Talk:Modal logic

Anyone know the html codes for a square and a diamond? These are alternate symbols for L and M... Evercat 22:49, 2 Oct 2003 (UTC)

They don't seem to be in the HTML 4.01 entities; you'll probably need to use TeX. -- The Anome

Contents

1 Names for logic systems 2 To Dos 3 Remarks 4 A question 5 Subjunctive modalities?? 6 Formal rules

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)

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

• 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] (http://mally.stanford.edu/notes.pdf); 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)

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

<math>from \vdash P, infer \vdash \Box P<math>.

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)

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 (http://consc.net/mmm-papers.html). 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 (http://consc.net/mmm-papers.html) 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 (http://consc.net/papers/intension.html)

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 <math>\Box<math> 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)