and modal formulas, such that: We read w ⊩ A {\displaystyle w\Vdash A} as " w satisfies A ", " A is satisfied in w ", or " w forces A ". The relation ⊩ {\displaystyle \Vdash } is called the satisfaction relation , evaluation , or forcing relation . The satisfaction relation

2958-468: 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

3060-605: Is a matter of 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. Epistemic modalities (from the Greek episteme , knowledge), deal with the certainty of sentences. The □ operator is translated as "x is certain that…", and the ◇ operator is translated as "For all x knows, it may be true that…" In ordinary speech both metaphysical and epistemic modalities are often expressed in similar words;

3162-401: Is an intuitionistic Kripke frame, M w is a (classical) L -structure for each node w  ∈  W , and the following compatibility conditions hold whenever u  ≤  v : Given an evaluation e of variables by elements of M w , we define the satisfaction relation w ⊩ A [ e ] {\displaystyle w\Vdash A[e]} : Here e ( x → a ) is

Possible world - Misplaced Pages Continue

3264-425: Is called the relational semantics . In this approach, the truth of a formula is determined relative to a point which is often called a possible world . For a formula that contains a modal operator, its truth value can depend on what is true at other accessible worlds. Thus, the relational semantics interprets formulas of modal logic using models defined as follows. The set W {\displaystyle W}

3366-417: Is canonical. In general, it is undecidable whether a given axiom is canonical. We know a nice sufficient condition: H. Sahlqvist identified a broad class of formulas (now called Sahlqvist formulas ) such that: This is a powerful criterion: for example, all axioms listed above as canonical are (equivalent to) Sahlqvist formulas. A logic has the finite model property (FMP) if it is complete with respect to

3468-866: Is conventionally read aloud as "necessarily", and can be used to represent notions such as moral or legal obligation , knowledge , historical inevitability , among others. The latter is typically read as "possibly" and can be used to represent notions including permission , ability , compatibility with evidence . While well-formed formulas of modal logic include non-modal formulas such as P ∧ Q {\displaystyle P\land Q} , it also contains modal ones such as ◻ ( P ∧ Q ) {\displaystyle \Box (P\land Q)} , P ∧ ◻ Q {\displaystyle P\land \Box Q} , ◻ ( ◊ P ∧ ◊ Q ) {\displaystyle \Box (\Diamond P\land \Diamond Q)} , and so on. Thus,

3570-477: Is itself built on Richard Montague 's intensional logic . Contemporary research in semantics typically uses possible worlds as formal tools without committing to a particular theory of their metaphysical status. The term possible world is retained even by those who attach no metaphysical significance to them. In the field of database theory , possible worlds are also a notion used in the setting of uncertain databases and probabilistic databases , which serve as

3672-410: Is itself canonical. It follows from the preceding discussion that any logic axiomatized by a canonical set of formulas is Kripke complete, and compact . The axioms T, 4, D, B, 5, H, G (and thus any combination of them) are canonical. GL and Grz are not canonical, because they are not compact. The axiom M by itself is not canonical ( Goldblatt , 1991), but the combined logic S4.1 (in fact, even K4.1 )

3774-425: Is logically possible to accelerate beyond the speed of light , modern science stipulates that it is not physically possible for material particles or information. Philosophers 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 bodies and can experience

3876-440: Is not a great one. In any case, different answers to such questions yield different systems of modal logic. Adding axioms to K gives rise to other well-known modal systems. One cannot prove in K that if " p is necessary" then p is true. The axiom T remedies this defect: T holds in most but not all modal logics. Zeman (1973) describes a few exceptions, such as S1 . Other well-known elementary axioms are: These yield

3978-484: 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

4080-443: Is often called the universe . The binary relation R {\displaystyle R} is called an accessibility relation , and it controls which worlds can "see" each other for the sake of determining what is true. For example, w R u {\displaystyle wRu} means that the world u {\displaystyle u} is accessible from world w {\displaystyle w} . That

4182-458: Is permitted that p ) seems appropriate, but we should probably not include that p → ◻ ◊ p {\displaystyle p\to \Box \Diamond p} . In fact, to do so is to commit the naturalistic fallacy (i.e. to state that what is natural is also good, by saying that if p is the case, p ought to be permitted). The commonly employed system S5 simply makes all modal truths necessary. For example, if p

Possible world - Misplaced Pages Continue

4284-597: Is possible, then it is "necessary" that p is possible. Also, if p is necessary, then it is necessary that p is necessary. Other systems of modal logic have been formulated, in part because S5 does not describe every kind of modality of interest. 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

4386-416: Is said to be In classical modal logic, therefore, the notion of either possibility or necessity may be taken to be basic, where these other notions are defined in terms of it in the manner of De Morgan duality . Intuitionistic modal logic treats possibility and necessity as not perfectly symmetric. For example, suppose that while walking to the convenience store we pass Friedrich's house, and observe that

4488-408: Is simply the propositional calculus augmented by □, 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 □ p is true then □□ p is true, i.e., that necessary truths are "necessarily necessary". If such perplexities are deemed forced and artificial, this defect of K

4590-622: Is the largest class of frames such that L is sound wrt C . It follows that L is Kripke complete if and only if it is complete of its corresponding class. Consider the schema T  : ◻ A → A {\displaystyle \Box A\to A} . T is valid in any reflexive frame ⟨ W , R ⟩ {\displaystyle \langle W,R\rangle } : if w ⊩ ◻ A {\displaystyle w\Vdash \Box A} , then w ⊩ A {\displaystyle w\Vdash A} since w   R   w . On

4692-466: Is the set of all L - MCS , and the relations R and ⊩ {\displaystyle \Vdash } are as follows: The canonical model is a model of L , as every L - MCS contains all theorems of L . By Zorn's lemma , each L -consistent set is contained in an L - MCS , in particular every formula unprovable in L has a counterexample in the canonical model. The main application of canonical models are completeness proofs. Properties of

4794-401: Is to say, should □ P → □□ P be an axiom in these systems? While the answer to this question is unclear, 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 ): Saul Kripke Saul Aaron Kripke ( / ˈ k r ɪ p k i / ; November 13, 1940 – September 15, 2022)

4896-441: Is to say, the state of affairs known as u {\displaystyle u} is a live possibility for w {\displaystyle w} . Finally, the function V {\displaystyle V} is known as a valuation function . It determines which atomic formulas are true at which worlds. Then we recursively define the truth of a formula at a world w {\displaystyle w} in

4998-446: Is true at a world if P {\displaystyle P} is true at every accessible possible world. A variety of proof systems exist which are sound and complete with respect to the semantics one gets by restricting the accessibility relation. For instance, the deontic modal logic D is sound and complete if one requires the accessibility relation to be serial . While the intuition behind modal logic dates back to antiquity,

5100-416: Is uniquely determined by its value on propositional variables. A formula A is valid in: We define Thm( C ) to be the set of all formulas that are valid in C . Conversely, if X is a set of formulas, let Mod( X ) be the class of all frames which validate every formula from X . A modal logic (i.e., a set of formulas) L is sound with respect to a class of frames C , if L  ⊆ Thm( C ). L

5202-476: Is used. For this reason, modal logicians sometimes talk about frames , which are the portion of a relational model excluding the valuation function. The different systems of modal logic are defined using frame conditions . A frame is called: The logics that stem from these frame conditions are: The Euclidean property along with reflexivity yields symmetry and transitivity. (The Euclidean property can be obtained, as well, from symmetry and transitivity.) Hence if

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5304-726: The CUNY Graduate Center , and in 2003 he was appointed a distinguished professor of philosophy there. Kripke has received honorary degrees from the University of Nebraska , Omaha (1977), Johns Hopkins University (1997), University of Haifa , Israel (1998), and the University of Pennsylvania (2005). He was a member of the American Philosophical Society and an elected Fellow of the American Academy of Arts and Sciences , and in 1985

5406-456: The GL - tautology ◻ A → ◻ ◻ A {\displaystyle \Box A\to \Box \Box A} . For any normal modal logic L , a Kripke model (called the canonical model ) can be constructed, which validates precisely the theorems of L , by an adaptation of the standard technique of using maximal consistent sets as models. Canonical Kripke models play

5508-534: The Zombie Argument , and physicalism and supervenience in the philosophy of mind . Many debates in the philosophy of religion have been reawakened by the use of possible worlds. The idea of possible worlds is most commonly attributed to Gottfried Leibniz , who spoke of possible worlds as ideas in the mind of God and used the notion to argue that our actually created world must be "the best of all possible worlds ". Arthur Schopenhauer argued that on

5610-469: The accessibility relation . Depending on the properties of the accessibility relation ( transitivity , reflexivity, etc.), the corresponding frame is described, by extension, as being transitive, reflexive, etc. A Kripke model is a triple ⟨ W , R , ⊩ ⟩ {\displaystyle \langle W,R,\Vdash \rangle } , where ⟨ W , R ⟩ {\displaystyle \langle W,R\rangle }

5712-418: The argument from ways depends on these assumptions and may be challenged by casting doubt on the quantifier-method of ontology or on the reliability of natural language as a guide to ontology. The ontological status of possible worlds has provoked intense debate. David Lewis famously advocated for a position known as modal realism , which holds that possible worlds are real, concrete places which exist in

5814-558: The descriptivist theory found in Gottlob Frege 's concept of sense and Bertrand Russell 's theory of descriptions . Kripke is often seen in opposition to the other great late-20th-century philosopher to eschew logical positivism: W. V. O. Quine . Quine rejected essentialism and modal logic. Kripke also gave an original reading of Ludwig Wittgenstein , known as " Kripkenstein ", in his Wittgenstein on Rules and Private Language . The book contains his rule-following argument,

5916-871: The language L {\displaystyle {\mathcal {L}}} of basic propositional logic can be defined recursively as follows. Modal operators can be added to other kinds of logic by introducing rules analogous to #4 and #5 above. Modal predicate logic is one widely used variant which includes formulas such as ∀ x ◊ P ( x ) {\displaystyle \forall x\Diamond P(x)} . In systems of modal logic where ◻ {\displaystyle \Box } and ◊ {\displaystyle \Diamond } are duals , ◻ ϕ {\displaystyle \Box \phi } can be taken as an abbreviation for ¬ ◊ ¬ ϕ {\displaystyle \neg \Diamond \neg \phi } , thus eliminating

6018-420: 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

6120-445: 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: The weakest normal modal logic , named " K " in honor of Saul Kripke ,

6222-530: The English word "actual" is an indexical, that doesn't mean that other worlds exist. For comparison, one can use the indexical "I" without believing that other people actually exist. Some philosophers instead endorse the view of possible worlds as maximally consistent sets of propositions or descriptions, while others such as Saul Kripke treat them as purely formal (i.e. mathematical) devices. At least since Aristotle, philosophers have been greatly concerned with

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6324-426: The accessibility relation R {\displaystyle R} , which allows us to express the relative nature of possibility. For example, we might say that given our laws of physics it is not possible for humans to travel faster than the speed of light, but that given other circumstances it could have been possible to do so. Using the accessibility relation we can translate this scenario as follows: At all of

6426-415: The accessibility relation R is reflexive and Euclidean, R is provably symmetric and transitive as well. Hence for models of S5, R is an equivalence relation , because R is reflexive, symmetric and transitive. We can prove that these frames produce the same set of valid sentences as do the frames where all worlds can see all other worlds of W ( i.e. , where R is a "total" relation). This gives

6528-425: The area in 1912. 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

6630-569: The benefits of possible worlds semantics "on the cheap".) Modal realism is controversial. W.V. Quine rejected it as "metaphysically extravagant". Stalnaker responded to Lewis's arguments by pointing out that a way things could have been is not itself a world, but rather a property that such a world can have. Since properties can exist without them applying to any existing objects, there's no reason to conclude that other worlds like ours exist. Another of Stalnaker's arguments attacks Lewis's indexicality theory of actuality . Stalnaker argues that even if

6732-407: The canonical model construction often work, using tools such as filtration or unravelling. As another possibility, completeness proofs based on cut-free sequent calculi usually produce finite models directly. Most of the modal systems used in practice (including all listed above) have FMP. In some cases, we can use FMP to prove Kripke completeness of a logic: every normal modal logic is complete wrt

6834-437: The canonical model of K immediately imply completeness of K with respect to the class of all Kripke frames. This argument does not work for arbitrary L , because there is no guarantee that the underlying frame of the canonical model satisfies the frame conditions of L . We say that a formula or a set X of formulas is canonical with respect to a property P of Kripke frames, if A union of canonical sets of formulas

6936-825: 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 ⟹ ◻ ◊ P {\displaystyle P\implies \Box \Diamond P} , ◻ P ⟹ ◻ ◻ P {\displaystyle \Box P\implies \Box \Box P} and ◻ P ⟹ P {\displaystyle \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

7038-598: The characters' planned or fantasized alternative action series." This application has become especially useful in the analysis of hyperfiction . Kripke semantics for intuitionistic logic follows the same principles as the semantics of modal logic, but uses a different definition of satisfaction. An intuitionistic Kripke model is a triple ⟨ W , ≤ , ⊩ ⟩ {\displaystyle \langle W,\leq ,\Vdash \rangle } , where ⟨ W , ≤ ⟩ {\displaystyle \langle W,\leq \rangle }

7140-680: The contrary our world must be the worst of all possible worlds, because if it were only a little worse it could not continue to exist. Scholars have found implicit earlier traces of the idea of possible worlds in the works of René Descartes , a major influence on Leibniz, Al-Ghazali ( The Incoherence of the Philosophers ), Averroes ( The Incoherence of the Incoherence ), Fakhr al-Din al-Razi ( Matalib al-'Aliya ), John Duns Scotus and Antonio Rubio ( Commentarii in libros Aristotelis Stagiritae de Coelo ). The modern philosophical use of

7242-416: The corresponding modal graph which is total complete ( i.e. , no more edges (relations) can be added). For example, in any modal logic based on frame conditions: If we consider frames based on the total relation we can just say that 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 this does not have to be

7344-453: The definition of ⊩ {\displaystyle \Vdash } . T corresponds to the class of reflexive Kripke frames. It is often much easier to characterize the corresponding class of L than to prove its completeness, thus correspondence serves as a guide to completeness proofs. Correspondence is also used to show incompleteness of modal logics: suppose L 1  ⊆  L 2 are normal modal logics that correspond to

7446-457: The evaluation which gives x the value a , and otherwise agrees with e . The three lectures that form Naming and Necessity constitute an attack on the descriptivist theory of names . Kripke attributes variants of descriptivist theories to Frege , Russell , Wittgenstein , and John Searle , among others. According to descriptivist theories, proper names either are synonymous with descriptions, or have their reference determined by virtue of

7548-930: The exact same sense that the actual world exists. On Lewis's account, the actual world is special only in that we live there. This doctrine is called the indexicality of actuality since it can be understood as claiming that the term "actual" is an indexical , like "now" and "here". Lewis gave a variety of arguments for this position. He argued that just as the reality of atoms is demonstrated by their explanatory power in physics, so too are possible worlds justified by their explanatory power in philosophy. He also argued that possible worlds must be real because they are simply "ways things could have been" and nobody doubts that such things exist. Finally, he argued that they could not be reduced to more "ontologically respectable" entities such as maximally consistent sets of propositions without rendering theories of modality circular. (He referred to these theories as "ersatz modal realism" which try to get

7650-786: The first modal axiomatic systems were developed by C. I. Lewis in 1912. The now-standard relational semantics emerged in the mid twentieth century from work by Arthur Prior , Jaakko Hintikka , and Saul Kripke . Recent developments include alternative topological semantics such as neighborhood semantics as well as applications of the relational semantics beyond its original philosophical motivation. Such applications include game theory , moral and legal theory , web design , multiverse-based set theory , and social epistemology . Modal logic differs from other kinds of logic in that it uses modal operators such as ◻ {\displaystyle \Box } and ◊ {\displaystyle \Diamond } . The former

7752-400: 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

7854-521: The former written when he was a teenager, were on modal logic . The most familiar logics in the modal family are constructed from a weak logic called K, named after Kripke. Kripke introduced the now-standard Kripke semantics (also known as relational semantics or frame semantics) for modal logics. Kripke semantics is a formal semantics for non-classical logic systems. It was first made for modal logics, and later adapted to intuitionistic logic and other non-classical systems. The discovery of Kripke semantics

7956-516: The formula ◊ P {\displaystyle \Diamond P} (read as "possibly P") is actually true if and only if P {\displaystyle P} is true in some world which is accessible from the actual world. Possible worlds play a central role in the work of both linguists and/or philosophers working in formal semantics . Contemporary formal semantics is couched in formal systems rooted in Montague grammar , which

8058-417: The lights are off. On the way back, we observe that they have been turned on. (Of course, this analogy does not apply alethic modality in a truly rigorous fashion; for it to do so, it would have to axiomatically make such statements as "human beings cannot rise from the dead", "Socrates was a human being and not an immortal vampire", and "we did not take hallucinogenic drugs which caused us to falsely believe

8160-437: The lights were on", ad infinitum . Absolute certainty of truth or falsehood exists only in the sense of logically constructed abstract concepts such as "it is impossible to draw a triangle with four sides" and "all bachelors are unmarried".) For those having difficulty with the concept of something being possible but not true, the meaning of these terms may be made more comprehensible by thinking of multiple "possible worlds" (in

8262-455: The logical statuses of propositions, e.g. necessity, contingency, and impossibility. In the twentieth century, possible worlds have been used to explicate these notions. In modal logic, a proposition is understood in terms of the worlds in which it is true and worlds in which it is false . Thus, equivalences like the following have been proposed: Possible worlds play a central role in many other debates in philosophy. These include debates about

8364-406: The modal formula ◊ P {\displaystyle \Diamond P} can be read as "possibly P {\displaystyle P} " while ◻ P {\displaystyle \Box P} can be read as "necessarily P {\displaystyle P} ". In the standard relational semantics for modal logic, formulas are assigned truth values relative to

8466-456: The name's being associated with a description or cluster of descriptions that an object uniquely satisfies. Kripke rejects both these kinds of descriptivism. He gives several examples purporting to render descriptivism implausible as a theory of how names get their references determined (e.g., surely Aristotle could have died at age two and so not satisfied any of the descriptions we associate with his name, but it would seem wrong to deny that he

8568-724: The need for a separate syntactic rule to introduce it. However, separate syntactic rules are necessary in systems where the two operators are not interdefinable. Common notational variants include symbols such as [ K ] {\displaystyle [K]} and ⟨ K ⟩ {\displaystyle \langle K\rangle } in systems of modal logic used to represent knowledge and [ B ] {\displaystyle [B]} and ⟨ B ⟩ {\displaystyle \langle B\rangle } in those used to represent belief. These notations are particularly common in systems which use multiple modal operators simultaneously. For instance,

8670-475: The notion was pioneered by David Lewis and Saul Kripke . Modal logic Modal logic is a kind of logic used to represent statements about necessity and possibility . It plays a major role in philosophy and related fields as a tool for understanding concepts such as knowledge , obligation , and causation . For instance, in epistemic modal logic , the formula ◻ P {\displaystyle \Box P} can be used to represent

8772-488: The other hand, a frame which validates T has to be reflexive: fix w  ∈  W , and define satisfaction of a propositional variable p as follows: u ⊩ p {\displaystyle u\Vdash p} if and only if w   R   u . Then w ⊩ ◻ p {\displaystyle w\Vdash \Box p} , thus w ⊩ p {\displaystyle w\Vdash p} by T , which means w   R   w using

8874-545: The other, weaker logics. Modal logic has also been interpreted using topological structures. For instance, the Interior Semantics interprets formulas of modal logic as follows. A topological model is a tuple X = ⟨ X , τ , V ⟩ {\displaystyle \mathrm {X} =\langle X,\tau ,V\rangle } where ⟨ X , τ ⟩ {\displaystyle \langle X,\tau \rangle }

8976-433: 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. Metaphysical possibility has been thought to be more restricting than bare logical possibility (i.e., fewer things are metaphysically possible than are logically possible). However, its exact relation (if any) to logical possibility or to physical possibility

9078-473: The plausible (1) is interpreted in a way that involves quantification over "ways". Many philosophers, following Willard Van Orman Quine , hold that quantification entails ontological commitments , in this case, a commitment to the existence of possible worlds. Quine himself restricted his method to scientific theories, but others have applied it also to natural language, for example, Amie L. Thomasson in her paper entitled Ontology Made Easy . The strength of

9180-456: The same class of frames, but L 1 does not prove all theorems of L 2 . Then L 1 is Kripke incomplete. For example, the schema ◻ ( A ≡ ◻ A ) → ◻ A {\displaystyle \Box (A\equiv \Box A)\to \Box A} generates an incomplete logic, as it corresponds to the same class of frames as GL (viz. transitive and converse well-founded frames), but does not prove

9282-436: The sense of Leibniz ) or "alternate universes"; something "necessary" is true in all possible worlds, something "possible" is true in at least one possible world. Something is physically, or nomically, possible if it is permitted by the laws of physics . For example, current theory is thought to allow for there to be an atom with an atomic number of 126, even if there are no such atoms in existence. In contrast, while it

9384-403: The set of accessible possible worlds is non-empty). Regardless of notation, each of these operators is definable in terms of the other in classical modal logic: Hence □ and ◇ form a dual pair of operators. In many modal logics, the necessity and possibility operators satisfy the following analogues of de Morgan's laws from Boolean algebra : Precisely what axioms and rules must be added to

9486-600: The set of its necessity operators consists of a non-empty set W equipped with binary relations R i for each i  ∈  I . The definition of a satisfaction relation is modified as follows: A simplified semantics, discovered by Tim Carlson, is often used for polymodal provability logics . A Carlson model is a structure ⟨ W , R , { D i } i ∈ I , ⊩ ⟩ {\displaystyle \langle W,R,\{D_{i}\}_{i\in I},\Vdash \rangle } with

9588-437: The statement that P {\displaystyle P} is known. In deontic modal logic , that same formula can represent that P {\displaystyle P} is a moral obligation. Modal logic considers the inferences that modal statements give rise to. For instance, most epistemic modal logics treat the formula ◻ P → P {\displaystyle \Box P\rightarrow P} as

9690-404: The systems (axioms in bold, systems in italics): K through S5 form a nested hierarchy of systems, making up the core of normal modal logic . But specific rules or sets of rules may be appropriate for specific systems. For example, in deontic logic , ◻ p → ◊ p {\displaystyle \Box p\to \Diamond p} (If it ought to be that p , then it

9792-429: The world-relative approach was to represent the possibility that objects in one world may fail to exist in another. But if standard quantifier rules are used, every term must refer to something that exists in all the possible worlds. This seems incompatible with our ordinary practice of using terms to refer to things that exist contingently. Kripke's response to this difficulty was to eliminate terms. He gave an example of

9894-408: The worlds accessible to our own world, it is not the case that humans can travel faster than the speed of light, but at one of these accessible worlds there is another world accessible from those worlds but not accessible from our own at which humans can travel faster than the speed of light. The choice of accessibility relation alone can sometimes be sufficient to guarantee the truth or falsity of

9996-810: Was a Corresponding Fellow of the British Academy . He won the Schock Prize in Logic and Philosophy in 2001. Kripke was married to philosopher Margaret Gilbert . Kripke died of pancreatic cancer on September 15, 2022, in Plainsboro, New Jersey, at the age of 81. Kripke's contributions to philosophy include: He has also contributed to recursion theory (see admissible ordinal and Kripke–Platek set theory ). Two of Kripke's earlier works, "A Completeness Theorem in Modal Logic" (1959) and "Semantical Considerations on Modal Logic" (1963),

10098-407: Was a breakthrough in the making of non-classical logics, because the model theory of such logics was absent before Kripke. A Kripke frame or modal frame is a pair ⟨ W , R ⟩ {\displaystyle \langle W,R\rangle } , where W is a non-empty set, and R is a binary relation on W . Elements of W are called nodes or worlds , and R is known as

10200-680: Was an American analytic philosopher and logician . He was Distinguished Professor of Philosophy at the Graduate Center of the City University of New York and emeritus professor at Princeton University . From the 1960s until his death, he was a central figure in a number of fields related to mathematical and modal logic , philosophy of language and mathematics , metaphysics , epistemology , and recursion theory . Kripke made influential and original contributions to logic , especially modal logic. His principal contribution

10302-410: 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

10404-462: Was labeled a prodigy , teaching himself Ancient Hebrew by the age of six, reading Shakespeare 's complete works by nine, and mastering the works of Descartes and complex mathematical problems before finishing elementary school. He wrote his first completeness theorem in modal logic at 17, and had it published a year later. After graduating from high school in 1958, Kripke attended Harvard University and graduated summa cum laude in 1962 with

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