25:
7904:
8072:
Possible world semantics is a broader term encompassing various approaches, including Kripke semantics. It generally refers to the idea of analyzing modal statements by considering alternative possible worlds where different propositions are true or false. While Kripke semantics is a specific type of
7998:, developed a translation of sentential modal logic into classical predicate logic that, if he had combined it with the usual model theory for the latter, would have produced a model theory equivalent to Kripke models for the former. But his approach was resolutely syntactic and anti-model-theoretic.
7850:
The main defect of Kripke semantics is the existence of Kripke incomplete logics, and logics which are complete but not compact. It can be remedied by equipping Kripke frames with extra structure which restricts the set of possible valuations, using ideas from algebraic semantics. This gives rise to
7987:
and Tarski established the representability of
Boolean algebras with operators in terms of frames. If the two ideas had been put together, the result would have been precisely frame models, which is to say Kripke models, years before Kripke. But no one (not even Tarski) saw the connection at the
8073:
possible world semantics, there are other ways to model possible worlds and their relationships. Kripke semantics is a specific form of possible world semantics that employs relational structures to represent the relationships between possible worlds and propositions in modal logic.
8015:
gave a semantics in his papers introducing epistemic logic that is a simple variation of Kripke's semantics, equivalent to the characterisation of valuations by means of maximal consistent sets. He doesn't give inference rules for epistemic logic, and so cannot give a completeness
1142:, are valid in every Kripke model). However, the converse does not hold in general: while most of the modal systems studied are complete of classes of frames described by simple conditions, Kripke incomplete normal modal logics do exist. A natural example of such a system is
7975:
for the modalities of necessity and possibility by means of giving the valuation function a parameter that ranges over
Leibnizian possible worlds. Bayart develops this idea further, but neither gave recursive definitions of satisfaction in the style introduced by
7488:
7877:
Blackburn et al. (2001) point out that because a relational structure is simply a set together with a collection of relations on that set, it is unsurprising that relational structures are to be found just about everywhere. As an example from
3612:
2854:
8022:
had many of the key ideas contained in Kripke's work, but he did not regard them as significant, because he had no completeness proof, and so did not publish until after Kripke's papers had created a sensation in the logic
8004:
gave a rather more complex approach to the interpretation of modal logic, but one that contains many of the key ideas of Kripke's approach. He first noted the relationship between conditions on accessibility relations and
4116:
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
1934:
4607:
2674:
4484:
4403:
7609:
7182:
6724:
7257:
5284:
2603:
2036:
1567:
2395:
3408:
3246:
3028:
2525:
1748:
2769:
2718:
3066:
6920:
6518:
4667:
2176:
3113:
5572:
2334:
2448:
294:
8221:
7548:
7118:
6671:
3795:
993:
886:
423:
7983:
developed an approach to modeling modal logics that is still influential in modern research, namely the algebraic approach, in which
Boolean algebras with operators are used as models.
4282:
2293:
5316:
4699:
4109:
A union of canonical sets of formulas is itself canonical. It follows from the preceding discussion that any logic axiomatized by a canonical set of formulas is Kripke complete, and
6256:
6123:
2931:
7351:
7057:
5645:
1975:
1860:
1610:
6761:
6555:
2969:
2246:
2214:
2074:
6878:
6476:
1258:
955:
455:
342:
6830:
5701:
4538:
4340:
5820:
7890:. Blackburn et al. thus claim because of this connection that modal languages are ideally suited in providing "internal, local perspective on relational structures." (p. xii)
6415:. That is, the 'local' aspect of existence for sections of a sheaf was a kind of logic of the 'possible'. Though this development was the work of a number of people, the name
5939:
4851:
4612:
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.
4937:
3315:
2123:
7686:
5023:
3376:
2097:
1408:
1287:
670:
587:
527:
7335:
6073:
5130:
3378:, which means that for every possible world in the model, there is always at least one possible world accessible from it (which could be itself). This implicit implication
7715:
3911:
2889:
1786:
1218:
6315:
6156:
3353:
3193:
613:
553:
7836:
7810:
7741:
7302:
7021:
6790:
6365:
6206:
6035:
6000:
5890:
5855:
5771:
5736:
5499:
5101:
5075:
4989:
4963:
4903:
4877:
4809:
4783:
3965:
3136:
1434:
1367:
1313:
912:
774:
696:
639:
228:
8029:
presented a semantics of intuitionistic logic based on trees, which closely resembles Kripke semantics, except for using a more cumbersome definition of satisfaction.
6282:
5965:
5049:
4757:
4723:
3862:
3831:
1819:
1470:
1013:
818:
744:
475:
3991:
3937:
3170:
252:
208:
188:
168:
4191:
There are various methods for establishing FMP for a given logic. Refinements and extensions of the canonical model construction often work, using tools such as
3882:
716:
3530:
8149:
modal formulae can be meaningfully 'understood'. Thus: whereas the notion of 'has a model' in classical non-modal logic refers to some individual formula
4133:
2776:
1123:). It is vital to know which modal logics are sound and complete with respect to a class of Kripke frames, and to determine also which class that is.
8001:
8451:
The Search for
Certainty : A Philosophical Account of Foundations of Mathematics: A Philosophical Account of Foundations of Mathematics
1620:
The following table lists common modal axioms together with their corresponding classes. The naming of the axioms often varies; Here, axiom
120:
and other non-classical systems. The development of Kripke semantics was a breakthrough in the theory of non-classical logics, because the
7067:
The key property which follows from this definition is that bisimulations (hence also p-morphisms) of models preserve the satisfaction of
1870:
4543:
4220:
modal algebra can be transformed into a Kripke frame. As an example, Robert Bull proved using this method that every normal extension of
2609:
3422:
The following table lists several common normal modal systems. Frame conditions for some of the systems were simplified: the logics are
4212:
In some cases, we can use FMP to prove Kripke completeness of a logic: every normal modal logic is complete with respect to a class of
8141:
a specific 'something' that makes a specific modal formula true; in Kripke semantics a 'model' must rather be understood as a larger
4419:
4345:
1483:
than to prove its completeness, thus correspondence serves as a guide to completeness proofs. Correspondence is also used to show
7191:
5225:
4238:
Kripke semantics has a straightforward generalization to logics with more than one modality. A Kripke frame for a language with
2546:
1985:
1525:
124:
of such logics was almost non-existent before Kripke (algebraic semantics existed, but were considered 'syntax in disguise').
8557:
8500:
8410:
8384:
8360:
8334:
8306:
8285:
8262:
2344:
231:
8638:
3381:
3200:
1633:
8713:
8676:
2979:
2471:
1694:
4158:
This is a powerful criterion: for example, all axioms listed above as canonical are (equivalent to) Sahlqvist formulas.
2736:
2685:
8578:
8431:
3039:
8620:
7951:
7925:
4634:
2133:
68:
46:
7933:
7561:
7134:
6676:
3073:
39:
5507:
2304:
2406:
261:
7515:
7085:
6638:
4624:
follows the same principles as the semantics of modal logic, but it uses a different definition of satisfaction.
3762:
960:
853:
390:
4241:
2256:
8529:
8459:
7929:
7494:
but many applications need the reflexive and/or transitive closure of this relation, or similar modifications.
3411:
1143:
1115:
7483:{\displaystyle \langle w_{0},w_{1},\dots ,w_{n}\rangle \;R'\;\langle w_{0},w_{1},\dots ,w_{n},w_{n+1}\rangle }
1110:
8695:
5289:
4672:
3710:
6213:
6080:
2896:
8756:
8746:
8054:
7879:
6883:
6481:
5577:
1945:
1830:
1580:
2942:
2219:
2187:
2047:
8741:
8690:
6851:
6449:
1231:
928:
428:
315:
5659:
4507:
4309:
8114:
from the notion of 'model' in classical non-modal logics: In classical logics we say that some formula
6408:
5778:
4172:(FMP) if it is complete with respect to a class of finite frames. An application of this notion is the
4036:
The main application of canonical models are completeness proofs. Properties of the canonical model of
5897:
4824:
7883:
6729:
6523:
4910:
3264:
3139:
2102:
8009:-style axioms for modal logic. Kanger failed, however, to give a completeness proof for his system;
7842:
As in the case of unravelling, the definition of the accessibility relation on the quotient varies.
7662:
4996:
3358:
2079:
1384:
1263:
646:
560:
503:
8685:
7914:
7307:
6795:
6043:
5109:
4154:
there is an algorithm that computes the corresponding frame condition to a given
Sahlqvist formula.
33:
8595:
7918:
7868:
7691:
7026:
3887:
2865:
1762:
1194:
6287:
6128:
5198:
Intuitionistic logic is sound and complete with respect to its Kripke semantics, and it has the
4184:
which has FMP is decidable, provided it is decidable whether a given finite frame is a model of
3335:
3175:
592:
532:
8514:
7815:
7789:
7720:
7281:
7000:
6769:
6407:, it was realised around 1965 that Kripke semantics was intimately related to the treatment of
6320:
6161:
6005:
5970:
5860:
5825:
5741:
5706:
5469:
5080:
5054:
4968:
4942:
4882:
4856:
4788:
4762:
3944:
3706:
3118:
1413:
1346:
1292:
891:
753:
675:
618:
377:
213:
143:
50:
8604:
8348:
8751:
8736:
8185:
6261:
5944:
5199:
5028:
4736:
4708:
4168:
3839:
3816:
1796:
1455:
998:
830:
803:
721:
460:
255:
8487:
8273:
3970:
3916:
3152:
835:. The satisfaction relation is uniquely determined by its value on propositional variables.
8731:
8704:
8320:
4621:
4173:
3501:
1081:
237:
193:
173:
117:
153:
8:
8179:
8039:
7628:
preserves the accessibility relation, and (in both directions) satisfaction of variables
7075:
4110:
1979:
101:
3607:{\displaystyle \forall w\,\exists u\,(w\,R\,u\land \forall v\,(u\,R\,v\Rightarrow u=v))}
8399:
8049:
8044:
3867:
2723:
2338:
2250:
1790:
1225:
1135:
701:
147:
8480:
8616:
8574:
8553:
8539:
8525:
8496:
8455:
8427:
8406:
8380:
8356:
8330:
8302:
8281:
8258:
8026:
5215:
4409:
4177:
4148:
4137:
3632:
3322:
7984:
8510:
8476:
8248:
8019:
7887:
7864:
4233:
4203:
4129:
3493:
2973:
1653:
8667:
8420:
Gasquet, Olivier; Herzig, Andreas; Said, Bilal; Schwarzentruber, François (2013).
8153:
that logic, the notion of 'has a model' in modal logic refers to the logic itself
4209:
Most of the modal systems used in practice (including all listed above) have FMP.
8708:
8671:
8642:
8568:
8547:
8449:
8421:
8374:
8370:
8324:
8296:
8252:
8012:
4200:
4014:
3669:
2849:{\displaystyle w\,R\,u\land w\,R\,v\Rightarrow \exists x\,(u\,R\,x\land v\,R\,x)}
2127:
1637:
353:
8651:
8088:
1569:
generates an incomplete logic, as it corresponds to the same class of frames as
8394:
8344:
7995:
7872:
6986:
4408:
A simplified semantics, discovered by Tim
Carlson, is often used for polymodal
3514:
1864:
1669:
93:
109:
8725:
8655:
8543:
8316:
7980:
7968:
7852:
5181:
4213:
3656:
1645:
1184:
is Kripke complete if and only if it is complete of its corresponding class.
1106:
139:
8355:. Handbook of Philosophical Logic. Vol. 3. Springer. pp. 225–339.
7991:
7760:
6841:
6435:
6431:, there are methods for constructing a new Kripke model from other models.
6428:
6412:
6404:
4816:
4188:. In particular, every finitely axiomatizable logic with FMP is decidable.
3642:
3624:
3318:
2459:
1573:(viz. transitive and converse well-founded frames), but does not prove the
121:
8495:. Handbook of the History of Logic. Vol. 7. Elsevier. pp. 1–98.
7964:
Similar work that predated Kripke's revolutionary semantic breakthroughs:
8280:. Handbook of Philosophical Logic. Vol. 2. Springer. pp. 1–88.
6985:
A bisimulation of models is additionally required to preserve forcing of
4132:
whether a given axiom is canonical. We know a nice sufficient condition:
1665:
1625:
1501:
are normal modal logics that correspond to the same class of frames, but
133:
113:
105:
8570:
Multiagent
Systems: Algorithmic, Game-Theoretic, and Logical Foundations
8589:
8006:
7619:
6446:, but the latter term is rarely used). A p-morphism of Kripke frames
3258:
1059:
97:
8137:. In the Kripke semantics of modal logic, by contrast, a 'model' is
8121:
a 'model' if there exists some 'interpretation' of the variables of
7903:
4303:. The definition of a satisfaction relation is modified as follows:
1929:{\displaystyle w\,R\,u\Rightarrow \exists v\,(w\,R\,v\land v\,R\,u)}
296:("possibly A" is defined as equivalent to "not necessarily not A").
8549:
Sheaves in
Geometry and Logic: A First Introduction to Topos Theory
4702:
4602:{\displaystyle \forall u\in D_{i}\,(w\;R\;u\Rightarrow u\Vdash A).}
4044:
with respect to the class of all Kripke frames. This argument does
3486:
3426:
with respect to the frame classes given in the table, but they may
2669:{\displaystyle w\,R\,u\land w\,R\,v\Rightarrow u\,R\,v\lor v\,R\,u}
1479:
It is often much easier to characterize the corresponding class of
7971:
seems to have been the first to have the idea that one can give a
4284:
as the set of its necessity operators consists of a non-empty set
3724:
if no contradiction can be derived from it using the theorems of
8515:"A Kripke-Joyal Semantics for Noncommutative Logic in Quantales"
8419:
8094:
3701:) can be constructed that refutes precisely the non-theorems of
8652:"4.4 Constructive Propositional Logic — Kripke Semantics"
8524:. Vol. 6. London: College Publications. pp. 209–225.
8232:
Quasi-historical
Interlude: the Road from Vienna to Los Angeles
3709:
as models. Canonical Kripke models play a role similar to the
1046:) be the class of all frames which validate every formula from
4479:{\displaystyle \langle W,R,\{D_{i}\}_{i\in I},\Vdash \rangle }
4398:{\displaystyle \forall u\,(w\;R_{i}\;u\Rightarrow u\Vdash A).}
8423:
Kripke's Worlds: An Introduction to Modal Logics via Tableaux
8076:
8591:
The proof theory and semantics of intuitionistic modal logic
8157:(i.e.: the entire system of its axioms and deduction rules).
4147:
the class of frames corresponding to a Sahlqvist formula is
1138:(in particular, theorems of the minimal normal modal logic,
5337:, and the following compatibility conditions hold whenever
7252:{\displaystyle s=\langle w_{0},w_{1},\dots ,w_{n}\rangle }
5279:{\displaystyle \langle W,\leq ,\{M_{w}\}_{w\in W}\rangle }
7508:
be a set of formulas closed under taking subformulas. An
4056:
of the canonical model satisfies the frame conditions of
2598:{\displaystyle \Box (\Box A\to B)\lor \Box (\Box B\to A)}
138:
The language of propositional modal logic consists of a
8520:. In Governatori, G.; Hodkinson, I.; Venema, Y. (eds.).
8194:, p. 20, 2.2 The semantics of intuitionistic logic.
2031:{\displaystyle w\,R\,v\wedge v\,R\,u\Rightarrow w\,R\,u}
1562:{\displaystyle \Box (A\leftrightarrow \Box A)\to \Box A}
8246:
4199:. As another possibility, completeness proofs based on
2390:{\displaystyle w\,R\,u\land w\,R\,v\Rightarrow u\,R\,v}
1022:
of frames or models, if it is valid in every member of
3705:, by an adaptation of the standard technique of using
1339:, and define satisfaction of a propositional variable
1105:
Semantics is useful for investigating a logic (i.e. a
8209:
7818:
7792:
7723:
7694:
7665:
7564:
7518:
7354:
7310:
7284:
7194:
7137:
7088:
7029:
7003:
6886:
6854:
6798:
6772:
6732:
6679:
6641:
6526:
6484:
6452:
6323:
6290:
6264:
6216:
6164:
6131:
6083:
6046:
6008:
5973:
5947:
5900:
5863:
5828:
5781:
5744:
5709:
5662:
5580:
5510:
5472:
5292:
5228:
5112:
5083:
5057:
5031:
4999:
4971:
4945:
4913:
4885:
4859:
4827:
4791:
4765:
4739:
4711:
4675:
4637:
4546:
4510:
4422:
4348:
4312:
4244:
3973:
3947:
3919:
3890:
3870:
3842:
3819:
3765:
3533:
3412:
existential quantifier on the range of quantification
3384:
3361:
3338:
3267:
3203:
3178:
3155:
3121:
3076:
3042:
2982:
2945:
2899:
2868:
2779:
2739:
2688:
2612:
2549:
2474:
2409:
2347:
2307:
2259:
2222:
2190:
2136:
2105:
2082:
2050:
1988:
1948:
1873:
1833:
1799:
1765:
1697:
1583:
1528:
1476:
corresponds to the class of reflexive Kripke frames.
1458:
1416:
1387:
1349:
1295:
1266:
1234:
1197:
1001:
963:
931:
894:
856:
806:
756:
724:
704:
678:
649:
621:
595:
563:
535:
506:
463:
431:
393:
318:
264:
240:
216:
196:
176:
156:
104:
systems created in the late 1950s and early 1960s by
8294:
8160:
6926:, which satisfies the following “zig-zag” property:
4615:
4052:, because there is no guarantee that the underlying
3403:{\displaystyle \Diamond A\rightarrow \Diamond \top }
3241:{\displaystyle \forall w\,\forall u\,\neg (w\,R\,u)}
8481:"Mathematical Modal Logic: a View of its Evolution"
5205:
3023:{\displaystyle w\,R\,u\land w\,R\,v\Rightarrow u=v}
2520:{\displaystyle \Box (\Box (A\to \Box A)\to A)\to A}
1743:{\displaystyle \Box (A\to B)\to (\Box A\to \Box B)}
1113:relation reflects its syntactical counterpart, the
8566:
8398:
8110:of 'model' in the Kripke semantics of modal logic
8082:
7830:
7804:
7735:
7709:
7680:
7603:
7542:
7482:
7329:
7296:
7251:
7176:
7112:
7051:
7015:
6914:
6872:
6824:
6784:
6755:
6718:
6665:
6549:
6512:
6470:
6359:
6309:
6276:
6250:
6200:
6150:
6117:
6067:
6029:
5994:
5959:
5933:
5884:
5849:
5814:
5765:
5730:
5695:
5639:
5566:
5493:
5310:
5278:
5124:
5095:
5069:
5043:
5017:
4983:
4957:
4931:
4897:
4871:
4845:
4803:
4777:
4751:
4717:
4693:
4661:
4601:
4532:
4478:
4397:
4334:
4276:
3985:
3959:
3931:
3905:
3876:
3856:
3825:
3789:
3606:
3402:
3370:
3347:
3309:
3240:
3187:
3164:
3130:
3107:
3060:
3022:
2963:
2925:
2883:
2848:
2764:{\displaystyle \Diamond \Box A\to \Box \Diamond A}
2763:
2713:{\displaystyle \Box \Diamond A\to \Diamond \Box A}
2712:
2668:
2597:
2519:
2442:
2389:
2328:
2287:
2240:
2208:
2170:
2117:
2091:
2068:
2030:
1969:
1928:
1854:
1813:
1780:
1742:
1604:
1561:
1464:
1428:
1402:
1361:
1307:
1281:
1252:
1212:
1100:
1034:) to be the set of all formulas that are valid in
1007:
987:
949:
906:
880:
812:
768:
738:
710:
690:
664:
633:
607:
581:
547:
521:
469:
449:
417:
336:
288:
246:
222:
202:
182:
162:
8489:Logic and the Modalities in the Twentieth Century
4136:identified a broad class of formulas (now called
4097:, the underlying frame of the canonical model of
8723:
8538:
7858:
3061:{\displaystyle \Diamond A\leftrightarrow \Box A}
16:Formal semantics for non-classical logic systems
8315:
7341:. The definition of the accessibility relation
4662:{\displaystyle \langle W,\leq ,\Vdash \rangle }
2171:{\displaystyle \forall w\,\exists v\,(w\,R\,v)}
8611:. In Zamuner, Edoardo; Levy, David K. (eds.).
8401:Intuitionistic Logic, Model Theory and Forcing
8351:. In Gabbay, Dov M.; Guenthner, Franz (eds.).
7604:{\displaystyle \langle W',R',\Vdash '\rangle }
7177:{\displaystyle \langle W',R',\Vdash '\rangle }
6719:{\displaystyle \langle W',R',\Vdash '\rangle }
4501:for each modality. Satisfaction is defined as
3108:{\displaystyle \forall w\,\exists !u\,w\,R\,u}
1615:
1522:is Kripke incomplete. For example, the schema
8605:"The architecture of meaning: Wittgenstein's
8271:
7500:is a useful construction which uses to prove
5567:{\displaystyle w\Vdash P(t_{1},\dots ,t_{n})}
4033:has a counterexample in the canonical model.
2329:{\displaystyle \Diamond A\to \Box \Diamond A}
1327:. On the other hand, a frame which validates
7751:preserves satisfaction of all formulas from
7598:
7565:
7537:
7519:
7477:
7413:
7400:
7355:
7246:
7201:
7171:
7138:
7107:
7089:
7071:formulas, not only propositional variables.
6909:
6887:
6867:
6855:
6713:
6680:
6660:
6642:
6564:preserves the accessibility relation, i.e.,
6507:
6485:
6465:
6453:
5305:
5293:
5273:
5258:
5244:
5229:
4688:
4676:
4656:
4638:
4473:
4452:
4438:
4423:
4271:
4245:
4029:, in particular every formula unprovable in
3784:
3766:
2443:{\displaystyle \Box (\Box A\to A)\to \Box A}
1247:
1235:
982:
964:
944:
932:
875:
857:
444:
432:
412:
394:
331:
319:
289:{\displaystyle \Diamond A:=\neg \Box \neg A}
8702:
8649:
8573:. Cambridge University Press. p. 397.
8230:, See the last two paragraphs in Section 3
8204:
8129:true; this specific interpretation is then
7932:. Unsourced material may be challenged and
7543:{\displaystyle \langle W,R,\Vdash \rangle }
7113:{\displaystyle \langle W,R,\Vdash \rangle }
6726:is a p-morphism of their underlying frames
6666:{\displaystyle \langle W,R,\Vdash \rangle }
4180:that a recursively axiomatized modal logic
3790:{\displaystyle \langle W,R,\Vdash \rangle }
988:{\displaystyle \langle W,R,\Vdash \rangle }
881:{\displaystyle \langle W,R,\Vdash \rangle }
418:{\displaystyle \langle W,R,\Vdash \rangle }
127:
8567:Shoham, Yoav; Leyton-Brown, Kevin (2008).
8509:
8475:
8215:
8203:See a slightly different formalization in
7893:
7845:
7501:
7412:
7403:
6403:As part of the independent development of
5144:, could be defined as an abbreviation for
4577:
4573:
4373:
4362:
4277:{\displaystyle \{\Box _{i}\mid \,i\in I\}}
3850:
3846:
3410:is similar to the implicit implication by
2288:{\displaystyle w\,R\,v\Rightarrow v\,R\,w}
732:
728:
8486:. In Gabbay, Dov M.; Woods, John (eds.).
8447:
8343:
8276:. In Gabbay, D.M.; Guenthner, F. (eds.).
8166:
7952:Learn how and when to remove this message
6398:
6232:
6099:
4566:
4355:
4261:
3582:
3578:
3571:
3558:
3554:
3547:
3540:
3231:
3227:
3217:
3210:
3101:
3097:
3093:
3083:
3004:
3000:
2990:
2986:
2907:
2903:
2839:
2835:
2825:
2821:
2814:
2801:
2797:
2787:
2783:
2662:
2658:
2648:
2644:
2634:
2630:
2620:
2616:
2383:
2379:
2369:
2365:
2355:
2351:
2281:
2277:
2267:
2263:
2161:
2157:
2150:
2143:
2024:
2020:
2010:
2006:
1996:
1992:
1919:
1915:
1905:
1901:
1894:
1881:
1877:
1807:
1803:
1172:is the largest class of frames such that
69:Learn how and when to remove this message
8454:. Oxford University Press. p. 256.
8272:Bull, Robert A.; Segerberg, K. (2012) .
4206:usually produce finite models directly.
4196:
4192:
4161:
1053:A modal logic (i.e., a set of formulas)
32:This article includes a list of general
8615:. London: Routledge. pp. 211–244.
8602:
8587:
8393:
8369:
8295:Chagrov, A.; Zakharyaschev, M. (1997).
8227:
8191:
7074:We can transform a Kripke model into a
5311:{\displaystyle \langle W,\leq \rangle }
4694:{\displaystyle \langle W,\leq \rangle }
4082:is valid in every frame that satisfies
3417:
8724:
8703:Moschovakis, Joan (16 December 2022).
8665:
6422:
6251:{\displaystyle w\Vdash (\forall x\,A)}
6118:{\displaystyle w\Vdash (\exists x\,A)}
5466:, we define the satisfaction relation
2926:{\displaystyle w\,R\,v\Rightarrow w=v}
481:and modal formulas, such that for all
6915:{\displaystyle \langle W',R'\rangle }
6513:{\displaystyle \langle W',R'\rangle }
5640:{\displaystyle P(t_{1},\dots ,t_{n})}
4486:with a single accessibility relation
3713:construction in algebraic semantics.
1970:{\displaystyle \Box A\to \Box \Box A}
1855:{\displaystyle \Box \Box A\to \Box A}
1605:{\displaystyle \Box A\to \Box \Box A}
7930:adding citations to reliable sources
7897:
7345:varies; in the simplest case we put
5366:realizations of function symbols in
4725:satisfies the following conditions:
4227:
2964:{\displaystyle \Diamond A\to \Box A}
2241:{\displaystyle \Diamond \Box A\to A}
2209:{\displaystyle A\to \Box \Diamond A}
2069:{\displaystyle \Box A\to \Diamond A}
299:
210:("necessarily"). The modal operator
18:
8714:Stanford Encyclopedia of Philosophy
8677:Stanford Encyclopedia of Philosophy
8662:N.B: Constructive = intuitionistic.
8636:
7755:. In typical applications, we take
7188:is the set of all finite sequences
6873:{\displaystyle \langle W,R\rangle }
6471:{\displaystyle \langle W,R\rangle }
5318:is an intuitionistic Kripke frame,
4021:-consistent set is contained in an
3744:-consistent set that has no proper
3688:
1253:{\displaystyle \langle W,R\rangle }
950:{\displaystyle \langle W,R\rangle }
450:{\displaystyle \langle W,R\rangle }
337:{\displaystyle \langle W,R\rangle }
13:
8657:Introduction to Mathematical Logic
7994:, building on unpublished work of
6840:P-morphisms are a special kind of
6419:is often used in this connection.
6226:
6093:
6053:
5696:{\displaystyle w\Vdash (A\land B)}
5119:
4547:
4533:{\displaystyle w\Vdash \Box _{i}A}
4349:
4335:{\displaystyle w\Vdash \Box _{i}A}
4040:immediately imply completeness of
3997:The canonical model is a model of
3682:transitive, serial, and Euclidean
3565:
3541:
3534:
3397:
3365:
3218:
3211:
3204:
3182:
3122:
3084:
3077:
2808:
2144:
2137:
2112:
2106:
2086:
1888:
1063:with respect to a class of frames
513:
280:
274:
230:("possibly") is (classically) the
177:
38:it lacks sufficient corresponding
14:
8768:
8666:Garson, James (23 January 2023).
8630:
8613:Wittgenstein's Enduring Arguments
8379:(2nd ed.). Clarendon Press.
8326:A New Introduction to Modal Logic
7059:, for any propositional variable
6832:, for any propositional variable
5815:{\displaystyle w\Vdash (A\lor B)}
4616:Semantics of intuitionistic logic
4224:has FMP, and is Kripke complete.
4144:a Sahlqvist formula is canonical,
3864:if and only if for every formula
8596:Edinburgh Research Archive (ERA)
7902:
6383:) is the evaluation which gives
5934:{\displaystyle w\Vdash (A\to B)}
5206:Intuitionistic first-order logic
4846:{\displaystyle w\Vdash A\land B}
23:
8353:Alternatives to Classical Logic
6756:{\displaystyle f\colon W\to W'}
6550:{\displaystyle f\colon W\to W'}
6438:in Kripke semantics are called
6125:if and only if there exists an
4932:{\displaystyle w\Vdash A\lor B}
4288:equipped with binary relations
4063:We say that a formula or a set
3310:{\displaystyle \Box \to \Box B}
2118:{\displaystyle \neg \Box \bot }
1508:does not prove all theorems of
1101:Correspondence and completeness
477:is a relation between nodes of
348:is a (possibly empty) set, and
258:in terms of necessity like so:
8426:. Springer. pp. XV, 198.
8257:. Cambridge University Press.
8197:
8172:
8100:
8083:Shoham & Leyton-Brown 2008
8066:
7681:{\displaystyle u\Vdash \Box A}
6924:B ⊆ W × W’
6808:
6802:
6742:
6635:A p-morphism of Kripke models
6536:
6354:
6351:
6345:
6339:
6333:
6245:
6239:
6236:
6223:
6195:
6192:
6186:
6180:
6174:
6112:
6106:
6103:
6090:
6062:
6056:
6024:
6018:
5989:
5983:
5928:
5922:
5919:
5913:
5907:
5879:
5873:
5844:
5838:
5809:
5803:
5800:
5788:
5760:
5754:
5725:
5719:
5690:
5684:
5681:
5669:
5634:
5631:
5625:
5603:
5597:
5584:
5561:
5555:
5552:
5520:
5488:
5482:
5018:{\displaystyle w\Vdash A\to B}
5009:
4593:
4581:
4567:
4389:
4377:
4356:
3601:
3598:
3586:
3572:
3548:
3391:
3371:{\displaystyle \Diamond \top }
3317:, which logically establishes
3298:
3295:
3286:
3280:
3274:
3271:
3235:
3221:
3049:
3008:
2952:
2911:
2872:
2843:
2815:
2805:
2749:
2698:
2638:
2592:
2586:
2577:
2568:
2562:
2553:
2511:
2508:
2502:
2499:
2490:
2484:
2478:
2431:
2428:
2422:
2413:
2373:
2314:
2271:
2232:
2194:
2165:
2151:
2092:{\displaystyle \Diamond \top }
2057:
2014:
1955:
1923:
1895:
1885:
1843:
1772:
1737:
1728:
1719:
1716:
1713:
1707:
1701:
1590:
1550:
1547:
1538:
1532:
1403:{\displaystyle w\Vdash \Box p}
1282:{\displaystyle w\Vdash \Box A}
1204:
1042:is a set of formulas, let Mod(
665:{\displaystyle w\Vdash \Box A}
582:{\displaystyle w\Vdash A\to B}
573:
522:{\displaystyle w\Vdash \neg A}
157:
1:
8278:Extensions of Classical Logic
8240:
7859:Computer science applications
7337:for a propositional variable
7330:{\displaystyle w_{n}\Vdash p}
6825:{\displaystyle f(w)\Vdash 'p}
6068:{\displaystyle w\Vdash \bot }
5356:is included in the domain of
5125:{\displaystyle w\Vdash \bot }
4733:is a propositional variable,
3697:, a Kripke model (called the
3430:to a larger class of frames.
112:. It was first conceived for
7880:theoretical computer science
6391:, and otherwise agrees with
5459:of variables by elements of
5218:language. A Kripke model of
3693:For any normal modal logic,
995:for all possible choices of
146:, a set of truth-functional
7:
8691:Encyclopedia of Mathematics
8033:
7759:as the projection onto the
7710:{\displaystyle \Box A\in X}
7052:{\displaystyle w'\Vdash 'p}
4629:intuitionistic Kripke model
4089:for any normal modal logic
4071:with respect to a property
3906:{\displaystyle \Box A\in X}
2884:{\displaystyle A\to \Box A}
1781:{\displaystyle \Box A\to A}
1616:Common modal axiom schemata
1213:{\displaystyle \Box A\to A}
1144:Japaridze's polymodal logic
10:
8773:
8650:Detlovs, V.; Podnieks, K.
8448:Giaquinto, Marcus (2002).
7884:labeled transition systems
7862:
6409:existential quantification
6310:{\displaystyle a\in M_{u}}
6151:{\displaystyle a\in M_{w}}
4231:
4176:question: it follows from
3348:{\displaystyle \Diamond A}
3188:{\displaystyle \Box \bot }
2532:reflexive and transitive,
1752:holds true for any frames
608:{\displaystyle w\nVdash A}
548:{\displaystyle w\nVdash A}
190:), and the modal operator
131:
92:, and often confused with
7831:{\displaystyle v\Vdash A}
7805:{\displaystyle u\Vdash A}
7736:{\displaystyle v\Vdash A}
7297:{\displaystyle s\Vdash p}
7016:{\displaystyle w\Vdash p}
6785:{\displaystyle w\Vdash p}
6360:{\displaystyle u\Vdash A}
6258:if and only if for every
6201:{\displaystyle w\Vdash A}
6030:{\displaystyle u\Vdash B}
5995:{\displaystyle u\Vdash A}
5885:{\displaystyle w\Vdash B}
5850:{\displaystyle w\Vdash A}
5766:{\displaystyle w\Vdash B}
5731:{\displaystyle w\Vdash A}
5494:{\displaystyle w\Vdash A}
5329:-structure for each node
5096:{\displaystyle u\Vdash B}
5070:{\displaystyle u\Vdash A}
4984:{\displaystyle w\Vdash B}
4958:{\displaystyle w\Vdash A}
4898:{\displaystyle w\Vdash B}
4872:{\displaystyle w\Vdash A}
4804:{\displaystyle u\Vdash p}
4778:{\displaystyle w\Vdash p}
4009:contains all theorems of
3960:{\displaystyle X\Vdash A}
3711:Lindenbaum–Tarski algebra
3325:in every possible world.
3140:uniqueness quantification
3131:{\displaystyle \exists !}
1487:of modal logics: suppose
1429:{\displaystyle w\Vdash p}
1362:{\displaystyle u\Vdash p}
1331:has to be reflexive: fix
1308:{\displaystyle w\Vdash A}
907:{\displaystyle w\Vdash A}
769:{\displaystyle w\Vdash A}
691:{\displaystyle u\Vdash A}
634:{\displaystyle w\Vdash B}
223:{\displaystyle \Diamond }
8603:Stokhof, Martin (2008).
8376:Elements of Intuitionism
8125:which makes the formula
8060:
7973:possible world semantics
3730:maximal L-consistent set
1452:using the definition of
128:Semantics of modal logic
94:possible world semantics
8660:. University of Latvia.
8522:Advances in Modal Logic
7894:History and terminology
7869:state transition system
7846:General frame semantics
7778:if and only if for all
7512:-filtration of a model
6277:{\displaystyle u\geq w}
5960:{\displaystyle u\geq w}
5941:if and only if for all
5044:{\displaystyle u\geq w}
5025:if and only if for all
4752:{\displaystyle w\leq u}
4718:{\displaystyle \Vdash }
3857:{\displaystyle X\;R\;Y}
3826:{\displaystyle \Vdash }
3707:maximal consistent sets
1814:{\displaystyle w\,R\,w}
1465:{\displaystyle \Vdash }
1008:{\displaystyle \Vdash }
813:{\displaystyle \Vdash }
739:{\displaystyle w\;R\;u}
470:{\displaystyle \Vdash }
457:is a Kripke frame, and
144:propositional variables
116:, and later adapted to
53:more precise citations.
8705:"Intuitionistic Logic"
8588:Simpson, Alex (1994).
8371:Dummett, Michael A. E.
8349:"Intuitionistic Logic"
8251:; Venema, Yde (2002).
7832:
7806:
7737:
7711:
7682:
7605:
7544:
7484:
7331:
7298:
7253:
7178:
7114:
7053:
7017:
6916:
6874:
6826:
6786:
6757:
6720:
6667:
6551:
6514:
6472:
6417:Kripke–Joyal semantics
6399:Kripke–Joyal semantics
6361:
6311:
6278:
6252:
6202:
6152:
6119:
6069:
6031:
5996:
5961:
5935:
5886:
5851:
5816:
5767:
5732:
5697:
5641:
5568:
5495:
5312:
5280:
5126:
5097:
5071:
5045:
5019:
4985:
4959:
4933:
4899:
4873:
4847:
4805:
4779:
4753:
4719:
4695:
4663:
4603:
4534:
4480:
4399:
4336:
4278:
3987:
3986:{\displaystyle A\in X}
3961:
3933:
3932:{\displaystyle A\in Y}
3907:
3878:
3858:
3827:
3791:
3748:-consistent superset.
3728:, and Modus Ponens. A
3608:
3404:
3372:
3349:
3311:
3242:
3189:
3166:
3165:{\displaystyle \Box A}
3132:
3109:
3062:
3024:
2965:
2927:
2885:
2850:
2765:
2714:
2670:
2599:
2521:
2444:
2391:
2330:
2289:
2242:
2210:
2172:
2119:
2093:
2070:
2032:
1971:
1930:
1856:
1815:
1782:
1744:
1670:symbolic logic systems
1606:
1563:
1466:
1430:
1404:
1363:
1309:
1283:
1254:
1214:
1130:of Kripke frames, Thm(
1009:
989:
951:
908:
882:
814:
770:
740:
712:
692:
666:
635:
609:
583:
549:
523:
471:
451:
419:
378:accessibility relation
338:
290:
248:
224:
204:
184:
164:
140:countably infinite set
8609:and formal semantics"
8143:universe of discourse
8055:Muddy Children Puzzle
7833:
7807:
7738:
7712:
7683:
7606:
7545:
7504:for many logics. Let
7485:
7332:
7299:
7254:
7179:
7115:
7054:
7018:
6917:
6875:
6827:
6787:
6758:
6721:
6668:
6552:
6515:
6473:
6362:
6312:
6279:
6253:
6203:
6153:
6120:
6070:
6032:
5997:
5962:
5936:
5887:
5852:
5817:
5768:
5733:
5698:
5642:
5569:
5496:
5380:agree on elements of
5313:
5281:
5200:finite model property
5127:
5098:
5072:
5046:
5020:
4986:
4960:
4934:
4900:
4874:
4848:
4806:
4780:
4754:
4720:
4696:
4664:
4620:Kripke semantics for
4604:
4535:
4481:
4400:
4337:
4279:
4169:finite model property
4162:Finite model property
4075:of Kripke frames, if
3988:
3962:
3934:
3908:
3879:
3859:
3828:
3792:
3716:A set of formulas is
3609:
3405:
3373:
3350:
3312:
3243:
3190:
3167:
3133:
3110:
3063:
3025:
2966:
2928:
2886:
2851:
2766:
2715:
2671:
2600:
2522:
2445:
2392:
2331:
2290:
2243:
2211:
2173:
2120:
2094:
2071:
2033:
1972:
1931:
1857:
1816:
1783:
1745:
1607:
1564:
1467:
1431:
1405:
1364:
1310:
1284:
1255:
1215:
1156:to a class of frames
1149:A normal modal logic
1116:syntactic consequence
1010:
990:
952:
909:
883:
822:satisfaction relation
815:
771:
741:
713:
693:
667:
636:
610:
584:
550:
524:
472:
452:
420:
339:
291:
249:
247:{\displaystyle \Box }
225:
205:
203:{\displaystyle \Box }
185:
183:{\displaystyle \neg }
165:
7979:J.C.C. McKinsey and
7926:improve this section
7816:
7790:
7721:
7692:
7663:
7562:
7516:
7352:
7308:
7282:
7192:
7135:
7131:, we define a model
7086:
7027:
7001:
6884:
6852:
6796:
6770:
6730:
6677:
6639:
6524:
6482:
6450:
6442:(which is short for
6321:
6288:
6262:
6214:
6162:
6129:
6081:
6044:
6006:
5971:
5945:
5898:
5861:
5826:
5779:
5742:
5707:
5660:
5578:
5508:
5470:
5455:Given an evaluation
5290:
5226:
5110:
5081:
5055:
5029:
4997:
4969:
4943:
4911:
4883:
4857:
4825:
4789:
4763:
4737:
4709:
4673:
4635:
4622:intuitionistic logic
4544:
4508:
4420:
4346:
4310:
4242:
3971:
3945:
3917:
3888:
3868:
3840:
3817:
3809:, and the relations
3763:
3643:strict partial order
3531:
3502:equivalence relation
3446:
3418:Common modal systems
3382:
3359:
3336:
3328:Note that for axiom
3265:
3201:
3176:
3153:
3119:
3074:
3040:
2980:
2943:
2897:
2866:
2777:
2737:
2686:
2610:
2547:
2472:
2407:
2345:
2305:
2257:
2220:
2188:
2134:
2103:
2080:
2048:
1986:
1946:
1871:
1831:
1797:
1763:
1695:
1581:
1526:
1456:
1414:
1385:
1347:
1293:
1264:
1232:
1195:
1187:Consider the schema
1111:semantic consequence
999:
961:
957:, if it is valid in
929:
892:
854:
804:
754:
722:
702:
676:
647:
619:
593:
561:
533:
504:
461:
429:
391:
316:
262:
238:
214:
194:
174:
163:{\displaystyle \to }
154:
118:intuitionistic logic
86:relational semantics
8757:Non-classical logic
8747:Philosophical logic
8301:. Clarendon Press.
8274:"Basic Modal Logic"
8180:Andrzej Grzegorczyk
8095:Gasquet et al. 2013
8040:Alexandrov topology
6423:Model constructions
5444:, then it holds in
4048:work for arbitrary
3355:implicitly implies
1664:are named based on
1632:is named after the
1168:). In other words,
489:and modal formulas
102:non-classical logic
8742:Mathematical logic
8540:Mac Lane, Saunders
8205:Moschovakis (2022)
8050:Two-dimensionalism
8045:Normal modal logic
7828:
7802:
7767:over the relation
7733:
7707:
7678:
7601:
7540:
7480:
7327:
7294:
7249:
7174:
7110:
7049:
7013:
6963:u’ R’ v’
6952:u’ R’ v’
6912:
6870:
6822:
6782:
6763:, which satisfies
6753:
6716:
6663:
6547:
6510:
6468:
6444:pseudo-epimorphism
6357:
6307:
6274:
6248:
6198:
6148:
6115:
6065:
6027:
5992:
5957:
5931:
5882:
5847:
5812:
5763:
5728:
5693:
5637:
5564:
5491:
5308:
5276:
5122:
5093:
5067:
5041:
5015:
4981:
4955:
4929:
4895:
4869:
4843:
4801:
4775:
4749:
4715:
4705:Kripke frame, and
4691:
4659:
4599:
4530:
4476:
4410:provability logics
4395:
4332:
4274:
4138:Sahlqvist formulas
4128:In general, it is
3983:
3957:
3929:
3903:
3874:
3854:
3823:
3801:is the set of all
3787:
3759:is a Kripke model
3604:
3424:sound and complete
3400:
3368:
3345:
3307:
3238:
3185:
3162:
3128:
3105:
3058:
3020:
2961:
2923:
2881:
2846:
2761:
2710:
2666:
2595:
2517:
2440:
2387:
2326:
2285:
2238:
2206:
2168:
2115:
2089:
2066:
2028:
1967:
1926:
1852:
1811:
1778:
1740:
1602:
1559:
1462:
1426:
1400:
1359:
1305:
1279:
1250:
1210:
1180:. It follows that
1136:normal modal logic
1038:. Conversely, if
1005:
985:
947:
904:
878:
810:
766:
736:
708:
688:
662:
631:
605:
579:
545:
519:
467:
447:
415:
334:
286:
244:
220:
200:
180:
160:
8637:Burgess, John P.
8559:978-1-4612-0927-0
8511:Goldblatt, Robert
8502:978-0-08-046303-2
8477:Goldblatt, Robert
8412:978-0-444-53418-7
8405:. North-Holland.
8386:978-0-19-850524-2
8362:978-94-009-5203-4
8336:978-1-134-80028-5
8308:978-0-19-853779-3
8287:978-94-009-6259-0
8264:978-1-316-10195-7
8097:, pp. 14–16.
8027:Evert Willem Beth
7962:
7961:
7954:
7888:program execution
7120:and a fixed node
5325:is a (classical)
5148:→ ⊥. If for all
4228:Multimodal logics
3877:{\displaystyle A}
3740:for short) is an
3686:
3685:
3652:Grz or T, 4, Grz
3323:rule of inference
3251:
3250:
2536:−Id well-founded
1164: = Mod(
1107:derivation system
1093: ⊇ Thm(
1071: ⊆ Thm(
711:{\displaystyle u}
300:Basic definitions
150:(in this article
79:
78:
71:
8764:
8718:
8709:Zalta, Edward N.
8699:
8681:
8672:Zalta, Edward N.
8661:
8646:
8641:. Archived from
8626:
8599:
8584:
8563:
8535:
8519:
8506:
8494:
8485:
8472:
8470:
8468:
8444:
8442:
8440:
8416:
8404:
8390:
8366:
8340:
8312:
8291:
8268:
8235:
8225:
8219:
8213:
8207:
8201:
8195:
8189:
8183:
8176:
8170:
8164:
8158:
8104:
8098:
8092:
8086:
8080:
8074:
8070:
8020:Richard Montague
7957:
7950:
7946:
7943:
7937:
7906:
7898:
7865:Kripke structure
7837:
7835:
7834:
7829:
7811:
7809:
7808:
7803:
7747:It follows that
7742:
7740:
7739:
7734:
7716:
7714:
7713:
7708:
7687:
7685:
7684:
7679:
7610:
7608:
7607:
7602:
7597:
7586:
7575:
7549:
7547:
7546:
7541:
7489:
7487:
7486:
7481:
7476:
7475:
7457:
7456:
7438:
7437:
7425:
7424:
7411:
7399:
7398:
7380:
7379:
7367:
7366:
7336:
7334:
7333:
7328:
7320:
7319:
7303:
7301:
7300:
7295:
7274: <
7258:
7256:
7255:
7250:
7245:
7244:
7226:
7225:
7213:
7212:
7183:
7181:
7180:
7175:
7170:
7159:
7148:
7119:
7117:
7116:
7111:
7082:. Given a model
7058:
7056:
7055:
7050:
7045:
7037:
7022:
7020:
7019:
7014:
6995:w B w’
6975:v B v’
6959:u B u’
6948:v B v’
6932:u B u’
6921:
6919:
6918:
6913:
6908:
6897:
6879:
6877:
6876:
6871:
6844:. In general, a
6831:
6829:
6828:
6823:
6818:
6791:
6789:
6788:
6783:
6762:
6760:
6759:
6754:
6752:
6725:
6723:
6722:
6717:
6712:
6701:
6690:
6672:
6670:
6669:
6664:
6556:
6554:
6553:
6548:
6546:
6519:
6517:
6516:
6511:
6506:
6495:
6477:
6475:
6474:
6469:
6427:As in classical
6366:
6364:
6363:
6358:
6316:
6314:
6313:
6308:
6306:
6305:
6283:
6281:
6280:
6275:
6257:
6255:
6254:
6249:
6207:
6205:
6204:
6199:
6157:
6155:
6154:
6149:
6147:
6146:
6124:
6122:
6121:
6116:
6074:
6072:
6071:
6066:
6036:
6034:
6033:
6028:
6001:
5999:
5998:
5993:
5966:
5964:
5963:
5958:
5940:
5938:
5937:
5932:
5891:
5889:
5888:
5883:
5856:
5854:
5853:
5848:
5821:
5819:
5818:
5813:
5772:
5770:
5769:
5764:
5737:
5735:
5734:
5729:
5702:
5700:
5699:
5694:
5646:
5644:
5643:
5638:
5624:
5623:
5596:
5595:
5573:
5571:
5570:
5565:
5551:
5550:
5532:
5531:
5500:
5498:
5497:
5492:
5317:
5315:
5314:
5309:
5285:
5283:
5282:
5277:
5272:
5271:
5256:
5255:
5136:The negation of
5131:
5129:
5128:
5123:
5102:
5100:
5099:
5094:
5076:
5074:
5073:
5068:
5050:
5048:
5047:
5042:
5024:
5022:
5021:
5016:
4990:
4988:
4987:
4982:
4964:
4962:
4961:
4956:
4938:
4936:
4935:
4930:
4904:
4902:
4901:
4896:
4878:
4876:
4875:
4870:
4852:
4850:
4849:
4844:
4810:
4808:
4807:
4802:
4784:
4782:
4781:
4776:
4758:
4756:
4755:
4750:
4724:
4722:
4721:
4716:
4700:
4698:
4697:
4692:
4668:
4666:
4665:
4660:
4608:
4606:
4605:
4600:
4565:
4564:
4539:
4537:
4536:
4531:
4526:
4525:
4485:
4483:
4482:
4477:
4466:
4465:
4450:
4449:
4404:
4402:
4401:
4396:
4372:
4371:
4341:
4339:
4338:
4333:
4328:
4327:
4283:
4281:
4280:
4275:
4257:
4256:
4234:Multimodal logic
4166:A logic has the
4134:Henrik Sahlqvist
4125:) is canonical.
3992:
3990:
3989:
3984:
3966:
3964:
3963:
3958:
3938:
3936:
3935:
3930:
3912:
3910:
3909:
3904:
3883:
3881:
3880:
3875:
3863:
3861:
3860:
3855:
3833:are as follows:
3832:
3830:
3829:
3824:
3796:
3794:
3793:
3788:
3689:Canonical models
3613:
3611:
3610:
3605:
3498:T, 5 or D, B, 4
3442:Frame condition
3433:
3432:
3409:
3407:
3406:
3401:
3377:
3375:
3374:
3369:
3354:
3352:
3351:
3346:
3316:
3314:
3313:
3308:
3247:
3245:
3244:
3239:
3194:
3192:
3191:
3186:
3171:
3169:
3168:
3163:
3137:
3135:
3134:
3129:
3114:
3112:
3111:
3106:
3067:
3065:
3064:
3059:
3029:
3027:
3026:
3021:
2974:partial function
2970:
2968:
2967:
2962:
2932:
2930:
2929:
2924:
2890:
2888:
2887:
2882:
2855:
2853:
2852:
2847:
2770:
2768:
2767:
2762:
2719:
2717:
2716:
2711:
2675:
2673:
2672:
2667:
2604:
2602:
2601:
2596:
2526:
2524:
2523:
2518:
2449:
2447:
2446:
2441:
2396:
2394:
2393:
2388:
2335:
2333:
2332:
2327:
2294:
2292:
2291:
2286:
2247:
2245:
2244:
2239:
2215:
2213:
2212:
2207:
2177:
2175:
2174:
2169:
2124:
2122:
2121:
2116:
2098:
2096:
2095:
2090:
2075:
2073:
2072:
2067:
2037:
2035:
2034:
2029:
1976:
1974:
1973:
1968:
1935:
1933:
1932:
1927:
1861:
1859:
1858:
1853:
1820:
1818:
1817:
1812:
1787:
1785:
1784:
1779:
1749:
1747:
1746:
1741:
1684:Frame condition
1675:
1674:
1668:'s numbering of
1654:L. E. J. Brouwer
1611:
1609:
1608:
1603:
1568:
1566:
1565:
1560:
1471:
1469:
1468:
1463:
1435:
1433:
1432:
1427:
1409:
1407:
1406:
1401:
1368:
1366:
1365:
1360:
1314:
1312:
1311:
1306:
1288:
1286:
1285:
1280:
1259:
1257:
1256:
1251:
1224:is valid in any
1219:
1217:
1216:
1211:
1014:
1012:
1011:
1006:
994:
992:
991:
986:
956:
954:
953:
948:
913:
911:
910:
905:
887:
885:
884:
879:
819:
817:
816:
811:
800:”. The relation
788:is satisfied in
775:
773:
772:
767:
745:
743:
742:
737:
717:
715:
714:
709:
697:
695:
694:
689:
671:
669:
668:
663:
640:
638:
637:
632:
614:
612:
611:
606:
588:
586:
585:
580:
554:
552:
551:
546:
528:
526:
525:
520:
476:
474:
473:
468:
456:
454:
453:
448:
424:
422:
421:
416:
376:is known as the
343:
341:
340:
335:
295:
293:
292:
287:
253:
251:
250:
245:
229:
227:
226:
221:
209:
207:
206:
201:
189:
187:
186:
181:
169:
167:
166:
161:
82:Kripke semantics
74:
67:
63:
60:
54:
49:this article by
40:inline citations
27:
26:
19:
8772:
8771:
8767:
8766:
8765:
8763:
8762:
8761:
8722:
8721:
8686:"Kripke models"
8684:
8639:"Kripke Models"
8633:
8623:
8581:
8560:
8532:
8517:
8503:
8492:
8483:
8466:
8464:
8462:
8438:
8436:
8434:
8413:
8395:Fitting, Melvin
8387:
8363:
8345:Van Dalen, Dirk
8337:
8317:Cresswell, M.J.
8309:
8288:
8265:
8247:Blackburn, P.;
8243:
8238:
8226:
8222:
8216:Goldblatt 2006b
8214:
8210:
8202:
8198:
8190:
8186:
8177:
8173:
8165:
8161:
8105:
8101:
8093:
8089:
8081:
8077:
8071:
8067:
8063:
8036:
8013:Jaakko Hintikka
7958:
7947:
7941:
7938:
7923:
7907:
7896:
7875:
7863:Main articles:
7861:
7848:
7817:
7814:
7813:
7812:if and only if
7791:
7788:
7787:
7775:
7722:
7719:
7718:
7693:
7690:
7689:
7664:
7661:
7660:
7590:
7579:
7568:
7563:
7560:
7559:
7517:
7514:
7513:
7465:
7461:
7452:
7448:
7433:
7429:
7420:
7416:
7404:
7394:
7390:
7375:
7371:
7362:
7358:
7353:
7350:
7349:
7315:
7311:
7309:
7306:
7305:
7304:if and only if
7283:
7280:
7279:
7268:
7264:
7240:
7236:
7221:
7217:
7208:
7204:
7193:
7190:
7189:
7163:
7152:
7141:
7136:
7133:
7132:
7126:
7087:
7084:
7083:
7038:
7030:
7028:
7025:
7024:
7023:if and only if
7002:
6999:
6998:
6987:atomic formulas
6979:u R v
6965:, there exists
6938:, there exists
6936:u R v
6901:
6890:
6885:
6882:
6881:
6853:
6850:
6849:
6848:between frames
6811:
6797:
6794:
6793:
6792:if and only if
6771:
6768:
6767:
6745:
6731:
6728:
6727:
6705:
6694:
6683:
6678:
6675:
6674:
6640:
6637:
6636:
6617:u R v
6566:u R v
6539:
6525:
6522:
6521:
6499:
6488:
6483:
6480:
6479:
6451:
6448:
6447:
6425:
6401:
6322:
6319:
6318:
6301:
6297:
6289:
6286:
6285:
6263:
6260:
6259:
6215:
6212:
6211:
6163:
6160:
6159:
6142:
6138:
6130:
6127:
6126:
6082:
6079:
6078:
6045:
6042:
6041:
6007:
6004:
6003:
5972:
5969:
5968:
5946:
5943:
5942:
5899:
5896:
5895:
5862:
5859:
5858:
5827:
5824:
5823:
5822:if and only if
5780:
5777:
5776:
5743:
5740:
5739:
5708:
5705:
5704:
5703:if and only if
5661:
5658:
5657:
5652:
5619:
5615:
5591:
5587:
5579:
5576:
5575:
5574:if and only if
5546:
5542:
5527:
5523:
5509:
5506:
5505:
5471:
5468:
5467:
5464:
5449:
5442:
5435:
5429:
5417:
5410:
5404:
5394:-ary predicate
5385:
5378:
5371:
5361:
5354:
5323:
5291:
5288:
5287:
5261:
5257:
5251:
5247:
5227:
5224:
5223:
5208:
5190:
5176:
5166:
5111:
5108:
5107:
5082:
5079:
5078:
5056:
5053:
5052:
5030:
5027:
5026:
4998:
4995:
4994:
4970:
4967:
4966:
4944:
4941:
4940:
4939:if and only if
4912:
4909:
4908:
4884:
4881:
4880:
4858:
4855:
4854:
4853:if and only if
4826:
4823:
4822:
4815:condition (cf.
4790:
4787:
4786:
4764:
4761:
4760:
4738:
4735:
4734:
4710:
4707:
4706:
4674:
4671:
4670:
4636:
4633:
4632:
4618:
4560:
4556:
4545:
4542:
4541:
4540:if and only if
4521:
4517:
4509:
4506:
4505:
4495:
4455:
4451:
4445:
4441:
4421:
4418:
4417:
4416:is a structure
4367:
4363:
4347:
4344:
4343:
4342:if and only if
4323:
4319:
4311:
4308:
4307:
4293:
4252:
4248:
4243:
4240:
4239:
4236:
4230:
4204:sequent calculi
4164:
4121:(in fact, even
4067:of formulas is
3972:
3969:
3968:
3967:if and only if
3946:
3943:
3942:
3918:
3915:
3914:
3889:
3886:
3885:
3869:
3866:
3865:
3841:
3838:
3837:
3818:
3815:
3814:
3764:
3761:
3760:
3753:canonical model
3699:canonical model
3691:
3532:
3529:
3528:
3420:
3383:
3380:
3379:
3360:
3357:
3356:
3337:
3334:
3333:
3266:
3263:
3262:
3202:
3199:
3198:
3177:
3174:
3173:
3154:
3151:
3150:
3120:
3117:
3116:
3075:
3072:
3071:
3041:
3038:
3037:
2981:
2978:
2977:
2944:
2941:
2940:
2898:
2895:
2894:
2867:
2864:
2863:
2778:
2775:
2774:
2738:
2735:
2734:
2722:(a complicated
2687:
2684:
2683:
2611:
2608:
2607:
2548:
2545:
2544:
2473:
2470:
2469:
2408:
2405:
2404:
2346:
2343:
2342:
2306:
2303:
2302:
2258:
2255:
2254:
2221:
2218:
2217:
2189:
2186:
2185:
2135:
2132:
2131:
2104:
2101:
2100:
2081:
2078:
2077:
2049:
2046:
2045:
1987:
1984:
1983:
1947:
1944:
1943:
1872:
1869:
1868:
1832:
1829:
1828:
1798:
1795:
1794:
1764:
1761:
1760:
1696:
1693:
1692:
1652:is named after
1644:is named after
1638:epistemic logic
1624:is named after
1618:
1582:
1579:
1578:
1527:
1524:
1523:
1521:
1514:
1507:
1500:
1493:
1457:
1454:
1453:
1415:
1412:
1411:
1386:
1383:
1382:
1369:if and only if
1348:
1345:
1344:
1294:
1291:
1290:
1265:
1262:
1261:
1233:
1230:
1229:
1196:
1193:
1192:
1103:
1000:
997:
996:
962:
959:
958:
930:
927:
926:
893:
890:
889:
855:
852:
851:
805:
802:
801:
755:
752:
751:
723:
720:
719:
703:
700:
699:
677:
674:
673:
672:if and only if
648:
645:
644:
620:
617:
616:
594:
591:
590:
589:if and only if
562:
559:
558:
534:
531:
530:
529:if and only if
505:
502:
501:
462:
459:
458:
430:
427:
426:
392:
389:
388:
354:binary relation
317:
314:
313:
302:
263:
260:
259:
239:
236:
235:
215:
212:
211:
195:
192:
191:
175:
172:
171:
155:
152:
151:
136:
130:
90:frame semantics
84:(also known as
75:
64:
58:
55:
45:Please help to
44:
28:
24:
17:
12:
11:
5:
8770:
8760:
8759:
8754:
8749:
8744:
8739:
8734:
8720:
8719:
8700:
8682:
8663:
8647:
8645:on 2004-10-20.
8632:
8631:External links
8629:
8628:
8627:
8621:
8600:
8585:
8580:978-0521899437
8579:
8564:
8558:
8544:Moerdijk, Ieke
8536:
8530:
8507:
8501:
8473:
8460:
8445:
8433:978-3764385033
8432:
8417:
8411:
8391:
8385:
8367:
8361:
8341:
8335:
8313:
8307:
8292:
8286:
8269:
8263:
8242:
8239:
8237:
8236:
8220:
8208:
8196:
8184:
8171:
8167:Giaquinto 2002
8159:
8106:Note that the
8099:
8087:
8075:
8064:
8062:
8059:
8058:
8057:
8052:
8047:
8042:
8035:
8032:
8031:
8030:
8024:
8017:
8010:
7999:
7996:C. A. Meredith
7989:
7985:Bjarni Jónsson
7977:
7960:
7959:
7910:
7908:
7901:
7895:
7892:
7886:, which model
7873:model checking
7860:
7857:
7847:
7844:
7840:
7839:
7827:
7824:
7821:
7801:
7798:
7795:
7773:
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7732:
7729:
7726:
7706:
7703:
7700:
7697:
7677:
7674:
7671:
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7596:
7593:
7589:
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7492:
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7479:
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7455:
7451:
7447:
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7407:
7402:
7397:
7393:
7389:
7386:
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7378:
7374:
7370:
7365:
7361:
7357:
7326:
7323:
7318:
7314:
7293:
7290:
7287:
7266:
7265: R w
7262:
7248:
7243:
7239:
7235:
7232:
7229:
7224:
7220:
7216:
7211:
7207:
7203:
7200:
7197:
7173:
7169:
7166:
7162:
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7147:
7144:
7140:
7124:
7109:
7106:
7103:
7100:
7097:
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7091:
7065:
7064:
7048:
7044:
7041:
7036:
7033:
7012:
7009:
7006:
6983:
6982:
6955:
6922:is a relation
6911:
6907:
6904:
6900:
6896:
6893:
6889:
6869:
6866:
6863:
6860:
6857:
6838:
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6814:
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6807:
6804:
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6751:
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6741:
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6708:
6704:
6700:
6697:
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6689:
6686:
6682:
6662:
6659:
6656:
6653:
6650:
6647:
6644:
6633:
6632:
6627:) =
6607:’, there is a
6589:
6545:
6542:
6538:
6535:
6532:
6529:
6509:
6505:
6502:
6498:
6494:
6491:
6487:
6467:
6464:
6461:
6458:
6455:
6424:
6421:
6400:
6397:
6369:
6368:
6356:
6353:
6350:
6347:
6344:
6341:
6338:
6335:
6332:
6329:
6326:
6304:
6300:
6296:
6293:
6273:
6270:
6267:
6247:
6244:
6241:
6238:
6235:
6231:
6228:
6225:
6222:
6219:
6209:
6197:
6194:
6191:
6188:
6185:
6182:
6179:
6176:
6173:
6170:
6167:
6145:
6141:
6137:
6134:
6114:
6111:
6108:
6105:
6102:
6098:
6095:
6092:
6089:
6086:
6076:
6064:
6061:
6058:
6055:
6052:
6049:
6038:
6026:
6023:
6020:
6017:
6014:
6011:
5991:
5988:
5985:
5982:
5979:
5976:
5956:
5953:
5950:
5930:
5927:
5924:
5921:
5918:
5915:
5912:
5909:
5906:
5903:
5893:
5881:
5878:
5875:
5872:
5869:
5866:
5846:
5843:
5840:
5837:
5834:
5831:
5811:
5808:
5805:
5802:
5799:
5796:
5793:
5790:
5787:
5784:
5774:
5762:
5759:
5756:
5753:
5750:
5747:
5727:
5724:
5721:
5718:
5715:
5712:
5692:
5689:
5686:
5683:
5680:
5677:
5674:
5671:
5668:
5665:
5655:
5650:
5636:
5633:
5630:
5627:
5622:
5618:
5614:
5611:
5608:
5605:
5602:
5599:
5594:
5590:
5586:
5583:
5563:
5560:
5557:
5554:
5549:
5545:
5541:
5538:
5535:
5530:
5526:
5522:
5519:
5516:
5513:
5490:
5487:
5484:
5481:
5478:
5475:
5462:
5453:
5452:
5447:
5440:
5433:
5427:
5415:
5408:
5402:
5388:
5383:
5376:
5369:
5364:
5359:
5352:
5349:the domain of
5321:
5307:
5304:
5301:
5298:
5295:
5275:
5270:
5267:
5264:
5260:
5254:
5250:
5246:
5243:
5240:
5237:
5234:
5231:
5207:
5204:
5188:
5182:vacuously true
5174:
5164:
5134:
5133:
5121:
5118:
5115:
5104:
5092:
5089:
5086:
5066:
5063:
5060:
5040:
5037:
5034:
5014:
5011:
5008:
5005:
5002:
4992:
4980:
4977:
4974:
4954:
4951:
4948:
4928:
4925:
4922:
4919:
4916:
4906:
4894:
4891:
4888:
4868:
4865:
4862:
4842:
4839:
4836:
4833:
4830:
4820:
4800:
4797:
4794:
4774:
4771:
4768:
4748:
4745:
4742:
4714:
4690:
4687:
4684:
4681:
4678:
4658:
4655:
4652:
4649:
4646:
4643:
4640:
4617:
4614:
4610:
4609:
4598:
4595:
4592:
4589:
4586:
4583:
4580:
4576:
4572:
4569:
4563:
4559:
4555:
4552:
4549:
4529:
4524:
4520:
4516:
4513:
4493:
4490:, and subsets
4475:
4472:
4469:
4464:
4461:
4458:
4454:
4448:
4444:
4440:
4437:
4434:
4431:
4428:
4425:
4406:
4405:
4394:
4391:
4388:
4385:
4382:
4379:
4376:
4370:
4366:
4361:
4358:
4354:
4351:
4331:
4326:
4322:
4318:
4315:
4291:
4273:
4270:
4267:
4264:
4260:
4255:
4251:
4247:
4229:
4226:
4214:modal algebras
4178:Post's theorem
4163:
4160:
4156:
4155:
4152:
4145:
4107:
4106:
4093:that contains
4087:
3995:
3994:
3982:
3979:
3976:
3956:
3953:
3950:
3940:
3928:
3925:
3922:
3902:
3899:
3896:
3893:
3873:
3853:
3849:
3845:
3822:
3786:
3783:
3780:
3777:
3774:
3771:
3768:
3690:
3687:
3684:
3683:
3680:
3677:
3673:
3672:
3667:
3664:
3660:
3659:
3653:
3650:
3646:
3645:
3639:
3636:
3629:
3628:
3622:
3619:
3615:
3614:
3603:
3600:
3597:
3594:
3591:
3588:
3585:
3581:
3577:
3574:
3570:
3567:
3564:
3561:
3557:
3553:
3550:
3546:
3543:
3539:
3536:
3525:
3522:
3518:
3517:
3515:total preorder
3512:
3509:
3505:
3504:
3499:
3496:
3490:
3489:
3484:
3481:
3477:
3476:
3473:
3470:
3466:
3465:
3462:
3459:
3455:
3454:
3451:
3448:
3444:
3443:
3440:
3437:
3419:
3416:
3399:
3396:
3393:
3390:
3387:
3367:
3364:
3344:
3341:
3306:
3303:
3300:
3297:
3294:
3291:
3288:
3285:
3282:
3279:
3276:
3273:
3270:
3249:
3248:
3237:
3234:
3230:
3226:
3223:
3220:
3216:
3213:
3209:
3206:
3195:
3184:
3181:
3161:
3158:
3148:
3144:
3143:
3127:
3124:
3104:
3100:
3096:
3092:
3089:
3086:
3082:
3079:
3068:
3057:
3054:
3051:
3048:
3045:
3035:
3031:
3030:
3019:
3016:
3013:
3010:
3007:
3003:
2999:
2996:
2993:
2989:
2985:
2971:
2960:
2957:
2954:
2951:
2948:
2938:
2934:
2933:
2922:
2919:
2916:
2913:
2910:
2906:
2902:
2891:
2880:
2877:
2874:
2871:
2861:
2857:
2856:
2845:
2842:
2838:
2834:
2831:
2828:
2824:
2820:
2817:
2813:
2810:
2807:
2804:
2800:
2796:
2793:
2790:
2786:
2782:
2771:
2760:
2757:
2754:
2751:
2748:
2745:
2742:
2732:
2728:
2727:
2720:
2709:
2706:
2703:
2700:
2697:
2694:
2691:
2681:
2677:
2676:
2665:
2661:
2657:
2654:
2651:
2647:
2643:
2640:
2637:
2633:
2629:
2626:
2623:
2619:
2615:
2605:
2594:
2591:
2588:
2585:
2582:
2579:
2576:
2573:
2570:
2567:
2564:
2561:
2558:
2555:
2552:
2542:
2538:
2537:
2527:
2516:
2513:
2510:
2507:
2504:
2501:
2498:
2495:
2492:
2489:
2486:
2483:
2480:
2477:
2467:
2463:
2462:
2450:
2439:
2436:
2433:
2430:
2427:
2424:
2421:
2418:
2415:
2412:
2402:
2398:
2397:
2386:
2382:
2378:
2375:
2372:
2368:
2364:
2361:
2358:
2354:
2350:
2336:
2325:
2322:
2319:
2316:
2313:
2310:
2300:
2296:
2295:
2284:
2280:
2276:
2273:
2270:
2266:
2262:
2248:
2237:
2234:
2231:
2228:
2225:
2205:
2202:
2199:
2196:
2193:
2183:
2179:
2178:
2167:
2164:
2160:
2156:
2153:
2149:
2146:
2142:
2139:
2125:
2114:
2111:
2108:
2088:
2085:
2065:
2062:
2059:
2056:
2053:
2043:
2039:
2038:
2027:
2023:
2019:
2016:
2013:
2009:
2005:
2002:
1999:
1995:
1991:
1977:
1966:
1963:
1960:
1957:
1954:
1951:
1941:
1937:
1936:
1925:
1922:
1918:
1914:
1911:
1908:
1904:
1900:
1897:
1893:
1890:
1887:
1884:
1880:
1876:
1862:
1851:
1848:
1845:
1842:
1839:
1836:
1826:
1822:
1821:
1810:
1806:
1802:
1788:
1777:
1774:
1771:
1768:
1758:
1754:
1753:
1750:
1739:
1736:
1733:
1730:
1727:
1724:
1721:
1718:
1715:
1712:
1709:
1706:
1703:
1700:
1690:
1686:
1685:
1682:
1679:
1617:
1614:
1601:
1598:
1595:
1592:
1589:
1586:
1558:
1555:
1552:
1549:
1546:
1543:
1540:
1537:
1534:
1531:
1519:
1512:
1505:
1498:
1491:
1485:incompleteness
1461:
1440:, which means
1425:
1422:
1419:
1399:
1396:
1393:
1390:
1358:
1355:
1352:
1304:
1301:
1298:
1278:
1275:
1272:
1269:
1249:
1246:
1243:
1240:
1237:
1209:
1206:
1203:
1200:
1126:For any class
1109:) only if the
1102:
1099:
1030:We define Thm(
1028:
1027:
1016:
1004:
984:
981:
978:
975:
972:
969:
966:
946:
943:
940:
937:
934:
923:
903:
900:
897:
877:
874:
871:
868:
865:
862:
859:
820:is called the
809:
765:
762:
759:
748:
747:
735:
731:
727:
707:
687:
684:
681:
661:
658:
655:
652:
642:
630:
627:
624:
604:
601:
598:
578:
575:
572:
569:
566:
556:
544:
541:
538:
518:
515:
512:
509:
466:
446:
443:
440:
437:
434:
414:
411:
408:
405:
402:
399:
396:
360:. Elements of
333:
330:
327:
324:
321:
301:
298:
285:
282:
279:
276:
273:
270:
267:
256:may be defined
243:
219:
199:
179:
159:
132:Main article:
129:
126:
96:) is a formal
77:
76:
31:
29:
22:
15:
9:
6:
4:
3:
2:
8769:
8758:
8755:
8753:
8750:
8748:
8745:
8743:
8740:
8738:
8735:
8733:
8730:
8729:
8727:
8716:
8715:
8710:
8706:
8701:
8697:
8693:
8692:
8687:
8683:
8679:
8678:
8673:
8669:
8668:"Modal Logic"
8664:
8659:
8658:
8653:
8648:
8644:
8640:
8635:
8634:
8624:
8622:9781134107070
8618:
8614:
8610:
8608:
8601:
8597:
8593:
8592:
8586:
8582:
8576:
8572:
8571:
8565:
8561:
8555:
8551:
8550:
8545:
8541:
8537:
8533:
8527:
8523:
8516:
8512:
8508:
8504:
8498:
8491:
8490:
8482:
8478:
8474:
8463:
8457:
8453:
8452:
8446:
8435:
8429:
8425:
8424:
8418:
8414:
8408:
8403:
8402:
8396:
8392:
8388:
8382:
8378:
8377:
8372:
8368:
8364:
8358:
8354:
8350:
8346:
8342:
8338:
8332:
8329:. Routledge.
8328:
8327:
8322:
8318:
8314:
8310:
8304:
8300:
8299:
8293:
8289:
8283:
8279:
8275:
8270:
8266:
8260:
8256:
8255:
8250:
8245:
8244:
8233:
8229:
8224:
8217:
8212:
8206:
8200:
8193:
8188:
8181:
8175:
8168:
8163:
8156:
8152:
8148:
8145:within which
8144:
8140:
8136:
8132:
8128:
8124:
8120:
8117:
8113:
8109:
8103:
8096:
8091:
8084:
8079:
8069:
8065:
8056:
8053:
8051:
8048:
8046:
8043:
8041:
8038:
8037:
8028:
8025:
8021:
8018:
8014:
8011:
8008:
8003:
8000:
7997:
7993:
7990:
7986:
7982:
7981:Alfred Tarski
7978:
7974:
7970:
7969:Rudolf Carnap
7967:
7966:
7965:
7956:
7953:
7945:
7935:
7931:
7927:
7921:
7920:
7916:
7911:This section
7909:
7905:
7900:
7899:
7891:
7889:
7885:
7881:
7874:
7870:
7866:
7856:
7854:
7853:general frame
7843:
7825:
7822:
7819:
7799:
7796:
7793:
7785:
7782: ∈
7781:
7777:
7770:
7769:
7768:
7766:
7762:
7758:
7754:
7750:
7730:
7727:
7724:
7704:
7701:
7698:
7695:
7675:
7672:
7669:
7666:
7658:
7654:
7650:
7646:
7642:
7638:
7635:
7632: ∈
7631:
7627:
7624:
7621:
7617:
7614:
7613:
7612:
7594:
7591:
7587:
7583:
7580:
7576:
7572:
7569:
7557:
7553:
7550:is a mapping
7534:
7531:
7528:
7525:
7522:
7511:
7507:
7503:
7499:
7495:
7472:
7469:
7466:
7462:
7458:
7453:
7449:
7445:
7442:
7439:
7434:
7430:
7426:
7421:
7417:
7408:
7405:
7395:
7391:
7387:
7384:
7381:
7376:
7372:
7368:
7363:
7359:
7348:
7347:
7346:
7344:
7340:
7324:
7321:
7316:
7312:
7291:
7288:
7285:
7277:
7273:
7269:
7241:
7237:
7233:
7230:
7227:
7222:
7218:
7214:
7209:
7205:
7198:
7195:
7187:
7167:
7164:
7160:
7156:
7153:
7149:
7145:
7142:
7130:
7127: ∈
7123:
7104:
7101:
7098:
7095:
7092:
7081:
7077:
7072:
7070:
7062:
7046:
7042:
7039:
7034:
7031:
7010:
7007:
7004:
6996:
6992:
6991:
6990:
6988:
6980:
6976:
6972:
6969: ∈
6968:
6964:
6960:
6956:
6953:
6949:
6945:
6942: ∈
6941:
6937:
6933:
6929:
6928:
6927:
6925:
6905:
6902:
6898:
6894:
6891:
6864:
6861:
6858:
6847:
6843:
6842:bisimulations
6835:
6819:
6815:
6812:
6805:
6799:
6779:
6776:
6773:
6766:
6765:
6764:
6749:
6746:
6739:
6736:
6733:
6709:
6706:
6702:
6698:
6695:
6691:
6687:
6684:
6657:
6654:
6651:
6648:
6645:
6630:
6626:
6622:
6618:
6614:
6611: ∈
6610:
6606:
6602:
6598:
6594:
6590:
6587:
6583:
6579:
6575:
6571:
6567:
6563:
6560:
6559:
6558:
6543:
6540:
6533:
6530:
6527:
6520:is a mapping
6503:
6500:
6496:
6492:
6489:
6462:
6459:
6456:
6445:
6441:
6437:
6436:homomorphisms
6432:
6430:
6420:
6418:
6414:
6410:
6406:
6396:
6394:
6390:
6386:
6382:
6378:
6374:
6348:
6342:
6336:
6330:
6327:
6324:
6302:
6298:
6294:
6291:
6271:
6268:
6265:
6242:
6233:
6229:
6220:
6217:
6210:
6189:
6183:
6177:
6171:
6168:
6165:
6143:
6139:
6135:
6132:
6109:
6100:
6096:
6087:
6084:
6077:
6059:
6050:
6047:
6039:
6021:
6015:
6012:
6009:
5986:
5980:
5977:
5974:
5954:
5951:
5948:
5925:
5916:
5910:
5904:
5901:
5894:
5876:
5870:
5867:
5864:
5841:
5835:
5832:
5829:
5806:
5797:
5794:
5791:
5785:
5782:
5775:
5757:
5751:
5748:
5745:
5722:
5716:
5713:
5710:
5687:
5678:
5675:
5672:
5666:
5663:
5656:
5653:
5628:
5620:
5616:
5612:
5609:
5606:
5600:
5592:
5588:
5581:
5558:
5547:
5543:
5539:
5536:
5533:
5528:
5524:
5517:
5514:
5511:
5504:
5503:
5502:
5485:
5479:
5476:
5473:
5465:
5458:
5450:
5443:
5436:
5426:
5422:
5418:
5412: ∈
5411:
5401:
5398:and elements
5397:
5393:
5389:
5386:
5379:
5372:
5365:
5362:
5355:
5348:
5347:
5346:
5344:
5341: ≤
5340:
5336:
5333: ∈
5332:
5328:
5324:
5302:
5299:
5296:
5268:
5265:
5262:
5252:
5248:
5241:
5238:
5235:
5232:
5221:
5217:
5213:
5203:
5201:
5196:
5194:
5187:
5183:
5179:
5173:
5169:
5163:
5159:
5155:
5151:
5147:
5143:
5139:
5116:
5113:
5105:
5090:
5087:
5084:
5064:
5061:
5058:
5038:
5035:
5032:
5012:
5006:
5003:
5000:
4993:
4978:
4975:
4972:
4952:
4949:
4946:
4926:
4923:
4920:
4917:
4914:
4907:
4892:
4889:
4886:
4866:
4863:
4860:
4840:
4837:
4834:
4831:
4828:
4821:
4818:
4814:
4798:
4795:
4792:
4772:
4769:
4766:
4746:
4743:
4740:
4732:
4728:
4727:
4726:
4712:
4704:
4685:
4682:
4679:
4653:
4650:
4647:
4644:
4641:
4630:
4625:
4623:
4613:
4596:
4590:
4587:
4584:
4578:
4574:
4570:
4561:
4557:
4553:
4550:
4527:
4522:
4518:
4514:
4511:
4504:
4503:
4502:
4500:
4497: ⊆
4496:
4489:
4470:
4467:
4462:
4459:
4456:
4446:
4442:
4435:
4432:
4429:
4426:
4415:
4414:Carlson model
4411:
4392:
4386:
4383:
4380:
4374:
4368:
4364:
4359:
4352:
4329:
4324:
4320:
4316:
4313:
4306:
4305:
4304:
4302:
4299: ∈
4298:
4294:
4287:
4268:
4265:
4262:
4258:
4253:
4249:
4235:
4225:
4223:
4219:
4215:
4210:
4207:
4205:
4202:
4198:
4194:
4189:
4187:
4183:
4179:
4175:
4171:
4170:
4159:
4153:
4150:
4146:
4143:
4142:
4141:
4139:
4135:
4131:
4126:
4124:
4120:
4114:
4112:
4104:
4100:
4096:
4092:
4088:
4085:
4081:
4078:
4077:
4076:
4074:
4070:
4066:
4061:
4059:
4055:
4051:
4047:
4043:
4039:
4034:
4032:
4028:
4024:
4020:
4016:
4012:
4008:
4004:
4000:
3980:
3977:
3974:
3954:
3951:
3948:
3941:
3926:
3923:
3920:
3900:
3897:
3894:
3891:
3871:
3851:
3847:
3843:
3836:
3835:
3834:
3820:
3812:
3808:
3804:
3800:
3781:
3778:
3775:
3772:
3769:
3758:
3754:
3749:
3747:
3743:
3739:
3735:
3731:
3727:
3723:
3719:
3714:
3712:
3708:
3704:
3700:
3696:
3681:
3678:
3675:
3674:
3671:
3668:
3665:
3662:
3661:
3658:
3657:partial order
3654:
3651:
3648:
3647:
3644:
3640:
3637:
3634:
3631:
3630:
3626:
3623:
3620:
3617:
3616:
3595:
3592:
3589:
3583:
3579:
3575:
3568:
3562:
3559:
3555:
3551:
3544:
3537:
3526:
3523:
3520:
3519:
3516:
3513:
3510:
3507:
3506:
3503:
3500:
3497:
3495:
3492:
3491:
3488:
3485:
3482:
3479:
3478:
3474:
3471:
3468:
3467:
3463:
3460:
3457:
3456:
3452:
3449:
3445:
3441:
3438:
3435:
3434:
3431:
3429:
3425:
3415:
3413:
3394:
3388:
3385:
3362:
3342:
3339:
3331:
3326:
3324:
3320:
3304:
3301:
3292:
3289:
3283:
3277:
3268:
3260:
3256:
3232:
3228:
3224:
3214:
3207:
3196:
3179:
3159:
3156:
3149:
3146:
3145:
3141:
3125:
3102:
3098:
3094:
3090:
3087:
3080:
3069:
3055:
3052:
3046:
3043:
3036:
3033:
3032:
3017:
3014:
3011:
3005:
3001:
2997:
2994:
2991:
2987:
2983:
2975:
2972:
2958:
2955:
2949:
2946:
2939:
2936:
2935:
2920:
2917:
2914:
2908:
2904:
2900:
2892:
2878:
2875:
2869:
2862:
2859:
2858:
2840:
2836:
2832:
2829:
2826:
2822:
2818:
2811:
2802:
2798:
2794:
2791:
2788:
2784:
2780:
2772:
2758:
2755:
2752:
2746:
2743:
2740:
2733:
2730:
2729:
2725:
2721:
2707:
2704:
2701:
2695:
2692:
2689:
2682:
2679:
2678:
2663:
2659:
2655:
2652:
2649:
2645:
2641:
2635:
2631:
2627:
2624:
2621:
2617:
2613:
2606:
2589:
2583:
2580:
2574:
2571:
2565:
2559:
2556:
2550:
2543:
2540:
2539:
2535:
2531:
2528:
2514:
2505:
2496:
2493:
2487:
2481:
2475:
2468:
2465:
2464:
2461:
2458:
2454:
2451:
2437:
2434:
2425:
2419:
2416:
2410:
2403:
2400:
2399:
2384:
2380:
2376:
2370:
2366:
2362:
2359:
2356:
2352:
2348:
2340:
2337:
2323:
2320:
2317:
2311:
2308:
2301:
2298:
2297:
2282:
2278:
2274:
2268:
2264:
2260:
2252:
2249:
2235:
2229:
2226:
2223:
2203:
2200:
2197:
2191:
2184:
2181:
2180:
2162:
2158:
2154:
2147:
2140:
2129:
2126:
2109:
2083:
2063:
2060:
2054:
2051:
2044:
2041:
2040:
2025:
2021:
2017:
2011:
2007:
2003:
2000:
1997:
1993:
1989:
1981:
1978:
1964:
1961:
1958:
1952:
1949:
1942:
1939:
1938:
1920:
1916:
1912:
1909:
1906:
1902:
1898:
1891:
1882:
1878:
1874:
1866:
1863:
1849:
1846:
1840:
1837:
1834:
1827:
1824:
1823:
1808:
1804:
1800:
1792:
1789:
1775:
1769:
1766:
1759:
1756:
1755:
1751:
1734:
1731:
1725:
1722:
1710:
1704:
1698:
1691:
1688:
1687:
1683:
1680:
1677:
1676:
1673:
1671:
1667:
1663:
1659:
1656:; and axioms
1655:
1651:
1647:
1646:deontic logic
1643:
1639:
1635:
1631:
1627:
1623:
1613:
1599:
1596:
1593:
1587:
1584:
1576:
1572:
1556:
1553:
1544:
1541:
1535:
1529:
1518:
1511:
1504:
1497:
1494: ⊆
1490:
1486:
1482:
1477:
1475:
1459:
1451:
1447:
1443:
1439:
1423:
1420:
1417:
1397:
1394:
1391:
1388:
1380:
1376:
1372:
1356:
1353:
1350:
1342:
1338:
1335: ∈
1334:
1330:
1326:
1322:
1318:
1302:
1299:
1296:
1276:
1273:
1270:
1267:
1244:
1241:
1238:
1227:
1223:
1207:
1201:
1198:
1190:
1185:
1183:
1179:
1176:is sound wrt
1175:
1171:
1167:
1163:
1159:
1155:
1152:
1147:
1145:
1141:
1137:
1133:
1129:
1124:
1122:
1118:
1117:
1112:
1108:
1098:
1096:
1092:
1088:
1084:
1083:
1078:
1074:
1070:
1066:
1062:
1061:
1056:
1051:
1049:
1045:
1041:
1037:
1033:
1025:
1021:
1017:
1002:
979:
976:
973:
970:
967:
941:
938:
935:
924:
921:
918: ∈
917:
901:
898:
895:
872:
869:
866:
863:
860:
849:
848:
847:
845:
841:
836:
834:
832:
827:
823:
807:
799:
795:
791:
787:
783:
779:
763:
760:
757:
733:
729:
725:
705:
685:
682:
679:
659:
656:
653:
650:
643:
628:
625:
622:
602:
599:
596:
576:
570:
567:
564:
557:
542:
539:
536:
516:
510:
507:
500:
499:
498:
496:
492:
488:
485: ∈
484:
480:
464:
441:
438:
435:
409:
406:
403:
400:
397:
386:
381:
379:
375:
371:
367:
363:
359:
355:
351:
347:
328:
325:
322:
311:
307:
297:
283:
277:
271:
268:
265:
257:
241:
233:
217:
197:
149:
145:
141:
135:
125:
123:
119:
115:
111:
107:
103:
99:
95:
91:
87:
83:
73:
70:
62:
52:
48:
42:
41:
35:
30:
21:
20:
8752:Sheaf theory
8737:Model theory
8712:
8689:
8675:
8656:
8643:the original
8612:
8606:
8590:
8569:
8552:. Springer.
8548:
8521:
8488:
8465:. Retrieved
8450:
8437:. Retrieved
8422:
8400:
8375:
8352:
8325:
8321:Hughes, G.E.
8297:
8277:
8253:
8249:de Rijke, M.
8231:
8228:Stokhof 2008
8223:
8211:
8199:
8192:Simpson 1994
8187:
8174:
8162:
8154:
8150:
8146:
8142:
8138:
8134:
8130:
8126:
8122:
8118:
8115:
8111:
8107:
8102:
8090:
8078:
8068:
7992:Arthur Prior
7972:
7963:
7948:
7942:October 2009
7939:
7924:Please help
7912:
7882:, they give
7876:
7849:
7841:
7783:
7779:
7771:
7764:
7756:
7752:
7748:
7746:
7656:
7652:
7648:
7644:
7640:
7633:
7629:
7625:
7615:
7555:
7551:
7509:
7505:
7497:
7496:
7493:
7342:
7338:
7275:
7271:
7260:
7185:
7128:
7121:
7079:
7073:
7068:
7066:
7060:
6994:
6984:
6978:
6974:
6970:
6966:
6962:
6958:
6951:
6947:
6943:
6939:
6935:
6931:
6923:
6846:bisimulation
6845:
6839:
6833:
6634:
6628:
6624:
6620:
6616:
6612:
6608:
6604:
6600:
6596:
6592:
6585:
6581:
6577:
6573:
6569:
6565:
6561:
6443:
6439:
6434:The natural
6433:
6429:model theory
6426:
6416:
6413:topos theory
6405:sheaf theory
6402:
6392:
6388:
6384:
6380:
6376:
6372:
6370:
5648:
5460:
5456:
5454:
5445:
5438:
5431:
5424:
5420:
5413:
5406:
5399:
5395:
5391:
5381:
5374:
5367:
5357:
5350:
5342:
5338:
5334:
5330:
5326:
5319:
5222:is a triple
5219:
5211:
5209:
5197:
5192:
5185:
5177:
5171:
5167:
5161:
5157:
5153:
5149:
5145:
5141:
5137:
5135:
4817:monotonicity
4812:
4730:
4631:is a triple
4628:
4626:
4619:
4611:
4498:
4491:
4487:
4413:
4407:
4300:
4296:
4289:
4285:
4237:
4221:
4217:
4211:
4208:
4190:
4185:
4181:
4174:decidability
4167:
4165:
4157:
4140:) such that
4127:
4122:
4118:
4115:
4108:
4102:
4098:
4094:
4090:
4083:
4079:
4072:
4068:
4064:
4062:
4057:
4053:
4049:
4045:
4041:
4037:
4035:
4030:
4026:
4022:
4018:
4015:Zorn's lemma
4010:
4006:
4002:
3998:
3996:
3810:
3806:
3802:
3798:
3756:
3752:
3750:
3745:
3741:
3737:
3733:
3729:
3725:
3721:
3717:
3715:
3702:
3698:
3694:
3692:
3638:GL or 4, GL
3427:
3423:
3421:
3329:
3327:
3319:modus ponens
3257:can also be
3254:
3252:
2773:convergent:
2724:second-order
2533:
2529:
2460:well-founded
2456:
2455:transitive,
2452:
1661:
1657:
1649:
1641:
1629:
1621:
1619:
1574:
1570:
1516:
1509:
1502:
1495:
1488:
1484:
1480:
1478:
1473:
1449:
1445:
1441:
1437:
1378:
1374:
1370:
1343:as follows:
1340:
1336:
1332:
1328:
1324:
1320:
1316:
1221:
1188:
1186:
1181:
1177:
1173:
1169:
1165:
1161:
1157:
1153:
1150:
1148:
1139:
1131:
1127:
1125:
1121:derivability
1120:
1114:
1104:
1094:
1090:
1086:
1080:
1076:
1072:
1068:
1064:
1058:
1054:
1052:
1047:
1043:
1039:
1035:
1031:
1029:
1023:
1019:
919:
915:
843:
839:
837:
829:
825:
821:
797:
793:
789:
785:
781:
777:
749:
494:
490:
486:
482:
478:
387:is a triple
385:Kripke model
384:
382:
373:
369:
365:
361:
357:
349:
345:
309:
306:Kripke frame
305:
303:
137:
122:model theory
114:modal logics
89:
85:
81:
80:
65:
56:
37:
8732:Modal logic
8467:24 December
8439:24 December
8298:Modal Logic
8254:Modal Logic
8002:Stig Kanger
7855:semantics.
7558:to a model
7080:unravelling
6440:p-morphisms
5437:) holds in
5216:first-order
4813:persistency
4197:unravelling
4149:first-order
4130:undecidable
4001:, as every
3649:Grz, S4Grz
3475:transitive
3453:all frames
1666:C. I. Lewis
1634:truth axiom
1626:Saul Kripke
1577:-tautology
1154:corresponds
364:are called
310:modal frame
148:connectives
134:Modal logic
110:André Joyal
106:Saul Kripke
51:introducing
8726:Categories
8594:(Thesis).
8531:1904987206
8461:019875244X
8241:References
8155:as a whole
8131:a model of
8023:community;
7620:surjection
7611:such that
7498:Filtration
7259:such that
6973:such that
6946:such that
6615:such that
6557:such that
6387:the value
6284:and every
6158:such that
5152:such that
4703:preordered
4232:See also:
4193:filtration
4151:definable,
4101:satisfies
3722:consistent
3527:preorder,
3464:reflexive
3428:correspond
3070:function:
2893:discrete:
2726:property)
1980:transitive
1119:relation (
838:A formula
826:evaluation
780:satisfies
718:such that
312:is a pair
59:April 2013
34:references
8696:EMS Press
8607:Tractatus
8546:(2012) .
8513:(2006b).
8479:(2006a).
8347:(2013) .
8323:(2012) .
8135:formula F
7913:does not
7823:⊩
7797:⊩
7728:⊩
7702:∈
7696:◻
7673:◻
7670:⊩
7599:⟩
7592:⊩
7566:⟨
7538:⟩
7535:⊩
7520:⟨
7478:⟩
7443:…
7414:⟨
7401:⟩
7385:…
7356:⟨
7322:⊩
7289:⊩
7247:⟩
7231:…
7202:⟨
7172:⟩
7165:⊩
7139:⟨
7108:⟩
7105:⊩
7090:⟨
7040:⊩
7008:⊩
6910:⟩
6888:⟨
6868:⟩
6856:⟨
6813:⊩
6777:⊩
6743:→
6737::
6714:⟩
6707:⊩
6681:⟨
6661:⟩
6658:⊩
6643:⟨
6591:whenever
6537:→
6531::
6508:⟩
6486:⟨
6466:⟩
6454:⟨
6346:→
6328:⊩
6295:∈
6269:≥
6227:∀
6221:⊩
6187:→
6169:⊩
6136:∈
6094:∃
6088:⊩
6054:⊥
6051:⊩
6013:⊩
5978:⊩
5952:≥
5914:→
5905:⊩
5868:⊩
5833:⊩
5795:∨
5786:⊩
5749:⊩
5714:⊩
5676:∧
5667:⊩
5647:holds in
5610:…
5537:…
5515:⊩
5477:⊩
5390:for each
5306:⟩
5303:≤
5294:⟨
5274:⟩
5266:∈
5239:≤
5230:⟨
5120:⊥
5117:⊩
5088:⊩
5062:⊩
5036:≥
5010:→
5004:⊩
4976:⊩
4950:⊩
4924:∨
4918:⊩
4890:⊩
4864:⊩
4838:∧
4832:⊩
4796:⊩
4770:⊩
4744:≤
4713:⊩
4689:⟩
4686:≤
4677:⟨
4657:⟩
4654:⊩
4648:≤
4639:⟨
4588:⊩
4582:⇒
4554:∈
4548:∀
4519:◻
4515:⊩
4474:⟩
4471:⊩
4460:∈
4424:⟨
4384:⊩
4378:⇒
4350:∀
4321:◻
4317:⊩
4295:for each
4266:∈
4259:∣
4250:◻
4069:canonical
3978:∈
3952:⊩
3924:∈
3898:∈
3892:◻
3821:⊩
3785:⟩
3782:⊩
3767:⟨
3627:preorder
3587:⇒
3566:∀
3563:∧
3542:∃
3535:∀
3398:⊤
3395:◊
3392:→
3386:◊
3366:⊤
3363:◊
3340:◊
3302:◻
3299:→
3290:∧
3281:→
3269:◻
3259:rewritten
3219:¬
3212:∀
3205:∀
3183:⊥
3180:◻
3157:◻
3123:∃
3085:∃
3078:∀
3053:◻
3050:↔
3044:◊
3009:⇒
2995:∧
2956:◻
2953:→
2947:◊
2912:⇒
2876:◻
2873:→
2830:∧
2809:∃
2806:⇒
2792:∧
2756:◊
2753:◻
2750:→
2744:◻
2741:◊
2705:◻
2702:◊
2699:→
2693:◊
2690:◻
2653:∨
2639:⇒
2625:∧
2587:→
2581:◻
2575:◻
2572:∨
2563:→
2557:◻
2551:◻
2512:→
2503:→
2494:◻
2491:→
2482:◻
2476:◻
2435:◻
2432:→
2423:→
2417:◻
2411:◻
2374:⇒
2360:∧
2339:Euclidean
2321:◊
2318:◻
2315:→
2309:◊
2272:⇒
2253: :
2251:symmetric
2233:→
2227:◻
2224:◊
2201:◊
2198:◻
2195:→
2145:∃
2138:∀
2113:⊥
2110:◻
2107:¬
2087:⊤
2084:◊
2061:◊
2058:→
2052:◻
2015:⇒
2001:∧
1962:◻
1959:◻
1956:→
1950:◻
1910:∧
1889:∃
1886:⇒
1847:◻
1844:→
1838:◻
1835:◻
1791:reflexive
1773:→
1767:◻
1732:◻
1729:→
1723:◻
1717:→
1708:→
1699:◻
1597:◻
1594:◻
1591:→
1585:◻
1554:◻
1551:→
1542:◻
1539:↔
1530:◻
1460:⊩
1421:⊩
1395:◻
1392:⊩
1354:⊩
1300:⊩
1274:◻
1271:⊩
1248:⟩
1236:⟨
1226:reflexive
1205:→
1199:◻
1003:⊩
983:⟩
980:⊩
965:⟨
945:⟩
933:⟨
899:⊩
876:⟩
873:⊩
858:⟨
808:⊩
761:⊩
683:⊩
657:◻
654:⊩
626:⊩
600:⊮
574:→
568:⊩
540:⊮
514:¬
511:⊩
465:⊩
445:⟩
433:⟨
413:⟩
410:⊩
395:⟨
332:⟩
320:⟨
281:¬
278:◻
275:¬
266:◊
242:◻
218:◊
198:◻
178:¬
158:→
98:semantics
8397:(1969).
8373:(2000).
8034:See also
7772:u ≡
7761:quotient
7688:, where
7595:′
7584:′
7573:′
7409:′
7270:for all
7184:, where
7168:′
7157:′
7146:′
7043:′
7035:′
6906:′
6895:′
6816:′
6750:′
6710:′
6699:′
6688:′
6568:implies
6544:′
6504:′
6493:′
6002:implies
5286:, where
5077:implies
4669:, where
4216:, and a
4201:cut-free
3797:, where
3679:D, 4, 5
3625:directed
3621:T, 4, G
3524:T, 4, M
3511:T, 4, H
3487:preorder
1648:; axiom
1640:; axiom
1628:; axiom
1191: :
1082:complete
1018:a class
925:a frame
914:for all
850:a model
833:relation
750:We read
698:for all
425:, where
344:, where
8711:(ed.).
8698:, 2001
8674:(ed.).
8112:differs
7976:Tarski;
7934:removed
7919:sources
7776: v
7717:, then
7647:)
6997:, then
6599:)
6576:)
5180:→ ⊥ is
5170:, then
4785:, then
4111:compact
4017:, each
3655:finite
3641:finite
3197:empty:
3138:is the
1515:. Then
1410:, thus
1381:. Then
1289:, then
1134:) is a
831:forcing
796:forces
792:”, or “
47:improve
8619:
8577:
8556:
8528:
8499:
8458:
8430:
8409:
8383:
8359:
8333:
8305:
8284:
8261:
8178:After
8151:within
8108:notion
8016:proof;
7871:, and
7659:) and
7651:
7278:, and
7078:using
6603:
6580:
5160:, not
4759:, and
4218:finite
3670:serial
3635:, K4W
3439:Axioms
3253:Axiom
2128:serial
1448:
1444:
1377:
1373:
1323:
1319:
1315:since
1228:frame
372:, and
370:worlds
36:, but
8707:. In
8670:. In
8518:(PDF)
8493:(PDF)
8484:(PDF)
8061:Notes
8007:Lewis
7988:time.
7618:is a
7554:from
6371:Here
5430:,...,
5419:: if
5405:,...,
5214:be a
5184:, so
4701:is a
4054:frame
4013:. By
3913:then
3884:, if
3618:S4.2
3521:S4.1
3508:S4.3
3483:T, 4
3321:as a
1865:dense
1681:Axiom
1260:: if
1160:, if
1067:, if
1060:sound
888:, if
844:valid
828:, or
366:nodes
352:is a
8617:ISBN
8575:ISBN
8554:ISBN
8526:ISBN
8497:ISBN
8469:2014
8456:ISBN
8441:2014
8428:ISBN
8407:ISBN
8381:ISBN
8357:ISBN
8331:ISBN
8303:ISBN
8282:ISBN
8259:ISBN
8133:the
7917:any
7915:cite
7851:the
7076:tree
6977:and
6961:and
6950:and
6934:and
6880:and
6673:and
6619:and
6478:and
6411:in
6040:not
5738:and
5373:and
5210:Let
5106:not
4879:and
4412:. A
4222:S4.3
4123:K4.1
4119:S4.1
3813:and
3751:The
3732:(an
3676:D45
3436:Name
2466:Grz
1678:Name
1660:and
1085:wrt
846:in:
784:”, “
776:as “
493:and
254:and
232:dual
170:and
108:and
100:for
8147:any
8139:not
8119:has
7928:by
7763:of
7639:if
7502:FMP
7267:i+1
7069:all
6993:if
6957:if
6930:if
5857:or
5140:, ¬
4965:or
4819:)),
4729:if
4627:An
4195:or
4046:not
4027:MCS
4007:MCS
3807:MCS
3755:of
3738:MCS
3480:S4
3469:K4
3261:as
3172:or
2976::
2401:GL
2216:or
2099:or
2076:or
1636:in
1436:by
1097:).
1089:if
1079:is
1075:).
1057:is
842:is
615:or
368:or
356:on
308:or
234:of
142:of
88:or
8728::
8694:,
8688:,
8654:.
8542:;
8319:;
8234:..
7867:,
7786:,
7649:R’
7343:R’
7186:W’
6989::
6944:W’
6940:v’
6631:’.
6601:R’
6588:),
6578:R’
6395:.
6317:,
5967:,
5501::
5345::
5202:.
5195:.
5156:≤
5051:,
4113:.
4060:.
3666:D
3663:D
3633:GL
3494:S5
3472:4
3461:T
3458:T
3450:—
3447:K
3414:.
3332:,
3147:-
3142:)
3034:-
2937:-
2860:-
2731:G
2680:M
2541:H
2341::
2299:5
2182:B
2130::
2042:D
1982::
1940:4
1867::
1825:-
1793::
1757:T
1689:K
1672:.
1612:.
1575:GL
1571:GL
1472:.
1220:.
1146:.
1050:.
824:,
497::
383:A
380:.
304:A
272::=
8717:.
8680:.
8625:.
8598:.
8583:.
8562:.
8534:.
8505:.
8471:.
8443:.
8415:.
8389:.
8365:.
8339:.
8311:.
8290:.
8267:.
8218:.
8182:.
8169:.
8127:F
8123:F
8116:F
8085:.
7955:)
7949:(
7944:)
7940:(
7936:.
7922:.
7838:.
7826:A
7820:v
7800:A
7794:u
7784:X
7780:A
7774:X
7765:W
7757:f
7753:X
7749:f
7743:.
7731:A
7725:v
7705:X
7699:A
7676:A
7667:u
7657:v
7655:(
7653:f
7645:u
7643:(
7641:f
7636:,
7634:X
7630:p
7626:f
7622:,
7616:f
7588:,
7581:R
7577:,
7570:W
7556:W
7552:f
7532:,
7529:R
7526:,
7523:W
7510:X
7506:X
7490:,
7473:1
7470:+
7467:n
7463:w
7459:,
7454:n
7450:w
7446:,
7440:,
7435:1
7431:w
7427:,
7422:0
7418:w
7406:R
7396:n
7392:w
7388:,
7382:,
7377:1
7373:w
7369:,
7364:0
7360:w
7339:p
7325:p
7317:n
7313:w
7292:p
7286:s
7276:n
7272:i
7263:i
7261:w
7242:n
7238:w
7234:,
7228:,
7223:1
7219:w
7215:,
7210:0
7206:w
7199:=
7196:s
7161:,
7154:R
7150:,
7143:W
7129:W
7125:0
7122:w
7102:,
7099:R
7096:,
7093:W
7063:.
7061:p
7047:p
7032:w
7011:p
7005:w
6981:.
6971:W
6967:v
6954:,
6903:R
6899:,
6892:W
6865:R
6862:,
6859:W
6836:.
6834:p
6820:p
6809:)
6806:w
6803:(
6800:f
6780:p
6774:w
6747:W
6740:W
6734:f
6703:,
6696:R
6692:,
6685:W
6655:,
6652:R
6649:,
6646:W
6629:v
6625:v
6623:(
6621:f
6613:W
6609:v
6605:v
6597:u
6595:(
6593:f
6586:v
6584:(
6582:f
6574:u
6572:(
6570:f
6562:f
6541:W
6534:W
6528:f
6501:R
6497:,
6490:W
6463:R
6460:,
6457:W
6393:e
6389:a
6385:x
6381:a
6379:→
6377:x
6375:(
6373:e
6367:.
6355:]
6352:)
6349:a
6343:x
6340:(
6337:e
6334:[
6331:A
6325:u
6303:u
6299:M
6292:a
6272:w
6266:u
6246:]
6243:e
6240:[
6237:)
6234:A
6230:x
6224:(
6218:w
6208:,
6196:]
6193:)
6190:a
6184:x
6181:(
6178:e
6175:[
6172:A
6166:w
6144:w
6140:M
6133:a
6113:]
6110:e
6107:[
6104:)
6101:A
6097:x
6091:(
6085:w
6075:,
6063:]
6060:e
6057:[
6048:w
6037:,
6025:]
6022:e
6019:[
6016:B
6010:u
5990:]
5987:e
5984:[
5981:A
5975:u
5955:w
5949:u
5929:]
5926:e
5923:[
5920:)
5917:B
5911:A
5908:(
5902:w
5892:,
5880:]
5877:e
5874:[
5871:B
5865:w
5845:]
5842:e
5839:[
5836:A
5830:w
5810:]
5807:e
5804:[
5801:)
5798:B
5792:A
5789:(
5783:w
5773:,
5761:]
5758:e
5755:[
5752:B
5746:w
5726:]
5723:e
5720:[
5717:A
5711:w
5691:]
5688:e
5685:[
5682:)
5679:B
5673:A
5670:(
5664:w
5654:,
5651:w
5649:M
5635:)
5632:]
5629:e
5626:[
5621:n
5617:t
5613:,
5607:,
5604:]
5601:e
5598:[
5593:1
5589:t
5585:(
5582:P
5562:]
5559:e
5556:[
5553:)
5548:n
5544:t
5540:,
5534:,
5529:1
5525:t
5521:(
5518:P
5512:w
5489:]
5486:e
5483:[
5480:A
5474:w
5463:w
5461:M
5457:e
5451:.
5448:v
5446:M
5441:u
5439:M
5434:n
5432:a
5428:1
5425:a
5423:(
5421:P
5416:u
5414:M
5409:n
5407:a
5403:1
5400:a
5396:P
5392:n
5387:,
5384:u
5382:M
5377:v
5375:M
5370:u
5368:M
5363:,
5360:v
5358:M
5353:u
5351:M
5343:v
5339:u
5335:W
5331:w
5327:L
5322:w
5320:M
5300:,
5297:W
5269:W
5263:w
5259:}
5253:w
5249:M
5245:{
5242:,
5236:,
5233:W
5220:L
5212:L
5193:A
5191:¬
5189:⊩
5186:w
5178:A
5175:⊩
5172:w
5168:A
5165:⊩
5162:u
5158:u
5154:w
5150:u
5146:A
5142:A
5138:A
5132:.
5114:w
5103:,
5091:B
5085:u
5065:A
5059:u
5039:w
5033:u
5013:B
5007:A
5001:w
4991:,
4979:B
4973:w
4953:A
4947:w
4927:B
4921:A
4915:w
4905:,
4893:B
4887:w
4867:A
4861:w
4841:B
4835:A
4829:w
4811:(
4799:p
4793:u
4773:p
4767:w
4747:u
4741:w
4731:p
4683:,
4680:W
4651:,
4645:,
4642:W
4597:.
4594:)
4591:A
4585:u
4579:u
4575:R
4571:w
4568:(
4562:i
4558:D
4551:u
4528:A
4523:i
4512:w
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