Talk:Modal logic

Kripke semantics and possible world semantics

This article says these are equivalent, but the entry for Kripke semantics specifically says they are not in the very first sentence of its entry. — Preceding unsigned comment added by 70.117.17.114 (talk) 19:40, 5 June 2016 (UTC)

Names for logic systems

Question: Who created the system of names like "S4" and "t" for logic systems? Is there a central registry somewhere? -- The Anome 10:44, 17 Jul 2004 (UTC)

List of modal logic systems is the most complete list I know of, and covers the names of most of the common ones, linked to what they refer to. The lack of an "official" list means that some names have been used for several different systems. That page tries to make some sense of this mess, with references to where the various systems were discussed. In the case of several systems having the same name in the literature, or the same name refering to different systems, pointers to the appropriate literature are given. And yes, S1 - S5 were named by Lewis. Others in the S series were named (often inconsistantly) by others. Note, contrary to the statement below, the system T is not always written in lower case. A quick check of some of the books on the bibliography that I have on my shelf had "T" ususally. Disclaimer: The link above is to one of my pages. -- User:Nahaj

I can't remember, though I used to know. There is no central registry beyond continuity of discussion in journals, classrooms, etc. There is a good story behind "t", which includes why it is always written in lower-case, but I've forgotten it. Radgeek, below, is correct about the "S" systems. They were originally developed by Lewis, whose central concern was not modality but finding a correct logical interpretation of implication. He proposed five systems, named (creatively), S1, S2, etc. Once semantics were developed for modal logics, the first two were found to be non-normal, and are rarely used now. S3 is, I tihnk, equivalent to one of the others in modern use. S4 and S5 are the only frequently used Lewis logics to retain their old names.

I could be mistaken, but I think that the central modal logic systems (up to S5) were named by C.I. Lewis. I don't know if there is anyone who is keeping track of all the systems on tap, though... part of the question, of course, would be what you count as a modal logic system. (E.G. does Prior's tense logic count?) Radgeek 19:25, 17 Jul 2004 (UTC)

I think that its an accepted convention that all semantically intensional logics are modal logics. A semantically intensional logic or language being one containing semantically intensional, or "opaque" operators, operators for which semantic compositionality doesn't hold. Simply, a modal logic is any logic some of whose (non-atomic) expressions are such that the semantic value of the whole is not a function of the semantic values of the parts.

Although there is of course a narrower sense in which modal logics are just those ones dealing with metaphysical possibility and necessity. But that doesn't make for a good definition, since you can formalize a whole logic without ever indicating whether its operators are to be interpreted as "possibly" and "necessarily" or as "permissible" and "obligitory", etc. Yes, Prior's tense logic is a modal logic under the larger definition, and I tihnk that's the definition the article ought to adhere to given current usage.

To Dos

The following is mostly a note to myself, but comments are welcome.

To add Wikipedia currently says nothing about the following items, which should go either here or in kripke semantics:

• The TRIV modality (where p <=> []p);
• Standard non-normal modal logics: S1-S3, E1-E5.
• S5 is a maximal modality;
• Bimodal logics, eg. S4*S4, S5*S5;
• Proof theory of modal logic: sequent systems, need for other systems, link to geometric theories
• Taxonomy of normal modal logics: Lemmon-Scott axioms.
• I think hybrid logic is well enough established to deserve a treatment
• Quantified modal logic, Barcan axiom, problem of quantifying-in
• Dynamic logic is important enough to deserve a short discussion on the main page
• Provability logic needs more discussion

To change

• Introduction of frame semantics very disjointed: should have normal modal logics introduced in this article, and frame semantics introduced the section following the idea of possible worlds interpretations.
• Temporal logic and tense logic are not the same thing
• The system most commonly used today is modal logic S5 -- how on earth can one tell? S4 is pretty heavily used, and sees the most comp sci applications.
• Link to semantics of logic

Enough for now... ---- Charles Stewart 08:51, 16 Sep 2004 (UTC)

Remarks

• The intro should start off with Lewis' initial formalization of systems in an effort to better "define" strict implication. (Historically, Carnap's system is of great interesting, contrasted with current alethic systems which don't stipulate a vacuous condition on the set of worlds.) And then give an overview of what modality (or context) is and how it applies to logic. Then introduce the various operators (just the two and avoiding multi-modal logic until later) without providing an interpretation of them (i.e. as 'it is necessary that').

Agreed ---- Charles Stewart 08:26, 14 Oct 2004 (UTC)

Disagree about giving the historical detail first. Strict implication, while very interesting, is quite extraneous to an intial introduction to modality. The article would be better off connecting modal logics with the modal verbs/adverbs of English, preferably just possibly and necessarily to start. AGREE with respect to your next point. Half agree regarding the last: I think it is better explained by using some interpretation, and necessary/possible is the simplest. ---- User:136.142.20.124

Agree Lewis' distinction betweeen the two [Material being ~(p&~q) and strict being ~M(p&~q) ] I believe strikes to the heart of the matter, and is easily explained. -- Nahaj 17:41 25 Aug 2005 (UTC)

• Introduce the semantics under their own heading. There is more than just possible world semantics.

Agreed ---- Charles Stewart 08:26, 14 Oct 2004 (UTC)

Agreed ---- User:136.142.20.124

• The idea of necessity by means of possible worlds was first introduced by Leibniz, IIRC. This is more of a question.

Aristotle was the first to note that necessity and possibility could be defined in terms of each other, according to Edward Zalta [1]; I don't feel comfortable writing about Leibniz, don't know enouygh about him, but I think his innovations were mostly to do with the relation of ethics and theology to necessity; he did talk about possibility in terms of possible worlds, which is suggestive, but doesn't really achieve much by itself. ---- Charles Stewart 08:26, 14 Oct 2004 (UTC)

(Instead of the Zalta quote, a more direct citation is book one of de Interpretatione, passim). Most, or many glosses of possible-world semantics refer to Kripke's work as a "reployment" or "elaboration" or some such of Leibniz' idea. So that attribution of the origin of the idea is in common currency, anyway. ---- User:136.142.20.124

• This page needs a lot of work and sorting out. There is no clear organization of sections or information within section, even. And why isn't Hintikka, the pioneer of doxastic and epistemic logic, mentioned?

He's mentioned in the history section of Kripke Semantics; he certainly should be mentioned here. But then, he certainly should have a page... Carnap also deserves a mention here. ---- Charles Stewart 08:26, 14 Oct 2004 (UTC)

• I have made a couple of changes to this page for correctness. Firstly, I changed the claim that 'Nec P' and 'not Poss not P' have the same meaning to the claim that they are logically equivalent (and likewise for `Poss P' with `not Nec not P') since the claim that they *mean* the same thing is highly controversial. Secondly, it was claimed that K does not determing whether `Nec P' implies `Nec Nec P'. That's just wrong, it does: according to K there is no such implication, since counter-examples are available in K. I change the passage to point out that K does not guarantee that propositions which are necessary are necessarily necessary. Thirdly, it was written that S5 was intuitive because "if P is true at all possible worlds, then it seems that there can be no possible world at which it is true that there is some possible world where P is false". That claim is not guaranteed by S5, since all S5 does is ensure that world are split into equivalence classes. So P can be necessary at a world w and there be a world w* at which P is false which is not accessible from nor accesses any world which is either accessible from or accesses w. So I removed that sentence and replaced it with something true which I hope captures the original intention. ---- Ross Cameron 11:33, 17 Feb 2005 (GMT)

This section ("Formal rules") is problematic anyway, since it assumes that we would want to interpret K as an alethic modal logic. For some applications of modal logic, eg. deontic modal logic, the logic of obligations, one definitely does not want the T-axiom. I'll change this. ---- Charles Stewart 12:10, 17 Feb 2005 (UTC)

• The section on "Controversies" states that Bertrand Russell's rejection of Modal Logic was unreasoned (which is a pretty strong claim for a mere wikipedia page to make); it should probably claim something about the cited argument against Russell's view (i.e. that the cited author viewed Russell's position as unreasoned). I am not familiar with the cited work, so I can't make the change myself. This error seems ironic in an article on modal logic. Michael Witbrock

A question

Is there a more formal version of the Necessitation Rule (from K), or can someone explain to me in a strict way what it means? I'm having a bit of trouble understanding just *why* this statement is not the same as "A -> []A". (It's obvious that it is not, but I can't seem to find a good reason, outside of "if it were, then the axiom (4) from S4 would immediately follow from it", which really is not an argument. On a side note, I should also say that I know *nothing* about modal logic, but reading the article made me wonder about this). -- Schnee (cheeks clone) 17:12, 23 Mar 2005 (UTC)

The rule of necessitation is a rule, not a theorem. It states that, in symbols

$from\vdash P,infer\vdash \Box P$.

If P is a valid theorem of classical logic (and hence any normal modal logic since they all have as theorems classical validities), then it is true in every model at every world, regardless of the accessibility relation. But the definition of [] is exactly that -- true at every world. So for any model, if P is a valid theorem it is true at every world and hence []P is also true at every world (and so on through iterations of the rule [][]P, [][][]P, etc. also being valid at every world).

On the other hand, P -> []P is much different since it states that for *any* formula P, not only validities, then []P, and that is clearly wrong (and hence not a theorem). Nortexoid 23:19, 23 Mar 2005 (UTC)

Possibly a differet (non-modal) example might clairify... Assuming some reasonable sense of a "Provable" predicate:

The RULE: from "$\vdash P$ infer $\vdash Provable(P)$" would say roughly: "If you have a P, P is provable", and is somewhat plausable.

The AXIOM:" P $\to$ Provable(P)", says roughtly "Everything is Provable", which isn't plausable.

From your example it seems that you are not using the standard interpretation of "$\to$". If we interpret "$\to$" as material implication then "P $\to$ Provable(P)" is equivalent to "~P $\lor$ Provable(P)" which means P is either false or provable - not that it is provable. —Preceding unsigned comment added by 79.72.63.62 (talk) 14:57, August 26, 2007 (UTC)

From a semantic point of view, K is the minimal modal loic in the sense that he axiom holds in all classes of frames. Other modal logics place additional constraints (e.g., transitivity) on the accessibility relation in Kripke semantics. See Blackburn et al., "Modal Logic" for a detailed discussion. Greg Woodhouse 22:51, 30 November 2006 (UTC)

@Schneelocke, Nortexoid, and GregWoodhouse: As presented in the page now, Necessitation is not a rule, it is a meta-rule. A rule is something saying "from A, B, ... derive C". Provability is defined in terms of rules, hence a rule cannot invoke provability. Henceforth Necessitation should be a (meta)-theorem, which, of course, might hold or might not hold according to the logic under consideration. It is fine to (meta)-define a logic L normal if Necessitation holds for L, but Necessitation cannot enter the very definition of the specific L. Am I missing something? 160.80.2.38 (talk) 19:16, 14 August 2022 (UTC)

Subjunctive modalities??

I have not once come across the term "subjunctive" modality in any literature on the subject of modal logic. It is a grammatical term referring to mood, or possibly made with reference to conditionals (e.g. counterfactuals). But of course alethic modality includes more than mere counterfactuals. They are certainly not synonyms. At any rate, the term should be removed in which it occurs interchangeably or as a synonym for 'alethic'. (I suggest it should simply be removed altogether in favor of the term that is actually used -- alethic.) Nortexoid 08:26, 24 Mar 2005 (UTC)

User:Radgeek seems to be keen on the usage, but didn't respond to a question about the usage when I asked him about it back in February. Subjunctives are not the way that alethic mode is usually expressed in english, so the whole treatment is a bit strange. I'd be happy to see the terminology dropped in this article. I believe that David Chalmers likes to talk about subjunctive modality, so it's not completely an unused terminology. --- Charles Stewart 14:21, 24 Mar 2005 (UTC)

Thanks Charles. That is all the more reason to drop it. Chalmers probably uses the term to refer to one's intensional, conscious states -- though I cannot be sure. Nortexoid 23:12, 24 Mar 2005 (UTC)

Howdy,

I'm sorry to see that the reply I put up about the use of "subjunctive"/"alethic" back

in February didn't come through. It was at the very bottom of a long talk page that

got cleared into archives shortly thereafter, though, so it was probably easy to miss.

I don't actually have much to add that Charles hasn't mentioned above, but here is

what I wrote:

Thanks for your question about the usage of "subjunctive" and "alethic." The quick answer is that the "subjunctive" / "epistemic" contrast comes from David Chalmers. The longer answer is that Chalmers frames the distinction as a distinction between subjunctive and epistemic (modalities, content) because he approaches the difference from the difference between indicative-mood conditionals and past-subjunctive-mood conditionals, which lines up with the metaphysical/epistemic modality distinction. ("If I am the King of France, then I'll have their heads cut off" vs. "If I were the King of France, then I'd have their heads cut off"). For myself, I usually prefer framing the distinction in terms of metaphysical vs. epistemic modalities; but "subjunctive" vs. "epistemic" currently seems to be the most widely used across WikiPedia articles (see e.g. Logical possibility), so it seems as good a candidate as any for adoption.

Of course, I'm hardly wedded to this; if you think there are strong reasons not to frame it this way, let me know and we'll see what we can work out.

—Radgeek 13:48, 28 Feb 2005 (UTC)

The connection with ordinary language usage in English, as I mentioned above, is that

Chalmers is especially concerned with the role that these modalities play in

counterfactual conditionals, which are in turn usually (or at least properly)

expressed in past subjunctive mood in English. His fondness for the term is

complicated a bit by the fact that *present* subjunctive mood is often used to express

*epistemic* modalities in conditionals; but oh well.

There is a large number of logicians in the field of modal logic who do not speak English. I don't think many people care that English has a certain grammatical term corresponding to a subclass of modal expressions. It sounds like Chalmers coined the term to suit a specific need of his. I don't think it applies at all to modal logic generally. Nortexoid 00:07, 29 Mar 2005 (UTC)

The comments about English grammar and the use of the subjunctive mood were in response to Charles Stewart's comments on English usage above. They weren't intended as having any broad import about what term should be used, just as a response to one proposed worry about the specific choice. On the other hand, it's also worth noting that the use of past subjunctive-conditional constructions to express counterfactuals is by no means limited to English. It's a pretty common construction. (Also that this is an article in English for the English-language edition of WikiPedia, so it's not very clear to me why

As for Chalmers' purposes, it's quite clear from the text of the relevant papers that the term is intended to make a specific distinction between two kinds of modality, and specifically the kind of inferences that each kind of modality permits. He applies it in order to develop his theory of two-dimensional semantics, but the points about the content of propositional attitudes are derived from the arguments about modal inferences, not vice versa. Specifically, he's arguing that Kripke's conclusions about necessity and contingency apply to one kind of modal term and not another:

What an epistemic intension does not do, if Kripke's arguments are correct, is determine an expression's extension when evaluated in explicitly counterfactual scenarios. When we consider these scenarios, we are not considering them as epistemic possibilities: as ways things might be. Rather, we are acknowledging that the character of the actual world is fixed, and are considering these possibilities in the subjunctive mood: as ways things might have been. That is, rather than considering the possibilities as actual (as with epistemic possibilities), we are considering them as counterfactual. If Kripke is right, then evaluation in this sort of explicitly counterfactual context works quite differently from the evaluation of epistemic possibilities. This point still needs explaining.

It is striking that all of Kripke's conclusions concerning modality are grounded in claims concerning what might have been the case, or what could have been the case, or what would have been the case had something else been the case. Kripke is explicit (1980, pp. 36-37) in tying his notion of necessity to these formulations, and almost all of his arguments for modal claims proceed via these claims. What all these formulations have in common is that they involve scenarios that are acknowledged not to be actual, and that are explicitly considered as counterfactual scenarios.

All these claims are subjunctive claims, not in the syntactic sense, but in the semantic sense: they involve hypothetical situations that are considered as counterfactual. The paradigm of such a claim is a subjunctive conditional: 'if P had been the case, Q would have been the case'. We can say that all these claims involve a subjunctive context, where a subjunctive context is one that invokes counterfactual consideration. Such contexts include those created by 'might have', 'would have', 'could have', or 'should have' (on the non-epistemic readings of these phrases), subjunctive conditionals involving 'if/were/would be' or 'if/had/would have', and other phrases. In Kripke's sense of 'possible' and 'necessary', where 'it is possible that P' is equivalent to 'it might have been the case that P', then modal contexts such as 'It is possible that' are themselves subjunctive contexts.

...

Just as the epistemic intension mirrors the way that we describe and evaluate epistemic possibilities, the subjunctive intension captures the way that we describe and evaluate subjunctive possibilities. To evaluate the subjunctive intension of a sentence S in a world W, one can ask questions such as: if W had obtained, would S be the case?

--David Chalmers, On Sense and Intension

You might not think that the terminology popularized by Chalmers is the best for the task. That's fine; I don't think that it is either. Most discussions that I've read on the matter other than Chalmers' and those by people writing in response to Chalmers use "metaphysical modality" / "metaphysical possibility" to include the kind of modalities that have been variously described on WikiPedia as metaphysical, subjunctive, alethic, etc. This usage has the nice feature that it is common in the literature, and also that it also (rightly) ties the distinction to broader discussions of metaphysical/epistemological distinctions in philosophy. Alethic would also be fine (although I, for one, have never encountered the use anywhere other than on WikiPedia). The main thing is just to pick one and run with it across articles. Which had not been done heretofore. Whatever the consensus seems to be on the best term to use, I'll be glad to sign on with. Radgeek 23:14, 31 Mar 2005 (UTC)

It's worth noting that Chalmer's use of the term is semantic, *not* primarily

psychological. He uses it to look at some questions in philosophy of mind (e.g.,

he uses it to split or sidestep debates about broad and narrow content of

propositional attitudes), but it's developed to solve semantic puzzles and it has to

do with what modal relations a particular proposition supports. (Subjunctive content

grounds counterfactuals; epistemic content grounds alternative scenarios.) It's not

intrinsically tied to conscious states (indeed, he thinks externalist considerations

apply specifically to *subjunctive* content that don't apply to *epistemic* content

-- Twin Earth cases and the like) any more than any other notion having to do with

propositional content does.

That said, I mention above, I'm not a huge fan of "subjunctive" as the contrast phrase

to "epistemic", but there were at least three distinct usages on the pages on modal

logic and related topics--"subjunctive," "aletheic," and "metaphysical," and possibly

others--but "subjunctive modality" seemed to be used a bit more frequently than the

others were, so I tried to standardize a bit across articles. If there is a consensus

in favor of some other term to contrast with epistemic modalities, rather than

"subjunctive," I wouldn't have any objections to using *that* instead. The main thing

is just to try to find one that we can stick with.

Cheers, Radgeek 06:58, 28 Mar 2005 (UTC)

I'm not sure what you mean by "contrast phrase to "epistemic"", but I suggest using alethic as a general term for the notions of necessity and possibility. Finer distinctions may be made under metaphysical necessity, logical necessity, the necessity of logical validity and demonstrability, and so on. Nortexoid 00:07, 29 Mar 2005 (UTC)

That's fine by me, if that's what most people want. What do other folks think?

My only question about the usage is that I'd be interested to have some references where "alethic" has been used in the literature as the term for this class of modalities.

Cheers, Radgeek 23:14, 31 Mar 2005 (UTC)

Some sources include Logical Options: An introduction to classical and alternative logics (Bell, DeVidi, et al); First-order Modal Logic (Fitting, Mendelsohn); The Worlds of Possibility: Modal Realism and the Semantics of Modal Logic (Chihara); Modal logic: An introduction (Chellas); and I'm sure many many more.

If "subjunctive modality" is common, then it is common among non-logic texts/journals. The term "alethic" is popular in the literature on modal logic, but not necessarily popular in other literature (e.g. in the philosophy of mind or non-logical work on necessity, metaphysics or epistemology). Since this article is about modal logic, not epistemology or metaphysics or whatever, I think we should stick to alethic. You appear to agree provided others cast some sort of vote, but a vote shouldn't matter while the article sits around collecting dust using obscure terminology until enough people agree to use "alethic". Nortexoid 00:26, 1 Apr 2005 (UTC)

Thanks for the references.

As for the rest, I think there's some things to disagree with here but it's not very important to me to hash it all out. As I said before I don't care very much which term is picked for use in articles about modality in philosophy, just as long as some term is picked for primary use rather than the hodgepodge that has been used heretofore. I'm sorry if I've been holding up edits by being unclear: I'm not sticking to any one term, either absolutely or conditionally, and I don't think any kind of head-counting is necessary or desirable before edits are done. If you think the edit needs to be made, I say make it; I was throwing out a question about what others think to try to solicit discussion, but I didn't mean to suggest that edits should be held up unless / until others chime in.

Cheers, Radgeek 02:23, 2 Apr 2005 (UTC)

Fair enough, but removing that section and replacing it with one having a more logical feel would require an extensive edit. The philosophical discussion is good and a link to subjunctive possibility would be sure to be included, but I don't have any time at the moment. Nortexoid 02:08, 11 Apr 2005 (UTC)

(edit: changed subjective -> subjunctive modality in my previous entry) Chalmer's is very involved with the critique of Kripke's argument against Fregean semantics, so he's particularly concerned with how one is anchored to one's worldview. A google search for "two-dimensional semantics" yields most of the relevant literature; our absence of a treatment of two-dimensional semantics is a bit of a lacuna, especially given how much of this stuff is web accessible. --- Charles Stewart 08:26, 26 Mar 2005 (UTC)

Formal rules

It's strange that the formal rules are specifically about an alethic modal logic when they shouldn't be, and at the same time, all they describe are axioms for K instead of axioms for what are generally considered alethic modal logics -- i.e. S4 and S5 (though they briefly discuss them).

The $\Box$ is not always given an alethic reading and it should be given a neutral one throughout the article. (It might represent temporal modalities, states of affairs, etc.) A thorough rewrite of this section seems fitting, doesn't it? Nortexoid 00:19, 29 Mar 2005 (UTC)

What's the deal with the first paragraph and the redirect? Intensional logic and modal logic are not the same thing. Someone needs to fix this. KSchutte 4 July 2005 10:35 (UTC)

Intensional logic may be subsumed under Modal logic. Nortexoid 5 July 2005 11:17 (UTC)

Certainly not vice versa. I also disagree that every system of modal logic need be intensional insofar as a system can lack semantic content. The definition of intensional logic is "any system that distinguishes an expression's intension from its extension." KSchutte 5 July 2005 16:36 (UTC)

I'm not even sure what intensional logic even means! Okay, that's a bit dramatic, but isn't any intensional meaning we associate with modal logic something that falls outside the scope of modal logic? The former belongs to the domain of philsophy or psychology, and the latter to mathematical logic. Greg Woodhouse 22:58, 30 November 2006 (UTC)

Recent changes:

I removed the comment that "A new Introduction to Modal Logic" superseeds the other two Hughes and Cresswell works. That was the original intension, but there is a lot in "An introduction to Modal Logic" that never made it into the New Introduction.

Re: Publisher of Zeman's modal logic: The copy of the book in front of me lists D. Reidel Publishing as the publisher. Note that the web page Dr. Zeman maintains says the Original publisher was Oxford.

Might this mean that there were different publishers in different countries, with OUP being who Zeeman gave his manuscript to? --- Charles Stewart 21:28, 25 October 2005 (UTC)

Possibly, that is not a subject we've cooresponded on. But note that the book is copyright 1966, and the web pages have his copyright 1973-200, so I'm sure that there is an interesting story there.

The link to dynamic logic needed to be disambiguated. I haven't tested the other links to see if they go to the right place. Greg Woodhouse 22:46, 30 November 2006 (UTC)

History.

There are some interesting statments of history there. I think there is some confusion, and an odd bias here.

Lewis's 1918 work "A survey of Symbolic Logic" (University of California Press, Berkeley) was a followup on his articles in "Mind" and "Journal of Philosophy" (as his 1918 itself work points out). He even gave an axiom set in one earlier paper. (So the 1918 paper shouldn't be cited as the start.) Note that it is a survey of Symbolic Logic, not a survey of Modal Logics. It introduces one recognisably modal system, the "Logic of Strict Implimentation, that was generally refered to as "Lewis' S system".

But even then, he wasn't the only one working on a calculus of strict implication... Others were also working on the same problem in responce to Russell and Whitehead's "Principia Mathematica". The journals of the time have many articles on the efforts to find a "real" implicational logic by people unhappy with Russell and Whiteheads "material implication".

[And of course, Lewis admited in his "Strict Implication, An Emendation" in the "Journal of Philosophy, and Scientific Method" (XVII [1920, p300])) that the axiom set in 1918 was flawed (in Responce to E.L. Post's proof that it had the generated theorem "possibly p is equivalent to p". (And Lewis provided a fix for the problem.)

On page 292 of the 1918 work credited the basic ideas to MacColl's "Symbolic Logic and its Applications."

Note that as late as 1920 paper it was just "Lewis's system S". The series S1 through S5 were introduced in Lewis and Langfords 1932 "Symbolic Logic". At that point the "System of Strict Implication" of the 1918 paper was renamed to be S3. [Still the most interesting of the set to my mind.]

In 1918 his system was based on an indivisible "impossiblity" operator. By 1932 he was using possibility as an operator, with impossibility being the negation of that operator, and he had the basic Lewis Systems.

Kurt Gödels papers were key in changing the view from Lewis' "impossiblilty" based axioms to the current "Necessity" based axioms. [Edit: And Gödel introduced using a basis of PC + modal axioms, instead of modal axioms from which PC could also be proved.]

Note that the main history from 1936 on is on line at the various Journal's web pages. Some history of the 1918 system is covered on my page http://www.cc.utah.edu/~nahaj/logic/structures/systems/s.html which I'm currently working on.

If I hear no objections, I'll make a major edit adding the historical context that Lewis' work grew out of. [The goal was, after all, a legitimate calculus of strict entailment, not a modal logic... those just grew out of the effort. Nahaj 03:24, 28 October 2005 (UTC)

If you are calling it a calculus of strict *entailment*, then I already object with any changes you might make. Nortexoid 05:16, 28 October 2005 (UTC)

I agree, strict implication is the normal term. --- Charles Stewart 14:25, 28 October 2005 (UTC)

I had trouble figuring out what the "it" in your sentence refered to. I'm guessing you mean it to mean Modal Logic. I don't call modal logic such a calculus. But modal logic *DID* come out of attempts to develop such a calculus. The original Lewis systems *WERE* by his direct statement such calculi. Nahaj 13:27, 28 October 2005 (UTC)

It was supposed to formally capture a notion of *implication*, not entailment. Nortexoid 14:20, 28 October 2005 (UTC)

Please go ahead. This material sounds interesting. --- Charles Stewart 04:04, 28 October 2005 (UTC)

I'm still researching MacColl's work. If Lewis thinks the main ideas of his work trace there (as he says he does), then at least a mention is needed. Nahaj 13:27, 28 October 2005 (UTC)

I'm researching MacColl at the moment for an Algebraic Logic account, and MacColl's influence on Lewis is something that might go into the story there. Maybe we should exchange notes. Most of the sources I am using are in the algebraic logic point at User:Chalst/tasks. --- Charles Stewart 14:25, 28 October 2005 (UTC)

I'm reading his series on "Symbolic Reasoning" in "Mind" (A series that appeared from the 1890's to 1900's). I'm availiable through email if you want to discuss the topic in a non-public forum. Nahaj 15:57, 28 October 2005 (UTC) [His "Symbolic Logic and its Applications" is mostly a collection of his articles in "Mind"] Aside: I notice that B. Russel said in his review of MacCall's book: "... will be found highy instructive by beginners, and stimulating by all readers." Nahaj 16:18, 28 October 2005 (UTC)

Speed of Light

Is it logically possible to travel faster than light?

Most philosophers would say that it is logically possible to travel faster than the speed of light but not physically possible. Saying "I'm going faster than the speed of light!" doesn't seem to involve a logical contradiction but it is physically impopssible.

Timothy J Scriven 11:09, 4 September 2006 (UTC)

The physical possibilities section was a bit confused, now somewhat less so. Current physics limits information and matter to <= c, with or without mass. Atomic weight 150 is not proven physically possible until there is an example, although it is possible according to current theory. A nucleus made of cheese is neither logically nor physically possible, so there is no need to specify in which sense it is not possible. Current theory, BTW, depends on some mathematical entities (infinities)that are only approximately possible, so some things that are considered physically possible in theory are only approximately so, and thus may be logically impossible. Fairandbalanced (talk) 05:30, 4 February 2009 (UTC)

Actually it is not true that travel of matter or information at speeds over c is scientifically impossible. It's only impossible according to general relativity, but according to quantum mechanics light is described by a wavefunction that has a non-zero value in all points in space and therefore it is possible to encounter any photon at any point in space, which implies that photons can travel at any speed. According to quantum mechanics, it's only on average that light travels at c in vacuum. Actually faster than light speed transmission of signals has already been experimentally verified (they transmitted Mozart's 40th symphony at over 4c over a barrier of some 10cm, see: http://www.npl.washington.edu/AV/altvw75.html or http://www.sciencenews.org/articles/20000610/fob7.asp). Being this an encyclopedia we should correct the main article on this point.

The larger issue is that as scientific knowledge is contingent (and provisional), and as "physical possibility" refers to what we know via science, there can't it seems to me be any "physical impossibilities" but only "logical impossibilities". After all even if something is physically impossible in our universe, or in a world where the laws of our physical universe apply, it does not follow that there aren't possible worlds where such laws do not obtain.Dianelos (talk) 23:34, 10 July 2009 (UTC)

The measurement of Nucleus decay shows that particles do travel the barrier faster than light. Students are conciliated by the fact that decay carries no information. So only information rather than particles cannot and the statement "modern science stipulates that it is not physically possible for material particles or information" that we see in the article is obviously false and contradicts to the Quantum tunneling article. Quantum non-locality also says that information also travels faster then light. We just cannot control it to exploit this effect for communication. Is it good to have such a false statement in the article? --Javalenok (talk) 10:00, 3 August 2012 (UTC)

PLTL and LTL

Is there a difference between LTL and PLTL? -- Neatlittleeraser 13:17, 6 July 2006 (UTC)

LTL usually means the propositional variant, although there is a quantified version, QLTL. — Charles Stewart (talk) 10:18, 17 March 2009 (UTC)

Logically necessary example

Many logicians also hold that mathematical truths are logically necessary: it is impossible that 2+2 ≠ 4.

Is this a correct example? Let me explain why I am in doubt: it's very well possible that 2 + 2 = 0. We can either define a number system in which the same symbols as we're used to represent different quantities and the identity might hold. Or we could work in the numbers modulo 4. Or we could work in some group where "2 + 2" is the group operation, which happens to be denoted by the symbol +, applied to some element, which happens to be denoted by the symbol 2. My point is: the statement is only necessarily true given the proper axioms and definitions (or: given the appropriate context) in which case it is necessarily true because if follows from those axioms and definitions (which is exactly against the point the original author is trying to make, if I understand correctly). Without such context, one could argue both "2 + 2 = 4" and "2 + 2 = 6" to be logically necessary. --CompuChip 14:44, 15 November 2006 (UTC)

Recent developments in modal logic attempt to address these concerns (e.g., 2-dimensionalism) and should be included in this article. Your right, we have to say, "given our semantics for 2, +, = and 4, "2 + 2 = 4" could never be false, in any possible world. But there could conceivably be a possible world (even a possible language in this world) where these symbols have different meanings, and where "2 + 2 = 4" comes out false. Thus, there are multiple propositions expressed by any given sentence (esp. when you consider possible worlds), but when we assume one of these propositions as expressed, we can talk meaningfully about necessity (see Chalmers, David. The Conscious Mind pp. 56-69).--Heyitspeter (talk) 19:51, 16 March 2009 (UTC)

I agree about the indeterminacy regarding the relation of expressions to propositions, but (i) when one says P is necessarily true, one talks about the proposition that we understand the expression to be, not whatever proposition one might counterfactually conceive to be be understood. And (ii) there is of necessity some "charity of interpretation" going on here, where symbols should be interpreted by us in the usual way; one is not a constructive participant in the conversation if one says "But when I was using those alethic modalities, I was talking of the degenerate interpretation of possibility, where the only possibility is actuality". If one wishes to use symbols in a nonstandard way without being a nuisance, one must advertise the fact. One is violating more than one of Grice's maxims if one does not. — Charles Stewart (talk) 10:32, 17 March 2009 (UTC)

Eh, the charity of interpretation goes both ways. CompuChip is talking about, for example, a vector space where "1+1=1." You don't advertise that this is a vector space where the symbols are used in a non-standard way. You just deduce it. "1+1=1" is a true proposition in some mathematical environment (so "1+1≠1" is false in some mathematical environment). The 1s can be interpreted (really deduced) to be 0-vectors. But "1+1≠1" sure seems necessarily true when we take the symbols to have their usual contents. That is to say, given one usage of the terms, "1+1=1" is a statement that expresses a necessarily true proposition. However, given another usage, "1+1=1" expresses a false proposition. That's 2-dimensionalism in a nutshell (clumsily). There are two levels of necessity: deep and superficial. "1+1≠1" is deeply necessary when we fix its meaning. But "1+1≠1" is superficially contingent, because it can be false under some possible interpretation. --Heyitspeter (talk) 11:02, 17 March 2009 (UTC)

Sure, but outside of a context where we don't draw attention to the possibilities for rival interpretations, there is a strong convention that we are talking about the proposition determined by the meaning conventionally used by the relevant language community. Sure one can say "look what happens when I vary the interpretation", however to say "that's nonsense!" because of the possibility of rival interpretations is problematic. — Charles Stewart (talk) 11:18, 17 March 2009 (UTC)

Ahh, yeah. I see what you mean. So the two-dimensionalist (usually) would agree with you, that the cognitive content of a proposition is its primary content (the fixed meaning, not the possible one). But he or she would also claim that the secondary content is important, and not trivial. It's supposed to explain how "2+2=4" is necessary, but appears contingent, because of the possible propositions it can be taken to express. It sounds like you would agree with this. I'm sorry if I've been really muddled in my explications, it's late here...--Heyitspeter (talk) 11:35, 17 March 2009 (UTC)

Right. I guess the 2d semantics is more relevant to the rigid designator article than here, though this article should have something about that, since it is so crucial to the interpretation of mode in natural language by modal logic. — Charles Stewart (talk) 14:51, 17 March 2009 (UTC)

Haha yeah I think you're right. It is kind of secondary to this article. I think I just saw this guy's question and was like, "wow, that is something important to discuss (in general), we should talk about that!" without really thinking about relevancy. --Heyitspeter (talk) 19:13, 17 March 2009 (UTC)

And they would not be necessary if we chose to use 'necessary' in some way other than it is usually used, even if all of the numerals and the entire symbolism of arithmetic is given its intended interpretation. But that does not make '2+2=4' not necessary; it's called changing the subject. The sentence is necessary under the standard interpretation of the symbolism of arithmetic. That's what we mean when we say it's necessary. We don't mean under every interpretation of the symbolism, as that is obviously false, and not very interesting. Nortexoid 20:57, 15 December 2006 (UTC)

It really isn't trivial, though. e.g., suppose "water = watery stuff"; that this is just what water is. That's an a priori truth. We don't have to look at the world to know it. But now suppose there is another world where the watery stuff isn't the same as the watery stuff in our world (where mercury has watery qualities, and our water [H20] doesn't). This world is possible, which means that "water = watery stuff" isn't necessarily true, because the stuff we call water isn't necessarily watery stuff. There is some possible world where our fixed meaning of water isn't watery. So "water = watery stuff" is an example of an a priori contingent truth, and it is so because different interpretations can be had for the same symbols. --Heyitspeter (talk) 20:09, 16 March 2009 (UTC)

The 'diamond' is more formally called the lozenge, is it not?

That's a typical ascii representation, yes, but many texts and articles actually use a square rotated 45% and not a thin rhombus. Simões (^talk/_contribs) 17:51, 30 March 2007 (UTC)

Back to the original query: I think it's strange to talk of logical necessity wrt. mathematical propositions: if we say that X-al necessity means necessity modulo what X gives (eg. conceptual, physical, metaphysical, etc.), then it looks as if the logically necessary propositions are only the logical tautologies. Since the sentence has been dropped, the whole issue is moot. — Charles Stewart (talk) 14:51, 17 March 2009 (UTC)

Doxastic logic

I'd like to see a section or page for Doxastic logic. Gregbard 00:06, 29 June 2007 (UTC)

Be bold, add one. :) If you need them, I can probably dig up a non-web print reference two I could point you at. Send email, I have frequently gone months without signing on here.] And for web references, have you checked out Professor Peter Suber's site? He (among other things) collects information of many many different types of logics. Nahaj 21:10, 20 September 2007 (UTC)

If I understand it correctly there is the logic of beliefrevision, see http://plato.stanford.edu/contents.html, and the Hintikka system in Knowledge and belief. SEP has for instance no entry on doxastic logic. --RickardV 20:36, 2 August 2007 (UTC)

Lower case T

I have changed the system name T in the article to upper case. No modal logic book in my collection lists it as lower case. (For example, Hughes and Cresswell's "Introduction to Modal Logic", Brian Challas' Introduction, Fey's Introduction... [fuller list upon request]) In addition the articles in "The Journal of Symbolic Logic" and in the "Notra Dame Journal of Formal Logic" also have it upper case. [Specific examples upon request]. In the two years since I challenged the lower case claim above, nobody has yet given a single reference that supports the claim that it is usually lowercased. So, I think it is fair to say that the lower case usage is NOT standard, and the upper case usage is. Nahaj 23:38, 18 September 2007 (UTC) [ReEdited] Nahaj 20:35, 20 September 2007 (UTC)

A contradiction?

From the article:

Metaphysical possibility is generally thought to be stronger than bare logical possibility (i.e., fewer things are metaphysically possible than are logically possible). Its exact relation to physical possibility is a matter of some dispute. Philosophers also disagree over whether metaphysical truths are necessary merely "by definition", or whether they reflect some underlying deep facts about the world, or something else entirely.

If philosophers disagree about whether or not metaphysically true is the same as true by definition, it seems unlikely that they have come to a consensus about whether metaphysical possibility differs from logical possibility. Phrenophobia 07:15, 7 November 2007 (UTC)

Implementations and applications?

There seems to be little coverage of implementations and applications of modal logic.

1. Are there any modal logic reasoning engines?
2. Are they efficient and complete?
3. Can modal logic reasoning be added to existing logic reasoning systems (e.g. prolog)?
4. Is modal logic being used to solve any real problems?

Pgr94 (talk) 11:48, 16 April 2008 (UTC)

Found a promising article: I. Horrocks, U. Hustadt, U. Sattler, and R. Schmidt. Computational modal logic. In P. Blackburn, J. van Benthem, and F. Wolter, editors, Handbook of Modal Logic, chapter 4, pages 181-245. Elsevier, 2006.[2]

Also part 4 of the above handbook covers applications: [3]

Pgr94 (talk) 13:14, 28 April 2008 (UTC)

See also the resources here: http://www.cs.man.ac.uk/~schmidt/tools/ pgr94 (talk) 17:59, 12 February 2010 (UTC)

Schemata

I've uploaded an english version of a schema used on the french article. You may want to use it, or not. There are Kripke models schemata in the Kripke semantics article as well, tell me (here, because I don't watch pages on the english WP) if you want SVG versions in english. --Eusebius en (talk) 10:07, 21 September 2008 (UTC)

A change concerning contingent propositions, etc.

The article included the claim that all contingent propositions are actually true, but this does not accord with current usage. I have therefore changed the wording to this:

A proposition is said to be

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

Among the copious evidence for this usage are these sources: [4], [5], [6], [7], [8], and [9].

I have also changed a piped wikilink to read "physical laws", since physical possibility is defined in terms of physical laws and not natural laws. A minority of philosophers, such as David Chalmers, hold that there is a difference. While we may not agree (I don't!), the notional distinction ought to be maintained. After all, we maintain other distinctions, between metaphysical and logical necessity for example – which Chalmers himself claims amount to the same.

Elsewhere I have changed wording for strict propriety, so that the term possible is not itself used in examples to show the difference between alethic possibility and epistemic possibility. That is bound to confuse people needlessly.

–_⊥^{¡ɐɔıʇǝo}Noetica!^T– 10:24, 4 February 2009 (UTC)

Little conflict resolution

• (v.1) It seems reasonable to say that possibly it will rain tomorrow, and possibly it won't; on the other hand, seeing as how we can't change the past, if it rained yesterday, it cannot be quite correct to say "It may not have rained yesterday."

• (v.2) It seems reasonable to say that possibly it will rain tomorrow, and possibly it won't; on the other hand, if we know that it rained yesterday, then it cannot be quite correct to say "It may not have rained yesterday."

So I changed v.2 to v.1, and it was reverted because it was asserted that v.1 was problematic, as it does not explicitly state that we know something about the past. Just want to point out that this is a conditional (if it rained yesterday, it cannot be quite correct to say "It may not have rained yesterday"), so it doesn't matter whether we know that the antecedent is true or not. For example, "if P and not-P, then every proposition is true." We know this conditional to be true, even though we can never know that the antecedent is true, because the antecedent is never true. So it doesn't matter whether we know anything about the past. The point is, if x happened (note the past tense), then x necessarily happened. We don't need to know anything about the past to see that this statement is true (of course, it may not be true, but our knowledge of the past has nothing to do with it). Possible problems crop up because the statement is ambiguous as to whether or not the consequent involves a subjective statement or a proposition. I'll change it a bit to avoid this. I figure:

• (v.3) It seems reasonable to say that possibly it will rain tomorrow, and possibly it won't; on the other hand, seeing as how we can't change the past, if it rained yesterday, it probably isn't true that it may not have rained yesterday.

Happy editing :) ---Heyitspeter (talk) 21:54, 14 March 2009 (UTC)

yesterday's rain

I don't feel strongly about this and in fact I have barely looked at the article. It's just that I think talking about facts in the past that we may not know about introduces unnecessary complications that distract from the main point. It's quite possible that I don't know whether it rained yesterday, so in natural language "It may not have rained yesterday." is a perfectly valid sentence. --Hans Adler (talk) 22:01, 14 March 2009 (UTC)

Ahh I see what you're saying. You're trying to include sentential operators (For all x knows that: ). "For all I know, it didn't rain yesterday" can be true even if it actually did rain yesterday, so you would need actual knowledge of the antecedent to force the consequent (i.e., I know it didn't rain yesterday, so it actually didn't rain yesterday). But when you drop these epistemic modal operators you get a different story. If it rained yesterday, it is not possible that it did not rain yesterday (whether we know it rained or not). Your analysis is correct if you assume that these statements are epistemic, but it isn't if we don't assume that they're epistemic. And we don't want to, because we want to talk about the world, and not about our conception of it. That's kind of cheating, and one could argue that there are no non-epistemic propositions, but that's kind of not in question here. What do you think? EDIT: I think we posted responses to each other at the same time, so they ended up as two different sections (see above). --Heyitspeter (talk) 22:12, 14 March 2009 (UTC)

I stopped thinking. I am not at all familiar with modal logic, and it seems it's too late for me (hourwise) to learn what "epistemic" means so I have a chance to respond in a meaningful way. Sorry for having bothered you. I will simply trust that you know what you are doing. --Hans Adler (talk) 23:03, 14 March 2009 (UTC)

Hey it's not a bother or anything, in any way. The point you made was right, even. For now, get some sleep, haha, but please read this over tomorrow and see if you still think I've made a bad move. I'd be happy to hear about it.--Heyitspeter (talk) 23:19, 14 March 2009 (UTC)

p.s. the section on 'epistemic logic' in this article clarifies a lot of these issues.--Heyitspeter (talk) 23:27, 14 March 2009 (UTC)

Redundant section

The article currently has sections Brief history and Development of modal logic. In view of the latter, the former seems redundant. Is there any reason not to delete it? — Charles Stewart (talk) 12:38, 3 May 2009 (UTC)

Done.--Heyitspeter (talk) 05:35, 26 July 2009 (UTC)

Catholic theology

It might be of some value if the article could mention the role that modal logic has played in the development of Catholic theology. For example, Gödel's ontological proof mentions the extent to which modal logic was used in Saint Anselm's ontological argument for the existence of God. ADM (talk) 14:36, 6 November 2009 (UTC)

Want to add a section? Worst comes to worst someone will delete it. I believe Alvin Plantinga has made a modal argument for the necessary existence of God as well...--Heyitspeter (talk) 23:38, 6 November 2009 (UTC)

Deontic killing

In what deontic universe is killing a victim a moral obligation? Aren't there better examples that better avoid the emotional distress of reading the example overshadowing the underlying logic? Rursus dixit. (^mbork³!) 20:58, 6 March 2010 (UTC)

Hmm. I mean, it's supposed to be the case that there isn't such a deontic universe, for the sake of the reductio. But I see what you mean about the example being graphic to some extent. I figured it would be good to use an instance of an action deemed more or less universally evil, and it is matched in the citation, but I see what you mean. Care to propose an alternative reprehensible act that isn't so emotionally distressful?--Heyitspeter (talk) 00:03, 8 March 2010 (UTC)

Intuitive problems with deontic logic

This section is not making the point that it is trying to make, but I don't know what that is so I don't know what needs to change. The supposed problem is that the neither (1) or (2) express the desired statement, but the given explanation of (1) is identical to the original statement, I'm indenting everything so that's easy to see:

    Now suppose we want to express the thought that
         "if you have killed the victim,     it ought to be the case that you have killed him quickly".

    But (1) says that
         if you have killed the victim, then it ought to be the case that you have killed him quickly. This surely isn't right, because you ought not to have killed him at all.

Ohspite (talk) 18:34, 9 December 2011 (UTC)

I also find the quality of this section quite low!

I agree with the person above: given $K\Rightarrow \Box Q$ and K as true, then the result Q is not the truth, the whole truth, and nothing but the truth, which is $K,\Box Q,...$ see naturalistic phallacy on the same page...

Could the conclusions which are described in words be more formally made, showing which rule is used at each step?

this is for both the conclusions made for (1) and (2)... — Preceding unsigned comment added by 83.134.181.100 (talk) 17:58, 22 January 2012 (UTC)

I also cannot parse this section: "(1) says that if you have killed the victim, then it ought to be the case that you have killed him quickly. This surely isn't right, because you ought not to have killed him at all". Why isn't this translation to logic right? In a possible world where you kill, you ought to kill humanely. Perhaps you shouldn't have entered this possible world by killing in the first place, but that's irrelevant to the statement at hand. Because the problem with this section is apparently long-standing, I've decided to be bold and delete. -150.203.209.134 (talk) 06:35, 10 August 2012 (UTC)

The problem that there is supposed to be with interpretation (1) is that "you ought to kill him quickly" entails "you ought to kill him" simpliciter, so that interpretation means that if you have killed someone, then you ought to have killed them. That it goes on to specify the manner in which you ought to have killed them (quickly) is immaterial to the objection that this interpretation logically renders the statement a claim that any murders which happen ought to have happened.

It's the objection to (2) here which doesn't make any sense to me. "You kill him slowly" isn't equivalent to ~Q; "you do not kill him quickly" is. That could be accomplished either by killing him slowly, or by not killing him at all. And how []~K together with (2) is supposed to somehow give [](K -> ~Q) is not at all clear either; (2) just says "either you must kill him quickly, or you must not kill him". ([]Q or []~K) = ([]K -> []Q) = [](K -> Q).

The standard objection to that interpretation that I am aware of is that it seems to give permission to kill so long as it is done quickly. I've never thought that objection made any sense: the statement does not tell you that you may not kill, but neither does it tell you that you may; all it really says is that you may not kill slowly, if at all. It is perfectly consistent with the stronger claim that you may not kill at all.

I am going to revert the deletion, but I'm not going to be so bold as to change the objection to (2) just yet as I don't have any sources handy to back up the above paragraph, that's just memory from my class on metaethics (which is where we covered deontic logic in my course). --Pfhorrest (talk) 07:27, 10 August 2012 (UTC)

[]~K entails [] (K -> ~Q) because, if in every permissible world, there is no killing, then every permissible world that does contain a killing contains a slow killing. It's a standard modal argument (and presumably, a standard objection to modal logic really; we could just as well describe this as "The king is immortal" entailing "When the king dies, the queen will not be happy", which looks just as odd).Ben Standeven (talk) 01:07, 22 October 2012 (UTC)

Satisfaction symbol

Why does this article use ⊨ instead of $\Vdash$ (which is used in Kripke semantics, for example) as the symbol for the satisfaction relation? --Spug (talk) 16:30, 10 May 2010 (UTC)

Because the former is the symbol used in the text referenced by that section (viz. Fitting & Mendelsohn's 'First-Order Modal Logic'). That's the only rationale. Do you think it's worth changing?--Heyitspeter (talk) 18:45, 10 May 2010 (UTC)

Aha! Well, I do think it creates confusion that two different (but similar) symbols are used for the same relation. Especially when articles on other logics such as FOL use ⊨, and this article does as well, as the only (?) article on ML. Isn't $\Vdash$ used almost ubiquitously in (newer?) literature (especially introductions to ML such as Blackburn, Sally Popkorn, etc.)? --Spug (talk) 21:17, 10 May 2010 (UTC)

I don't know! Is it used in all those texts?--Heyitspeter (talk) 01:17, 11 May 2010 (UTC)

If the symbol used for Modal logic really is used by FOL and other logics, it might make more sense to adjust the Kripke semantics article to align with them...--Heyitspeter (talk) 01:38, 11 May 2010 (UTC)

Well, ⊨ is usually a binary relation between models (or theories) and sentences, while $\Vdash$ is a ternary relation between a (Kripke) model and a point in that model (that is, a world) and sentences. Seeing as they take different arguments I think it's smart to differentiate them, and change to $\Vdash$ in this article as well. --Spug (talk) 09:08, 12 May 2010 (UTC)

Hmm. I'm not particularly opposed to your proposal; as far as I'm concerned, feel free to be bold. But I may as well tell you what I'm thinking: Given F&M's manner of defining truth in a model, where ⊨ is a binary relation between worlds and propositional letters, they're actually using the satisfaction symbol correctly by your considerations. Trading it for $\Vdash$ while leaving everything else in the article alone would give us a ternary relation between only two elements..--Heyitspeter (talk) 18:58, 12 May 2010 (UTC)

Actually, it looks like the $\Vdash$ used over at Kripke semantics is a binary relation too...--Heyitspeter (talk) 22:01, 12 May 2010 (UTC)

Okay. I've looked over a few of Kripke's early articles. "Semantical analysis of modal logic I" and "II (non-normals)," and his first paper on completeness. Kripke does not use $\Vdash$ to define truth in a model. He uses $\phi$, which he identifies as the model. As per the first paper:

"We say a formula A is true in a model $\phi$ associated with a model structure (G, K, R) if $\phi$(A, G) = T; false if $\phi$(A, G) = F. We say A is valid if it is true in all its models; satisfiable if it is true in at least one of them."

Here A is a formula, obviously. K is a non-empty set (the possible worlds) and R is the accessibility relation. G is "a distinguished 'real world'. (Interesting that Kripke distinguishes this world where later authors do not!) I imagine this is hard to follow. Kripke spends a couple pages explaining what I just detailed in a few sentences, so don't feel weird asking for clarification. But maybe it's not important. The point is that not only does Kripke not use $\Vdash$ or ⊨ to define satisfiability, he doesn't even use relations that have the same -arity! His is monadic, where the two we're discussing are ternary and binary, respectively.--Heyitspeter (talk) 22:32, 12 May 2010 (UTC)

Hahah honestly, what I just wrote was more interesting than relevant. I don't see any reason not to alter this article to match Kripke semantics. I say "do it."--Heyitspeter (talk) 03:03, 13 May 2010 (UTC)

SOS

Will someone create a lead that will alow me to understand what the hell this article is talking about. Thanks 198.85.219.5 (talk) 23:13, 22 July 2010 (UTC)

Classical Systems and Minimal Models

Chellas's "Modal Logic an Introduction", CUP, 1980, mentions classical systems of modal logic and minimal models for them. The article could well discuss these matters (with references to Montague and Scott), and so much else besides: it seems to me to be a mere sketch. —Preceding unsigned comment added by 213.122.58.155 (talk) 14:10, 11 August 2010 (UTC)

WP:BOLD --Heyitspeter (talk) 04:09, 13 August 2010 (UTC)

Not all model logics are Intensional

Not all modal operators are intensional, for example, Łukasiewicz and Tarksi worked on extensional modal logics, most notably Łukasiewicz's many-valued logics. Gödel's many-valued (and the fuzzy variant also known as Dummett logic) also has an extensional modal operator.

The section on Other Modal Logics confuses this, and is wrong. —Preceding unsigned comment added by 94.173.19.247 (talk) 17:04, 3 November 2010 (UTC)

⊭ needs a definition

The phrase "w ⊨ ¬P if and only if w ⊭ P" currently appears in the article. The symbol "⊭" is only used one time in the article and is not defined. As "w ⊨ ¬P" means something like "not P is true for world w", I assume that "w ⊭ P" means "P is false for world w". However, as I am a novice here (I am learning modal logic from the article), I very easily could have something subtle wrong in my semantics. So, it'd be good for an expert in the subject to define the "⊭" symbol. Jason Quinn (talk) 14:30, 2 December 2010 (UTC)

There is a standard convention in mathematics that adding a slash over a relation symbol negates it. So $x\not <y$ means x is not less than y, and $w\not \models P$ means that w is not a model of P.

In this case, "w ⊨ ¬P" literally means that "w is a model of ¬P", that is, that P is false in w. On the other hand w ⊭ P means that P is not true in w. It may seem strange to claim that these are equivalent, but remember that if the semantics were sufficiently strange then maybe there would be some sentence P so that neither P nor ¬P is true in w. Also remember that "truth" here is a defined notion, it is not the informal "truth" in the real world. — Carl (CBM · talk) 15:21, 2 December 2010 (UTC)

Thanks for the reply, Carl. I realize now I didn't make myself sufficiently clear in my question. My comment was more concerned about the formal symbolic definition of "w ⊭ P". My guess is that "w ⊭ P" is, by definition, identically equivalent to the symbolic formula "¬(w ⊨ P)", which I think agrees with what we have both said prior in terms of an natural language interpretation of the logic. Do you agree? This seems to be the obvious way to interpret it, but I'm aware aware that nothing obvious can be taken for granted in this field, as subtle "gotchas" abound. Jason Quinn (talk) 14:12, 3 December 2010 (UTC)

I took the liberty of replacing "w ⊭ P" with "¬(w ⊨ P)", to which it is equivalent. This is both because, as Jason Quinn notes, "⊭" is not defined, and the visual difference between "⊨" and "⊭" is almost imperceptible in my browser. I spent quite a while staring in perplexity at the axiom, not seeing the difference between "⊨" and "⊭". CarlFeynman (talk) 03:51, 3 January 2011 (UTC)

Except someone changed it back.linas (talk) 02:16, 8 July 2012 (UTC)

Soundness and completeness of axiomatic systems

I feel it is worth mentioning the soundness and completeness results for axiomatic systems, at least for K. The approach of canonical models is also very important.

Going forward, finite and bounded model properties, decidability and complexity are worth a mention.

j.t. (talk) 19:52, 29 May 2011 (UTC)

Modal translation?

Consider ROBDD used for satisfiability tests, or formal verification on software, while these typically use combinational / sequential logic, their prime intention is modal, i.e. does something meet a specification? it should meet the specification, it may not violate specific requirements,...

Now to phrase my question in theoretical computer science terms: can modal logics be translated into each other? like skolemnization: can quantifiers as letters in words be treated as propositions? Im pretty new to modal logics... can reasonings in modal logic be described as predicate logic? to what extents are modal logic syntactic sugar and to what extent impossible to rephrase in another logic? — Preceding unsigned comment added by 83.134.181.100 (talk) 18:29, 22 January 2012 (UTC)

Euclidean?

Why bother defining Euclidean? The article currently defines:

euclidean iff, for every u,v and w, w R u and w R v implies u R v (note that it also implies: v R u)

and also:

S5 := reflexive, symmetric, transitive and Euclidean

but if a relation is reflexive symmetric and transitive, then it is also euclidean, and this property brings nothing new to the discussion. Nowhere else in the article is the word 'eucliden' used, so why bother? linas (talk) 02:22, 8 July 2012 (UTC)

Euclidean =/= Transitive

I had the same confusion, transitive means if one have aRb and bRc, then aRc also exists, while Euclidean says if one have aRb and aRc, then bRc. A subtle difference, but here it is important to know which is the implied segment.

I am some reluctant to understand this subtle difference, what makes me accept this differentiation, is the relation between the form of the formula (axiom) and the accessibility relation. — Preceding unsigned comment added by 189.178.233.172 (talk) 11:13, 4 November 2012 (UTC)

Yes, of course, Euclidean is not the same thing as transitive. but transitive plus symmetric relation is, and in the intervening decade, the article has been altered to say "its all the same thing". From what I can tell, the urge to define "euclidean" is a historical artifact. 67.198.37.16 (talk) 21:59, 31 May 2024 (UTC)

Transitive and symmetric is the same as Euclidean and symmetric (Prop.3 at Euclidean relation). However, transitive and symmetric is not the same as Euclidean alone (see 2nd picture for a relation that is Euclidean, but not symmetric). - Jochen Burghardt (talk) 17:39, 2 June 2024 (UTC)

Interderivation of necessity and possibility

Section 2.2, on axiomatic systems, states outright that possibility and necessity are interderivable through negation. This isn't true of all modal logics, only classical ones; intuitionistic logic, as usual, excludes some de Morgan duals, and this is true of intuitionistic modal logic.

I might not quibble, but the distinction is made in the introduction (where the duality is qualified as existing in classical modal logic) and in section 3 on alethic logic (where intuitionistic logic is mentioned), so there is an issue of consistency. Luke Maurer (talk) 18:47, 16 July 2012 (UTC)

the page Axiom S5 should be merged or erased the reasons in that discussion page

I asked to delete the page axiom S5 but a supervisor removed the petition, suggesting me to change that to merge. I see that the part about S5, here, seems to be more complete than the Axiom S5 page, which must be removed. I have not read the S5 (modal logic) page, but I think it may be merged with this one if it contains something not included here, if not just need to be deleted. In synthesis, the other pages should be deleted, for the reasons written in the discussion page of axiom S5, starting with the wrong title. — Preceding unsigned comment added by 189.178.233.172 (talk) 11:03, 4 November 2012 (UTC)

Last paragraph of "Epistemic logic" seems to be incorrect OR

But this is disconcerting, because with K, we can prove that we know all the logical consequences of our beliefs: If q is a logical consequence of p, then L(p->q). And if so, then we can deduce that (Lp->Lq) using K. When we translate this into epistemic terms, this says that if q is a logical consequence of p, then we know that it is, and if we know p, we know q.

This starts off by assuming "If q is a logical consequence of p, then L(p->q)". The assumption that seems to be here is drastic and unwarranted. This is saying that a->La—that is, there is only one possible world. Or, in epistemic terms, we know everything that is true. That's far more disconcerting than the fact that we know the logical consequences of our own beliefs, so who cares that the latter follows (via K) from the former?

It's possible that the assumption was actually N, put in alethic terms, and then used as an argument in epistemic terms. But with epistemic N, the fact that "q is a logical consequence of p" (which presumably means "it is a provable theorem that p->q") does not mean L(p->q). The fact that the same paragraph refers to belief rather than knowledge shows the same kind of confusion. Then it's not outrageously silly, just a simple mistake.

At any rate, people usually do not use the same K axiom in epistemic logic in the first place—instead, they use Lp ^ L(p->q) -> Lq. So, the whole argument would fall apart even if it didn't rest on a broken premise.

Stalnaker and Lewis have both given correct arguments for why epistemic N and K, with or without possible worlds, imply that we know the logical consequences of our knowledge. Either the argument here is a misremembered version of one of their arguments, or it's OR that happens to reach the same conclusion by accident. --70.36.140.230 (talk) 10:38, 20 November 2012 (UTC)

I'm moving it off the main page and onto this page. It's uncited and incoherent. I agree that the argument appears to be against N rather than K. If you believe it should be restored, then explicitly explain this claim: "When we translate this into epistemic terms, this says that if q is a logical consequence of p, then we know that it is ..." The entire objection depends on that, and otherwise we're wasting a lot of paragraph space knocking it down. Besides being ridiculously wordy, the refutation is fine, but there's no reason for it without the objection.

The removed section:

Some schools of thought have found this to be disconcerting, noting that with K, we can prove that we know all the logical consequences of our beliefs: If q is a logical consequence of p, then \Box (p \rightarrow q). And if so, then we can deduce that (\Box p \rightarrow \Box q) using K. When we translate this into epistemic terms, this says that if q is a logical consequence of p, then we know that it is, and if we know p, we know q. That is to say, we know all the logical consequences of our beliefs. This must be true for all possible Kripkean modal interpretations of epistemic cases where \Box is translated as "knows that". But then, for example, if x knows that prime numbers are divisible only by themselves and the number one, then x knows that 231 − 1 is prime (since this number is only divisible by itself and the number one). That is to say, under the modal interpretation of knowledge, anyone who knows the definition of a prime number knows that this number is prime. This shows that epistemic modal logics that are based on normal modal systems provide an idealized account of knowledge, and explain objective, rather than subjective knowledge (if anything).

However, the deduction that K implies that we know all the logical consequences of our beliefs is heavily dependent on semantics (much to the word "all") and may be faulty. We can read the distribution axiom as, "If it is necessary that p implies q, then necessarily p implies that q is necessary." which does not imply that p or q are true in either case, but just shows a relationship (if, then). However, including "x knows" in the statement might make it seem to lose some of that ambiguity. "If x knows that 'p implies q', then 'x knows that p' implies 'x knows that q'.". This can be further rewritten as, "x knows that 'p implies q', implies, 'x knows that p' implies 'x knows that q'.". To make this more readable, we can say, "if x knows that p implies q, then whenever x knows that p, x also knows that q.". The prime number example above is faulty. A correct application of the Distribution Axiom K would be worded: "If x knows that prime numbers are divisible only by themselves and the number one, then whenever x knows that a number is prime, x also knows that the number is divisible only by itself and the number 1". That is, "if x knows that 231 − 1 is divisible only by itself and the number 1, then x knows that 231 − 1 is a prime number.". Those statements can NOT be used to say that "If x knows that a prime number is divisible by itself and the number 1, then x knows that 231 − 1 is a prime number." Because x just simply doesn't have enough information to draw that conclusion. Therefore, for all x knows, K might still be disconcerting."

173.25.54.191 (talk) 07:50, 2 August 2014 (UTC)

Possible copyright problem

This article has been revised as part of a large-scale clean-up project of multiple article copyright infringement. (See the investigation subpage) Earlier text must not be restored, unless it can be verified to be free of infringement. For legal reasons, Wikipedia cannot accept copyrighted text or images borrowed from other web sites or printed material; such additions must be deleted. Contributors may use sources as a source of information, but not as a source of sentences or phrases. Accordingly, the material may be rewritten, but only if it does not infringe on the copyright of the original or plagiarize from that source. Please see our guideline on non-free text for how to properly implement limited quotations of copyrighted text. Wikipedia takes copyright violations very seriously. Moonriddengirl ^(talk) 00:10, 17 November 2013 (UTC)

Prior and Barcan

Was Ruth Barcan Arthur Norman Prior's protégé? Maybe so, but why does the author think so? — Preceding unsigned comment added by 86.185.217.237 (talk) 12:25, 8 September 2014 (UTC)

Does not meet Wiki standards.

Check it out:
WP:SLANG http://wiki.riteme.site/wiki/Argot

"Wikipedia articles, and other encyclopedic content, should be written in a formal tone. ... Formal tone means that the article should not be written using unintelligible argot, slang, colloquialisms, doublespeak, legalese, or jargon; it means that the English language should be used in a businesslike manner."

Even worse, this crap is in the intro paragraph, quote:
"("Possibly, p", "It is possible that p"), necessity ("Necessarily, p", "It is necessary that p"), and impossibility ("Impossibly, p", "It is impossible that p")."

Yer shittin me,right? And I've studied logic. Looks promising and keystone, try it in English. Also see: MOS:LEAD:

"The lead should contain no more than four paragraphs, must be carefully sourced as appropriate, and should be written in a clear, accessible style with a neutral point of view to invite a reading of the full article....." 

and MOS:INTRO:

"...In general, specialized terminology and symbols should be avoided in an introduction. ....Where uncommon terms are essential to describing the subject, they should be placed in context, briefly defined, and linked.  The subject should be placed in a context with which many readers could be expected to be familiar."

First sentence Wikipedia:MOSINTRO#First_sentence

"The article should begin with a declarative sentence telling the nonspecialist reader what (or who) is the subject.  
 ...If its subject is amenable to definition, then the first sentence should give a concise definition: where possible, one that puts the article in context for the nonspecialist. Similarly, if the title is a specialised term, provide the context as early as possible...."

Inappropriate jargon is always bad writing. Guess what? Good writing ain't easy, takes work. Wiki's about Communication, it's no longer about impressing people that you really read the book...that is assumed. —Cheers!
--2602:306:CFCE:1EE0:3044:A2C3:2683:987B (talk) 16:02, 21 November 2018 (UTC)Doug Bashford

Necessitation

Am I right in thinking that the necessitation rule is logically equivalent to the axiom:

$\neg \Diamond \bot$

In which case, it would be useful to include it as an axiom. Dezaxa (talk) 02:15, 15 August 2022 (UTC)

Isn't the example in “Syntax for modal operators” wrong?

◊(x²=1 ∧ 0=y) is NOT logically equivalent to (◊(x²=1)) ∧ ◊(0=y). The first formula says that x²=1 ∧ 0=y is true at some accessible possible world, whereas the second formula says that x²=1 is true at some accessible possible world and 0=y is true at some accessible possible world. In the first formula, the worlds for the subformulas x²=1 and 0=y are the same, whereas in the second formula, they may differ.

Thanks. I went to address the problem and somehow ended up rewriting the section. Botterweg14 (talk) 01:43, 10 November 2022 (UTC)