Talk:Kripke semantics

639:(distributed widely by hand and finally published as "An Introduction to Modal Logic" by E.J.Lemon in collaboration with Dana Scott ) use "G" for your "2", and "M" (McKinsey) for your "1". I've seen *lots* of folk following Lemmon and using G for 2. Usage of "M" is more mixed, but seems to be very popular. I've been using "MS" for "1" on my pages (Although I appear to have not given my source on that name, I'll have to hunt down where I picked it up, If I can't find it I'll probably drop back to calling it M... The problem being that Chellas use M for something else, and is a very popular introductory text.) I don't seem to have the axiom referred to as "3" in any of my index, so I wasn't worried about it for now. And all the others have MANY sources using them by the names here. 409:, which discusses the precedence of rival claims to give the first relational semantics on pages 25 to 41. Hintikka's work defined a notion of model that is almost exactly the usual definition of maximal consistent set for modal logic, and applies it to deontic logic; Kripke acknowledges it as similar in his first work. Kanger's work has important technically differences, but the idea of an accessibility relation is there and he does note the correspondence between conditions on accessibility relations and axioms characterising modal systems (reflexivity <-: --> 84: 731:

distribution of different modalities the Church-Rosser property and "axiom chr" for short: axiom 2 is a particular instance of this scheme (2nd edition, p189), and it is commonly used in dynamic logic to mean this (see eg. p13 of Schmidt&Tishkovsky, Multi-agent dynamic logics with informational test, which uses CR for the formula and chr for the frame condition). I didn't find any direct uses of CR to name 2, but I think they must have been in conference proceedings, which I don't have here. ---

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memory or my choice of reading material. (If you have a specific reference that would be nice.) Whatever it is called, if taken from the literature, is going to conflict with something else in the literature. CR does have two nice properties if you have a source... 1) There is a non-wikipedian tradition to quote, and 2) It doesn't seem to conflict with anything indexed in my catalog, which is usually a sign that it doesn't conflict with anything particularly common.

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theorem is that the semantic and the syntactic definitions of consistency are equivalent. And vice versa, the completeness theorem is an easy conclusion of the consistency theorem (the equivalency of the semantic and the syntactic definition of consistency). This is particularly easy in propositional logic, to which the trick with elimination of necessitation rule is apparently reducing the modal one.

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to each other and v and w have edges to each other, and let some A hold in u and v but not in w. Now T and 5 hold, but it is not transitive, as there is no edge from u to w (and axiom 4 does not hold). To see that D,B,4 is not reflexive look at a frame with two nodes and two edges, one in each direction and let some A hold in both.

534:, different people use different names. I tried to use standard names where available, but the rest of the table is only a local definition for the purposes of Knowledge (or rather, just this article: I'm unaware of any other Knowledge article referring to these axioms), whence the warning above the table. 2162:

of a Kripke model, not true in the whole model. Defining consistent sets semantically makes no sense, as the whole point of the construction is to prove the completeness theorem, we need to start with a syntactically consistent set, and show that it is satisfiable in a model, not the other way round.

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I have changed the axioms in the list of S5. I was unsure what the list T,5 or D,B,4 meant, but if it means either the two axioms T,5 or the three eaxioms D,B,4 it was wrong. To see that T,5 is not transitive, try a frame with three nodes, u,v,w, where each has a reflexive edge and u and v have edges

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I was somewhat confused as to 1 being the "main axiom" for S4.1. Then I realized that this must be McKinsey's S4.1, and not Sobocinski's S4.1. McKinsey's 4.1 is generally (to my experience) called S4M nowadays (The "M" being your "1"). And Sobocinski's 4.1 seems to me the one more often meant when

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B axiom, and so on), which makes the work pretty sophisticated in my view. The Goldblatt article also names the work of Jonsson-Tarski as a possible predecessor, although they never applied their ideas to modal logic. Goldblatt also discusses work of Carnap, Bayart and Montague, the names maybe are

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In short: if Hintikka and Kanger found relational semantics independently, it might be worthwhile to mention it in the introductory paragraph. If what they found was just something vaguely similar to Kripke semantics, I'd advise either to not mention it at all, or to put it in a "history" subsection,

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a primitive notion. Also, contrary to what you wrote, you defined a logic as a set of formulas without mentioning any "closure properties" (quote: "A modal logic (i.e., a set of formulas) L"). Although you defined the concept of Thm(C) (quote: "We define Thm(C) to be the set of all formulas that are

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is a ubiquitous logical term with - given a logic in question - pretty unambiguous meaning, notwithstanding the existence of its different definitions. Your characterization of the one I gave before as "some other notion" is incorrect. It was your original definition (corrected later) of consistency

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and apparently two different notations for equivalence), but in principle this would not do any harm. Finally, including alternative formulations of axioms is redundant and unhelpful unless there is a very good reason for it, such as when an axiom is commonly used in two different forms. The D axiom

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I would have voted for McK, CR and H, but I missed EJ's post and I won't rock the boat now. Just a question: since there are at least three distinct axioms that when added to S4 give S4.3, should we have different names for them? This article probably doesn't need them, but if we want to get into

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Perhaps it should be joined up with intuitionistic logic. Note that the Tarski-Jonsson topological semantics ofr intuitionistic logic predates Kripke/frame semantics, and is really closer in spirit to what is going on in sheaf semantics anyhow. (I'm not really on top of sheaves, but I have a rough

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Should we call this Kripke semantics at all, since Kripke was not the first to discover them: both Hintikka and Kanger gave semantics of this form before Kripke. One sees the usage "Frame semantics" in the literature; how about we move the article to "Frame semantics", and let Kripke semantics be a

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Text for the second table mentions "the logics are complete with respect to the frame classes given in the table, but they may correspond to a larger class of frames". Is it possible to add a column to show which logics actually correspond to the listed classes of frames? I would also be interested

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Your stipulation that "defining consistent sets semantically makes no sense" is false. That's what some skilled logicians (e.g. late Barwise) did/do in order to prove the completeness theorem (which incorporates both completeness and soundness). One of the immediate consequences of the completeness

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So, if/when it becomes necessary to discuss these axioms on the temporal logic pages, the names will have to be distinguished somehow. I do not have a good idea how this should be done (being no expert on temporal logic, I've never seen a source which would need more than one .3 variant at the same

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It's just a matter of technicalities which definition is used, unless, that is, one uses the syntactic definition without properly formulating it. So, it would be better if you used semantic definition taking into account your reluctance to admit that rules of inference are part of (the definition

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in the end ("1" and "2" have are serving the function). But I think it would be nice to use a "standard" notation that people would be likely to see else where, without having to scare them with the extent of the current naming mess. I don't remember seeing CR for 2, but that might be either my

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Another issue: where does Kripke-Joyal semantics belong? I originally moved it to the section on intuitionistic logic, as I vaguely remembered that topoi have something to do with int. log., rather than modal logic. After reading the Goldblatt paper I'm not so sure about it. It might be better to

3254:, and therefore it is not necessary to point out at the very beginning the obvious convention that since modal logic extends classical logic, other Boolean connectives than those indicated are introduced as abbreviations the same way as in classical logic (while you are making so much fuss about 882:

As for S4.3: obviously, the same name cannot be used for distinct axioms. (I concur that the other .3-like variants are not needed in this article - the point of the list here is not to give a comprehensive survey of modal axioms, but to give a few concrete examples for correspondence theory and

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relevant to naming conventions; the article should be named "Kripke semantics" because that's the name most of the world uses. For example, the article on Zorn lemma is named "Zorn's lemma", even though Hausdorff and then Kuratowski discovered it independently before Zorn. Or consider "Ramanujan

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Your claim that "we need to start with a syntactically consistent set, and show that it is satisfiable in a model, not the other way round" is false, to. The "other way round" is typically used to prove the soundness (which is part of the completeness theorem in the standard sense). Besides, it

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I chose the names "1", "2", and "3" for their simplicity, and didactical clarity: as the names of the other logics are loosely composed of names of their axioms, it is easily understandable for someone who sees the logics for the first time to denote the main axiom of S4.1 as "1". If you have a

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You have been working with no compensation writing articles for Knowledge, for the benefit of others. This is highly commendable. (I often feel that scholars and resaerchers - creators of intellectual property, if you will - are being mercilessly exploited by the rest of the humanity) For that

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I checked a few refs and discovered that while the axiom if fairly often described as capturing either the Church-Rosser property or confluence (see, eg. Blackburn/de Rijke/Venema p160), it is less often talked about as "axiom CR". The Handbook of Philosophical Logic calls the axiom allowing

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of satisfaction are essentially the same in temporal logic as in modal logic, but the authors are using different symbols. (Of course, it could be that one author wrote one chapter whereas the second wrote the other, and no editor enforced notational consistency.) Here in Knowledge, articles

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I've applied quite a major change, mostly a new section on History&Terminology, with a precis of the above work before Kripke, and a discussion of the rival terminology. I could have pointed out that the Blackburn et al book, now the standard ref. on modal logic, talks about relational

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apparently hoping to get away without any rules of inference, based on the fact that if a logic in question is sound and complete (in the classical sense and not in the article's sense) then its semantic consequence relation is equal to its syntactic consequence relation. (That is what the

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The axiom naming issue is a mess. But I'd personally prefer some historic name (with that flimsy justificaton of history) over giving it a new name. (Which is why I asked the question, I had hoped that you had such a reference for "1", that I might add to my list...) The Lemmon papers

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Another omission is the use of "proves" in "L1 does not prove all theorems of L2" and "but does not prove the GL-tautology" in the article without a definition of "proves" there (specifically, what axioms beyond L1 and what rules of inference are allowed in your definition of "proves"?).

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OK, I've changed the TODOs to source comments. As for the naming: I've never heard about Hintikka/Kanger giving Kripke semantics before Kripke, but it doesn't surprise me. It happens all the time in mathematics that important things get unnoticed and then rediscovered. This is

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Axiom names: I can find many many sources (in agreement) for most axioms in the table (such as K, 5, D, B, T, etc.) But so far I'm unable to locate sources for the axiom names "1", and "2". Does someone have an actual reference to these that they could point me at?

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semantics, but I didn't, as does Goldblatt, but I didn't. Possible world semantics is also widely used, perhaps as much as Kripke semantics, but it marks you as ignorant to use it: someone more diplomatic than me should put that in. And I added four references. ----

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Whatever the "modal logic textbooks" you know of are doing, can most likely be done differently. It's not that mathematical logic serves at pleasure of modal logic. You may wish to take a note of it. (A side comment: the rest of the world uses the term

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It is called Kripke semantics by mathematicians, logicians, computer scientists, and philosophers, even though it is known by many than Hintikka came up with the idea earlier. So it's best to keep the name but cite references to earlier literature.

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Maybe .1, .2, .3 is better? .3 as the axiom characterising no right branching on transitive frames is pretty standard (see, eg. the Blackburn/de Rijke/Venema book), and I can't suppress the spurious association between axiom 1 and S1, etc. ---

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doesn't hold. Where is the mistake? I cannot figure it out. I suppose the point is that this frame is not euclidean, but this means that there is something in the definition that I'm not able to catch... Thanks! -- Stefano 11:53, 7 June 2010

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While I'm at it: CR (for Church-Rosser) is another widely used name for 2. I'd rather not use M for 1 since it is fairly often used instead of T for reflexivity: wouldn't McK work as well? I have the idea that I've seen that somewhere. ---

3023:, not about consistency. As it happens, the construction of canonical models involves a notion of consistent sets. The definition that makes the construction of canonical models work is the one in the article, not any of those you suggest. 3959:

It appears that the authors of the Style of Writing you quoted preferred form over contents. They apparently gave themselves an authority that they didn't earn, and the results are, sometimes, detrimental to precision and correctness.

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using only axioms K and Modus Ponens) injects the concept of "the theorem of L" which you haven't defined in your article. It replaced a major flaw with an undefined concept. (Also "if" needs to be replaced with "if, and only if,").

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However: The Kripke-Joyal semantics is more general than applying just to intuitionistic logic since it may be employed in Boolean categories as well as Heyting categories (or, in fact, in categories which are neither).

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redurect---but I don't think that should change the name of the article. There is an idea, called "Kripke Semantics", and whether that name is just or unjust the encyclopedia should treat the topic with that name. --

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Apparently, none of these is simultaneously widely used, historic, and unambiguous, thus no choice will be completely satisfactory, and we must make a decision at some point. My order of preferences is currently:

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L-consistent (in the sense of the above definition) for any propositional variable p and logic L with no axioms (or only with axiom scheme K). Also, because propositional axioms are not incorporated, the theory

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is not consistent, because it derives a contradiction already in classical propositional logic (though this admittedly wasn’t clear when the definition was originally worded with axioms in place of theorems).

1415: 608:, which he attributes without reference to Lemmonn, Hintikka and Geach. If you are after the locus solum for the .3 notation, I don't have it, but it's used in the Blackburn/de Rijke/Venema book, p193. --- 1873:"Semantics is useful for investigating a logic (i.e. a derivation system) only if the semantic consequence relation reflects its syntactical counterpart, the syntactic consequence relation (derivability)." 1729: 3775:"Since you still do not get it: the section “Canonical models” is about canonical models, not about consistency. As it happens, the construction of canonical models involves a notion of consistent sets." 3851:(as if one flaw or omission was an excuse for another; it seems very likely that the article also has other flaws/omissions, some of them may become apparent after you properly fix those that I noted) 362:

graphs": the name is simply an honor to a famous mathematician, it doesn't imply anything about authorship. Mathematical concepts (usu not theorems) are often named after people who didn't invent them.

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I would speculate that the reason for the above flaws is that the author of the article attempted to use semantically-defined notion of "derivable", despite this statement (quoted from the article):

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L-consistent for any propositional variable p and logic L with no axioms (or only with axiom scheme K) since in the language of propositional modal logic defined in the article one cannot derive

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as it does not connect anything nor is it connected to anything.) You used it in section on intuitionistic logic without including it in the alphabet of the language (just like you did with the

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All of this is described in any modal logic textbook (e.g., Sections 4.1–2 of Blackburn, De Rijke, and Venema, or Section 5.1 of Chagrov and Zakharyaschev; a preemptive warning, both books use

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This is not a Euclidean frame, as 1 → 2 and 1 → 2 would force 2 → 2. By the same token, 3 → 3 (and in general, the target of every arrow has to be reflexive). When you fix the frame like this,

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Can an analog of truth tables be made? In combinational logic you list the possible states of variables (for each variable either true or false) and the result of logic gates, but also =: -->

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So, what shall we do with the article? The names so far mentioned in the discussion, plus Chagrov & Zakharyaschev's names (which did not appear here yet), give the following options:

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is used intentionally together with certain, usually implicit, minimality assumption. It has been demonstrated (for instance, in semantics of logic programming) that the replacement of

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My understanding is that "possible world semantics" refers to the philosophical interpretation of intensional logics by possible worlds, rather than the formal relational semantics. --

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time.) BTW, there is nothing special about S4.3 here: at least the axioms G, Grz, M also have variants in use which are equivalent over S4, but differ in a nonreflexive context. --

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arogant, and yet you continue to make baseless claims about “major omissions” and “flaws” in the article, which are actually due to your poor understanding of the subject.

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on Knowledge. If you would like to support the project, please visit the project page, where you can get more details on how you can help, and where you can join the

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consequence relation of the logic), but the construction of canonical models. The maximal consistent sets correspond to sets of formulas satisfiable in a particular

2050:. This would allow to simplify the set of propositional axioms that are needed for the correctness of the syntactical definition of L-consistency (mentioned above). 383:

about the name of the article. I think it's important to credit earlier discoverers where possible---I was the person who added the discussion of Kuratowski to the

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Agreed. I was a bit taken aback to read that relational semantics was not possible-world semantics as I've been using the two interchangeably for decades. --

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I almost forgt to say: OK about not changing the name, the above is just some notes before I figure out how best to incorporate them into the article. ----

4204:), or is there some technical nuance that this choice seeks to express? I'm not suggesting changing the article, just seeking clarification of whether this 4349: 3763:"you continue to make baseless claims about “major omissions” and “flaws” in the article, which are actually due to your poor understanding of the subject" 253: 243: 3944:

is concerned, up until some time ago, it was believed by some that it is an unambiguous convention. This, however, is not true in certain contexts, where

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incorporated in it (whether syntactically or semantically defined) that the author, apparently, hoped to avoid up until his definition of L-consistency.

130: 3418:, so nitpicking on trivial syntactic matters such as what exactly is taken as the set of basic connectives is distracting and unhelpful to the readers.— 348:

There are lots of @@To Do:...@@ tags strew about the article: I haven't time right now to delete them, but if they are not gone tomorrow I shall. ----

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which would also mention Beth semantics. In any case, the name of the article should stay. BTW, there is already a redirect "Relational semantics".

959: 764:"M", "G", "H" -- traditional, short and nice, but conflicting with other systems (renaming the "GL" axiom to "W" might be worthwhile in this case) 339: 2474:. This will land you right away on a completeness (soundness included) theorem. Or - to make it even more clear - you could use this definition: 4359: 3919:

Sadly, it is not going anywhere, indeed. If you consider fixing errors and omissions a waste of time then the quality of your article suffers.

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Is there software that lets you compute on relatively short expressions? perhaps the easiest way to learn these things is by many examples...

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A normal modal logic is defined as a set of formulas with certain closure properties. Those are its theorems. Thus, being a theorem is a

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I do not like the second choice. The first choice (MGH) is my first choice. G is commonly seen when people are building system names.

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completeness theorem usually states.) This didn't work quite well, because of some details (like the ones above) that need to be fixed.

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in this context, it cannot be defined in terms of anything simpler. As for “if and only if”, it shouldn’t be used for definitions, per

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The above would explain a simplistic definition of (modal) logic as "a set of formulas". Usually, a logic has a provability relation

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When you have finished reviewing my changes, you may follow the instructions on the template below to fix any issues with the URLs.

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I also note that despite the unsourced claims in this article that possible world semantics is deprecated (as a term), SEP uses it

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may be labelled '@'", but I see nothing in this article to support that claim. Is it true? If so, does anyone have a citation? --

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in the language's alphabet together with variables, connectives and modalities, or else your article will have one more flaw..

3855:"The definition that makes the construction of canonical models work is the one in the article, not any of those you suggest." 2557:

Or if you prefer yours (which I bet you do regardless of the consequences), at least make sure it's correct and does not make

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Also, why is there no discussion of forcing here? Forcing, afterall, is really a special case of the Kripke-Joyal semantics.

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appears naive to believe that one can or cannot prove the completeness theorem depending on which definition one begun with.

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Is it really different than this? I know that Kripke semantics do not involve an ontological commitment to possible worlds:

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The reason is that it is a formal semantics for more than modal logics. For example, it is used with intuitionistic logic.

3743:"It is completely irrelevant that some other notions of consistent sets may or may not be useful in a different context." 1778:"A set of formulas is L-consistent if no contradiction can be derived from them using the axioms of L, and Modus Ponens." 4101: 3933:

Is the article fixable? Perhaps. But even if so, this is not a good excuse to not "waste" one's time on cleaning it up.

3452:"A set of formulas is L-consistent if no contradiction can be derived from them using the axioms of L, and Modus Ponens" 3444:"A set of formulas is L-consistent if no contradiction can be derived from it using the theorems of L, and Modus Ponens" 1240: 33: 3025:

It is completely irrelevant that some other notions of consistent sets may or may not be useful in a different context.

290: 168: 3903:. I could go on, but I’m not going to waste any more time on this discussion, as it’s obviously not going anywhere.— 4312: 3967:"The articles should be accessible, as much as possible, to readers not already familiar with the subject matter." 1062: 604:

Reference for what: the axiom or the name? Prior talks about these axioms (and a third) characterisation S4.3 in

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to delete these "External links modified" talk page sections if they want to de-clutter talk pages, but see the

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is the falsehood constant and hardly just a "basic connective of intuitionistic logic". (It's silly to call it

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the nitty gritty of temporal logic, we might well want some extended discussion of the axiomatics of S4.3. ---

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is commonly used in elegant formulations (like Lyndon's) of propositional logic, and your trying to confine

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is a nullary connective, whether you like it or not. The language of intuitionistic logic is defined in the

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etc can be written this way... what are the modal values of the completely parsed statements? true/false?

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The Handbook of Philosophical Logic uses H for the variant .3 I gave (I guess H stands for Hintikka) ---

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leave it as a top-level section after all. Can someone who actually understands sheaves confirm this? --

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If you have discovered URLs which were erroneously considered dead by the bot, you can report them with

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is one of the basic connectives of intuitionistic logic, so it needs no more “formal introduction” than

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throughout, except in the chapter about modal logic, where instead the definition of Kripke model uses

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But if this does not bother you and you have decided to follow the Guide, here is a quotation from it:

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L-consistent for any propositional variable p and logic L with no axioms (or only with axiom scheme K).

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I’m afraid you’ll need to learn at least the basics of modal logic before offering such “fixes”. Yes,

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2R3 and 3R2), right? Then the axiom 5 must hold. Now suppose that A holds only on 2. Then, on 1 holds

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or 8: true, necessary, possible true, necessary, not possible, true, not necessary, possible, ... ?

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is a formal semantics for modal logic systems, created in late 1950's and early 1960's by Saul Kripke

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is a formal semantics created in late 1950's and early 1960's by Saul Kripke for modal logic systems

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worth mentioning, although they don't develop what I would call a Kripke-style model theory. ----

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on Knowledge. If you would like to participate, please visit the project page, where you can join

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is L-consistent" problem go away. (I am sure there are may ways of fixing flaws of this article.)

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working out details of the proof of the completeness theorem like many do in mathematical logic.

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is incomplete. The article neglects to incorporate the necessitation rule. This makes the theory

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p&q). If I am not mistakened, the two variants give the same S4.3, but different K4.3s. ---

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before doing mass systematic removals. This message is updated dynamically through the template

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because the former looks more like the rest of them. If you use the latter, you need to include

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is used in the section Semantics of intuitionistic logic but never formally introduced. Having

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valid in C."), it is a function of a class of structures/frames and not of sets of formulas.

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reason, I am sorry for my harshness, but, perhaps, it's in my nature and I couldn't help it.

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https://web.archive.org/web/20041020014707/http://www.princeton.edu/~jburgess/Kripke1.doc

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I'm sorry that you missed the post, we can of course continue the discussion if you wish.

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Some references to the precedence of Kanger & Hintikka's work: Goldblatt has written

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for the satisfaction relation; this becomes doubly odd considering that the definitions

3148:, but even if there were, this would concern the first part of the article dealing with 4043:, "External links modified" talk page sections are no longer generated or monitored by 2289:

Anyway, I didn't make false statements. You did. And the article you defend has flaws.

969: 4083:

If you found an error with any archives or the URLs themselves, you can fix them with

3502: 797:

Good. As Charles apparently does not object, I'll change the names to M, G, H now. --

282: 4328: 1904:

I can offer here a quick fix to all these flaws. Just define L-consistency this way:

1738:

or true and necessary, true and possible, false and necessary false and possible (4)

1659: 1642: 1621: 1604: 384: 3736:

to intuitionistic logic in this respect suggest your unfamiliarity with that fact.

2297:

sense, the definition of L-consistency is still incomplete. It declares the theory

393: 2607:

L-consistent would be more honest as some of them disobey the necessitation rule.

1961:

Also, the presentation may use some polishing. For instance, the falsehood symbol

4030: 3751:

is relevant to this article shows only your inability or unwilligness to see it.

4281: 4050: 2714:, at the beginning of the article. It shows poor workmanship to have all three: 837: 732: 706: 675: 609: 579: 553: 495: 464: 429: 414: 349: 102: 4049:. No special action is required regarding these talk page notices, other than 3904: 3419: 2257: 1558: 815: 785: 720: 684: 658: 642: 618: 596: 517: 213: 190: 79: 3105:

article, so there is no point in repeating it here. There is no problem with

1421:

where the first implication is by 4, the second by D, and the third by B. --

998:

The equivalence of S5 with T+5 and D+B+4 is a quite standard fact, see e.g.

1635: 1146:

by more Euclideanness. A syntactic derivation of 4 in KT5 goes as follows:

4320: 4293: 4106: 3995: 3910: 3888: 3425: 3003: 2263: 2188:

I have no idea what you are talking about vis a vis intuitionistic logic.

2068: 1763: 1682: 1667: 1650: 1629: 1612: 1584: 1564: 1425: 1359:

is reflexive. Alternatively, here is a syntactic derivation of T in KDB4:

992: 976: 954: 909: 840: 818: 801: 788: 778: 735: 723: 709: 687: 678: 661: 645: 621: 612: 599: 582: 556: 542: 520: 334: 3900: 2150:). The purpose of the definition is not consequence relations (though it 570:: Actually the .3 axiom I was thinking of is different, namely: <: --> 3865:

I tried to tone down an earlier version on this post, and need to add a

4304: 3019:

Since you still do not get it: the section “Canonical models” is about

1422: 973: 906: 798: 775: 539: 486: 472: 446: 369: 212:, a collaborative effort to improve the coverage of content related to 1958:

But then axiom K (as well as the rules of inference) are superfluous.

3747:

Wrong, again. The fact that you don't see how the classic concept of

1603:. So, please explain what's the difference claimed in the lead here. 3859:(you are so wrong on "not any of those you suggest" - or prove it!) 2654:

truth constant in the scheme D), although you did bother to include

1126:). In general, every reflexive Euclidean relation is transitive: if 184: 157: 3664:

is a fallacy as it derives a false conclusion from a true premise.

3101:

article where it should be, it is quite tangential to the topic of

1655:

Dunn and H say pretty much the same thing, perhaps not as clearly

4215:

Logic in Computer Science – Modelling and Reasoning about Systems

3980:

in a few more examples, maybe here or a reference if available.

3027:

In any case, the definitions and terminology in this section are

1191:{\displaystyle \Box A\to \Diamond \Box A\to \Box \Diamond \Box A} 3152:, it would have nothing to do whatsoever with the definition of 4164:) for the satisfaction relation merely a glyph variation (like 1436:

Can please someone explain the axiom 5 on the following frame?

1410:{\displaystyle \Box A\to \Box \Box A\to \Diamond \Box A\to A,} 15: 657:

I've seen S4.1 in papers lately. I'll have to investigate.

1745:

Can these symbols be seen as logic gates with value tables?

1724:{\displaystyle \Box p\leftrightarrow \lnot \Diamond \lnot p} 595:

For future reference, what is a good reference for this .3?

445:

Thank you for the link, it is quite interesting reading. --

281: 2410:

true in (satisfied by) a possible world in a Kripke model",

4021:

for additional information. I made the following changes:

1331:, if you name the points in the obvious way). Indeed, let 2527:

satisfied by a truth assignment that treats each formula

1027:, hence the relation is not Euclidean. More concretely, 3847:

and apparently two different notations for equivalence"

617:

Both, and thanks, that's everything I was looking for.

530:, there is no generally agreed standard for most modal 4284:

use different commands for the satisfaction relation.

4257: 4237: 4190: 4170: 4146: 4122: 3833: 3813: 3793: 3722: 3696: 3673: 3631: 3608: 3576: 3525: 3505: 3462: 3376: 3353: 3321: 3300: 3280: 3260: 3236: 3216: 3181: 3158: 3111: 3083: 3037: 2974: 2951: 2916: 2869: 2843: 2820: 2800: 2780: 2760: 2740: 2720: 2700: 2680: 2660: 2640: 2616: 2563: 2533: 2507: 2484: 2460: 2440: 2420: 2390: 2367: 2303: 2234: 2214: 2194: 2169: 2127: 2081: 2030: 2007: 1987: 1967: 1934: 1914: 1887: 1829: 1788: 1699: 1537: 1491: 1468: 1367: 1287: 1243: 1206: 1154: 1100: 1065: 1033: 101:, a collaborative effort to improve the coverage of 4053:using the archive tool instructions below. Editors 1271:{\displaystyle \Box \Diamond \Box A\to \Box \Box A} 264: 4263: 4243: 4196: 4176: 4152: 4128: 3930:And there are many more glitches in your article. 3839: 3819: 3799: 3728: 3708: 3682: 3652: 3617: 3594: 3543: 3511: 3491: 3397: 3362: 3339: 3306: 3286: 3266: 3242: 3222: 3193: 3167: 3140: 3089: 3064: 2980: 2960: 2937: 2898: 2855: 2829: 2806: 2786: 2766: 2746: 2726: 2706: 2686: 2666: 2646: 2622: 2592: 2542: 2519: 2490: 2466: 2446: 2426: 2402: 2373: 2332: 2240: 2220: 2200: 2183:to denote local consequence, so no necessitation). 2175: 2142: 2105: 2042: 2016: 1993: 1973: 1946: 1920: 1893: 1858: 1814: 1723: 1549: 1503: 1477: 1409: 1308: 1270: 1227: 1190: 1118: 1086: 1051: 3448:with which you replaced the original definition: 3250:in the modal part, I think that readers are not 1866:is L-consistent for the same logic L as above. 1335:be a serial symmetric transitive relation, and 1007:Your first frame does not satisfy 5: the point 4039:This message was posted before February 2018. 4031:http://www.princeton.edu/~jburgess/Kripke1.doc 3759:Some more examples of flaws in your comments: 3660:, so including the latter does not make sense" 3405:, so including the latter does not make sense. 1461:This is an euclidean frame (1R2 and 1R3 -: --> 4208:anything. One reason this puzzles me is that 3827:, modal formulas in the article also include 3294:, modal formulas in the article also include 1641:should replace the current unsourced claims. 1087:{\displaystyle v\nVdash \Box \Diamond \neg A} 8: 3781:X is not about Y, X only happens to define Y 3538: 3526: 3486: 3463: 3410:On a general note, this is an article about 3135: 3112: 3059: 3038: 2893: 2870: 2587: 2564: 2327: 2304: 2100: 2082: 1853: 1830: 1809: 1789: 4009:I have just modified one external link on 3874: 2989: 2054: 261: 152: 47: 4256: 4236: 4189: 4169: 4145: 4121: 3832: 3812: 3792: 3787:"while you are making so much fuss about 3721: 3695: 3672: 3630: 3607: 3575: 3524: 3504: 3461: 3375: 3352: 3320: 3299: 3279: 3259: 3235: 3215: 3180: 3157: 3110: 3082: 3036: 2973: 2950: 2938:{\displaystyle \Diamond (A\rightarrow A)} 2915: 2868: 2842: 2819: 2799: 2779: 2759: 2739: 2719: 2699: 2679: 2659: 2639: 2615: 2562: 2532: 2506: 2483: 2459: 2439: 2419: 2389: 2366: 2302: 2233: 2213: 2193: 2168: 2126: 2080: 2043:{\displaystyle \varphi \rightarrow \bot } 2029: 2024:may be introduced as an abbreviation of 2006: 1986: 1966: 1933: 1913: 1886: 1828: 1787: 1698: 1536: 1490: 1467: 1366: 1286: 1242: 1228:{\displaystyle \Diamond \Box A\to \Box A} 1205: 1153: 1099: 1064: 1032: 538:better idea, feel free to suggest it. -- 3492:{\displaystyle \{p,\,\neg \neg \neg p\}} 3141:{\displaystyle \{p,\,\neg \neg \neg p\}} 2899:{\displaystyle \{p,\,\neg \neg \neg p\}} 2593:{\displaystyle \{p,\,\neg \neg \neg p\}} 2333:{\displaystyle \{p,\,\neg \neg \neg p\}} 1859:{\displaystyle \{p,\,\neg \neg \neg p\}} 1119:{\displaystyle u\nVdash \Diamond \neg A} 526:No. The trouble is that unlike names of 3756:that deserved such a characterization. 3472: 3121: 2879: 2814:in your axiom schemes. You may declare 2573: 2313: 2053:A link to forcing needs an explanation. 1839: 1798: 1569:Woa, thank you soooo much! -- Stefano 1052:{\displaystyle v\Vdash \Diamond \neg A} 883:subsequent topics, preferrably simple.) 154: 49: 19: 3065:{\displaystyle \{p,\neg \neg \neg p\}} 683:Thank you. (For including the source) 222:about philosophy content on Knowledge. 3901:MOS:MATH#Writing style in mathematics 3456:(that improperly declared the theory 2270:Rebuttal to replay to Major Omissions 1323:Your second frame is not transitive ( 1309:{\displaystyle \Box A\to \Box \Box A} 714:I suspect almost anything would work 7: 3625:, but it is not commonly written as 3595:{\displaystyle \Box p\to \Diamond p} 3570:"The D axiom is commonly written as 3370:, but it is not commonly written as 3340:{\displaystyle \Box p\to \Diamond p} 2694:, much more common connectives than 206:This article is within the scope of 95:This article is within the scope of 2945:to your list of axioms rather than 2434:is L-consistent iff L can't derive 1815:{\displaystyle \{p,\,\neg \Box p\}} 1634:Perhaps a sourced explanation from 38:It is of interest to the following 4350:Low-importance Philosophy articles 4213:Huth, Michael; Ryan, Mark (2004). 3814: 3794: 3723: 3709:{\displaystyle A\rightarrow \bot } 3703: 3674: 3612: 3532: 3529: 3480: 3477: 3474: 3357: 3281: 3261: 3237: 3217: 3188: 3159: 3129: 3126: 3123: 3084: 3053: 3050: 3047: 2975: 2955: 2910:And an aesthetic comment. I added 2887: 2884: 2881: 2856:{\displaystyle A\rightarrow \bot } 2850: 2821: 2761: 2721: 2701: 2681: 2641: 2617: 2581: 2578: 2575: 2508: 2485: 2461: 2441: 2421: 2391: 2368: 2321: 2318: 2315: 2195: 2091: 2037: 2008: 1988: 1968: 1935: 1915: 1847: 1844: 1841: 1800: 1715: 1709: 1693:OK the only thing I can follow is 1110: 1078: 1043: 494:idea about what is going on) ---- 14: 4340:Mid-priority mathematics articles 4013:. Please take a moment to review 3653:{\displaystyle \Diamond (p\to p)} 3398:{\displaystyle \Diamond (p\to p)} 2414:and then prove, if you can, that 2117:-consistent set (that is, unless 2106:{\displaystyle \{p,\neg \Box p\}} 1774:The definition of L-consistency: 115:Knowledge:WikiProject Mathematics 3563:And the list does not end here. 2863:, which will make at least the " 606:K1, K2 and related modal systems 410:T axiom, transitivity <-: --> 228:Knowledge:WikiProject Philosophy 193: 183: 156: 118:Template:WikiProject Mathematics 82: 72: 51: 20: 3544:{\displaystyle \{\neg \neg p\}} 2774:, particularly if you use both 1550:{\displaystyle \Box \Diamond A} 1504:{\displaystyle \Box \Diamond A} 248:This article has been rated as 231:Template:WikiProject Philosophy 135:This article has been rated as 4217:. Cambridge University Press. 3700: 3647: 3641: 3635: 3618:{\displaystyle \Diamond \top } 3583: 3392: 3386: 3380: 3363:{\displaystyle \Diamond \top } 3328: 3185: 2961:{\displaystyle \Diamond \top } 2932: 2926: 2920: 2847: 2741: 2661: 2235: 2154:be in fact used to define the 2131: 2034: 1706: 1426:12:26, 18 September 2007 (UTC) 1398: 1386: 1374: 1294: 1256: 1216: 1173: 1161: 993:09:33, 18 September 2007 (UTC) 968:Makes sense. And BTW, you can 724:22:08, 30 September 2005 (UTC) 710:21:28, 30 September 2005 (UTC) 688:20:43, 30 September 2005 (UTC) 679:19:20, 30 September 2005 (UTC) 662:00:54, 30 September 2005 (UTC) 646:00:54, 30 September 2005 (UTC) 622:20:43, 30 September 2005 (UTC) 613:19:20, 30 September 2005 (UTC) 600:00:54, 30 September 2005 (UTC) 583:14:11, 29 September 2005 (UTC) 557:13:45, 29 September 2005 (UTC) 543:12:48, 29 September 2005 (UTC) 521:15:13, 25 September 2005 (UTC) 1: 4360:Low-importance logic articles 4307:, "a 'designated' world in a 4294:12:42, 10 November 2019 (UTC) 4107:20:18, 12 December 2017 (UTC) 3940:in the definitions replacing 2600:L-consistent for L as above. 2543:{\displaystyle \Box \varphi } 2520:{\displaystyle \Sigma \cup L} 2403:{\displaystyle \Sigma \cup L} 2282:where you and few others use 2017:{\displaystyle \neg \varphi } 1947:{\displaystyle \Sigma \cup L} 1683:20:36, 23 November 2012 (UTC) 910:16:40, 12 December 2005 (UTC) 789:02:50, 22 November 2005 (UTC) 779:02:42, 22 November 2005 (UTC) 109:and see a list of open tasks. 4335:C-Class mathematics articles 4112:Satisfaction relation symbol 2747:{\displaystyle \rightarrow } 2667:{\displaystyle \rightarrow } 2550:as a propositional variable, 1764:02:10, 23 January 2012 (UTC) 841:22:53, 8 December 2005 (UTC) 819:22:37, 8 December 2005 (UTC) 802:18:25, 7 December 2005 (UTC) 411:4 axiom, symmetry <-: --> 4345:C-Class Philosophy articles 2143:{\displaystyle p\to \Box p} 948:the preceding comment is by 736:14:54, 3 October 2005 (UTC) 407:a history (gzipped PS file) 328:the preceding comment is by 4381: 4321:00:01, 25 March 2021 (UTC) 4070:(last update: 5 June 2024) 4006:Hello fellow Wikipedians, 3996:15:38, 12 April 2014 (UTC) 3194:{\displaystyle p\to \bot } 1668:18:21, 20 April 2011 (UTC) 1651:17:43, 20 April 2011 (UTC) 1630:17:24, 20 April 2011 (UTC) 1613:17:19, 20 April 2011 (UTC) 1478:{\displaystyle \Diamond A} 1355:by transitivity, that is, 254:project's importance scale 4365:Logic task force articles 4303:According to the article 3077:In intuitionistic logic, 2807:{\displaystyle \Diamond } 2603:Also, calling these sets 2357:of) logic, for instance, 2069:19:49, 2 July 2013‎ (UTC) 977:18:11, 29 July 2006 (UTC) 955:12:32, 24 July 2006 (UTC) 489:14:56, 26 Aug 2004 (UTC) 467:16:47, 25 Aug 2004 (UTC) 352:10:31, 23 Aug 2004 (UTC) 335:12:30, 24 July 2006 (UTC) 289: 260: 247: 178: 134: 67: 46: 4244:{\displaystyle \models } 4197:{\displaystyle \varphi } 4153:{\displaystyle \models } 3911:18:54, 9 July 2013 (UTC) 3889:17:33, 6 July 2013 (UTC) 3867:comment for completeness 3440:The "fixed" definition: 3436:injects another omission 3426:11:04, 4 July 2013 (UTC) 3004:04:31, 4 July 2013 (UTC) 2264:11:52, 3 July 2013 (UTC) 2248:. This is an article on 1595:Possible world semantics 1585:12:29, 7 June 2010 (UTC) 1565:11:26, 7 June 2010 (UTC) 1339:any point. There exists 1142:by Euclideanness, hence 498:08:29, 16 Sep 2004 (UTC) 475:14:56, 26 Aug 2004 (UTC) 449:10:48, 25 Aug 2004 (UTC) 432:16:59, 24 Aug 2004 (UTC) 417:16:25, 24 Aug 2004 (UTC) 396:13:35, 23 Aug 2004 (UTC) 372:11:31, 23 Aug 2004 (UTC) 141:project's priority scale 4264:{\displaystyle \Vdash } 4129:{\displaystyle \Vdash } 4002:External links modified 3315:is commonly written as 2491:{\displaystyle \Sigma } 2467:{\displaystyle \Sigma } 2427:{\displaystyle \Sigma } 2374:{\displaystyle \Sigma } 2176:{\displaystyle \vdash } 1921:{\displaystyle \Sigma } 1894:{\displaystyle \vdash } 1557:does indeed hold in 1.— 265:Associated task forces: 98:WikiProject Mathematics 4355:C-Class logic articles 4265: 4245: 4198: 4178: 4154: 4130: 3841: 3821: 3801: 3730: 3710: 3690:as an abbreviation of 3684: 3683:{\displaystyle \neg A} 3654: 3619: 3596: 3545: 3513: 3493: 3399: 3364: 3341: 3308: 3288: 3268: 3244: 3224: 3195: 3169: 3168:{\displaystyle \neg p} 3142: 3091: 3066: 2982: 2962: 2939: 2900: 2857: 2837:is an abbreviation of 2831: 2830:{\displaystyle \neg A} 2808: 2788: 2768: 2748: 2728: 2708: 2688: 2668: 2648: 2624: 2594: 2544: 2521: 2492: 2468: 2448: 2428: 2404: 2375: 2334: 2242: 2222: 2221:{\displaystyle \land } 2202: 2177: 2144: 2107: 2044: 2018: 1995: 1975: 1948: 1922: 1895: 1860: 1816: 1725: 1638:p. 90 (2nd paragraph) 1551: 1505: 1479: 1411: 1310: 1272: 1229: 1192: 1138:by reflexivity, hence 1120: 1088: 1053: 960:Please sign your posts 576:(p&q) \/ <: --> 480:Kripke-Joyal semantics 340:Please sign your posts 286: 209:WikiProject Philosophy 28:This article is rated 4313:John Maynard Friedman 4266: 4246: 4199: 4179: 4177:{\displaystyle \phi } 4155: 4131: 3842: 3840:{\displaystyle \lor } 3822: 3820:{\displaystyle \bot } 3802: 3800:{\displaystyle \top } 3731: 3729:{\displaystyle \bot } 3711: 3685: 3655: 3620: 3597: 3546: 3514: 3494: 3400: 3365: 3342: 3309: 3307:{\displaystyle \lor } 3289: 3287:{\displaystyle \bot } 3269: 3267:{\displaystyle \top } 3245: 3243:{\displaystyle \bot } 3225: 3223:{\displaystyle \top } 3196: 3170: 3143: 3092: 3090:{\displaystyle \bot } 3067: 2983: 2981:{\displaystyle \top } 2963: 2940: 2901: 2858: 2832: 2809: 2789: 2787:{\displaystyle \Box } 2769: 2767:{\displaystyle \neg } 2749: 2729: 2727:{\displaystyle \bot } 2709: 2707:{\displaystyle \bot } 2689: 2687:{\displaystyle \neg } 2669: 2649: 2647:{\displaystyle \top } 2625: 2623:{\displaystyle \bot } 2595: 2545: 2522: 2493: 2469: 2449: 2447:{\displaystyle \bot } 2429: 2405: 2376: 2335: 2243: 2223: 2203: 2201:{\displaystyle \bot } 2178: 2145: 2108: 2045: 2019: 1996: 1994:{\displaystyle \bot } 1976: 1974:{\displaystyle \bot } 1949: 1923: 1896: 1861: 1817: 1726: 1552: 1506: 1480: 1412: 1311: 1273: 1230: 1193: 1121: 1089: 1054: 936:should be changed to 389:Kuratowski-Zorn lemma 387:page and created the 285: 4255: 4235: 4188: 4168: 4144: 4120: 4051:regular verification 3831: 3811: 3791: 3720: 3694: 3671: 3629: 3606: 3574: 3566:Your statement that 3523: 3503: 3460: 3374: 3351: 3319: 3298: 3278: 3258: 3234: 3214: 3210:As for the usage of 3203:intuitionistic logic 3179: 3156: 3109: 3099:intuitionistic logic 3081: 3035: 2972: 2949: 2914: 2867: 2841: 2818: 2798: 2778: 2758: 2738: 2718: 2698: 2678: 2658: 2638: 2614: 2561: 2531: 2505: 2482: 2458: 2438: 2418: 2388: 2365: 2301: 2274:Don't be arrogant. 2254:intuitionistic logic 2241:{\displaystyle \to } 2232: 2212: 2192: 2167: 2125: 2079: 2028: 2005: 1985: 1965: 1954:has a Kripke model." 1932: 1928:is L-consistent iff 1912: 1885: 1827: 1786: 1697: 1535: 1489: 1466: 1365: 1347:by seriality, hence 1285: 1278:by Nec and K from 1, 1241: 1204: 1152: 1098: 1063: 1031: 121:mathematics articles 4041:After February 2018 2478:"A set of formulas 2361:"A set of formulas 1908:"A set of formulas 1735:necesary,possible? 1351:by symmetry, hence 234:Philosophy articles 4261: 4241: 4194: 4174: 4150: 4126: 4095:InternetArchiveBot 4046:InternetArchiveBot 3837: 3817: 3797: 3726: 3706: 3680: 3667:The definition of 3650: 3615: 3592: 3541: 3509: 3489: 3473: 3395: 3360: 3337: 3304: 3284: 3264: 3240: 3220: 3191: 3165: 3138: 3122: 3087: 3062: 3029:perfectly standard 2978: 2958: 2935: 2896: 2880: 2853: 2827: 2804: 2784: 2764: 2744: 2724: 2704: 2684: 2664: 2644: 2620: 2590: 2574: 2540: 2517: 2488: 2464: 2444: 2424: 2400: 2371: 2330: 2314: 2238: 2218: 2198: 2173: 2140: 2103: 2040: 2014: 1991: 1971: 1944: 1918: 1891: 1856: 1840: 1812: 1799: 1721: 1547: 1501: 1475: 1407: 1306: 1268: 1225: 1188: 1116: 1084: 1049: 752:.2, G, CR, chr, ga 749:.1, M, MS, McK, ma 287: 219:general discussion 90:Mathematics portal 34:content assessment 4223:978-0-521-54310-1 4071: 3986:comment added by 3971:Have a nice day. 3891: 3879:comment added by 3512:{\displaystyle p} 3006: 2994:comment added by 2071: 2059:comment added by 2001:aboard, negation 1754:comment added by 1575:comment added by 1518:comment added by 964: 344: 308: 307: 304: 303: 300: 299: 296: 295: 201:Philosophy portal 151: 150: 147: 146: 4372: 4278:Kripke semantics 4270: 4268: 4267: 4262: 4250: 4248: 4247: 4242: 4227: 4203: 4201: 4200: 4195: 4183: 4181: 4180: 4175: 4163: 4159: 4157: 4156: 4151: 4139: 4135: 4133: 4132: 4127: 4105: 4096: 4069: 4068: 4047: 4011:Kripke semantics 3998: 3975:Frame conditions 3897:primitive notion 3846: 3844: 3843: 3838: 3826: 3824: 3823: 3818: 3806: 3804: 3803: 3798: 3735: 3733: 3732: 3727: 3715: 3713: 3712: 3707: 3689: 3687: 3686: 3681: 3659: 3657: 3656: 3651: 3624: 3622: 3621: 3616: 3601: 3599: 3598: 3593: 3550: 3548: 3547: 3542: 3518: 3516: 3515: 3510: 3498: 3496: 3495: 3490: 3404: 3402: 3401: 3396: 3369: 3367: 3366: 3361: 3346: 3344: 3343: 3338: 3313: 3311: 3310: 3305: 3293: 3291: 3290: 3285: 3273: 3271: 3270: 3265: 3249: 3247: 3246: 3241: 3229: 3227: 3226: 3221: 3200: 3198: 3197: 3192: 3174: 3172: 3171: 3166: 3147: 3145: 3144: 3139: 3096: 3094: 3093: 3088: 3071: 3069: 3068: 3063: 3021:canonical models 2987: 2985: 2984: 2979: 2967: 2965: 2964: 2959: 2944: 2942: 2941: 2936: 2905: 2903: 2902: 2897: 2862: 2860: 2859: 2854: 2836: 2834: 2833: 2828: 2813: 2811: 2810: 2805: 2793: 2791: 2790: 2785: 2773: 2771: 2770: 2765: 2753: 2751: 2750: 2745: 2733: 2731: 2730: 2725: 2713: 2711: 2710: 2705: 2693: 2691: 2690: 2685: 2673: 2671: 2670: 2665: 2653: 2651: 2650: 2645: 2629: 2627: 2626: 2621: 2599: 2597: 2596: 2591: 2549: 2547: 2546: 2541: 2526: 2524: 2523: 2518: 2498:is L-consistent 2497: 2495: 2494: 2489: 2473: 2471: 2470: 2465: 2453: 2451: 2450: 2445: 2433: 2431: 2430: 2425: 2409: 2407: 2406: 2401: 2381:is L-consistent 2380: 2378: 2377: 2372: 2339: 2337: 2336: 2331: 2247: 2245: 2244: 2239: 2227: 2225: 2224: 2219: 2207: 2205: 2204: 2199: 2182: 2180: 2179: 2174: 2149: 2147: 2146: 2141: 2112: 2110: 2109: 2104: 2049: 2047: 2046: 2041: 2023: 2021: 2020: 2015: 2000: 1998: 1997: 1992: 1980: 1978: 1977: 1972: 1953: 1951: 1950: 1945: 1927: 1925: 1924: 1919: 1900: 1898: 1897: 1892: 1865: 1863: 1862: 1857: 1821: 1819: 1818: 1813: 1766: 1730: 1728: 1727: 1722: 1587: 1556: 1554: 1553: 1548: 1527: 1510: 1508: 1507: 1502: 1484: 1482: 1481: 1476: 1416: 1414: 1413: 1408: 1315: 1313: 1312: 1307: 1277: 1275: 1274: 1269: 1234: 1232: 1231: 1226: 1197: 1195: 1194: 1189: 1125: 1123: 1122: 1117: 1093: 1091: 1090: 1085: 1058: 1056: 1055: 1050: 970:edit it yourself 945: 574:(p&<: --> 325: 319:redirect to it. 272: 262: 236: 235: 232: 229: 226: 203: 198: 197: 196: 187: 180: 179: 174: 171: 160: 153: 123: 122: 119: 116: 113: 92: 87: 86: 76: 69: 68: 63: 55: 48: 31: 25: 24: 16: 4380: 4379: 4375: 4374: 4373: 4371: 4370: 4369: 4325: 4324: 4301: 4253: 4252: 4233: 4232: 4224: 4212: 4186: 4185: 4166: 4165: 4161: 4142: 4141: 4137: 4118: 4117: 4114: 4099: 4094: 4062: 4055:have permission 4045: 4019:this simple FaQ 4004: 3981: 3977: 3829: 3828: 3809: 3808: 3789: 3788: 3739:Then you claim 3718: 3717: 3692: 3691: 3669: 3668: 3627: 3626: 3604: 3603: 3572: 3571: 3521: 3520: 3501: 3500: 3458: 3457: 3438: 3432:The "fix" with 3372: 3371: 3349: 3348: 3317: 3316: 3296: 3295: 3276: 3275: 3256: 3255: 3232: 3231: 3212: 3211: 3177: 3176: 3154: 3153: 3107: 3106: 3079: 3078: 3033: 3032: 2970: 2969: 2947: 2946: 2912: 2911: 2865: 2864: 2839: 2838: 2816: 2815: 2796: 2795: 2776: 2775: 2756: 2755: 2736: 2735: 2716: 2715: 2696: 2695: 2676: 2675: 2656: 2655: 2636: 2635: 2612: 2611: 2605:propositionally 2559: 2558: 2529: 2528: 2503: 2502: 2480: 2479: 2456: 2455: 2436: 2435: 2416: 2415: 2386: 2385: 2363: 2362: 2299: 2298: 2272: 2230: 2229: 2210: 2209: 2190: 2189: 2165: 2164: 2123: 2122: 2077: 2076: 2026: 2025: 2003: 2002: 1983: 1982: 1963: 1962: 1930: 1929: 1910: 1909: 1883: 1882: 1825: 1824: 1784: 1783: 1772: 1770:Major Omissions 1749: 1695: 1694: 1691: 1598: 1570: 1533: 1532: 1513: 1487: 1486: 1464: 1463: 1434: 1363: 1362: 1283: 1282: 1239: 1238: 1202: 1201: 1150: 1149: 1096: 1095: 1061: 1060: 1029: 1028: 985: 926: 924:First paragraph 838:Charles Stewart 733:Charles Stewart 707:Charles Stewart 676:Charles Stewart 610:Charles Stewart 580:Charles Stewart 575:q) \/ <: --> 571:p&<: --> 554:Charles Stewart 513: 496:Charles Stewart 482: 465:Charles Stewart 460: 430:Charles Stewart 415:Charles Stewart 350:Charles Stewart 313: 270: 233: 230: 227: 224: 223: 199: 194: 192: 172: 166: 120: 117: 114: 111: 110: 88: 81: 61: 32:on Knowledge's 29: 12: 11: 5: 4378: 4376: 4368: 4367: 4362: 4357: 4352: 4347: 4342: 4337: 4327: 4326: 4300: 4297: 4286:95.195.195.123 4260: 4240: 4229: 4228: 4222: 4193: 4173: 4149: 4140:) rather than 4125: 4116:Is the use of 4113: 4110: 4089: 4088: 4081: 4034: 4033: 4025:Added archive 4003: 4000: 3976: 3973: 3969: 3968: 3936:As far as the 3916: 3914: 3913: 3862: 3857: 3856: 3849: 3848: 3836: 3816: 3796: 3777: 3776: 3765: 3764: 3745: 3744: 3725: 3705: 3702: 3699: 3679: 3676: 3662: 3661: 3649: 3646: 3643: 3640: 3637: 3634: 3614: 3611: 3591: 3588: 3585: 3582: 3579: 3540: 3537: 3534: 3531: 3528: 3508: 3488: 3485: 3482: 3479: 3476: 3471: 3468: 3465: 3454: 3453: 3446: 3445: 3437: 3430: 3429: 3428: 3407: 3406: 3394: 3391: 3388: 3385: 3382: 3379: 3359: 3356: 3336: 3333: 3330: 3327: 3324: 3303: 3283: 3263: 3239: 3219: 3207: 3206: 3190: 3187: 3184: 3164: 3161: 3137: 3134: 3131: 3128: 3125: 3120: 3117: 3114: 3086: 3074: 3073: 3061: 3058: 3055: 3052: 3049: 3046: 3043: 3040: 3016: 3015: 2977: 2957: 2954: 2934: 2931: 2928: 2925: 2922: 2919: 2908: 2907: 2895: 2892: 2889: 2886: 2883: 2878: 2875: 2872: 2852: 2849: 2846: 2826: 2823: 2803: 2783: 2763: 2743: 2723: 2703: 2683: 2663: 2643: 2619: 2589: 2586: 2583: 2580: 2577: 2572: 2569: 2566: 2552: 2551: 2539: 2536: 2516: 2513: 2510: 2487: 2463: 2443: 2423: 2412: 2411: 2399: 2396: 2393: 2370: 2354: 2353: 2348: 2347: 2342: 2341: 2329: 2326: 2323: 2320: 2317: 2312: 2309: 2306: 2271: 2268: 2267: 2266: 2237: 2217: 2197: 2185: 2184: 2172: 2139: 2136: 2133: 2130: 2102: 2099: 2096: 2093: 2090: 2087: 2084: 2039: 2036: 2033: 2013: 2010: 1990: 1970: 1956: 1955: 1943: 1940: 1937: 1917: 1890: 1875: 1874: 1855: 1852: 1849: 1846: 1843: 1838: 1835: 1832: 1811: 1808: 1805: 1802: 1797: 1794: 1791: 1780: 1779: 1771: 1768: 1756:83.134.181.100 1744: 1720: 1717: 1714: 1711: 1708: 1705: 1702: 1690: 1687: 1686: 1685: 1597: 1592: 1591: 1590: 1589: 1588: 1546: 1543: 1540: 1500: 1497: 1494: 1474: 1471: 1459: 1458: 1457: 1456: 1452: 1448: 1444: 1433: 1430: 1429: 1428: 1419: 1418: 1417: 1406: 1403: 1400: 1397: 1394: 1391: 1388: 1385: 1382: 1379: 1376: 1373: 1370: 1320: 1319: 1318: 1317: 1305: 1302: 1299: 1296: 1293: 1290: 1279: 1267: 1264: 1261: 1258: 1255: 1252: 1249: 1246: 1236: 1224: 1221: 1218: 1215: 1212: 1209: 1199: 1187: 1184: 1181: 1178: 1175: 1172: 1169: 1166: 1163: 1160: 1157: 1115: 1112: 1109: 1106: 1103: 1083: 1080: 1077: 1074: 1071: 1068: 1048: 1045: 1042: 1039: 1036: 1004: 1003: 984: 981: 980: 979: 942: 941: 934: 933: 925: 922: 921: 920: 919: 918: 917: 916: 915: 914: 913: 912: 893: 892: 891: 890: 889: 888: 887: 886: 885: 884: 871: 870: 869: 868: 867: 866: 865: 864: 863: 862: 850: 849: 848: 847: 846: 845: 844: 843: 826: 825: 824: 823: 822: 821: 807: 806: 805: 804: 792: 791: 772: 771: 768: 765: 757: 756: 753: 750: 743: 742: 741: 740: 739: 738: 728: 727: 726: 697: 696: 695: 694: 693: 692: 691: 690: 667: 666: 665: 664: 651: 650: 649: 648: 633: 632: 631: 630: 629: 628: 627: 626: 625: 624: 588: 587: 586: 585: 562: 561: 560: 559: 546: 545: 535: 512: 509: 501: 500: 499: 481: 478: 477: 476: 459: 456: 455: 454: 453: 452: 451: 450: 438: 437: 436: 435: 434: 433: 421: 420: 419: 418: 400: 399: 398: 397: 374: 373: 367: 363: 346: 345: 312: 309: 306: 305: 302: 301: 298: 297: 294: 293: 288: 278: 277: 275: 273: 267: 266: 258: 257: 250:Low-importance 246: 240: 239: 237: 205: 204: 188: 176: 175: 173:Low‑importance 161: 149: 148: 145: 144: 133: 127: 126: 124: 107:the discussion 94: 93: 77: 65: 64: 56: 44: 43: 37: 26: 13: 10: 9: 6: 4: 3: 2: 4377: 4366: 4363: 4361: 4358: 4356: 4353: 4351: 4348: 4346: 4343: 4341: 4338: 4336: 4333: 4332: 4330: 4323: 4322: 4318: 4314: 4310: 4306: 4298: 4296: 4295: 4291: 4287: 4283: 4279: 4274: 4258: 4238: 4225: 4220: 4216: 4211: 4210: 4209: 4207: 4191: 4171: 4147: 4123: 4111: 4109: 4108: 4103: 4098: 4097: 4086: 4082: 4079: 4075: 4074: 4073: 4066: 4060: 4056: 4052: 4048: 4042: 4037: 4032: 4028: 4024: 4023: 4022: 4020: 4016: 4012: 4007: 4001: 3999: 3997: 3993: 3989: 3985: 3974: 3972: 3966: 3965: 3964: 3961: 3957: 3955: 3951: 3947: 3943: 3939: 3934: 3931: 3928: 3925: 3920: 3917: 3912: 3909: 3906: 3902: 3898: 3894: 3893: 3892: 3890: 3886: 3882: 3881:‎76.95.183.31 3878: 3870: 3868: 3863: 3860: 3854: 3853: 3852: 3834: 3786: 3785: 3784: 3782: 3774: 3773: 3772: 3770: 3762: 3761: 3760: 3757: 3754: 3750: 3742: 3741: 3740: 3737: 3697: 3677: 3665: 3644: 3638: 3632: 3609: 3589: 3586: 3580: 3577: 3569: 3568: 3567: 3564: 3561: 3557: 3554: 3535: 3506: 3483: 3469: 3466: 3451: 3450: 3449: 3443: 3442: 3441: 3435: 3431: 3427: 3424: 3421: 3417: 3413: 3409: 3408: 3389: 3383: 3377: 3354: 3334: 3331: 3325: 3322: 3301: 3253: 3209: 3208: 3204: 3182: 3162: 3151: 3132: 3118: 3115: 3104: 3100: 3076: 3075: 3056: 3044: 3041: 3030: 3026: 3022: 3018: 3017: 3013: 3009: 3008: 3007: 3005: 3001: 2997: 2996:‎76.95.183.31 2993: 2952: 2929: 2923: 2917: 2890: 2876: 2873: 2844: 2824: 2801: 2781: 2633: 2610: 2609: 2608: 2606: 2601: 2584: 2570: 2567: 2555: 2537: 2534: 2514: 2511: 2501: 2477: 2476: 2475: 2397: 2394: 2384: 2360: 2359: 2358: 2350: 2349: 2344: 2343: 2324: 2310: 2307: 2296: 2292: 2291: 2290: 2287: 2285: 2281: 2275: 2269: 2265: 2262: 2259: 2255: 2251: 2215: 2187: 2186: 2170: 2161: 2157: 2153: 2137: 2134: 2128: 2120: 2116: 2097: 2094: 2088: 2085: 2074: 2073: 2072: 2070: 2066: 2062: 2058: 2051: 2031: 2011: 1959: 1941: 1938: 1907: 1906: 1905: 1902: 1888: 1879: 1872: 1871: 1870: 1867: 1850: 1836: 1833: 1806: 1803: 1795: 1792: 1777: 1776: 1775: 1769: 1767: 1765: 1761: 1757: 1753: 1746: 1742: 1739: 1736: 1733: 1718: 1712: 1703: 1700: 1689:hard to grasp 1688: 1684: 1680: 1676: 1675:Vaughan Pratt 1672: 1671: 1670: 1669: 1665: 1661: 1657: 1653: 1652: 1648: 1644: 1640: 1637: 1632: 1631: 1627: 1623: 1619: 1615: 1614: 1610: 1606: 1602: 1596: 1593: 1586: 1582: 1578: 1577:95.244.18.126 1574: 1568: 1567: 1566: 1563: 1560: 1544: 1541: 1538: 1530: 1529: 1528: 1525: 1521: 1520:95.244.18.126 1517: 1498: 1495: 1492: 1472: 1469: 1453: 1449: 1445: 1441: 1440: 1439: 1438: 1437: 1431: 1427: 1424: 1420: 1404: 1401: 1395: 1392: 1389: 1383: 1380: 1377: 1371: 1368: 1361: 1360: 1358: 1354: 1350: 1346: 1342: 1338: 1334: 1330: 1326: 1322: 1321: 1316:from 1 and 3. 1303: 1300: 1297: 1291: 1288: 1280: 1265: 1262: 1259: 1253: 1250: 1247: 1244: 1237: 1222: 1219: 1213: 1210: 1207: 1200: 1185: 1182: 1179: 1176: 1170: 1167: 1164: 1158: 1155: 1148: 1147: 1145: 1141: 1137: 1133: 1129: 1113: 1107: 1104: 1101: 1081: 1075: 1072: 1069: 1066: 1046: 1040: 1037: 1034: 1026: 1023:does not see 1022: 1018: 1014: 1010: 1006: 1005: 1001: 997: 996: 995: 994: 991: 982: 978: 975: 971: 967: 966: 965: 963: 961: 956: 952: 951:62.30.147.160 949: 939: 938: 937: 931: 930: 929: 923: 911: 908: 903: 902: 901: 900: 899: 898: 897: 896: 895: 894: 881: 880: 879: 878: 877: 876: 875: 874: 873: 872: 860: 859: 858: 857: 856: 855: 854: 853: 852: 851: 842: 839: 834: 833: 832: 831: 830: 829: 828: 827: 820: 817: 813: 812: 811: 810: 809: 808: 803: 800: 796: 795: 794: 793: 790: 787: 783: 782: 781: 780: 777: 774:Opinions? -- 769: 766: 763: 762: 761: 754: 751: 748: 747: 746: 737: 734: 729: 725: 722: 717: 713: 712: 711: 708: 703: 702: 701: 700: 699: 698: 689: 686: 682: 681: 680: 677: 673: 672: 671: 670: 669: 668: 663: 660: 655: 654: 653: 652: 647: 644: 641: 637: 636: 635: 634: 623: 620: 616: 615: 614: 611: 607: 603: 602: 601: 598: 594: 593: 592: 591: 590: 589: 584: 581: 569: 566: 565: 564: 563: 558: 555: 550: 549: 548: 547: 544: 541: 536: 533: 529: 525: 524: 523: 522: 519: 510: 508: 505: 497: 492: 491: 490: 488: 479: 474: 470: 469: 468: 466: 458:Major changes 457: 448: 444: 443: 442: 441: 440: 439: 431: 427: 426: 425: 424: 423: 422: 416: 408: 404: 403: 402: 401: 395: 390: 386: 382: 379:I agree with 378: 377: 376: 375: 371: 368: 364: 360: 355: 354: 353: 351: 343: 341: 336: 332: 331:62.30.147.160 329: 322: 321: 320: 316: 310: 292: 284: 280: 279: 276: 274: 269: 268: 263: 259: 255: 251: 245: 242: 241: 238: 221: 220: 215: 211: 210: 202: 191: 189: 186: 182: 181: 177: 170: 165: 162: 159: 155: 142: 138: 132: 129: 128: 125: 108: 104: 100: 99: 91: 85: 80: 78: 75: 71: 70: 66: 60: 57: 54: 50: 45: 41: 35: 27: 23: 18: 17: 4309:Kripke model 4302: 4272: 4230: 4214: 4205: 4115: 4093: 4090: 4065:source check 4044: 4038: 4035: 4008: 4005: 3988:173.74.57.70 3982:— Preceding 3978: 3970: 3962: 3958: 3953: 3949: 3945: 3941: 3937: 3935: 3932: 3929: 3923: 3921: 3918: 3915: 3896: 3875:— Preceding 3871: 3866: 3864: 3861: 3858: 3850: 3780: 3778: 3766: 3758: 3752: 3748: 3746: 3738: 3666: 3663: 3565: 3562: 3558: 3552: 3455: 3447: 3439: 3433: 3415: 3411: 3251: 3202: 3149: 3102: 3028: 3024: 3020: 3011: 2990:— Preceding 2909: 2631: 2604: 2602: 2556: 2553: 2499: 2413: 2382: 2355: 2294: 2293:Even in the 2288: 2283: 2279: 2276: 2273: 2249: 2159: 2155: 2151: 2118: 2114: 2061:76.95.183.31 2055:— Preceding 2052: 1960: 1957: 1903: 1880: 1876: 1868: 1781: 1773: 1750:— Preceding 1747: 1743: 1740: 1737: 1734: 1692: 1654: 1633: 1616: 1599: 1460: 1435: 1356: 1352: 1348: 1344: 1340: 1336: 1332: 1328: 1324: 1143: 1139: 1135: 1131: 1127: 1024: 1020: 1016: 1012: 1008: 986: 958: 947: 943: 935: 927: 773: 758: 744: 715: 640: 567: 531: 527: 514: 506: 502: 483: 461: 385:Zorn's Lemma 358: 347: 338: 327: 317: 314: 249: 217: 207: 137:Mid-priority 136: 96: 62:Mid‑priority 40:WikiProjects 4282:Modal logic 3922:Theorem is 3753:Consistency 3749:consistency 3150:modal logic 1571:—Preceding 1514:—Preceding 1198:by T and 5, 990:Dag Hovland 928:The phrase 814:Thank you. 511:Axiom names 112:Mathematics 103:mathematics 59:Mathematics 4329:Categories 4102:Report bug 3769:projection 3553:vice versa 3031:. And no, 2632:connective 1343:such that 1327:, but not 577:(<: --> 225:Philosophy 214:philosophy 164:Philosophy 4231:is using 4085:this tool 4078:this tool 3412:semantics 3010:You call 2256:article.— 2250:semantics 755:.3, H, sc 573:<: --> 4273:op. cit. 4091:Cheers.— 3984:unsigned 3877:unsigned 3434:theorems 2992:unsigned 2057:unsigned 1752:unsigned 1660:Tijfo098 1643:Tijfo098 1622:Tijfo098 1605:Tijfo098 1573:unsigned 1516:unsigned 1454:3 -: --> 1450:2 -: --> 1446:1 -: --> 1442:1 -: --> 572:q =: --> 315:Issues: 311:Priority 4305:at sign 4184:versus 4162:\models 4015:my edit 2121:proves 1134:, then 394:Dominus 252:on the 139:on the 30:C-class 4299:@ sign 4138:\Vdash 3602:or as 3416:syntax 3414:, not 3347:or as 3252:idiots 2280:theory 2113:is an 1636:Béziau 1485:, but 1281:hence 1059:, but 1019:, but 532:axioms 528:logics 36:scale. 4206:means 3952:with 3519:from 2454:from 2295:local 2284:logic 2160:point 2156:local 1325:xRyRx 1235:by 5, 1011:sees 972:. -- 816:Nahaj 786:Nahaj 721:Nahaj 685:Nahaj 659:Nahaj 643:Nahaj 619:Nahaj 597:Nahaj 518:Nahaj 291:Logic 169:Logic 4317:talk 4290:talk 4280:and 4219:ISBN 3992:talk 3905:Emil 3885:talk 3807:and 3551:nor 3420:Emil 3274:and 3230:and 3103:this 3000:talk 2794:and 2754:and 2674:and 2258:Emil 2065:talk 1760:talk 1679:talk 1664:talk 1647:talk 1626:talk 1609:talk 1581:talk 1559:Emil 1524:talk 1130:and 1094:(as 1015:and 716:here 4059:RfC 4029:to 3954:iff 3942:iff 3924:not 3201:in 3175:as 2500:iff 2383:iff 2286:.) 2228:or 2152:can 1353:xRx 1349:yRx 1345:xRy 1329:xRx 1144:xRz 1140:yRx 1136:xRx 1132:yRz 1128:xRy 1000:SEP 359:not 244:Low 131:Mid 4331:: 4319:) 4292:) 4259:⊩ 4239:⊨ 4192:φ 4172:ϕ 4148:⊨ 4124:⊩ 4072:. 4067:}} 4063:{{ 3994:) 3950:if 3946:if 3938:if 3908:J. 3887:) 3869:. 3835:∨ 3815:⊥ 3795:⊤ 3783:) 3771:) 3724:⊥ 3704:⊥ 3701:→ 3675:¬ 3642:→ 3633:◊ 3613:⊤ 3610:◊ 3587:◊ 3584:→ 3578:◻ 3533:¬ 3530:¬ 3481:¬ 3478:¬ 3475:¬ 3423:J. 3387:→ 3378:◊ 3358:⊤ 3355:◊ 3332:◊ 3329:→ 3323:◻ 3302:∨ 3282:⊥ 3262:⊤ 3238:⊥ 3218:⊤ 3189:⊥ 3186:→ 3160:¬ 3130:¬ 3127:¬ 3124:¬ 3085:⊥ 3054:¬ 3051:¬ 3048:¬ 3012:me 3002:) 2976:⊤ 2956:⊤ 2953:◊ 2927:→ 2918:◊ 2888:¬ 2885:¬ 2882:¬ 2851:⊥ 2848:→ 2822:¬ 2802:◊ 2782:◻ 2762:¬ 2742:→ 2734:, 2722:⊥ 2702:⊥ 2682:¬ 2662:→ 2642:⊤ 2618:⊥ 2582:¬ 2579:¬ 2576:¬ 2538:φ 2535:◻ 2512:∪ 2509:Σ 2486:Σ 2462:Σ 2442:⊥ 2422:Σ 2395:∪ 2392:Σ 2369:Σ 2322:¬ 2319:¬ 2316:¬ 2261:J. 2236:→ 2216:∧ 2196:⊥ 2171:⊢ 2135:◻ 2132:→ 2095:◻ 2092:¬ 2067:) 2038:⊥ 2035:→ 2032:φ 2012:φ 2009:¬ 1989:⊥ 1969:⊥ 1939:∪ 1936:Σ 1916:Σ 1889:⊢ 1848:¬ 1845:¬ 1842:¬ 1804:◻ 1801:¬ 1762:) 1716:¬ 1713:◊ 1710:¬ 1707:↔ 1701:◻ 1681:) 1666:) 1658:. 1649:) 1628:) 1620:. 1611:) 1583:) 1562:J. 1542:◊ 1539:◻ 1526:) 1496:◊ 1493:◻ 1470:◊ 1423:EJ 1399:→ 1393:◻ 1390:◊ 1387:→ 1381:◻ 1378:◻ 1375:→ 1369:◻ 1301:◻ 1298:◻ 1295:→ 1289:◻ 1263:◻ 1260:◻ 1257:→ 1251:◻ 1248:◊ 1245:◻ 1220:◻ 1217:→ 1211:◻ 1208:◊ 1183:◻ 1180:◊ 1177:◻ 1174:→ 1168:◻ 1165:◊ 1162:→ 1156:◻ 1111:¬ 1108:◊ 1105:⊮ 1079:¬ 1076:◊ 1073:◻ 1070:⊮ 1044:¬ 1041:◊ 1038:⊩ 983:S5 974:EJ 957:: 953:- 907:EJ 799:EJ 776:EJ 568:PS 540:EJ 487:EJ 473:EJ 447:EJ 381:EJ 370:EJ 337:: 333:- 271:/ 167:: 4315:( 4288:( 4226:. 4160:( 4136:( 4104:) 4100:( 4087:. 4080:. 3990:( 3883:( 3698:A 3678:A 3648:) 3645:p 3639:p 3636:( 3590:p 3581:p 3539:} 3536:p 3527:{ 3507:p 3487:} 3484:p 3470:, 3467:p 3464:{ 3393:) 3390:p 3384:p 3381:( 3335:p 3326:p 3205:. 3183:p 3163:p 3136:} 3133:p 3119:, 3116:p 3113:{ 3060:} 3057:p 3045:, 3042:p 3039:{ 2998:( 2933:) 2930:A 2924:A 2921:( 2894:} 2891:p 2877:, 2874:p 2871:{ 2845:A 2825:A 2588:} 2585:p 2571:, 2568:p 2565:{ 2515:L 2398:L 2328:} 2325:p 2311:, 2308:p 2305:{ 2138:p 2129:p 2119:L 2115:L 2101:} 2098:p 2089:, 2086:p 2083:{ 2063:( 1942:L 1854:} 1851:p 1837:, 1834:p 1831:{ 1810:} 1807:p 1796:, 1793:p 1790:{ 1758:( 1719:p 1704:p 1677:( 1662:( 1645:( 1624:( 1607:( 1579:( 1545:A 1522:( 1499:A 1473:A 1455:2 1451:3 1447:3 1443:2 1432:5 1405:, 1402:A 1396:A 1384:A 1372:A 1357:R 1341:y 1337:x 1333:R 1304:A 1292:A 1266:A 1254:A 1223:A 1214:A 1186:A 1171:A 1159:A 1114:A 1102:u 1082:A 1067:v 1047:A 1035:v 1025:w 1021:u 1017:w 1013:u 1009:v 962:! 946:— 342:! 326:— 256:. 143:. 42::

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